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--- abstract: 'The gradient of the Casimir force between a Si-SiO${}_2$-graphene substrate and an Au-coated sphere is measured by means of a dynamic atomic force microscope operated in the frequency shift technique. It is shown that the presence of graphene leads to up to 9% increase in the force gradient at the shortest separation considered. This is in qualitative agreement with the predictions of Lifshitz theory using the dielectric permittivities of Si and SiO${}_2$ and the Dirac model of graphene.' author: - 'A. A. Banishev' - 'H. Wen' - 'J. Xu' - 'R. K. Kawakami' - 'G. L. Klimchitskaya' - 'V. M. Mostepanenko' - 'U. Mohideen' title: 'Measuring the Casimir force gradient from graphene on a [$\mbox{SiO}_2$]{} substrate' --- Introduction ============ In the last few years graphene has attracted considerable attention as a material of much promise for nanotechnology due to its unique mechanical, electrical and optical properties.[@1; @2] Noting that at short separations between test bodies the fluctuation-induced dispersion interactions, such as the van der Waals and Casimir forces, become dominant,[@2a] it is important to investigate them in the presence of a graphene sheet. In this connection much theoretical work has been done on the calculation of dispersion forces between two graphene sheets,[@3; @3a; @4; @5; @6; @7a] a graphene sheet and a metallic, dielectric or semiconductor plate,[@4; @5; @6; @7; @8; @9; @10; @11; @12] a graphene sheet and an atom or a molecule[@13; @14; @15; @16] etc. The calculations were performed using phenomenological density-functional methods,[@17; @18; @19; @20] second order perturbation theory,[@21] and the Lifshitz theory with some specific form for the reflection coefficients of electromagnetic oscillations on graphene.[@3; @5; @8; @9] However, in spite of the impressive progress in measurements of the Casimir force in configurations with metallic, dielectric and semiconductor test bodies (see reviews in Refs. [@22; @23; @24; @25] and more recent experiments [@26; @27; @28]), there is yet no previous measurement of dispersion forces acting on graphene. In the present paper we report measurements of the gradient of the Casimir force acting between a graphene sheet deposited on a SiO${}_2$ film covering a Si plate and an Au-coated sphere. Our measurements are performed by means of dynamic atomic force microscope (AFM) operated in the frequency-shift technique described in detail in Refs. [@27; @28]. We demonstrate significant increase in the gradient of the Casimir force in comparison with that between a Si plate covered with a SiO${}_2$ film and an Au-coated sphere, i.e., in the absence of graphene sheet. At short separations this increase is up to a factor 4-5 larger than the total experimental error in the measurement of the force gradient determined at a 67% confidence level. We also compare the experimental results with an approximate theory where the gradients of the Casimir force between a Si-SiO${}_2$ system and Au-coated sphere and between a graphene described by the Dirac model and the same sphere are computed independently of one another using the Lifshitz theory and then are added. Some excess of the theoretical force gradient over the experimental one is attributed to the screening of the Si-SiO${}_2$ surface by a graphene sheet. The paper is organized as follows. In Sec. II we briefly describe the detection system, the measurement scheem and the sample preparation. Section III contains the measurement results and their comparison with theory. Section IV contains our conclusions. Experimental setup ================== The detection system used in our measurements consists of an AFM cantilever with attached hollow glass microsphere coated with Au, piezoelectric actuators, fiber interferometers, light source, and phase locked loop (PLL). The thickness of the Au coating and the radius of the coated sphere were measured to be 280nm and $54.10\pm 0.09\,\mu$m using an AFM and a scanning electron microscope, respectively. A turbo-pump, oil-free dry scroll mechanical-pump and ion-pump were used to achieve high vacuum down to $10^{-9}\,$Torr (see Refs. [@27; @28] for detail of the setup). In the dynamic measurement scheme the total force $F_{\rm tot}(a)=F_{\rm el}(a)+F(a)$ acting on the sphere \[where $F_{\rm el}(a)$ and $F(a)$ are the electric and Casimir force, respectively, and $a$ is the separation distance between the sphere and graphene\] modifies the resonant natural frequency of the oscillator. The change in the frequency $\Delta\omega=\omega_r-\omega_0$, where $\omega_r$ and $\omega_0$ are the resonance frequencies in the presence and in the absence of external force $F_{\rm tot}(a)$, respectively, was recorded by the PLL. This was done at every 0.14nm while the plate was moved towards the grounded sphere starting at the maximum separation. This was repeated with one of 10 different voltages $V_i$ in the range from –38.5 to 58.4mV for the first graphene sample and from –5.2 to 97.4mV for the second graphene sample applied to the graphene sheet while the sphere remained grounded. The application of voltages and respective measurements were repeated for two times resulting in 20 sets of $\Delta\omega$ as a function of separation for each graphene sample. Large area graphene used in our experiment was obtained through a two-step Chemical Vapor Deposition (CVD) process described.[@29] In this process $25\,\mu$m thick polycrystalline copper foil (99.8% purity) was cleaned by diluted HCl solution followed by deionized water rinse. Then the copper foil was placed into $\sim 5\,\mbox{cm}\times 3\,$cm copper bag which had undergone the same clean process as above. The copper bag was loaded into a ceramic tube furnace for the CVD process. First the copper bag was annealed at $1000^{\circ}$C under continuous Ar/H${}_2$ (69sccm/10sccm) flow. Graphene was grown on the copper foil by introducing methane/hydrogen gas of 1.3sccm/4sccm for one hour and 35sccm/4sccm for another hour. Then the furnace was cooled down to room temperature under a continuous flow of Ar/H${}_2$ (69sccm/10sccm). Finally, the grown graphene was transferred from the copper foil to 300nm SiO${}_2$ layer on a B-doped Si layer of $500\,\mu$m thickness on the bottom by using poly-metil methacrylate (PMMA) as the graphene support layer and ammonium persulfate solution as the copper etchant. We have examined the quality of the graphene layer through Raman spectroscopy [@30a; @30b] and quantum Hall effect measurements,[@30c; @30d] which show single layer graphene characteristics. Measurements of 2D-mobility for a large area graphene onto SiO${}_2$ substrates performed in our laboratory demonstrate mobility above $3000\,\mbox{cm}^2/\mbox{Vs}$. A roughly estimate for the concentration of impurities would be $1.2\times 10^{10}\,\mbox{cm}^{-2}$, if we consider that each impurity adsorbs one electron. The gradients of the total and Casimir forces were found from the measured frequency shifts using electrostatic calibration. To perform this calibration of the setup, we used the expression for the electric force in sphere-plate geometry [@23] $$F_{\rm el}(a)=X(a,R)(V_i-V_0)^2. \label{eq1}$$ Here $X(a,R)$ is a known function and $V_0$ is the residual potential difference between a sphere surface and a graphene sheet which is nonzero even when both surfaces are grounded. In the linear regime which is realized in our setup [@27] the gradient of the Casimir force is given by $$F^{\prime}(a)\equiv\frac{\partial F(a)}{\partial a} = -\frac{1}{C}\Delta\omega -\frac{\partial X(a,R)}{\partial a} (V_i-V_0)^2, \label{eq2}$$ where $C=\omega_0/(2k)$ and $k$ is the spring constant of the cantilever. Note that the absolute separations between the zero level of the roughness on the sphere and graphene are found from $a=z_{\rm piezo}+z_0$, where $z_{\rm piezo}$ is the plate movement due to the piezoelectric actuator and $z_0$ is the closest approach between the Au sphere and graphene (in dynamic experiments the latter is much larger than the separation on contact of the two surfaces). From the position of a maximum in the parabolic dependence of $\Delta\omega$ on $V_i$ in Eq. (\[eq2\]), one can determine $V_0$ with the help of a $\chi^2$-fitting procedure. From the curvature of the parabola with the help of the same fit it is possible to determine $z_0$ and $C$. This was done at different separations for the two graphene samples used in our experiment. In Fig. \[fg1\] we present the values of $V_0$ as a function of separation determined from the fit for the first and second graphene samples (the lower and upper sets of dots, respectively). The obtained values were corrected for mechanical drift of the frequency-shift signal, as discussed in Ref. [@27]. As can be seen from Fig. \[fg1\], the resulting $V_0$ do not depend on separation. To check this observation, we have performed the best fit of $V_0$ to the straight lines shown in Fig. \[fg1\] leaving their slopes as free parameters. It was found that the slopes are $-4.96\times 10^{-6}\,$mV/nm and $6.2\times 10^{-4}\,$mV/nm for the first and second samples, respectively, i.e., the independence of $V_0$ on $a$ was confirmed to a high accuracy. This finally leads to the mean values $V_0=18.4\pm 0.9\,$mV and $V_0=65.7\pm 0.9\,$mV for the first and second samples, respectively, where errors are determined at a 67% confidence level. Note that different graphene sheets may lead to different $V_0$ due to occasional impurities. The possible impurities could be organic, H${}_2$, O${}_2$, N${}_2$ and H${}_2$O. All these may become dopants of graphene and change its work function. Next the quantities $z_0$ and $C$ were determined from the fit at different separations and found to be separation-independent. For the first and second samples the mean values are equal to $z_0=222.5\pm 0.4\,$nm, $C=58.7\pm 0.17\,$kHzm/N and $z_0=222.2\pm 0.4\,$nm, $C=58.9\pm 0.17\,$kHzm/N, respectively. From the measured resonant frequency we have confirmed that the obtained value of $C$ results in the spring constant $k$ consistent with the estimated value provided by the cantilever fabricator. Measurement results and comparison with theory ============================================== For each graphene sample the gradients of the Casimir force $F^{\prime}(a)$ as a function of $a$ were obtained from the measured $\Delta\omega$ in two ways: by applying 10 different voltages $V_i$ with subsequent subtraction of the electric forces (2 repetitions) and by applying the compensating voltage $V_i=V_0$ (22 repetitions). In these cases 20 and 22 force-distance relations were obtained, the mean force gradients were computed and their total experimental errors were determined at a 67% confidence level as a combination of random and systematic errors (see Ref. [@27] for details). In Fig. \[fg2\](a,b) the mean gradients of the Casimir force and their errors measured for the first sample with applied compensating voltage are shown as crosses with a step of 1nm. Table  1 presents the values of mean $F^{\prime}(a)$ at several separations measured in the two different ways for the first (columns a,b) and second (columns c,d) samples. As can be seen in Table 1, the measurement results for the two graphene samples obtained in two different ways are in very good mutual agreement. Now we compare the experimental results with theoretical predictions. At the moment there is no theory allowing rigorous calculation of the Casimir force between a graphene deposited on a Si-SiO${}_2$ substrate and an Au sphere. The problem is that Si and SiO${}_2$ layers are described by their dielectric permittivities and the reflection properties of graphene in the Dirac model are described by the polarization tensor. This does not allow direct application of the Lifshitz theory for layered structures [@23; @30]. Because of this, here we restrict ourselves to the approximate approach, where the contributions of Si-SiO${}_2$ substrate and graphene sheet to the Casimir interaction with an Au sphere are computed separately using the Lifshitz theory and are then added together. In the framework of the proximity force approximation (PFA), the Lifshitz formula for the gradient of the Casimir force between an Au sphere and any planar structure takes the form $$F^{\prime}(a)=2k_BTR\sum_{l=0}^{\infty}{\vphantom{\sum}}^{\prime} \!\!\int_{0}^{\infty}\!\!\!\! q_lk_{\bot}dk_{\bot}\sum_{\alpha} \frac{r_{\alpha}^{(1)}r_{\alpha}^{(2)}}{e^{2q_la}-r_{\alpha}^{(1)}r_{\alpha}^{(2)}}. \label{eq3}$$ Here $k_B$ is the Boltzmann constant, $T=300\,$K is the laboratory temperature, $k_{\bot}$ is the projection of the wave vector on a planar structure, $q_l^2=k_{\bot}^2+\xi_l^2/c^2$, and $\xi_l=2\pi k_BTl/\hbar$ with $l=0,\,1,\,2,\,\ldots$ are the Matsubara frequencies. The prime near the summation sign multiplies the term with $l=0$ by 1/2, and $\alpha={\rm TM,TE}$ denotes the transverse magnetic and transverse electric polarizations of the electromagnetic field. Note that an error arising from the application of PFA was recently found [@31; @32; @33] using the exact theory for the sphere-plate geometry and was shown to be less than $a/R$, i.e., of about 0.5% in our experiment. The quantity $r_{\alpha}^{(1)}=r_{\alpha}^{(1)}(i\xi_l,k_{\bot})$ in Eq. (\[eq3\]) is the standard Fresnel reflection coefficient for an Au surface calculated at the imaginary frequencies (an Au layer can be considered as a semispace). It is expressed in terms of the dielectric permittivity $\varepsilon^{\,\rm Au}(i\xi_l)$ using the tabulated optical data for Au [@34] extrapolated to zero frequency either by the Drude or by the plasma models.[@22; @23] Unlike the case when a graphene layer is present, the Casimir interaction of the Si-SiO${}_2$ substrate with an Au sphere is described by the well tested fundamental Lifshitz theory. Here the quantity $r_{\alpha}^{(2)}=r_{\alpha}^{(2)}(i\xi_l,k_{\bot})$ has the meaning of the reflection coefficient on the two-layer (Si-SiO${}_2$) structure [@23; @30; @35] where Si can be considered as a semispace. It is expressed in terms of $\varepsilon^{\,\rm Si}(i\xi_l)$ and $\varepsilon^{\,{\rm SiO}_2}(i\xi_l)$. In our computations we used $\varepsilon^{\,\rm Si}(i\xi_l)$ obtained [@36] from the optical data [@37] for Si extrapolated to zero frequency either by the Drude or by the plasma models (Si plate used has the resistivity between 0.001 and $0.005\,\Omega\,$cm which corresponds to a plasma frequency between $5\times 10^{14}$ and $11\times 10^{14}\,$rad/s and the relaxation parameter $\gamma\approx1.1\times 10^{14}\,$rad/s). A sufficiently accurate expression for $\varepsilon^{\,{\rm SiO}_2}(i\xi_l)$ from Ref. [@38] was used in the computations. The r.m.s. roughness on the surfaces of sphere and graphene was measured by means of AFM and found to be equal to 1.6nm and 1.5nm, respectively. It was taken into account using the multiplicative approach,[@22; @23] and its maximum contribution to the force gradient is equal to only 0.1% at the shortest separation. The computational results for $F^{\prime}(a)$ between a Si-SiO${}_2$ substrate and an Au sphere are shown by the solid band in Fig. \[fg2\]. The width of the band indicates the uncertainty in the value of $\omega_p$ and a difference between the predictions of the Drude and plasma model approaches to the description of Au and Si which is small in this experiment. The latter is illustrated in columns e and f of Table 1. Figure \[fg2\] and Table 1 indicate conclusively that within the separation region from 224 to 320nm the measured gradients of the Casimir force are larger than that for a Si-SiO${}_2$ substrate interacting with an Au sphere. This demonstrates the influence of the graphene sheet on the Casimir force. The reflection coefficients for a suspended graphene described by the Dirac model are represented in the form [@9; @11; @16] $$\begin{aligned} && r_{\rm TM}^{(2)}= \frac{q_l\Pi_{00}}{q_l\Pi_{00}+2\hbar k_{\bot}^2}, \label{eq4} \\ && r_{\rm TE}^{(2)}= -\frac{k_{\bot}^2\Pi_{\rm tr}- q_l^2\Pi_{00}}{k_{\bot}^2(\Pi_{\rm tr}+2\hbar q_l)-q_l^2\Pi_{00}}, \nonumber\end{aligned}$$ where $\Pi_{mn}$ are the components of the polarization tensor in 3-dimensional space-time and the trace stands for the sum of spatial components. The computational results for the gradient of the Casimir force between the suspended graphene with the mass gap parameter $\Delta=0$ and $\Delta=0.1\,$eV and an Au sphere as a function of $a$ are shown in Fig. \[fg3\] by the upper and lower lines, respectively (here the results do not depend on whether the Drude or the plasma model approach for Au is used [@11]). In Fig. \[fg2\] the dashed band shows the sum of the force gradients between a Si-SiO${}_2$ substrate and an Au sphere and between graphene and the same sphere. The width of the band takes into account the respective width for a substrate interacting with a sphere and also differences in predictions of the Dirac model of graphene with the mass gap parameter varying from 0 to 0.1eV. It can be seen in Fig. \[fg2\] that the used approximate approach overestimates the measured force gradient, as it should, keeping in mind that it does not take into account the screening of the SiO${}_2$ surface by the graphene layer. Thus our results also illustrate nonadditivity of the van der Waals and Casimir interactions in multilayer structures.[@40] Note that at short separations our approximate approach (dashed line in Fig. \[fg2\]) is in better agreement with the data than the approach which disregards the graphene layer (solid line in Fig. \[fg2\]). Thus, at $a=224\,$nm the relative difference between the prediction of the approach disregarding graphene and the measured force gradient is equal to –10.1% of the measurement result and between the prediction of our approximate approach taking graphene into account and the same force gradient is equal to 7.1%. It is quite natural, however, that at large separations the influence of the graphene layer is overestimated by our approximate approach. Conclusions =========== To conclude, we have demonstrated the influence of a graphene layer on the Casimir force between a Si-SiO${}_2$ substrate and an Au sphere. At the shortest separation measured the relative excess in the force gradient due to the presence of graphene deposited on a substrate reaches 9% and decreases with increasing separation. Our experimental results are found to be in qualitative agreement with an approximate theoretical approach describing the reflection coefficients on graphene via the polarization tensor in 3-dimensional space-time, whereas the layers of the substrate are described by means of the dielectric permittivity. The standard Lifshitz theory for layered structures is not applicable to such cases. A more exact theoretical description than the one used in this work remains a challenge to theory. The present work will serve as a motivation in this direction. The Casimir interaction of graphene should be taken into account in future applications of carbon nanostructures in nanotechnology. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the DOE Grant No. DEF010204ER46131 (A.B., G.L.K., V.M.M., U.M.), by the NRI-NSF Grant No. NEB–1124601 (graphene film, H.W., R.K.K.) and by the NSF Grant No. PHY0970161 (equipment, G.L.K., V.M.M., U.M.). [99]{} A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. [**81**]{}, 109 (2009). M. S. Dresselhaus, Physica Status Solidi (b) [**248**]{}, 1566 (2011). R. H. French, V. A. Parsegian, R. Podgornik et al., Rev. Mod. Phys. [**82**]{}, 1887 (2010). M. Bordag, J. Phys. A: Math. Gen. [**39**]{}, 6173 (2006). G. Gómes-Santos, Phys. Rev. B [**80**]{}, 245424 (2009). D. Drosdoff and L. M. Woods, Phys. Rev. B [**82**]{}, 155459 (2010). B. E. Sernelius, Europhys. Lett. [**95**]{}, 57003 (2011). J. Sarabadani, A. Naji, R. Asgari, and R. Podgornik, Phys. Rev. B [**84**]{}, 155407 (2011). G. L. Klimchitskaya and V. M. Mostepanenko, [Phys. Rev.]{} B [**87**]{}, 075439 (2013). M. Bordag, B. Geyer, G. L. Klimchitskaya, and V. M. Mostepanenko, [Phys. Rev.]{} B [**74**]{}, 205431 (2006). M. Bordag, I. V. Fialkovsky, D. M. Gitman, and D. V. Vassilevich, [Phys. Rev. B]{} [**80**]{}, 245406 (2009). I. V. Fialkovsky, V. N. Marachevsky, and D. V. Vassilevich, [Phys. Rev. B]{} [**84**]{}, 035446 (2011). D. Drosdoff and L. M. Woods, Phys. Rev. A [**84**]{}, 062501 (2011). M. Bordag, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. B [**86**]{}, 165429 (2012). A. D. Phan, L. M. Woods, D. Drosdoff, I. V. Bondarev, and N. A. Viet, Appl. Phys. Lett. [**101**]{}, 113118 (2012). E. V. Blagov, G. L. Klimchitskaya, and V. M. Mostepanenko, [Phys. Rev.]{} B [**75**]{}, 235413 (2007). Yu. V. Churkin, A. B. Fedortsov, G. L. Klimchitskaya, and V. A. Yurova, [Phys. Rev. B]{} [**82**]{}, 165433 (2010). T. E. Judd, R. G. Scott, A. M. Martin, B. Kaczmarek, and T. M. Fromhold, New. J. Phys. [**13**]{}, 083020 (2011). M. Chaichian, G. L. Klimchitskaya, V. M. Mostepanenko, and A. Tureanu, Phys. Rev. A [**86**]{}, 012515 (2012). A. Bogicevic, S. Ovesson, P. Hyldgaard, B. I. Lundqvist, H. Brune, and D. R. Jennison, [Phys. Rev. Lett.]{} [**85**]{}, 1910 (2000). E. Hult, P. Hyldgaard, J. Rossmeisl, and B. I. Lundqvist, [Phys. Rev. B]{} [**64**]{}, 195414 (2001). J. F. Dobson, A. White, and A. Rubio, [Phys. Rev. Lett.]{} [**96**]{}, 073201 (2006). A. N. Rudenko, F. J. Keil, M. I. Katsnelson, and A. I. Lichtenstein, [Phys. Rev. B]{} [**84**]{}, 085438 (2011). I. V. Bondarev and Ph. Lambin, Phys. Rev. B [**70**]{}, 035407 (2004). G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Rev. Mod. Phys. [**81**]{}, 1827 (2009). M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, [*Advances in the Casimir Effect*]{} (Oxford University Press, Oxford, 2009). A. W. Rodriguez, F. Capasso, and S. G. Johnson, Nature Photon. [**5**]{}, 211 (2011). G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Int. J. Mod. Phys. B [**25**]{}, 171 (2011). C.-C. Chang, A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. Lett. [**107**]{}, 090403 (2011). C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. B [**85**]{}, 165443 (2012). A. A. Banishev, C.-C. Chang, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. B [**85**]{}, 195422 (2012). X. Li, C. W. Magnuson, A. Venugopal, R. M. Tromp, J. B. Hannon, E. M. Vogel, L. Colombo, and R. S. Ruoff, J. Amer. Chem. Soc. [**133**]{}, 2816 (2011). A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, and A. K. Geim, Phys. Rev. Lett. [**97**]{}, 187401 (2006). D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, and L. Wirtz, Nano Lett. [**7**]{}, 238 (2007). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature [**438**]{}, 197 (2005). Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature [**438**]{}, 201 (2005). M. S. Tomaš, Phys. Rev. A [**66**]{}, 052103 (2002). C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Phys. Rev. D [**84**]{}, 105031 (2011). G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, Europhys. Lett. [**97**]{}, 50001 (2012). G. Bimonte, T. Emig, and M. Kardar, Appl. Phys. Lett. [**100**]{}, 074110 (2012). , ed. E. D. Palik (Academic, New York, 1985). G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, J. Phys.: Condens. Matter [**24**]{}, 424202 (2012). F. Chen, U. Mohideen, G. L. Klimchitskaya, and V. M. Mostepanenko, [Phys. Rev. A]{} [**72**]{}, 020101(R) (2005); [**74**]{}, 022103 (2006). , vol II, ed. E. D. Palik (Academic, New York, 1991). L. Bergström, Adv. Colloid Interface Sci. [**70**]{}, 125 (1997). R. Podgornik, R. H. French, and V. A. Parsegian, J. Chem. Phys. [**124**]{}, 044709 (2006). ![\[fg1\]The residual potential difference between an Au-coated sphere and the first (lower dots) and second (upper dots) graphene sheets on a Si-SiO${}_2$ substrate as a function of separation. The mean values of $V_0$ are shown by the gray lines. ](figAuGr-1.ps)  The experimental data for the gradient of the Casimir force $F^{\prime}$ at (a) short and (b) long separations are shown as crosses plotted at a 67% confidence level (measured with the applied compensating voltage for the first sample). The theoretical $F^{\prime}$ between an Au-coated sphere and a Si-SiO${}_2$ substrate calculated using the Lifshitz theory and between an Au-coated sphere and graphene deposited on this substrate calculated using an additive approach are shown as the solid and dashed bands, respectively. ](figAuGr-2.ps)  The gradient of the Casimir force between an Au-coated sphere and a suspended graphene sheet calculated using the Dirac model with the mass gap parameter equal to 0.1eV (lower line) and 0 (upper line) as a function of separation. ](figAuGr-3.ps) ------------------- ------------------- ------------------- ------------------- ------------------- ----------- ----------- \[1mm\] $a\,$(nm) a[    ]{}[    ]{} b[    ]{}[    ]{} c[    ]{}[    ]{} d[    ]{}[    ]{} e[    ]{} f[    ]{} 224 $34.27\pm 0.64$ $33.58\pm 0.65$ $34.12\pm 0.64$ $33.76\pm 0.65$ 30.90 30.70 250 $22.62\pm 0.64$ $22.27\pm 0.64$ $22.72\pm 0.64$ $22.42\pm 0.64$ 20.67 20.51 300 $11.50\pm 0.64$ $11.19\pm 0.64$ $11.65\pm 0.64$ $11.53\pm 0.64$ 10.66 10.54 350 $6.52\pm 0.64$ $6.28\pm 0.64$ $6.30\pm 0.64$ $6.60\pm 0.64$ 6.12 6.03 400 $3.98\pm 0.64$ $3.67\pm 0.64$ $3.99\pm 0.64$ $3.70\pm 0.64$ 3.81 3.73 500 $1.90\pm 0.64$ $1.76\pm 0.64$ $1.78\pm 0.64$ $1.60\pm 0.64$ 1.73 1.68 ------------------- ------------------- ------------------- ------------------- ------------------- ----------- ----------- : The mean values of the gradient of the Casimir force together with their total experimental errors at different separations (first column) measured in this work with applied compensating voltage (column a) and with different applied voltages (column b) for the first graphene sample (columns c and d, respectively, for the second graphene sample). Columns e and f contain theoretical values for the gradients of the Casimir force between the Au sphere and Si-SiO${}_2$ substrate calculated when Au and Si are described by the plasma and Drude model approaches, respectively.

--- abstract: 'Using the Maxwell-Bloch equations for resonantly absorbing and amplifying media, we study the temporal dynamics of light propagation through the $\mathcal{PT}$-symmetric structures with alternating loss and gain layers. This approach allows us to precisely describe the response of the structure near the exceptional points of $\mathcal{PT}$-symmetry breaking phase transition and, in particular, take into account the nonlinear effect of loss and gain saturation in the $\mathcal{PT}$-symmetry broken state. We reveal that in this latter state the multilayer system possesses a lasing-like behavior releasing the pumped energy in the form of powerful pulses. We predict locking of pulse direction due to the $\mathcal{PT}$-symmetry breaking, as well as saturation-induced irreversibility of phase transition and nonreciprocal transmission.' author: - 'Denis V. Novitsky$^{1,2,3}$' - 'Alina Karabchevsky$^{4,5,6}$' - 'Andrei V. Lavrinenko$^7$' - 'Alexander S. Shalin$^2$' - 'Andrey V. Novitsky$^{7,8}$' title: '$\mathcal{PT}$-symmetry breaking in multilayers with resonant loss and gain locks light propagation direction' --- Introduction ============ $\mathcal{PT}$-symmetric structures and $\mathcal{PT}$-symmetry breaking seem to be one of the most intensively studied fields in optics and photonics of active systems [@Zyablovsky2014; @Feng2017; @El-Ganainy2018]. It dates back to the seminal works of Bender and Boettcher [@Bender1998; @Bender2007], who discovered the real-valued spectra of non-Hermitian Hamiltonians in quantum mechanics provided these Hamiltonians are parity-time ($\mathcal{PT}$) symmetric, i.e., invariant with respect to simultaneous parity change and time reversal. Such ideas can be straightforwardly transferred to optics using the spatial ordering of passive and active components, thus introducing the concept of an optical $\mathcal{PT}$-symmetric system. The simplest optical $\mathcal{PT}$-symmetric structure can be realized by means of one-dimensional multilayers — similar to photonic crystals — with proper spatial variation of the complex permittivity as $\varepsilon(z) = \varepsilon^\ast(-z)$ (here asterisk stands for complex conjugation). This condition implies that the real part of the permittivity or refractive index is an even function of the coordinate, whereas the imaginary part is an odd function. Change of the sign of the imaginary part of the permittivity ${\rm Im} \varepsilon(z) = -{\rm Im} \varepsilon(-z)$ apparently requires amplifying materials. It can be, for instance, an alternation of loss and gain in periodic multilayers. Two principal schemes, longitudinal and transverse ones, are usually employed. In the former system, light propagates directly through the multilayer. In the latter scheme light propagates perpendicularly to the permittivity distribution, along the layers boundaries, being analogous to a system of interconnected waveguides (an optical grating). First theoretical [@El-Ganainy2007; @Makris2008] and experimental [@Ruter2010] results on optical $\mathcal{PT}$ symmetry were reported exactly for the coupled waveguides. $\mathcal{PT}$ symmetry allows new ways for controlling radiation fluxes both in optics and plasmonics [@Yang2015]. A number of effects can be highlighted as a fingerprint of the $\mathcal{PT}$ symmetry: nonreciprocity of light transmission and beam power oscillations [@Makris2008], anisotropic transmission resonances [@Ge2012], unidirectional “invisibility” [@Lin2011], negative refraction, and focusing of light [@Fleury2014]. $\mathcal{PT}$ symmetry governs light localization in disordered structures [@Kartashov2016]. The usage of gain media raises questions concerning available nonlinear phenomena, such as optical switching and generation of new types of solitons [@Suchkov2016; @Konotop2016]. Apodization of the refractive index spatial profiles can be used to facilitate switching conditions in $\mathcal{PT}$-symmetric Bragg gratings [@Lupu2016]. Some nonlinear effects connected to $\mathcal{PT}$ symmetry have been recently observed in experiments with coherent atomic gases [@Hang2017]. More prospects are opened by the fact that $\mathcal{PT}$-symmetric optical gratings are capable of supporting topologically protected bound states [@Weimann2017]. It should be noted that the effects mentioned above may be observable in loss-gain structures lacking the $\mathcal{PT}$-symmetry. A typical example is the reflectionless transmission and unidirectional “invisibility” for both normal [@Shen2014; @Ramirez2017] and oblique [@Novitsky2017] incidence. However, the symmetry breaking at the exceptional points belongs exclusively to the $\mathcal{PT}$ symmetry domain. There is a number of phenomena associated with violation of the $\mathcal{PT}$ symmetry. First, a sharp change in polarization response of the system is possible at the exceptional points [@Lawrence2014]. Later an omnipolarizer was designed for converting any light polarization into a given one [@Hassan2017]. Second, enhanced sensitivity of such kind of systems to external perturbations near exceptional points provides a new approach for sensing [@Chen2017; @Hodaei2017]. Third, $\mathcal{PT}$-symmetry breaking plays an important role in laser physics offering new types of lasers [@Feng2014; @Hodaei2014; @Gu2016] and anti-lasers [@Wong2016] based on the effect of coherent perfect absorption [@Chong2010; @Longhi2010]. Finally, the possibility of light stopping at the exceptional point was recently reported [@Goldzak2018]. In this paper, we study the phenomenon of $\mathcal{PT}$-symmetry breaking in one-dimensional multilayers composed of resonantly absorbing and amplifying media. We describe propagation of light in the time domain using the Maxwell-Bloch equations and taking into account loss and gain saturation. The influence of the saturable nonlinearity on nonreciprocity and bistability of $\mathcal{PT}$-symmetric structures was previously reported both for transverse [@Ramezani2010] and longitudinal geometries [@Phang2014; @Liu2014; @Barton2017; @Witonski2017]. However, those investigations introduced the saturation phenomenologically via the permittivity. Our approach is based on self-consistent description of temporal dynamics for both light field and medium loss/gain exhibiting a more realistic treatment of $\mathcal{PT}$-symmetric optical systems. Therefore, we have a deep insight into peculiarities of the $\mathcal{PT}$-symmetry breaking and related effects. In particular, we study the lasing-like regime in the $\mathcal{PT}$-symmetry broken state, where the saturation effects lead to the unique phase transition in the parameter space and nonreciprocal transmission of generated pulses. Pulse-direction locking by the $\mathcal{PT}$-symmetry breaking in the lasing-like regime is related to the strong light confinement and resembles light polarization locking to its propagation direction in quantum optics [@Lodahl17]. The paper is organized as follows. Section \[eqpars\] is devoted to the description of the theoretical model of the loss-gain multilayer and parameters used in calculations. We discuss light behavior in the $\mathcal{PT}$-symmetric phase in Sec. \[stat\] by comparing numerical solution of the Maxwell-Bloch equations and stationary transfer-matrix calculations. In Sec. \[trans\], the $\mathcal{PT}$-symmetry breaking as a phase transition to the lasing-like regime is studied with emphasis on temporal dynamics of the light propagating through the multilayers. Section \[concl\] summarizes the article. \[eqpars\] Resonant loss and gain media ======================================= A periodic planar structure composed of $2N$ alternating loss and gain layers shown in Fig. \[fig1\] is illuminated by monochromatic light of angular frequency $\omega$ at normal incidence. In this study, both loss and gain are described in the similar manner, using the model of a homogeneously-broadened two-level medium. Excluding the rapidly varying factors $\exp(-i\omega t)$ in polarization, population difference, and electric field, we write the Maxwell-Bloch equations [@Novitsky2011] for slowly varying amplitudes of these quantities as $$\begin{aligned} \frac{d\rho}{d\tau}&=& i l \Omega w + i \rho \delta - \gamma_2 \rho, \label{dPdtau} \\ % \frac{dw}{d\tau}&=&2 i (l^* \Omega^* \rho - \rho^* l \Omega) - \gamma_1 (w-w_{eq}), \label{dNdtau} \\ % \frac{\partial^2 \Omega}{\partial \xi^2}&-& n_d^2 \frac{\partial^2 \Omega}{\partial \tau^2}+2 i \frac{\partial \Omega}{\partial \xi}+2 i n_d^2 \frac{\partial \Omega}{\partial \tau} + (n_d^2-1) \Omega \nonumber \\ &&=3 \alpha l \left(\frac{\partial^2 \rho}{\partial \tau^2}-2 i \frac{\partial \rho}{\partial \tau}-\rho\right), \label{Maxdl}\end{aligned}$$ where $\tau=\omega t$ and $\xi=kz$ are respectively the dimensionless time and distance, $\Omega=(\mu/\hbar \omega) A$ is the normalized Rabi frequency, $A$ is the electric field strength, $\omega$ is the light circular frequency, $k = \omega/c$ is the wavenumber in vacuum, $c$ is the speed of light, $\hbar$ is the reduced Planck constant, and $\mu$ is the dipole moment of the quantum transition. Rabi frequency is dynamically coupled to the characteristics of the two-level system – complex amplitude of microscopic (atomic) polarization $\rho$ and difference between populations of ground and excited states $w$. Efficiency of the light-matter coupling is given by the dimensionless parameter $\alpha= \omega_L / \omega = 4 \pi \mu^2 C/3 \hbar \omega$, where $\omega_L$ is the Lorentz frequency and $C$ is the concentration (density) of active (two-level) atoms. In general, light frequency $\omega$ is detuned from frequency $\omega_0$ of the atomic resonance as described by $\delta=(\omega_0-\omega)/\omega$. The normalized relaxation rates of population $\gamma_1=1/(\omega T_1)$ and polarization $\gamma_2=1/(\omega T_2)$ are expressed by means of the longitudinal $T_1$ and transverse $T_2$ relaxation times. The influence of the polarization of the background dielectric having real-valued refractive index $n_d$ on the embedded active particles is taken into account by the local-field enhancement factor $l=(n_d^2+2)/3$ [@Crenshaw2008; @Bloembergen]. Equilibrium population difference $w_{eq}$ will allow us to describe both gain and loss materials with the same Maxwell-Bloch equations (\[dPdtau\])-(\[Maxdl\]). When external pump is absent, the two-level atoms are in the ground state ($w_{eq}=1$), and the medium is lossy. In the case of gain, the equilibrium population difference can be referred to as a pumping parameter. In the fully inverted medium, all atoms are excited by the pump ($w_{eq}=-1$). For the saturated medium with both levels populated equally in the equilibrium, there are no transitions between the levels ($w_{eq}=0$). In the steady-state approximation, when the amplitudes of population difference, polarization, and field are time-independent, one can use the effective permittivity of a two-level medium [@Novitsky2017] $$\begin{aligned} \varepsilon_{eff} &=& n_d^2+4 \pi \mu C \rho_{st}/E % = n_d^2+\frac{K(-\delta+i\gamma_2)}{1+|\Omega|^2/\Omega^2_{sat}}, \label{epsTLM}\end{aligned}$$ where $\Omega_{sat}=\sqrt{\gamma_1 (\gamma_2^2+\delta^2)/4l^2 \gamma_2}$ sets the level of saturation intensity and $K=3 \omega_L l^2 w_{eq}/[\omega (\gamma_2^2+\delta^2)]$. At the exact resonance $\delta=0$ and in approximation of low-intensity external radiation $|\Omega| \ll\Omega_{sat}$, Eq. (\[epsTLM\]) transforms to $\varepsilon_{eff} \approx n_d^2+3 i l^2 \omega_L T_2 w_{eq}$. From this equation it is clear that gain and loss correspond to negative and positive $w_{eq}$, respectively. In the stationary approximation, it is straightforward to obtain a $\mathcal{PT}$-symmetric structure composed of alternating layers with balanced loss ($\varepsilon_{eff+}$) and gain ($\varepsilon_{eff-}$), where $$\begin{aligned} \varepsilon_{eff\pm} \approx n_d^2 \pm 3 i l^2 \omega_L T_2 |w_{eq}|. \label{epsPT}\end{aligned}$$ $\mathcal{PT}$ symmetry holds true, because the necessary condition $\varepsilon(z) = \varepsilon^\ast(-z)$ is fulfilled, providing even (odd) function of $z$ for the real (imaginary) part of the permittivity. In Supplemental Material [@Supp], $\mathcal{PT}$-symmetry conditions are derived straight from the Maxwell-Bloch equations. It is shown that the system is $\mathcal{PT}$-symmetric only in a steady state established after some transient period. Identity of the absolute values of the imaginary parts of permittivities $\varepsilon_{eff+}$ and $\varepsilon_{eff-}$ can be achieved in different ways. From a practical point of view, it would be convenient to take unexcited absorbing layers ($w_{eq,L}=1$) and pump only the amplifying layers to the level $w_{eq,G} = - \alpha_{L}/\alpha_{G}$, where $\alpha_{L}$ and $\alpha_{G}$ are the light-matter coupling coefficients for loss and gain layers, respectively. Tuning of these coefficients can be properly carried out by affecting the concentration of active particles in both types of layers. Without imposing any restrictions, it is fair to claim within this theoretical investigation that the loss and gain layers have equal concentrations $C$ (hence, equal couplings $\alpha$) and absolute values of the pumping parameter $|w_{eq}|$. Some additional data on the variant with completely unexcited absorbing layers are given in Supplemental Material [@Supp]. Eqs. (\[dPdtau\])–(\[Maxdl\]) are solved numerically using the FDTD approach developed in our previous publication [@Novitsky2009] and recently adapted to study loss-gain structures [@Novitsky2017]. As an initial value of the population difference, we employ the pumping parameter, i.e., $w(t=0)=w_{eq}$. Comparison of the results of numerical simulations with those of the transfer-matrix method with Eq. (\[epsPT\]) for permittivities of loss and gain layers will unveil the limitations of applicability of the latter approach. In this paper, we use semiconductor doped with quantum dots as an active material and assume the condition of exact resonance $\delta=0$ is valid. It can be characterized by the following parameters [@Palik; @Diels]: $n_d=3.4$, $\omega_L=10^{11}$ s$^{-1}$, $T_1=1$ ns, and $T_2=0.5$ ps. Gain coefficient $g=4 \pi \textrm{Im}(\sqrt{\varepsilon_{eff-}})/\lambda \leq 10^4$ cm$^{-1}$ is estimated according to Eq. (\[epsPT\]) for $\lambda \sim 1.5$ $\mu$m and $|w_{eq}| \leq 0.2$ can be realized in practice [@Babicheva12]. The multilayer structure contains $N=20$ unit cells. Both loss and gain layers have the same thickness $d=1$ $\mu$m. The pumping scheme similar to that realized by Wong *et al.* [@Wong2016] can be used in our system. It is also worth noting that the choice of materials is not unique, but the multilayer parameters and light wavelength may need to be appropriately adjusted to obtain similar results with different materials. \[stat\]Temporal dynamics of light in $\mathcal{PT}$-symmetric phase ==================================================================== We start our study analysing the stationary characteristics of one-dimensional $\mathcal{PT}$-symmetric structures. In the stationary mode, the transfer-matrix method for the wave propagation through multilayers with permittivities (\[epsPT\]) is exploited. Owing to reciprocity of the system, the transmission of oppositely propagating (forward and backward) waves is the same, but the reflection is different. We denote these two directions of wave propagation with subscripts LG and GL (see Fig. \[fig1\]) originating from the order of layers in the unit cell of the structure. In the case of $w_{eq} = 0$ (homogeneous dielectric slab of thickness $2Nd$), the reflections are equal, $R_{LG}=R_{GL}$ \[Fig. \[fig2\](a)\]. Divergence of the curves for $R_{LG}$ and $R_{GL}$ in Figs. \[fig2\](b)-\[fig2\](d) indicates existence of the $\mathcal{PT}$-symmetry in the full accordance with the properties of transfer matrices of such systems [@Ge2012]. In these figures, one can notice another well-known feature – the so-called anisotropic transmission resonance (ATR) [@Ge2012]. It emerges under the following conditions: transmission $T=1$ and one of the reflections is zero. Thus, the ATR arises for the two wavelengths marked with arrows in Fig. \[fig2\] corresponding to $R_{GL} > R_{LG}=0$ and $R_{LG} > R_{GL}=0$. Transmission exceeds unity between these points. The increase of the pumping parameter (hence, the absolute value of the imaginary part of slabs’ permittivities) results in the red shift of the ATR. This means that the ATR can be reached by changing $|w_{eq}|$ at any fixed wavelength near the transmission peak. In order to examine this prediction of the transfer-matrix method, we perform numerical simulations of the Maxwell-Bloch equations (\[dPdtau\])-(\[Maxdl\]) for a monochromatic wave ($\lambda=1.513$ $\mu$m) propagating through the $\mathcal{PT}$-symmetric multilayer with different pumping parameters. To keep the correspondence with the transfer-matrix approach, the saturation should be neglected. Here it is realized for as low incident wave amplitude as $\Omega_0=10^{-5} \gamma_2 \ll \Omega_{sat}$. The results of numerical calculations shown with symbols in Fig. \[fig3\] agree well with those of matrix method (lines in Fig. \[fig3\]). The ATR at $|w_{eq}| \sim 0.125$ is also confirmed in the FDTD calculations. Temporal dynamics shown in Fig. \[fig4\] demonstrates the transient process to the steady-state formation in the dielectric slab \[in the absence of loss and gain, panel (a)\] and $\mathcal{PT}$-symmetric structure in conditions of the ATR \[panel (b)\]. Since the steady state is rapidly established, the transfer-matrix approach is well applicable in this no-saturation regime. Absence of saturation is directly demonstrated by the inset in Fig. \[fig4\], where the population difference does not change during establishment of the steady state. The initial transient regime is unavoidable in realistic systems. It can be studied with the FDTD method, and $\mathcal{PT}$ symmetry is unreachable in this mode due to impossibility to change the sign of relaxation rates $\gamma_1$ and $\gamma_2$ (see Ref. [@Supp]). \[trans\]Phase transition dynamics ================================== $\mathcal{PT}$-symmetry breaking can be considered as a peculiar phase transition. It can be realized either by changing wavelength $\lambda$ of light for a given value of pumping parameter $|w_{eq}|$, or, conversely, by changing the pumping parameter at a fixed wavelength. The latter variant is analyzed in this paper. The former one deserves a separate study, since it implies the necessity to consider the effects of frequency detuning. The criterion of the $\mathcal{PT}$-symmetry breaking is usually formulated in terms of the eigenvalues $s_1$ and $s_2$ of the scattering matrix [@Ge2012]. The scattering matrix connects the left and right input fields with left and right output fields. A $\mathcal{PT}$-symmetric system has unimodular eigenvalues $|s_1|=|s_2|=1$ of the scattering matrix. When $\mathcal{PT}$ symmetry is violated, the modules of the eigenvalues are inverse as $|s_1|>1$ and $|s_2| = 1/|s_1| < 1$. At the points of phase transition called exceptional points, the eigenvectors of the scattering matrix coincide. Using the definition of the scattering matrix [@Ge2012] $$\begin{aligned} S=\left( \begin{array}{cc}{r_{LG} \qquad t_{GL}} \\ {t_{LG} \qquad r_{GL}} \end{array} \right), \label{scat}\end{aligned}$$ we calculate both eigenvalues and eigenvectors for pumping parameter $|w_{eq}|$ swept through the whole interval from $0$ to $1$. Here $t_{LG}$, $t_{GL}$, $r_{LG}$, and $r_{GL}$ are the transmission and reflection coefficients, which can be expressed through the respective elements of transfer matrix $M$: $t_{LG}=1/M_{11}$, $t_{GL}=\det[M]/M_{11}=t_{LG}$ (since $\det[M]=1$), $r_{LG}=M_{21}/M_{11}$, and $r_{GL}=-M_{12}/M_{11}$. Stationary transmission and reflection are calculated as $T=|t_{LG}|^2=|t_{GL}|^2$, $R_{LG}=|r_{LG}|^2$, and $R_{GL}=|r_{GL}|^2$. Results of transfer-matrix calculations for the multilayer structure with the permittivities (\[epsPT\]) are shown in Fig. \[fig5\]. At the first exceptional point $|w_{eq}| \approx 0.222$, the phase transition occurs and eigenvalues cease to be unimodular \[Fig. \[fig5\](a)\], whereas the difference between eigenvectors vanishes \[Fig. \[fig5\](b)\]. Unimodularity is violated and, therefore, $\mathcal{PT}$ symmetry is broken up to the second exceptional point. The latter returns the system into the $\mathcal{PT}$-symmetric state. All in all, there is a number of ranges of $|w_{eq}|$ with broken symmetry and a corresponding number of exceptional points. Transfer-matrix approach used so far cannot describe dynamics of the wave propagation in resonant media by its definition. Now we will employ the rigorous Maxwell-Bloch equations for studying light dynamics near an exceptional point. There is a dramatic discrepancy between the two calculation techniques, when the pumping parameter approaches the exceptional point. Relative difference in transmission calculated with help of the Maxwell-Bloch and transfer-matrix approaches, $|T_{MB}-T_{TM}|/T_{MB} \approx 0.13$, results in the satisfactory agreement for $|w_{eq}|=0.22$, but its value $0.48$ at $|w_{eq}|=0.23$ is unacceptably large. Such a large discrepancy stems from the qualitatively different behaviors of the system at $|w_{eq}|=0.23$: the system is above the exceptional point according to the transfer-matrix method, whereas it is still in the $\mathcal{PT}$-symmetric state according to the FDTD simulations. In fact, temporal dynamics with an established stationary state in Fig. \[fig6\](a) is distinctive for the $\mathcal{PT}$ symmetry (cf. Fig. \[fig4\]). Formation of the steady state after a rather short time corroborates existence of the balance between gain and loss at $|w_{eq}|=0.23$. The general tendency is that the closer to the exceptional point, the longer the transient period is. At the exceptional point \[Fig. \[fig6\](b)\], the field rapidly grows changing population difference $w$, this grow being limited by saturation. Breaking of $\mathcal{PT}$ symmetry is expected to result in the strong (exponential) amplification of propagating waves due to the fact that gain can not be compensated with losses in this case. In Fig. \[fig6\](b), instability exhibiting very strong light amplification is observed at $|w_{eq}|=0.24$. The energy pumped in the system is promptly released as a high-intensity pulse. In concordance with Ref. [@Novitsky2017], this regime can be called a lasing-like (or quasilasing) mode. It can be treated as a dynamical feature of the phase state of broken $\mathcal{PT}$ symmetry. In contrast to the true lasing the pulse is generated not by small field fluctuations, but rather in response to the incident wave (though the intensity of input wave can be taken much lower, as evidenced by Supplemental Material Fig. S4 [@Supp]). At the exceptional point, we have a very long transient period and the stationary levels of reflection and transmission are again expected to occur in the long-time limit [@Novitsky2017]. These levels strongly differ from those calculated with the transfer-matrix method due to saturation development discussed further. In other words, the resulting population difference will be no more determined by pumping parameter $w_{eq}$ as assumed in Eq. (\[epsPT\]). The pulse gets shorter and more powerful with increasing $|w_{eq}|$ as evidenced by comparison of Fig. \[fig6\](b) and Supplemental Material Fig. S1 [@Supp]. The spectra of transmitted radiation shown in Fig. \[fig7\] indicate the shift of maximal amplification from the resonant wavelength $\lambda=1.513$ $\mu$m (below the exceptional point) to longer wavelengths (above the exceptional point). This redshift in the lasing-like regime can be explained by two factors: (i) higher light absorption on the resonant wavelength in the loss layers, so that amplification at the neighboring wavelengths becomes prevalent, (ii) intensity modulation due to incomplete stationary-state establishment (see Supplemental Material Fig. S5 [@Supp]). We should stress that although the matrix method with permittivities (\[epsPT\]) is able to approximately determine an exceptional point, it fails in adequate description of the phase transition and in describing temporal dynamics of light-structure interaction. In other words, the full system of Maxwell-Bloch equations should be exploited in the vicinity of the points of $\mathcal{PT}$-symmetry breaking. Strong amplification of a signal in the non-$\mathcal{PT}$-symmetric phase \[Fig. \[fig6\](b)\] plays important role in loss and gain saturation. Indeed, normalized light amplitude $\Omega$ inside the system is not much less than $\Omega_{sat}$ anymore. Population difference preserves its initial value $w(t)=w_{eq}$ in the $\mathcal{PT}$-symmetric state below the exceptional point (see the dashed lines in Fig. \[fig8\], for $|w_{eq}|=0.23$). However, fluctuations of the population difference due to saturation occur above the exceptional point, at $|w_{eq}|=0.24$, in both LG (in the loss layer) and GL (in the gain layer) configurations. Saturation imposes constraint on further increase of light intensity as evidenced by the coincidence of the intensity peak in Fig. \[fig6\](b) and saturation development in time in Fig. \[fig8\]. Fluctuations indicate that the $\mathcal{PT}$ symmetry is broken through violation of the necessary condition $\varepsilon(z) = \varepsilon^\ast(-z)$ and loss and gain are not balanced anymore. Saturation also leads to *the irreversible phase transition* in the system: the return of the system to the $\mathcal{PT}$-symmetric state predicted by the stationary theory at larger pumping parameters (see Fig. \[fig5\]) is impossible, since Eq. (\[epsPT\]) is not valid anymore. Direct FDTD calculations for $|w_{eq}| > 0.24$ (see Supplemental Material Fig. S1 [@Supp]) do support this conclusion resulting in the lasing-like dynamics similar to that shown in Fig. \[fig6\](b). Broken $\mathcal{PT}$ symmetry drastically affects dynamics of the transmitted and reflected waves. Owing to the nonlinear process of saturation, the transmission becomes asymmetric, $T_{LG} \neq T_{GL}$, i.e., the multilayer structure is *nonreciprocal*. Usually the saturation-induced nonreciprocity is introduced through the nonlinear permittivity Eq. (\[epsTLM\]) [@Liu2014; @Barton2017], but solution of dynamic Eqs. (\[dPdtau\])-(\[Maxdl\]) is more accurate and informative. In the saturation regime the system is non-Hermitian, but it can be linearized to a $\mathcal{PT}$-symmetric multilayer [@Barton2017]. Intensities of the pulses escaping the system do not depend on the direction of incident light: almost the same pulses are emitted from the gain and loss ends of the multilayer after reversing the input light direction ($T_{LG} = R_{GL}$ and $T_{GL} = R_{LG}$) as shown in Fig. \[fig6\](b). In other words, direction of the output pulses is *locked by $\mathcal{PT}$-symmetry breaking*. This locking can be presumably caught only within the dynamical calculations, because it has not been reported earlier. Nonreciprocal transmission is accompanied by the propagation direction locking at higher pumping as well, what is demonstrated in Supplemental Material Fig. S1 [@Supp] for $|w_{eq}|=0.3$ and $0.4$. Locking of the light propagation directions can be viewed as a possible basis for peculiar all-optical diodes and transistors. It should be emphasized that the saturation is not the reason for this locking. In order to demonstrate this, we consider wave propagation in saturation regime (for initial amplitude $\Omega_0=10^{-2} \gamma_2$) which breaks $\mathcal{PT}$ symmetry at every value of the pumping parameter. Nonreciprocity of transmission due to saturation is clearly seen in Fig. \[fig9\], but the direction locking of output pulses is missing. Therefore, the $\mathcal{PT}$-symmetry breaking is necessary for this locking to occur. The behavior similar to that described above generally occurs near the exceptional points as evidenced by Figs. S2, S3, and S4 in Supplemental Material [@Supp]. In particular, the number of layers controls position of the exceptional point: when the structure length is decreased, higher pumping is needed for $\mathcal{PT}$-symmetry breaking (Fig. S2 of Supplemental Material [@Supp]) and vice versa. Although system’s response is generally very complex due to interplay of the loss/gain and multilayer resonances [@Witonski2017], the results similar to discussed above are expected for other parameters of the structure and proper operating frequency. Such scalability is encouraging for designing realistic realizations of $\mathcal{PT}$-symmetric multilayers. We would like to emphasize that another model of $\mathcal{PT}$-symmetric multilayer with unexcited absorbing layers mentioned in Section \[eqpars\] provides the similar results (see Supplemental Material Fig. S6 [@Supp]). \[concl\]Conclusion =================== We have analyzed temporal dynamics of light in $\mathcal{PT}$-symmetric periodic multilayers, the gain and loss slabs being modeled as a resonant medium. Light-matter interactions in the resonant media are described by the Maxwell-Bloch equations, which are simulated numerically to provide deeper insight into transition dynamics between the $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phases. In particular, predictions of the stationary transfer-matrix method are shown to be inadequate in the vicinity of the exceptional points. We feature the so-called lasing-like regime in the $\mathcal{PT}$-symmetry broken state characterized by emission of powerful pulses of radiation and development of saturation. The latter is the reason for phase transition irreversibility – that is, the system cannot return to the $\mathcal{PT}$-symmetric state for the pumping parameters above the exceptional point. In the $\mathcal{PT}$-broken phase, the direction of pulses escaping the system is found to be locked by the $\mathcal{PT}$-symmetry breaking, meaning that the intensities of two output waves are independent of the direction of the incident radiation. The approach based on the Maxwell-Bloch equations seems to be rather general and applicable to structures with other geometries, e.g., coupled ring resonators [@Feng2014]. We envisage its application to investigation of other effects near the exceptional points, such as coherent perfect absorption (anti-lasing) [@Wong2016; @Chong2010; @Longhi2010]. Intricate interplay between loss and gain in $\mathcal{PT}$-symmetric systems opens up new opportunities for constructing photonic devices for optical communications, computing, and sensing. The approach proposed here is expected to be useful to realize some of these diverse functionalities. [0]{} A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, [Phys. Usp.]{} **57**, 1063 (2014). L. Feng, R. El-Ganainy, and L. Ge, [Nat. Photon.]{} **11**, 752 (2017). R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, [Nat. Phys.]{} **13**, 11 (2018). C. M. Bender and S. Boettcher, [Phys. Rev. Lett.]{} **80**, 5243 (1998). C. M. Bender, [Rep. Prog. Phys.]{} **70**, 947 (2007). R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, [Opt. Lett.]{} **32**, 2632 (2007). K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, [Phys. Rev. Lett.]{} **100**, 103904 (2008). C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, [Nat. Phys.]{} **6**, 192 (2010). F. Yang and Z. L. Mei, [Sci. Rep.]{} **5**, 14981 (2015). L. Ge, Y. D. Chong, and A. D. Stone, [Phys. Rev. A]{} **85**, 023802 (2012). Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, [Phys. Rev. Lett.]{} **106**, 213901 (2011). R. Fleury, D. L. Sounas, and A. Alù, [Phys. Rev. Lett.]{} **113**, 023903 (2014). Ya. V. Kartashov, C. Hang, V. V. Konotop, V. A. Vysloukh, G. Huang, and L. Torner, [Laser Photon. Rev.]{} **10**, 100 (2016). S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Yu. S. Kivshar, [Laser Photon. Rev.]{} **10**, 177 (2016). V. V. Konotop, J. Yang, and D. A. Zezyulin, [Rev. Mod. Phys.]{} **88**, 035002 (2016). A. T. Lupu, H. Benisty, and A. V. Lavrinenko, [IEEE J. Sel. Top. Quant. Electr.]{} **22**, 4402807 (2016). C. Hang and G. Huang, [Adv. Phys. X]{} **2**, 737 (2017). S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, [Nat. Mater.]{} **16**, 433 (2017). Y. Shen, X. H. Deng, and L. Chen, [Opt. Express]{} **22**, 19940 (2014). J. Ramirez-Hernandez, F. M. Izrailev, and N. N. Makarov, [Phys. Rev. A]{} **96**, 013856 (2017). D. V. Novitsky, V. R. Tuz, S. L. Prosvirnin, A. V. Lavrinenko, and A. V. Novitsky, [Phys. Rev. B]{} **96**, 235129 (2017). M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, [Phys. Rev. Lett.]{} **113**, 093901 (2014). A. U. Hassan, B. Zhen, M. Soljacic, M. Khajavikhan, and D. N. Christodoulides, [Phys. Rev. Lett.]{} **118**, 093002 (2017). W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, [Nature (London)]{} **548**, 192 (2017). H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, [Nature (London)]{} **548**, 187 (2017). L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, [Science]{} **346**, 972 (2014). H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, [Science]{} **346**, 975 (2014). Z. Gu, N. Zhang, Q. Lyu, M. Li, S. Xiao, and Q. Song, [Laser Photon. Rev.]{} **10**, 588 (2016). Z. J. Wong, Y.-L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, [Nat. Photon.]{} **10**, 796 (2016). Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, [Phys. Rev. Lett.]{} **105**, 053901 (2010). S. Longhi, [Phys. Rev. A]{} **82**, 031801(R) (2010). T. Goldzak, A. A. Mailybaev, and N. Moiseyev, [Phys. Rev. Lett.]{} **120**, 013901 (2018). H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, [Phys. Rev. A]{} **82**, 043803 (2010). S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, [Opt. Lett.]{} **39**, 2603 (2014). X. Liu, S. Dutta Gupta, and G. S. Agarwal, [Phys. Rev. A]{} **89**, 013824 (2014). D. R. Barton III, H. Alaeian, M. Lawrence, and J. Dionne, [Phys. Rev. B]{} **97**, 045432 (2018). P. Witoński, A. Mossakowska-Wyszyńska, and P. Szczepański, [IEEE J. Quant. Electron.]{} **53**, 2100111 (2017). P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, [Nature (London)]{} **541**, 473 (2017). D. V. Novitsky, [Phys. Rev. A]{} **84**, 013817 (2011). M. E. Crenshaw, [Phys. Rev. A]{} **78**, 053827 (2008). N. Bloembergen, *Nonlinear Optics* (Benjamin, New York, 1965). See Supplemental Material at \[URL\] for derivation of the conditions for $\mathcal{PT}$ symmetry; additional calculations of temporal dynamics of light in $\mathcal{PT}$-symmetry broken phases; calculations for the case of completely unexcited absorbing layers. D. V. Novitsky, [Phys. Rev. A]{} **79**, 023828 (2009). E. D. Palik (ed.), *Handbook of Optical Constants of Solids* (Academic Press, San Diego, 1998). J.-C. Diels and W. Rudolph, *Ultrashort Laser Pulse Phenomena* (Academic Press, San Diego, 2nd edn, 2006). V. E. Babicheva, I. V. Kulkova, R. Malureanu, K. Yvind, and A. V. Lavrinenko, [Photon. Nanostruct. – Fund. Appl.]{} **10**, 389 (2012).

--- abstract: | Our main result (Theorem \[main theorem\]) suggests a possible dividing line ($\mu$-superstable $+$ $\mu$-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. \[main theorem\] Let ${\mathcal{K}}$ be an abstract elementary class with no maximal models of cardinality $\mu^+$ which satisfies the joint embedding and amalgamation properties. Suppose $\mu\geq\operatorname{LS}({\mathcal{K}})$. If ${\mathcal{K}}$ is $\mu$- and $\mu^+$-superstable and satisfies $\mu^+$-symmetry, then for any increasing sequence $\langle M_i\in{\mathcal{K}}_{\geq\mu^{+}}\mid i<\theta<(\sup\|M_i\|)^+\rangle$ of $\mu^+$-saturated models, ${\bigcup}_{i<\theta}M_i$ is $\mu^+$-saturated. We also apply results of [@Va3-SS] and use towers to transfer symmetry from $\mu^+$ down to $\mu$ in abstract elementary classes which are both $\mu$- and $\mu^+$-superstable: \[symmetry transfer\] Suppose ${\mathcal{K}}$ is an abstract elementary class satisfying the amalgamation and joint embedding properties and that ${\mathcal{K}}$ is both $\mu$- and $\mu^+$-superstable. If ${\mathcal{K}}$ has symmetry for non-$\mu^+$-splitting, then ${\mathcal{K}}$ has symmetry for non-$\mu$-splitting. address: | Department of Mathematics\ Robert Morris University\ Moon Township, PA, 15108, USA author: - 'M.M. VanDieren' bibliography: - 'union-of-sat-transfer-12.18.bib' title: Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes --- saturated models ,abstract elementary classes ,superstability ,splitting ,limit models 03C48 ,03C45 ,03C50 In first-order logic, the statement, the union of any increasing sequence $\langle M_i\mid i<\theta\rangle$ of saturated models is saturated, is a consequence of superstability ([@Ha] and [@She Theorem III.3.11]). In fact, the converse is also true [@AG]. Our paper provides a new first-order proof of Theorem III.3.11 of [@She] when $\kappa(T)=\aleph_0$. In abstract elementary classes (AECs), there are several approaches to generalizing superstability, and there is not yet a consensus on the correct notion. In fact it could be that superstability breaks down into several distinct dividing lines. Shelah suggests the existence of superlimits of every sufficiently large cardinality [@Sh-AECbook Chapter N Section 2] as the definition of superstability. Elsewhere he uses frames, but in his categoricity transfer results (e.g. [@Sh394]) he makes use of a localized notion more similar to $\mu$-superstability (Definition \[ss assm\]). In this paper we examine how the statement, that the union of any increasing sequence $\langle M_i\mid i<\theta\rangle$ of saturated models is saturated, and $\mu$-superstability interact in abstract elementary classes. There has been much progress in understanding the interaction. We refer the reader to the introduction of [@BoVa-tame] for an extensive review of the history of the union of saturated models and the various proposals for a definition of superstability in AECs. The most general result to date is due to Boney and Vasey for tame AECs. They prove that a version of superstability and tameness imply that the union of an increasing chain of $\mu$-saturated models is $\mu$-saturated for $\mu>\beth_\lambda=\lambda>\operatorname{LS}({\mathcal{K}})$ [@BoVa-tame Theorem 0.1]. We prove a related result here. Our result differs from [@BoVa-tame] in both assumptions and methodology. We do not assume tameness, nor the existence of arbitrarily large models, and $\mu$ does not need to be large. Our methods involve limit models (and implicitly towers) and non-splitting instead of the machinery of averages and forking. Additionally our proof is shorter. Underlying the proof of Theorem \[main theorem\] are towers. A tower is a relatively new model-theoretic concept unique to abstract elementary classes. Towers were introduced by Shelah and Villaveces [@ShVi] as a tool to prove the uniqueness of limit models and later used by VanDieren [@Va1], [@Va2] and by Grossberg, VanDieren, and Villaveces [@GVV]. A *tower* is a sequence of length $\alpha$ of limit models, denoted by $\bar M=\langle M_i\in{\mathcal{K}}_\mu\mid i<\alpha\rangle$, along with a sequence of designated elements $\bar a=\langle a_{i}\in M_{i+1}\backslash M_i\mid i+1<\alpha\rangle$ and a sequence of designated submodels $\bar N=\langle N_{i}\mid i+1<\alpha\rangle$ for which $M_i\prec_{{\mathcal{K}}}M_{i+1}$, $\operatorname{ga-tp}(a_i/M_i)$ does not $\mu$-split over $N_i$, and $M_i$ is universal over $N_i$ (see for instance Definition I.5.1 of [@Va1]). Unlike many of the model-theoretic concepts in the literature of abstract elementary classes, the concept of a tower does not have a pre-established first-order analog. Therefore there is a need to understand the applications and limitations of this concept. In [@Va3-SS], VanDieren establishes that the statement that reduced towers are continuous is equivalent to symmetry for $\mu$-superstable abstract elementary classes (see Fact \[symmetry theorem\]). Here we further explore the connection between reduced towers and symmetry by using reduced towers in the proof of Theorem \[symmetry transfer\]. We can use Theorem \[symmetry transfer\] to weaken the assumptions of Corollary 1 of [@Va3-SS] by replacing categoricity in $\mu^+$ with categoricity in $\mu^{+n}$ for some $n<\omega$ to conclude symmetry for non-$\mu$-splitting (see Corollary \[categoricity corollary\] in Section \[downward section\]). Additionally, we make progress on improving the work of [@ShVi], [@Va1], [@Va2], [@GVV], and [@Va3-SS] by proving the uniqueness of limit models of cardinality $\mu$ follows from categoricity in $\mu^{+n}$ for some $n<\omega$ without requiring tameness. The uniqueness of limit models has been explored by others, assuming tameness (e.g. [@BoGr]). On its own, transferring symmetry is an interesting property that has been studied by others. For instance, Shelah and separately Boney and Vasey transfer symmetry in a frame between cardinals under set-theoretic assumptions [@She Section II] or using some level of tameness [@BoVa-tame Section 6], respectively. Our paper differs from this work in a few ways. First, we do not assume tameness nor set-theoretic assumptions, and we do not work within the full strength of a frame. The methods of this paper include reduced towers whereas the other authors use the order property as one of many mechanisms to transfer symmetry. This line of work is further extended in [@VV]. One of the main questions surrounding this work is the interaction between the hypothesis of $\mu$-superstability, $\mu$-symmetry, the uniqueness of limit models of cardinality $\mu$, and the statement that the union of an increasing chain of $\mu$-saturated models is $\mu$-saturated. Theorem \[main theorem\] compliments [@Va3-SS] where the statement, that the union of an increasing sequence $\langle M_i\in{\mathcal{K}}_{\mu^+}\mid i<\theta\rangle$ of saturated models is saturated, implies $\mu$-symmetry. The following combination of Theorem 4 and Theorem 5 of [@Va3-SS] is close to, but not, the converse of Theorem \[main theorem\]. \[transfer theorem\] Let ${\mathcal{K}}$ be an abstract elementary class satisfying the amalgamation and joint embedding properties. Suppose ${\mathcal{K}}$ is $\mu$- and $\mu^+$-superstable. If, in addition, ${\mathcal{K}}$ satisfies the property that the union of any chain of saturated models of cardinality $\mu^+$ is saturated, then ${\mathcal{K}}$ has $\mu$-symmetry. In fact combining the results from [@Va3-SS] with the work here we get the implications depicted in Figure \[fig:ss\]. (ss) [$\mu^+$-symmetry]{}; (unique) [Uniqueness of limit models of cardinality $\mu^+$]{}; (mu-sym) [$\mu$-symmetry]{}; (ss) to node\[pos=0.5,above\] [Theorem \[symmetry transfer\]]{} node\[pos=0.5, below\][Special case of Theorem 1.1 of [@VV]]{} (mu-sym); (ss) to node\[pos=0.5,right\] [Theorem \[uniqueness thm\]]{}(unique); (union) [Union of increasing chain of $\mu^+$-saturated models is $\mu^+$-saturated]{}; (uniqueness) [Uniqueness of limit models of cardinality $\mu$]{}; (unique) to node\[pos=0.5,right\] [Theorem \[saturated theorem\]]{}(union); (union) to node\[pos=0.5,right\][Theorem 1 of [@Va3-SS]]{}(uniqueness); (mu-sym) to node\[pos=0.5,right\][Theorem \[uniqueness thm\]]{}(uniqueness); (union) to node\[pos=0.8,left\][Theorem 4 and 5 of [@Va3-SS]]{} (mu-sym); This diagram suggests several questions including: does the uniqueness of limit models of cardinality $\mu$ imply $\mu^+$-symmetry (or even $\mu$-symmetry) in $\mu$-superstable classes? There are also many questions that remain open concerning the non-structure side of any of the proposed definitions for superstability for AECs. In fact, very little is known about the implications of the failure of $\mu$-superstability. However VanDieren and Vasey have shown that with $\mu$-superstability holding in sufficiently many cardinals, failure of $\mu$-symmetry would imply the order property [@VV2], which Shelah has claimed implies many models [@Sh394]. The paper is structured as follows. Section \[sec:background\] provides some of the pre-requisite material. The subsequent section contains an observation about how saturated models and limit models are related which is key in being able to construct towers of cardinality $\mu^+$ from towers of cardinality $\mu$. This construction is the basis for the proof of Theorem \[symmetry transfer\] which appears in Section \[downward section\]. Then in Section \[sec:warm-up\] we prove a weaker result than Theorem \[main theorem\] to highlight the structure of the proof of Theorem \[main theorem\] since the construction in the proof of Theorem \[main theorem\] is more complicated requiring a directed system instead of an increasing chain. Finally, in Section \[sec:main theorem\] we prove Theorem \[main theorem\]. We finish the paper with a summary of how this work fits into the recently growing body of research on superstability in abstract elementary classes. At the suggestion of the referees, this paper is the synthesis of two preprints [@Vold-union] and [@Vold-transfer] which were disseminated in July of 2015. Background {#sec:background} ========== For the remainder of this paper we will assume that ${\mathcal{K}}$ is an abstract elementary class with no maximal models of cardinality $\mu^+$ satisfying the joint embedding and amalgamation properties. Many of the pre-requisite definitions and notation can be found in [@GVV]. Here we recall the more specialized concepts that we will be using explicitly in the proofs of Theorem \[main theorem\] and Theorem \[symmetry transfer\]. We will use the following definition of $\mu$-superstability: \[ss assm\] ${\mathcal{K}}$ is *$\mu$-superstable* if ${\mathcal{K}}$ is Galois-stable in $\mu$ and $\mu$-splitting satisfies the property: for all infinite $\alpha$, for every sequence $\langle M_i\mid i<\alpha\rangle$ of limit models of cardinality $\mu$ with $M_{i+1}$ universal over $M_i$, and for every $p\in{\operatorname{ga-S}}(M_\alpha)$, where $M_\alpha=\bigcup_{i<\alpha}M_i$, we have that there exists $i<\alpha$ such that $p$ does not $\mu$-split over $M_i$. Other definitions of $\mu$-superstability for AECs appear in the literature. For instance Vasey introduces a very similar definition of superstability with the additional requirement of no maximal models of cardinality $\mu$ [@V1 Definition 10.1]. We choose to separate this condition out to be consistent with the presentation in [@GVV], [@Va3-SS], etc. In [@Sh-AECbook Chapter N Section 2], Shelah discusses the problem of generalizing first-order superstability to AECs. There Shelah suggests using the existence of a superlimit model in every sufficiently large cardinality as a dividing line. Here we take a different, more local approach where an AEC may exhibit superstable-like properties in small cardinalities but not necessarily in larger cardinalities. This helps to classify, for instance, those classes such as the Hart-Shelah example [@HS] which have structural properties in small cardinalities but non-structural attributes in larger cardinalities. In [@VV] and [@VV2] we consider how Theorem \[main theorem\] and Theorem \[symmetry transfer\] color the global picture of superstability when one assumes categoricity or tameness. Guided by the first-order characterization of superstability that the union of an increasing chain of saturated models is saturated, Theorem \[main theorem\] provides evidence that Definition \[ss assm\] along with $\mu$-symmetry may be a reasonable generalization of superstability. \[sym defn\] We say that an abstract elementary class exhibits *symmetry for non-$\mu$-splitting* if whenever models $M,M_0,N\in{\mathcal{K}}_\mu$ and elements $a$ and $b$ satisfy the conditions \[limit sym cond\]-\[last\] below, then there exists $M^b$ a limit model over $M_0$, containing $b$, so that $\operatorname{ga-tp}(a/M^b)$ does not $\mu$-split over $N$. See Figure \[fig:sym\]. We will abbreviate this concept by *$\mu$-symmetry* when it is clear that the dependence relation is $\mu$-splitting. 1. \[limit sym cond\] $M$ is universal over $M_0$ and $M_0$ is a limit model over $N$. 2. \[a cond\] $a\in M\backslash M_0$. 3. \[a non-split\] $\operatorname{ga-tp}(a/M_0)$ is non-algebraic and does not $\mu$-split over $N$. 4. \[last\] $\operatorname{ga-tp}(b/M)$ is non-algebraic and does not $\mu$-split over $M_0$. (0,1.25) rectangle (.75,.5); (.25,.75) node [$N$]{}; (0,0) rectangle (3,1.25); (0,1.25) rectangle (1,0); (.85,.25) node [$M_0$]{}; (3.2, .25) node [$M$]{}; (0,1.25) rectangle (1.5, -.5); at (1.1,-.25)\[circle, fill, draw, label=45:$b$\] ; at (2,.75)\[circle, fill, draw, label=45:$a$\] ; (1.75,-.25) node [$M^{b}$]{}; This concept of $\mu$-symmetry was introduced in [@Va3-SS] and shown to be equivalent to a property about reduced towers (see Fact \[symmetry theorem\]). Before stating this result, let us recall a bit of terminology regarding towers. The collection of all towers $(\bar M,\bar a,\bar N)$ made up of models of cardinality $\mu$ and sequences indexed by $\alpha$ is denoted by ${\mathcal{K}}^*_{\mu,\alpha}$. For $(\bar M,\bar a,\bar N)\in{\mathcal{K}}^*_{\mu,\alpha}$, if $\beta<\alpha$ then we write $(\bar M,\bar a,\bar N)\restriction\beta$ for the tower made of the subseqences $\bar M\restriction\beta=\langle M_i\mid i<\beta\rangle$, $\bar a\restriction\beta=\langle a_i\mid i+1<\beta\rangle$, and $\bar N\restriction\beta=\langle N_i\mid i+1<\beta\rangle$. We sometimes abbreviate the tower $(\bar M,\bar a,\bar N)$ by ${\mathcal{T}}$. For towers $(\bar M,\bar a,\bar N)$ and $(\bar M',\bar a',\bar N')$ in ${\mathcal{K}}^*_{\mu,\alpha}$, we say $$(\bar M,\bar a,\bar N)\leq (\bar M',\bar a',\bar N')$$ if for all $i<\alpha$, $M_i\preceq_{{\mathcal{K}}}M'_i$, $\bar a=\bar a'$, $\bar N=\bar N'$ and whenever $M'_i$ is a proper extension of $M_i$, then $M'_i$ is universal over $M_i$. If for each $i<\alpha$, $M'_i $ is universal over $M_i$ we will write $(\bar M,\bar a,\bar N)< (\bar M',\bar a',\bar N')$. \[reduced defn\] A tower $(\bar M,\bar a,\bar N)\in{\mathcal{K}}^*_{\mu,\alpha}$ is said to be *reduced* provided that for every $(\bar M',\bar a,\bar N)\in{\mathcal{K}}^*_{\mu,\alpha}$ with $(\bar M,\bar a,\bar N)\leq(\bar M',\bar a,\bar N)$ we have that for every $i<\alpha$, $$(*)_i\quad M'_i\cap{\bigcup}_{j<\alpha}M_j = M_i.$$ The following result from [@Va3-SS] links together symmetry and reduced towers: \[symmetry theorem\] Assume ${\mathcal{K}}$ is an abstract elementary class satisfying superstability properties for $\mu$. Then the following are equivalent: 1. \[sym\] ${\mathcal{K}}$ has symmetry for non-$\mu$-splitting. 2. \[red\] If $(\bar M,\bar a,\bar N)\in{\mathcal{K}}^*_{\mu,\alpha}$ is a reduced tower, then $\bar M$ is a continuous sequence (i.e. for every limit ordinal $\beta<\alpha$, we have $M_\beta={\bigcup}_{i<\beta}M_i$). There are a few facts about reduced towers known to hold under the assumption of $\mu$-superstability. The following appears in [@Va1] as Theorem III.11.2. \[density of reduced\] Suppose ${\mathcal{K}}$ is an abstract elementary class satisfying the joint embedding and amalgamation properties. If ${\mathcal{K}}$ is $\mu$-superstable, then there exists a reduced $<$-extension of every tower in ${\mathcal{K}}^*_{\mu,\alpha}$. The next lemma is Lemma III.11.5 in [@Va1]. \[monotonicity\] Suppose ${\mathcal{K}}$ is a $\mu$-superstable abstract elementary class satisfying the joint embedding and amalgamation properties. Suppose that $(\bar M,\bar a,\bar N)\in{\mathcal{K}}^*_{\mu,\alpha}$ is reduced. If $\beta<\alpha$, then $(\bar M,\bar a,\bar N)\restriction \beta$ is reduced. Before moving onto the proofs of Theorem \[main theorem\] and Theorem \[symmetry transfer\], we state a fact about direct limits that we will use in Section \[sec:main theorem\]. It is implicit in the proof of Lemma 2.12 of [@GV]. \[direct limit lemma\] Suppose that $\theta$ is a limit ordinal and $\langle M_i\in{\mathcal{K}}_{\mu}\mid i<\theta\rangle$ and $\langle f_{i,j}\mid i\leq j<\theta\rangle$ form a directed system. If $\langle N_i\mid i<\theta\rangle$ is an increasing and continuous sequence of models so that for every $i<\theta$, $N_i\prec_{{\mathcal{K}}}M_i$ and $f_{i,i+1}\restriction N_i=\operatorname{id}_{N_i}$, then there is a direct limit $M^*$ of the system and ${\mathcal{K}}$-embeddings $\langle f_{i,\theta}\mid i<\theta\rangle$ so that ${\bigcup}_{i<\theta}N_i\preceq_{{\mathcal{K}}}M^*$ and $f_{i,\theta}\restriction N_i=id_{N_i}$. Limit and Saturated Models {#sec:limit and sat} ========================== In this section we establish that for $\mu$-superstable and $\mu$-symmetric abstract elementary classes, limit models are in fact saturated. We begin by noticing that a $(\mu,\mu^+)$-limit model[^1] is isomorphic to a $(\mu^+,\mu^+)$-limit model in $\mu^+$-stable abstract elementary classes. \[mu-plus-limit\] If ${\mathcal{K}}$ is $\mu^+$-stable and does not have a maximal model of cardinality $\mu^+$, then any $(\mu,\mu^+)$-limit model is a $(\mu^+,\mu^+)$-limit model. Let $M_{\mu^+}$ be a $(\mu,\mu^+)$-limit model witnessed by $\langle M_i\mid i\leq\mu^+\rangle$. Without loss of generality $M_{i+1}$ is a $(\mu,\omega)$-limit over $M_i$. By $\mu^+$-stability, we can fix $N$ a $(\mu^+,\mu^+)$-limit model witnessed by $\langle N'_i\mid i\leq\mu^+\rangle$ so that $M_0\prec_{{\mathcal{K}}}N_0$. Fix $\{a_i\mid i<\mu^+\}$ to be an enumeration of $N$. We will define an increasing and continuous sequence $\langle f_i\mid i\leq\mu^+\rangle$ so that 1. for $i<j\leq\mu^+$, $f_i=f_j\restriction M_i$ 2. for $i\leq \mu^+$ limit, $f_i={\bigcup}_{j<i}f_j\restriction M_j$ 3. $f_i:M_i\rightarrow N$ 4. $a_i\in\text{range}(f_{i+1}\restriction M_{i+1})$ Take $f_0=\operatorname{id}$. For $i$ limit, by the continuity of $\bar M$, we can take $f_i:={\bigcup}_{j<\mu^+}f_j\restriction M_j$. For the successor case $i=j+1$, fix $\grave f_j\in\operatorname{Aut}({\mathfrak{C}})$ an extension of $f_j$. Let $k<\mu^+$ be such that $a_j\in N_k$ and $N_k$ is universal over $\grave f_j(M_j)$. Let $M'_{j+1}\prec_{{\mathcal{K}}}\grave f^{-1}_j(N_k)$ be a $(\mu,\omega)$-limit over $M_i$ containing $\grave f^{-1}_j(a_j)$. This is possible since $N_k$ is universal over $\grave f_j(M_j)$. By the uniqueness of $(\mu,\omega)$-limit models, there exists $g:M'_{j+1}\cong_{M_j}M_{j+1}$. Now take $f_{j+1}:=\grave f_j\circ g^{-1}\restriction M_{j+1}$. Notice $f_{j+1}\restriction M_{j}=f_j\restriction M_j$ since $g^{-1}$ fixes $M_j$. Also, by our choice of $g$, $a_j\in f_{j+1}[M_{j+1}]$ as required. Notice that $f_{\mu^+}$ is an isomorphism between $M_{\mu^+}$ and $N$. The direct approach of constructing a saturated model is to realize all the relevant types. Another method is to show that the model is a limit model and depending on the context, there are times when limit models are saturated. Trivially, a $(\mu,\mu^+)$-limit model is saturated. Moreover, if the class ${\mathcal{K}}$ satisfies the condition > for every $l\in\{1,2\}$, and every pair of limit ordinals $\theta_l<\mu^+$, and pair of $(\mu,\theta_l)$-limit models $M_l$, we have $M_1\cong M_2$, then any limit model of cardinality $\mu$ is also saturated. To see this, suppose $M$ is a $(\mu,\theta)$-limit model and fix $\chi<\mu$ and $N\in{\mathcal{K}}_\chi$ with $N\prec_{{\mathcal{K}}}M$. By uniqueness of limit models, we can think of $M$ as $(\mu,\chi^+)$-limit model witnessed by $\langle M_i\mid i<\chi^+\rangle$. The model $N$ appears in one of the $M_i$, so $M_{i+1}$ will realize all the types over $M_i$, and hence over $N$. In our context, under the hypothesis of Theorem \[main theorem\], we have uniqueness of limit models of cardinality $\mu^+$: \[uniqueness thm\] Let ${\mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\mu$ is a cardinal $\geq\operatorname{LS}({\mathcal{K}})$ and $\theta_1$ and $\theta_2$ are limit ordinals $<\mu^+$. If ${\mathcal{K}}$ is $\mu$-superstable and satisfies $\mu$-symmetry, then for $M_1$ and $M_2$ which are $(\mu, \theta_1)$ and $(\mu,\theta_2)$-limit models over $N$, respectively, we have that $M_1$ is isomorphic to $M_2$ over $N$. Moreover the limit model of cardinality $\mu$ is saturated. This is just a restatement of Theorem 5 of [@Va3-SS] and the proof of Theorem 1.9 of [@GVV]. Combining Theorem \[uniqueness thm\] with Proposition \[mu-plus-limit\], we get the following corollary. \[limit is sat\] Let ${\mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\kappa$ is a cardinal $\geq\operatorname{LS}({\mathcal{K}})$, and $\theta$ is limit ordinal $<\kappa^{++}$. If ${\mathcal{K}}$ is $\kappa$-stable, $\kappa^+$-superstable and satisfies $\kappa^+$-symmetry, then any $(\kappa^+,\theta)$-limit model is also a $(\kappa,\kappa^+)$-limit model. Downward Symmetry Transfer {#downward section} ========================== In this section we provide the proof of Theorem \[symmetry transfer\]. While the result follows from Theorem 4 and 5 of [@Va3-SS], we include the proof here for completeness since [@Va3-SS] is currently under review and has not yet been published. Additionally, the proof of Theorem \[symmetry transfer\] serves as the blueprint for the successor step for a more general result of transferring symmetry downward that appears in the unpublished work [@VV]. In the proof of Theorem \[symmetry transfer\], we will be using towers composed of models of cardinality $\mu$ and other towers composed of models of cardinality $\mu^+$. These towers will be based on the same sequence of elements $\langle a_\beta\mid \beta<\delta\rangle$. To distinguish the towers of models of size $\mu^+$ from those of size $\mu$, we will use different notation. The models of cardinality $\mu^+$ will be decorated with an asterisk ($*$), accent ($\grave{}$), or a $\mu^+$ in the superscript. All other models in this proof will have cardinality $\mu$. Suppose ${\mathcal{K}}$ does not have symmetry for $\mu$-non-splitting. By Fact \[symmetry theorem\] and the $\mu$-superstability assumption, ${\mathcal{K}}$ has a reduced discontinuous tower. Let $\alpha$ be the minimal ordinal such that ${\mathcal{K}}$ has a reduced discontinuous tower of length $\alpha$. By Fact \[monotonicity\], we may assume that $\alpha=\delta+1$ for some limit ordinal $\delta$. Fix ${\mathcal{T}}=(\bar M,\bar a,\bar N)\in{\mathcal{K}}^*_{\mu,\alpha}$ a reduced discontinuous tower with $b\in M_\delta\backslash {\bigcup}_{\beta<\delta}M_\beta$. By Fact \[density of reduced\] and minimality of $\alpha$, we can build an increasing and continuous chain of reduced, continuous towers $\langle{\mathcal{T}}^i\mid i<\mu^+\rangle$ extending ${\mathcal{T}}\restriction\delta$. For each $\beta<\delta$, set $M^{\mu^+}_\beta:={\bigcup}_{i<\mu^+}M^i_\beta$. Notice that for each $\beta<\delta$ $$\label{non-split} \operatorname{ga-tp}(a_\beta/M^{\mu^+}_\beta)\text{ does not }\mu\text{-split over }N_\beta.$$ If $\operatorname{ga-tp}(a_\beta/M^{\mu^+}_\beta)$ did $\mu$-split over $N_\beta$, it would be witnessed by models inside some $M^i_\beta$, contradicting the fact that $\operatorname{ga-tp}(a_\beta/M^{i}_\beta)$ does not $\mu$-split over $N_\beta$. We will construct a tower in ${\mathcal{K}}^*_{\mu^+,\delta}$ from $\bar M^{\mu^+}$. Notice that by construction, each $M^{\mu^+}_\beta$ is a $(\mu,\mu^+)$-limit model. Therefore by Proposition \[mu-plus-limit\], each $M^{\mu^+}_\beta$ is a $(\mu^+,\mu^+)$-limit model. Fix $\langle \grave M^i_\beta\mid i<\mu^+\rangle$ witnessing that $M^{\mu^+}_\beta$ is a $(\mu^+,\mu^+)$-limit model. Without loss of generality we can assume that $N_\beta\prec_{{\mathcal{K}}}\grave M^0_\beta$. By $\mu^+$-superstability we know that for each $\beta<\delta$ there is $i(\beta)<\mu^+$ so that $\operatorname{ga-tp}(a_\beta/M^{\mu^+}_\beta)$ does not $\mu^+$-split over $\grave M^{i(\beta)}_\beta$. Set $N^{\mu^+}_\beta:=\grave M^{i(\beta)}_\beta$. Notice that $(\bar M^{\mu^+},\bar a,\bar N^{\mu^+})$ is a tower in ${\mathcal{K}}^*_{\mu^+,\delta}$. Extend $(\bar M^{\mu^+},\bar a,\bar N^{\mu^+})$ to a tower ${\mathcal{T}}^{\mu^+}\in{\mathcal{K}}^*_{\mu^+,\alpha}$ by appending to $\bar M^{\mu^+}$ a $\mu^+$-limit model universal over $M_\delta$ which contains ${\bigcup}_{\beta<\delta}M^{\mu^+}_\beta$. Since ${\mathcal{T}}^{\mu^+}$ is discontinuous, by Fact \[symmetry theorem\] and our $\mu^+$-symmetry assumption, we know that it is not reduced. However, by our $\mu^+$-symmetry assumption, Fact \[symmetry theorem\] and Fact \[density of reduced\] imply that there exists a reduced, continuous tower ${\mathcal{T}}^*\in{\mathcal{K}}^*_{\mu^+,\alpha}$ extending ${\mathcal{T}}^{\mu^+}$. By multiple applications of Fact \[density of reduced\], we may assume that in ${\mathcal{T}}^*$ each $M^*_\beta$ is a $(\mu^+,\mu^+)$-limit over $M^{\mu^+}_\beta$. See Fig. \[fig:tower\]. (0,1.5) rectangle (.75,.5); (0,1.5) rectangle (1.75,1); (.25,.65) node [$N_0$]{}; (1.25,1.1) node [$N_\beta$]{}; (0, 1.5) –(0,.75)– (1,-1) – (1.5,-1) – (1.75,1)–(1.75,1.5)– cycle; (1.85,.7) node [$N^{\mu^+}_\beta$]{}; (0,0) rectangle (4,1.5); (.85,.25) node [$M_0$]{}; (1.4,.25) node [$M_1$]{}; (1.8,.25) node [$\dots M_\beta$]{}; (2.35,.25) node [$M_{\beta+1}$]{}; (3.15,.2) node [$\dots\displaystyle{{\bigcup}_{k<\delta}M_k}$]{}; (3.85, .25) node [$M_\delta$]{}; (-.5,.25) node [$(\bar M,\bar a,\bar N)$]{}; (0,1.5) rectangle (3.5, -.4); (0, 1.5) – (0,-2) – (3.6,-2) – (4,0)–(4,1.5) – cycle; (0, 1.5) – (0,-1.35) – (3.5,-1.35) – (4,0)–(4,1.5) – cycle; (.85,-.15) node [$M^{i}_0$]{}; (1.8,-.15) node [$\dots M^{i}_\beta$]{}; (2.35,-.15) node [$M^{i}_{\beta+1}$]{}; (1.4,-.15) node [$M^{i}_1$]{}; (3.15,-.2) node [$\dots\displaystyle{{\bigcup}_{l<\delta}M^{i}_l}$]{}; (-.5,-.15) node [${\mathcal{T}}^i$]{}; (.85,-.6) node [$\vdots$]{}; (1.75,-.6) node [$\vdots$]{}; (2.35,-.6) node [$\vdots$]{}; (3.2,-.6) node [$\vdots$]{}; (1.35,-.6) node [$\vdots$]{}; (0,1.5) rectangle (3.5, -1.35); (0,1.5) rectangle (1,-2); (0,1.5) rectangle (1.5, -2); (0,1.5) rectangle (2.5, -2); (0,1.5) rectangle (2,-2); (.8,-1.15) node [$M^{\mu^+}_0$]{}; (1.8,-1.15) node [$ M^{\mu^+}_\beta$]{}; (2.3,-1.15) node [$M^{\mu^+}_{\beta+1}$]{}; (1.35,-1.15) node [$M^{\mu^+}_1$]{}; (3.1,-1.2) node [$\dots\displaystyle{{\bigcup}_{l<\delta}M^{\mu^+}_l}$]{}; (-.5,-1.15) node [${\mathcal{T}}^{\mu^+}$]{}; (-.5,-1.75) node [${\mathcal{T}}^{*}$]{}; (.8,-1.75) node [$M^{*}_0$]{}; (1.4,-1.75) node [$M^*_1$]{}; (1.8,-1.75) node [$ M^{*}_\beta$]{}; (2.3,-1.75) node [$M^{*}_{\beta+1}$]{}; at (3.75,.75)\[circle, fill, draw, label=90:$b$\] ; at (2.25,.65)\[circle, fill, draw, label=290:$a_\beta$\] ; at (1.1,.65)\[circle, fill, draw, label=290:$a_1$\] ; (3.65, -.6) node [$M^{\mu^+}_\delta$]{}; (3.1,-1.8) node [$\dots\displaystyle{{\bigcup}_{\beta<\delta}M^*_\beta}=M^*_\delta$]{}; \[star non-split\] For every $\beta<\alpha$, $\operatorname{ga-tp}(a_\beta/M^*_\beta)$ does not $\mu$-split over $N_\beta$. Since $M^*_\beta$ and $M^{\mu^+}_\beta$ are both $(\mu^+,\mu^+)$-limit models over $N^{\mu^+}_\beta$, there exists $f:M^*_\beta\cong_{N^{\mu^+}_\beta}M^{\mu^+}_\beta$. Since ${\mathcal{T}}^*$ is a tower extending ${\mathcal{T}}^{\mu^+}$, we know that $\operatorname{ga-tp}(a_\beta/M^*_\beta)$ does not $\mu^+$-split over $N^{\mu^+}_\beta$. Therefore by the definition of non-splitting, it must be the case that $\operatorname{ga-tp}(f(a_\beta)/M^{\mu^+}_\beta)=\operatorname{ga-tp}(a_\beta/M^{\mu^+}_\beta)$. From this equality of types we can fix $g\in\operatorname{Aut}_{M^{\mu^+}_\beta}({\mathfrak{C}})$ with $g(f(a_\beta))=a_\beta$. An application of $(g\circ f)^{-1}$ to $(\ref{non-split})$ yields the statement of the claim. Since ${\mathcal{T}}^*$ is continuous and extends ${\mathcal{T}}^{\mu^+}$ which contains $b$, there is $\beta<\delta$ such that $b\in M^*_\beta$. Fix such a $\beta$. We now will define a tower ${\mathcal{T}}^b\in{\mathcal{K}}^*_{\mu,\alpha}$ extending ${\mathcal{T}}$. For $\gamma<\beta$, take $M^b_\gamma:=M_\gamma$. For $\gamma=\beta$, let $M^b_\gamma$ be a $(\mu,\mu)$-limit model over $M_\gamma$ inside $M^*_\gamma$ so that $b\in M^b_\gamma$. For $\gamma>\beta$, take $M^b_\gamma$ to be a $(\mu,\mu)$-limit model over $M_\gamma$ so that ${\bigcup}_{\xi<\gamma}M^b_{\xi}\prec_{{\mathcal{K}}}M^b_\gamma$. Notice that by Claim \[star non-split\] and monotonicity of non-splitting, the tower ${\mathcal{T}}^b$ defined as $(\bar M^b,\bar a,\bar N)$ is a tower extending ${\mathcal{T}}$ with $b\in (M^b_\beta\backslash M_\beta)\bigcap M_\alpha$. This contradicts our assumption that ${\mathcal{T}}$ was reduced. The following is a strengthening of Corollary 1 from [@Va3-SS]. In particular, here we replace the assumption that ${\mathcal{K}}$ is categorical in $\mu^+$ with the statement: ${\mathcal{K}}$ is categorical in $\mu^{+n}$ for some $n<\omega$. \[categoricity corollary\] Suppose that ${\mathcal{K}}$ satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Fix $\mu$ a cardinal $\geq\operatorname{LS}({\mathcal{K}})$. If ${\mathcal{K}}$ is categorical in $\lambda=\mu^{+n}$, then ${\mathcal{K}}$ has symmetry for non-$\mu$-splitting. Notice that categoricity in $\lambda$ and the existence of arbitrarily large models allows us to make use of EM-models. These assumptions imply stability in $\kappa$ for $\kappa=\mu^{+k}$ with $0\leq k<n$ (see for instance Theorem 8.2.1 of [@Ba]). Also, $\kappa$-superstability for $\kappa=\mu^{+k}$ for $0\leq k<n$ follows from categoricity by the argument of Theorem 2.2.1 of [@ShVi]. While [@ShVi] uses the assumption of GCH, it can be eliminated here because we are assuming the amalgamation property [@GVas Theorem 6.3]. By Corollary 1 of [@Va3-SS], we get symmetry for non-$\mu^{+(n-1)}$-splitting. Then, Theorem \[symmetry transfer\] gives us symmetry for non-$\mu^{k}$-splitting for the remaining $0\leq k<n-1$. Using Corollary \[categoricity corollary\], we add to the line of work on the uniqueness of limit models by deriving a relative of the main result, Theorem 1.9, of [@GVV] and Theorem 1 of [@Va3-SS]. Suppose that ${\mathcal{K}}$ satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Fix $\mu$ a cardinal $\geq\operatorname{LS}({\mathcal{K}})$. If ${\mathcal{K}}$ is categorical in $\mu^{+n}$, then for each $0<k<n$, and limit ordinals $\theta_1,\theta_2<\mu^{+(k+1)}$, if $M_1$ and $M_2$ are $(\mu^{+k},\theta_1)$- and $(\mu^{+k},\theta_2)$-limit models over $N$, respectively, then $M_1$ is isomorphic to $M_2$ over $M$. This follows from Corollary \[categoricity corollary\], Fact \[symmetry theorem\], and the arguments of [@GVV] which show that superstability plus the statement that reduced towers are continuous is enough to get uniqueness of limit models in a given cardinality. Union of Saturated Models: warm-up {#sec:warm-up} ================================== The goal of this section is to prove the following warm-up to Theorem \[main theorem\]. \[limit thm\] Let ${\mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose that $\lambda$ and $\mu$ are cardinals $\geq\operatorname{LS}({\mathcal{K}})$ with $\lambda\geq\mu^{++}$ and that $\theta$ is a limit ordinal $<\lambda^+$. If ${\mathcal{K}}$ is $\mu^+$-superstable and satisfies $\mu^+$-symmetry, then for any increasing sequence $\langle M_i\mid i<\theta\rangle$ of $\mu^{++}$-saturated models of cardinality $\lambda$, $M={\bigcup}_{i<\theta}M_i$ is $\mu^+$-saturated. Notice that the statement of Theorem \[limit thm\] differs from Theorem \[main theorem\] in two ways. The cardinality, $\lambda$, of the saturated models in the chain is greater than or equal to the level of saturation, $\mu^{++}$, of the models $M_i$. Also, the level of saturation that we get in the union is only $\mu^+$. The proof of this theorem will prepare us for a similar construction used in the proof Theorem \[main theorem\] with the addition of a directed system. Given $N\prec_{{\mathcal{K}}}{\bigcup}_{i<\theta}M_i$ of cardinality $\mu$, the structure of the proof is to construct an increasing chain $\langle M^*_i\mid i<\theta\rangle$ of models of cardinality $\mu^+$ inside ${\bigcup}_{i<\theta}M_i$ so that $M^*:={\bigcup}_{i<\theta}M^*_i$ contains $N$ and so that $M^*_{i+1}$ is universal over $M^*_i$. Then by definition of limit models, $M^*$ is a $(\mu^+,\theta)$-limit model. By Theorem \[uniqueness thm\], $M^*$ is saturated, and every type over $N$ is realized in $M^*$ and hence in ${\bigcup}_{i<\theta}M_i$. First observe that we may assume that the sequence $\langle M_i\mid i<\theta\rangle$ is continuous. Otherwise, we could consider $\langle M_i\mid i<\theta\rangle$ a counter-example of the theorem of minimal length and proceed to prove the theorem by contradiction using the argument below. Fix $N\in{\mathcal{K}}_\mu$ with $N\prec_{{\mathcal{K}}}M$ and $p\in {\operatorname{ga-S}}(N)$. We will show that $p$ is realized in $M$. Notice that if $\operatorname{cf}(\theta)\geq\mu^+$, the result follows easily. If $\operatorname{cf}(\theta)\geq\mu^+$, then $N\prec_{{\mathcal{K}}}M_\alpha$ for some $i<\theta$. Because $M_i$ is $\mu^{++}$-saturated, $p$ is realized in $M_i$. So, let us consider the more interesting case that $\operatorname{cf}(\theta)<\mu^+$. Our goal is to define a sequence of models $\langle M^*_i\mid i<\operatorname{cf}(\theta)\rangle$ inside $M$ so that $M^*_{i+1}$ is universal over $M^*_i$ and so that $M^*:={\bigcup}_{i<\operatorname{cf}(\theta)}M^*_i$ contains $N$. Suppose for the sake of contradiction that $p$ is omitted in $M$. Then we can, by increasing the universe of $N$ if necessary, use the Downward Löwenheim-Skolem axiom to find $\langle N_i\in{\mathcal{K}}_\mu\mid i<\theta\rangle $ an increasing and continuous resolution of $N$ so that $N_i\prec_{{\mathcal{K}}}M_i$ for each $i<\theta$. We define an increasing and continuous sequence $\langle M^*_{i}\mid i<\theta\rangle$ so that for $i<\theta$: 1. $M^*_{i}\in{\mathcal{K}}_{\mu^+}$ is a limit model. 2. $N_i\prec_{{\mathcal{K}}}M^*_{i}$. 3. $M^*_i\prec_{{\mathcal{K}}}M_i$. 4. \[univ over condition\] $M^*_{i+1}$ is a universal over $M^*_{i}$. This construction is straightforward since each $M_i$ is $\mu^{++}$-saturated and hence universal over every submodel of cardinality $\mu^+$. We are assuming $\mu^+$-stability, so limit models of cardinality $\mu^+$ exist. Therefore $M_0$ contains a $(\mu^+,\omega)$-limit model containing $N_0$. Let this be $M^*_0$. Suppose $M^*_i$ has been defined. Let $M^{**}$ be a submodel of $M_{i+1}$ of cardinality $\mu^+$ containing $N_{i+1}{\bigcup}M^*_i$. Because $M^*_{i+1}$ is $\mu^{++}$ saturated, it is $\mu^+$-universal over $M^{**}$, and therefore it contains a model $M^*_{i+1}$ of cardinality $\mu^+$ universal over $M^{**}$. At limit ordinals $i$, we can take unions since both the sequences $\bar M$ and $\bar N$ are continuous. Let $M^*:={\bigcup}_{i<\theta}M^*_i$. By condition \[univ over condition\] of the construction, $M^*$ is a $(\mu^+,\theta)$-limit model. Since we assume $\mu^+$-symmetry and $\mu^+$-superstability, we can apply Theorem \[uniqueness thm\] to conclude that this $(\mu^+,\theta)$-limit model is $\mu^+$-saturated. Thus $p$ is realized in $M^*$, and consequently in $M$ as required. A similar proof to Theorem \[limit thm\] for a result related to Corollary \[limit cor\] are found in [@Ba Theorem 10.22]. \[limit cor\] Let ${\mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\lambda>\operatorname{LS}({\mathcal{K}})$ is a limit cardinal and $\theta$ is a limit ordinal $<\lambda^+$. If ${\mathcal{K}}$ is $\mu^+$-superstable and satisfies $\mu^+$-symmetry for unboundedly many $\mu<\lambda$, then for any increasing and continuous sequence $\langle M_i\mid i<\theta\rangle$ of $\lambda$-saturated models, ${\bigcup}_{i<\theta}M_i$ is $\lambda$-saturated. Union of Saturated Models {#sec:main theorem} ========================= In this section we prove Theorem \[main theorem\], by proving a slightly stronger statement. Notice that Theorem \[uniqueness thm\] and Theorem \[saturated theorem\] together imply Theorem \[main theorem\]. \[saturated theorem\] Let ${\mathcal{K}}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $\mu\geq\operatorname{LS}({\mathcal{K}})$ is a cardinal. If ${\mathcal{K}}$ is $\mu$- and $\mu^+$-superstable and satisfies the property that all limit models of cardinality $\mu^+$ are isomorphic, then for any increasing sequence $\langle M_i\in{\mathcal{K}}_{\geq\mu^{+}}\mid i<\theta<(\sup\|M_i\|)^+\rangle$ of $\mu^+$-saturated models, ${\bigcup}_{i<\theta}M_i$ is $\mu^+$-saturated. The proof is similar to the proof of Theorem \[limit thm\], only here the construction of $\langle M^*_i\mid i<\theta\rangle$ inside $M:={\bigcup}_{i<\theta}M_i$ is a little more nuanced since the cardinality of $M^*_i$ and the cardinality of the saturated models $M_i$ may be the same. We will be using directed limits, and while we won’t arrange that the limit of the directed system of $\langle M^*_i\mid i<\theta\rangle$ lies in $M$, we will get the most critical part, the realization of the type, to lie in $M$. As in the first paragraphs of the proof of Theorem \[limit thm\], we may assume without loss of generality that the sequence $\langle M_i\in{\mathcal{K}}_{\geq\mu^{+}}\mid i<\theta<(\sup\|M_i\|)^+\rangle$ is continuous and that $\operatorname{cf}(\theta)=\theta<\mu^+$. Fix $N\in{\mathcal{K}}_\mu$ with $N \prec_{{\mathcal{K}}}{\bigcup}_{i<\theta}M_i$ and suppose $p\in{\operatorname{ga-S}}(N)$ is omitted in $M:={\bigcup}_{i<\theta}M_i$. Then, because each $M_{i+1}$ is $\mu^+$-saturated, we may assume without loss of generality that $N$ is a $(\mu,\theta)$-limit model witnessed by $\langle N_i\mid i<\theta\rangle$ with $N_i\prec_{{\mathcal{K}}}M_i$, if necessary by expanding $N$. Furthermore by $\mu$-superstability we may assume that $p$ does not $\mu$-split over some $\check N$ with $N_0$ a limit model over $\check N$, by renumbering the sequences $\bar N$ and $\bar M$ if necessary. For each $i<\theta$, because $M_i$ is $\mu^+$-saturated, we can find a sequence $\langle \mathring M_i^\alpha\in{\mathcal{K}}_{\mu}\mid \alpha<\mu^+\rangle$ so that $\mathring M^0_i=N_i$ , $\mathring M^\alpha_i\prec_{{\mathcal{K}}}M_i$, and $\mathring M^{\alpha+1}_i$ is $\mu$-universal over $\mathring M_i^\alpha$. Therefore $M_i$ contains a $(\mu,\mu^+)$-limit model, which is isomorphic to a $(\mu^+,\mu^+)$-limit model by Proposition \[mu-plus-limit\]. So, inside each $M_i$ we can find a $(\mu^+,\mu^+)$-limit model witnessed by a sequence that we will denote by $\langle\grave M_i^\alpha\in{\mathcal{K}}_{\mu^+}\mid \alpha<\mu^+\rangle$, and we may arrange the enumeration so that $N_i\prec_{{\mathcal{K}}}\grave M^0_i$. We will build a directed system of models $\langle M^*_i\mid i<\theta\rangle$ with mappings $\langle f_{i,j}\mid i\leq j<\theta\rangle$ so that the following conditions are satisfied: 1. $M^*_i\in{\mathcal{K}}_{\mu^+}$. 2. $M^*_i\preceq_{{\mathcal{K}}}{\bigcup}_{\alpha<\mu^+}\grave M_i^\alpha\preceq_{{\mathcal{K}}}M_i$. 3. for $i\leq j<\theta$, $f_{i,j}:M^*_i\rightarrow M^*_j$. 4. \[identity condition\] for $i\leq j<\theta$, $f_{i,j}\restriction N_i=id_{N_i}$. 5. \[univ condition direct limit\] $M^*_{i+1}$ is universal over $f_{i,i+1}(M^*_i)$. Refer to Figure \[fig:T\*\]. (0,0) rectangle (4.5,-.5); (0,0) rectangle (4.5,-2); (0,0) rectangle (1,-2); (0,0) rectangle (2,-2); (0,0) rectangle (3,-2); (.85,-.4) node [$N_0$]{}; (1.75,-.4) node [$\dots N_j$]{}; (2.8,-.4) node [$N_{j+1}$]{}; (3.85,-.4) node [$\dots {\bigcup}_{i<\theta}N_i=N$]{}; (.8,-1.9) node [$M_0$]{}; (1.7,-1.9) node [$\dots M_j$]{}; (2.7,-1.9) node [$M_{j+1}$]{}; (3.85,-1.9) node [$\dots {\bigcup}_{i<\theta}M_i=M$]{}; (m01-in) at (0,-.9); (m01-out) at (1,-.9); (m01-in) to\[out=-20,in=200\] coordinate\[pos=0.7\](A1) (m01-out); (.6,-.9) node [$M^*_{0}$]{}; (m10-in) at (0,-.5); (m10-out) at (2,-.5); (m10-in) to\[out=-20,in=240\] coordinate\[pos=0.7\](Ai) coordinate\[pos=.8\](ai)(m10-out); (1.4,-.7) node [$M^*_{j}$]{}; (m1t-in) at (0,-.4); (m1t-out) at (3,-0.5); (m1t-in) to\[out=-80,in=240\] coordinate\[pos=0.8\](Ai1) (m1t-out); (2.6,-.7) node [$\grave M^1_{j+1}$]{}; (A1) to \[bend right=65\] node\[pos=0.7,below\] [$f_{0,j}$]{}(Ai); (ai) to \[bend right=25\] node\[pos=0.7,above\] [$f_{j,j+1}$]{}(Ai1); (m1t-in) to (.2, -1.5) to (2.2,-1.8) to coordinate\[pos=0.8\](Ai1) (m1t-out); (1.5,-1.55) node [$\grave M^2_{j+1}=M^*_{j+1}$]{}; The construction is possible. Take $M^*_0$ to be $\grave M_0^1$ and $f_{0,0}=\operatorname{id}$. At limit stages take $M^{**}_i$ and $\langle f^{**}_{k,i}\mid k<i\rangle$ to be a direct limit as in Fact \[direct limit lemma\]. We do not immediately get that $M^{**}_i\preceq_{{\mathcal{K}}}M_i$; we just know we can choose $M^{**}_i$ to contain $N_i$ by the continuity of $\bar N$ and condition \[identity condition\] of the construction. We also know by condition \[univ condition direct limit\] that $M^{**}_i$ is a $(\mu^+,i)$-limit model witnessed by $\langle f_{k,i}(M^*_k)\mid k<i\rangle$. By our assumption of the uniqueness of limit models of cardinality $\mu^+$, $M^{**}_i$ is a $(\mu^+,\mu^+)$-limit model. Since $N_i$ has cardinality $\mu$, being able to write $M^{**}_i$ as a $(\mu^+,\mu^+)$-limit model tells us that $M^{**}_i$ is $\mu^+$-universal over $N_i$. Recall that ${\bigcup}_{\alpha<\mu^+}\grave M^\alpha_i$ is also a $(\mu^+,\mu^+)$-limit model containing $N_i$. Therefore, we can find an isomorphism $g$ from $M^{**}_i$ to ${\bigcup}_{\alpha<\mu^+}\grave M^\alpha_i$ fixing $N_i$. Now take $M^*_i:=g(M^{**}_i)={\bigcup}_{\alpha<\mu^+}\grave M^\alpha_i$, $f_{k,i}:=g\circ f^{**}_{k,i}$ for $k<i$, and $f_{i,i}=\operatorname{id}$. For the successor stage of the construction, assume that $M^*_j$ and $\langle f_{k,j}\mid k\leq j\rangle$ have been defined. Since $M^*_j$ is a model of cardinality $\mu^+$ containing $N_j$ and because $\grave M^{1}_{j+1}$ is $\mu^+$-universal over $N_{j+1}$ we can find a embedding $g:M^*_j\rightarrow \grave M^1_{j+1}$ with $g\restriction N_j=\operatorname{id}_{N_j}$. Take $M^*_{j+1}:=\grave M^2_{j+1}$, set $f_{k,j+1}:=g\circ f_{k,j}$ for all $k\leq j$, and define $f_{j+1,j+1}:=\operatorname{id}$. This completes the construction. Take $M^*$ with mappings $\langle f_{i,\theta}\mid i<\theta\rangle$ to be the direct limit of the system as in Fact \[direct limit lemma\]. While $M^*$ may not be inside $M$, we can arrange that $f_{i,\theta}\restriction N_i=\operatorname{id}_{N_i}$ and that $N\prec_{{\mathcal{K}}}M^*$. Notice that by condition \[univ condition direct limit\] of the construction, $M^*$ is a $(\mu^+,\theta)$-limit model. From our assumption of the uniqueness of $\mu^+$-limit models and Proposition \[mu-plus-limit\], we can conclude that $M^*$ is saturated. For each $i<\theta$, let $f^*_{i,\theta}\in\operatorname{Aut}({\mathfrak{C}})$ extend $f_{i,\theta}$ so that $f^*_{i,\theta}(N)\preceq_{{\mathcal{K}}}M^*$. This is possible since we know that $M^*$ is $\mu^+$-universal over $f_{i,\theta}(M^*_i)$ by condition \[univ condition direct limit\] of the construction. Let $N^*\prec_{{\mathcal{K}}}M^*$ be a model of cardinality $\mu$ extending $N$ and ${\bigcup}_{i<\theta} f^*_{i,\theta}(N)$. By the extension property for non-$\mu$-splitting, we can find $p^*\in{\operatorname{ga-S}}(N^*)$ extending $p$ so that $$\label{p*} p^*\text{ does not }\mu\text{-split over }\check N.$$ Since $M^*$ is a saturated model of cardinality $\mu^+$ containing the domain of $p^*$, we can find $b^*\in M^*$ realizing $p^*$. By the definition of a direct limit, there exists $i<\theta$ and $b\in M^*_i$ so that $f_{i,\theta}(b)=b^*$. Because $f_{i,\theta}\restriction N_i=id_{N_i}$, we know that $b\models p\restriction N_i$. Suppose for sake of contradiction that there is some $j>i$ so that $\operatorname{ga-tp}(b/N_j)\neq p\restriction N_j$. Then, by the uniqueness of non-splitting extensions, it must be the case that $\operatorname{ga-tp}(b/N_j)$ $\mu$-splits over $\check N$. By invariance, $$\label{non-split equation} \operatorname{ga-tp}(f_{i,\theta}(b)/f^*_{i,\theta}(N_j)) \;\mu\text{-splits over }\check N.$$ By monotonicity of non-splitting, the definition of $b$, and choice of $N^*$ containing $f^*_{i,\theta}(N)$, $(\ref{non-split equation})$ implies $\operatorname{ga-tp}(b^*/N^*)$ $\mu$-splits over $\check N$. This contradicts $(\ref{p*})$. Since $b\models p\restriction N_j$ for all $j<\theta$ and $p\restriction N_j$ does not $\mu$-split over $\check N$, $\mu$-superstability implies that $\operatorname{ga-tp}(b/N)$ does not $\mu$-split over $\check N$. By uniqueness of non-$\mu$-splitting extensions $\operatorname{ga-tp}(b/N)=p$. Since $b\in M_i$, we are done. Concluding Remarks ================== The characterization of $\mu$-symmetry by reduced towers in [@Va3-SS] spawned many results during the summer of 2015, including the work here. While these new results deal with some of the same concepts (towers, superstability, limit models, union of saturated models), the contexts and methods differ. The focus here is in local properties of the classes ${\mathcal{K}}_\mu$ and ${\mathcal{K}}_{\mu^+}$ without assuming categoricity, tameness, or sufficiently large cardinals. In this section, we summarize how some of the other results relate to Theorem \[main theorem\] and Theorem \[symmetry transfer\]. Most closely related to Theorem \[symmetry transfer\] is [@VV] where the authors develop a more nuanced technology of towers. The structure of the proof of Theorem \[symmetry transfer\] involves taking a tower ${\mathcal{T}}\in{\mathcal{K}}^*_{\mu,\alpha}$ and building from it a tower in ${\mathcal{K}}^*_{\mu^+,\alpha}$. VanDieren and Vasey show that it is possible to carry out this kind of construction to produce a tower in ${\mathcal{K}}^*_{\lambda,\alpha}$ for $\lambda>\mu^+$ [@VV] if one assumes $\kappa$-superstability for an interval of cardinals. The consequent improvements of Theorem \[symmetry transfer\] and its corollaries to more global properties of the class are explored in [@VV]. Another paper using this technology of towers is [@Bo-Van] in which the authors, Boney and VanDieren, study the implications of Theorem \[main theorem\] and Theorem \[symmetry transfer\] in classes that are $\mu$-stable but not $\mu$-superstable. In just a few months after the introduction of $\mu$-symmetry and its equivalent formulation and the announcement of Theorem \[main theorem\], several advances have been made. Theorem \[main theorem\] has broken down a door in the development of a classification theory for abstract elementary classes assuming additional properties on the class like tameness or additional structural properties like categoricity. VanDieren and Vasey examine Theorem \[main theorem\] in *tame* abstract elementary classes and use it to show the existence of a unique type-full good $\mu^+$-frame in a $\mu$-superstable, $\mu$-tame AEC [@VV]. This analysis is then used by VanDieren and Vasey to improve structural results for AECs categorical in a sufficiently large cardinality. For example, they show that for ${\mathcal{K}}$ an AEC with no maximal models and $\mu$ is a cardinal $\geq\operatorname{LS}({\mathcal{K}})$, if ${\mathcal{K}}$ is categorical in a $\lambda\geq h(\mu^+)$, then the model of size $\lambda$ is $\mu^+$-saturated [@VV2]. The union of saturated models is saturated is employed by Vasey to prove the equivalence of the existence of prime models and categoricity in a tail of cardinals in categorical, tame, and short AECs [@V-prime]. Furthermore, Vasey in [@V-downward] uses Theorem \[main theorem\] in a crucial way to lower the bound, from the second Hanf number down to the first, on the categoricity cardinal in Shelah’s seminal Downward Categoricity Theorem for AECs [@Sh394]. Additionally, VanDieren has examined the proofs of Theorem \[symmetry transfer\] and Theorem \[main theorem\] in categorical AECs in which the amalgamation property is not assumed [@Va-char], providing additional insight into Shelah and Villaveces’ original exploration of limit models [@ShVi]. Acknowledgement {#acknowledgement .unnumbered} =============== The author is grateful to Sebastien Vasey for email correspondence about [@Va3-SS] during which he asked her about the union of saturated models. She is also thankful to Rami Grossberg, William Boney, Sebastien Vasey, and the referees for suggestions and comments that improved the clarity of this paper. [^1]: $M$ is a $(\mu,\mu^+)$-limit model if $M={\bigcup}_{i<\mu^+}M_i$ for some increasing and continuous sequence of models $\langle M_i\in{\mathcal{K}}_{\mu}\mid i<\mu^+\rangle$ where $M_{i+1}$ is universal over $M_i$ for each $i<\mu^+$.

--- abstract: 'We study Floquet topological transition in irradiated graphene when the polarization of incident light changes randomly with time. We numerically confirm that the noise averaged time evolution operator approaches a steady value in the limit of exact Trotter decomposition of the whole period where incident light has different polarization at each interval of the decomposition. This steady limit is found to coincide with time-evolution operator calculated from the noise-averaged Hamiltonian. We observe that at the six corners (Dirac($K$) point) of the hexagonal Brillouin zone of graphene random Gaussian noise strongly modifies the phaseband structure induced by circularly polarized light whereas in zone-center ($\Gamma$ point) even a strong noise isn’t able to do the same. This can be understood by analyzing the deterministic noise averaged Hamiltonian which has a different Fourier structure as well as lesser no of symmetries compared to the noise-free one. In 1D systems noise is found to renormalize the drive amplitude only.' author: - 'Bhaskar Mukherjee$^{1}$' title: Floquet topological transition by unpolarized light --- Introduction {#I} ============ Realizing topological phenomenon in solid state system has been one of the major topic in condensed matter physics since the discovery of IQHE in 2D semiconductor devices[@iqhe1]. These materials are model system for 2D non-interacting electron gas which under the application of strong magnetic field forms highly gapped Landau levels at low temperature. This results in very precise quantization of Hall conductance[@iqhe2; @iqhe3] and supports robust conducting chiral states at the edges[@iqhe4; @iqhe5]. Later it was shown that the magnetic field is not necessary and one can also observe such phenomenon in systems described by tight-binding Hamiltonians[@haldane]. The so called Haldane Modeldescribe electrons hopping in a honeycomb lattice threaded by periodic magnetic flux with zero net flux. The resulting complex hopping is difficult to implement experimentally and it is only recently that the advancement in ultra-cold atomic systems have made such experiments possible[@expt1]. To avoid such complicated implementation of the Haldane model and thus realize Chern insulating states more easily, a possible alternative way, namely irradiation of electromagnetic wave on graphene, is proposed recently to achieve the essential goal of time reversal symmetry breaking. Graphene is a gapless 2D Dirac system which can open up a gap at the Dirac Point under irradiation of circularly polarized light[@jap1; @jap2]. This resulting new state, termed as Floquet topological insulator was found later in many other systems[@gil; @roderich]. It is also detectable by various transport signatures [@takashi; @arijit; @gil2]. These are steady states of periodically driven non-equilibrium systems[@rigol1; @rigol2; @das1; @das2] which recently gained tremendous attention because of it’s potential to create new phases. These phases can hardly be found in their equilibrium counterparts. Traditional bulk-boundary correspondence was extended to Floquet topological systems taking into account the periodicity of the Floquet spectrum[@rudner1; @rudner2]. Experimental verification of such states has already been achieved using both time and angle resolved photoemission spectroscopy(PES)[@gedik; @gavensky] and also in photonic systems[@rechtsman; @sebabrata]. Throughout the last decade a large number of studies of real time dynamics in closed quantum systems have extended the notions of universality from equilibrium to non-equilibrium via Kibble-Zurek scaling[@ksengupta]. Further studies show that the qualitative nature of these scalings can be completely reversed by introducing noise in the drive[@anirban]. In these studies the Heisenberg equation of motion picks up a dephasing term due to averaging over different noise realizations which leads to non-unitary dynamics. Recently in equilibrium systems it has been shown that periodicity in space (i.e the crystal structure) is not necessary to get topological behavior and one can also see it in amorphous systems[@adhip]. Analogously one can ask at this point that what would happen in Floquet systems if time periodicity of the Hamiltonian is broken due to the presence of noise in the drive. Several studies in this direction in models decomposable in free fermions have already revealed that the nature of the asymptotic steady state depends on the type of aperiodic protocol[@sourav]. Further some analytical studies show that disorder-averaging can be avoided for a special class of protocols[@utso]. Influenced by this kind of works we plan to study the fate of the Floquet topological systems when the smooth time variation of incident electromagnetic wave is broken by the insertion of a random phase in one of the component of vector potential. This kind of noise is always there in a typical experiment if the setup to produce polarized light isn’t calibrated properly. Moreover such noise can also be generated artificially using synthetic gauge fields. We term this kind of monochromatic wave as unpolarized light in the sense that the associated Lissajous figures keeps on changing with time. The central results of this work can be summarized as follows. We show that depending on the spatial dimension of the problem Floquet topological transitions can be influenced by the random change in polarization of incident light. For graphene we find that the transitions at Dirac(K) point are significantly modified compared to $\Gamma$ point. The origin of this effect can be understood to be due to a fundamental change in Fourier structure of the noise-averaged time-dependent Hamiltonian at K point. At low frequencies of the incident radiation, it is well known that symmetries of the underlying Hamiltonian is crucial for topological transition[@us]. In the presence of noise, we find such symmetries to be broken. Interestingly, in contrast to standard expectation, we find that few of these symmetries are restored in the noise-averaged Hamiltonian. This symmetry restoration has impact on the self-averaging limit in this parameter regime. Finally for a 1D model($p$-wave superconducting wire), using a non-trivial drive protocol, we show that even a strong noise (large standard deviation) can’t prohibit the transition. The rest of the paper is planned as follows. In Sec.\[II\] we introduce our protocol for irradiated graphene and plot the results(phasebands) for numerical disorder averaging. In Sec.\[IIA\] we establish the existence of self-averaging limit which suggests numerical averaging is meaningful and can be mimicked by the ensemble averaged Hamiltonian. This is followed by possible explanation of the deviation from noise free (circularly polarized case) behavior separately in high and low frequency regime in Sec.\[IIB\] and Sec.\[IIC\] respectively. Next, in Sec.\[III\], we shows results for 1D systems. Finally we conclude and discuss possible experimental scenarios in Sec.\[IV\]. Irradiated Graphene {#II} =================== We consider graphene irradiated by electromagnetic wave defined by the vector potential $\bold{A}=A_0(\cos(\omega t+\phi(t)), \sin(\omega t))$. One have to further assume it to be space independent in graphene plane to keep the integrability of the problem intact. The $\phi=0$(circularly polarized) case is well studied in the literature[@arijit2]. We allow $\phi$ to be a normally distributed random variable with mean $\mu$ and standard deviation $\sigma$ at each instant of time which gives rise to its unpolarized nature. If one wish to produce this vector potential in lab then this kind of noise will be inherently present as random experimental error. The normalized probability distribution of $\phi$ at each time instant t is given by $$\label{(1)} P(\phi)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(\phi-\mu)^2}{2\pi \sigma^2}}$$ $\mu$ can be any real number within the interval ($-\pi \le \mu \le \pi$). Here we will concentrate on the special value $\mu=0$ (i.e this is the value of $\mu$ in all plot). This will allow us to directly compare the result with circularly polarized case. The time-dependent graphene Hamiltonian(for each k-mode) after Peierls’s substitution with this protocol becomes $$H(\bold{k},t)=\left(\begin{array}{cc} 0 & Z(\bold{k},t)\\ Z^*(\bold{k},t) & 0 \end{array}\right)\nonumber$$ where $Z(\bold{k},t)=-\gamma (2e^{i\frac{\tilde{k_x}}{2}}\cos(\frac{\sqrt{3}\tilde{k_y}}{2})+e^{-i\tilde{k_x}})$ and $\bold{\tilde{k}}=\bold{k}+e\bold{A}$ Next we calculate the time-evolution operator over one time period($T$) for each k-mode by dividing the period in N parts $$\begin{aligned} \label{(2)} U_k(T,0)&=&T_te^{-i\int_0^TH_k(t')dt'}\nonumber\\ &=&e^{-iH_k(T-\delta t)\delta t}e^{-iH_k(T-2\delta t)\delta t}.....e^{-iH_k(2\delta t)\delta t}\nonumber\\ &&e^{-iH_k(\delta t)\delta t}\end{aligned}$$ where $T_t$ denotes time-ordered product and $\delta t=T/N$ is a very small but fixed time interval. Such decomposition introduces Trotter error which gets reduced with increasing N and reproduces the exact U for the chosen continuous drive in the $N\rightarrow \infty$ limit. We calculate the time-dependent Hamiltonian at each partition by drawing $\phi$ from a normal distribution and using Eq.\[(2)\] get $U(T,0)$ for one particular noise realization. We then average over several such realizations numerically and get the noise averaged time evolution operator $$\label{(3)} \langle U_k(T,0)\rangle=\langle T_te^{-i\int_0^TH_k(t')dt'} \rangle.$$ Eq.\[(3)\] has a self-averaging limit[@Lobejko], in the sense that all four elements of $\langle U(T,0) \rangle$ goes to some steady value with increasing no of partitions (N). We shall discuss this in more details in the next sub-section. In Fig.1 we plot the phasebands ($\Phi(T)$) obtained using $\cos(\Phi(T))=Re[\langle U(T) \rangle_{11}]$. {width="0.49\linewidth"} {width="0.49\linewidth"} One can see with increasing magnitude of random noise the phasebands gets modified but we recover the results for pure circularly polarized light in $\sigma\rightarrow0$ limit as expected. We find that the phasebands remain almost unchanged for $\Gamma$ point for a broad range of parameter values; however at $K$ point, they are strongly modified by the noise. We calculate Chern number of the lower Floquet band using the eigenfunctions of $\langle U(T) \rangle$ in a discretized Brillouin zone. The plot is shown in Fig.2. We find that the transitions (position of integer jump in Chern number) can sustain an appreciable amount of temporal noise and merely gets shifted in parameter space but very strong noise (large $\sigma$) abolish them. {width="0.49\linewidth"} {width="0.49\linewidth"} Ensemble averaged Hamiltonian {#IIA} ----------------------------- In this subsection we explore the possibility of constructing a deterministic Hamiltonian such that time-evolution operator constructed using it resembles the noise averaged time-evolution operator. In a recent work[@Lobejko] Lobejko [*et al*]{} have showed rigorously that the difference of ensemble averaged time-evolution operator and the time-evolution operator constructed by the ensemble averaged Hamiltonian scales as $O(\frac{1}{N})$ for a certain class of protocols. For these protocols the ensemble averaged Hamiltonian at two different time commutes which they have termed as commutation in statistical sense. They further extends the applicability of above theorem to some simple non-commuting Hamiltonian by numerical simulations. But unlike those cases irradiated graphene contains the noise term within the argument of complicated trigonometric functions. Hence the ensemble averaged Hamiltonian can not be obtained here simply by substituting $\phi$ by it’s mean value. Therefore we explicitly calculate the ensemble-averaged Hamiltonian for irradiated graphene at time t $$\label{(4)} \langle H_k(t) \rangle=\int_{-\infty}^{\infty}P(\phi)H_k(\phi,t)d\phi$$ with $P(\phi)$ in Eq.\[(1)\] we get using Jacobi-Anger relations[@google]. $$\begin{aligned} \label{(5)} \langle Z(\bold{k},t) \rangle&=&-\gamma (2e^{i\frac{k_x}{2}}\cos(\frac{\sqrt{3}(k_y+\alpha \sin(\omega t))}{2})[J_0(\frac{\alpha}{2}) +2\sum_{n=1}^{\infty}i^nJ_n(\frac{\alpha}{2})e^{-\frac{n^2\sigma^2}{2}}\cos(n(\omega t+\mu))]+e^{-ik_x}[J_0(\alpha)+\nonumber\\ &&2\sum_{n=1}^{\infty}(-i)^nJ_n(\alpha)e^{-\frac{n^2\sigma^2}{2}}\cos(n(\omega t+\mu))])\end{aligned}$$ Using this we numerically calculate the Frobenius norm of the distance between $\langle U(H(t))\rangle$ and $U(\langle H(t)\rangle)$ $$\label{(6)} D_N=\parallel\langle Te^{-i\int_0^TH(t')dt'}\rangle-Te^{-i\int_0^T\langle H(t') \rangle dt'}\parallel$$ and the same norm for the corresponding variance matrix $$\label{(7)} S_N=\parallel\langle(Te^{-i\int_0^TH(t')dt'}-Te^{-i\int_0^T\langle H(t') \rangle dt'})^2\rangle\parallel$$ where N is the no of partitions used to calculate (using Eq.\[(2)\] and \[(3)\]) each quantities inside the norm. These are two appropriate quantities to measure the deviation of the time-evolution operator in different noise realizations. We see power law fall of both $D_N$ and $S_N$ in no of partitions(N)(see Fig.3) which suggest self-averaging limit exists here. It is only in this limit that the disorder averaging is meaningful in dynamical systems. This is in close analogy to equilibrium disordered systems where for each disorder realization some amount of deviation (from the mean) is introduced in all physical observable due to the finite size of the system but these deviations get canceled when averaged out over several disorder realizations and thus helps to achieve the thermodynamic result fast. Here in dynamical system finite no of partition(N) play the role of finite system size and the thermodynamic limit corresponds to the continuous drive ($N \rightarrow \infty$). Vanishing of $S_N$ in large N also implies the equivalence $$\label{(8)} \cos(\langle\Phi(T)\rangle)\equiv\langle\cos(\Phi(T))\rangle$$ which we have used throughout the paper. In Fig.3 note that $D_N$ and $S_N$ have larger values at $K$ point compared to $\Gamma$ point for small N. This is related to the fact that time dependent Hamiltonian of irradiated graphene at $K$ point is more complicated than at $\Gamma$ point due to the presence of lesser no of symmetries[@us]. Larger the complexity larger N one need to use to reduce these errors. \ \ This power law fall suggests that the time consuming numerical disorder averaging can be avoided by the use of ensemble averaged Hamiltonian to calculate $U(T,0)$ with a sufficiently large no of partitions of whole period. We further demonstrate this by explicitly comparing the phasebands from both this way in Fig.4. {width="0.49\linewidth"} {width="0.49\linewidth"} Our next target is to understand better why in some cases a weak noise is sufficient to abolish all the transition (as in $K$ point) where as in some other cases(as in $\Gamma$ point) even a strong noise just causes a shift of the crossing positions and nothing more than that. We will do it by analyzing the ensemble averaged Hamiltonian (Eq.\[(5)\]) in two different frequency regime. High frequency, Floquet formalism {#IIB} --------------------------------- The Floquet formalism allows one to treat a periodic time-dependent problem as a time-independent eigenvalue problem. The cost of this is to deal with an infinite dimensional Hilbert space which is a vector space of T periodic functions and also known as Sambe space. The representation of Floquet Hamiltonian (related to $U(T,0)$ by $U(T,0)=e^{-iH_FT}$) in this basis is defined by the following matrix elements $$\label{(9)} H_{i,j}^{m,n}=m\omega \delta_{mn} \delta_{ij}+\frac{1}{T}\int_0^Te^{-i(m-n)\omega t'}H_{ij}(t')dt'$$ where $(m,n)$ is row and column index of different square blocks each of size $(H_1\times H_1)$ where $H_1$ is the Hilbert space dimension of the equilibrium problem (2 for each k-mode in our case) and $(i,j)$ denotes position of each matrix element within one such block. For numerical purposes one can truncate this matrix after some order which depends on details of the problem especially the absolute value of maximum order of the Fourier components (of time-dependent Hamiltonian) with non-vanishing coefficient. One also needs to increase the truncation dimension with decreasing frequency. Following this prescription one can safely truncate the Floquet Hamiltonian in zero-th order at $\Gamma$ point(where one has a $2\times2$ $H_F$) and in 1st order at $K$ point( where one has a $6\times6$ $H_F$) for high frequencies and low Amplitude of radiation[@jap1; @arijit2]. Thus one gets expressions of Floquet conduction band ($\Phi(T)$) in 1st quasi-energy BZ for the noise free (circularly polarized) case with hopping-amplitude($\gamma$) set to unity $$\label{(10)} \hskip -2.7cm \Phi(\Gamma,T)=3J_0(\alpha)T$$ $$\label{(11)} \Phi(K,T)=\frac{\sqrt{4\pi^2+36J_1^2(\alpha)T^2}-2\pi}{2}$$ Next we aim to calculate some simplified expression of phaseband for the unpolarized light using the ensemble averaged Hamiltonian in some suitable parameter regime. We can sufficiently simplify Eq.\[(5)\] for strong noise. Note that though $\phi$ appears as argument of trigonometric functions due to it’s random nature at each instant of time $\phi[\mu,\sigma]$ and $\phi[\mu+2n\pi,\sigma+2p\pi]$ will not give same time evolution operator. Using $e^{-\frac{n^2\sigma ^2}{2}}\approx0$ for large $\sigma$ in Eq.\[(5)\] we get $$\begin{aligned} \label{(12)} \langle Z(\bold{k},t)\rangle\mid_{\sigma\gg0}&\approx&-\gamma(2J_0(\frac{\alpha}{2})e^{ikx}\cos(\frac{\sqrt{3}}{2}(ky+\nonumber\\ &&\alpha \sin(\omega t)))+J_0(\alpha)e^{-ikx})\end{aligned}$$ for $\Gamma$ point this gives a Hamiltonian proportional to $\sigma_x$ only and hence one simply gets the phaseband $$\label{(13)} \Phi(\Gamma,T)=\int_0^T\langle Z(\Gamma,t')\rangle dt'$$ the integrand is difficult but again using Jacobi-Anger relations we get(taking $\gamma=1$) $$\begin{aligned} \label{(14)} \Phi(\Gamma,T)&=&(2J_0(\frac{\alpha}{2})J_0(\frac{\sqrt{3}\alpha}{2})+J_0(\alpha))T+\nonumber\\ &&4J_0(\frac{\alpha}{2})\sum_{n=1}^{\infty}J_{2n}(\frac{\sqrt{3}\alpha}{2})\int_0^T\cos(2n\omega t')dt'\nonumber\\ &=&(2J_0(\frac{\alpha}{2})J_0(\frac{\sqrt{3}\alpha}{2})+J_0(\alpha))T\end{aligned}$$ similarly for $K$ point we get $$\label{(15)} \Phi(K,T)=(J_0(\alpha)-J_0(\frac{\alpha}{2})J_0(\frac{\sqrt{3}\alpha}{2}))T$$ we compare cosines of Floquet bands for circularly polarized($\sigma=0$) and unpolarized($\sigma\gg0$) case in Fig.5. The functional behavior of these two bands do not change much for $\Gamma$ point whereas for $K$ point they show drastically different behavior. This huge change for $K$ point is due to the fact that strong noise (highly unpolarized light) changes the lowest non-vanishing Fourier component of $\langle H_K(t) \rangle$ from 1 to 0 and thus reduces the effective Sambe space dimension from 6 to 2. These changes make the Floquet band at $K$ point to depend on $J_0$ s only abolishing $J_1$ s. Note that $J_0$ and $J_1$ has completely different behavior when the argument is small, the former is a decreasing function but the later is an increasing function of the argument.  and \[(14)\](red-dashed) for $\Gamma$ point (left panel) and of Eq.\[(11)\](black-solid) and \[(15)\](red-dashed) for $K$ point (right panel). All parameters are same as before.](high_freq_gamma "fig:"){width="0.49\linewidth"}  and \[(14)\](red-dashed) for $\Gamma$ point (left panel) and of Eq.\[(11)\](black-solid) and \[(15)\](red-dashed) for $K$ point (right panel). All parameters are same as before.](high_freq_K "fig:"){width="0.49\linewidth"} Low frequency {#IIC} ------------- At low frequencies (and also at high radiation amplitudes) one need to take into account the higher Fourier components of the time-dependent Hamiltonian and consequently the truncation dimension of the Floquet Hamiltonian increases. This is why at low frequencies one can’t have simple analytical expression of Floquet bands in terms of Bessel functions and one needs to consider other methods like the adiabatic-impulse which gives good matching with numerics in low to moderate frequencies and high amplitudes[@us]. Symmetries of $H(t)$ also play a crucial role in predicting the existence of phaseband crossings at different high symmetry points. But before going into the details of that we investigate the behavior of $D_N$ and $S_N$ as a function of N at low frequencies. Generally low $\omega$ and hence a high period ($T$) necessitates a proportional increase of no of partitions but numerics suggests that the convergence of these quantities to zero is much slower than that in this parameter regime. In Fig.6(a)-(c) we demonstrate this. We see for a typical high $\sigma$ one needs to increase N nearly quadratically (instead of linearly) with T to make the value of $D_N$ go below some particular threshold. We, therefore to reduce the numerical cost, keep our all calculations confined within small $\sigma$ values at low frequencies. {width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"} {width="0.49\linewidth"} It was shown in ref\[33\][@us] that there exists 6 fold symmetries at $\Gamma$ point of graphene irradiated by circularly polarized light. This was shown to be responsible for phaseband crossing simultaneously at $T/3$, $2T/3$ and $T$. But here for unpolarized light typically all these symmetries are absent for any disorder-realization. Consequently, disorder averaging also leads to avoided crossing. Here also the ensemble averaged Hamiltonian can capture the essential physics but interestingly two of the symmetries get restored in it. We chart out the symmetries of $\Gamma$ point under the irradiation of CP and unpolarized(ensemble averaged $H(t)$) light in detail in Table.1. This kind of symmetry mismatch between the two quantities inside the norm of Eq.\[(6)\] has significant impact on fall of $D_N$ at low frequencies. We find that $D_N$ falls very slowly with N (see Fig.6) here. ------------------------------------------- ---------- ------------- 0.1cm 0.8cm Symmetries 0.5cm CP Unpolarized $H(T-t)=H(t)$ 0.6cm 0.8cm $H(\frac{T}{2}\pm t)=\tau_x H(t) \tau_x$ 0.6cm 0.8cm $H(\frac{T}{6}\pm t)=\tau_x H(t) \tau_x$ 0.6cm 0.8cm $H(\frac{T}{3}\pm t)=H(t)$ 0.6cm 0.8cm $H(\frac{2T}{3}\pm t)=H(t)$ 0.6cm 0.8cm $H(\frac{5T}{6}\pm t)=\tau_x H(t) \tau_x$ 0.6cm 0.8cm ------------------------------------------- ---------- ------------- : Symmetries of $\Gamma$ point for circularly polarized and unpolarized light. In Fig.7 we show this symmetry mismatch between CP and unpolarized light pictorially (a large $\sigma$ is used for this purpose in Fig.7(a)) and its consequences. Fig.7(b) shows for exact numerical disorder averaging a small $\sigma$ is sufficient to abolish the crossing at $T/3$. Fig.7(c)-(d) shows for ensemble averaged Hamiltonian the crossings at $T/3$ and $2T/3$ gets increasingly avoided with increasing $\sigma$. \ \ 1D systems {#III} ========== One-dimensional interacting spin chains whose Hamiltonian can be expressed in terms of free fermions via Jordan-Wigner transformation have attracted a lot of theoretical attention in last decades due to their integrable structure, existence of topological transition as well as possibility of experimental realization using ion-traps and ultracold atom systems. Non-equilibrium dynamics in these models is equally interesting because non-trivial topology can be induced by periodic drive of different terms in the Hamiltonian[@manisha]. This can be independently done using multiple lasers with different amplitudes and frequency. In these experiments phase differences between different drive terms can be randomly changed in a time scale $t_0\ll 1/\omega$ where $\omega$ is the frequency of drive. This constitutes a 1D platform to study similar physics as studied in previous section for 2D systems using unpolarized light. The survival of the topological transition under such noisy drive is the key issue we would like to address. To this end, we consider a p-wave superconductor described by the following Hamiltonian[@amit1; @amit2] $$\label{(16)} H=\sum_{i=1}^{L-1}[(\gamma c_i^\dagger c_i+H.c)+\Delta (c_ic_{i+1}+H.c)]-\mu\sum_{i=1}^L(2c_i^\dagger c_i-1)$$ This model is equivalent to a spin-$\frac{1}{2}$ XY chain in perpendicular magnetic field via Jordan-Wigner transformation[@1961]. After a Fourier transformation defined by $c_k=\frac{1}{L}\sum_{j=1}^Lc_ie^{ikj}$ we can write this as $$\label{(17)} H=2\sum_{0\leq k \leq \pi}\psi_k^\dagger H_k \psi_k$$ where $\psi_k=(c_k, c^\dagger_{-k})^T$ is a two component vector. Thus each k-mode of such systems can be described by the following Hamiltonian(we scale everything by $\gamma$) $$\label{(18)} H(k,t)=(\mu-\cos(k))\sigma_z+\Delta \sin(k)\sigma_x$$ and we use the following drive protocol $\mu=A\cos(\omega t+\phi(t))$ and $\Delta=\cos(r\omega t)$ where $r$ is an integer and $\phi$ is as usual a random variable at each time t. The dynamics of this model is non-trivial for $r>1$ due to the non-removable time dependence in both diagonal and off-diagonal element[@sau; @satyaki]. This model(with $\phi(t)=0$) has a phaseband crossing for $k=\pi/2$ at $t=T/2$ which exists at all frequencies. We study here what happens to this crossing if at each instant of time $\phi$ is a random Gaussian variable with zero mean. Below we mention the scheme for partitioning a full period to calculate the noise averaged $U(t,0)$ now at any time $t\leq T$ $$\label{(19)} \delta t=\frac{t}{N}=const$$ i.e we increase no of partitions proportionally as the time $t$ gets closer to $T$ keeping the duration of constant time evolution($\delta t$) fixed. Thus we calculate noise averaged phaseband at all time t within a period for different noise strength ($\sigma$) and compare it with noise free case in Figure.8(a). Interestingly noise modifies the phaseband at all times except at $t=T/2$ which is the phaseband crossing point for noise free drive. This shows that the transition at $t=T/2$ is immune to any amount of temporal disorder. As a routine task we calculate the noise averaged instantaneous Hamiltonian for the chosen protocol $$\label{(20)} \langle H(k=\frac{\pi}{2},t) \rangle=A \cos(\omega t) e^{-\sigma^2/2}\sigma_z+\cos(r \omega t)\sigma_x$$ ![Phasebands from numerically averaged U operator (continuous line) and from the averaged Hamiltonians(dots) for 1D model(in Eq.\[(15)\]) in left panel. A=1.5, $\omega=1.0$, r=3. Right panel shows the change of instantaneous energies with the insertion of noise.](1d "fig:"){width="0.49\linewidth"} ![Phasebands from numerically averaged U operator (continuous line) and from the averaged Hamiltonians(dots) for 1D model(in Eq.\[(15)\]) in left panel. A=1.5, $\omega=1.0$, r=3. Right panel shows the change of instantaneous energies with the insertion of noise.](Et_1d "fig:"){width="0.49\linewidth"} In Fig.8(left panel) we see time evolution governed by this averaged $H$ mimics the numerically disorder averaged U operator as like before. We note that this numerical agreement leads to the following statement The effect of random noise is just to renormalize the laser amplitude $$\label{(21)} \tilde{A}=Ae^{-\sigma^2/2}$$ The robustness of the transition at $t=T/2$ also follows from the symmetry of Eq.\[(18)\]. Note that the symmetry of the noise free Hamiltonian for $k=\pi/2$ and for odd $r$ (namely $H(T/2-t)=-H(t)$ )is not destroyed by the insertion of noise here (see Fig.8(right panel)). This can be used together with the Trotter like decomposition of U operator (as in Eq.\[(2)\]) to show $U^{-1}(T/2)=U^\dagger(T/2)=U(T/2)$ signifying that a crossing through Floquet zone-center will always be there at $t=T/2$ for all parameter values ($A$, $\omega$, $\phi$ etc). Further right panel of Fig.8 demands that the same adiabatic-impulse method (as done for the noise free case in ref.\[33\]) can be used to show the existence of the crossing at $t=T/2$ in spite of the change in sizes of different adiabatic regions. Discussion {#IV} ========== In this work we have studied the existence of self-averaging limit in graphene irradiated by unpolarized light. We see the limit holds in high-frequency regime and can be captured by the noise-averaged Hamiltonian. In low frequencies the limit is achieved very slowly as a possible consequence of retaining two of the symmetries in noise-averaged Hamiltonian. This opens up an opportunity to search for some other deterministic Hamiltonian for speeding up the convergence to asymptotic limit. We hardly found any steady limit at extremely low frequencies to the best of our numerical ability. Floquet topological transitions are found to be modified by the insertion of noise to various degrees depending on the k-point in BZ. These range from a small shift in crossing positions to complete abolition of the transition depending on the amount of disorder. We find that certain k-points are more affected as a consequence of a change in Fourier structure of their time-dependent Hamiltonian induced by the noise. The presence of a 6-fold symmetry at $\Gamma$ point plays a crucial role for the existence of a special type of crossings which simultaneously happens at $T/3$,$2T/3$,$T$[@us]. This kind of crossings are ubiquitous in low frequencies but ceases to exist in high frequency(scanning the whole parameter regime as much as possible we found they are absent below $T\approx11$). Now breaking of 4 out of those 6-symmetries by the noise abolishes these transitions confirming again the importance of symmetries in low frequencies. In 1D systems due to the simplicity of the BZ, noise obeys all symmetries of the clean time-dependent Hamiltonian and as a consequence crossings persist at all noise strengths. It merely renormalizes the drive amplitude. In typical experiments one needs to keep the optical axis of a quarter wave plate exactly at $45{^\circ}$ with the plane of vibration of the incident plane polarized light to extract pure circularly polarized light. Now if this angle changes randomly (which is always present in small amount if the experiment is not performed carefully such as a small vibration of the table on which the set up lies may cause it) then the polarization of the outgoing light will also fluctuate. One can also use synthetic gauge fields to produce such noisy vector potential. This kind of perturbation is very common in an interference experiment if incoherent sources are used. The quantitatively different noise-response from various k-points can be experimentally verified by measuring the photoinduced gap in a momentum resolved manner using pump-probe spectroscopy as done in ref\[22\]. The abolition of transition and hence a change in topological structure of the Floquet bands can be detected by analyzing the intensity and angular dependence of ARPES spectra[@gavensky]. In conclusion we have shown random noise in the vector potential of incident light has significant impact on Floquet topological transition in graphene. One can analyze the symmetries and Fourier structure of the noise-averaged Hamiltonian to understand the modifications done by the noise. In 1D systems such noisy drives has no effect on the transitions. Acknowledgements {#acknowledgements .unnumbered} ================ Author thanks K. Sengupta for support. [99]{} K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494(1980). R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). R. E. Prange, Phys. Rev. B 23, 4802(R) (1981). B. I. Halperin, Phys. Rev. B 25, 2185 (1982). F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). Gregor Jotzu et al, Nature volume 515, pages 237–240 (2014). T Oka, H Aoki, Physical Review B 79 (8), 081406 (2009). Jun-ichi Inoue, Akihiro Tanaka, Phys. Rev. Lett. 105, 017401 (2010). NH Lindner, G Refael, V Galitski, Nature Physics 7 (6), 490 (2011). J Cayssol, B Dóra, F Simon, R Moessner, physica status solidi (RRL)-Rapid Research Letters 7 (1‐2), 101-108 (2013). T Kitagawa, T Oka, A Brataas, L Fu, E Demler, Physical Review B 84 (23), 235108 (2011). A Kundu, B Seradjeh, Physical Review Letters 111 (13), 136042 (2013). I Esin, MS Rudner, G Refael, NH Lindner, Physical Review B 97 (24), 245401 (2018). L D’Alessio, M Rigol, Physical Review X 4 (4), 041048 (2014). L D’Alessio, M Rigol, Nature communications 6, 8336 (2015). A Lazarides, A Das, R Moessner, Physical Review E 90 (1), 012110 (2014). A Lazarides, A Das, R Moessner, Physical review letters 112 (15), 150401 (2014). Achilleas Lazarides, Arnab Das, Roderich Moessner, Physical review letters 115 (3), 030402 (2015). T Kitagawa, E Berg, M Rudner, E Demler, Physical Review B 82 (23), 235114 (2010). Mark S. Rudner, Netanel H. Lindner, Erez Berg, Michael Levin, Phys. Rev. X 3, 031005 (2013). Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, Science 342, 453 (2013). L. P. Gavensky, G. Usaj, and C. A. Balseiro, Sci. Rep. 6, 36577 (2016). Mikael C.Rechtsman et al, Nature volume 496, pages 196–200 (2013). Sebabrata Mukherjee et al, Nature Communications volume 8, Article number: 13918 (2017). A Polkovnikov, K Sengupta, A Silva, M Vengalattore, Reviews of Modern Physics 83 (3), 863 (2011). Anirban Dutta, Armin Rahmani, Adolfo del Campo, Phys. Rev. Lett. 117, 080402 (2016). Adhip Agarwala and Vijay B. Shenoy, Phys. Rev. Lett. 118, 236402 (2017). Sourav Nandy, Arnab Sen, and Diptiman Sen, Phys. Rev. X 7, 031034 (2017). Utso Bhattacharya, Somnath Maity, Uddipan Banik, Amit Dutta, Phys. Rev. B 97, 184308(2018). A Kundu, HA Fertig, B Seradjeh, Physical review letters 113 (23), 236803 (2014). Marcin Lobejko, Jerzy Dajka and Jerzy Luczka, arxiv:1805.02871 (2018). Bhaskar Mukherjee, Priyanka Mohan, Diptiman Sen and K. Sengupta, Physical Review B 97, 205415 (2018) J. D. Sau and K. Sengupta, Phys. Rev. B 90, 104306 (2014). S. Kar, B. Mukherjee, and K. Sengupta, Phys. Rev. B 94, 075130 (2016). A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosen- baum, and D. Sen, Quantum phase transitions in transverse field spin models: from statistical physics to quantum information (Cambridge University Press, Cambridge, 2015). V. Mukherjee, U. Divakaran, A. Dutta, and D. Sen, Phys. Rev. B 76, 174303 (2007); C. De Grandi and A. Polkovnikov, in Quantum Quenching, Annealing, and Computation, edited by A. K. Chan- dra, A. Das, and B. K. Chakrabarti, Lecture Notes in Physics, Vol. 802 (Springer, Heidelberg, 2010), p. 75. M. Thakurathi, A. A. Patel, D. Sen, and A. Dutta, Phys. Rev. B 88, 155133 (2013); M. Thakurathi, K. Sengupta, and D. Sen, Phys. Rev. B 89, 235434 (2014). E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. (NY) 16, 407 (1961). Wikipedia.

--- abstract: 'For a finite-dimensional associative algebra $A$, we introduce the notion of a self-injective core of $A$. Self-injective cores give rise to $2$-subcategories of the $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ defined in [@MM1]. $2$-representations of a particular such $2$-subcategory were studied in [@Zi2]. We classify the simple transitive $2$-representations of a wide class of such $2$-subcategories. We also construct a family of non-cell simple transitive $2$-representations of a certain $2$-semicategory of projective functors. The existence of such $2$-representations for a closely related $2$-category was conjectured in [@Zi2].' author: - 'Mateusz Stroi['' n]{}ski' title: 'Simple transitive $2$-representations of $2$-categories associated to self-injective cores' --- Introduction ============ The systematic study of $2$-representations of finitary $2$-categories started with the series of papers by Mazorchuk and Miemietz ([@MM1], [@MM2], [@MM3], [@MM4], [@MM5], [@MM6]). The article [@MM5] first introduced the notion of a simple transitive $2$-representation, which is analogous to that of a simple module in a classical setting. Classification of simple transitive $2$-representations of a given finitary $2$-category has since become central to the study of $2$-representation theory that followed said series of articles, see [@Zi2], [@KM], [@MMZ2], [@MMMTZ], [@Jo] for recent examples. A natural choice of a finitary $2$-category to study is that of projective bimodules over a given finite-dimensional associative algebra $A$, which we denote by ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. In this case, the aforementioned classification problem was solved completely in [@MMZ1]: up to equivalence, simple transitive $2$-representations of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ are exhausted by cell $2$-representations. These were defined already in [@MM1] for an arbitrary finitary $2$-category, by endowing it with a cell structure, in analogy to representation theory of Hecke algebras. The cell structure of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is relatively simple, so we may construct $2$-full $2$-subcategories of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ simply by removing (equivalently, keeping) some of its right or left cells. This gives us two different kinds of $2$-subcategories of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, which are the main object of study of this paper. Particular cases of such $2$-subcategories associated to a specific family of associative algebras, denoted by $\Lambda_{n}$, were studied in [@Zi2] - we generalize some of the results obtained in that paper, and also give a partial answer to a problem very closely related to a conjecture posed therein. If $A$ is self-injective, then ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is a fiat category, which in the setting of [@EGNO] corresponds to having left and right duals. The $2$-representation theory of fiat $2$-categories is much better understood than the general finitary case. The $2$-subcategories of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ we study break the symmetry of the cell structure of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, and hence are never fiat: however, for one of our main results we assume that said $2$-subcategory corresponds to a [*self-injective core*]{} for $A$, that is, a subset $S$ of a system of primitive, mutually orthogonal idempotents for $A$, such that for every $e \in S$ there is $f \in S$ satisfying $(eA)^{*} \simeq Af$. The self-injective core then provides our $2$-category of interest with a further $2$-subcategory, which is fiat, so that we remain close to the fiat case - a part of the cell structure admits the kind of symmetry found in the case of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. From this perspective, the $2$-subcategories of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ we consider can be viewed as stepping just outside the realm of fiat categories, in an attempt to gain a better understanding of the non-fiat finitary $2$-categories. Let $S$ be a self-injective core for $A$ and let ${{\sc\mbox{D}\hspace{1.0pt}}}_{R}$ be the $2$-subcategory of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ formed by the right cells of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ corresponding either to the identity $1$-morphism or elements of $S$. The first of our two main results (Theorem \[MainThm4\]) can be formulated as \[IntroThm1\] Up to equivalence, the simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{R}$ are given by cell $2$-representations. Moreover, up to equivalence, there are only two cell $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{R}$. The second of our main results concerns the following conjecture posed in [@Zi2]: given a zig-zag algebra $\Lambda_{n}$ on a star-shaped graph, consider the $2$-subcategory ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!\Lambda_{n}}$ formed by the left cell $\mathcal{L}_{0}$ and the two-sided cell given by the identity $1$-morphism. Equivalence classes of simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ are in bijection with set partitions of ${\left\{ 1,2,\ldots, n \right\}}$. Our result is formulated in the setting of $2$-representations of $2$-semicategories, recently developed in [@KMZ]: consider again the algebra $\Lambda_{n}$ and consider the $2$-subsemicategory ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\Lambda_{n}}$ formed by the left cell $\mathcal{L}_{0}$ (alternatively, formed by removing the identity $1$-morphism from ${{\sc\mbox{D}\hspace{1.0pt}}}_{L}$). The result (Theorem \[MainThm2\]) now gives the following: \[IntroThm2\] There is a family of pairwise non-equivalent simple transitive $2$-representations of ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ indexed by set partitions of ${\left\{ 1,2,\ldots, n \right\}}$. A particularly interesting aspect of this result is that ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ admits only one cell $2$-representation (and ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ admits only two), so the construction behind Theorem \[IntroThm2\] goes beyond cell $2$-representations, while in the setting of projective bimodules and $2$-categories of the form ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, no non-cell simple transitive $2$-representations were known before (even though the conjecture above predicts their existence). The document is organized as follows: in Section $2$ we collect the preliminaries necessary to state our main results. In Section $3$ we introduce the notion of a self-injective core, define the $2$-subcategories of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ obtained by removing cells and give a complete description of their cell structures, together with examples. In Section $4$ we prove Theorem \[MainThm4\], stated above as Theorem \[IntroThm1\]. Section $5$ establishes a result analogous to Theorem \[MainThm4\], in a particular case of a more difficult setting, where we no longer have a self-injective core to work with. Section $6$ moves to a “gentle” problem for the left cell case, where simple transitive $2$-representations are determined as cell $2$-representations, using methods earlier employed in [@MZ1] and [@Zi1] among others. The final section of the paper proves the second main result (Theorem \[MainThm2\]), stated above as Theorem \[IntroThm2\], crucially using a result of Power ([@Po]), which allows us to consider pseudofunctors rather than $2$-functors. [**Acknowledgments.**]{} The author would like to thank his supervisor Volodymyr Mazorchuk for numerous stimulating discussions, from which many of the questions addressed in this work emerged. Preliminaries ============= Setup ----- Throughout let $\Bbbk$ denote an algebraically closed field. By a finite-dimensional algebra $A$ we mean an associative, unital, finite-dimensional algebra over $\Bbbk$, and by a complete system of idempotents of $A$ we mean a set ${\left\{ e_{i} \in A \; | \; i = 1, 2, \ldots, r \right\}}$ of primitive, mutually orthogonal idempotents of $A$ such that $\sum_{i=1}^{r} e_{i} = 1$. Finitary $2$-categories and their $2$-representations ----------------------------------------------------- We say that a category $\mathcal{C}$ is [*finitary over $\Bbbk$*]{} if it is additive, idempotent split, $\Bbbk$-linear, and has finitely many isomorphism classes of indecomposable objects. Equivalently, $\mathcal{C}$ is finitary if there is a finite-dimensional algebra $Q$ such that $\mathcal{C}$ is equivalent to $Q\!\on{-proj}$. A $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}$ is [*finitary over $\Bbbk$*]{} if it has finitely many objects such that for every $\mathtt{i,j} \in {{\sc\mbox{C}\hspace{1.0pt}}}$, the category ${{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j})$ is finitary, horizontal composition is biadditive and $\Bbbk$-bilinear, and for any object $\mathtt{i} \in {{\sc\mbox{C}\hspace{1.0pt}}}$, the identity $1$-morphism $\mathbb{1}_{\mathtt{i}}$ is an indecomposable object of ${{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,i})$. Given a connected, basic finite-dimensional algebra $A$, fix a small category $\mathcal{A}$ equivalent to $A\!\on{-mod}$. The $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ consists of - a single object $\mathtt{i}$; - endofunctors of $\mathcal{A}$ isomorphic to tensoring with $A$-$A$-bimodules in $\on{add}((A \otimes_{\Bbbk} A) \oplus A)$ as $1$-morphisms (in other words, so-called projective functors of $\mathcal{A}$); - as $2$-morphisms, all natural transformations between such functors. In particular, ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is finitary. Consider also the following $2$-categories: - $\mathfrak{A}_{\Bbbk}$ - the $2$-category whose objects are small finitary categories over $\Bbbk$, are additive $\Bbbk$-linear functors between such categories and whose $2$-morphisms are all natural transformations between such functors; - $\mathfrak{R}_{\Bbbk}$ - the $2$-category whose objects are small abelian $\Bbbk$-linear categories, are right exact $\Bbbk$-linear functors between such categories and whose $2$-morphisms are all natural transformations between such functors. For the rest of this section, let ${{\sc\mbox{C}\hspace{1.0pt}}}$ be a finitary $2$-category, unless otherwise stated. A [*finitary $2$-representation*]{} of ${{\sc\mbox{C}\hspace{1.0pt}}}$ is then a $2$-functor $\mathbf{M}: {{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \mathfrak{A}_{\Bbbk}$ such that $\mathbf{M}_{\mathtt{i,j}}$ is additive and $\Bbbk$-linear for all $\mathtt{i,j} \in {{\sc\mbox{C}\hspace{1.0pt}}}$. An [*abelian $2$-representation*]{} of ${{\sc\mbox{C}\hspace{1.0pt}}}$ is such a $2$-functor whose codomain instead is $\mathfrak{R}_{\Bbbk}$. Finitary $2$-representations together with non-strict $2$-transformations and modifications form a $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}\!\on{-afmod}$; abelian $2$-representations similarly form a $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}\!\on{-mod}$; see [@MM3 Subsection 2.3] for details. In particular, by [@MM3 Proposition 2], a non-strict $2$-transformation of $2$-representations, $\Phi: \mathbf{M} \rightarrow \mathbf{N}$, such that $\Phi_{\mathtt{i}}$ is an equivalence of categories for all $\mathtt{i}$, is invertible. We say that two $2$-representations $\mathbf{M},\mathbf{N}$ are [*equivalent*]{} if such $2$-transformation exists. Note that the action of $1$-morphisms in an abelian representation is necessarily naturally isomorphic to taking the tensor product with a bimodule, and similarly the action of $2$-morphisms is represented by bimodule morphisms. If ${{\sc\mbox{C}\hspace{1.0pt}}}$ only has one object $\mathtt{i}$, we define the [*rank*]{} of a finitary $2$-representation $\mathbf{M}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}$ as the number of isoclasses of indecomposable objects of $\mathbf{M}(\mathtt{i})$. If ${{\sc\mbox{C}\hspace{1.0pt}}}$ has more than one object, one can fix an ordering of its objects and define the rank of a finitary $2$-representation of ${{\sc\mbox{C}\hspace{1.0pt}}}$ as a suitable tuple of positive integers. However, in this document we will only consider $2$-categories with a single object. We say that ${{\sc\mbox{C}\hspace{1.0pt}}}$ is [*weakly fiat*]{} if it is finitary and has a weak antiautomorphism $(-)^{\ast}$ of finite order and adjunction morphisms, see [@MM6 Subsection 2.5]. If $(-)^{\ast}$ is involutive, we say that ${{\sc\mbox{C}\hspace{1.0pt}}}$ is [*fiat*]{}. The existence of left and right adjoints suffices to conclude weak fiatness: taking right (alternatively left) adjoints is functorial and gives the desired weak $2$-equivalence; see [@EGNO]. A $2$-category of the form ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is weakly fiat if and only if $A$ is self-injective. This is an immediate consequence of [@MM1 Lemma 45]: \[AdjSelf\] Let $f,e$ be primitive mutually orthogonal idempotents of $A$. Then $$\big( \left(Ae \otimes_{\Bbbk} fA\right) \otimes_{A} -, \left( (fA)^{*} \otimes_{\Bbbk} eA \right) \otimes_{A} - \big)$$ is an adjoint pair of endofunctors of $A\!\on{-mod}$. Abelianization -------------- Given a finitary $2$-representation $\mathbf{M}$, we may consider its [*abelianization*]{} $\overline{\mathbf{M}}$, an abelian $2$-representation of ${{\sc\mbox{C}\hspace{1.0pt}}}$, defined in [@MMMT Section 3]. The abelianization defined therein is a significant improvement of that given in [@MM2 Subsection 4.2], however the main features we will use are shared by both constructions: we may recover $\mathbf{M}$ via a canonical embedding and the action of $1$-morphisms in $\overline{\mathbf{M}}$ is exact if ${{\sc\mbox{C}\hspace{1.0pt}}}$ is (weakly) fiat. Further, abelianization gives a $2$-functor $\overline{\,\,\cdot\,\,}: {{\sc\mbox{C}\hspace{1.0pt}}}\!\on{-afmod} \rightarrow {{\sc\mbox{C}\hspace{1.0pt}}}\!\on{-mod}$. Cells and cell $2$-representations ---------------------------------- We now introduce certain relations on the set of isomorphism classes of indecomposable $1$-morphisms of a finitary $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}$, which are analogous to Green’s relations for semigroups. For indecomposable $1$-morphisms $F,G \in {{\sc\mbox{C}\hspace{1.0pt}}}$ we write $F \geq_{L} G$ if there is a $1$-morphism $H \in {{\sc\mbox{C}\hspace{1.0pt}}}$ such that $F$ is isomorphic to a direct summand of $H \circ G$. This gives the [*left preorder*]{} $L$ on the set of indecomposable $1$-morphisms of ${{\sc\mbox{C}\hspace{1.0pt}}}$; the [*right preorder*]{} $R$ and the [*two-sided preorder*]{} $J$ are defined similarly. The equivalence classes of the induced equivalence relations are called the left, right and two-sided [*cells*]{} respectively. (Alternatively, $L$-cells, $R$-cells and $J$-cells.) Let $\mathcal{L}$ be a left cell of ${{\sc\mbox{C}\hspace{1.0pt}}}$ and let $\mathtt{i}$ be the object such that $F \in \mathcal{L}$ implies $F \in {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j})$ for some $\mathtt{j} \in {{\sc\mbox{C}\hspace{1.0pt}}}$. Consider the representable $2$-functor $\mathbf{P}_{\mathtt{i}} := {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i},-)$. This $2$-functor gives a finitary $2$-representation of ${{\sc\mbox{C}\hspace{1.0pt}}}$, and the additive closure of $1$-morphisms $F$ such that $F \geq_{L} \mathcal{L}$ gives a $2$-subrepresentation of $\mathbf{P}_{\mathtt{i}}$. This $2$-subrepresentation is [*transitive*]{}, that is, for any indecomposable $X \in \mathbf{M}(\mathtt{j}), Y \in \mathbf{M}(\mathtt{k})$ there is a $1$-morphism $G$ of ${{\sc\mbox{C}\hspace{1.0pt}}}$ such that $Y$ is isomorphic to a direct summand of $\mathbf{M}(F)X$. Every transitive $2$-representation admits a unique [*simple transitive*]{} quotient - a $2$-representation with no ${{\sc\mbox{C}\hspace{1.0pt}}}$-stable ideals (see [@MM5]). The [*cell $2$-representation $\mathbf{C}_{\mathcal{L}}$*]{} is the simple transitive subquotient of $\mathbf{P}_{\mathtt{i}}$ associated to $\mathcal{L}$. A two-sided cell $\mathcal{J}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}$ is said to be [*idempotent*]{} if there are $F,G,H \in \mathcal{J}$ such that $H$ is isomorphic to a direct summand of $G \circ F$. Given a transitive $2$-representation $\mathbf{M}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}$, there is a unique maximal two-sided cell not annihilated by $\mathbf{M}$ (see [@CM Lemma 1]), called the [*apex*]{} of $\mathbf{M}$. The apex of a transitive $2$-representation must be idempotent, and it coincides with the apex of its unique simple transitive quotient (See [@CM Lemma 3]). ### Simple transitive $2$-representations of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ {#CACells} Let $A$ be a basic, connected, finite-dimensional algebra. Fix a complete system of idempotents $E = {\left\{ e_{1},\ldots,e_{r} \right\}}$ of $A$. A complete list of representatives of isoclasses of indecomposable $1$-morphisms of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is given by the collection ${\left\{ A \right\}} \cup {\left\{ Ae_{i} \otimes_{\Bbbk} e_{j}A \; | \; 1 \leq i,j \leq r \right\}}$. Denote $Ae_{i} \otimes_{\Bbbk} e_{j}A$ by $F_{ij}$. Similarly to cells coming from Green’s relations for semigroups, the cell structure of a finitary $2$-category is often represented by eggbox diagrams. For ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, we have the following: $$\begin{array}{|c|c|c|c|} \hline F_{11}&F_{12}&\cdots &F_{1r}\\ \hline F_{21}&F_{22}&\cdots&F_{2r}\\ \hline \vdots&\vdots&\ddots &\vdots\\ \hline F_{r1}&F_{r2}&\cdots &F_{rr}\\ \hline \end{array}$$ $$\begin{array}{|c|} \hline A \\ \hline \end{array}$$ Where the two eggboxes represent the two $J$-cells $\mathcal{J}_{0},\mathcal{J}_{1}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, satisfying $\mathcal{J}_{1} >_{J} \mathcal{J}_{0}$, columns of the eggboxes represent the left cells, and rows represent the right cells. By construction we then have $r+1$ cell $2$-representations: the apex of a cell $2$-representation $\mathbf{C}_{\mathcal{L}}$ here is the $J$-cell containing $\mathcal{L}$. Decategorification and action matrices -------------------------------------- Define the [*decategorification of ${{\sc\mbox{C}\hspace{1.0pt}}}$*]{} as the category $[{{\sc\mbox{C}\hspace{1.0pt}}}]$ with the same objects as those of ${{\sc\mbox{C}\hspace{1.0pt}}}$, and for objects $\mathtt{i}, \mathtt{j} \in [{{\sc\mbox{C}\hspace{1.0pt}}}]$, the group of morphisms $[{{\sc\mbox{C}\hspace{1.0pt}}}](\mathtt{i}, \mathtt{j})$ is the split Grothendieck group $[{{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i}, \mathtt{j})]$. Similarly, we may decategorify a finitary $2$-representation $\mathbf{M}$ to a functor $[{{\sc\mbox{C}\hspace{1.0pt}}}] \rightarrow \mathbf{Ab}$. We then get the [*action matrices*]{} of $1$-morphisms of ${{\sc\mbox{C}\hspace{1.0pt}}}$ with respect to $\mathbf{M}$: for $F \in {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j})$, choose complete sets of representatives of isoclasses of indecomposables ${\left\{ X_{1},\ldots, X_{n} \right\}},{\left\{ Y_{1},\ldots, Y_{m} \right\}}$ for $\mathbf{M}(\mathtt{i})$ and $\mathbf{M}(\mathtt{j})$ respectively. With respect to the induced basis in split Grothendieck groups, the action matrix $[F]$ is the $m \times n$ matrix such that $[F]_{ij}$ is the multiplicity of $Y_{i}$ in $\mathbf{M}FX_{j}$. Self-injective cores ==================== Let $A$ be a basic, connected, finite-dimensional algebra, and fix a complete system of idempotents for $A$, $E = {\left\{ e_{i} \in A \; | \; i = 1, 2, \ldots, r \right\}}$. A [*self-injective core*]{} $S$ for $A$ is a non-empty subset of $E$ such that for every $e \in S$ there is $f \in S$ satisfying $(eA)^{*} \simeq Af$. Note that an algebra does not need to admit a self-injective core, and if it does, a self-injective core is not necessarily unique. Additionally, observe that a union of self-injective cores is a self-injective core. Alternatively, one could define a self-injective core on the level of the category $A\!\on{-proj}$ as a non-zero idempotent-split subcategory $\mathcal{S} \subseteq A\!\on{-proj}$ such that for the duality $(-)^{\ast} = \on{Hom}_{\Bbbk}(-,\Bbbk)$ one has $\on{Hom}_{A\!\on{-mod}}(\mathcal{S},A)^{\ast} \subseteq \mathcal{S}$. As a consequence one obtains $\on{Hom}_{A\!\on{-mod}}(\mathcal{S},A)^{\ast} \simeq \mathcal{S}$. If $A$ is weakly symmetric, then every non-empty subset of $E$ gives a self-injective core for $A$, so that $A$ admits $2^{r}-1$ self-injective cores. More generally, if $A$ is self-injective with Nakayama permutation $\nu \in S_{r}$, then self-injective cores of $A$ are unions of orbits of the action of $\nu$ on $E$. For instance, if $A=\Bbbk Q/\on{Rad}^{3}\Bbbk Q$ for $$Q = \begin{tikzcd} 1 \arrow[r, "a_{1}"] & 2 \arrow[d, "a_{2}"] \\ 4 \arrow[u, "a_{4}" ] & 3 \arrow[l, "a_{3}"] \end{tikzcd}$$ then the self-injective cores for $A$ are ${\left\{ e_{1},e_{3} \right\}}, {\left\{ e_{2},e_{4} \right\}}$ and ${\left\{ e_{1},e_{2},e_{3},e_{4} \right\}}$. If we further take the quotient $B$ of $A$ by relation $a_{2}a_{1} = 0$, the algebra no longer is self-injective, but ${\left\{ \overline{e_{2}},\overline{e_{4}} \right\}}$ is a self-injective core for $B$. Finally, a simple example of an algebra not admitting a single self-injective core is the path algebra of the Kronecker quiver $Q = \begin{tikzcd}[sep = small] 1 \arrow[r, shift left] \arrow[r, shift right] & 2 \end{tikzcd}$, which has no projective-injective modules. Fix some $A$ as above, admitting a self-injective core, and fix a complete system of idempotents $E$ and a self-injective core $S$. We now define the main objects of study in this document. In the setting above, using the description of indecomposable $1$-morphisms of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ given in Section \[CACells\], we define the following $2$-full $2$-subcategories thereof: - ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$, given by the additive closure of ${\left\{ A \right\}} \cup {\left\{ Ae_{i} \otimes_{\Bbbk} e_{j}A \; | \; e_{i},e_{j} \in S \right\}}$; - ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$, given by the additive closure of ${\left\{ A \right\}} \cup {\left\{ Ae_{i} \otimes_{\Bbbk} e_{j}A \; | \; e_{j} \in S \right\}}$; - ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$, given by the additive closure of ${\left\{ A \right\}} \cup {\left\{ Ae_{i} \otimes_{\Bbbk} e_{j}A \; | \; e_{i} \in S \right\}}$. Note that we define those $2$-categories entirely by specifying what part of $\mathcal{J}_{1}$ we choose to keep: ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is obtained by keeping the left cells associated to idempotents in $S$; alternatively, of course, we may say that we remove all the other left cells. Similarly, ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ is defined by keeping or removing right cells, and ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ by removing first left and then right cells, or vice versa. Observe also that in this setting, ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is a $2$-subcategory of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ as well as of $ {{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ - we even have ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J} = {{\sc\mbox{D}\hspace{1.0pt}}}_{\!L} \cap {{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$, but we will not use this fact. Before we continue with an example, we introduce some notation. Soon we will show that all indecomposable non-identity $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ lie in the same $J$-cell: we will denote that $J$-cell by $\mathcal{J}_{1}^{J}$. Viewing ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ as a $2$-subcategory of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}, {{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ or ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, we will denote the same colletion of $1$-morphisms by the same symbol, even if said collection does not necessarily form a $J$-cell in that case. Note that all of $\mathcal{J}_{1}^{J}$ still necessarily lies in the same $J$-cell of any of said $2$-categories containing it. In the respective cases ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}, {{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$, we will denote this $J$-cell by $\mathcal{J}_{1}^{L},\mathcal{J}_{1}^{R}$ respectively. In the case of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, this is $\mathcal{J}_{1}$. In the same spirit we will sometimes denote the $J$-cell consisting of the identity $1$-morphism by $\mathcal{J}_{0}^{J},\mathcal{J}_{0}^{L},\mathcal{J}_{0}^{R}$ respectively. \[FirstSight\] Let $\Lambda_{2}$ be the quotient of the path algebra of $$\begin{tikzcd} 2 \arrow[r, bend right, swap, "b_{2}"] & \color{lessred}0 \arrow[r, bend left, "a_{1}"] \arrow[l, swap, bend right, "a_{2}"] & \color{lessred}1 \arrow[l, bend left, "b_{1}"] \end{tikzcd}$$ by the relations $b_{2}a_{2} = b_{1}a_{1}$ and $a_{2}b_{1} = a_{1}b_{2} = 0$. Then the shaded part of the eggbox diagram of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, $$\begin{array}{|c|c|c|} \hline {\tikz[overlay,remember picture] \node [above = 2] (left2) {};}F_{00}&F_{01}&F_{02}\\ \hline F_{10}&F_{11}&F_{12}{\tikz[overlay,remember picture] \node (right2) {};}\\ \hline F_{20}&F_{21}&F_{22}\\ \hline \end{array}$$ [ ]{}$$\begin{array}{|c|} \hline {\tikz[overlay,remember picture] \node [above = 2] (left) {};} \Lambda_{2} {\tikz[overlay,remember picture] \node (right) {};}\\ \hline \end{array}$$ [ ]{}corresponds to the “right-cell” $2$-subcategory ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$, associated to the self-injective core $S = {\left\{ e_{0}, e_{1} \right\}}$, indicated in the quiver above. The following is the crucial property of self-injective cores: The $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is weakly fiat. Every simple transitive of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is equivalent to a cell $2$-representation. Moreover, a $2$-full $2$-subcategory of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is weakly fiat if and only if it is of the form ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ for some self-injective core $S$ of $A$. In particular, ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is weakly fiat if and only if $A$ is self-injective. From the adjunction in Lemma \[AdjSelf\] we see that ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is by definition constructed so that its $1$-morphisms admit and is closed under left and right adjoints in ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. Thus ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is weakly fiat. Moreover, its $J$-cells clearly are strongly regular in the sense of [@MM5 Section 6] so by [@MM6 Proposition 1], ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ satisfies the numerical condition, and by [@MM5 Theorem 18] its simple transitive $2$-representations are cell $2$-representations. A $2$-full subcategory ${{\sc\mbox{D}\hspace{1.0pt}}}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ is determined by the collection of indecomposable $1$-morphisms of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ it contains. For this collection to be closed under composition, it must be obtained by removing some left respectively right cells of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. For ${{\sc\mbox{D}\hspace{1.0pt}}}$ to be weakly fiat, each of those indecomposables must admit both left and right adjoints in ${{\sc\mbox{D}\hspace{1.0pt}}}$. Since the weak involution swaps the left and right preorders, ${{\sc\mbox{D}\hspace{1.0pt}}}$ must have equally many left and right cells, and both those must be associated to the same collection of idempotents of $A$. In other words there is $S \subseteq {\left\{ e_{1}, \ldots, e_{r} \right\}}$ such that $${{\sc\mbox{D}\hspace{1.0pt}}}\cap \mathcal{J}_{1} = {\left\{ Ae \otimes_{\Bbbk} fA \; | \; e,f \in S \right\}}.$$ It remains to show that $S$ is a self-injective core. Again using Lemma \[AdjSelf\], we see that for $Ae_{i} \otimes_{\Bbbk} e_{j}A$ in ${{\sc\mbox{D}\hspace{1.0pt}}}$ to admit a right adjoint, there must be a $1$-morphism $Af \otimes_{\Bbbk} e_{i}A$ in ${{\sc\mbox{D}\hspace{1.0pt}}}$ which is (as a bimodule) isomorphic to $(e_{j}A)^{*} \otimes_{\Bbbk} e_{i}A$. In particular, for the left module structure to be isomorphic we need $(e_{j}A)^{*} \simeq Af$, which proves that statement. The result above will be very useful, since we will use ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ as an auxiliary fiat $2$-subcategory in the study of the non-fiat $2$-categories of the form ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ or ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$. To that end, we give a detailed study of $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ in the following subsection. $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ ------------------------------------------------------------ \[CellsDJ\] The cell structure of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is the restriction of that of ${{\sc\mbox{C}\hspace{1.0pt}}}_{A}$. For ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, one uses $F_{ij} \circ F_{kl} \simeq F_{il}^{\oplus \on{dim}e_{j}\!Ae_{k}}$, from which one infers the left (or right) incomparability of the various left (right) cells in $\mathcal{J}_{1}$, and using $\on{dim}e_{k}Ae_{k} > 0$ and $$\label{Calculated} F_{ik} \circ F_{kj} \simeq F_{ij}^{\oplus \on{dim}e_{k}Ae_{k}}$$ one shows $F_{kj} \sim_{L} F_{ij}$ (similarly one treats the right cells). For a general $2$-full $2$-subcategory ${{\sc\mbox{D}\hspace{1.0pt}}}\subseteq {{\sc\mbox{C}\hspace{1.0pt}}}_{A}$, the cell preorders of ${{\sc\mbox{C}\hspace{1.0pt}}}_{A}$ must be refinements of those of ${{\sc\mbox{D}\hspace{1.0pt}}}$. Hence, the incomparability statement is preserved, being caused by the lack of suitable $1$-morphisms in ${{\sc\mbox{C}\hspace{1.0pt}}}_{A}$. To show that also left (or right) equivalence is preserved, note that If $F_{ij}, F_{kj} \in {{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$, then $e_{i}, e_{k} \in S$ and thus also $F_{ik} \in {{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$, so we may again use . Our next aim is to show that, similarly to ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, cell $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ with the same apex are equivalent. Towards that, we first formulate and give a proof of a statement often implicitly used in the literature, for instance in [@MM5 Proposition 9], [@MZ1], [@Zi2]: \[StdArg\] Let ${{\sc\mbox{C}\hspace{1.0pt}}}$ be a finitary $2$-category, $\mathcal{L} = {\left\{ F_{1},\ldots,F_{r} \right\}}$ be a left cell of ${{\sc\mbox{C}\hspace{1.0pt}}}$, $\mathcal{J}$ be the $J$-cell containing $\mathcal{L}$ , and let $\mathbf{M}$ be a simple transitive $2$-representation with apex $\mathcal{J}$. Consider an ordering $X_{1},\ldots,X_{k}$ of isomorphism classes of indecomposable objects of $\mathbf{M}(\mathtt{i})$, yielding in particular $\mathbf{M}(\mathtt{i}) \simeq Q\!\on{-proj}$ for $Q = \on{End}(\bigoplus_{l=1}^{k}X_{l})^{\on{op}}$ with a complete system of idempotents induced by ${\left\{ \on{id}_{X_{l}} \right\}}_{l=1}^{k}=: {\left\{ f_{l} \right\}}_{l=1}^{k}$. If there is an ordering as above such that - The Cartan matrix of $\mathbf{M}(\mathtt{i})$ is equal to that of $\on{add}(\mathcal{L})$; - There is an index $j \in {\left\{ 1,\ldots,k \right\}}$ such that $\mathbf{M}F_{i} \simeq Qf_{i} \otimes_{\Bbbk} f_{j}Q$ for then $\mathbf{M}$ is equivalent to $\mathbf{C}_{\mathcal{L}}$. Let $L_{1}, \ldots, L_{k}$ be a complete list of simple objects of $Q\!\on{-mod} \simeq \overline{\mathbf{M}}(\mathtt{i})$ associated to the idempotents $f_{1},\ldots,f_{k}$, and let $\mathbf{K}_{\mathcal{L}}$ be the action of ${{\sc\mbox{D}\hspace{1.0pt}}}$ on $\on{add}(F \; | \; F \geq_{L} \mathcal{L})$ by composition. Similarly to the the Yoneda lemma for $\mathbf{P}_{\mathtt{i}}$ given in [@MM2 Lemma 9], the functor induced by the map $$F_{i} \rightarrow\overline{\mathbf{M}}F_{i}\left(L_{j}\right)$$ is a $2$-transformation from $\mathbf{K}_{\mathcal{L}}$ to $\mathbf{M}$. And since $\mathcal{J}$ is the apex of $\mathbf{M}$, the identity $2$-morphism of any $1$-morphism $F$ satisfying $F >_{L} \mathcal{L}$ is sent to a zero map. Hence there is an induced $2$-transformation from the transitive quotient $\mathbf{N}_{\mathcal{L}}$ of $\mathbf{K}_{\mathcal{L}}$, which acts on $\on{add}(F \; | \; F \in \mathcal{L})$. We denote that $2$-transformation by $\sigma$. Note that the image of $F_{i}$ under $\sigma$ is the indecomposable object $Qf_{i}$ of $Q\!\on{-proj}$. This shows that all the isomorphism classes of indecomposable objects of $Q\!\on{-proj}$ are reached by $\sigma$, and so $\sigma$ is essentially surjective. The kernel of $\sigma$ is an ideal of $\mathbf{N}_{\mathcal{L}}$, which doesn’t contain any identity $2$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}$, since $F_{i}L_{j} \neq 0$ for all $i$. Thus it is contained in the maximal ideal $\mathbf{I}$ of $\mathbf{N}_{\mathcal{L}}$. We claim that also $\mathbf{I} \subseteq \on{Ker}\sigma$. This is because $\sigma$ factors through $\mathbf{N}_{\mathcal{L}}/\on{Ker}(\sigma)$, and the induced morphism $$\widetilde{\sigma}: \mathbf{N}_{\mathcal{L}}/\on{Ker}(\sigma) \rightarrow \mathbf{M}$$ is faithful, so that for all $i,j$, the linear map $$\widetilde{\sigma}_{ij}: \on{Hom}_{\mathbf{N}_{\mathcal{L}}/\on{Ker}(\sigma)}(F_{i}, F_{j}) \rightarrow \on{Hom}_{Q\!\on{-proj}}(Qe_{i}, Qe_{j})$$ is injective, hence $$\on{dim}\on{Hom}_{\mathbf{N}_{\mathcal{L}}/\on{Ker}(\sigma)}(F_{i}, F_{j}) \leq \on{dim}\on{Hom}_{Q\!\on{-proj}}(Qe_{i}, Qe_{j}).$$ Since $\on{Ker}(\sigma) \subseteq \mathbf{I}$, we have $$\on{dim}\on{Hom}_{\mathbf{C}_{\mathcal{L}}}(F_{i},F_{j}) \leq \on{dim}\on{Hom}_{\mathbf{N}_{\mathcal{L}}/\on{Ker}(\sigma)}(F_{i}, F_{j}).$$ But the equality of Cartan matrices for $\mathbf{C}_{\mathcal{L}}$ and $\mathbf{M}$ implies that the lower and the upper bound for $\on{dim}\on{Hom}_{\mathbf{N}_{\mathcal{L}}/\on{Ker}(\sigma)}(F_{i}, F_{j})$ coincide, so $$\on{dim}\mathbf{I}(F_{i},F_{j}) = \on{dim}\on{Ker}(\sigma)(F_{i},F_{j}), \text{ and } \on{Ker}(\sigma) = \mathbf{I}.$$ So we can take the cell $2$-representation as the domain of the induced morphism: $$\widetilde{\sigma}: \mathbf{C}_{\mathcal{L}} \rightarrow \mathbf{M}.$$ Now for all $i,j$, $\widetilde{\sigma}_{ij}$ is an injective linear map between equidimensional spaces, and thus is an isomorphism. This shows that $\widetilde{\sigma}$ is full and faithful. It is also essentially surjective, since so is $\sigma$. So $\widetilde{\sigma}$ is an equivalence of $2$-representations. \[EqCellsJ\] Two cell $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ are equivalent if and only if they have the same apex. First, observe that equivalent $2$-representations clearly have the same apex, so we only need to prove is that cell $2$-representations with the same apex are equivalent. This is trivial if the apex is $\mathcal{J}_{0}$. The case where the apex is $\mathcal{J}_{1}$ is similar to [@MM5 Proposition 9]: the maximal ideal of transitive subrepresentation used to construct the left cell $\mathcal{L}_{j} = {\left\{ Ae_{i} \otimes_{\Bbbk} e_{j}A \; | \; e_{i} \in S \right\}}$ corresponds to the ideal $Ae_{S} \otimes_{\Bbbk} \on{Rad}e_{j}A$ of $\on{End}_{A\text{-}A\!\on{-bimod}}(\bigoplus_{i} Ae_{i} \otimes_{\Bbbk} e_{j}A) \simeq Ae_{S} \otimes_{\Bbbk} e_{j}A$, where $e_{S} = \sum_{e_{i} \in S} e_{i}$. Thus the cell $2$-representation acts on $Ae_{S} \otimes_{\Bbbk} \Bbbk \simeq Ae_{S}$, independently of $j$, and since the action by definition satisfies the latter condition of Lemma \[StdArg\], we have the sought equivalence. The next statement concerns discrete extensions of $2$-representations, studied in [@CM] - we will mainly use the corollary that follows it, so we don’t give details here. The statement itself is a consequence of the results of [@CM Section 6]: \[DExtJ\] Let $\mathcal{L}_{0} := \mathcal{J}_{0}^{J}$ and let $\mathcal{L}_{1} \subset \mathcal{J}_{1}^{J}$ be two left cells of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$. Then 1. $\on{Dext}(\mathbf{C}_{\mathcal{L}_{0}}, \mathbf{C}_{\mathcal{L}_{0}}) = \varnothing$ 2. $\on{Dext}(\mathbf{C}_{\mathcal{L}_{1}}, \mathbf{C}_{\mathcal{L}_{1}}) = \varnothing$ 3. $\on{Dext}(\mathbf{C}_{\mathcal{L}_{1}}, \mathbf{C}_{\mathcal{L}_{0}}) = \varnothing$. \[Jext\] Let $F^{J}:= \bigoplus_{F_{ij} \in \mathcal{J}_{1}^{J}} F_{ij}$ and let $\mathtt{C}$ be the matrix associated to $F$ under the $2$-representation $\mathbf{C}_{\mathcal{L}_{1}}$. Given a finitary $2$-representation $\mathbf{M}$ of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ such that not all of its composition factors correspond to $\mathbf{C}_{\mathcal{L}_{0}}$, there is an ordering of the indecomposable objects of $\mathbf{M}(\mathtt{i})$ such that the matrix $[F^{J}]$ is of the form $$[F^{J}]= \left( \begin{array}{c|c} \mathtt{K} & * \\ \hline 0 & 0 \end{array} \right)$$ where $\mathtt{K}$ is of the form $$\left( \begin{array}{c|c|c} \mathtt{C} & 0 & 0 \\ \hline 0 & \ddots & 0 \\ \hline 0 & 0 & \mathtt{C} \end{array} \right)$$ Cell structures and cell $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}, {{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ -------------------------------------------------------------------------------------------------------------------------- In this subsection we provide partial variants of results for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ from the preceding subsection, in the cases ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}, {{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$. To summarize, cell $2$-representations with same apex still are equivalent like in Proposition \[EqCellsJ\], ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ inherits its right cell structure from ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ and ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ similarly inherits its left cell structure from ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. Obstructions may occur in left cell structure of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ and right cell structure of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ - we describe exactly what differences one may find in comparison to ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. We only prove our statements for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ - the proofs regarding the cell structure of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ are very similar, and ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is the more difficult case when considering its cell $2$-representations: this is because it is the left cell structure that governs these, which may be pathological for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$, whereas for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ one may just mimic the approach for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$. First, an example of aforementioned obstruction: Let $A = \Lambda_{2}$, the algebra introduced in Example \[FirstSight\]. Choose the self-injective core $S = {\left\{ e_{2} \right\}}$, and consider the associated $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. Since $F_{i2} \circ F_{12} = 0$ for $i = 0,1,2$, we don’t have $F_{02} \geq_{L} F_{12}$ nor $F_{22} \geq_{L} F_{12}$. As a consequence, ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ has three $J$-cells: $$\mathcal{J}_{0} = {\left\{ \Lambda_{2} \right\}}, \mathcal{J}_{1} = {\left\{ F_{02},F_{22} \right\}}, \mathcal{J}_{2} = {\left\{ F_{12} \right\}} \text{ with } \mathcal{J}_{0} <_{J} \mathcal{J}_{1} <_{J} \mathcal{J}_{2}.$$ Its left preorder is equal to its two-sided preorder, and its right preorder is inherited from ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. Since ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is a $2$-subcategory of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, in view of Proposition \[CellsDJ\], one sees that the cell structure of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is preserved under inclusion into ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ - no new relations are introduced. In particular, by $\mathcal{J}_{1}$ we will denote the two-sided cell of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ containing all the non-identity indecomposable $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$. \[BadCells\] The right preorder of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is the restriction of that of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. A non-identity indecomposable $1$-morphism of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ lies outside $\mathcal{J}_{1}^{L}$ if and only if it is of the form $F_{kl}$ with $k$ such that $e_{i}Ae_{k} = 0$ for all $e_{i} \in S$. In that case, every $1$-morphism of its right cell also lies outside of $\mathcal{J}_{1}^{L}$ and constitutes a maximal, non-idempotent $J$-cell. We refer to such $J$-cells as bad cells. The first claim follows by noting that the collection $\mathcal{R}_{k} = {\left\{ F_{kl} \; | \; l \in S \right\}}$, coming from the corresponding right cell of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$, is a right cell of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$: given $F_{kl},F_{kl'}$ in $\mathcal{R}_{k}$, the $1$-morphism $F_{kl}$ is a direct summand of $F_{kl'} \circ F_{l'l}$, where the latter lies in ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. If there is $e_{i} \in S$ such that $e_{i}Ae_{k} \neq 0$, then $F_{kl} \sim_{L} F_{il}$, being a direct sum of $F_{ii} \circ F_{kl}$, so the left preorder on $\mathcal{J}_{1}^{J} \cup \mathcal{R}_{k}$ is inherited from ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. On the other hand, if, given $k$, there is no $i$ as above, then for any $F_{kj} \in {{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ we have $G \circ F_{kj} = 0$ for any indecomposable $1$-morphism of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ not isomorphic to the identity $1$-morphism. So every such $F_{kj}$ is $L$-maximal and constitutes its own left cell, and the right cell $\mathcal{R}_{k}$ becomes a $J$-cell, which is $J$-maximal by $L$-maximality of $F_{kj}$ and $R$-incomparability of non-identity right cells of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. It is also non-idempotent, since any composition inside it evaluates to zero. \[EqCellsLR\] Two cell $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ are equivalent if and only if they have the same apex. From the characterization of bad cells in Proposition \[BadCells\] it follows that the cell $2$-representation associated to such a cell has apex $\mathcal{J}_{0}$, consisting of the identity morphism. One may view such $2$-representation as a simple transitive $2$-representation of the $2$-category generated only by the regular bimodule, and any two simple transitive $2$-representations of such a $2$-category are necessarily equivalent (see [@MM5],[@MM6]). For cell $2$-representations with apex $\mathcal{J}_{1}^{L}$, the proof is very similar to that for ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ (see [@MM5]), with one minor difference. Choose a left cell $\mathcal{L}_{j}$ in $\mathcal{J}_{1}^{L}$. The maximal ideal of the transitive action used to construct $\mathbf{C}_{\mathcal{L}_{j}}$ can be represented by an ideal $I$ of the algebra $\on{End}\left( \bigoplus_{i} F_{ij} \right) \simeq A \otimes_{\Bbbk} e_{j}A$. From our earlier calculations and the results for ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ we see that this ideal must contain $A \otimes \on{Rad}e_{j}A$. The difference in the case of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is that $I$ is not necessarily equal to that ideal, so ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is not $\mathcal{J}_{1}^{L}$-simple in the sense of [@MM2 Section 6.2]. Nonetheless, since the homspaces between indecomposable $1$-morphisms are $\Bbbk$-split (in the same sense as the $1$-morphisms are represented by $\Bbbk$-split bimodules), and $I$ contains $A \otimes_{\Bbbk} \on{Rad}e_{j}A$, the quotient $A \otimes_{\Bbbk}e_{j}A/I$ is independent of $j$, and in each case the indecomposable $1$-morphisms act as suitable projective bimodules over $A \otimes_{\Bbbk}e_{j}A/I$ to establish the equivalence using Lemma \[StdArg\]. \[DeltaSeen\] Consider again the algebra $\Lambda_{2}$ defined in Example \[FirstSight\], choose the self-injective core $S = {\left\{ 0 \right\}}$ and consider the associated $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. Consider the morphism $\varphi_{c_{1}} \in \on{End}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!L}(\mathtt{i,i})}(F_{10})$ given by the bimodule morphism defined by sending $e_{1} \otimes e_{0}$ to $a_{1}b_{1} \otimes e_{0}$. The action of $F_{j0}$ on $\varphi_{c_{1}}$ by tensor product is zero for $j = 0,1,2$, so $\Bbbk[\varphi_{c_{1}}]$ is a proper ideal in the transitive action used to define the cell $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ with apex $\mathcal{J}_{1}^{L}$, and thus is removed in the quotient defining said cell $2$-representation. The same holds for $\varphi_{c_{2}} \in \on{End}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!L}(\mathtt{i,i})}(F_{20})$. As a consequence, the cell $2$-representation acts on a category equivalent to $\Delta_{2}\!\on{-proj}$, where we let $\Delta_{2}$ be the quotient of $\Lambda_{2}$ by the ideal generated by relations $a_{1}b_{1} = 0 = a_{2}b_{2}$. The following combines [@Zi2 Theorem 3.1] with [@MZ1 Lemma 8]: \[ProjAct\] Let $\mathbf{M}$ be a simple transitive $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$, ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ or ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. The $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ act as projective functors. If $F$ is a $1$-morphism acting as a projective functor, and $G \sim_{L} F$, then $G$ also acts as a projective functor. As a consequence, $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ act as projective functors. Simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ {#s4} ============================================================================== The main goal of this section is to prove the following: \[MainThm4\] Let $A$ be a basic, connected, finite-dimensional algebra with a self-injective core $S$. Consider the right-cell $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ associated to $S$. Every simple transitive $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ is equivalent to a cell $2$-representation. Given a finitary $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}$ and a $2$-subcategory ${{\sc\mbox{D}\hspace{1.0pt}}}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}$, let $\mathbf{M}$ be a finitary $2$-representation of ${{\sc\mbox{C}\hspace{1.0pt}}}$, and denote by $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}}$ the restriction of $\mathbf{M}$ to a $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}$. Let $X_{1}, \ldots, X_{n}$ be a complete list of indecomposable objects of $\mathbf{M}(\mathtt{i})$ and let $X_{1}, \ldots, X_{s}$ be the objects of that list associated to a transitive subquotient $\mathbf{N}$ of $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}}$. Let $\mathbf{I}$ be the ideal of $\mathbf{N}$ such that $\mathbf{N/I}$ is simple transitive. In that setting, we make the following observation, which is a direct consequence of the definitions of transitive and simple transitive $2$-representations: \[StateIneq\] Under the conditions described above, let $$Q:= \on{End}_{\mathbf{M}(\mathtt{i})}\left( \bigoplus_{i=1}^{s} X_{i} \right),$$ so that $\mathbf{N}(\mathtt{i}) \simeq Q\!\on{-proj}$. The ideal $\mathbf{I}$ corresponds to an ideal $I$ of $Q$ such that $$\mathbf{N/I}(\mathtt{i}) \simeq Q/I\!\on{-proj}.$$ In particular, if we denote the Cartan matrix of $\mathbf{M}(\mathtt{i})$ by $\mathtt{C}^{\mathbf{M}}$ and similarly by $\mathtt{C}^{\mathbf{N/I}}$ denote the Cartan matrix of $\mathbf{N/I}(\mathtt{i})$, then for all $i,j \in {\left\{ 1,\ldots, s \right\}}$ we have $$\label{Ineq} \mathtt{C}^{\mathbf{M}}_{ij} \geq \mathtt{C}^{\mathbf{N/I}}_{ij}.$$ In this section, the observation above is particularly useful due to the following: \[TransRes\] Let $\mathbf{M}$ be a simple transitive $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$. The restriction $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ of $\mathbf{M}$ to ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ is transitive. Let $\mathcal{L}_{1}$ be a left cell of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ contained in $\mathcal{J}_{1}^{J}$. Then $\mathcal{L}_{1}$ also is a left cell of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$. Hence, using action notation, we may write $\on{add}({{\sc\mbox{D}\hspace{1.0pt}}}_{\!R} \mathcal{L}_{1}) = \on{add}(\mathcal{L}_{1})$. Similarly, we then also have $$\on{add}\!\big({{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}(\mathcal{L}_{1}\mathbf{M}(\mathtt{i})) \big) = \on{add}\!\big( ({{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}\mathcal{L}_{1})\mathbf{M}(\mathtt{i}) \big) = \on{add}\!\big( \mathcal{L}_{1}\mathbf{M}(\mathtt{i}) \big).$$ So $\mathcal{L}_{1}\mathbf{M}(\mathtt{i})$ is ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$-stable, and thus, since $\mathbf{M}$ is simple transitive, must equal all of $\mathbf{M}(\mathtt{i})$. Thus, if we let $F^{J} := \bigoplus_{F_{ij} \in \mathcal{J}_{1}^{J}} F_{ij}$, we see that all the rows of the matrix $[F^{J}]$ are non-zero; we conclude that no factors in the Jordan-H[" o]{}lder decomposition of $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ are equivalent to $\mathbf{C}_{\mathcal{J}_{0}}$. From Corollary \[Jext\] we see that $[F^{J}]$ is of the form $$\left( \begin{array}{c|c|c} \mathtt{C} & 0 & 0 \\ \hline 0 & \ddots & 0 \\ \hline 0 & 0 & \mathtt{C} \end{array} \right),$$ where the entries of $C$ are positive integers. Now let $F := \bigoplus_{F_{ij} \in \mathcal{J}_{1}^{R}} F_{ij}$. By transitivity of $\mathbf{M}$, the entries of $[F]$ also are positive integers. As we have just established, we have $$\on{add}\left((F \circ F^{J})\mathbf{M}(\mathtt{i})\right) = \on{add}\left(F^{J}\mathbf{M}(\mathtt{i})\right).$$ So there are matrices $\mathtt{C}_{1}, \ldots, \mathtt{C}_{k}$ whose entries are positive integers, satisfying $$[F] \cdot [F^{J}] = \left( \begin{array}{c|c|c} \mathtt{C}_{1} & 0 & 0 \\ \hline 0 & \ddots & 0 \\ \hline 0 & 0 & \mathtt{C}_{k} \end{array} \right),$$ but since all entries of $[F]$ are positive, and all entries of $[F^{J}]$ are non-negative with the diagonal entries positive, all entries of $[F] \cdot [F^{J}]$ must be positive. Thus $k=1$ and so $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ is transitive. A direct consequence of that statement is that we know the rank of $\mathbf{M}$, which is equal to $p := |S|$. Let $\mathbf{M}$ be a simple transitive $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$, let $Q$ be a finite-dimensional algebra such that $\mathbf{M}(\mathtt{i}) \simeq Q\!\on{-proj}$, let $f_{1}, \ldots, f_{p}$ be a system of idempotents for $Q$, and by $G_{ij}$ denote the indecomposable projective functor $Qf_{i} \otimes_{\Bbbk} f_{j}Q$. Let $\mathtt{C}^{Q}$ denote the Cartan matrix of $\mathbf{M}(\mathtt{i})$. We will show that $\mathtt{C}^{Q}$ coincides with the Cartan matrix of the target category of $\mathbf{C}_{\mathcal{L}}$, for $\mathcal{L}$ a left cell inside $\mathcal{J}_{1}^{R}$, which is easily verified to be that of $eAe\!\on{-proj}$ for $e = \sum_{e_{i} \in S} e_{i}$. At the same time we will also show that there is an ordering of a complete list of indecomposable objects of $\mathbf{M}(\mathtt{i})$ such that for $F_{ij} \in \mathcal{J}_{1}^{J}$, the action of $F_{ij}$ is given by $\mathbf{M}F_{ij} \simeq G_{ij}$. From this, as shown in Lemma \[StdArg\], Theorem \[MainThm4\] will follow. Recall from Proposition \[ProjAct\] that a $1$-morphism $F_{ij} \in \mathcal{J}_{1}^{J}$ acts as a projective functor. To show that this projective functor is naturally isomorphic to $G_{ij}$, we need to introduce the [*$X$-sets*]{} and [*$Y$-sets*]{} of [@MZ2]: \[XYSets\] Given $e_{i}, e_{j} \in S$: - Let $X_{ij}$ be the set of $s \in {\left\{ 1, \ldots, p \right\}}$ such that for some $t \in {\left\{ 1, \ldots, p \right\}}$, $G_{st}$ is isomorphic to a direct summand of $\mathbf{M}F_{ij}$. - Let $Y_{ij}$ be the set of $t \in {\left\{ 1, \ldots, p \right\}}$ such that for some $s \in {\left\{ 1, \ldots, p \right\}}$, $G_{st}$ is isomorphic to a direct summand of $\mathbf{M}F_{ij}$. Note that $X_{ij}$ is exactly the set of indices of non-zero rows of $[F_{ij}]$. \[Talk\] From Proposition \[TransRes\] we know the matrix $[F_{ij}]$ - it is the same as the matrix of $F_{ij}$ associated to the simple transitive quotient of $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}}$, which is equal to the matrix of the action of $eAe_{i} \otimes_{\Bbbk} e_{j}Ae$ on $eAe\!\on{-proj}$, for $e = \sum_{e_{i} \in S} e_{i}$. The only non-zero row of $[F_{ij}]$ is the $i$th one. This means that our choice of enumeration of idempotents in $S$ gives us an ordering on indecomposables of $\mathbf{M}(\mathtt{i})$ such that $X_{ij} = {\left\{ i \right\}}$ for all $j$. Since $F_{iq}$ admits a left adjoint in ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ for all $q$, the proof of [@MZ2 Lemma 22] yields: \[Catch22\] Under the ordering of indecomposables described above, given $e_{q} \in S$, $Y_{iq} = {\left\{ q \right\}}$ for all $e_{i} \in S$. We conclude that $\mathbf{M}F_{ij} \simeq G_{ij}^{\oplus m_{ij}}$ for some positive integer $m_{ij}$ depending on $i,j$. We are now ready to prove the main result: \[PfThm4\] Let $\mathtt{C}^{eAe}$ denote the Cartan matrix of $eAe$ and similarly let $\mathtt{C}^{Q}$ denote the Cartan matrix of $Q$. Let $e_{i}, e_{j} \in S$, so that $F_{ij} \in \mathcal{J}_{1}^{J}$. Observe that $$G_{ij}Qe_{l} \simeq Qe_{i}^{\oplus \mathtt{C}^{Q}_{jl}},\text{ so } [G_{ij}]_{il} = \mathtt{C}^{Q}_{jl}.$$ And as $\mathbf{M}F_{ij} \simeq G_{ij}^{\oplus m_{ij}}$, we have $[F_{ij}] = m_{ij}[G_{ij}]$, and also $$[F_{ij}]_{il} = m_{ij}\mathtt{C}^{Q}_{jl}$$ By the discussion following Definition \[XYSets\], the matrix $[F_{ij}]$ is the same as the matrix of the corresponding $1$-morphism of ${{\sc\mbox{C}\hspace{1.0pt}}}_{eAe}$ in the cell $2$-representation thereof. Thus we know that $$[F_{ij}]_{il} = \mathtt{C}_{jl}^{eAe}.$$ It now follows that $$m_{ij}\mathtt{C}_{jl}^{Q} = \mathtt{C}_{jl}^{eAe}.$$ But in the proof of Proposition \[EqCellsJ\] we showed that the Cartan matrix of the simple transitive quotient of $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ is exactly $\mathtt{C}^{eAe}$. So by the inequality \[Ineq\] stated in Observation \[StateIneq\], we have $$\mathtt{C}_{jl}^{Q} \geq \mathtt{C}_{jl}^{eAe},$$ which, since $m_{ij}$ is to be a positive integer, implies $m_{ij} = 1$ and $\mathtt{C}_{jl}^{Q} = \mathtt{C}_{jl}^{eAe}$. This implies both that $\mathbf{M}F_{ij} \simeq G_{ij}$ and that $\mathtt{C}^{Q} = \mathtt{C}^{eAe}$, from which, by Proposition \[StdArg\], the result follows. Removing right cells for $A=\Delta_{n}$ {#s5} ======================================= In this section, we focus on the two families of algebras ${\left\{ \Lambda_{n} \; | \; n \geq 1 \right\}}$ and ${\left\{ \Delta_{n} \; | \; n \geq 1 \right\}}$ generalizing the algebras $\Lambda_{2}, \Delta_{2}$ introduced in Examples \[FirstSight\] and \[DeltaSeen\] respectively. The first family was studied in [@Zi2]: it was established there that for the self-injective core $S = {\left\{ e_{0} \right\}}$, the simple transitive $2$-representations of the $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ are equivalent to cell $2$-representations. This is generalized by Theorem \[MainThm4\]. In Example \[DeltaSeen\] we have found that the cell $2$-representation connects $\Lambda_{2}$ with a quotient $\Delta_{2}$ thereof. In general, $\Delta_{n}$ is a quotient of $\Lambda_{n}$ for which only one of its $n+1$ indecomposable projectives is injective, giving a unique self-injective core ${\left\{ e_{0} \right\}}$. In this section we show that removing some cells of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!\Delta_{n}}$, even without requiring those remaining to correspond to a self-injective core, gives a $2$-category whose simple transitive $2$-representations are equivalent to its cell $2$-representations. \[DefStar\] Given an integer $n \geq 1$, the [*star algebra*]{} $\Lambda_{n}$ is the quotient of the path algebra of $$\begin{tikzcd}[row sep = small] & 0 \arrow[ddl, shift right, bend right, swap, "a_{1}"] \arrow[dd, dotted] \arrow[ddr, shift left, bend left, "a_{n}"] \\ \\ 1 \arrow[uur, swap, "b_{1}"] & \ldots \arrow[uu, dotted] & n \arrow[uul, "b_{n}"] \end{tikzcd}$$ by the ideal generated by relations - $b_{i}a_{i} = b_{j}a_{j} \text{ for all }i,j \in {\left\{ 1,\ldots,n \right\}}$; - $a_{i}b_{j} = 0 \text{ for }i\neq j$. For $n > 1$ these relations imply $\on{Rad}^{3}\Lambda_{n} = 0$. For $\Lambda_{1}$ we explicitly require that to be the case: $$b_{1}a_{1}b_{1} = a_{1}b_{1}a_{1} = 0.$$ The algebra $\Delta_{n}$ is defined as the quotient of $\Lambda_{n}$ by the relations $$a_{i}b_{i} = 0 \text{ for }i=1,\ldots,n.$$ Fix $n \geq 1$ and denote $\Delta_{n}$ by $\Delta$. The algebra $\Lambda_{n}$ is the so-called zig-zag algebra on the underlying graph of the quiver of $\Lambda_{n}$. For more on general zig-zag algebras, see e.g. [@ET]. \[Wstar\] Let $E = {\left\{ e_{0},\ldots,e_{n} \right\}}$ be the complete system of idempotents for $\Delta$ induced by the quiver above, and consider a subset $S \subseteq E$. We let - ${{\sc\mbox{D}\hspace{1.0pt}}}_{J}^{\Delta}$ be the $2$-full $2$-subcategory of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\Delta}$ given by the additive closure of ${\left\{ \Delta \right\}} \cup {\left\{ \Delta e_{i} \otimes_{\Bbbk} e_{j}\Delta \; | \; e_{i}, e_{j} \in S \right\}}$; - ${{\sc\mbox{D}\hspace{1.0pt}}}_{R}^{\Delta}$ be the $2$-full $2$-subcategory of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\Delta}$ given by the additive closure of ${\left\{ \Delta \right\}} \cup {\left\{ \Delta e_{i} \otimes_{\Bbbk} e_{j}\Delta \; | \; e_{i} \in S \right\}}$. Note that given a permutation $\sigma \in S_{n}$, the algebra morphism induced by $$e_{0} \mapsto e_{0}, \; e_{i} \mapsto e_{\sigma(i)} \text{ for } i=1,\ldots,n$$ is an automorphism of $\Delta$. So for $$S = {\left\{ e_{0}, e_{1}, \ldots, e_{m} \right\}} \text{ and } S' = {\left\{ e_{0}, e_{\sigma(1)}, \ldots, e_{\sigma(m)} \right\}},$$ the resulting $2$-categories are biequivalent. And so if $S$ contains $m$ idempotents among $e_{1},\ldots, e_{n}$ for some $m \leq n$, we will without loss of generality always assume that those idempotents are exactly $e_{1}, \ldots, e_{m}$. Moreover, it is easy to verify that if $e_{0} \not\in S$, then all the non-identity left cells of ${{\sc\mbox{D}\hspace{1.0pt}}}_{R}^{\Delta}$ outside of $\mathcal{J}_{1}^{J}$ are bad cells, in which case ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}^{\Delta}$ reduces to ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$. In this document we will only consider the case $e_{0} \in S$. Under this assumption, we will show that simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$, as well as those of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}^{\Delta}$, are equivalent to cell $2$-representations. The proof of Propositon \[EqCellsLR\] also applies to ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ and ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}^{\Delta}$, so cell $2$-representations with the same apex are equivalent in both cases. \[53Prop\] Let $S = {\left\{ e_{0}, e_{1}, \ldots, e_{m} \right\}}$. Then ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ is biequivalent to ${{\sc\mbox{C}\hspace{1.0pt}}}_{\Delta_{m}}$. Similarly to Example \[DeltaSeen\], one determines that the cell $2$-representation with apex $\mathcal{J}_{1}$ of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ acts on a category equivalent to $\Delta_{m}\!\on{-proj}$. The $1$-morphisms act as respective projective functors, since the projection of $\Delta \otimes_{\Bbbk} \Delta^{\on{op}}$ onto $\Delta_{m}\otimes_{\Bbbk} \Delta_{m}^{\on{op}}$ sends $\Delta e_{i} \otimes_{\Bbbk} e_{j}\Delta$ to $\Delta_{m}e_{i} \otimes_{\Bbbk} e_{j}\Delta_{m}$. Similarly to [@MM1 Proposition 46], the above action gives rise to a $2$-functor $\mathbf{R}: {{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta} \rightarrow {{\sc\mbox{C}\hspace{1.0pt}}}_{\Delta_{m}}$ sending a $1$-morphism of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ to the functor it acts by in the cell $2$-representation, and similarly for $2$-morphisms. This $2$-functor is locally essentially surjective, since it reaches all isoclasses of indecomposable $1$-morphisms. It is locally faithful, since the cell $2$-representation is $2$-faithful: explicit calculation proves that there is no $2$-morphism $\varphi$ of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ such that $\varphi F_{ij}=0$ for all $i,j$. To show that it is locally full, we show $$\on{dim} \on{Hom}_{{\scc\mbox{D}\hspace{0.5pt}}_{J}^{\Delta}}(F,G) =\on{dim} \on{Hom}_{{\scc\mbox{C}\hspace{0.5pt}}_{\Delta_{m}}}(\mathbf{R}F,\mathbf{R}G)$$ so that the maps between such spaces induced by $\mathbf{R}$ are surjective due to being injective between equidimensional spaces. Using the fact that an indecomposable $1$-morphism $F_{ij}$ is represented by a bimodule $\Delta e_{i} \otimes_{\Bbbk} e_{j}\Delta$, which is cyclic on $e_{i} \otimes e_{j}$, and similarly the regular bimodule is cyclic on $1 \in \Delta$, we describe the Hom-spaces above using the images of cyclic vectors. On the level of $2$-morphisms inside $\mathcal{J}_{1}^{J}$, the equidimensionality above reduces to the observation that $$e_{i}\Delta e_{j} = e_{i}(e_{0}+\cdots + e_{m})\Delta(e_{0}+\ldots + e_{m})e_{j} \simeq e_{i}\Delta_{m} e_{j}.$$ The same holds for morphisms from $\mathcal{J}_{1}^{J}$ to the identity $1$-morphism $\Delta$, we have: $$\on{Hom}_{{\scc\mbox{D}\hspace{0.5pt}}_{J}^{\Delta}}(F_{ij}, \Delta) = {\left\{ \varphi \; | \; \varphi(e_{i} \otimes e_{j}) \in e_{i}\Delta e_{j} \right\}} \simeq e_{i}\Delta e_{j}.$$ Finally, $$\on{End}_{{\scc\mbox{D}\hspace{0.5pt}}_{J}^{\Delta}}(\Delta) \simeq Z(\Delta)$$ and $$\begin{aligned} &\on{Hom}_{{\scc\mbox{C}\hspace{0.5pt}}_{\Delta_{m}}}(\Delta, F_{ij}) \\ &= {\left\{ \varphi \; | \; \varphi(1) = x \text{ for } x \in \Delta_{m}e_{i} \otimes_{\Bbbk} e_{j}\Delta_{m} \text{ such that } ax = xa \text{ for all }a \in \Delta_{m} \right\}}. \end{aligned}$$ Elementary calculation shows $Z(\Delta) = \Bbbk[1,c]$ for $c = b_{j}a_{j}$ for any $j$, and, denoting by $\varphi_{x}$ the morphism sending $1$ to $x$, we have $$\begin{aligned} \on{Hom}_{{\scc\mbox{C}\hspace{0.5pt}}_{\Delta}(\mathtt{i,i})}(\Delta, F_{ij}) = \begin{cases} \Bbbk[\varphi_{b_{i} \otimes a_{j}}] \text{ for }i,j \neq 0 \\ \Bbbk[\varphi_{c \otimes a_{j}}] \text{ for }i=0, j\neq 0 \\ \Bbbk[\varphi_{b_{i} \otimes c}] \text{ for }i\neq0, j=0 \\ \Bbbk \left[\varphi_{c \otimes c}, \varphi_{z}\right] \text{ for }i=j=0 \text{ and } z= e_{0} \otimes c + c \otimes e_{0} +\sum_{k=1}^{n} a_{k} \otimes b_{k}. \end{cases} \end{aligned}$$ This shows that the dimension is independent of the number of vertices in the underlying graph, proving the claimed equidimensionality, and concluding the proof. The biequivalence above yields a bijection between the equivalence classes of simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}, {{\sc\mbox{C}\hspace{1.0pt}}}_{\Delta_{m}}$. The $2$-representations of the latter are equivalent to cell $2$-representations, so we conclude that also the simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ must be equivalent to cell $2$-representations. Next, we want to establish that the restriction of $\mathbf{M}$ to a $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ is transitive, similarly to Proposition \[TransRes\]. To that end, we need a statement concerning discrete extensions, similar to Proposition \[DExtJ\]. Let $A$ be a finite-dimensional algebra admitting a non-zero projective-injective module. Let $\mathbf{C}_{\mathcal{L}_{0}}$ be the cell $2$-representation of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ with apex $\mathcal{J}_{0}$ and let $\mathbf{C}_{\mathcal{L}_{1}}$ be the cell $2$-representation with apex $\mathcal{J}_{1}$. Then we have the following: 1. $\on{Dext}(\mathbf{C}_{\mathcal{L}_{0}}, \mathbf{C}_{\mathcal{L}_{0}}) = \varnothing$ 2. $\on{Dext}(\mathbf{C}_{\mathcal{L}_{1}}, \mathbf{C}_{\mathcal{L}_{1}}) = \varnothing$ 3. $\on{Dext}(\mathbf{C}_{\mathcal{L}_{1}}, \mathbf{C}_{\mathcal{L}_{0}}) = \varnothing$. Parts $(1)$ and $(2)$ are shown in [@CM Theorem 22]. For part $(3)$, let $e_{1},\ldots, e_{r}$ be a complete system of idempotents and let ${\left\{ F_{ij} \right\}}_{i,j \in {\left\{ 1,\ldots, r \right\}}}$ label the indecomposable $1$-morphisms of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. Let $i,j \in {\left\{ 1,\ldots,r \right\}}$ be such that $Ae_{i} \simeq (e_{j}A)^{\ast}$. Then our assumption and [@MM1 Lemma 45] show that for any $k \in {\left\{ 1,\ldots, r \right\}}$ we have a pair $(F_{kj}, F_{ik})$ of adjoint $1$-morphisms. Let $\mathbf{M}$ be a $2$-representation with a transitive $2$-subrepresentation $\mathbf{N}$ with simple transitive quotient equivalent to $\mathbf{C}_{\mathcal{L}_{0}}$ and such that the quotient $\mathbf{K}$ is transitive, with simple transitive quotient equivalent to $\mathbf{C}_{\mathcal{L}_{1}}$, so that we have a short exact sequence $$0 \rightarrow \mathbf{N} \rightarrow \mathbf{M} \rightarrow \mathbf{K} \rightarrow 0$$ in the sense of [@CM]. Let $X_{1},\ldots, X_{r}, Y$ be a complete list of representatives of isoclasses of indecomposable objects of $\mathbf{M}(\mathtt{i})$, with $X_{1},\ldots, X_{r}$ corresponding to $\mathbf{K}$ and $Y$ corresponding to $\mathbf{N}$. We aim to show that there is no $1$-morphism $F$ of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ and $k \in {\left\{ 1,\ldots,r \right\}}$ such that $Y$ is isomorphic to a direct summand of $\mathbf{M}F(X_{k})$. Clearly, we can argue on the level of indecomposable $1$-morphisms. Let $X := \bigoplus_{l=1}^{r} X_{l}$. Consider a $1$-morphism of the form $F_{kj}$. Using $\mathbf{M}F_{ik}(Y) = 0$ and the adjunction above, we get $$\on{Hom}_{\mathbf{M}(\mathtt{i})}(\mathbf{M}F_{kj}(X), Y) \simeq \on{Hom}_{\mathbf{M}(\mathtt{i})}(X, \mathbf{M}F_{ik}(Y)) = 0$$ showing the sought statement for $F_{kj}$. Similarly, we have $$0 = \on{Hom}_{\mathbf{M}(\mathtt{i})}(\mathbf{M}F_{kj}(Y), X) \simeq \on{Hom}_{\mathbf{M}(\mathtt{i})}(Y, \mathbf{M}F_{ik}(X)),$$ which shows it for $F_{ik}$. Denote by $L_{k}$ the simple $A$-module associated to $e_{k}$. Observe that $$\on{dim}e_{j}Ae_{i} = [Ae_{i}: L_{j}] = [(e_{j}A)^{\ast} : L_{j}] > 0,$$ where $[M: L_{j}]$ denotes the composition multiplicity of $L_{j}$ in $M$, for $M \in A\!\on{-mod}$. Now let $k,l \in {\left\{ 1,\ldots, r \right\}}$. We have $F_{kl}^{\on{dim}e_{j}Ae_{i}} \simeq F_{kj} \circ F_{il}$, so in view of the above we see that $F_{kl}$ is a direct summand of $F_{ki} \circ F_{jl}$. And clearly $Y$ is not isomorphic to a direct summand of $\mathbf{M}(F_{ki}\circ F_{jl})(X)$, by what we have shown before. Now, just as in Proposition \[TransRes\], the restriction of $\mathbf{M}$ to a $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ is transitive, and so we see that the number of indecomposable objects of $Q\!\on{-proj}$ - the rank of $\mathbf{M}$ - is equal to $m+1$. In particular, the Cartan matrix $\mathtt{C}^{Q}$ of $Q$ is an $(m+1) \times (m+1)$ matrix. Let ${\left\{ f_{1}, \ldots, f_{m} \right\}}$ be a system of idempotents for $Q$, and by $G_{ij}$ denote an indecomposable projective functor isomorphic to tensoring with $Qf_{i} \otimes_{\Bbbk} f_{j}Q$ over $Q$. Finally, let $L_{i}$ denote the simple object of $Q\!\on{-mod}$ associated to $f_{i}$. As a consequence of Proposition \[ProjAct\] and $F_{00}$ being self-adjoint, the following holds: The $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}^{\Delta}$ of the form $F_{i0}$ act as projective functors. Since the restriction of $\mathbf{M}$ to ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$ is transitive, we know the matrices of those $1$-morphisms, and so, using $X$-sets and $Y$-sets as in Definition \[XYSets\], we see that $X_{i0} = {\left\{ i \right\}}$. Moreover, since all of those morphisms are $L$-equivalent, their $Y$-sets coincide; denote that set by $Y_{0}$. The self-adjointness of $F_{00}$ yields $Y_{0} = {\left\{ 0 \right\}}$, as in Lemma \[Catch22\]. So $\mathbf{M}F_{i0} \simeq G_{i0}^{\oplus m_{i}}$. But now the considerations from the proof of theorem \[MainThm4\] apply, since we do have transitive restriction and we do know that the simple transitive quotient of that restriction is the cell $2$-representation. So $m_{i} = 1$. In other words, we have: \[TheNiceCell\] For $i=0,\ldots,n$, there is a natural isomorphism $\mathbf{M}F_{i0} \simeq G_{i0}$. In view of Proposition \[StdArg\], we now only need to show that the Cartan matrix of a simple transitive $2$-representation is the same as that of a cell $2$-representation, i.e. that of $\Delta_{m}$. Again mimicking the approach from Section \[s4\], we observe that $$G_{i0} \circ G_{j0} \simeq G_{i0}^{\oplus \mathtt{C}^{Q}_{0j}}$$ and $$F_{i0} \circ F_{j0} \simeq \begin{cases} F_{i0} \text{ for }j \neq 0 \\ F_{i0}^{\oplus 2} \text{ for j = 0} \end{cases}$$ yield $\mathtt{C}^{Q}_{0j} = 1$ for $j = 1,\ldots, m$ and $\mathtt{C}^{Q}_{00} = 2$. Furthermore, the self-adjointness of $F_{00}$ yields $$\begin{aligned} & 2\mathtt{C}^{Q}_{i0} = \on{dim} \on{Hom}_{Q\!\on{-proj}}(Qf_{i}, Qf_{0}^{\oplus 2}) = \on{dim} \on{Hom}_{Q\!\on{-proj}}(Qf_{i}, F_{00}Q_{0}) \\ &= \on{dim}\on{Hom}_{Q\!\on{-proj}}(F_{00}Qf_{i},Q_{0}) = \on{dim}\on{Hom}_{Q\!\on{-proj}}(Q_{0},Q_{0}) = \mathtt{C}^{Q}_{00} = 2. \end{aligned}$$ So $\mathtt{C}^{Q}_{i0} = 1$ for $i \neq 0$. It remains to show that - $\mathtt{C}_{jj} = 1$ for $j= 1,\ldots, m$; - $\mathtt{C}_{ij} = 0$ for $i \neq j$ and $i,j \in {\left\{ 1,\ldots, m \right\}}$. We will show these statements in that order. Given a module $M \in Q\!\on{-mod}$ and a simple $Q$-module $L$, denote the composition multiplicity of $L$ in $M$ by $[M:L]$. \[CompositionRight\] A $1$-morphism of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}^{\Delta}$ of the form $F_{ij}$ with $j \neq 0$ acts isomorphically to tensoring with a split $Q$-bimodule of the form $Qf_{i} \otimes_{\Bbbk} M$, for some right $Q$-module $M$ satisfying $[M:L_{0}] = 1 = [M:L_{j}]$ and $[M:L_{k}] = 0$ otherwise. Since $F_{ij}$ acts as a right exact functor on the abelianization $\overline{\mathbf{M}}(\mathtt{i})$, and $\mathbf{M}$ embeds into the abelianization as described in Proposition \[ProjAct\], $F_{ij}$ acts by tensoring with a $Q$-bimodule $X$ in both $\mathbf{M}$ and $\overline{\mathbf{M}}$. Using $F_{ij} \sim_{R} F_{0j}$, in particular $F_{ij} \simeq F_{ij} \circ F_{0j}$, we find that $$X \simeq Qf_{i} \otimes_{\Bbbk} f_{j}Q \otimes_{Q} X,$$ which proves the first part of the statement for $M = f_{j}Q \otimes_{Q}X$. The remaining part is due to the fact that, as we have found earlier, we know the matrix $[F_{ij}]$ - its only two non-zero entries are both equal to $1$ and correspond to $\mathbf{M}F_{ij}(Q_{0})$ and $\mathbf{M}F_{ij}(Q_{j})$. In other words, $$Qf_{i} \simeq Qf_{i} \otimes_{\Bbbk} M \otimes_{Q} Qe_{i} \text{ for }i=0,1$$ and zero otherwise, which concludes the proof. \[1sOnDiagonal\] For $j = 1,\ldots,m$, we have $\mathtt{C}^{Q}_{jj} = 1$ . By definition, $\mathtt{C}^{Q}_{jj} = \on{dim}\on{End}_{Q\!\on{-proj}}Qf_{j}$. Hence in particular $\mathtt{C}^{Q}_{jj} \geq 1$ and since $Qf_{j}$ is indecomposable, $\mathtt{C}^{Q}_{jj} > 1$ if and only if $\on{Rad}\on{End}_{Q\!\on{-proj}}Qf_{j} \neq 0$. We will show that this is not the case: let $\alpha \in \on{Rad}\on{End}_{Q\!\on{-proj}}Qf_{j}$. The only $1$-morphisms that act on $Qf_{j}$ in a non-zero way are those of the form $F_{i0}$ or $F_{ij}$, for $i=0,\ldots,m$. In the first case, $F_{i0}\alpha$ is given by $$Qf_{i} \otimes_{\Bbbk} f_{0}Q \otimes_{Q} Qf_{j} \xrightarrow{\on{id}_{Qf_{i}} \otimes \widetilde{\alpha}} Qf_{i} \otimes_{\Bbbk} f_{0}Q \otimes_{Q} Qf_{j},$$ and using $f_{0}Q \otimes_{Q} Qf_{j} \simeq f_{0}Qf_{j}$, we see that $\widetilde{\alpha}$ corresponds to the $\Bbbk$-linear endomorphism of $f_{0}Qf_{j}$ induced by the right multiplication by $\alpha$. Recall that $\mathtt{C}^{Q}_{0j} = 1$ and so $\on{dim}f_{0}Qf_{j} = 1$. Since $\alpha \in \on{Rad}\on{End}_{Q\!\on{-proj}}Qf_{j}$, $\widetilde{\alpha}$ must be a nilpotent endomorphism. A nilpotent endomorphism of a one-dimensional space is zero. So $F_{i0}\alpha = 0$. For the other case, let $M$ be such that $F_{ij} \simeq Qe_{i} \otimes_{\Bbbk} M$. The morphism $F_{ij}\alpha$ can be written as $$Qe_{i} \otimes_{\Bbbk} Me_{j} \xrightarrow{\on{id} \otimes \widetilde{\alpha}} Qe_{i} \otimes_{\Bbbk} Me_{j},$$ where again $Me_{j}$ is one-dimensional and $\widetilde{\alpha} \in \on{End}_{Q\!\on{-mod}}(Me_{j})$ is nilpotent, and so $F_{ij}\alpha = 0$. Furthermore, acting on $\on{Rad}\on{End}_{Q\!\on{-proj}}Qf_{j}$ by the identity functor or its endomorphism again corresponds to multiplication with central elements of $Q$ and sends $\on{Rad}\on{End}_{Q\!\on{-proj}}Qf_{j}$ to itself. We have thus shown that the ideal of $Q\!\on{-proj}$ generated by $\on{Rad}\on{End}_{Q\!\on{-proj}}Qf_{j}$ is ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}^{\Delta}$-invariant, and since $\mathbf{M}$ is simple transitive, this ideal must be zero. In particular, $\alpha = 0$. The final statement we need to show is: For $i,j \in {\left\{ 1,\ldots,m \right\}}$ such that $i\neq j$, we have $\mathtt{C}^{Q}_{ij} = 0$ . In analogy to previous lemma, we will show that $\on{Hom}_{Q\!\on{-proj}}(Qf_{i},Qf_{j})$ is an ideal of $\mathbf{M}$. Assume $\alpha \in \on{Hom}_{Q\!\on{-proj}}(Qf_{i},Qf_{j})$. From Lemma \[CompositionRight\] and Lemma \[TheNiceCell\] we see that the $1$-morphisms of $\mathcal{J}_{1}^{R}$ that act on $Qf_{i}$ in a non-zero way are those of the form $F_{li}, F_{l0}$ for $l = 0,\ldots,m$, and so those are the only ones we need to be concerned with (as the identity functor maps $\on{Hom}_{Q\!\on{-proj}}(Qf_{i},Qf_{j})$ to itself). For $F_{li}$ however we have $F_{li}Q_{j} = 0$, as $F_{li} \simeq Qf_{l} \otimes_{\Bbbk} M$ with $[M: L_{j}] = 0$ by Lemma \[CompositionRight\]. Recall from Observation \[StateIneq\] that if we consider the restriction of $\mathbf{M}$ to ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}^{\Delta}$, which, as we have shown earlier, is transitive, and take the simple transitive quotient $\mathbf{N}$ of that $2$-representation, then the algebra $B$ such that $\mathbf{N}(\mathtt{i}) \simeq B\!\on{-proj}$ is a quotient of $Q$. As observed earlier, we know that $\mathbf{N}$ is the cell $2$-representation of $\Delta_{m}$ and thus $B= \Delta_{m}$. Let $I$ be the ideal such that $\Delta_{m} = Q/I$. Note that since for $i,j \in {\left\{ 1,\ldots,m \right\}}$ such that $i \neq j$ we have $\mathtt{C}^{\Delta_{m}}_{ij} = 0$, $\alpha$ must vanish under the quotient - in other words, $\alpha \in I$. Let $i \neq 0$. By definition of $\Delta_{m}$, $\on{Hom}_{\Delta_{m}\!\on{-proj}}(\Delta_{m}e_{i}, \Delta_{m}e_{0}) = \Bbbk[\varphi_{b_{i}}]$, where $\varphi_{b_{i}}$ is the unique morphism sending $e_{i}$ to $b_{i}$. By $\beta_{i}$ we will denote the element of $e_{0}Qe_{i}$ which under the projection $Q \rightarrow Q/I = \Delta_{m}$ maps to $b_{i}$. It is unique due to the fact that for $i \neq 0$, $\mathtt{C}_{i0}^{Q} = 1 = \mathtt{C}_{i0}^{\Delta_{n}}$. Also due to that equality, if we write $F_{l0}\alpha$ as $$Qf_{l} \otimes_{\Bbbk} f_{0}Qf_{i} \xrightarrow{\on{id}_{Qf_{l}} \otimes \widetilde{\alpha}} Qf_{l} \otimes_{\Bbbk} f_{0}Qf_{j},$$ where $\widetilde{\alpha}$ is induced by right multiplication with $\alpha$ (this is analogous to the calculation in the proof of Lemma \[1sOnDiagonal\]), we see that $\widetilde{\alpha}$ is a linear map between one-dimensional spaces and hence must be given by multiplication with a scalar. In other words, $\beta_{i} \alpha = \lambda \beta_{j}$ for some $\lambda \in \Bbbk$. Let $\pi: Q \rightarrow Q/I = \Delta_{m}$ be the canonical projection. As we have observed earlier, $\pi(\beta_{i}) = b_{i}$ and $\pi(\alpha) = 0$. Since $\pi$ is an algebra morphism, we must have $$0 = b_{i}\cdot 0 = \pi(\beta_{i})\pi(\alpha) = \pi(\beta_{i}\alpha) = \pi(\lambda \beta_{j}) = \lambda \pi(\beta_{j}) = \lambda b_{j},$$ which implies that $\lambda = 0$. But this shows that the ideal of $\mathbf{M}$ generated by $\on{Hom}_{Q\!\on{-proj}}(Qe_{i},Qe_{j})$ does not contain all the morphisms of $\mathbf{M}(\mathtt{i})$, and hence must be zero. So $$\on{Hom}_{Q\!\on{-proj}}(Qe_{i},Qe_{j}) = 0,$$ which concludes the proof. We have thus shown the main theorem of this section: Let ${\left\{ e_{0}, \ldots, e_{n} \right\}}$ be the system of idempotents induced by the definition of $\Delta_{n}$. Let $S \subseteq {\left\{ e_{0}, \ldots, e_{n} \right\}}$ be such that $e_{0} \in S$. The simple transitive $2$-representations of the $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}^{\Delta}$ associated to $S$ are equivalent to cell $2$-representations. ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ for leaf quotients of type $A$ zig-zag algebras ===================================================================================== From the perspective of this document, the main difference between ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ and ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ is that the left cells of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$ do not coincide with those of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$, as is the case for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$. In particular, the analogous statement to Proposition \[TransRes\] for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ doesn’t necessarily hold, and so we cannot determine the rank of a simple transitive $2$-representation of a general $2$-category of the form ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ using our methods for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$. This is a significant difficulty in [@Zi2 Section 5], where ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is considered for an algebra of the form $\Lambda_{n}$, as described in Definition \[DefStar\], and self-injective core $S = {\left\{ e_{0} \right\}}$. This indicates that we should not expect our methods for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!R}$ to work as well for ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ - and this is mirrored in this section, where we only consider a particular family of algebras together with a particular choice of a self-injective core, and in that setting use weak variants of results of Section \[s4\] to facilitate solving explicit numerical problems, reminiscent of those in [@MZ1]. \[BnDef\] Given a positive integer $n$, let $B_{n}$ denote the quotient of the path algebra of $$\begin{tikzcd} n \ar[r, bend left, "a_{n}"] & n-1 \arrow[l, bend left, "b_{n}"] \ar[r, bend left, "a_{n-1}"] & \cdots \arrow[l, bend left, "b_{n-1}"] \arrow[r, bend left, "a_{2}"] & 1 \arrow[l, bend left, "b_{2}"] \arrow[r, bend left, "a_{1}"] & 0 \arrow[l, bend left, "b_{1}"] \end{tikzcd}$$ by the ideal generated by relations - $a_{j-1}a_{j} = 0$ and $b_{j}b_{j-1} = 0$ for $j=1,\ldots,n$; - $b_{j}a_{j} = a_{j+1}b_{j+1}$ for $j= 1,\ldots,n-1$; - $a_{1}b_{1} = 0$. All the indecomposable projectives of $B_{n}$ except for the one associated to $e_{0}$ are projective-injective with socle and top isomorphic, so the self-injective cores for $B_{n}$ are exactly the non-empty subsets of ${\left\{ e_{1},\ldots, e_{n} \right\}}$ $B_{n}$ is the quotient of the zig-zag algebra on the Dynkin diagram $A_{n+1}$ by the ideal generated by $a_{1}b_{1}$. This element is the cycle at the leaf we labelled by zero, which motivates the name [*leaf quotient of type $A$ zig-zag algebras*]{}, used in [@PW], where these algebras are studied in terms of their (generalized) tilting modules and exceptional sequences. For the rest of this section we will study the simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ associated to the self-injective core $S = {\left\{ e_{1},\ldots, e_{n} \right\}}$ of $B_{n}$, for some $n \geq 1$. The main result of this section is of the same flavour as those of preceding two sections: \[MainThm6\] Let $\mathbf{M}$ be a simple transitive $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. $\mathbf{M}$ is equivalent to a cell $2$-representation. Under the ordering of the system of idempotents indicated by the quiver above, The Cartan matrix of $B_{n}$ is the $(n+1) \times (n+1)$ matrix \[CartanB\] $$\begin{pmatrix} 1 & 1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 \\ 0 & 1 & \ddots & \ddots & 0\\ 0 & 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 1 & 2 \end{pmatrix}.$$ To keep consistent with the said ordering of idempotents, we will index the rows and columns of this matrix starting from $0$ rather than from $1$. We have $$\on{dim}B_{n} = (n+1) + \on{dim}(\on{Rad}/\on{Rad}^{2})B_{n} + \on{dim}(\on{Rad}^{2}B_{n}) = (n+1) + 2n + n = 4n+1.$$ As in the earlier sections, we will also be interested in the algebra $eB_{n}e$ for $e = e_{1} + \ldots + e_{n}$. The dimension of this algebra is $$\on{dim}eB_{n}e = \on{dim}B_{n} - 3 = 4n-2.$$ Its Cartan matrix is the $n \times n$ lower diagonal block of that of $B_{n}$ given above. For the remainder of this section, choose $n \geq 1$ and let $B := B_{n}$. First, we remark that in this case, ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ has no bad cells: the only possible such cell would be $\mathcal{R}_{0} = {\left\{ F_{0j} \; | \; j = 1,\ldots, n \right\}}$, but $F_{i1} \circ F_{0j} \neq 0$ shows that $\mathcal{R}_{0}$ is not bad. As we have discussed in the proof of Proposition \[EqCellsJ\], another difficulty with ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ is that although all cell $2$-representations with apex $\mathcal{J}_{1}^{L}$ are equivalent, the Cartan matrix of a cell $2$-representation $\mathbf{C}_{\mathcal{L}_{j}}$ of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ for $B$ may be different from the Cartan matrix of $B\!\on{-proj}$ (in contrast to ${{\sc\mbox{C}\hspace{1.0pt}}}_{B}$). However, here this is not the case: \[Target6\] Let $\mathcal{L}_{j}$ be a left cell of $\mathcal{J}_{1}^{L}$. The Cartan matrix of the target category of $\mathbf{C}_{\mathcal{L}_{j}}$ coincides with that of $B\!\on{-proj}$. As explained in the proof of Proposition \[EqCellsJ\], the Cartan matrices in question differ if and only if there are $1$-morphisms $F_{ij},F_{kj}$ of $\mathcal{J}_{1}^{L}$ and a non-zero $2$-morphism $\alpha: F_{ij} \rightarrow F_{kj}$ with $\alpha(e_{i} \otimes e_{j}) \in Be_{k} \otimes_{\Bbbk} \Bbbk[e_{j}]$ such that $F_{lm}\alpha = 0$ for all $F_{lm} \in {{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. Assume that $\alpha$ is such a $2$-morphism. Also from the discussion in Proposition \[EqCellsJ\], we see that in our particular case, if $\alpha \in \on{Hom}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!L}(\mathtt{i,i})}(F_{ij},F_{kj})$ is non-zero and $i \neq 0$, then an identity can be recovered from $F_{ii}\alpha$. So the domain of $\alpha$ is $F_{0j}$. The only $1$-morphisms of the form $F_{lj}$ such that $\on{Hom}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!L}(\mathtt{i,i})}(F_{0j},F_{lj}) \neq 0$ are $F_{0j},F_{1j}$. However, $\on{End}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!L}(\mathtt{i,i})}(F_{0j}) = \Bbbk[\on{id}_{F_{0j}}]$, so an identity morphism is immediately found if $\alpha \in \on{End}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!L}(\mathtt{i,i})}(F_{0j})$. We are left with the case $\alpha \in \on{Hom}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!L}(\mathtt{i,i})}(F_{0j},F_{1j})$. Since we assume $$\alpha(e_{0} \otimes e_{j}) \in Be_{1} \otimes_{\Bbbk} \Bbbk[e_{j}]$$ and $e_{0}Be_{1} = \Bbbk[a_{1}]$, up to scalar multiple we must have $\alpha(e_{0} \otimes e_{j}) = a_{1} \otimes e_{j}$. Now $F_{i1}\alpha$ is represented by the map $$\widetilde{\alpha}: Be_{i} \otimes_{\Bbbk} e_{1}Be_{0} \otimes_{\Bbbk} e_{j}B \rightarrow Be_{i} \otimes_{\Bbbk} e_{1}Be_{1} \otimes_{\Bbbk} e_{j}B$$ given by sending $e_{i} \otimes b_{1} \otimes e_{j}$ to $e_{i} \otimes b_{1}a_{1} \otimes e_{j} \neq 0$, so $\widetilde{\alpha} \neq 0$, and hence $F_{i1}\alpha \neq 0$, which concludes the proof. Using the above, a short calculation yields: \[QIdempotentF\] Let $F:= \bigoplus_{F_{ij} \in \mathcal{J}_{1}^{L}} F_{ij}$. We have $$F \simeq F^{J} \oplus F^{\mathcal{R}_{0}}$$ for $F^{J} := \bigoplus_{F_{ij}\in \mathcal{J}_{1}^{J}} F_{ij}$ and $F^{\mathcal{R}_{0}} := \bigoplus_{i \in S} F_{i0}$. We also have $F \circ F \simeq F^{\oplus (4n-1)}$. Let $\mathbf{M}$ be a simple transitive $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. Our next objective is to find the matrix $[\mathbf{M}F]$, and hence also determine the rank of $\mathbf{M}$. The first observation we make is that $F\circ F \simeq F^{\oplus 4n-1}$ implies that $[\mathbf{M}F]^{2} = (4n-1)[\mathbf{M}F]$ and that $F \simeq F^{J} \oplus F^{\mathcal{R}_{0}}$ shows $[\mathbf{M}F] = [\mathbf{M}F^{J}] + [\mathbf{M}F^{\mathcal{R}_{0}}]$. Consider the restriction $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ of $\mathbf{M}$ to ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$. Since we asume the apex of $\mathbf{M}$ to be $\mathcal{J}_{1}^{L}$, $[\mathbf{M}F^{J}]$ is non-zero, and so the Jordan-H[" o]{}lder decomposition of $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ must contain a transitive subquotient whose simple transitive quotient is $\mathbf{C}_{\mathcal{L}_{1}}$ - the cell $2$-representation with apex $\mathcal{J}_{1}^{J}$. Direct computation shows that under the ordering of objects induced by the quiver in Definition \[BnDef\], $[\mathbf{C}_{\mathcal{L}_{1}}F^{J}]$ is equal to the $n \times n$ matrix $$\mathtt{C} = \begin{pmatrix} 3 & 4 & \ldots & 4 & 3 \\ 3 & 4 & \ldots & 4 & 3 \\ & & \vdots \\ 3 & 4 & \ldots & 4 & 3 \end{pmatrix}.$$ However, we permute the basis vectors to get a canonical form of $\mathtt{C}$ with respect to a classification result we are about to use, and instead assume an ordering of the indecomposable objects of $\mathbf{M}(\mathtt{i})$ such that $[\mathbf{M}F^{J}] = [\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}F^{J}]$ contains $\mathtt{C}'$ as a diagonal block, rather than $\mathtt{C}$, for $$\mathtt{C'} = \begin{pmatrix} 4 & \ldots & 4 & 3 & 3 \\ 4 & \ldots & 4 & 3 & 3 \\ & & \vdots \\ 4 & \ldots & 4 & 3 & 3 \end{pmatrix}.$$ Since we have $[\mathbf{M}F]^{2} = (4n-1)\mathbf{M}F$, the trace of $[\mathbf{M}F]$ must be $4n-1$. The trace of $\mathtt{C'}$ equals $4n-2$. Since $[\mathbf{M}F^{J}], [\mathbf{M}F^{\mathcal{R}_{0}}]$ have non-negative integer entries, we must have $\on{tr}[\mathbf{M}F^{J}] \leq 4n-1$. Hence $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ admits exactly one subquotient associated to $\mathbf{C'}_{\mathcal{L}_{1}}$ and from Theorem \[Jext\] we see that there is an ordering of indecomposable objects of $\mathbf{M}(\mathtt{i})$ so that $[\mathbf{M}F^{J}]$ is of the form $$\left( \begin{array}{c|c} \mathtt{C'} & * \\ \hline 0 & 0 \end{array} \right).$$ However, in our case we will reorder the indecomposables so that it will be of the form $$\left( \begin{array}{c|c} 0 & 0 \\ \hline * & \mathtt{C'} \end{array} \right).$$ There is at most one zero row on the top of this matrix. This is because the transitivity of $\mathbf{M}$ implies that all the entries of $[\mathbf{M}F]$ must be positive, and we have established that $$\on{tr}[\mathbf{M}F^{J}] = 4n-2 \text{ and } \on{tr}[\mathbf{M}F] = 4n-1.$$ Clearly, if there was more than one zero row in $[\mathbf{M}F^{J}]$, there would be a diagonal entry of $[\mathbf{M}F]$ equal to zero. $[\mathbf{M}F]$ is a quasi-idempotent matrix: it satisfies a relation of the form $\mathtt{T}^{2} = a\mathtt{T}$ for some $a \in \mathbb{N}$. Such matrices with positive integer entries were classified in [@TZ], and using this classification together with $[\mathbf{M}F] = [\mathbf{M}F^{J}] + [\mathbf{M}F^{\mathcal{R}_{0}}]$ and the form of $[\mathbf{M}F^{J}]$ we found above, we conclude that $[\mathbf{M}F]$ for $n>2$ must be one of the $n\times n$ matrices $$\begin{aligned} & \begin{pmatrix} 4 & 4 & \ldots & 4 & 3 \\ 4 & 4 & \ldots & 4 & 3 \\ & & \vdots \\ 4 & 4 & \ldots & 4 & 3 \\ \end{pmatrix} = \mathtt{C'} + \begin{pmatrix} 0 & 0 & \ldots & 1 & 0 \\ 0 & 0 & \ldots & 1 & 0 \\ & & \vdots \\ 0 & 0 & \ldots & 1 & 0 \\ \end{pmatrix}, \\ & \qquad \\ & \begin{pmatrix} 5 & 4 & \ldots & 3 & 3 \\ 5 & 4 & \ldots & 3 & 3 \\ & & \vdots \\ 5 & 4 & \ldots & 3 & 3 \\ \end{pmatrix} = \mathtt{C'} + \begin{pmatrix} 1 & 0 & \ldots & 0 & 0 \\ 1 & 0 & \ldots & 0 & 0 \\ & & \vdots \\ 1 & 0 & \ldots & 0 & 0 \\ \end{pmatrix}, \end{aligned}$$ or the $n+1 \times n+1$ matrix $$\begin{pmatrix} 4 & \ldots & 4 & 3 & 3 & 1 \\ 4 & \ldots & 4 & 3 & 3 & 1 \\ & & & \vdots & \vdots \\ 4 & \ldots & 4 & 3 & 3 & 1 \\ \end{pmatrix}.$$ For $n = 2$, the possible matrices are $$\begin{pmatrix} 4 & 3 \\ 4 & 3 \end{pmatrix}, \begin{pmatrix} 4 & 4 \\ 3 & 3 \end{pmatrix}, \begin{pmatrix} 3 & 3 & 1 \\ 3 & 3 & 1 \\ 3 & 3 & 1 \end{pmatrix}.$$ For $n=1$ we must have $\begin{pmatrix} 3 \end{pmatrix}$, $\begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix}$ or $\begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}$. In this case, we use $B_{1} = \Delta_{1}$, where $\Delta_{1}$ is as defined in the preceding section, and the category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ for $S = {\left\{ e_{1} \right\}}$ is studied in [@Zi2 Section 4]. In fact, all the homological arguments used there to eliminate the matrices $\begin{pmatrix} 3 \end{pmatrix}, \begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix}$ apply here, and so in this case $$[\mathbf{M}F] = \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}.$$ We now eliminate the rank $n$ case for $n \geq 2$. In such cases, $[\mathbf{M}F]$ equals $\mathtt{C'} + \mathtt{P}$ where there is a $k \in {\left\{ 1,\ldots,n \right\}}$ such that $\mathtt{P}$ is given by $$\mathtt{P}_{ij} = \begin{cases} 1 \text{ if }j=k; \\ 0 \text{ else.} \end{cases}$$ Recall that $\mathtt{C'}$ corresponds to the action of $F^{J}$ and $\mathtt{P}$ corresponds to the action of $F^{\mathcal{R}_{0}}$. The latter has $n$ indecomposable summands, each acting in a non-zero way. So the matrix of each summand has exactly one non-zero entry. The only idempotent summand of $F^{\mathcal{R}_{0}}$ is $F_{01}$, and so its matrix must satisfy $[\mathbf{M}F_{01}]^{2} = [\mathbf{M}F_{01}]$, hence why its non-zero entry must lie on the diagonal. Using our knowledge of the matrices of $1$-morphisms lying in ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$, we remark that the matrix of $F_{11}$ only has one non-zero row, and two non-zero entries in that row. On the other hand, since $F_{01} \circ F_{11} \simeq F_{01}^{\oplus 2}$, the equation $[F_{01}][F_{11}] = 2[F_{01}]$ must be satisfied, and from what we have established about $[F_{01}]$, we see that for that to be the case, $[F_{11}]$ would need to have a row with exactly one non-zero entry. This is a contradiction, which allows us to eliminate all matrices where $n=r$. Hence we have shown the following: Let $\mathbf{M}$ be a simple transitive $2$-representation of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ and consider the $1$-morphism $F = \bigoplus_{F_{ij} \in \mathcal{J}_{1}^{L}} F_{ij}$. There is an ordering of the isomorphism classes of indecomposable objects of $\mathbf{M}(\mathtt{i})$ such that for $n>2$, the matrix $[\mathbf{M}F]$ is equal to the $n+1 \times n+1$ matrix $$\begin{pmatrix} 4 & \ldots & 4 & 3 & 3 & 1 \\ 4 & \ldots & 4 & 3 & 3 & 1 \\ & & & \vdots & \vdots \\ 4 & \ldots & 4 & 3 & 3 & 1 \\ \end{pmatrix}.$$ For $n=2$, this matrix is of the form$$\begin{pmatrix} 3 & 3 & 1 \\ 3 & 3 & 1 \\ 3 & 3 & 1 \end{pmatrix}.$$ In each case, $\mathtt{C'}$ is the upper $n \times n$ diagonal block of $\mathtt{M'}$, all the rows of $\mathtt{M'}$ are equal and all the entries of the last column of $\mathtt{M'}$ are equal to $1$. This implies that under the ordering of indecomposable objects of $\mathbf{M}(\mathtt{i})$ such that $[\mathbf{M}F] = \mathtt{M'}$, the first $n$ indecomposables belong to the transitive subquotient of $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}_{\!J}}$ corresponding to the cell $2$-representation $\mathbf{C}_{\mathcal{L}_{1}}$ of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$, and the last object to the trivial subquotient of $\mathbf{M}_{{\scc\mbox{D}\hspace{0.5pt}}}$. The cell $2$-representation $\mathbf{C}_{\mathcal{L}_{1}}$ induces a different ordering of the first $n$ indecomposable objects, under which the upper $n \times n$ block of $[\mathbf{M}F]$ equals $\mathtt{C}$ rather than $\mathtt{C'}$. From now on we will choose the ordering which imposes this ordering on these $n$ objects, indexing them from $1$ to $n$, and moves the last object to the top of the list, indexing it by $0$. As a consequence of that, we have $$[\mathbf{M}F] = \begin{pmatrix} 1 & 3 & 4 & \ldots & 4 & 3 \\ 1 & 3 & 4 & \ldots & 4 & 3 \\ & & & \vdots & \vdots \\ 1 & 3 & 4 & \ldots & 4 & 3 \\ \end{pmatrix},$$ this motivates our earlier introduced convention for this section: we enumerate the rows and columns of this matrix (and generally speaking the matrices of $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$) starting from $0$, rather than from $1$, as is common practice. Let $Q$ be the finite-dimensional algebra $Q$ such that $Q\!\on{-proj} \simeq \mathbf{M}(\mathtt{i})$ and let $f_{0}, f_{1}, \ldots, f_{n}$ be the complete system of idempotents of $Q$ induced by our ordering of the isoclasses of indecomposables of $\mathbf{M}(\mathtt{i})$. Similarly to the preceding sections, let $G_{ij}$ denote an endofunctor of $Q\!\on{-proj}$ naturally isomorphic to tensoring with $Qf_{i} \otimes_{\Bbbk} f_{j}Q$ over $Q$. We will now work towards establishing equivalence between $\mathbf{M}$ and a cell $2$-representation using the standard argument described in Proposition \[StdArg\], and applied in preceding sections. First, we want to show $\mathbf{M}F_{ij} \simeq G_{ij}$. From Proposition \[ProjAct\] we know that $\mathbf{M}F_{ij}$ is a projective functor, for all $F_{ij} \in \mathcal{J}_{1}^{L}$. Let $X_{ij}, Y_{ij}$ be defined analogously to Definition \[XYSets\]. We prove a slight modification of [@MZ2 Lemma 20]: For all $j,j'$ we have $X_{ij} = X_{ij'}$. Similarly $Y_{ij} = Y_{i'j}$ for all $i,i'$. For any $j,j'$, $F_{ij'}$ is a direct summand of $F_{ij} \circ F_{jj'}$, so that also $\mathbf{M}F_{ij'}$ is a direct summand of $\mathbf{M}F_{ij} \circ \mathbf{M}F_{jj'}$, and the $X$-set of composition of projective functors is equal to the $X$-set of the left factor whenever the result of composition is non-zero. The composition $F_{ij} \circ F_{jj'}$ is never zero; in this case we know that $F_{ij} \circ F_{jj'} \simeq F_{ij'}^{\oplus 2}$. So $X_{ij'} \subseteq X_{ij}$. But we can also change the roles of $j,j'$ to find $X_{ij} \subseteq X_{ij'}$. The second statement follows in a similar fashion: the $Y$-set is inherited from the right factor of a composition and so we want to use the fact that $F_{i'j}$ is a direct summand of $F_{i'i} \circ F_{ij}$. However, if $i = 0$, then $F_{i'i}$ is not in ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. The easy fix to that is to note that $F_{i'1} \circ F_{0j} \simeq F_{i'j}$ is non-zero, and the statement follows. Let $X_{i}$ denote the common value of $X_{ij}$ for all $j$, and define $Y_{j}$ similarly. Note the $X$-sets are indexed by the right cells and the $Y$-sets by the left cells of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$, so there is $n+1$ $X$-sets and $n$ $Y$-sets. For $q = 1, \ldots, n$, we have $X_{q} = Y_{q}$. Observe that in each left cell of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ there is a morphism of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ admitting a left adjoint. The $Y$-set of that morphism is then equal to the $X$-set of that of its left adjoint: see the proof of Lemma \[Catch22\]. Recall that for $F_{ij} \in {{\sc\mbox{D}\hspace{1.0pt}}}_{\!J}$, we know everything about $[F_{ij}]$ except the entries of its leftmost column outside of the top row; the top row of $[F_{ij}]$ is zero. Recall also that $X_{ij}$ is exactly the set of indices of non-zero rows of $[F_{ij}]$. From this we immediately find - $i \in X_{i}$ for $i = 1,\ldots, n$; - $0 \not\in X_{i}$ for $i = 1,\ldots,n$; - $0 \in X_{0}$. If, given $i,j \in {\left\{ 1,\ldots,n \right\}}$, there is $k \neq i$ such that the $k$th row of $[F_{ij}]$ is non-zero, then from the form of $[F_{ij}]$ we know that that row is equal to $$\begin{pmatrix} 1 & 0 & \ldots & 0 & 0 \end{pmatrix}.$$ This row being non-zero means that $\mathbf{M}F_{ij}$ has some indecomposable direct summands of the form $G_{kq}$ for some $q \in {\left\{ 0,\ldots,n \right\}}$. Here it is clearly exactly one summand. We claim that this summand is of the form $G_{k0}$. This is because $[G_{kq}]_{kq} \neq 0$ as $$Qf_{k} \otimes_{\Bbbk} f_{q}Q \otimes_{Q} Qf_{q} \simeq Qf_{k} \otimes_{\Bbbk} f_{q}Qf_{q}.$$ The claim follows from the fact that $[F_{ij}]_{kq} = 0$ for all $q \neq 0$. So $0 \in Y_{j}$. But then also $0 \in X_{j}$, which implies $j = 0$ and contradicts the assumption that $F_{ij} \in {{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. We have thus shown the following: For $i = 1,\ldots, n$, we have $X_{i} = {\left\{ i \right\}} = Y_{i}$. Hence, for $i,j \in {\left\{ 1,\ldots,n \right\}}$, $$\mathbf{M}F_{ij} \simeq G_{ij}^{\oplus m_{ij}}.$$ for some positive integers $m_{ij}$ depending both on $i$ and $j$. To complete this statement we also need to consider the case $i = 0$: $X_{0} = {\left\{ 0 \right\}}$. Hence we have $$\mathbf{M}F_{0j} \simeq G_{0j}^{m_{0j}} \text{ for }j=1,\ldots,n \text{ and positive integers }m_{0j} \text{ depending on }j.$$ The $1$-morphism $F^{\mathcal{R}_{0}} = \bigoplus_{j=1}^{n} F_{0j}$ is idempotent: $F^{\mathcal{R}_{0}} \circ F^{\mathcal{R}_{0}} \simeq F^{\mathcal{R}_{0}}$. Hence $[F^{\mathcal{R}_{0}}]^{2} = [F^{\mathcal{R}_{0}}]$. We know that $[F^{\mathcal{R}_{0}}]$ is of the form $$\begin{pmatrix} 1 & 3 & 4 & \ldots & 4 & 3 \\ x_{1} & 0 & 0 & \ldots & 0 & 0 \\ x_{2} & 0 & 0 & \ldots & 0 & 0 \\ \vdots & & \vdots & & \vdots \\ x_{n} & 0 & 0 & \ldots & 0 & 0 \end{pmatrix}$$ for some non-negative integers $x_{1},\ldots,x_{n}$. If there is some $k \in {\left\{ 1,\ldots,n \right\}}$ such that $x_{k} \neq 0$, then all the entries of the $k$th row of $[F^{\mathcal{R}_{0}}]^{2}$ are non-zero, contradicting $[F^{\mathcal{R}_{0}}]$ being idempotent. The result now follows, since the indices of non-zero rows of the matrix of a projective functor form its $X$-set, $[F^{\mathcal{R}_{0}}] = \sum_{j=1}^{n} [F_{0j}]$, and all the matrices have non-negative integer entries. \[Action6\] $m_{ij} = 1$ for $i = 0,\ldots,n$ and $j = 1, \ldots, n$. From Observation \[StateIneq\] we see that $$\mathtt{C}_{ij}^{Q} \geq \mathtt{C}_{ij}^{eAe} \text{ for } i,j \in {\left\{ 1,\ldots,n \right\}}.$$ In particular, $\mathtt{C}_{jj}^{Q} \geq 2$. But now $F_{ij} \simeq G_{ij}^{\oplus m_{ij}}$ implies that $$F_{ij}Q_{j} \simeq Q_{i}^{\oplus m_{ij} \cdot \mathtt{C}^{Q}_{jj}}$$ and on the other hand, from the definition of $[F_{ij}]$ and the fact that $[F_{ij}]_{ij} = 2$, we get $$F_{ij}Q_{j} \simeq Q_{i}^{\oplus 2},$$ so $m_{ij} \cdot \mathtt{C}^{Q}_{jj} = 2$, which together with the bound on $\mathtt{C}_{jj}^{Q}$ implies $m_{ij} = 1$ for all $i,j \in {\left\{ 1,\ldots,n \right\}}$. We now turn to the case where $i = 0$. For that we use the fact that $F_{jj} \simeq F_{j1} \circ F_{0j}$ for all $j$. This yields $$G_{jj} \simeq G_{j1} \circ G_{0j}^{\oplus m_{0j}}.$$ On the other hand, by the law of composition of projective functors, we have $$G_{j1} \circ G_{0j}^{\oplus m_{0j}} \simeq G_{jj}^{\oplus m_{0j} \cdot \mathtt{C}^{Q}_{10}}.$$ Thus, $m_{0j} \cdot \mathtt{C}^{Q}_{10} = 1$, and so $m_{0j} = 1$, which concludes the proof. We have thus established one of the two sufficient conditions we found in Proposition \[StdArg\]; it remains to show that the Cartan matrices of $\mathbf{M}(\mathtt{i})$ and that of the target category of a cell $2$-representation with apex in $\mathcal{J}_{1}^{L}$ coincide. From Lemma \[Target6\] we see that the latter target category is equivalent to $B\!\on{-proj}$. In analogy to the preceding sections, let $\mathtt{C}^{Q}$ be the Cartan matrix of $\mathbf{M}(\mathtt{i})$. $\mathtt{C}^{Q}_{ij} = \mathtt{C}^{B}_{ij}$ for $i,j \in {\left\{ 1,\ldots,n \right\}}$ such that $i \neq j$. We have $$F_{ii} \circ F_{jj} \simeq F_{ij}^{\oplus \mathtt{C}^{B}_{ij}},$$ so also $$\mathbf{M}F_{ii} \circ \mathbf{M}F_{jj} \simeq \mathbf{M}F_{ij}^{\oplus \mathtt{C}^{B}_{ij}}.$$ Due to $\mathbf{M}F_{ij} \simeq G_{ij}$, this yields $$G_{ii} \circ G_{jj} \simeq G_{ij}^{\oplus \mathtt{C}_{ij}^{B}}.$$ On the other hand, the law of composition for projective functors gives us $$G_{ii} \circ G_{jj} \simeq G_{ij}^{\oplus \mathtt{C}_{ij}^{Q}},$$ which goes to show that $\mathtt{C}^{Q}_{ij} = \mathtt{C}^{B}_{ij}$. $\mathtt{C}^{Q}_{0j} = \mathtt{C}^{B}_{0j}$ and $\mathtt{C}^{Q}_{j0} = \mathtt{C}^{B}_{j0}$ for $j = 1,\ldots,n$. The entries of the form $\mathtt{C}^{Q}_{j0}$ we may find using the exact same method we employed for the preceding statement: $$F_{jj} \circ F_{0j} \simeq F_{jj}^{\oplus \mathtt{C}^{B}_{j0}}$$ gives $\mathbf{M}F_{jj} \circ \mathbf{M}F_{0j} \simeq \mathbf{M}F_{jj}^{\oplus \mathtt{C}^{B}_{j0}}$, and using $\mathbf{M}F_{ij} \simeq G_{ij}$ and comparing with the composition of projective functors $$G_{jj} \circ G_{0j} \simeq G_{jj}^{\oplus \mathtt{C}^{Q}_{j0}},$$ we obtain the sought equality. In particular, $$\mathtt{C}^{Q}_{j0} = \begin{cases} 1 \text{ for } j=1 \\ 0 \text{ for } j=2,\ldots,n. \end{cases}$$ Since the indecomposable $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ act as functors $G_{ij}$ with $j \neq 0$, this method is not applicable for entries of the form $\mathtt{C}^{Q}_{0j}$. In that case, provided $j \neq 0$, we use the fact that $F_{jj}$ is a self-adjoint $1$-morphism of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ for $j = 1,\ldots,n$. Hence it must act as a self-adjoint functor, and the adjunction yields $$\begin{aligned} & 2\mathtt{C}^{Q}_{0i} = \on{dim}\on{Hom}_{Q\!\on{-proj}}(Qf_{0},Qf_{i}^{\oplus 2}) = \on{dim}\on{Hom}_{Q\!\on{-proj}}(Qf_{0},F_{ii}Qf_{i}) \\ & = \on{dim}\on{Hom}_{Q\!\on{-proj}}(F_{ii}Qf_{0},Qf_{i}) = \on{dim}\on{Hom}_{Q\!\on{-proj}}(Qf_{i}^{\oplus \mathtt{C}^{Q}_{i0}}, Qf_{i}) \\ & = \mathtt{C}^{Q}_{i0} \cdot \on{dim}\on{Hom}_{Q\!\on{-proj}}(Qf_{i},Qf_{i}) = 2\mathtt{C}^{Q}_{i0} = \begin{cases} 2 \text{ if } i=1 \\ 0 \text{ else.} \end{cases} \end{aligned}$$ In other words, for $i=1,\ldots,n$, we have $$\mathtt{C}^{Q}_{0i} = \begin{cases} 1 \text{ if } i=1, \\ 0 \text{ if } i=2,\ldots,n. \end{cases}$$ Comparing with the Cartan matrix for $B$, given in the discussion following Theorem \[MainThm6\], we infer the equality. \[Matrix6\] The Cartan matrices $\mathtt{C}^{B}, \mathtt{C}^{Q}$ coincide. Clearly what is left to show is that $\mathtt{C}^{Q}_{00} = 1$, since we know that $\mathtt{C}^{B}_{00} = 1$. The proof of this statement is analogous to that of Lemma \[1sOnDiagonal\]. Since $\mathtt{C}^{Q}_{00} = \on{dim}\on{End}(Qf_{0})$, we must have $\mathtt{C}^{Q}_{00} \geq 1$, and since $Qf_{0}$ is indecomposable, $\mathtt{C}^{Q}_{00} > 1$ if and only if $\on{Rad}\on{End}Qf_{0} \neq 0$. Let $\alpha \in \on{Rad}\on{End}Qf_{0}$; abusing notation we will also denote the element $\alpha(e_{0})$ of $Qf_{0}$ by $\alpha$. We will show that the ideal of $\mathbf{M}$ generated by $\alpha$ does not contain all the morphisms of $\mathbf{M}(\mathtt{i})$, and thus must contain of zero morphisms only; in particular, $\alpha = 0$, and thus $\on{Rad}\on{End}Qf_{0} = 0$. To that end, we show that the morphisms on form $F_{ij}\alpha$ necessarily are zero. If $F_{ij}Qf_{0} = 0$, then clearly also $F_{ij}\alpha = 0$. As we have observed when determining entries $\mathtt{C}^{Q}_{i0}$ of $\mathtt{C}^{Q}$, the only indecomposable $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ such that $\mathbf{M}F_{ij}\left(Qf_{0}\right)$ is non-zero are those of the form $F_{i1}$. In that case, using the $\Bbbk$-linear isomorphism $f_{1}Q \otimes_{Q} Qf_{0} \simeq f_{1}Qf_{0}$, $F_{i1}\alpha$ is represented by $$Qf_{i} \otimes_{\Bbbk} f_{1}Qf_{0} \xrightarrow{\on{id}_{Qf_{i}} \otimes \widetilde{\alpha}} Qf_{i} \otimes_{\Bbbk} f_{1}Qf_{0},$$ where $\widetilde{\alpha}$ is the $\Bbbk$-linear endomorphism of $f_{1}Qf_{0}$ given by right multiplication with $\alpha$; since $\alpha \in \on{Rad}\on{End}Qf_{0}$, the endomorphism $\widetilde{\alpha}$ must be nilpotent. But $\on{dim}f_{1}Qf_{0}$ is one-dimensional; hence $\widetilde{\alpha}$ must be zero, and as a consequence also $F_{i1}\alpha = 0$. This shows that acting on $\alpha$ with non-identity $1$-morphisms of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ gives a zero morphism. Acting by the identity maps $\on{Rad}\on{End}Qf_{0}$ to itself, and so the ideal generated by it lies in the radical of $Q\!\on{-proj}$, hence does not contain all the morphisms of that category. As described earlier, this ideal must then be zero, and so $\alpha = 0$ and $\on{Rad}\on{End}Qf_{0} = 0$. The result follows. As we have commented earlier, in view of Proposition \[StdArg\], Lemma \[Action6\] and Lemma \[Matrix6\] imply Theorem \[MainThm6\]. Non-cell $2$-representations of $2$-semicategories for star algebras ==================================================================== Finitary $2$-semicategories and weak $2$-representations -------------------------------------------------------- In a recent article ([@KMZ]) by Ko, Mazorchuk and Zhang, instead of finitary $2$-categories, so-called finitary $2$-semicategories were studied. These can be described as finitary $2$-categories without identity $1$-morphisms, similarly to how a semigroup can be viewed as a monoid without an identity element. As is remarked in the introduction to that article, given a finite-dimensional algebra $A$, the identity $1$-morphism $_{A}A_{A}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$ can be viewed as artificially added to ${{\sc\mbox{C}\hspace{1.0pt}}}_{\!A}$. In this section, for an algebra of the form $\Lambda_{n}$, as described in Definition \[DefStar\], and the choice of self-injective core $S= {\left\{ e_{0} \right\}}$ for $\Lambda_{n}$, we will consider a $2$-semicategory given by the non-identity $1$-morphisms of the corresponding $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$, and construct a family of non-cell $2$-representations of that $2$-semicategory. First we give the necessary definitions found in or deduced from [@KMZ]. A [*$2$-semicategory*]{} ${{\sc\mbox{C}\hspace{1.0pt}}}$ consists of - a class $\on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$ of objects; - for each $\mathtt{i,j} \in \on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$, a category ${{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j})$, whose objects are called [*$1$-morphisms*]{}, morphisms are called $2$-morphisms, and the composition of $2$-morphisms is called the [*vertical composition*]{}, denoted by $\circ_{v}$; - for each $\mathtt{i,j,k} \in \on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$, a functor (called the [*horizontal composition*]{}) $$h_{\mathtt{i,j,k}}: {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{j,k}) \times {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j}) \rightarrow {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,k})$$ which is strictly associative, that is, $$h_{\mathtt{i,k,l}} \circ \left(\on{Id}_{{\scc\mbox{C}\hspace{0.5pt}}(\mathtt{k,l})} \times \, h_{\mathtt{i,j,k}} \right) = h_{\mathtt{i,j,l}} \circ \left( h_{\mathtt{j,k,l}} \times \on{Id}_{{\scc\mbox{C}\hspace{0.5pt}}(\mathtt{i,j})} \right).$$ Similarly to earlier chapters, we will write $GF := h(G,F)$ for composition of $1$-morphisms. A $2$-semicategory ${{\sc\mbox{C}\hspace{1.0pt}}}$ is said to be [*finitary*]{} if 1. $\on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$ is a finite set; 2. For $\mathtt{i,j} \in \on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$, the category ${{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j})$ is finitary; 3. The horizontal composition functor is $\Bbbk$-bilinear and biadditive. Let ${{\sc\mbox{C}\hspace{1.0pt}}}, {{\sc\mbox{D}\hspace{1.0pt}}}$ be $2$-semicategories. A [*$2$-semifunctor*]{} $\mathbf{M}$ from ${{\sc\mbox{C}\hspace{1.0pt}}}$ to ${{\sc\mbox{D}\hspace{1.0pt}}}$ consists of - a function $\mathbf{M}: \on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \on{Ob} {{\sc\mbox{D}\hspace{1.0pt}}}$; - a functor $\mathbf{M}_{\mathtt{i,j}}: {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j}) \rightarrow {{\sc\mbox{C}\hspace{1.0pt}}}(\mathbf{M}(\mathtt{i}), \mathbf{M}(\mathtt{j}))$ for each pair of objects $\mathtt{i,j} \in \on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$, such that $\mathbf{M}(GF) = \mathbf{M}G\mathbf{M}F$ for all $1$-morphisms $G,F$ such that $GF$ is well-defined. We will also be interested in non-strict $2$-semifunctors, which, following [@Le], we will call homomorphisms. \[Weak2RepDef\] Let ${{\sc\mbox{C}\hspace{1.0pt}}}$ and ${{\sc\mbox{D}\hspace{1.0pt}}}$ be $2$-semicategories. A [*homomorphism of $2$-semicategories*]{} $\mathbf{M}: {{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow {{\sc\mbox{D}\hspace{1.0pt}}}$ consists of - A function $\on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \on{Ob}{{\sc\mbox{D}\hspace{1.0pt}}}$; - Functors $\mathbf{M}_{\mathtt{i,j}}: {{\sc\mbox{C}\hspace{1.0pt}}}(\mathtt{i,j}) \rightarrow {{\sc\mbox{D}\hspace{1.0pt}}}(\mathtt{i,j})$ for all objects $\mathtt{i,j} \in \on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$; - For $\mathtt{i,j,k} \in \on{Ob}{{\sc\mbox{C}\hspace{1.0pt}}}$, natural isomorphisms $$\alpha_{\mathtt{i,j,k}}: \left( - \circ_{{{\sc\mbox{D}\hspace{1.0pt}}}} - \right) \circ \mathbf{M}_{\mathtt{j,k}} \times \mathbf{M}_{\mathtt{i,j}} \rightarrow \mathbf{M}_{i,k} \circ \left( - \circ_{{{\sc\mbox{C}\hspace{1.0pt}}}} - \right),$$ that is, an invertible $2$-morphism $\alpha_{G,F}: \mathbf{M}F\mathbf{M}G$ such that for any $2$-morphisms $\tau_{G}: G \rightarrow G'$ and $\tau_{F}: F \rightarrow F'$ the diagram $$\begin{tikzcd}[column sep = huge] \mathbf{M}G\mathbf{M}F \arrow[r, "\mathbf{M}\tau_{G} \circ_{h} \mathbf{M}\tau_{F}"] \arrow[d, "\alpha_{G,F}"] & \mathbf{M}G' \mathbf{M}F' \arrow[d, "\alpha_{G',F'}"] \\ \mathbf{M}(GF) \arrow[r, "\mathbf{M}(\tau_{G} \circ_{h} \tau_{F})"] & \mathbf{M}(G'F') \end{tikzcd}$$ commutes. Finally, we require the following diagram to commute: $$\begin{tikzcd} & \mathbf{M}H\mathbf{M}G \mathbf{M}F \arrow[dl, swap, "\alpha_{H,G}\mathbf{M}F"] \arrow[dr, "\mathbf{M}H\alpha_{G,F}"] \\ \mathbf{M}(HG) \mathbf{M}F \arrow[dr, swap, "\alpha_{HG,F}"] & & \mathbf{M}H \mathbf{M}(GF) \arrow[dl, "\alpha_{H,GF}"] \\ & \mathbf{M}(HGF) \end{tikzcd}$$ For the notions of $2$-transformations and modifications of semifunctors, similarly to above we follow the non-strict setup presented in [@Le], omitting the axioms concerning identity $1$-morphisms. Let ${{\sc\mbox{C}\hspace{1.0pt}}}$ be a $2$-semicategory. A [*finitary $2$-representation*]{} of ${{\sc\mbox{C}\hspace{1.0pt}}}$ is a $2$-semifunctor $\mathbf{M}: {{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \mathfrak{A}_{\Bbbk}$. An [*abelian $2$-representation*]{} of ${{\sc\mbox{C}\hspace{1.0pt}}}$ is a $2$-semifunctor $\mathbf{M}: {{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \mathfrak{R}_{\Bbbk}$. A [*weak finitary $2$-representation*]{} is a homomorphism of $2$-semicategories $\mathbf{M}: {{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \mathfrak{A}_{\Bbbk}$. A [*weak abelian $2$-representation*]{} is such a homomorphism whose codomain instead is $\mathfrak{R}_{\Bbbk}$. In each case we require that $\mathbf{M}_{\mathtt{i,j}}$ is additive and $\Bbbk$-linear for each pair $\mathtt{i,j}$ of objects. We say that two $2$-representations of a $2$-semicategory are [*equivalent*]{} if there exists a $2$-natural isomorphism between them, whose components also are additive and $\Bbbk$-linear. Similarly to finitary $2$-categories, the modified $2$-setup for $2$-semicategories produces a $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}\!\on{-afmod}$ of finitary $2$-representations of ${{\sc\mbox{C}\hspace{1.0pt}}}$, and a $2$-category ${{\sc\mbox{C}\hspace{1.0pt}}}\!\on{-amod}$ of abelian $2$-representations thereof. We remark that the proof of [@MM3 Proposition 2] does not use identity $1$-morphisms, and so the result still holds for $2$-semicategories: if a $2$-natural transformation $\Psi$ between two $2$-representations of a $2$-semicategory is such that $\Psi_{\mathtt{i}}$ is an equivalence of categories for all $\mathtt{i}$, then $\Psi$ is an equivalence of $2$-representations. The reason why we are interested in weak $2$-representations is the following result, which can be deduced from[@Po Theorem 3.4]: \[HardWork\] For any homomorphism of $2$-(semi)categories $\mathbf{M}: {{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \mathbf{Cat}$, there is a $2$-(semi)functor $\widehat{\mathbf{M}}: {{\sc\mbox{C}\hspace{1.0pt}}}\rightarrow \mathbf{Cat}$ together with a non-strict $2$-natural isomorphism $\Psi_{\mathbf{M}}: \mathbf{M} \rightarrow \widehat{\mathbf{M}}$. In particular this implies that the problem of classifying simple transitive $2$-representations of a $2$-(semi)category ${{\sc\mbox{C}\hspace{1.0pt}}}$ is equivalent to that of classifying weak simple transitive $2$-representations ${{\sc\mbox{C}\hspace{1.0pt}}}$: \[SemiWork\] Let ${{\sc\mbox{C}\hspace{1.0pt}}}$ be a finitary $2$-(semi)category and let $\mathbf{M}$ be a weak $2$-representation of ${{\sc\mbox{C}\hspace{1.0pt}}}$. There is a (strict) $2$-representation $\widehat{\mathbf{M}}$ of ${{\sc\mbox{C}\hspace{1.0pt}}}$ equivalent to $\mathbf{M}$. $2$-semicategories of projective functors ----------------------------------------- Let $A$ be a be a finite-dimensional, basic, connected algebra and let ${\left\{ e_{1},\ldots, e_{n} \right\}}$ be a complete set of idempotents of $A$. Fix a small category $\mathcal{A}$ equivalent to $A\!\on{-proj}$. The $2$-semicategory ${\sc\mbox{Z}\hspace{1.0pt}}_{\!A}$ is defined as follows: - $\on{Ob}{\sc\mbox{Z}\hspace{1.0pt}}_{\!A} = {\left\{ \mathtt{i} \right\}}$, where $\mathtt{i}$ can be identified with $\mathcal{A}$; - $1$-morphisms of ${\sc\mbox{Z}\hspace{1.0pt}}_{\!A}$ are endofunctors of $\mathcal{A}$ isomorphic to tensoring with the projective $A$-$A$-bimodules in $\on{add}(A \otimes_{\Bbbk} A)$; - $2$-morphisms are given by natural transformations between those functors. The $2$-semicategory ${\sc\mbox{Z}\hspace{1.0pt}}_{\!A}$ generally does not admit weak identity $1$-morphisms described in [@KMZ Section 2], and hence fails to be a bilax-unital $2$-category in the sense of [@KMZ]. Now we define the objects of study of the remaining part of this section: Consider the star algebra $\Lambda_{n}$ and its quotient $\Delta_{n}$, given in Definition \[DefStar\]. Choose the self-injective core $S = {\left\{ 0 \right\}}$ for both the algebras. We let ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ be the $2$-full $2$-subsemicategory of ${\sc\mbox{Z}\hspace{1.0pt}}_{\Lambda_{n}}$ generated by the $1$-morphisms $$\Lambda_{n}e_{i} \otimes_{\Bbbk} \Lambda_{n}e_{0} \text{ for } i=0,1,\ldots,n.$$ and similarly let ${\sc\mbox{G}\hspace{1.0pt}}_{L}$ be the $2$-semicategory generated by the $1$-morphisms $$\Delta_{n}e_{i} \otimes_{\Bbbk} \Delta_{n}e_{0} \text{ for } i=0,1,\ldots,n.$$ The main result --------------- The main reason for our interest in the $2$-semicategories above is that they closely connect to the following conjecture, formulated in [@Zi2]: For the algebra $\Lambda_{n}$ and the self-injective core $S = {\left\{ 0 \right\}}$, consider the $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$. Equivalence classes of simple transitive $2$-representations of ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ are in bijection with set partitions of ${\left\{ 1,2,\ldots, n \right\}}$. As was remarked in the introduction, the $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ above is just ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ with an identity $1$-morphism added. This shows the close connection between the above conjecture and the main theorem of this section: \[MainThm2\] There is a family of pairwise non-equivalent simple transitive $2$-representations of ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ indexed by set partitions of ${\left\{ 1,2,\ldots, n \right\}}$. Existence of non-cell $2$-representations ----------------------------------------- Similarly to \[53Prop\], we have the following: The cell $2$-representation of ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ acts on a category equivalent to $\Delta_{n}\!\on{-proj}$. This induces a $2$-semifunctor ${\sc\mbox{Z}\hspace{1.0pt}}_{L} \rightarrow {\sc\mbox{G}\hspace{1.0pt}}_{L}$ which on the level of isoclasses of indecomposables gives $$\Lambda_{n}e_{i} \otimes_{\Bbbk} \Lambda_{n}e_{0} \mapsto \Delta_{n}e_{i} \otimes_{\Bbbk} \Delta_{n}e_{0}.$$ Using the $2$-semifunctor above, we can construct $2$-representations of ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$ from $2$-representations of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$. This is indeed what we will do; it will then also be important to verify that the non-equivalent $2$-representations of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$ we construct do not give rise to equivalent $2$-representations of ${\sc\mbox{Z}\hspace{1.0pt}}_{L}$. The main idea for the construction is to define a functor ${\EuRoman}{P}_{\mu}: \Delta_{n}\!\on{-proj} \rightarrow \Delta_{k}\!\on{-proj}$ for a set partition $\mu$ partitioning ${\left\{ 1,\ldots,n \right\}}$ into $k$ subsets. In particular, we say that ${\EuRoman}{P}$ “collapses” the isoclasses of indecomposables as prescribed by $\mu$: the vertices corresponding to labels assigned to the same subset by $\mu$ are mapped to the same isomorphism class. Collapsing functors similar to those we study below exist also for some other algebras and self-injective cores, in particular for more general zigzag algebras. However, no abstract sufficient nor necessary conditions for a pair of an algebra and a self-injective core to admit a suitable collapsing functor are known to the author. Hence only the case of star algebras is presented in what follows. To verify that our construction yields a weak $2$-representation, we perform quite explicit computations; we reduce the complexity thereof by working with skeletal categories whenever possible. Fix $n \geq 1$ and let $\Delta := \Delta_{n}$. Let $\mathcal{A}$ be a skeletal category equivalent to $\Delta\!\on{-proj}$, which we parametrize as follows: - $Q_{0},Q_{1},\ldots, Q_{n}$ is the set of indecomposable objects of $\mathcal{A}$; - We choose the following bases for Hom-spaces between the indecomposables: $$\begin{cases} \on{Hom}_{\mathcal{A}}\left( Q_{0}, Q_{0} \right) = \Bbbk[\on{id}_{Q_{0}}, c = a_{i}b_{i}] \\ \on{Hom}_{\mathcal{A}}\left( Q_{0}, Q_{i} \right) = \Bbbk[b_{i}] \text{ for } i\neq 0; \\ \on{Hom}_{\mathcal{A}}\left( Q_{i}, Q_{0} \right) = \Bbbk[a_{i}] \text{ for } i\neq 0; \\ \on{Hom}_{\mathcal{A}}\left( Q_{i}, Q_{i} \right) = \Bbbk[\on{id}_{Q_{i}} ] \text{ for } i\neq 0 \\ \on{Hom}_{\mathcal{A}}\left( Q_{i}, Q_{j} \right) = 0 \text{ for } i,j \neq 0 \text{ and } i\neq j \end{cases}$$ with composition as indicated by the labels and Definition \[DefStar\]: $c = a_{i}b_{i}$ and $b_{i}a_{j} = 0$ for $i,j \in {\left\{ 1,\ldots, n \right\}}$. Note that applying any permutation $\sigma \in S_{n}$ on the indices $i,j$ gives an automorphism of $\mathcal{A}$, which comes from the automorphism of $\Delta$ given by permuting corresponding vertices of the underlying quiver, as observed in Definition \[DefStar\]. From now on, by ${\sc\mbox{G}\hspace{1.0pt}}_{L}$ we will mean the $2$-semicategory of projective endofunctors of $\mathcal{A}$ - we fix the underlying category. Different choices of that category give biequivalent constructions, so weak $2$-representations lift between such constructions. Fix a set partition $\mu$ of ${\left\{ 1,\ldots, n \right\}}$, subdividing ${\left\{ 1,\ldots,n \right\}}$ into $k$ disjoint subsets $M_{1},\ldots, M_{k}$. We define ${\EuRoman}{P}_{\mu}: \mathcal{A} \rightarrow \on{add}(Q_{0},Q_{1},\ldots, Q_{k}) \subseteq \mathcal{A}$ on the level of indecomposables by sending $Q_{0}$ to itself, and for $i \in M_{j}$, sending $Q_{i}$ to $Q_{j}$. On the level of morphisms, we send $b_{i}$ to $b_{j}$ and $a_{i}$ to $a_{j}$ accordingly. This determines an additive functor uniquely up to natural isomorphism, by letting ${\EuRoman}{P}_{\mu}$ act by diagonalizable matrices on Hom-spaces of non-indecomposable objects. For our calculations, we choose ${\EuRoman}{P}_{\mu}$ exactly as the functor that acts diagonally with respect to our fixed basis. Note that there is nothing canonical about choosing $\on{add}(Q_{0},Q_{1},\ldots,Q_{k})$ - we could choose any other set of $k$ indecomposables ${\left\{ Q_{\sigma(i)} \right\}}_{i=1}^{k}$ not isomorphic to $Q_{0}$ and define ${\EuRoman}{P}_{\mu}$ similarly. The two definitions then differ by postcomposing with the isomorphism $$\on{add}(Q_{0},Q_{1},\ldots, Q_{k}) \xrightarrow{\sim} \on{add}(Q_{0},Q_{\sigma(1)},\ldots, Q_{\sigma(k)})$$ constructed analogously to the automorphisms of $\mathcal{A}$ described earlier. \[BigDef\] Given $i \in {\left\{ 0,1,\ldots,n \right\}}$, let $F_{i0}$ be the endofunctor of $\mathcal{A}$ sending $Q_{j}$ to $Q_{i}$ for $j \neq 0$ and sending $Q_{0}$ to $Q_{i}^{\oplus 2}$, and on the level of morphisms given by $$\begin{cases} F_{i0}a_{j} = \begin{pmatrix} 0 \\ \on{id}_{Q_{i}} \end{pmatrix} ; \\ F_{i0}b_{j} = \begin{pmatrix} \on{id}_{Q_{i}} & 0 \end{pmatrix}; \\ F_{i0}c = \begin{pmatrix} 0 & 0 \\ \on{id}_{Q_{i}} & 0 \end{pmatrix} \end{cases}$$ and, similarly to ${\EuRoman}{P}_{\mu}$, continued diagonally (not only diagonalizably) with respect to our fixed basis to non-indecomposable objects. The functoriality follows from $F_{i0}a_{j}F_{i0}b_{j} = F_{i0}c$ and $F_{i0}b_{j}F_{i0}a_{k} = 0$. $F_{i0}$ is an indecomposable projective endofunctor of $\mathcal{A}$, by construction corresponding to $\Delta e_{i} \otimes_{\Bbbk} e_{0}\Delta$ acting on $\Delta\!\on{-proj}$. Given a collection ${\left\{ F_{i_{j}0} \right\}}_{j=1}^{k}$, by $\bigoplus_{j} F_{i_{j}0}$ we will denote the functor given the suitable direct sums on the level of functors, and again extending diagonally from the indecomposables ${\left\{ F_{i0} \right\}}$ on the level of matrices giving the action on Hom-spaces. To clarify what we mean by acting diagonally, we illustrate by a simple example: consider the morphism $\left( \begin{smallmatrix} b_{1} \\ 0 \end{smallmatrix} \right) \in \on{Hom}_{\mathcal{A}}(Q_{0},Q_{1}^{\oplus 2})$. We explicitly require $$F_{10} \begin{pmatrix} b_{1} \\ 0 \end{pmatrix} = \begin{pmatrix} \on{id}_{Q_{1}} & 0 \\ 0 & 0 \end{pmatrix} = \left(\begin{array}{c} F_{10}(b_{1})\\ \hline F_{10}(0) \end{array}\right)$$ although we could have a naturally isomorphic functor $\widetilde{F}_{i0}$ satisfying $$\widetilde{F}_{10} \begin{pmatrix} b_{1} \\ 0 \end{pmatrix} = \begin{pmatrix} \on{id}_{Q_{1}} & 0 \\ \on{id}_{Q_{1}} & 0 \end{pmatrix}.$$ Fix a set partition $\mu$ of ${\left\{ 1,\ldots, n \right\}}$ and let $${\EuRoman}{P} := {\EuRoman}{P}_{\mu}: \mathcal{A} \rightarrow \on{add}(Q_{0}, Q_{1},\ldots, Q_{k}).$$ Note that $F_{i0}a_{j}$ is independent of $j$, and similarly for $b_{j}$. On the other hand, all ${\EuRoman}{P}$ does is relabel such indices. Moreover, since the action of ${\EuRoman}{P}$ is diagonal as described above, the following clearly holds: \[PisNice\] Let ${\EuRoman}{P}$ be as above and let $G := \bigoplus_{j} F_{i_{j}0}$ be a projective endofunctor of $\mathcal{A}$ of the form described in Definition \[BigDef\]. Denote the inclusion functor of $\on{add}(Q_{0},Q_{1},\ldots,Q_{k})$ to $\mathcal{A}$ by ${\EuRoman}{J}$ and denote ${\EuRoman}{J} \circ {\EuRoman}{P}$ by $\Phi$. Then $$G = G\Phi.$$ We are now ready to construct the weak $2$-representation $\mathbf{M}_{\mu}$ of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$ associated to our chosen set partition $\mu$. We let $\mathbf{M}_{\mu}(\mathtt{i}) = \on{add}(Q_{0},Q_{1},\ldots, Q_{k})$; given a $1$-morphism $F$ of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$, we let $$\mathbf{M}_{\mu}F = {\EuRoman}{P}F{\EuRoman}{J}$$ and similarly for $2$-morphisms. Clearly, every $1$-morphism is realized as a $\Bbbk$-linear endofunctor of a finitary category, and every $2$-morphism as a natural transformation between such functors. Moreover, vertical composition of $2$-morphisms clearly is preserved, the action being given by composing with fixed functors. What needs to be verified is the coherence and naturality of the assignment on the level of $1$-morphisms. To that end we should first specify the structure morphisms $\alpha_{G,F}$ for any pair $(G,F)$ of $1$-morphisms in ${\sc\mbox{G}\hspace{1.0pt}}_{L}$. However, it turns out that for naturality in $F$ (the right argument) and coherence we have a great freedom of choice: \[AlphaConstraint\] Choose a natural isomorphism $\tau_{G}: G\Phi \xrightarrow{\sim} G$, for every $1$-morphism $G$ of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$. Let $\alpha_{G,F} = {\EuRoman}{P}\tau_{G}F{\EuRoman}{J}$. The collection ${\left\{ \alpha_{G,F} \right\}}_{F,G \in {\sc\mbox{G}\hspace{1.0pt}}_{L}(\mathtt{i,i})}$ satisfies the structural constraint of Definition \[Weak2RepDef\]. The condition in Definition \[Weak2RepDef\] requires the following diagram to commute: $$\begin{tikzcd}[sep = large] {\EuRoman}{P}H\Phi G \Phi F {\EuRoman}{J} \arrow[d, "{\EuRoman}{P}\tau_{H}G\Phi F{\EuRoman}{J}"] \arrow[r, "{\EuRoman}{P}H\Phi \tau_{G}F{\EuRoman}{J}"] & {\EuRoman}{P}H\Phi GF {\EuRoman}{J} \arrow[d, "{\EuRoman}{P}\tau_{H}GF{\EuRoman}{J}"] \\ {\EuRoman}{P}HG\Phi F {\EuRoman}{J} \arrow[r, "{\EuRoman}{P}H\tau_{G}F {\EuRoman}{J}"] & {\EuRoman}{P}HGF {\EuRoman}{J} \end{tikzcd}.$$ This is a direct consequence of the commutative square defining the horizontal composition of natural transformations. This is easiest to see by removing ${\EuRoman}{P,J},F$ from the diagram: $$\begin{tikzcd} (H\Phi)(G\Phi) \arrow[r, "H\Phi\tau_{G}"] \arrow[d, "\tau_{H}G\Phi"] & (H\Phi)G \arrow[d, "\tau_{H}G"] \\ H(G\Phi) \arrow[r, "H\tau_{G}"] & HG. \end{tikzcd}$$ This diagram commutes by naturality of $\tau_{H}$. From this the commutativity of the first diagram follows. Choose a natural isomorphism $\tau_{G}: G\Phi \xrightarrow{\sim} G$, for every $1$-morphism $G$ of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$. The collection ${\left\{ \alpha_{G,F} \right\}}_{F,G \in {\sc\mbox{G}\hspace{1.0pt}}_{L}(\mathtt{i,i})}$, as defined in Lemma \[AlphaConstraint\], is natural in $F$, i.e. given a $2$-morphism $\beta: F \rightarrow F'$, the following diagram commutes: $$\begin{tikzcd} {\EuRoman}{P}G \Phi F {\EuRoman}{J} \arrow[r, "{\EuRoman}{P} G \Phi \beta {\EuRoman}{J}"] \arrow[d, "{\EuRoman}{P} \tau_{G} F {\EuRoman}{J}"] & {\EuRoman}{P}G \Phi F' {\EuRoman}{J} \arrow[d, "{\EuRoman}{P} \tau_{G} F' {\EuRoman}{J}"] \\ {\EuRoman}{P}GF {\EuRoman}{J} \arrow[r, "{\EuRoman}{P}G \beta {\EuRoman}{J}"] & {\EuRoman}{P}GF' {\EuRoman}{J} \end{tikzcd}$$ commutes. Similarly to the proof of Lemma \[AlphaConstraint\], the diagram $$\begin{tikzcd} G \Phi F \arrow[r, "(G \Phi) \beta"] \arrow[d, "\tau_{G}F"] & G \Phi F' \arrow[d, "\tau_{G}F'"] \\ GF \arrow[r, "G\beta"] & GF' \end{tikzcd}$$ commutes because $\tau_{G}$ is a natural transformation, and that in turn implies that the diagram in the lemma commutes. What remains to show to establish that $\mathbf{M}_{\mu}$ is well-defined, is naturality in $G$. This is somewhat more difficult than the case of naturality in $F$ or coherence. Denote the collection of $1$-morphisms of the form given in Definition \[BigDef\] by $\mathcal{S}$. By Proposition \[PisNice\], for $G \in \mathcal{S}$, letting $\tau_{G} = \on{id}_{G}$ and following the definition in Lemma \[AlphaConstraint\] yields $\alpha_{G,F} = \on{id}_{\mathbf{M}_{\mu}(GF)}$. Thus obtained collection ${\left\{ \alpha_{G,F} \right\}}_{F,G \in \mathcal{S}}$ is then clearly natural in $G$. We show that we may lift this property from $\mathcal{S}$ to all of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$: For any $F \in {\sc\mbox{G}\hspace{1.0pt}}_{L}(\mathtt{i,i})$ and $G \in \mathcal{S}$, set $\alpha_{G,F} = {\EuRoman}{P}\on{id}_{G\Phi}F{\EuRoman}{J} = {\EuRoman}{P}\on{id}_{G}F{\EuRoman}{J}$. For a $1$-morphism $H \not\in \mathcal{S}$, let $G$ be the $1$-morphism of $\mathcal{S}$ isomorphic to $H$ and fix an isomorphism $H \xrightarrow{\gamma} G$. Consider the composition $H\Phi \xrightarrow{\gamma \Phi} G\Phi \xrightarrow{=} G \xrightarrow{\tau^{-1}} H$. Let $\alpha_{H,F} = {\EuRoman}{P}(\tau^{-1}\circ \tau\Phi)F {\EuRoman}{J}$. The collection ${\left\{ \alpha_{H,F} \right\}}_{H,F \in {\sc\mbox{G}\hspace{1.0pt}}_{L}(\mathtt{i,i})}$ is natural in $H$. Let $H,H'$ be $1$-morphisms of ${\sc\mbox{G}\hspace{1.0pt}}_{L}$ and consider a $2$-morphism $\beta: H \rightarrow H'$. Similarly to the proof of Lemma \[AlphaConstraint\], to establish sought naturality, it suffices to show the commutativity of the following diagram: $$\begin{tikzcd} H\Phi \arrow[r, "\beta \Phi"] \arrow[d, "\tau \Phi"] & H'\Phi \arrow[d, "\tau' \Phi"] \\ G\Phi \arrow[d, equal] & G'\Phi \arrow[d, equal] \\ G \arrow[d, "\tau^{-1}"] & G' \arrow[d, "\tau'^{-1}"] \\ H \arrow[r, "\beta"] & H' \end{tikzcd}$$ By the naturality we have established on $\mathcal{S}$, we have $$(\tau' \circ \beta \circ \tau^{-1})\Phi = \tau' \circ \beta \circ \tau^{-1}.$$ This implies $$\begin{aligned} &\tau' \Phi \circ \beta \Phi \circ \tau^{-1} \Phi = \tau' \circ \beta \circ \tau^{-1} \\ & \tau' \Phi \circ \beta \Phi = \tau' \circ \beta \circ \tau^{-1} \circ \tau \Phi \\ & \tau'^{-1} \circ \tau' \Phi \circ \beta\Phi = \beta \circ \tau^{-1} \circ \tau\Phi, \end{aligned}$$ which, as we can read from the diagram above, is what we wanted to show. This concludes the proof of $\mathbf{M}_{\mu}$ being well-defined. We remark that different choices we could make in the definition of the functor ${\EuRoman}{P}$ give equivalent $2$-representations, the equivalence being given in the discussion preceding Definition \[BigDef\]. Proof of the main result ------------------------ In the preceding subsection we have constructed weak $2$-representations of the $2$-semicategory ${\sc\mbox{G}\hspace{1.0pt}}_{n}$ associated to set partitions of ${\left\{ 1,\ldots,n \right\}}$. To prove Theorem \[MainThm2\], we will show that said weak $2$-representations are simple transitive and pairwise non-equivalent. From Proposition \[SemiWork\] it clearly follows that the same then holds for respective strictifications. \[FinalProp1\] Given a set partition $\mu$ of ${\left\{ 1,\ldots,n \right\}}$, the $2$-representation $\mathbf{M}_{\mu}$ is simple transitive. Observe that $$\on{End}\left(\bigoplus_{i=0}^{k}Q_{i}\right)^{\on{op}} \simeq \Delta_{k} \simeq (e_{0} + \cdots e_{k}) \Delta_{n} (e_{0} + \cdots e_{k}).$$ Denote $\on{add}(\Delta_{n}e_{0},\ldots, \Delta_{n}e_{k})$ by $\mathcal{B}$. The isomorphisms above yield $\mathcal{B} \simeq \Delta_{k}\!\on{-proj}$. Using $e_{i}\Delta_{n}e_{j} = 0$ provided $i,j \neq 0$ and $i \neq j$, we obtain a natural isomorphism $$\label{ElementsIso} (e_{0}\Delta_{n} \otimes_{\Delta_{n}}-)_{|\mathcal{B}} \simeq e_{0}\Delta_{k} \otimes_{\Delta_{k}} -$$ induced by the inclusion $(e_{0} + \cdots e_{k}) \Delta_{n} (e_{0} + \cdots e_{k}) \hookrightarrow \Delta_{n}$. In view of the discussion preceding Definition \[BigDef\], concerning the definition of the functor ${\EuRoman}{P}_{\mu}$, we may without loss of generality assume that ${\EuRoman}{P}_{\mu}(Q_{1}) = Q_{1}$. Then $\mathbf{M}_{\mu}F_{10} = {\EuRoman}{P}_{\mu}F_{10} {\EuRoman}{J} = {F_{10}}_{|\on{add}(Q_{0},\ldots, Q_{k})}$. In view of the equation , we see that $\mathbf{M}_{\mu}F_{10}$ is the projective endofunctor of $\on{add}(Q_{0},Q_{1},\ldots, Q_{k})$ corresponding to $\Delta_{k}e_{1} \otimes_{\Bbbk} e_{0}\Delta_{k}$. Since ${\EuRoman}{P}_{\mu}$ on the level of indecomposables gives a surjection $${\left\{ Q_{0},Q_{1},\ldots, Q_{n} \right\}} \twoheadrightarrow {\left\{ Q_{0},Q_{1},\ldots, Q_{k} \right\}}$$ we may argue similarly to see that ${\sc\mbox{G}\hspace{1.0pt}}_{n}$ acts by projective functors corresponding to functors of the form $\Delta_{k}e_{i} \otimes_{\Bbbk} e_{0}\Delta_{k}$ acting on $\Delta_{k}\!\on{-proj}$. Similarly to Examples \[FirstSight\] and \[DeltaSeen\], we observe that in the general case of a $2$-subsemicategory of a $2$-semicategory of the form ${\sc\mbox{Z}\hspace{1.0pt}}_{A}$, a cell $2$-representation can be viewed as a quotient of that action. In the case of $A = \Delta_{n}$, these two coincide. Hence there is no ${\sc\mbox{G}\hspace{1.0pt}}_{n}$-invariant ideal of $\on{add}(Q_{0},\ldots,Q_{k})$, and so $\mathbf{M}_{\mu}$ is simple transitive. \[FinalProp2\] Let $\mu, \mu'$ be set partitions of ${\left\{ 1,\ldots,n \right\}}$, such that $\mu \neq \mu'$. Then $\mathbf{M}_{\mu} \not\simeq \mathbf{M}_{\mu'}$. If $\mathbf{M}_{\mu}$ and $\mathbf{M}_{\mu'}$ are equivalent, then for $i,j \in {\left\{ 1,\ldots,n \right\}}$ we have $$\mathbf{M}_{\mu}F_{i0} \simeq \mathbf{M}_{\mu}F_{j0} \text{ if and only if } \mathbf{M}_{\mu'}F_{i0} \simeq \mathbf{M}_{\mu'}F_{j0}.$$ But as established in the proof of the preceding proposition, $\mathbf{M}_{\mu}F_{i0} \simeq \mathbf{M}_{\mu}F_{j0}$ if and only if $i$ and $j$ belong to the same subset in the partition $\mu$ of ${\left\{ 1,\ldots,n \right\}}$. Hence we may recover $\mu$ from $\mathbf{M}_{\mu}$, which proves the claim. Theorem \[MainThm2\] now follows from Proposition \[FinalProp1\] and Proposition \[FinalProp2\]. Let $\mu$ be a set partition of ${\left\{ 1,\ldots,n \right\}}$ into $k$ subsets. Abusing notation, denote by $\mu$ the surjection $${\left\{ 0,1,\ldots,n \right\}} \twoheadrightarrow {\left\{ 0,1,\ldots,k \right\}}$$ corresponding to the surjection ${\left\{ Q_{0},\ldots, Q_{n} \right\}} \twoheadrightarrow {\left\{ Q_{0},\ldots, Q_{k} \right\}}$ giving the action of ${\EuRoman}{P}_{\mu}$ on isoclasses of indecomposables. Under the ordering of indecomposables of $\on{add}(Q_{0},\ldots, Q_{k})$ indicated by our notation, we obtain the following action matrices: $$[\mathbf{M}_{\mu}F_{00}] = \begin{pmatrix} 2 & 1 & \cdots & 1 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{pmatrix} = [\mathbf{M}_{\mu'}F_{00}]$$ and $$[\mathbf{M}_{\mu}F_{i0}] = \begin{pmatrix} 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \\ 2 & 1 & \cdots & 1 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{pmatrix}$$ where the $(\mu(i)+1)$th row is non-zero. This shows that there is a fixed ordering of objects of $\on{add}(Q_{0},\ldots, Q_{k})$, with respect to which the action matrices of the various $2$-representations we have constructed exhaust the set of possible sets of action matrices for a simple transitive $2$-representation of the $2$-category ${{\sc\mbox{D}\hspace{1.0pt}}}_{\!L}$ associated to $\Lambda_{n}$ and the self-injective core ${\left\{ e_{0} \right\}}$, as determined in [@Zi2]. As we have seen earlier in this section, this $2$-category is closely related to the $2$-semicategories we study. In particular, the classification given in [@Zi2] applies also to ${\sc\mbox{G}\hspace{1.0pt}}_{n}$: the only modification we need to impose on the proofs there is a different justification of $F_{00}$ necessarily being realized as a self-adjoint functor. In [@Zi2], this is just a statement about adjoint $1$-morphisms, which in its simplest form requires the presence of an identity $1$-morphism. In our case, we use the fact that the $2$-subsemicategory of ${\sc\mbox{G}\hspace{1.0pt}}_{n}$ given by the additive closure of $F_{00}$ is a fiax $2$-category, in the sense of [@KMZ] (this fact is a special case of Proposition $4.1$ therein), which suffices to conclude said self-adjointness. Shortly put, the additive decategorifications of simple transitive $2$-representations of ${\sc\mbox{G}\hspace{1.0pt}}_{n}$ are classified as those of ${\sc\mbox{D}\hspace{1.0pt}}_{L}$ in [@Zi2], and the decategorifications of the $2$-representations we have constructed exhaust that list. [999999999]{} A. Chan, V. Mazorchuk. Diagrams and discrete extensions for finitary 2-representations. Mathematical Proceedings of the Cambridge Philosophical Society, 2017, 1-28. P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik. Tensor categories. Mathematical Surveys and Monographs [**205**]{}. American Mathematical Society, Providence, RI, 2015. M.Ehrig, D.Tubbenhauer. Algebraic properties of zigzag algebras. Communications in Algebra (2020), 48:1, 11-36. H. Jonsson. On Simple Transitive 2-representations of Bimodules over the Dual Numbers. Preprint arXiv: arXiv:2005.08488 H. Ko, V. Mazorchuk. $2$-representations of small quotients of Soergel bimodules in infinite types. Preprint arXiv: arXiv:1809.11113. To appear in Proc. Am. Math. Soc. H. Ko, V. Mazorchuk, X. Zhang. Adjunction in the absence of identity. Preprint arXiv:2003.04610v1. T. Leinster. Basic bicategories. Preprint arXiv:math/9810017. V. Mazorchuk, V. Miemietz. Cell $2$-representations of finitary $2$-categories. Compositio Math. **147** (2011), 1519–1545. V. Mazorchuk, V. Miemietz. Additive versus abelian $2$-representations of fiat $2$-categories. Mosc. Math. J. [**14**]{} (2014), no. 3, 595–615. V. Mazorchuk, V. Miemietz. Endomorphisms of cell $2$-representations. Int. Math. Res. Not. IMRN , Vol. [**2016**]{}, no. 24, 7471–7498. V. Mazorchuk, V. Miemietz. Morita theory for finitary $2$-categories. Quantum Topol. [**7**]{} (2016), no. 1, 1–28. V. Mazorchuk, V. Miemietz. Transitive $2$-representations of finitary $2$-categories. Trans. Amer. Math. Soc. [**368**]{} (2016), no. 11, 7623–7644. V. Mazorchuk, V. Miemietz. Isotypic faithful $2$-representations of $\mathcal{J}$-simple fiat $2$-categories. Math. Z. **282** (2016), no. 1-2, 411–434. M. Mackaay, V. Mazorchuk, V. Miemietz, D. Tubbenhauer. Simple transitive 2-representations via (co)algebra $1$-morphisms. Indiana Univ. Math. J., 68:1–33, 2019. M. Mackaay, V. Mazorchuk, V. Miemietz, D. Tubbenhauer, X. Zhang. $2$-representations of Soergel bimodules. Preprint arXiv:1906.11468 V. Mazorchuk, V. Miemietz, X. Zhang. Pyramids and $2$-representations. Rev. Mat. Iberoam. **36** (2020), no. 2, 387–405. V. Mazorchuk, V. Miemietz, X. Zhang. $2$-categories of symmetric bimodules and their $2$-representations Preprint arXiv:1904.05798. To appear in Pac. J. Math. V. Mazorchuk, X. Zhang. Simple Transitive 2-Representations for Two Nonfiat 2-Categories of Projective Functors. Ukr Math J **70**, 1873–1900 (2019) V. Mazorchuk, X. Zhang. Bimodules over uniformly oriented $A_n$ quivers with radical square zero. Preprint arXiv:1703.08377. To appear in J. Math. Kyoto Univ. A.J. Power. A general coherence result. J. Pure Appl. Algebra [**57**]{} (1989), no. 2, 165-173. E. Persson Westin. Tilting modules and exceptional sequences for leaf quotients of type A zig-zag algebras. Beitr Algebra Geom [**61**]{}, 189–207 (2020). E. Thörnblad, J. Zimmermann. Counting quasi-idempotent irreducible integral matrices. J. Integer Seq. 21 (2018), no. 4, Art. 18.4.8, 14 pp. J. Zimmermann. Simple transitive $2$-representations of Soergel bimodules in type $B_2$. J. Pure Appl. Algebra [**221**]{} (2017), no. 3, 666–690. J. Zimmermann. Simple transitive 2-representations of left cell 2-subcategories of projective functors for star algebras, Communications in Algebra (2019), 47:3, 1222-1237 Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, SWEDEN, email: [mateusz.stroinskimath.uu.se]{}

--- abstract: 'I review the current status of studies of the X-ray sources in Galactic old open clusters. Cataclysmic variables (CVs), magnetically-active binaries (ABs), and sub-subgiants (SSGs) dominate the X-ray emission of old open clusters. Surprisingly, the number of ABs detected inside the half-mass radius with $L_X\gtrsim1\times10^{30}$ erg s$^{-1}$ (0.3–7 keV) does not appear to scale with cluster mass. Comparison of the numbers of CVs, ABs, and SSGs per unit mass in old open and globular clusters shows that each of these classes is under-abundant in globulars. This suggests that dense environments suppress the frequency of even some of the hardest binaries.' author: - 'Maureen van den Berg$^{1,2}$' title: 'X-ray sources in Galactic old open star clusters' --- Introduction ============ The fate of star clusters and the binaries in them are closely intertwined. Dynamical encounters between single stars and binaries affect the stellar velocity distribution, and thus the evolution of the cluster as a whole; vice versa, the dense environment of a cluster can trigger interactions that affect the binary properties. Open clusters allow detailed studies of complete samples of binaries, with the advantage that their age, distance, and composition are known from cluster membership. Surveys like the WIYN Open Cluster Study [@math00] now enable a detailed comparison between observations and numerical models of the evolution of stellar clusters and their binary populations [e.g. @hurlea05]. Until the early 1990s, the study of old open clusters was not pursued with X-ray observations. The reason for this is that the expected contribution of single stars to the X-ray emission of old stellar populations is small. In low-mass stars X-rays are emitted by the hot gas of the corona surrounding the star. The level of X-ray emission is linked to the stellar rotation rate, as rotation powers the magnetic dynamo that is responsible for heating the corona; the lower the spin rate, the weaker the X-ray emission. The rotation of late-type single stars declines over time as a result of angular-momentum loss through wind outflows along the stars’ magnetic field. For stars that are part of a binary, a high level of X-ray emission can be maintained despite an advanced age. Systems in which mass is exchanged between the binary companions are bright in X-rays if one of the stars is a compact object. Examples are cataclysmic variables (CVs) where the compact star is a white dwarf, or low-mass X-ray binaries (LMXBs) that contain a neutron star or black hole. Strong tidal interaction between detached main-sequence stars in a close binary locks their rotation to the orbital period. The rapid rotation induced by the synchronization keeps the dynamo active, and thus the process responsible for the X-ray production. Such systems are called magnetically-active binaries, or ABs in short. For a study of the interacting binaries in an old stellar population, X-ray emission is an excellent tracer. X-ray observations are therefore crucial for studying how the environments of star clusters affect the binary numbers and binary properties. In this paper I review what we have learned from X-ray studies of old open clusters, a field to which Utrecht has contributed significantly. I start with an overview of X-ray studies of old open clusters in Sect. \[sec\_obs\]. In Sect. \[sec\_classes\] I describe the various classes of open-cluster X-ray sources. Sect. \[sec\_glob\] compares the X-ray properties of old open and globular clusters. The chapter by Frank Verbunt gives an overview of X-ray studies of globular clusters. An overview of X-ray observations of old open clusters {#sec_obs} ====================================================== [*ROSAT*]{} {#sec_rosat} ----------- The first old open cluster that was the target of a pointed X-ray observation is M67 (NGC2682). Belloni, Verbunt, and collaborators used the [*ROSAT*]{} Position Sensitive Proportional Counter (PSPC) to observe this 4-Gyr old cluster down to a limit of about $2 \times 10^{30}$ erg s$^{-1}$ (0.1–2.4 keV) in 1991. The photometric monitoring campaign by [@gillea], aimed at looking for solar-like oscillations in members of M67, had just serendipitously discovered the first CV in an open cluster (EUCnc). The light curve of this optically-faint object showed large-amplitude brightness variations at a period of 2.09 h that resembled the variability of so-called AMHer systems—CVs that contain a white dwarf with a strong magnetic field ($B \gtrsim 10$ MG). The main motive for the X-ray observation by Belloni et al. was to do X-ray follow-up for this specific object; as CVs with accurate distance and reddening estimates were (and still are) rare, this was an excellent opportunity for an accurate measurement of the intrinsic X-ray luminosity (assuming cluster membership). The X-ray counterpart was readily detected, and the softness of its X-ray spectrum in the [*ROSAT*]{} band agreed with its AMHer classification [@bellea93]. Later, the magnetic nature was also confirmed by its optical spectrum [@pasqbellea] and [ *ROSAT*]{} light curve [@bellea]. Although this initial observation of M67 was not optimally pointed at the cluster center, it did reveal that at least six more likely members of M67 were similarly bright in the [*ROSAT*]{} band as the CV. Thanks to the wealth of optical information available for this well-studied cluster, it quickly became clear that most of these (in fact: [*all*]{} of these, as we now know) are binaries with periods $\lesssim$45 d, including many with circular orbits. Since the theory of tidal interaction predicts that synchronization occurs before circularization [e.g. @zahn89], this suggests that in these systems the stellar rotation is coupled to the orbit. [@bellea93] therefore suggested that these sources are ABs in which the X-rays are the result of magnetic activity. A second [*ROSAT*]{} observation (this time centered on the cluster), combined with new optical results, detected many other binaries in M67 and refined the classification of some of the earlier detections [@bellea]. Surprisingly, this old cluster that at first seemed like an unexciting target for an X-ray study, revealed an enormous variety of X-ray sources. More details on the different source classes are given in Sect. \[sec\_classes\]. {width="4.9cm"} Several other old open clusters, both younger and older than M67, were studied with [*ROSAT*]{}. In order of increasing age, these are NGC6940 [0.6–1 Gyr; @belltagl], IC4651 [1–1.5 Gyr; @belltagl98], NGC752 [2 Gyr; @bellverb], and NGC188 [6.5 Gyr; @bellea]. Classification of the X-ray sources in each of these resulted in a similar picture as for M67: cross-correlation of the source lists with optical catalogs pointed at a high incidence of binaries among the candidate counterparts. An unexpected discovery was that some known binaries that were detected did not fit the profile of being circular, tidally-locked ABs. For example, some [*ROSAT*]{} sources were identified with eccentric systems having orbital periods of years, far too long for tidal coupling strong enough to induce rapid rotation. At present we still do not understand the X-rays of most of these long-period binaries (Sect. \[sec\_longperiod\]). Overviews of the [*ROSAT*]{} results can be found in [@bell97] and [@verb00]. None of the other clusters showed an X-ray source population so rich and varied as M67. However, as M67 was studied with the highest sensitivity (in part because of its proximity and low reddening) or with the largest fractional coverage, a systematic comparison between clusters based on the [*ROSAT*]{} data is difficult to make. Positional uncertainties of the [*ROSAT*]{} sources range from about 5 to 25, and in some cases multiple candidate counterparts lie inside the error circles. By repeating their X-ray/optical matching with artificial source lists, Belloni et al. estimated that the chances for random coincidences were relatively small, and in fact many proposed identifications were later confirmed with [*Chandra*]{} or [*XMM-Newton*]{} observations. Indeed, the fraction of known binaries among the possible counterparts is too large to be entirely coincidental. On the other hand, in individual cases it is wise to be wary of the possibility of a spurious match. An example is the source X45 in NGC752, which [@bellverb] identified with the blue straggler and long-period binary H209 in NGC752. This identification prompted a comparison with the detection of the blue straggler S1082 in M67, which was found to be a (possibly physically-bound) multiple system consisting of a long-period and short-period binary, in which the latter is responsible for the X-rays (see also Sect. \[sec\_longperiod\]). But extensive optical follow-up by [@vdberg2001c] did not reveal any sign of a close binary in H209. While it cannot be excluded that the parameters of H209 are unfavorable for finding optical signatures for S1082-like multiplicity, the option that the fainter optical source in the 90% error circle is the true counterpart should be seriously considered (Fig. \[fig\_h209\]). Low-resolution optical spectra are needed to classify this alternative counterpart; given the $\sim$17-separation from the bright blue straggler, such spectra should be easy to acquire. [*Chandra*]{} and [*XMM-Newton*]{} ---------------------------------- With [*Chandra*]{} and [*XMM-Newton*]{} more sensitive studies have been done of clusters that had already been observed with [ *ROSAT*]{}. [@vdbergea04] describe a detailed [*Chandra*]{} study of M67, NGC188 was observed with [*XMM-Newton*]{} [@gond05], and a deep study of NGC752 with both satellites—mainly aimed at studying the X-ray emission of single solar-type stars—can be found in [@giarea08]. The sensitivity and positional accuracy of the new instruments also allowed more distant, compact, or reddened open clusters to be studied, thus enabling exploration of the X-ray source populations over a broader range of cluster parameter space. So far, especially the addition of the old (8 Gyr), massive cluster NGC6791 [@vdbergea12ba] has been useful for gaining new insights thanks to the many X-ray sources detected and the extensive body of available optical data (deep photometry, proper motions, variability, follow-up spectroscopy). There are now two parallel ongoing efforts aimed at studying the X-ray properties of old and intermediate-age open clusters. The [*Chandra*]{} survey by van den Berg et al. focuses on the oldest clusters ($\gtrsim 3$ Gyr) while the survey by Pooley et al. with [*XMM-Newton*]{} also includes several younger ones. [*Chandra*]{} and [*XMM-Newton*]{} observations have not (yet) led to the discovery of any fundamentally different source classes than those already found with [*ROSAT*]{}, although, if validated by follow-up spectroscopy, the candidate quiescent low-mass X-ray binary (qLMXB) in NGC6819 reported by [@gosnea12] could turn out to be the first of its kind to be uncovered in an open cluster (Sect. \[sec\_other\]). However, the much-improved positional accuracy has significantly reduced the chances for spurious identifications, and the broader spectral response has facilitated source classification. X-ray source classes {#sec_classes} ==================== Cataclysmic variables --------------------- After the discovery of EUCnc in M67, four more open-cluster CVs were found, all in NGC6791 [@kaluea97; @vdbergea12ba]. EUCnc is quite faint ($\sim$4$\times$10$^{29}$ erg s$^{-1}$, 0.3–7 keV), while B8 in NGC6791 is two orders of magnitude brighter. Most were first identified as CV candidates through their optical colors or variability, while CX19 in NGC6791 is the first X-ray–selected CV in an open cluster. Its proposed optical counterpart was chosen as a high-priority target for follow-up spectroscopy based on the blue color; Balmer and He II emission lines confirm its CV nature. For CVs in globular clusters it is a point of discussion as to whether they have intrinsically different properties than those in the field. The on-average larger X-ray luminosities, higher X-ray–to–optical flux ratios, and possibly lower outburst rates for globular-cluster CVs could perhaps be explained if there is a prevalence of magnetic systems among them, or if the typical accretion rates are lower [@edmogillea03b]. These differences could then be related to their formation scenarios, which may have involved dynamical interactions. As there are only a handful of confirmed CVs in open clusters, it is difficult to make a comparison with either the field or globular-cluster CV population. Besides EUCnc, CX19 is possibly magnetic [@vdbergea12ba], which is in line with the finding that many field CVs that turn out to be magnetic are bright in X-rays and are first discovered in X-rays. What can be said is that scaling the number of CVs in NGC6791 by cluster mass results in a CV density that is consistent with estimates for the CV density in the field, which points at a primordial origin and does not require any dynamical formation or destruction processes. If the density of CVs in open clusters is indeed similar as in the field, the reason why some CV searches have not been successful is likely because of the relatively small number of cluster members surveyed [e.g. @kafkea04]. A few candidate CVs in open clusters have been identified through dwarf-nova–like outbursts (V57 in in the 2–3 Gyr old NGC2158, [@mochea06]; 15877\_2 in the $\sim$3.5 Gyr old NGC6253 [@demaea10]), or their X-ray spectral properties and UV/optical colors (X2 in NGC6819, @gosnea12). More follow-up is needed to establish the nature and membership status of these candidates. Some of the faint, blue candidate counterparts to [*Chandra*]{} or [*XMM-Newton*]{} sources may also turn out to be CVs (or other compact accreting binaries), although, at least for those in M67 and NGC6791, it is estimated that they are mainly background galaxies. Active binaries {#sec_abs} --------------- Active binaries, including both detached systems and contact binaries, are the most common open-cluster X-ray sources. They are among the brightest sources, but found down to the detection limit of the observations ($\sim$2$\times$10$^{28}$ erg s$^{-1}$, 0.3–7 keV, for M67). The study of coronal activity of M67 sources by [@pasqbell] found no relation between the [*ROSAT*]{} X-ray luminosity and stellar parameters like optical magnitude or orbital period. As we now know, this was the result of wrong or incomplete information on orbital or rotation periods, by the inclusion of sources with likely more complex evolutionary histories than regular ABs, and by the high limiting flux of the initial [*ROSAT*]{} pointing (which, as it turns out, was only sensitive enough to detect regular ABs with orbital periods between $\sim$0.5 and 1.5 d). Later studies with larger, and cleaner, AB samples [*do*]{} reveal an activity-rotation relationship in the sense that the coronal X-ray luminosity decreases as a function of orbital period up to the limiting period for tidal circularization [@vdbergea04; @vdbergea12ba]; for M67 this period lies around 12 d. The explanation is that the stellar rotation, and hence the level of activity, is lower in tidally-circularized (and thus synchronized) binaries with longer periods. As is seen for the X-ray emission of single stars [e.g. @rand97], saturation of the X-ray activity occurs for stars in binaries with the shortest orbital periods and highest rotation rates, such as contact binaries. Since the time scale for tidal synchronization is shorter than for circularization, it is possible that the rotation of stars in eccentric binaries is locked to the orbital period around periastron where the interaction is strongest (so-called pseudo-synchronization). If the resulting rotation is fast enough to generate activity, such binaries can also show up as ABs. An example is the $\sim$32-d period binary S1242 in M67, which has an eccentricity of 0.66 and whose photometric period of 4.88 d [@gillea] corresponds to the corotation period at periastron. [@vdbergea12ba] compared the number of ABs inside the half-mass radius with $L_X \gtrsim 1\times 10^{30}$ erg s$^{-1}$ (0.3–7 keV) for three old open clusters that were observed with [*Chandra*]{} or [*XMM-Newton*]{}. Surprisingly, they find that this number does not scale with cluster mass, as would be expected for a primordial population. NGC6791 is 4.5–6.4 times more massive than M67, but has 0.9–1.6 times the number of ABs. NGC6819 has 2.4 times the mass of M67, but only has a fraction ($\sim$0.13) of the number of ABs. At this point we have no explanation for this, and (deeper) studies of more clusters are required. Possibly, M67 has lost a higher fraction of its initial mass compared to the other two clusters. If it lost preferentially single, low-mass stars through evaporation while retaining more binary stars that sunk to the core due to mass segregation, the current binary population would appear as representative of a much more massive cluster. The small number of X-ray sources in NGC188 could therefore not just be the result of limited sensitivity of the [*ROSAT*]{} pointing, as was suggested by [@verb00]. Sub-subgiants ------------- Among the brightest (up to $\sim$10$^{31}$ erg s$^{-1}$, 0.3–7 keV) X-ray sources in old open clusters are binaries that lie below or to the red of the sub-giant branch, the so-called sub-subgiants or red stragglers. This name derives from the fact that their photometry cannot be explained by the combined light of two ordinary cluster members. Over a dozen sub-subgiants are known in open and globular clusters, and they are typically detected in X-rays. All show signs of binarity, but they include very distinct source classes (detached binaries, CVs, and at least one neutron star with an evolved companion). It is possible that they exist in the field as well, but the poorly constrained ages and distances of field stars make it difficult to recognize them as sub-subgiants. Given their X-ray spectral properties and signs of chromospheric activity (CaII H&K and H$\alpha$ emission), the X-ray emission of sub-subgiants in old open clusters is likely the result of coronal activity. This explanation is not without problems for all systems, though. While the optical variability of the sub-subgiant S1113 in M67 suggests that the rotation is synchronized to the orbit [@vdbergea02], the light curve of the other M67 sub-subgiant S1063 (an eccentric binary in a 18.4-d period) displays what looks like spot modulation on a period that is longer than both the pseudo-synchronous and orbital period (Fig. \[fig\_s1040\]). Tidal locking, which sets the stellar rotation in normal ABs, may not have been achieved yet, or another mechanism drives the spin rate in S1063. A recent or ongoing episode of mass transfer has been invoked to explain the optical photometry of CVs in 47Tuc that lie in the sub-subgiant region of the color-magnitude diagram [@albrea01]. But for S1063 the eccentric orbit argues against mass transfer in the recent past, as a (nearly) Roche-lobe filling star would have quickly circularized the orbit [@mathea03]. Sub-subgiants are not uncommon in old open clusters, and their numbers appear to scale with cluster mass [@vdbergea12ba see also Table \[tab\_glob\]]. This suggests that the explanation of their anomalous properties lies in a hitherto overlooked binary-evolution path, and is not the result of a short-lived perturbed state. N-body simulations by [@hurlea05] created one star below the sub-giant branch, which resulted from the merger of two stars in a binary after instable mass transfer. This star is single though, and some basic ingredient (primordial triples?) is still missing from the models. Long-period binaries, and blue and yellow stragglers {#sec_longperiod} ---------------------------------------------------- Another class of poorly-understood X-ray sources are those that are identified with wide binaries. In such systems tidal interaction is far too weak to lead to enhanced stellar rotation rates. In most cases the suggested optical counterparts also have anomalous optical properties: based on their photometry they are classified as blue or yellow stragglers, which is a sign of a complex past that may have involved mass transfer, a merger, or perhaps some kind of dynamical encounter. It is not clear if their X-ray emission is always intrinsically linked to their evolutionary histories. An example of a wide spectroscopic binary with “normal” colors is the [*ROSAT*]{} source \#6 (VR111) in NGC6940, which has a period of almost 3600 d [@belltagl]. Blue stragglers are found in most old clusters, and it is a matter of debate if they are formed through binary evolution, dynamical encounters, or both [see e.g. @mathgell09]. X-ray emission is not a common property of blue stragglers. As the X-rays indicate that some kind of binary interaction is currently active, tracking down their origin can also provide clues to what led to a system’s formation. The best example is S1082 in M67. Studies triggered by its X-ray detection showed that it contains two stars that are blue stragglers on their own account, and that a close and wide binary contribute to the optical light [@vdbergea2001ad; @sandea03]. If these are bound, at least six stars must have been involved in creating S1082, making a dynamical formation most likely [@leigsill11]. Other X-ray–emitting blue stragglers in wide orbits are S997 in M67 [@vdbergea04], H209 in NGC752 (Sect. \[sec\_rosat\]), and possibly L44 in IC4651 [@belltagl98]. M67 features three yellow stragglers, i.e. stars between the blue stragglers and the red-giant branch, which all have wide orbits ($\gtrsim$40 d) and secure [*Chandra*]{} detections. The eccentric binaries S1072 and S1237 show no signs of chromospheric activity [@vdbergea]; possibly a faint close binary in these systems is overwhelmed by the light of the primary. S1040 is an interesting system for which [@verbphin] predicted the presence of a white-dwarf secondary based on the circular, 42.8-d orbit. They argued that the radius of the yellow-straggler primary is too small to be responsible for the circularization, which must then be attributed to a former primary that was much larger in the past and filled its Roche lobe. The white dwarf was indeed discovered in a UV pointing of M67 [@landea]. Magnetic activity of the yellow straggler is manifested in the coronal X-ray emission, but also in chromospheric emission lines [e.g. @vdbergea], and is possibly a relic of the previous phase of mass transfer. The rotation rate as derived from photometric variability is lower than expected for synchronous rotation (see Fig. \[fig\_s1040\]). {width="5.5cm"} Other source classes {#sec_other} -------------------- Continued loss of angular momentum in a contact binary can lead to a merger of the two stars. It is expected that the result is a single, rapidly-rotating star. The brightest ($\sim$2$\times$10$^{31}$ erg s$^{-1}$, 0.1–2.4 keV) [*ROSAT*]{} source X29 in NGC188 was identified with such an FKCom-type giant [@bellea]. As the X-rays are driven by rotation, this class of coronal emitter is closely related to the ABs. A few X-rays sources have been identified with cluster giants that show no signs of binarity. Some are among the brightest sources in the cluster. Examples are the [*ROSAT*]{} sources \#13 (VR108) in NGC6940 [@belltagl] and X19 (S364) in M67 [@bellea], and the [*Chandra*]{} source CX9 in NGC6791 [@vdbergea12ba]. These could be truly single stars or very wide binaries; in both cases the X-rays remain a mystery. Alternatively, these stars could be spurious matches (although VR108 does show weak signs of Ca II H&K activity; [@vdberg2001c]). Optical spectroscopy of the faint blue counterpart to a very soft [ *ROSAT*]{} source revealed it to be a hot white dwarf [@pasqbellea]. There are no indications for binarity of this star, and given the estimated temperature of about 68000 K [@flemea97] it is most likely a thermal X-ray emitter. While Pasquini et al. argue that it is a member because single X-ray–emitting white dwarfs are rare, the proper-motion study by [@yadaea08] suggests a low probability for cluster membership. Based on its soft X-ray spectrum and limits on the X-ray–to–optical flux ratio, [@gosnea12] tentatively classify the brightest X-ray source detected in the field of NGC6819 as a qLMXBs. As the chances of finding any primordial LMXBs in a cluster the size of NGC6819 are low, they suggest that—if its qLMXB nature is confirmed—this object likely has a dynamical origin. So far, no qLMXBs have been found in open clusters. [@vdbergea04] suggested that the soft and highly variable X-ray source CX2 in M67 could be a good candidate, but follow-up spectroscopy showed that the candidate optical counterpart is an active galaxy. Comparison with globular clusters {#sec_glob} ================================= It has been known for a long time that as a result of their high central stellar densities, globular clusters are very efficient in forming objects that are rare in the field such as LMXBS and their descendants, the milli-second pulsars or MSPs (see the contribution by Frank Verbunt). [@verb00] pointed out first that, when bright ($\gtrsim 10^{36}$ erg s$^{-1}$) LMXBs in globular clusters are disregarded, the integrated [*ROSAT*]{} X-ray luminosities per unit mass of most globular clusters is lower than that of M67. This raised the questions of whether globular clusters are efficient at destroying those binaries that are responsible for most of the X-ray emission in old open clusters (i.e. ABs in the case of M67), or whether M67 contains an exceptionally high fraction of ABs. At the time, this problem could not be tackled directly by observations because of the lack of sensitivity and spatial resolution to detect and identify ABs in globular clusters. --------- ------- ----------------------- ----------- ------------- ------------ ------------- ----------------------- cluster age $M$ $N_X$ $N_{X, CV}$ $N_{X, S}$ $N_{X, AB}$ $\log (2~L_{30}$/$M$) (Gyr) ($M_{\odot}$) NGC6819 2–2.4 2600 [*6–7*]{} [*1?*]{} [*1?*]{} [*1?*]{} … M67 4 1100 12 0 1 7–8 28.9 NGC6791 8 (5–7)$\times$$10^{3}$ 15–19 3–4 3 7–11 28.6–28.8 47Tuc 11.2 $1.3\times10^{6}$ $\sim$200 30–119 10 42–131 28.0 NGC6397 13.9 $2.5\times10^{5}$ 15–18 11 2 0–2 27.7 --------- ------- ----------------------- ----------- ------------- ------------ ------------- ----------------------- With the excellent imaging capabilities of [*Chandra*]{} we can revisit this issue. Although almost 80 globular clusters have been studied with [*Chandra*]{}, only a handful have been observed with sufficient sensitivity to access a significant part of the AB X-ray luminosity function. [@vdbergea12ba] made a detailed comparison between the numbers of CVs, ABs, and sub-subgiants down to $L_X = 1 \times 10^{30}$ erg s$^{-1}$ in those old open and globular clusters for which the most comprehensive source classifications are available (see also Table \[tab\_glob\]). It turns out that both suggested scenarios appear to be relevant for explaining the integrated X-ray properties of old clusters. As discussed in Sec. \[sec\_abs\], among old open clusters M67 has a high AB frequency for its mass. On the other hand, not just M67 but also NGC6791 has a higher X-ray emissivity than the two globular clusters. All three main source classes are under-represented in globulars when scaling by mass. The specific frequency of CVs appears to be less disparate than that of ABs; a possible reason is that in globulars they are dynamically created as well [@poolhut06]. The X-ray emission from qLMXBs and MSPs in globular clusters, i.e. types of faint X-ray sources of which there are no confirmed cases in old open clusters, cannot make up for the lack of CVs and coronal sources. In a study of a larger sample of globular clusters, Heinke et al. (in preparation) find that their X-ray emissivity is lower than that of old low-density environments in general. The suppressed numbers of CVs and ABs is in line with the overall low binary frequency in globular clusters compared to open clusters [@miloea12]. This suggests that in the intricate process of forming and breaking up binaries in the cores of globulars, the balance is tipped in favor of binary destruction—not just for wide systems but also for close, interacting binaries that dominate the X-ray emission of old populations. Much remains to be done to classify, dissect, and compare the populations of faint X-ray sources in different settings in more detail. Old open clusters, with their rich X-ray source populations that are relatively accessible for optical follow-up work, can play a crucial part in these studies. I would like to thank Frank Verbunt, Josh Grindlay, Haldan Cohn, Phyllis Lugger, and Craig Heinke for many discussions on the topic of X-ray sources and close binaries in old open and globular clusters. This research was partly supported by various [*Chandra*]{} grants for GO programs. natexlab\#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{} , M. D., [Gilliland]{}, R. L., [Brown]{}, T. M., [Edmonds]{}, P. D., [Guhathakurta]{}, P., & [Sarajedini]{}, A. 2001, , 559, 1060. , T. 1997, Memorie della Società Astronomia Italiana, 68, 993 Belloni, T., & Tagliaferri, G. 1997, A&A, 326, 608 — 1998, A&A, 335, 517 Belloni, T., & Verbunt, F. 1996, A&A, 305, 806 Belloni, T., Verbunt, F., & Mathieu, R. D. 1998, A&A, 339, 431 Belloni, T., Verbunt, F., & Schmitt, J. H. M. M. 1993, A&A, 269, 175 , H. N., [Lugger]{}, P. M., [Couch]{}, S. M., [Anderson]{}, J., [Cool]{}, A. M., [van den Berg]{}, M., [Bogdanov]{}, S., [Heinke]{}, C. O., & [Grindlay]{}, J. E. 2010, , 722, 20. , F., [Poretti]{}, E., [Montalto]{}, M., [Desidera]{}, S., & [Piotto]{}, G. 2010, , 509, A17. , P. D., [Gilliland]{}, R. L., [Heinke]{}, C. O., & [Grindlay]{}, J. E. 2003, ApJ, 596, 1197. , T. A., [Liebert]{}, J., [Bergeron]{}, P., & [Beauchamp]{}, A. 1997, in White dwarfs, edited by J. [Isern]{}, M. [Hernanz]{}, & E. [Garcia-Berro]{}, vol. 214 of Astrophysics and Space Science Library, 91 , G., [Pillitteri]{}, I., [Favata]{}, F., & [Micela]{}, G. 2008, , 490, 113. Gilliland, R. L., Brown, T. M., Duncan, D. K., Suntzeff, N. B., Lockwood, G. W., Thompson, D. T., Schild, R. E., Jeffrey, W. A., & Penprase, B. E. 1991, AJ, 101, 541 , P. 2005, , 438, 291 , N. M., [Pooley]{}, D., [Geller]{}, A. M., [Kalirai]{}, J., [Mathieu]{}, R. D., [Frinchaboy]{}, P., & [Ramirez-Ruiz]{}, E. 2012, , 745, 57. , C. O., [Grindlay]{}, J. E., [Edmonds]{}, P. D., [Cohn]{}, H. N., [Lugger]{}, P. M., [Camilo]{}, F., [Bogdanov]{}, S., & [Freire]{}, P. C. 2005, ApJ, 625, 796 , J. R., [Pols]{}, O. R., [Aarseth]{}, S. J., & [Tout]{}, C. A. 2005, , 363, 293. , S., [Gibbs]{}, D. G., II, [Henden]{}, A. A., & [Honeycutt]{}, R. K. 2004, , 127, 1622 , J., [Stanek]{}, K. Z., [Garnavich]{}, P. M., & [Challis]{}, P. 1997, , 491, 153. Landsman, W., Aparicio, J., Bergeron, P., [di Stefano]{}, R., & Stecher, T. P. 1997, ApJ, 481, L93 , N., & [Sills]{}, A. 2011, , 410, 2370. , R. D. 2000, in Stellar Clusters and Associations: Convection, Rotation, and Dynamos, edited by R. [Pallavicini]{}, G. [Micela]{}, & S. [Sciortino]{}, vol. 198 of Astronomical Society of the Pacific Conference Series, 517 , R. D., & [Geller]{}, A. M. 2009, , 462, 1032 , R. D., [van den Berg]{}, M., [Torres]{}, G., [Latham]{}, D., [Verbunt]{}, F., & [Stassun]{}, K. 2003, , 125, 246. , A. P., [Piotto]{}, G., [Bedin]{}, L. R., [Aparicio]{}, A., [Anderson]{}, J., [Sarajedini]{}, A., [Marino]{}, A. F., [Moretti]{}, A., [Davies]{}, M. B., [Chaboyer]{}, B., [Dotter]{}, A., [Hempel]{}, M., [Mar[í]{}n-Franch]{}, A., [Majewski]{}, S., [Paust]{}, N. E. Q., [Reid]{}, I. N., [Rosenberg]{}, A., & [Siegel]{}, M. 2012, , 540, A16. , B. J., [Stanek]{}, K. Z., [Sasselov]{}, D. D., [Szentgyorgyi]{}, A. H., [Adams]{}, E., [Cooper]{}, R. L., [Foster]{}, J. B., [Hartman]{}, J. D., [Hickox]{}, R. C., [Lai]{}, K., [Westover]{}, M., & [Winn]{}, J. N. 2006, , 131, 1090. Pasquini, L., & Belloni, T. 1998, A&A, 336, 902 Pasquini, L., Belloni, T., & Abbott, T. M. C. 1994, A&A, 290, L17 , D., & [Hut]{}, P. 2006, ApJL, 646, L143. Randich, S. 1997, in Cool stars in clusters and associations: magnetic activity and age indicators, edited by G. Micela, R. Pallavicini, & S. Sciortino, vol. 68, 971 , E. L., [Latham]{}, D. W., [Shetrone]{}, M. D., & [Milone]{}, A. A. E. 2003, , 125, 810. van den Berg, M., Orosz, J., Verbunt, F., & Stassun, K. 2001, A&A, 375, 375 , M., [Stassun]{}, K. G., [Verbunt]{}, F., & [Mathieu]{}, R. D. 2002, , 382, 888. , M., [Tagliaferri]{}, G., [Belloni]{}, T., & [Verbunt]{}, F. 2004, A&A, 418, 509. , M., & [Verbunt]{}, F. 2001, , 375, 387 van den Berg, M., Verbunt, F., & Mathieu, R. D. 1999, A&A, 347, 866 , M., [Verbunt]{}, F., [Tagliaferri]{}, G., [Belloni]{}, T., [Bedin]{}, L., & [Platais]{}, I. 2012, ApJ (submitted) , F. 2000, in Stellar Clusters and Associations: Convection, Rotation, and Dynamos, edited by R. [Pallavicini]{}, G. [Micela]{}, & S. [Sciortino]{}, vol. 198 of Astronomical Society of the Pacific Conference Series, 421. Verbunt, F., & Phinney, E. S. 1995, A&A, 296, 709 , R. K. S., [Bedin]{}, L. R., [Piotto]{}, G., [Anderson]{}, J., [Cassisi]{}, S., [Villanova]{}, S., [Platais]{}, I., [Pasquini]{}, L., [Momany]{}, Y., & [Sagar]{}, R. 2008, , 484, 609. , J.-P. 1989, , 220, 112

--- abstract: 'We present an implementation of range separated functionals utilizing the Slater-function on grids in real space in the projector augmented waves method. The screened Poisson equation is solved to evaluate the necessary screened exchange integrals on Cartesian grids. The implementation is verified against existing literature and applied to the description of charge transfer excitations. We find very slow convergence for calculations within linear response time-dependent density functional theory and unoccupied orbitals of the canonical Fock operator. Convergence can be severely improved by using Huzinaga’s virtual orbitals instead. This combination furthermore enables an accurate determination of long-range charge transfer excitations by means of ground-state calculations.' author: - Rolf - Michael bibliography: - './Yukawa\_GPAW.bib' title: Charge transfer excitations with range separated functionals using improved virtual orbitals --- Introduction ============ The study of intra-molecular charge transfer excitations (CTE) is of interest in photovoltaics[@sariciftci_photoinduced_1992], organic electronics[@samori_epitaxial_2002] and molecular and organic magnetism[@blundell_organic_2004]. Within a single particle picture, the simplest CTE is the excitation of an electron from the highest molecular orbital (HOMO) of a donor to the lowest unoccupied orbital (LUMO) of a distant acceptor. Denoting the distance between donor and acceptor by $R$, Mulliken derived the energetics of this process in the asymptote of large distances $R$ to (atomic units are used throughout) $$\begin{aligned} E_\text{CTE} \approx \text{IP}_D - \text{EA}_A - \frac{1}{R},\label{eq:MullCTE}\end{aligned}$$ where $\text{IP}_D$ is the ionization potential of the donor, $\text{EA}_A$ the electron affinity of the acceptor and $\frac{1}{R}$ approximates the Coulomb energy between the excited electron and the hole it left behind[@mulliken_molecular_1969]. Density functional theory (DFT)[@hohenberg_inhomogeneous_1964] in the formulation of @kohn_self-consistent_1965[@kohn_self-consistent_1965] is the method of choice for *ab initio* calculations of electronic properties of condensed matter due to its advantageous cost to accuracy ratio. In contrast to wave-function based methods, DFT expresses the total energy of a given system as functional of the electron density $n$. Although DFT is exact in principle, the exact form of this functional itself is unknown and has to be approximated in practice[@livshits_well-tempered_2007; @chai_systematic_2008; @cohen_insights_2008]. In Kohn-Sham (KS) DFT the density is build from occupied non-interacting single-particle orbitals $\psi_i$ via $n = \sum_i f_i |\psi_i|^2$, where $f_i$ denotes the occupation number. The total energy is expressed as a sum of density functionals for the different contributions $$\begin{aligned} E_\text{tot} [n] = T_\text{S} [n] + V_\text{ext} [n] + U_\text{H}[n] + E_\text{xc} [n],\label{eq:E_KS_DFT}\end{aligned}$$ where the $T_\text{S}$ denotes the kinetic energy of the non-interacting system, $V_\text{ext}$ the energy of the density in the external potential and $U_\text{H}$ the classical Coulomb energy of the density with itself. These quantities can be calculated exactly. All other energy contributions are collected in the exchange-correlation energy $E_\text{xc}$ which is approximated. Within a generalized KS scheme[@baer_tuned_2010] $E_\text{xc}$ can be further split into the contributions from exchange as in Hartree-Fock theory (HFT) $$\begin{aligned} E_\text{x} = -\frac{1}{2} \int \int \frac{\left|\sum_i f_i\psi_i^\ast (\vec{r}_1) \psi_i (\vec{r}_2)\right|^2} {|\vec{r}_1 - \vec{r}_2|} \text{d}\vec{r}_1 \text{d}\vec{r}_2, \label{eq:Ex}\end{aligned}$$ and correlation $E_\text{c}$ that contains all energy contributions missing in the other terms[@kohn_self-consistent_1965]. Several types of approximations for $E_\text{xc}$ are in use. The local (LDA)[@vosko_accurate_1980; @perdew_self-interaction_1981; @perdew_accurate_1992] functional approximates $E_\text{xc}$ by the local values of the density, while semi-local GGA[@gill_new_1996; @perdew_generalized_1996] take local density gradients and MGGA[@adamo_meta-gga_2000; @tao_climbing_2003] local values of the kinetic electron density into account (we will call local and semi-local functionals as local functionals for brevity in the following). The accuracy of local functionals is sometimes improved by hybrid functionals that combine the exchange from local functionals with the “exact” exchange integrals from HFT eq. (\[eq:Ex\]) by a fixed ratio[@becke_new_1993]. While these approximations work fairly well for equilibrium properties[@baer_density_2005; @cohen_insights_2008], local functionals as well as hybrids fail badly in the description of CTEs because of their missing ability to describe non-local interactions correctly [@dreuw_long-range_2003; @baerends_kohnsham_2013; @kummel_charge-transfer_2017; @Maitra17]. Range separated functionals (RSF), that combine the exchange from the local functionals with the non-local exchange from HFT based on the spatial distance between two points $\vec{r}_1$ and $\vec{r}_2$ are able to predict the energetics and oscillator strengths of CTEs by linear response time-dependent DFT [@tawada_long-range-corrected_2004; @livshits_well-tempered_2007; @chai_systematic_2008; @akinaga_intramolecular_2009; @rohrdanz_long-range-corrected_2009; @stein_reliable_2009; @stein_prediction_2009; @baer_tuned_2010; @kronik_excitation_2012; @zhang_non-self-consistent_2012]. RSF use a separation function $\omega_\text{RSF}$ to split the Coulomb interaction kernel of the exchange integral (\[eq:Ex\]) into two parts[@yanai_new_2004] $$\begin{aligned} \frac{1}{r_{12}} =& \underbrace{\frac{1 - \left [ \alpha + \beta \left ( 1 - \omega_\text{RSF}(\gamma, r_{12})\right )\right]}{r_{12}}}_\text{SR, DFT}\notag \\ &+ \underbrace{\frac{\alpha + \beta \left (1- \omega_\text{RSF}(\gamma, r_{12})\right )}{r_{12}},}_\text{LR, HFT} \label{eq:CAM}\end{aligned}$$ where $\gamma$ is a separation parameter, and $\alpha$ and $\beta$ are mixing parameters for spatially fixed and range separated mixing, respectively. The exchange from the local functionals is usually used in the short-range (SR) part, while non-local exchange integrals from HFT are used for long-range (LR) exchange. There also exists a class of RSF that uses exact exchange at short-range and the exchange from a local functional for the long-range part and is very popular for the description of periodic systems[@heyd_hybrid_2003; @heyd_erratum:_2006], which is not the topic of our investigations, however. The correlation energy $E_\text{c}$ is approximated by a local functional globally[@iikura_long-range_2001] . The unoccupied states entering lrTDDFT using the canonical Fock operator of HFT approximate (exited) EAs[@stein_fundamental_2010; @kronik_excitation_2012]. Coulomb-repulsion and exchange interaction cancel for occupied states making them subject to the interaction with $N-1$ electrons in an $N$ electron system. This cancellation is absent for unoccupied (virtual) states, making them subject to the interaction with all $N$ electrons. Therefore these unoccupied states are rather inappropriate for neutral excited state calculations. The canonical Fock operator of HFT is not the only possible choice for the calculation of unoccupied states, as any Hermitian rotation within the space of virtual orbitals is allowed[@huzinaga_virtual_1970]. Therefore it is possible to create so called improved virtual orbitals (IVOs) that are able to approximate certain excitations already at the single particle level rather accurately[@kelly_correlation_1963; @kelly_many-body_1964; @huzinaga_virtual_1970; @huzinaga_virtual_1971] and we will show that these orbitals are also well suited for the calculation of charge transfer excitations within RSF. This work is organized as follows: The following section describes the numerical methods applied and section \[sec:RSF\] details the implementation of RSF in real space grids within the projector agmented wave method. Sec. \[sec:verify\] presents the verification of our implementation against existing literature and sec. \[sec:CTE\] applies RSF and its combination with IVOs to obtain CTE energies. The manuscript finally ends with conclusions. Methods ======= DFT using the real space grid implementation of the projector augmented waves (PAW)[@blochl_projector_1994] in the [[GPAW]{}]{} package[@mortensen_real-space_2005; @enkovaara_electronic_2010] was used for all calculations performed in this work. PAW is an all-electron method, which has shown to provide very similar results as converged basis sets for a test set of small molecules[@kresse_ultrasoft_1999] and transition metals[@valiev_calculations_2003; @Wurdemann2015]. With the exception of transition metals, where $3s$, $3p$, $3d$ and $4s$ shells were treated as valence electrons, all closed shells were subject to the frozen core approximation and (half-)open shells were treated as valence electrons. Relativistic effects were applied to the closed shells in the frozen cores in the scalar-relativistic approximation of @koelling_technique_1977[@koelling_technique_1977]. If not stated otherwise, a grid-spacing of $h=\unit[0.18]{\text{\AA}}$ was used for the smooth KS wave function and a simulation box which contains at least $\unit[6]{\text{\AA}}$ space around each atom was applied. Non-periodic boundary conditions were applied in all three directions and all calculations were done spin-polarized. Only collinear spin alignments were considered. The exchange correlation functionals PBE[@perdew_generalized_1996], the hybrid PBE0[@adamo_toward_1998] and the RSFs LCY-BLYP[@akinaga_range-separation_2008], LCY-PBE[@seth_range-separated_2012] and CAMY-B3LYP[@akinaga_range-separation_2008] were used. Linear response time-dependent density functional theory (lrTDDFT)[@casida_time-dependent_2009; @walter_time-dependent_2008] for RSF was implemented along the work of @tawada_long-range-corrected_2004[@tawada_long-range-corrected_2004] and @akinaga_intramolecular_2009[@akinaga_intramolecular_2009]. Twelve unoccupied bands were used in the lrTDDFT calculations unless stated otherwise. Implementation of RSF {#sec:RSF} ===================== Two functions are frequently used as separation function (\[eq:CAM\]) in literature. One is the complementary error-function[@iikura_long-range_2001; @tawada_long-range-corrected_2004; @yanai_new_2004; @baer_avoiding_2006; @peach_assessment_2006; @vydrov_tests_2007; @chai_systematic_2008; @rohrdanz_simultaneous_2008; @wong_absorption_2009] $\omega_\text{RSF}(\gamma, r_{12}) = \operatorname{erfc}(\gamma r_{12})$ that enables efficient evaluation when the KS orbitals are represented by Gaussian functions, and the other is the Slater-function[@baer_avoiding_2006; @akinaga_range-separation_2008; @akinaga_intramolecular_2009; @seth_range-separated_2012] $$\begin{aligned} \omega_\text{RSF}(\gamma, r_{12}) &= e^{ - \gamma r_{12}}. \label{eq:Yukawa}\end{aligned}$$ which we will apply in our work. The Slater-function is the natural choice for a screened Coulomb potential, as it leads to the Yukawa potential[@yukawa_interaction_1935] that can be derived to be the effective one-electron potential in many-electron systems[@seth_range-separated_2012]. Calculations utilizing the Slater-function were found to give superior results for the calculation of charge transfer and Rydberg excitations compared to the use of the complementary error-function [@akinaga_range-separation_2008; @akinaga_intramolecular_2009]. The calculation of the exact exchange in a RSF is straightforward in principle. Regarding only the long-range part in eq. (\[eq:CAM\]) and setting $\alpha = 0$ and $\beta =1$ for brevity, we have to evaluate the exchange integral $$\begin{aligned} K^\text{RSF}_{ij} &= \left(ij | 1 - \omega_{\text{RSF}} \left(\gamma, r_{12}\right) | ji \right) \label{eq:RSF_EXX_om}\end{aligned}$$ where a Mulliken-like notation $$\begin{aligned} \left( ij| \hat{x} | ji \right) &= \notag \\ \int \int& \frac{\psi_i^\ast (\vec{r}_1) \psi_j (\vec{r}_1) \, \hat{x}\, \psi_j^\ast (\vec{r}_2) \psi_i (\vec{r}_2)}{|\vec{r}_1 - \vec{r}_2|} \, \text{d}\vec{r}_1 \text{d}\vec{r}_2 \label{eq:Iex}\end{aligned}$$ with the one-particle functions $\psi_{i,j}$ and the operator $\hat{x}$ is used. The first part of eq. (\[eq:RSF\_EXX\_om\]) is the standard exchange integral $K_{ij}=\left(ij | ji \right)$ from HFT[@rostgaard_exact_2006; @enkovaara_electronic_2010] and the additional term is the screened exchange integral $$\begin{aligned} K^\gamma_{ij} &= \left(ij |\omega_\text{RSF} \left(r_{12}\right)| ji \right) \; . \label{eq:IRSF}\end{aligned}$$ In PAW the KS wave-function (WF) $\psi_i$ is represented as a combination of a soft pseudo WF, $\tilde{\psi}_i$ and atom-centered (local) corrections[@blochl_projector_1994; @mortensen_real-space_2005; @enkovaara_electronic_2010] $$\begin{aligned} \psi_i &= \tilde{\psi}_i + \sum_\alpha \sum_k \left ( | \phi_k^\alpha \rangle - | \tilde{\phi}_k^\alpha \rangle \right ) \mathcal{P}_{ik}^\alpha, \label{eq:PAW_Trans}\end{aligned}$$ where $\phi_k^\alpha$ and $\tilde{\phi}_k^\alpha$ denote atom centered all-electron and soft partial WFs, respectively, and $\mathcal{P}_{ik}^\alpha$ is a projection operator which maps the pseudo WF on the partial WF. The all-electron and soft partial WF match outside of the atom centered augmentation sphere. The band-indices $i$ and $k$ contain the main quantum numbers and $\alpha$ is the atomic index which runs over all atoms in the calculation. In our implementation, the pseudo WFs are evaluated on three dimensional Cartesian grids in real space while the partial WFs are evaluated on radial grids[@mortensen_real-space_2005; @enkovaara_electronic_2010]. Using the exchange density $n_{ij} = \psi_i^\ast \psi_j$ and the local exchange density $n^\alpha_{ij} = \sum_{k_1 k_2} \phi_{k_1}^\alpha \phi_{k_2}^\alpha \mathcal{P}_{ik_1}^{\alpha\ast}\mathcal{P}_{jk_2}^{\alpha}$ the integral (\[eq:IRSF\]) can be written as $$\begin{aligned} \left(\left(n_{ij}\right)\right)^\gamma &= \left(\left(\tilde{n}_{ij}\right)\right)^\gamma + \sum_\alpha \left [ \left(\left(n^\alpha_{ij}\right)\right)^\gamma - \left(\left(\tilde{n}^\alpha_{ij}\right)\right)^\gamma \right],\label{eq:ScrExPawNoComp}\end{aligned}$$ where $$\left(\left(n_{ij}\right)\right)^\gamma = \left(n_{ij}|n_{ji}\right)^\gamma = \left(n_{ij} | \exp \left( -\gamma r_{12}\right)| n_{ji} \right)$$ is used as shortcut for the screened exchange interaction of the exchange density with itself. Due to the non-locality of the screened exchange operator, a straight application of eq. (\[eq:ScrExPawNoComp\]) would lead to cross-terms between local functions located on different atomic sites and the need to integrate on incompatible grids[@blochl_projector_1994; @mortensen_real-space_2005]. To avoid these, compensation charges $\tilde{Z}_{ij}^{\alpha}$ are introduced[@rostgaard_exact_2006; @enkovaara_electronic_2010] such that using $\tilde{\varrho}_{ij} = \tilde{n}_{ij} + \sum_\alpha \tilde{Z}_{ij}^\alpha$ allows to write $$\begin{aligned} \left(\left(n_{ij}\right)\right)^\gamma &=\left(\left(\tilde{\varrho}_{ij}\right)\right)^\gamma + \sum_\alpha \Delta K_{ij}^{\alpha\gamma}, \label{eq:ScrExPawComp}\end{aligned}$$ where $\left(\left(\tilde{\varrho}_{ij}\right)\right)^\gamma$ is free of local contributions. The evaluation of the $\Delta K_{ij}^{\alpha\gamma}$ using the integration kernel from @rico_repulsion_2012[@rico_repulsion_2012] is detailed in supporting information (SI). Direct evaluation of the double-integral $$\begin{aligned} I &= \int \int \tilde{\varrho}_{ij}(1) \frac{e^{-\gamma r_{12}}}{r_{12}} \tilde{\varrho}_{ji}(2) \text{d}\vec{r}_1 \text{d}\vec{r}_2 \label{eq:IYukawa}\end{aligned}$$ on a three dimensional Cartesian grid is not only time-consuming, but also suffers from the singularity at $\vec{r}_1 = \vec{r}_2$. To circumvent this, the integral is solved using the method of the Green’s functions $$\begin{aligned} I &= \int \tilde{\varrho}_{ij}(\vec{r}_1) \tilde{v}_{ji}(\vec{r_1}) d\vec{r}_1 \; .\label{eq:IntGreen}\end{aligned}$$ The potential $\tilde{v}_{ij}$ is calculated by solving the screened Poisson or modified Helmholtz equation[@jackson_classical_1998; @greengard_new_2002] $$\begin{aligned} \left ( \nabla^2 - \gamma^2 \right ) \tilde{v}_{ij}(\vec{r_1}) &= - 4 \pi \tilde{\varrho}_{ij}(\vec{r_1}) \; . \label{eq:ScreenedPoisson2}\end{aligned}$$ A finite difference scheme together with a root finder[@mortensen_real-space_2005; @enkovaara_electronic_2010] is chosen, where a constant representing $\gamma$ is added to the central point of the finite difference stencil[@wajid_modified_2014]. The potential $\tilde{v}_{ij}$ of a charge system decays very slowly and the (screened) Poisson equation (\[eq:ScreenedPoisson2\]) is therefore applied for neutral charge distributions only[@enkovaara_electronic_2010]. Charged systems are neutralized by subtracting a Gaussian density for which the solution is known analytically (see SI for details). Range separated functionals also contain a contribution of the density functional exchange. @akinaga_range-separation_2008 derived an analytic expression for the exchange contribution of a GGA in the case of a Slater-function based RSF, $E_\text{x}^\text{GGA} \left (\gamma \right)$[@akinaga_range-separation_2008]. @seth_range-separated_2012 discovered, that this expression leads to numerical instabilities for very small densities and derived a superior expression for small densities based on a power series expansion[@seth_range-separated_2012]. Both expressions along with analytic expressions for the first, second, and third derivatives of $E_\text{x}^\text{GGA}(\gamma)$ based on the analytic expression derived by @akinaga_range-separation_2008 were implemented in *libxc*[@marques_libxc:_2012; @LehtolaRecentdevelopmentslibxc2018]. Verification of the implementation {#sec:verify} ================================== [lD[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[3]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[3]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[3]{}|D[.]{}[.]{}[2]{}]{} & & & &\ Functional & & & & & & & & & &\ PBE & & 7.56 & 7.53\^a & 0.03 & 3.74 & 3.73\^a & 0.01& 5.67 & 5.66\^a & 0.01\ PBE & & & 7.56\^b & 0.00 & & 3.67\^b & 0.07& & 6.07\^b & -0.4\ PBE0 & & 6.45 & 6.44\^a & 0.01 & 3.54 & 3.62\^a & -0.08& 4.24 & 4.28\^a & -0.04\ PBE0 & & & 6.47\^b & -0.02 & & 3.50\^b & 0.04& & 4.63\^b & -0.39\ LCY-BLYP & 0.70 & 6.24 & 6.33\^b & -0.09 & 3.44 & 3.69\^b & -0.25& 4.16 & 4.75\^b & -0.59\ & 0.75 & 6.12 & 6.21\^b & -0.09 & 3.43 & 3.68\^b & -0.25& 4.01 & 4.59\^b & -0.58\ LCY-PBE & 0.75 & 6.38 & 6.47\^b & -0.09 & 3.63 & 3.88\^b & -0.25& 3.93 & 4.74\^b & -0.81\ & 0.90 & 6.04 & 6.14\^b & -0.10& 3.60 & 3.84\^b & -0.24& 3.49 & 4.29\^b & -0.8\ CAMY-B3LYP & 0.34 & 6.32 & 6.38\^b & -0.06& 3.32 & 3.43\^b & -0.11& 4.34 & 4.89\^b & -0.55\ $\overline{\text{d}E}_\text{RSF}$& & & & -0.09& & & -0.22& & & -0.67\ exp. & && 6.62\^c &&& 3.81\^c & && 4.97\^c &\ A large amount of work was devoted to RSF in the literature[@iikura_long-range_2001; @tawada_long-range-corrected_2004; @toulouse_long-rangechar21short-range_2004; @yanai_new_2004; @baer_density_2005; @baer_avoiding_2006; @peach_assessment_2006; @vydrov_assessment_2006; @vydrov_importance_2006; @cohen_development_2007; @livshits_well-tempered_2007; @gerber_range_2007; @vydrov_tests_2007; @akinaga_range-separation_2008; @chai_systematic_2008; @henderson_range_2008; @livshits_density_2008; @rohrdanz_simultaneous_2008; @akinaga_intramolecular_2009; @livshits_deleterious_2009; @rohrdanz_long-range-corrected_2009; @stein_prediction_2009; @stein_reliable_2009; @wong_absorption_2009; @baer_tuned_2010; @stein_fundamental_2010; @refaely-abramson_fundamental_2011; @kronik_excitation_2012; @seth_range-separated_2012; @zhang_non-self-consistent_2012; @seth_modeling_2013; @autschbach_delocalization_2014; @cabral_do_couto_performance_2015]. In order to verify our implementation, we have re-calculated some of the published properties in our grid-based approach. @seth_range-separated_2012[@seth_range-separated_2012] calculated the mean ligand removal enthalpies defined as[@johnson_tests_2009; @seth_range-separated_2012] $$\begin{aligned} \bar{E}_{L} &= \frac{n E(L) + m E(M) - E(M_m L_n)}{n + m - 1}\end{aligned}$$ for a group of molecules including transition metals. We have calculated this quantity for the molecules TiO~2~, CuCl, and CrO~3~ and compared our results against the values published by @johnson_tests_2009[@johnson_tests_2009] for PBE and PBE0, as well as @seth_range-separated_2012[@seth_range-separated_2012] for PBE, PBE0 and a group of RSF in tab. \[tab:YukSeth\]. As the literature values were calculated without relativistic corrections, we have neglected relativistic effects in the reported values also. A grid spacing of $h = \unit[0.16]{\text{\AA}}$ was necessary to correctly describe $3d$-splitting[@Wurdemann2015] (see SI for details). Generally, our PBE and PBE0 values are in good agreement to the results obtained from both groups for TiO~2~ and CuCl. There is a difference of about $\unit[0.4]{eV}$ to the work of @seth_range-separated_2012 for CrO~3~, while our values are a in good agreement to the work of @johnson_tests_2009. These differences can attributed the different basis sets used. While @johnson_tests_2009 used the rather large 6–311++G(3df,3pd) basis set, @seth_range-separated_2012 use a smaller TZ2P basis set that is apparently not large enough. We have observed similar strong basis set effects in particular if chromium is involved already in prior studies for chromium[@Wurdemann2015]. RSF results using Slater functions are unfortunately only available from the TZ2P basis set. While our results are in good agreement to @seth_range-separated_2012 for TiO$_2$ ($\le \unit[-0.09]{eV}$ deviation), they already differ by up to $\unit[0.25]{eV}$ for CuCl. CAMY-B3LYP, which includes only a fraction of the screened exchange, generally leads to the smallest deviations. RSFs are obviously very sensitive to basis set limitations due to the long range of the effective single particle potential[@baer_tuned_2010]. We therefore trust the values obtained by our method which represent the large basis set limit. In comparison to experiment, PBE ligand removal energies are rather accurate for CuCl, but this functional over-binds TiO$_2$ and CrO$_3$. In the latter molecules $d$-orbitals contribute to binding and might be responsible for this overbinding. In contrast, the hybrid PBE0 as well as the RSFs tend to underbind in all three molecules, in particular if $d$-orbitals are involved. Interestingly, this trend is similar to the overestimation of $d$-binding in PBE and the lack of proper $d$-binding by hybrids we have observed in the Cr-dimer before[@Wurdemann2015]. The value of the separation parameter $\gamma$ can not be defined rigorously and its optimal choice is under discussion. A system dependence is to be expected[@iikura_long-range_2001; @baer_avoiding_2006; @livshits_well-tempered_2007; @chai_systematic_2008; @livshits_density_2008; @baer_tuned_2010; @stein_fundamental_2010; @refaely-abramson_fundamental_2011; @kronik_excitation_2012]. The group around Roi Baer devised schemes to optimize $\gamma$ without the use of empirical parameters[@livshits_well-tempered_2007; @livshits_density_2008; @stein_reliable_2009] by forcing the difference between the ionization potential (IP) calculated from total energy differences and the negative eigenvalue of the HOMO $-\epsilon_\text{HOMO}$ of an $N$ electron system to vanish[@livshits_well-tempered_2007] $$\begin{aligned} \underbrace{\left[ E_\text{gs}\left(N, \gamma\right) - E_\text{gs}\left(N-1, \gamma\right) \right]}_{\text{IP}(N)} \notag \\ \stackrel{\gamma = \gamma_\text{opt}}{\equiv} - \epsilon_\text{HOMO}(N, \gamma) \; . \label{eq:fitgammaIP}\end{aligned}$$ This condition is fulfilled for the exact functional[@almbladh_exact_1985], but is usually violated by local and hybrid approximations[@livshits_well-tempered_2007]. @livshits_density_2008 also devised an approach to determine $\gamma$ for the calculation of binding energy curves of symmetric bi-radical cations which imposes a match of the slopes of the energy curves for the charged and neutral molecule[@livshits_density_2008] $$\begin{aligned} E_\text{gs} \left(N-\frac{1}{2}, \gamma\right) - E_\text{gs} \left(N-1, \gamma\right) \notag\\ \quad \stackrel{\gamma = \gamma_\text{opt}}{\equiv} E_\text{gs} \left(N, \gamma\right) - E_\text{gs} \left(N-\frac{1}{2}, \gamma\right) \; .\label{eq:fitgammaFrac}\end{aligned}$$ Their group found that both approaches give almost identical values of $\gamma_\text{opt}$ for the same system[@baer_tuned_2010], which we confirm in the case of Cr$_2$. We will denote an RSF using an optimized value of $\gamma$ obtained by eqs. (\[eq:fitgammaIP\], \[eq:fitgammaFrac\]) by appending an asterisk, e.g. LCY-PBE${}^*$. [l|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}]{} & & & &\ & & & & & & & &\ exp.[@linstrom_ion_2016] & & 6.77 & & 6.4 & & 14.01 & & 15.58\ PBE & 3.70 & & 4.29 & & 9.09 & & 10.24 &\ PBE0 & 4.95 & & && 10.79 && 12.16 &\ BNL${}^*$[@livshits_well-tempered_2007] & & & && 14.3&& 16.6 &\ LCY-PBE${}^*$ & 6.79 && 6.85 && 14.32 &14.31 & 16.33 & 16.32\ $\gamma^\text{BNL}_\text{opt} (a_0^{-1})$[@livshits_well-tempered_2007] & & & &\ $\gamma^\text{LCY}_\text{opt} (a_0^{-1})$ & & & & We used eq. (\[eq:fitgammaIP\]) to obtain $\gamma_\text{opt}$ for Cr, Cr~2~, CO and N~2~ and verified that the eigenvalues for the HOMO as well as the experimental value of the ionization potential match in this case. The resulting screening parameters for the RSF BNL^\*^ and LCY-PBE^\*^ are listed in tab. \[tab:TuneGammaMol\]. BNL^\*^ is a LDA based RSF used by @livshits_well-tempered_2007 which utilizes the error-function instead of the Slater function[@livshits_well-tempered_2007]. For the gradient corrected PBE and the hybrid-functional PBE0 the eigenvalues of the HOMO doesn’t match the experimental ionization potential. This is different for the RSF: The eigenvalues for the HOMO are in quite good, for N~2~, to, in the case of Cr, perfect agreement to the experimental ionization potential. For the cases of CO and N~2~ also a good agreement between the values from @livshits_well-tempered_2007 and this work is achieved. For the values of the screening parameter a $\gamma^\text{LCY}_\text{opt} \approx \frac{3}{2} \gamma^\text{BNL}_\text{opt}$ dependency between the used screening functions, Slater vs. error-function, was stated[@shimazaki_band_2008]. The comparison between the values listed in tab. \[tab:TuneGammaMol\] supports this dependency. [l|D[.]{}[.]{}[2]{}|r |r|r|r]{} & &\ & & & & &\ 1b~1~ & 12.62 & 1% & 1% & -1% & -1%\ 3a~1~ & 14.74 & 0% & 4% & -3% & -3%\ 1b~2~ & 18.51 & 0% & 3% & -1% & -1%\ 2a~1~${}^a$ & 32.20 & 0% & 4% & -1% & 0% @baer_tuned_2010 also discussed the impact of the tuning of $\gamma$ on the inner ionization energies (ionization into an excited state of the cation). They stated, that by tuning $\gamma$ one is able to predict the inner ionization energies not only by the combination of a $\Delta$SCF and lrTDDFT calculation but also directly by the density of states of the neutral molecule in the sense of Koopmans theorem[@koopmans_uber_1934] from HFT[@baer_tuned_2010]. In this work their example, H~2~O was also verified. The deviations between the calculated IPs and the experimental values are shown in tab. \[tab:TuneGammaH2O\]. The calculated values are in a very good agreement to each other and to experiment, despite the issue, that @baer_tuned_2010 gave a ionization potential for the 2a~1~ state which differs from the value used as reference in both works. Charge transfer excitations {#sec:CTE} =========================== In this section we investigate the description of charge transfer excitations (CTE) within RSF. One of the frequently used model systems to study CTE is the ethylene-tetrafluoroethylene dimer[@dreuw_long-range_2003; @tawada_long-range-corrected_2004; @zhao_density_2006; @peach_assessment_2006; @livshits_well-tempered_2007; @chai_systematic_2008; @rohrdanz_long-range-corrected_2009; @zhang_non-self-consistent_2012]. This choice is not fortunate, as both constituents exhibit a negative EA[@chiu_temporary_1979], which leads to CTE that overlap with the continuum at least for infinite separation. Therefore we use the alternative Na~2~–NaCl complex, where Na~2~ is the donor and NaCl the acceptor with a positive EA (experimental adiabatic EA of 0.73 eV[@miller_electron_1986]). In order to catch the largely delocalized excited states we increased the amount of space within our simulation box to $x_\text{vac} = \unit[11]{\text{\AA}}$ around each atom and decreased the grid spacing to $h=\unit[0.2]{\text{\AA}}$ due to the higher computational effort. [l|D[.]{}[.]{}[9]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}|D[.]{}[.]{}[2]{}]{} & & & &\ Na~2~ & 4.8920.003 & 4.93 & 4.94 & -0.01\ NaCl^-^ & 0.7270.010 & 0.79 & 0.78 & 0.01 Using eq. (\[eq:fitgammaIP\]) to calculate $\gamma_\text{opt}$ for the individual molecules leads to $\gamma_\text{opt} = \unit[0.38]{a_0^{-1}}$ for Na~2~ and $\gamma_\text{opt} = \unit[0.40]{a_0^{-1}}$ for NaCl^-^. In order to obtain the optimal range separation parameter for the combined system, the Na~2~–NaCl complex, we use minimization of the function [@stein_reliable_2009] $$\begin{aligned} \label{eq:comp1} J(\gamma) = &\sum_{i=D^0,A^-} \Big| \epsilon_\text{HOMO}^{i} (\gamma) + \text{IP}_i (N, \gamma)\Big| \end{aligned}$$ with $\text{IP}_i (N, \gamma) = E_\text{gs}^i (N_i-1, \gamma) - E_\text{gs}^i (N_i, \gamma)$, where $D^0$ denotes the neutral donor, $A^-$ the acceptor anion and $N$ the number of electrons. The two molecules were considered separately where the experimental geometries of the neutral molecules from ref. were used. This treatment results in a value of $\gamma_\text{opt} = \unit[0.39]{a_0^{-1}}$, which leads to the energies in table \[tab:FitNa2NaCl\] that exhibit good agreement to experiment. The eigenvalue of the NaCl LUMO ($\epsilon_\text{LUMO} =\unit[-0.57]{eV}$) differs from the eigenvalue of the NaCl$^-$ HOMO which equals the NaCl EA through (\[eq:comp1\]) by $\approx \unit[0.2]{eV}$. This effect is known and can be attributed to the derivative discontinuity[@stein_fundamental_2010; @kronik_excitation_2012]. ![Linear response TDDFT excitations of the Na~2~–NaCl-system depending on the inverse distance between the two molecules $\nicefrac{1}{R}$. The colors indicate localization of the unoccupied states involved in the excitations on Na~2~ (yellow) or NaCl (purple). Solid line: excitation energy after eq. (\[eq:MullCTE\]). []{data-label="fig:CTE_Na2NaCl_lr"}](graphics/figure_01.pdf) In order to study CT excitations, the molecules were placed with their axes parallel to each other. We first consider the singlet excited state spectrum of the donor acceptor pair calculated by linear response TDDFT depending on the molecular separation as depicted in fig. \[fig:CTE\_Na2NaCl\_lr\]. The separation $R$ is given by the separation of the two parallel molecular axes. The excitations are colored by the weights of the involved unoccupied orbitals on the individual molecules. Only excitations with more than $\unit[50]{\%}$ contribution from the Na$_2$ HOMO and either an oscillator strength $\ge 10^{-2}$ or $\unit[90]{\%}$ weight on NaCl are considered. The excitation spectrum shows a clear CTE where the involved unoccupied states are clearly located on NaCl and its energy follows the expectation of Mullikens law eq. (\[eq:MullCTE\]) for larger separations ($1/R\le 0.08~a_0^{-1}$) as expected. There is a small constant deviation from the expectations of eq. (\[eq:MullCTE\]) that arises from the difference between the eigenvalue of the NaCl LUMO and the NaCl EA \[entering in eq. (\[eq:MullCTE\])\]. This is validated by the NaCl point at $1/R=0$ which is placed at the energy of IP$_D$ + $\epsilon_\text{LUMO, NaCl}$. There is more interaction between the two molecules for smaller distances. Therefore the excited states start to mix and their nature is harder to identify. The excitation energies exhibit a noticeable blur even for large $R$, where little interaction between the molecules would be expected. This is particularly visible in excitations localized on the Na~2~ molecule, which exhibit a large degree of delocalization, viewable by the coloring of excitations around $\unit[3]{eV}$. ![Density of electronic states (DOS) of the Na~2~–NaCl-system at distance $\unit[8]{\text{\AA}}$ for the canonical Fock operator (FO) and for improved virtual orbitals (IVO). The vacuum level $\epsilon_\text{vac}$ is indicated and used as energy reference. The Fermi level is set to $\epsilon_\text{F} = \frac{\epsilon_\text{HOMO} + \epsilon_\text{LUMO}}{2}$ (all states below $\epsilon_\text{F}$ are occupied, all states above are unoccupied). []{data-label="fig:DOS"}](graphics/figure_02.pdf) This numerical noise can be attributed to the fact that already the third unoccupied state in the Na$_2$ ground-state calculation is above the vacuum level and thus this state and all higher ones are influenced by eigenstates of the simulation box (fig. \[fig:DOS\]). This hinders the convergence of these excitations in the number of unoccupied states involved as shown below. In HFT the Coulomb- and exchange interaction cancel for an occupied state with itself, such that these states are subject to the interaction with $N-1$ electrons in an $N$ electron system. This cancellation is absent for unoccupied (virtual) states making them subject to the interaction with all $N$ electrons. Unoccupied states therefore “see” a neutralized core in a neutral system and thus approximate (exited) EAs in HFT and not excitations of the neutral system[@koopmans_uber_1934; @baerends_kohnsham_2013]. As many neutral closed shell molecules exhibit a negative EA [@baerends_kohnsham_2013], the eigenvalues of unoccupied states in HFT type calculations like RSF become positive, i.e. they reside in the continuum (see fig. \[fig:DOS\]). In calculations utilizing basis-sets, these states are stabilized by the finite size of the basis-set[@rosch_comment_1997; @tozer_computation_2005; @baerends_kohnsham_2013] and their properties thus get strongly basis set dependent. Similarly, these states strongly couple to eigenstates of the simulation box in grid-based calculations used here. Therefore RSF DFT calculations become numerically very demanding on grids due to numerical instabilities[@wurdemann_berechnung_2016]. Improved virtual orbitals ========================= Improved virtual orbitals are a remedy for the difficulties mentioned above. The canonical Hartree-Fock operator is not unique in case of unoccupied orbitals in HFT. These states do not contribute to the Slater determinant built from occupied states exclusively, any set of orbitals orthogonal to the occupied states could serve as valid set of unoccupied states. Following the work of @kelly_correlation_1963[@kelly_correlation_1963; @kelly_many-body_1964], @huzinaga_virtual_1970 devised a scheme to use this freedom to better approximate excitations in HFT[@huzinaga_virtual_1970; @huzinaga_virtual_1971]. In this scheme Coulomb and exchange interactions between a virtual hole in an initial occupied orbital $k$ and the virtual orbitals are described by a modified Fock-Operator [@huzinaga_virtual_1970] $$\begin{aligned} \hat{F}^\text{IVO}_i &= \hat{F}_i + \hat{V} \\ \hat{V} &= \left( 1 - \hat{P} \right) \label{eq:HuzProV} \hat{\Omega}_k\left( 1 - \hat{P} \right) \\ \hat {P} &= \sum_i^N \left | \psi_i \rangle \langle \psi_i \right |. \label{eq:HuzPro}\end{aligned}$$ $\hat{F}_i$ denotes the canonical Fock-Operator and $\psi_i$ the non-interacting single particle Hartree-Fock or, in the case of the present study, KS orbitals. $\hat{P}$ separates the space of the unoccupied orbitals from the occupied ones, circumventing slight changes in the eigenstates of the occupied states which occur otherwise[@kelly_many-body_1964; @huzinaga_virtual_1970]. The rotation operator $\hat{\Omega}_k$ can be chosen arbitrarily as long as it is Hermitian[@huzinaga_virtual_1970]. @huzinaga_virtual_1970 suggested to use[@huzinaga_virtual_1971] $$\begin{aligned} \hat{\Omega}_{k} &= - \hat{J}_k + \hat{K}_k \pm \hat{K}_k \label{eq:Huzi_Omega}\end{aligned}$$ for closed shell systems (as is the case discussed below) following the work of @hunt_excited_1969[@hunt_excited_1969]. $\hat{J}_k$ denotes the Coulomb-, $\hat{K}_k$ the exchange operator and $k$ the band-index of the orbital to excite from. The second exchange term can be used to approximate either singlet (“+”, $\hat{\Omega}_k^\text{S}$) and triplet (“-”, $\hat{\Omega}_k^\text{T}$) excitations, or can be omitted ($\hat{\Omega}_k^\text{A}$) to approximate their average. The initial orbital $k$ can be chosen arbitrarily [@huzinaga_virtual_1970; @huzinaga_virtual_1971] and determines the nature of the excitations to be described. Virtual orbitals subject to this scheme are called improved virtual orbitals (IVO) [@huzinaga_virtual_1970; @huzinaga_virtual_1971]. The IVO scheme (\[eq:HuzPro\]) can also be applied within RSF setting $\hat{K}_k = \hat{K}_k^\text{RSF}$ as in eq. (\[eq:RSF\_EXX\_om\]) and $\hat{J}_k = \hat{J}_k^\text{RSF}$ as the corresponding screened Coulomb counterpart. We have disregarded the orthogonalization through operator $\hat{P}$ due to numerical instabilities and $\hat{\Omega}_k$ was applied to the unoccupied states only. The matrix elements in the exact exchange part of the Hamiltonian mixing occupied and unoccupied states were set to zero consistently. We have verified that both approaches lead to virtually identical eigenvalues (see SI). As we will investigate the possibility of the combination of RSF and IVOs for the prediction of excitation energies by means of ground-state calculations we have used the singlet form: $\hat{\Omega}_k^\text{S} = - \hat{J}_k + 2 \hat{K}_k$, where $k$ is the quantum number of the HOMO located on Na$_2$. The resulting density of states is depicted in fig. \[fig:DOS\]. While the occupied state energies of the canonical Fock operator and the IVO operator agree by definition, the IVO operator leads to many more states below the vacuum level $\epsilon_\text{vac}$. There are infinitely many Rydberg states below $\epsilon_\text{vac}$ in principle, but these do not appear due to the finite box size in our calculations. It was shown, that lrTDHFT excitation energies in the Tamm-Dancoff approximation and IVO Eigenenergies agree in a two-orbital two electron model[@casida_progress_2012]. Therefore the IVOs can be expected to represent a good basis for lrTDDFT calculations. Their use involves slight modifications of the usual formalism in the formulation of lrTDDFT as generalized eigenvalue problem[@dreuw_long-range_2003; @casida_time-dependent_2009; @akinaga_intramolecular_2009] $$\begin{aligned} \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{B^\ast} & \mathbf{A^\ast} \end{pmatrix} \begin{pmatrix} \vec{X} \\ \vec{Y} \end{pmatrix} &= \omega \begin{pmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \end{pmatrix} \begin{pmatrix} \vec{X} \\ \vec{Y} \end{pmatrix},\end{aligned}$$ where $\vec{X}/\vec{Y}$ denote the excitations/de-excitations, and $\mathbf{1}$ the unity-matrix. The matrix elements of $\mathbf{A}$ and $\mathbf{B}$ are given by $$\begin{aligned} A_{ia\sigma,jb\tau} &= \delta_{ij}\delta_{ab} \delta_{\sigma\tau} \left( \epsilon_a - \epsilon_i \right) + K_{ia\sigma,jb\tau} \label{eq:lrA} \\ B_{ia\sigma,jb\tau} &= K_{ia\sigma,bj\tau}\label{eq:lrB}\end{aligned}$$ with the $K_{ia\sigma,jb\tau}$ written in the most general form (see SI) $$\begin{aligned} K_{ia\sigma,jb\tau} &= \left( i_\sigma a_\sigma|\nicefrac{1}{r_{12}}|j_\tau b_\tau \right) \notag \\ &\quad + \left( 1 - \alpha - \beta \right) \left(i_\sigma a_\sigma |f_\text{xc}|j_\tau b_\tau\right) \notag \\ & \quad + \beta \left(i_\sigma a_\sigma |f_\text{xc}^\text{RSF}|j_\tau b_\tau\right) \notag \\ & \quad - \delta_{\sigma\tau} \left(i j\left |\frac{\alpha + \beta \left ( 1 - \omega_\text{RSF}\right) }{r_{12}}\right |a b\right) \; .\label{eq:lrK}\end{aligned}$$ Mulliken notation $$\begin{aligned} \left( ab | \hat{x} | cd \right) =&\notag \\ \int \int a^\ast(\vec{r_1}) b(\vec{r_1})& \hat{x} (\vec{r_1}, \vec{r_2}, ...) c^\ast(\vec{r_2}) d(\vec{r_2}) \text{d}\vec{r_1} \text{d}\vec{r_2}\notag \end{aligned}$$ was used and $f_\text{xc} = \frac{\delta^2E_\text{xc}}{\delta\varrho\left(\vec{r_1}\right)\delta\varrho\left(\vec{r_2}\right)}$ is exchange-correlation kernel of the local functional and $f_\text{xc}^\text{RSF}$ the damped exchange-correlation kernel derived from $E_\text{x}^\text{GGA} (\gamma) + E_\text{c}^\text{GGA} = E_\text{xc}^\text{GGA} (\gamma)$. Occupied orbital indices are denoted by $i$ and $j$, unoccupied orbitals by $a$ and $b$, and $\sigma$ and $\tau$ are the spin-indices, while $\alpha$ and $\beta$ are the mixing parameters from the CAM scheme eq. (\[eq:CAM\]). The use of IVOs requires to modify the matrix $\mathbf{A}$ to[@berman_fast_1979] $$\begin{aligned} A_{ia\sigma,jb\tau}^\text{IVO} &= A_{ia\sigma,jb\tau} \notag \\ & \quad + \delta_{ab} \delta_{\sigma\tau}\left[ \left(aa|kk\right) - \left(ak|ka\right) \right . \notag \\ & \left .\quad \quad \mp \left(ak|ka\right)\right] \notag\\ &= A_{ia\sigma,jb\tau} - \delta_{ab} \delta_{\sigma\tau} \left\langle a \left| \hat{\Omega}_k \right| a \right\rangle\label{eq:LRIVO} \end{aligned}$$ where $k$ denotes the excitation orbital and $\hat{\Omega}_k$ is defined in (\[eq:Huzi\_Omega\]). The matrix $\mathbf{B}$ remains the same. ![ Convergence of the two lowest singlet excitation energies of the isolated NaCl molecule using the canonical Fock operator (FO) and improved virtual orbitals (IVO). []{data-label="fig:nacl_tddft_j"}](graphics/figure_03.pdf) The use of IVO indeed facilitates the calculation of excitations using lrTDDFT as shown in fig. \[fig:nacl\_tddft\_j\] for the two first singlet excitations of the isolated NaCl molecule. These are excitations mainly from the degenerated HOMO and HOMO-1 to the LUMO and converge rapidly in the IVO basis. In contrast, more than 200 unoccupied orbitals of the canonical HF operator are needed to arrive at converged energies, which shows that this basis is not very appropriate for the description of excitations. It is known, that excitations calculated by linear response time dependent HFT need a large linear combination of single particle excitations (i.e. a high number of unoccupied states), while excitations calculated by lrTDDFT using the kernels of local functionals can often be described by an individual excitation[@vanMeerPhysicalMeaningVirtual2014; @vanMeerNaturalexcitationorbitals2017]. The twelve unoccupied states that are used in the lrTDDFT calculations for the Na$_2$–NaCl-system presented in fig. \[fig:CTE\_Na2NaCl\_lr\] utilizing the canonical FO are therefore far from converged for the neutral excitations. This is seen by the CTE state and the first excitation of Na$_2$ in fig. \[fig:CTE\_Na2NaCl\_lr\], which are rather clear as these only involve states with negative eigenvalues which do not couple with the eigenstates of the simulation box. ![ Linear response TDDFT excitations of the Na~2~–NaCl-system depending on the inverse distance between the two molecules $\nicefrac{1}{R}$ as in fig. \[fig:CTE\_Na2NaCl\_lr\] using IVOs with $\hat{\Omega}_k^S$ and $k$ was chosen as the HOMO of the Na$_2$ molecule. Besides the use of the IVOs, everything like fig. \[fig:CTE\_Na2NaCl\_lr\]. []{data-label="fig:CTE_Na2NaCl"}](graphics/figure_04.pdf) The IVOs provide a better basis for the calculation of excitations as can be seen in the lrTDDFT spectrum for the Na$_2$–NaCl system utilizing IVOs depicted in fig. \[fig:CTE\_Na2NaCl\]. The depicted states are selected and colored as in fig. \[fig:CTE\_Na2NaCl\_lr\]. The “hole” is in the HOMO of the Na$_2$ molecule. The spectrum gets much clearer than in fig. \[fig:CTE\_Na2NaCl\_lr\] and the energy of the second excitation on Na$_2$ is clearly lower and it is strongly localized on Na$_2$ due to better convergence. Again, the calculated CTE energies are in perfect agreement to the behavior predicted by Mullikens law eq. (\[eq:MullCTE\]), except the offset due to the difference between the eigenvalue of the NaCl LUMO and its calculated EA. A second CTE state approximately $\unit[1]{eV}$ above the first can be identified. which is not the case in fig. \[fig:CTE\_Na2NaCl\_lr\]. This is an effect of the stronger stabilization due to the artificial hole. This suggests that RSFs might be also used to calculate the energetics of CTEs by means of ground-state calculations within the IVO formalism. This conjecture can be further rationalized in particular for CTEs, where HOMO and LUMO are spatially strongly separated such that most of the weight in eq. (\[eq:CAM\]) resides on the exchange integrals from HFT. By adding $\hat{V}$ from eq. (\[eq:HuzProV\]) to the Kohn-Sham-Hamiltonian and taking the HOMO as the hole-state $k$, the eigenvalue of the LUMO in a RSF becomes $\epsilon_\text{LUMO} \approx -\text{EA}_A^\text{calc} - \hat{J}_k$, where it was used that the HOMO and LUMO orbitals do not overlap which leads to vanishing exchange ($\hat{K}_k \to 0$). With $\epsilon_\text{HOMO}=-\text{IP}_D^\text{calc}$ the energetic difference between HOMO and LUMO results in $$\begin{aligned} \epsilon_\text{LUMO} - \epsilon_\text{HOMO} &\approx \text{IP}_D^\text{calc} -\text{EA}_A^\text{calc} - \frac{1}{R},\end{aligned}$$ which is equal to the desired result of eq. (\[eq:MullCTE\]). ![ Eigenvalues of the Na~2~–NaCl-system utilizing IVOs with $\hat{\Omega}_k^S$, where $k$ is the HOMO of the system located on Na$_2$. The Eigenvalues are given relative to the HOMO for $R\to\infty$ (dashed line) and their dependence on the inverse distance between the two molecules $\nicefrac{1}{R}$ is shown. The states are colored by their weight on the individual molecules. Solid line: eigenvalue of the LUMO of the NaCl molecule according to Mullikens law. The stars mark the CTE energies from TDDFT utilizing IVOs (fig. \[fig:CTE\_Na2NaCl\]). []{data-label="fig:CTE_Na2NaCl_IVO"}](graphics/figure_05.pdf) The possibility to calculate CTE by the combination of RSF and IVO by means of ground-state calculations is confirmed by fig. \[fig:CTE\_Na2NaCl\_IVO\]. It shows the eigenvalues of the HOMO as well as the eigenvalues of the unoccupied states of the Na$_{2}$–NaCl–System calculated by the utilization of the IVOs with $\hat{\Omega}_k^\text{S}$ as rotation operator in dependence of the intermolecular distance $R$ similar to figs. \[fig:CTE\_Na2NaCl\_lr\] and \[fig:CTE\_Na2NaCl\]. In the asymptote $R\to\infty$ ($\frac{1}{R} \to 0$) the eigenvalues from the eigenstates of the isolated Na~2~ molecule, which were subject to the modified Fock operator $\hat{F}^\text{IVO}$ with $\hat{\Omega}_k^\text{S}$, are shown. The eigenstates were colored by their weight on the individual molecules, where the projected local density of states was used. Similar to figs. \[fig:CTE\_Na2NaCl\_lr\] and \[fig:CTE\_Na2NaCl\] only states with an oscillator strength $\ge 10^{-2}$ for excitations from the Na$_2$-HOMO or $\unit[90]{\%}$ weight on NaCl are considered. For $R > \approx \unit[10]{\text{\AA}}$ ($\frac{1}{R} \le \unit[0.05]{a_0^{-1}}$) the eigenvalue of the LUMO located on the NaCl molecule, which corresponds to the CTE state, is clearly visible and follows the straight line defined by Mullikens law with a constant slight offset. In this region the calculated eigenvalues of the LUMO are in perfect agreement with the energies predicted by lrTDDFT utilizing IVOs. The offset to Mullikens law is based on the difference between the eigenvalue of the NaCl LUMO and its calculated EA, see above. As in fig. \[fig:CTE\_Na2NaCl\] a second CTE state approximately one $\unit{eV}$ above the first CTE state can be identified. While RSF open a way to calculate the energetics and oscillator strengths of Rydberg- and charge transfer excitations by lrTDDFT, the combination of RSF with IVOs opens a way to calculate the energetics of these excitations by means of ground-state calculations. Generalized lrTDDFT calculations using empty orbitals of the canonical FO need a much larger number of unoccupied states and thus are very hard to converge. This problem can be circumvented by utilizing IVOs that provide usable energies based on a rather small basis. But for lrTDDFT one needs to calculate Coulomb and exchange for every pair of excitation and de-excitation. Therefore lrTDDFT need a lot of computational resources compared to ground-state calculations. Thus utilizing the combination of RSF and IVOs one can calculate the energetics of CTEs on a light computational footprint. Conclusions =========== In this article the implementation of RSFs utilizing the Slater function using the method of PAW on real space grids was presented. The screened Poisson equation was used to calculate the RSF exchange integrals on Cartesian grids, while integrations on radial grids were performed using the integration kernel devised by @rico_repulsion_2012. The implementation was verified against literature, where excellent agreement has been found. Slight differences had been traced to the use of non-converged basis sets used in the literature, which underlines the importance of the basis set choice. Comparison between calculated and experimental mean ligand removal energies unveiled a poor description of binding situations including $d$-electrons by RSF and hybrid functionals that might be attributed of the poor treatment of $d$-binding in HFT. lrTDDFT within the generalized Kohn-Sham scheme using canonical unoccupied Fock-orbitals is hard to converge as the empty states approximate (excited) electron affinities. These orbitals are thus inappropriate to describe neutral excitations. As a remedy, we combined Huzinaga’s IVOs with RSF and extended the linear response TDDFT coupling matrix accordingly. The RSF IVO orbitals are a superior basis for the calculation of excitations with much better convergence properties than these of the canonical FO. These orbitals and their energies not only improve the calculation of lrTDDFT. Due to their construction, they also open a way to calculate the energetics of CTEs by means of ground-state calculations. The much smaller numerical footprint of a ground state calculation as compared to lrTDDFT might enable calculation of these energies for systems unreachable by lrTDDFT. Acknowledgements {#acknowledgements .unnumbered} ================ R. W. acknowledges funding by the Freiburger Materialforschungszentrum and also thanks Miguel Marques from the *libxc* project for a bug fix and further improvements on the code implemented in *libxc*. Computational resources of FZ-Jülich[@krause_jureca:_2016] (project HFR08) are thankfully acknowledged and the authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG. Supporting information available {#supporting-information-available .unnumbered} ================================ The evaluation of the local terms for the screened exchange, the influence of the grid-spacing and the amount of vacuum around each atom on the eigenvalues and derived energies, as well as the analytic expressions for the first, second and third derivative of the exchange term for an RSF with the use of the Slater function, the analytic solution of the screened Poisson equation for a Gaussian shaped density along with it’s derivation, the effects of dropping the projection operator $\hat{P}$ on the IVOs, excitation energies for the disodium molecule calculated by lrTDDFT utilizing RSF and IVOs with the three different forms of $\Omega_k$ and the rationale for the exchange terms in lrTDDFT are given in the supporting information.

--- abstract: | We compute the Kauffman bracket polynomial of *the three-lead Turk’s head*, *the chain sinnet* and *the figure-eight chain* shadow diagrams. Each of these knots can in fact be constructed by repeatedly concatenating the same $ 3 $-tangle, respectively, then taking the closure. The bracket is then evaluated by expressing the state diagrams of the concerned $ 3 $-tangle by means of the Kauffman monoid diagram’s elements. [Keywords:]{} bracket polynomial, tangle shadow, Kauffman state, flat sinnet. author: - | Franck Ramaharo\ Département de Mathématiques et Informatique\ Université d’Antananarivo\ 101 Antananarivo, Madagascar\ <franck.ramaharo@gmail.com>\ date: '\' title: '**A one-variable bracket polynomial for some Turk’s head knots**' --- Introduction ============ The present paper is a follow-up on our previous work which aims at collecting statistics on knot shadows [@Ramaharo]. We would like to establish the bracket polynomial for knot diagram generated by the $ 3 $-tangle shadows below: $$\label{eq:braid} \begin{array}{cccccccc} \protect\includegraphics[width=0.075\linewidth,valign=c]{genB1}&\qquad& \protect\includegraphics[width=0.075\linewidth,valign=c]{genB2}&\qquad& \protect\includegraphics[width=0.075\linewidth,valign=c]{genB3}\\ T &\qquad& C &\qquad& E \end{array}.$$ The knot diagrams under consideration are those obtained by repeatedly multiplying (or concatenating) the same $ 3 $-tangle, then taking the closure of the resulting $ 3 $-tangle (i.e., connecting the endpoints in a standard way, without introducing further crossings between the strands). Knots obtained from the $ 3 $-tangles pictured in belong to the Ashley’s *Turk’s head* family [@ABOK p. 226, Chap. 17]: the *three-lead Turk’s head* [@ABOK \#1305], the *chain sinnet* [@ABOK \#1374] and the *figure-eight chain* [@ABOK \#1376], respectively (e.g. see ). $\begin{array}{cccccccc} \protect\includegraphics[width=0.225\linewidth,valign=c]{braid1}&\quad& \protect\includegraphics[width=0.225\linewidth,valign=c]{braid2}&\quad& \protect\includegraphics[width=0.225\linewidth,valign=c]{braid3}\\ \mbox{Three-lead Turk's head} &\quad& \mbox{Chain sinnet} &\quad& \mbox{Figure-eight chain} \end{array}$ The remainder of this paper is arranged as follows. In , we establish the expression of the bracket polynomial for any $ 3 $-tangle shadow diagram. Then in , we apply those results to the flat sinnet Turk’s heads mentioned earlier. The Kauffman bracket of a 3-tangle shadow {#sec:bracket} ========================================= In this paper, the Kauffman bracket maps a shadow diagram $ D $ to $ \left< D\right>\in$ $\mathbb{Z}[x] $ and is constructed from the following rules: - $ \left<\bigcirc \right>=x $;\[it:i\] - $ \left<\bigcirc\sqcup D\right>=x\left< D\right>$;\[it:ii\] - $\left<\protect\includegraphics[width=.03\linewidth,valign=c]{crossing}\right>=\left<\protect\includegraphics[width=.03\linewidth,valign=c]{split_2}\right>+\left<\protect\includegraphics[width=.035\linewidth,valign=c]{split_1}\right>$. \[it:iii\] The diagram $ \bigcirc $ in $ (\mathbf{K1}) $ represents that of a single loop, and the symbol $ \sqcup $ in $ (\mathbf{K2}) $ denotes the disjoint union operation. Formula in $ (\mathbf{K3}) $ expresses the splitting of a crossing. Recall that the choice of such splittings for any single crossing is referred to as the so-called *Kauffman state*. Rules $ (\mathbf{K1}) $, $ (\mathbf{K2}) $ and $ (\mathbf{K3}) $ can be summarized by the summation which is taken over all the states for $ D $, namely $ \left<D\right>=\sum_{S}^{}x^{|S|} $ , where $ |S| $ gives the number of loops in the state $ S $. Kauffman shows that the states elements of a $ 3$-tangle diagram $ B :=\protect\includegraphics[width=0.05\linewidth,valign=c]{3tangleB}$ are generated by the product of a loop and the following $ 5 $ elements of the *$ 3 $-strand diagram monoid* $ \mathcal{D}_3 $ [@Kauffman1; @Stoimenow]: $$\begin{array}{ccccccccc} \protect\includegraphics[width=0.07\linewidth,valign=c]{1_3}&\qquad& \protect\includegraphics[width=0.075\linewidth,valign=c]{U_1}&\qquad& \protect\includegraphics[width=0.075\linewidth,valign=c]{U_2}&\qquad& \protect\includegraphics[width=0.075\linewidth,valign=c]{r}&\qquad& \protect\includegraphics[width=0.075\linewidth,valign=c]{s} \\ 1_3 &\qquad& U_1 &\qquad& U_2 &\qquad& r &\qquad& s \end{array}.$$ In other words, given a state $ S $, there exist a nonnegative integer $ k $ and an element $ U $ in $ \mathcal{D}_3 $ such that one writes $ S=\bigcirc^k\sqcup U $, where $ \bigcirc^k=\bigcirc\sqcup\bigcirc\sqcup\cdots\sqcup\bigcirc $ denotes the disjoint union of $ k $ loops [@Kauffman2 p. 100]. The bracket of the $ 3$-tangle $ B$ becomes $ \left<B\right>=\sum_{S}^{}\left<S\right>$, where $ \left<S\right>=x^{|S|}\left<U\right> $ for certain $U\in\mathcal{D}_3 $. Therefore $ \left<B\right> $ is a linear combination of the brackets $ \left<1_3\right> $, $\left<U_1\right> $, $\left<U_2\right> $,$\left<r\right> $ and $\left<s\right> $, i.e., there exist five polynomials $ a,b,c,d,e $ in $ \mathbb{Z}[x] $ such that $$\label{eq:bracket} \left<B\right>=a\left<1_3\right>+b\left<U_1\right>+c\left<U_2\right>+d\left<r\right>+e\left<s\right>.$$ \[lem:product\] Given two $ 3 $-tangles $ B$ and $ D $, we have $$\begin{aligned} \left<BD\right>&=a_Ba_D\left<1_3\right>+\left(b_Ba_D+\left(a_B+b_B x+d_B\right)b_D+\left(d_B x+b_B\right)e_D\right)\left<U_1\right>\\ &\hphantom{=}+\left(c_Ba_D+\left(a_B+c_B x+e_B\right)c_D+\left(c_B+e_B x\right)d_D\right)\left<U_2\right>\\ &\hphantom{=}+\left(d_Ba_D+\left(d_B x+b_B\right)c_D+\left(a_B+b_B x+d_B\right)d_D\right)\left<r\right>\\ &\hphantom{=}+\left(e_Ba_D+\left(c_B+e_B x\right)b_D+\left(a_B+c_B x+e_B\right)e_D\right)\left<s\right>.\end{aligned}$$. We first establish the states of $ B $ leaving $ D $ intact, and then in $ D $: $$\begin{aligned} \left<BD\right>&= a_B a_D \left<1_3^2\right>+ a_B b_D \left<1_3 U_1\right> + a_B c_D \left<1_3 U_2\right>+ a_B D \left<1_3 r\right>+ a_B e_D \left<1_3 s\right>\\ &\hphantom{=}+ b_B a_D \left<U_1 1_3\right>+ b_B b_D \left<U_1^2\right>+ b_B c_D \left<U_1 U_2\right>+ b_B d_D \left<U_1 r\right>+ b_B e_D \left<U_1 s\right>\\ &\hphantom{=}+ c_B a_D \left<U_2 1_3\right>+c_B b_D \left<U_2 U_1\right>+ c_B c_D \left<U_2^2\right>+ c_B d_D \left<U_2 r\right>+ c_B e_D \left<U_2 s\right>\\ &\hphantom{=}+ d_B a_D \left<r 1_3\right>+ d_B b_D \left<r U_1\right>+ d_B c_D \left<r U_2\right>+ d_B d_D \left<r^2\right>+ d_B e_D \left<r s\right>\\ &\hphantom{=}+ e_B a_D \left<s 1_3\right>+ e_B b_D \left<s U_1\right>+ e_B c_D \left<s U_2\right>+ e_B d_D \left<sr\right>+ e_B e_D \left<s^2\right>.\end{aligned}$$ The brackets for the pairs in the right-hand side can be evaluated by applying the following multiplication table. $.$ $1_3$ $U_{1}$ $U_{2}$ $r$ $s$ --------- --------- ------------------------ ------------------------ ------------------------- ------------------------ $1_3$ $1_3$ $U_{1}$ $U_{2}$ $r$ $s$ $U_{1}$ $U_{1}$ $\bigcirc\sqcup U_{1}$ $s$ $U_{1}$ $\bigcirc\sqcup s$ $U_{2}$ $U_{2}$ $r$ $\bigcirc\sqcup U_{2}$ $\bigcirc \sqcup r$ $U_{2}$ $r$ $r$ $\bigcirc\sqcup r$ $U_{2}$ $r$ $\bigcirc\sqcup U_{2}$ $s$ $s$ $U_{1}$ $\bigcirc \sqcup s$ $\bigcirc \sqcup U_{1}$ $s$ : Multiplication of elements in $ \mathcal{D}_3 $.[]{data-label="tab:product"} The proof is then completed by factoring with respect to the resulting brackets, eventually simplified according to $ (\mathbf{K2}) $. Let $ B_n:=BB\cdots B $ denote the $ 3 $-tangle obtained by multiplying the $ 3 $-tangle $ B $ $ n$ times, with $ B_0:=1_3 $. For convenience, we shall identify the bracket formal expression in by the $ 5 $-tuple $ [a,b,c,d,e]^T$. Similarly, assume that $\left<B_n\right>$ is identified by $ [a_n,b_n,c_n,d_n,e_n]^T $. The bracket $ 5 $-tuple for $ B_n $ is given by $$\label{eq:matrixdefinition} \begin{bmatrix} a_n\\b_n\\c_n\\d_n\\e_n \end{bmatrix}=\begin{bmatrix}a & 0 & 0 & 0 & 0\\ b & a+b x+d & 0 & 0 & d x+b\\ c & 0 & a+c x+e& c+e x & 0\\ d & 0 & d x+b & a+b x+d & 0\\ e & c+e x & 0 & 0 & a+c x+e\end{bmatrix}^n\begin{bmatrix} 1\\0\\0\\0\\0 \end{bmatrix}.$$ \[lem:lem1\] We write $ B_{n+1}=BB_n $, then from we have $$\label{eq:matrix} \begin{bmatrix} a_{n+1}\\b_{n+1}\\c_{n+1}\\d_{n+1}\\e_{n+1} \end{bmatrix}=\begin{bmatrix}a & 0 & 0 & 0 & 0\\ b & a+b x+d & 0 & 0 & d x+b\\ c & 0 & a+c x+e& c+e x & 0\\ d & 0 & d x+b & a+b x+d & 0\\ e & c+e x & 0 & 0 & a+c x+e\end{bmatrix}\begin{bmatrix} a_n\\b_n\\c_n\\d_n\\e_n \end{bmatrix}.$$ We conclude by unfolding the recurrence and taking into consideration the initial condition $ [a_0,b_0,c_0,d_0,e_0]^T=[1,0,0,0,0]^T$. We let $ M_B $ denote the $ 5\times5 $ matrix in , and we will later refer to it as the *states matrix* for the $ 3 $-tangle $ B$. Using the standard method for computing we obtain the characteristic polynomial for $ M_B $ $$\chi\left(M_B,\lambda\right)=-(\lambda-a)\left(\lambda-\dfrac{1}{2}\left(p-q\right)\right)^2\left(\lambda-\dfrac{1}{2}\left(p+q\right)\right)^2,$$ then $$\begin{aligned} a_n &= a^n,\label{eq:an}\\ b_n&=\dfrac{-1}{2q\left(x^2-1\right)}\left(2 a^n q x+\left(\dfrac{p-q}{2}\right)^n \left((b-c) x^2+(-d-e-q) x-2 b\right)\right.\nonumber\\ &\hphantom{=+\dfrac{-1}{2q(x^2-1)}}\left.+\left(\dfrac{p+q}{2}\right)^n \left((-b+c) x^2+(d+e-q) x+2 b\right) \right),\\ c_n&=\dfrac{-1}{2q\left(x^2-1\right)}\left(2 a^n q x+\left(\dfrac{p+q}{2}\right)^n \left((b-c) x^2+(d+e-q) x+2 c\right)\right.\nonumber\\ &\hphantom{=+\dfrac{-1}{2q(x^2-1)}}\left.+\left(\dfrac{p-q}{2}\right)^n \left((-b+c) x^2+(-d-e-q) x-2 c\right)\right),\\ d_n&=\dfrac{1}{2q\left(x^2-1\right)}\left(2 a^n q+\left(\dfrac{p-q}{2}\right)^n \left(-2 d x^2+(-b-c) x+d-e-q\right)\right.\nonumber\\ &\hphantom{=+\dfrac{-1}{2q\left(x^2-1\right)} }\left.+\left(\dfrac{p+q}{2}\right)^n \left(2 d x^2+\left(b+c\right) x-d+e-q\right)\right),\\ e_n&=\dfrac{1}{2q\left(x^2-1\right)}\left(2 a^n q+\left(\dfrac{p-q}{2}\right)^n \left(-2 e x^2+(-b-c) x-d+e-q\right)\right.\nonumber\\ &\hphantom{=+\dfrac{1}{2q\left(x^2-1\right)}}\left.+\left(\dfrac{p+q}{2}\right)^n \left(2 e x^2 +(b+c) x +d-e-q\right)\right),\label{eq:en}\end{aligned}$$ where $$\begin{aligned} p&:=(b+c)x+2 a+d+e\label{eq:p},\\ q&:=\sqrt{\left(b^2 -2 b c +c^2 +4 d e\right)x^2 + \left(2 b d +2 c d +2 b e +2 c e\right) x + 4 b c+d^2-2 d e+e^2}.\label{eq:q}\end{aligned}$$ Now let $ \overline{B_n} $ denote the tangle closure of $ B_n $. In order to evaluate $ \left<\overline{B_n}\right> $ from formula we need to apply the closure to the elements of $ \mathcal{D}_3 $. The expression of the bracket polynomial for the closure $ \overline{B_n} $ is given by $$\label{eq:closure} \left<\overline{B_n}\right>= x^3a_n+x^2\left(b_n+c_n\right)+x\left(d_n+e_n\right).$$ The splitting at each crossing do not conflict with the closing process, hence the only point remaining concerns the evaluation of the brackets to the closure of the elements of $ \mathcal{D}_3 $, namely $$\begin{array}{ccccccccc} \protect\includegraphics[width=0.1\linewidth,valign=c]{1_3_bar}&\qquad& \protect\includegraphics[width=0.1\linewidth,valign=c]{U_1_bar}&\qquad& \protect\includegraphics[width=0.1\linewidth,valign=c]{U_2_bar}&\qquad& \protect\includegraphics[width=0.1\linewidth,valign=c]{r_bar}&\qquad& \protect\includegraphics[width=0.1\linewidth,valign=c]{s_bar} \\ \left<\overline{ 1_3}\right>=x^3 &\qquad& \left<\overline{ U_1}\right>=x^2 &\qquad& \left<\overline{U_2} \right>=x^2 &\qquad& \left<\overline{r}\right>=x &\qquad& \left< \overline{s} \right>=x \end{array}.$$ Next, combining , – and , we obtain a better expression of the bracket: The bracket polynomial for the knot $ \overline{B_n} $ is given by $$\label{eq:bracketpolynomial} \left<\overline{B_n}\right>=xa^n\left(x^2-2\right)+x\left(\left(\dfrac{p-q}{2}\right)^n+\left(\dfrac{p+q}{2}\right)^n\right),$$ where $ p $ and $ q $ are expressions defined in and . Finally, we let $ \overline{B}(x;y):=\sum_{n\geq0}^{}\left<\overline{B_n}\right>y^n $ denote the generating function of $ \big(\left<\overline{B_n}\right>\big)_n $. By we deduce $$\begin{aligned} \overline{B}(x;y)&=\resizebox{.888\linewidth}{!}{$\frac{\left( (b+c)x+2 a+d+e\right) y-2}{\left( \left( d e-b c\right) {{x}^{2}}+\left( -a c-a b\right) x+\left( -d-a\right) e-a d+b c-{{a}^{2}}\right) {{y}^{2}}+\left( (b+c)x+2 a+d+e\right) y-1}$}\\ &\hphantom{=}+\frac{x \left( {{x}^{2}}-2\right) }{1-a y}.\end{aligned}$$ Application {#sec:application} =========== Throughout this section, let us refer to the $ 3 $-tangles in as *generators*. Recall that in the expression $ \left<\overline{B_n}\right> = \sum_{k>0}^{}s_B(n,k)x^k $ we have $ b_{n,k}=\#\{S\mid \textit{$ S $ is a state of $ B_n $ and $ |S|=k $}\} $, with $ B\in\{T,C,E\} $. For each flat sinnet Turk’s head below, we will give the corresponding distribution $ \left(s_B{(n,k)}\right)_{n,k}$ for small values of $ n $ and $ k $. 1. **Three-lead Turk’s head**. Let $ \sum_{k\geq 0}^{}s_T(n,k)x^k:=\left<\overline{T_n}\right> $. - Bracket for the generator $ T $: $$\begin{aligned} \left<\protect\includegraphics[width=0.045\linewidth,valign=c]{genB1}\right>&=\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T12}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T11}\right>\\ &=\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T24}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T23}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T22}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T21}\right>\\ \left<T\right>&=\left<1_3\right>+\left<U_1\right>+\left<U_2\right>+\left<s\right>.\end{aligned}$$ - States matrix: $$M_T=\begin{bmatrix}1 & 0 & 0 & 0 & 0\\ 1 & x+1 & 0 & 0 & 1\\ 1 & 0 & x+2 & x+1 & 0\\ 0 & 0 & 1 & x+1 & 0\\ 1 & x+1 & 0 & 0 & x+2\end{bmatrix}.$$ - Bracket for $ T_n $: $$\left<\overline{T_n}\right>=x\left(x^2-2 \right)+ x\left(\left(\dfrac{2 x+3-\sqrt{4 x+5}}{2}\right)^n+ \left(\dfrac{2x+3+\sqrt{4 x+5}}{2}\right)^n\right).$$ - Generating function: $$\overline{T}(x;y)=\dfrac{x \left( \left( -2 x-3\right) y+2\right) }{\left( {{x}^{2}}+2 x+1\right) {{y}^{2}}+\left( -2 x-3\right) y+1}+\dfrac{x \left( {{x}^{2}}-2\right) }{1-y}.$$ - Distribution of $ \left(s_T{(n,k)}\right)_{n,k}$: [@Sloane [[](http://oeis.org/A316659)]{}] 2. **Chain sinnet**. Let $ \sum_{k\geq 0}^{}s_C(n,k)x^k:=\left<\overline{C_n}\right> $. - Bracket for the generator $ C $: $$\begin{aligned} \label{key} \left<\protect\includegraphics[width=0.045\linewidth,valign=c]{genB2}\right>&=\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C12}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C11}\right>\\ &=\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C24}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C23}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C22}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C21}\right>\\ &=\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C38}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T24}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T24}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T23}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C34}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T22}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T22}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T21}\right>\\ \left<C\right>&=(x+2)\left<1_3\right>+(x+2)\left<U_1\right>+\left<U_2\right>+\left<s\right>.\end{aligned}$$ - States matrix: $$M_C=\begin{bmatrix}x+2 & 0 & 0 & 0 & 0\\ x+2 & {{x}^{2}}+3 x+2 & 0 & 0 & x+2\\ 1 & 0 & 2 x+3 & x+1 & 0\\ 0 & 0 & x+2 & {{x}^{2}}+3 x+2 & 0\\ 1 & x+1 & 0 & 0 & 2 x+3\end{bmatrix}.$$ - Bracket for $ C_n $: $$\begin{aligned} \left<\overline{C_n}\right>&=x\left(x^2-2\right)(x+2)^n+ x\left(\left(\dfrac{x^2+5x+5-\sqrt{{{x}^{4}}+2 {{x}^{3}}+3 {{x}^{2}}+10 x+9}}{2}\right)^n\right.\\ &\hphantom{=}+\left.\left(\dfrac{x^2+5x+5+\sqrt{{{x}^{4}}+2 {{x}^{3}}+3 {{x}^{2}}+10 x+9}}{2}\right)^n\right).\end{aligned}$$ - Generating function $$\overline{C}(x;y)=\frac{x \left( \left( -{{x}^{2}}-5 x-5\right) y+2\right) }{\left( 2 {{x}^{3}}+8 {{x}^{2}}+10 x+4\right) {{y}^{2}}+\left( -{{x}^{2}}-5 x-5\right) y+1}+\frac{x \left( {{x}^{2}}-2\right) }{1-\left( x+2\right) y}.$$ - Distribution of $ \left(s_C(n,k)\right)_{n,k}$: 3. **Figure-eight chain**. Let $ \sum_{k\geq 0}^{}s_E(n,k)x^k:=\left<\overline{E_n}\right> $. - Bracket for the generator $ E $: $$\begin{aligned} \label{key} \left<\protect\includegraphics[width=0.045\linewidth,valign=c]{genB3}\right>&=\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{E12}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{E11}\right>\\ &=\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{E22}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{E21}\right>+\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{genB2}\right>=(x+1)\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{E21}\right>+\left<C\right>\\ &=(x+1)\left(\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{C38}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T24}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T24}\right> +\left<\protect\includegraphics[width=0.045\linewidth,valign=c]{T23}\right>\right)+\left<C\right>\\ \left<E\right>&=\left(x^2+4x+4\right)\left<1_3\right>+(x+2)\left<U_1\right>+(x+2)\left<U_2\right>+\left<s\right>.\end{aligned}$$ - States matrix: $$M_E=\begin{bmatrix}{{x}^{2}}+4 x+4 & 0 & 0 & 0 & 0\\ x+2 & 2 {{x}^{2}}+6 x+4 & 0 & 0 & x+2\\ x+2 & 0 & 2 {{x}^{2}}+6 x+5 & 2 x+2 & 0\\ 0 & 0 & x+2 & 2 {{x}^{2}}+6 x+4 & 0\\ 1 & 2 x+2 & 0 & 0 & 2 {{x}^{2}}+6 x+5\end{bmatrix}.$$ - Bracket for $ \overline{E_n} $: $$\begin{aligned} \left<\overline{E_n}\right>&=x(x^2-2)\left(x^2+4x+4\right)^n+ x\left(\left(\dfrac{4 {{x}^{2}}+12 x+9-\sqrt{8 {{x}^{2}}+24 x+17}}{2}\right)^n\right.\\ &\hphantom{=}+\left. \left(\dfrac{4 {{x}^{2}}+12 x+9+\sqrt{8 {{x}^{2}}+24 x+17}}{2}\right)^n\right).\end{aligned}$$ - Generating function $$\begin{aligned} \overline{E}(x;y)&=\frac{x \left( \left( -4 {{x}^{2}}-12 x-9\right) y+2\right) }{\left( 4 {{x}^{4}}+24 {{x}^{3}}+52 {{x}^{2}}+48 x+16\right) {{y}^{2}}+\left( -4 {{x}^{2}}-12 x-9\right) y+1}\\ &\hphantom{=}+\frac{x \left( {{x}^{2}}-2\right) }{1-{{\left( x^2+4x+4\right) }} y}.\end{aligned}$$ - Distribution of $ \left(s_E(n,k)\right)_{n,k}$: Column $ 1 $ in is sequence [[](http://oeis.org/A004146)]{} in the OEIS [@Sloane], the sequence of alternate Lucas numbers minus $ 2 $, which is the determinant of the Turk’s Head Knots $ THK(3,n) $ [@KST]. Column $ 2 $ is the $ x $-coefficients of a generalized Jaco-Lucas polynomials for even indices [@Sun] (see column $ 1 $ in triangle [[](http://oeis.org/A122076)]{}) and is also a subsequence of a Fibonacci-Lucas convolution [[](http://oeis.org/A099920)]{} for odd indices. Column $ 1 $ in is [[](http://oeis.org/A060867)]{} with a leading $ 0 .$ Rows $ 1 $ in , , match the coefficients of the bracket for the $ 2 $-twist loop (see row $ 1 $ in [[](http://oeis.org/A300184)]{}, [[](http://oeis.org/A300192)]{} and row $ 0 $ in [[](http://oeis.org/A300454)]{}), the $ 3 $-twist loop and the $4 $-twist loop modulo planar isotopy and move on the $ 2 $-sphere [@Ramaharo], respectively (see , and ). Row $ 2 $ in gives those of the figure-eight knot (see and row $ 1 $ in [[](http://oeis.org/A300454)]{}). [99]{} Clifford W. Ashley, [*The Ashley Book of Knots*]{}, New York: Doubleday, 1944. Louis H. Kauffman, An invariant of regular isotopy, [*Trans. Amer. Math. Soc.*]{} [**318**]{} (1990), 417–471. Louis H. Kauffman, [*Knots and Physics*]{}, World Scientific, 1993. Seong Ju Kim, Ryan Stees, and Laura Taalman, Sequences of spiral knot determinants, [*J. Integer Seq.*]{} [**19**]{} (2016), 1–14. Franck Ramaharo, Statistics on some classes of knot shadows, arXiv preprint, <https://arxiv.org/abs/1802.07701v2>, 2018. Neil J. A. Sloane, [*The On-Line Encyclopedia of Integer Sequences*]{}, published electronically at <http://oeis.org>, 2018. Ydong Sun, Numerical triangles and several classical sequences, [*Fib. Quart.*]{} [**43**]{} (2005), 359–370. Alexander Stoimenow, Square numbers, spanning trees and invariants of achiral knots, [*Communications in Analysis and Geometry*]{} [**13**]{} (2005), 591–631. **2010 Mathematics Subject Classifications**: 05A19; 57M25.

ł IFT-UAM/CSIC-00-33\ hep-th/0011105\ [**Non-Critical Poincaré Invariant Bosonic String Backgrounds and Closed String Tachyons**]{} [**Enrique Álvarez**]{} [^1] [**César Gómez**]{} [^2] [**and Lorenzo Hernández** ]{} [^3]\ 0.4cm  [*Departamento de Física Teórica, C-XI, Universidad Autónoma de Madrid\ E-28049-Madrid, Spain\ and\ Instituto de Física Teórica, C-XVI, Universidad Autónoma de Madrid\ E-28049-Madrid, Spain*]{}[^4] [**Abstract**]{} A new family of non critical bosonic string backgrounds in arbitrary space time dimension $D$ and with $ISO(1,D-2)$ Poincaré invariance are presented. The metric warping factor and dilaton agree asymptotically with the linear dilaton background. The closed string tachyon equation of motion enjoys, in the linear approximation, an exact solution of “kink” type interpolating between different expectation values. A renormalization group flow interpretation ,based on a closed string tachyon potential of type $-T^{2}e^{-T}$, is suggested. Introduction and Summary ======================== In order for strings to behave as consistent quantum systems, a mimimal requirement seems to be the absence of anomalies. World sheet conformal invariance at tree level in the string perturbative expansion leads to vanishing sigma model beta functions. Denoting $D$ the dimension of the target space-time the relevant beta functions for the bosonic string are \[beta\] \^[g]{}\_[AB]{}&&=’R\_[AB]{}+ 2 ’\_A\_B -H\_[ACD]{} H\_[B]{}\^[CD]{}+ o(’\^2)\ \^[b]{}\_[AB]{}&&= -\^D H\_[DAB]{}+ ’\^DH\_[DAB]{} + o(’\^2)\ \^&&=-\^2 + ’\_D\^D-H\_[ABC]{}H\^[ABC]{}+ o(’\^2) The simplest solution is of course the critical string $D=26$ in flat Minkowski space time with constant dilaton and vanishing antisymmetric tensor. Another well known solution in $D\neq 26$ is the linear dilaton background [@myers] = q\_X\^ in flat $D$ dimensional Minkowski space-time, with q\_q\^ = The linear dilaton background solution is exact to all orders in $\alpha'$ and defines a good conformal field theory. Moreover for $D$ smaller or equal to two this background is tachyon free in the linear approximation, and is believed to develop a tachyonic barrier that prevents strings to enter in the string strong coupling region. Motivated by the [*curved Liouville*]{} approach of Polyakov [@polyakov] to non critical backgrounds we will present in this letter a new family of solutions to the sigma model beta function equations, at first order in $\alpha '$, for generic values of $D$ and with $ISO(1,D-2)$ Poincare invariance \[metric\] ds\^2= a(r)d\_[1,D-2]{}\^2 + dr\^2 Backgrounds of this type are typical in the holographic context once we identify the coordinate $r$ with the holographic direction or in curved Liouville if we interpret $r$ as the extra Liouville direction. To be specific, we shall find ( see Figure 1) that in $D=5$ spacetime dimensions (corresponding to one holographic plus the ordinary four minkowskian ones), the warp factor is: \[warp\] a(r) = th(r) and the dilaton \[dil\] = Notice from equations (\[warp\]) and (\[dil\]) that this solution in the asymptotic region $r \rightarrow \infty$ coincides with flat Minkowski space-time with the appropiated linear dilaton background. At $r=0$ the warp factor goes to zero inducing a naked singularity. Close to the origin we reproduce the metric of reference [@ag] with a logaritmic behavior for the dilaton. Moreover in the limit $\alpha' \rightarrow 0$ the metric becomes flat Minkowski except in a neighborhood of the origin [^5]. With respect to the problem of the tachyon stability we study the tachyon equation of motion in the linear approximation. The equation of motion becomes Riemann’s equation with three singular points. In the asymptotic region $r \rightarrow \infty$ we find the familiar $c=1$ barrier for linear dilaton background; namely for $D>2$ the tachyon goes to zero oscillating. At the origin the exact solution, preserving energy conservation, goes to a constant. Thus this solution can be temptatively interpreted as a kind of tachyonic kink ( see Fig 2) interpolating - in the sense of a renormalization group flow- the unstable linear dilaton asymptotic regime with vanishing vacuum expectation value of the tachyon and a stable background characterized by a non vanishing tachyon vacuum expectation value. This interpretation could be probably supported by a closed string tachyon potential of type $V(T)= - e^{-T} T^{2}$ as recently suggested in reference [@tseytlin]. In the rest of this paper we will present the technical details of our analysis. A new family of non-critical bosonic string backgrounds ======================================================= For vanishing Kalb-Ramond field the beta function equation $\beta^{g} =0$ for the Poincare invariant metric (\[metric\]) reads: \[beta1\] \_&&=a’’\_\ &&= 2 ” for $d=D-1$. The dilaton solving the first of these equations is given by =  log a’+log a and the general solution for the metric factor would be obtained by integrating the following equation: =c\_1 a\^[1+]{}+c\_2 a\^[1-]{} There are now several possibilities: - $c_1c_2<0$. The solution then reduces to: a(r) = \^ - $c_1c_2>0$. The solution reads : a(r) = \^ - $c_2=0$. The solution is then: a(r)=(r+c\_3)\^[-]{} (This is the dual $r \to \frac{1}{r}$ of the solution presented in [@ag]). - $c_1=0$. This leads to: a(r)=(r+c\_3)\^ (This is the confining background introduced in [@ag]). - $c_1=c_2=0$. This is the linear dilaton solution: a&&=a\_0\ &&=c\_3+c\_4 r with $c_3,c_4$ arbitrary constants. Notice that in this case equations (\[beta1\]) imply $\Phi'' =0$. The most important thing about this family of solutions is that they are non-critical; that is, if we compute the beta function of the dilaton (which is guaranteed to be a constant by virtue of the first two equations and Bianchi identities for the spacetime curvature,) it so happens that c\_1c\_2 = (1 - ) This clearly means that if $c_1c_2<0$, there is a solution in the family for any $D<26$; whereas for $c_1c_2>0$ the solution exists for any $D>26$. It is also worth remarking that as soon as one of the two arbitrary constants vanish, the solution must necessarily become critical (that is, with $D=26$). Let us now briefly describe some properties of the solutions in the physically most interesting case, namely for $D < 26$. The behavior in the asymptotic region $r\rightarrow \infty$ for the metric factor $a$ is a |[a]{}=()\^ whereas the dilaton yields - r where we easily recognize the precise behavior of the linear dilaton solution in flat space-time. In the region $r \rightarrow 0$ we get instead a ( r )\^ and the dilaton goes as ( -1) r Notice that all these solutions are singular at $r=0$. For instance the scalar curvature in d=4 is given by R = [-c\_1 c\_2]{} Given the fact that this is the [*general*]{} solution with our ansatz, this means that the existence of the naked singularity is somewhat embodied in the ansatz itself. A simple generalization of this background (\[metric\]) can be obtained by adding a certain number $d'$ of extra flat dimensions $(D=d+d'+1)$: ds\^2= a(r)d\_[1,d]{}\^2 + dr\^2 + d\_[d’]{} The only change in this case is on the values of the constants $c_1,c_2$ that are now fixed by the relation = Closed String Tachyons ====================== Bosonic string backgrounds suffer from an intrinsic source for instability (which sometimes has been taken as an indication of the existence of another, physically interesting and energetically favoured vacuum [@englert]), namely, a tachyonic scalar excitation in the closed string sector with m\^2 = -. It was first suggested by Polyakov [@polyakov] that in a metric of type (\[metric\]) the tachyon unstability could be tamed leading to a peaceful condensation. In addition we know that in a non trivial space time - AdS is a good example [@bf] - normalizability and energy conservation imposses strong restrictions on the allowed spectrum of masses. Following this last approach the first thing we will do would be to derive from energy conservation and normalizability, the boundary conditions on field configurations. Next we will consider - in the linear approximation - the exact solution to the tachyon equation of motion satisfying these boundary conditions. Finally we will interpret the solution as a kink describing tachyon condensation. Boundary Conditions ------------------- ### Normalization First of all, if the scalar is complex, there is a conserved number current: N\^Ae\^[-2]{}((f\_2)\^[\*]{} \^A f\_1 -f\_1 \^A (f\_2)\^[\*]{}) which gives rise to a conserved particle number Nd\^[d-1]{}x dr e\^[-2]{}(f\^[\*]{} \^0 f -f \^0 f\^[\*]{}) when this integral converges. The $a$-dependent factors in the measure behave as: N\~a\^[(d-1)/2]{} In the generic case, for $c_1c_2<0$ this behaves for $a\sim 0$ as: N\~a\^ On the other hand, when $a$ reached its asymptotic value, $a\rightarrow \bar{a}$, N\~ This last behaviour selects functions that vanish as a power when $a\rightarrow \bar{a}$. That is, lim\_[a|[a]{}]{}f = (|[a]{}-a)\^ where $\delta >\frac{1}{2}$. ### The definition of Conserved Energy Given an arbitrary scalar field with energy-momentum tensor $T_{AB}$, there is a [*Killing energy*]{} current, namely j\^[A]{}T\^[AB]{} k\_[B]{} which is covariantly conserved, i.e. \[div\] \_[A]{}j\^[A]{} = 0 In our case, the Killing corresponding to temporal translations is: k Applying Stokes’theorem to the integral over the $(d+1)$-dimensional region $\mathcal{R}$ defined in terms of a large distance $L$ by $ t_0<t< t_1$,and by $-L<x^i<L$,($i,j,\ldots =1,\ldots d-1$), $0<r<L$, of the $(d+1)$-form proportional to the first member of (\[div\]), i.e. of $d* j$, where $j$ is the one-form dual to the current, $j\equiv j_{A}dx^{A}$, we get 0 = \_ d\*j = \_ \*j = E(t\_2)-E(t\_1) + E where E(t)d\^[(d-1)]{} x dr j\^0(t,x\^i,r) and the flux over the boundary $\pd M$, E = (\_[x\^i =L]{}-\_[x\^i = -L]{})() (\_[i i\_1 …i\_[d-2]{}]{} dtdx\^[i\_1]{}…dx\^[i\_[d-2]{}]{}dr) j\^i + (\_[r=L]{}- \_[r=0]{}) d\^[d]{}x () j\^[d]{}. Only when the physical boundary conditions are such that E =0 there is actual energy conservation, =0 For a scalar field of mass $m$ it is readily found that T\_[AB]{} = e\^[-2]{}\[\_A \_B - g\_[AB]{} (g\^[CD]{}\_C \_D - m\^2 \^2)\] so that the $(d+1)$-dimensional energy is: E\[\]d\^[(d-1)]{}x dr T\^[0]{}\_B k\^B, i.e.,denoting the scalar perturbation by $T$, \[E\] E = dr d\^[d-1]{}x In all cases the important thing is the behaviour of the flux at the singular boundary, which in turn is dominated by the behaviour of $j^d$, when $a\rightarrow 0$, namely \[tacc\] j\^d\~- \_r T= - a\^[1-]{} T’ ### Positive-definiteness Combining the results of the preceding two paragraphs we can read the conditions for the scalar perturbations to be normalizable, as well as enjoy a conserved energy, that is, \[funcional\] f&&\~\_[a=0]{} 0(a\^[-u]{})\ f&&\~\_[a=|[a]{}]{}(|[a]{}-a)\^v with $0<u<1$ and $v>\frac{1}{2}$. If we want moreover the now well defined energy to be finite it is clearly necessary (owing to the $a^{\prime}$ factor in the denominator) that when $r\rightarrow \infty$ \[mu\] T\~e\^[- \^2 r]{} with \[finita\] \^2 &gt; which is much faster that required by (\[funcional\]).We can define another functional space in which energy is not only conserved, but also finite, and this space is essentially characterized by the behavior (\[finita\]). It is then plain that in the conserved-energy functional space defined by (\[funcional\]) there are fluctuations with arbitarily large negative energy, namely all those that at infinity behave as (\[mu\]) with $|\mu|< |m|$. In the more restrictive finite-energy space (\[finita\]), this behavior still persists as long as $d>1$. This is the same bound found for the linear dilaton background. The tachyon equation -------------------- Let us now examine the space of solutions of the linearized equation of motion and check whether on shell there are elements of our functional space of well-defined fluctuations. We shall write the wave equations (following the conventions of [@polchinski]) as: \[wave\] \^2 T - 2 \^A \_A T - m\^2 T = 0 For the Poincaré invariant ansatz the dilaton is given by =log a’ + log a where $a'\equiv\frac{da}{dr}$. Variables can be separated in the wave equation (\[wave\]) by means of: T(x\^A) T(a)(x\^) with the result: \[eqa\] \_d && =\ T” + T’&&= ()T (where we have used $\frac{dT}{dr} = \frac{dT}{da}a'\equiv T' a'$), and $\lambda$ is the separation constant. Now, for a plane wave $\Theta = e^{i kx}\equiv e^{-i(\omega t - \vec{k}\vec{x})}$, the first (Minkowskian) equation reads \^2 -\^2 =Thus, in order to be able to build wave packets of arbitrarily long wavelengths it is necessary that 0 Otherwise, these wave packets will get $Im\, \omega\neq 0$. The strategy is now to study the radial equation of the set (\[eqa\]) and check whether there is an aceptable set of solutions with $\lambda \geqslant 0$. ### Asymptotic behavior Our noncritical solution is generically characterized by a’= c\_1 a\^[1+]{}+ c\_2 a\^[1-]{} and $c_1 c_2 <0$. Specially interesting is the behavior of the preceding equation (\[eqa\]) in the neighborhood of the singularity ($a\rightarrow 0$) or at $r\rightarrow \infty $ ($a\rightarrow 0$). When $r\rightarrow \infty$ the behaviour is universal, (for any $D<26$, i.e. $c_1c_2<0$), the scale factor $a$ remains bounded, and equivalent to the behaviour when $x\sim 0$ (a regular singular point) of: T”- T’= T where $C\equiv \frac{m^2-\frac{\lambda}{\bar{a}}}{- c_2 c_1\, d} $. The behavior of the solution depends critically as to whether $1-d-\lambda > 0$, in which case there are two real powers $x^{\a_{\pm}}$. When the equality is saturated, that is, $d=1-\lambda$ there is a singular solution, $ x^{1/2}log x $ (which however vaniches in the limit $x\rightarrow 0$). When $1-d-\lambda<0$ there is an oscillatory behaviour with decreasing amplitude $\sim x^{1/2}$. In the oposite situation, that $c_1c_2>0$, $a\rightarrow \infty$, and the behaviour of the wave equation is then equivalent to the behaviour when $x\sim 0$ of: +=x\^[-2+]{}T When $a\rightarrow 0$ i.e., a neighborhood of the singularity (and this is now independent of the signs of the constants $c_1c_2$) the solutions of equation (\[eqa\]) behave as: - $d>1$. The behaviour of the solutions is, for $\lambda \neq 0$: T\_1&&\~1 - a\^[-1]{}\ T\_2&&\~ And for $\lambda =0$: T\_1&&\~1 + a\^\ T\_2&&\~ - $d=1$. In this case the solutions are replaced by:\ When $\lambda \neq 0$: T\_ \~a\^\ 2 And for $\lambda=0$: T\_1&&\~1- a\ T\_2&&\~ In summary for $D>2$ all solutions in the region $r \rightarrow \infty$ tend to zero oscillating. At the origin conservation of energy and normalizability force us to rule out the logaritmic behavior but they allow us to have a tachyon tending to a non vanishing constant. Next we will consider the exact solution in the special case of $\lambda = 0$. ### Exact Solution Denoting $q= \frac{\sqrt{d}}{2}$ and changing variables \[var\] a&&=e\^[z]{}\ x&&= th(qz) equation (\[eqa\]) for the tachyon (with $\lambda=0$) becomes (1-x\^[2]{})T” -2xT’ + T =0 for $T' = \frac{dT}{dx}$. This is the Riemann equation [@abramowitz] with three singular points at $x=1,-1,0$. The general solution is T = P( [cccc]{} -1 & 1 & 0 &\ 0 & 0 & & x\ 0 & 0 & 1-& ) =( ) \^ P( [cccc]{} 0 & & 1 &\ 0 & & 0 &\ 0 & & 1-2 & ) with = We have two linearly independent solutions to be denoted $T_{1}$ and $T_{2}$. Around the singular points they behave like: T\^[(1)]{}\_[x=-1]{}=c\_1 ( ) \^ F\[,,1; \] \~cte T\^[(1)]{}\_[x=0]{}=c\_1 ( ) \^ F\[,,2 ; \] \~x\^ and T\^[(2)]{}\_[x=-1]{} = c\_2 T\^[(1)]{}\_[x=-1]{} ( ) + c\_2S(x) \~log(1+x)\ T\^[(2)]{}\_[x=0]{}=c\_2 ( ) \^ ( ) \^[1-2 ]{} F\[1-,1-,-2 ; \] \~x\^[1- ]{}\ Where $S(x)$ is given by: S(x) = \_[n]{}\^( ( ) \^n ( 2(+n) -2() -2(n+1) + 2 (1) ) ) (where $\psi$ is the digamma function). Notice that $S(x) \rightarrow 0$ when $x\rightarrow -1$. The only one compatible with energy conservation at the origin is $T_{1}$ that precisely looks like a kink interpolating between $<T> = 0$ and $<T> = cte$ (see figure 2). This concludes our discussion of the exact tachyon solution. Discussion and Final Comments ============================= Starting with Sen’s conjecture [@sen] on open tachyon condensation, several candidates have recently appeared for the open tachyon potential (cf.[@kutasov], [@kutasov1]), and even for the closed tachyon potential (cf. [@tseytlin]). To be specific, in the open string case tachyon condensation can be understood on the basis of a tachyon potential of type $e^{-T}(1+T)$ (see Fig 3) derived from Witten’s [@witten] [@gerasimov] background independent open string field theory . In the case of the closed string tachyon the situation is still unclear, however as we have already said, there is a recent suggestion in [@tseytlin] ,based on $\sigma$ - model analysis, of a closed string tachyon potential of type $-T^{2}e^{-T}$ (see Fig 4). It is of course tempting to interpret the tachyon solution $T_{1}$ , we have just derived for the new family of backgrounds presented in this paper, as describing precisely the tachyon condensation for this particular closed string tachyon potential. A question that remains open is the dynamical meaning of the singularity, probably related to the particular way in which this background solution encodes the condensation of degrees of freedom. Incidentally, were this solution to be interpreted as a confining background, the static potential between heavy sources as derived from the lowest order Wilson loop computation yields a linear behavior [@manjarin]. Acknowledgments {#acknowledgments .unnumbered} =============== We are indebted to Pedro Resco for much help with the computations. Useful correspondence with Arkady Tseytlin is gratefully acknowledged. This work   has been partially supported by the European Union TMR program FMRX-CT96-0012 [*Integrability, Non-perturbative Effects, and Symmetry in Quantum Field Theory*]{} and by the Spanish grant AEN96-1655. The work of E.A. has also been supported by the European Union TMR program ERBFMRX-CT96-0090 [ *Beyond the Standard model*]{} and the Spanish grant AEN96-1664. L.H. is supported by the spanish predoctoral grant AP99 43367460. [99]{} M. Abramowitz and I. Stegun, [*Handbook of Mathematical Functions*]{}, (Dover University Press). E. Alvarez and C. Gómez, [*The Confining String from the Soft Dilaton theorem*]{} Nucl. Phys.  [**B566**]{} (2000) 363 [hep-th/9907158]{} E. Alvarez and J.J. Manjarín, [*Static Gauge Potential from Non-critical Strings*]{}, to appear. P. Breitenlohner and D. Freedman, [*Stability in Gauge Extended Supergravity*]{}, Ann. Phys. 144 (1982),249. A. Casher, F. Englert, H. Nicolai and A. Taormina, [*Consistent Superstrings As Solutions Of The D = 26 Bosonic String Theory*]{}, Phys. Lett.  [**B162**]{} (1985) 121. Anton A. Gerasimov, Samson L. Shatashvili, [*On Exact Tachyon Potential in Open String Field Theory*]{}, JHEP 0010 (2000) 034, [hep-th/0009103]{}.\ [*Stringy Higgs Mechanism and the Fate of Open Strings*]{}, [hep-th/0011009]{}. J. A. Harvey, D. Kutasov and E. J. Martinec, [*On the relevance of tachyons*]{}, [hep-th/0003101]{}. D. Kutasov, M. Marino and G. Moore, [*Some exact results on tachyon condensation in string field theory*]{}, [hep-th/0009148]{}. R. Myers, [*New Dimensions for old strings*]{}, Phys.Lett.B199:371,1987 J. Polchinski, [*String Theory*]{}, Cambridge University Press. A. M. Polyakov, [*The wall of the cave*]{}, Int. J. Mod. Phys.  [**A14**]{} (1999) 645 [hep-th/9809057]{}. A. Sen, [*Universality of the tachyon potential*]{}, JHEP [**9912**]{} (1999) 027 [hep-th/9911116]{}. A.A.Tseytlin, [*Sigma Model Approach to string theory effective actions with tachyons*]{}, [hep-th 0011033]{} E. Witten, [*On background independent open string field theory*]{}, Phys. Rev.  [**D46**]{} (1992) 5467 [hep-th/9208027]{}.\ [*Some computations in background independent off-shell string theory*]{}, Phys. Rev.  [**D47**]{} (1993) 3405 [hep-th/9210065]{}. [^1]: E-mail: [enrique.alvarez@uam.es]{} [^2]: E-mail: [cesar.gomez@uam.es ]{} [^3]: E-mail: [Lorenzo.Hernandez@uam.es ]{} [^4]: Unidad de Investigación Asociada al Centro de Física Miguel Catalán (C.S.I.C.) [^5]: This solution then seems to embody in a natural way stringy corrections to linear dilaton background

[**Coincidence Problem in an Oscillating Universe**]{} [Guangcan Yang [^1]\ ]{} [Anzhong Wang [^2]\ ]{} [**[Abstract]{}**]{} We analyze an oscillating universe model in brane world scenario. The oscillating universe cycles through a series of expansions and contractions and its energy density is dominated by dust matter at early-time expansion phase and by phantom dark energy at late-time expansion phase. We find that the period of the oscillating universe is not sensitive to the tension of the brane, but sensitive to the equation-of-state parameter $w$ of the phantom dark energy, and the ratio of the period to the current Hubble age approximately varies from $3$ to $9$ when the parameter $w$ changes from $-1.4$ to $-1.1$. The fraction of time that the oscillating universe spends in the coincidence state is also comparable to the period of the oscillating universe. This result indicates that the coincidence problem can be significantly ameliorated in the oscillating universe without singularity. One of remarkable discoveries over the past few years is that expansion of the universe is speeding up, rather than slowing down [@dark1; @dark2]. To explain the accelerated expansion, a so-called dark energy component with large negative pressure is needed in the energy density of the universe. A lot of evidence from astronomical observations indicates that our universe is spatial flat and consists of approximately $73\% $ dark energy, $27\%$ dust matter including cold dark matter and baryon matter, and negligible radiation. Although some proposals explaining dark energy have been suggested, it is fair to say that the dark energy problem is still a big challenge to theorists and cosmologists, for recent reviews see [@review] A simple candidate of dark energy is a tiny positive cosmological constant. In this case, one has to explain why the cosmological constant is so small, rather than its natural expectation, namely the Planck energy scale. This is the cosmological constant problem. Another puzzle associated with dark energy is why the dark energy density and dust matter energy density are comparable just now, or why the universe begins its accelerated expansion just only recently? This is called cosmic coincidence problem. To give a solution to the coincidence problem, some models like quintessence [@quit], k-essence [@k-essence; @k2], k-chameleon [@k-cham] etc., have been put forward. Of course, that the energy density of dark energy and that of dust matter are in the same order now might just be a coincidence, and does not imply any special meaning. Suppose that the dark energy has the equation of state, $p=w\rho$, where $p$ and $\rho$ are the pressure and energy density of dark energy, respectively. In order for the universe to accelerated expand, the equation-of-state parameter $w$ has to have $w<-1/3$. Current observation data give constraint: $-1.46 <w <-0.78$ [@obser]. The cosmological constant corresponds to a perfect fluid with $w=-1$. The quintessence model has $-1 <w <-1/3$, while the k-essence has $ -1<w<-1/3$ or $w<-1$, but cannot cross $w=-1$ [@vik]. It is well-known that if some matter has $w<-1$, it violates all energy conditions. In that case, some strange things will happen. The matter with $w<-1$ is called phantom matter, and dark energy with $w<-1$ is dubbed phantom dark energy [@phantom]. Indeed, in the phantom dark energy model, a remarkable feature is that the universe will end with a further singularity (big rip), where the scale factor of the universe, energy density of phantom matter etc. diverge. This implies that the lifetime of a phantom dominated universe is finite. Thus it is possible to ameliorate the coincidence problem in a phantom dominated universe if the fraction of the total lifetime of the universe in a state for which the dark energy and dark matter densities are roughly comparable is not so small [@Mic]. Indeed, lately Scherrer [@Sch] has carried out such a calculation by defining what means by that dark energy and dark matter energy densities are roughly comparable. Defining $r=\rho_{e}/\rho_{m}$, where $\rho_e$ and $\rho_m$ are energy densities of dark energy and dust matter, respectively, and setting a certain value $r_0$ of $r$, if $ 1/r_0 < r <r_0$, it is regarded as that the two energy densities are comparable and the universe is in the coincidence state. For $r_0=10$, Scherrer found that the fraction varies from $1/3$ to $1/8$ as $w$ changes from $-1.5$ to $-1.1$; the fraction is smaller for smaller values of $r_0$. Indeed the coincidence problem is significantly ameliorated in the sense that the fraction is not so small. Following [@Sch], Cai and Wang [@CW] studied the coincidence problem in an interacting phantom dark energy model with dark matter, and Avelino [@Ave] discussed the coincidence problem in a scalar field dark energy model with a linear effective potential. Note that the phantom dominated universe is characterized with a further singularity, while the universe with scalar field dark energy model with a linear effective potential will collapse to a big crunch. In this note we will investigate the coincidence problem for a cyclic universe model in brane world scenario. Over the past few years one of most important progresses in gravity theory is the proposal that our unverse is a 3-brane embedded in a higher dimensional spacetime, the so-called brane world scenario. In this scenario all standard model matters are confined on the brane, while gravity can propagate in the whole spacetime. Among brane world models, the RSII model [@RSII] is very attractive and is studied intensively in the literature. In the RSII model, the brane is embedded in a five dimensional anti-de Sitter space, due to a warped factor, the four dimensional general relativity on the brane is recovered in the low energy limit. And the Friedmann equation for a flat Friedmann-Robertson-Walker universe on the brane gets modified as follows [@Fre] $$\label{eq1} H^2= \frac{8\pi G}{3}\rho \left( 1+\frac{\rho}{2\sigma}\right),$$ where $H=\dot a/a $ denotes the Hubble parameter, $G$ is the Newtonian constant on the brane, $\sigma$ is the tension of the brane and $\rho$ is the energy density of matter on the brane. Clearly when $\rho \ll \sigma$, the four dimensional general relativity is recovered. The nucleosynthesis limit gives the constraint on the tension $\sigma > (1Mev)^4$, while the Newtonian law of gravity at distance $r\sim 1mm$ imposes the condition $\sigma > (10^3 Gev)^4$ [@Marrt]. Note that the extra dimension is spacelike in the RSII model. It is possible that the extra dimension is timelike [@SS]. In such a model, the Friedmann equation turns out to be [@SS] $$\label{eq2} H^2= \frac{8\pi G}{3}\rho \left( 1-\frac{\rho}{2\sigma}\right).$$ It is remarkable that one can build an oscillating universe model through phantom dark energy [@cyclic]. At the early-time expanding phase, the universe is dominated by dust matter. The phantom dark energy grows rapidly and dominates the late-time expanding phase. It can be seen from (\[eq2\]) that the universe start expansion at $\rho_{cr}=2\sigma$, where dust matter dominates the energy density of the universe and stop its expansion at the critical density $\rho_{cr}= 2\sigma$, where phantom dark energy dominates the energy density of the universe, and then start contraction until the energy density reaches to its critical value again. After this the universe will expand to the critical value again. In this way, the universe finishes a cycle. Clearly this cosmology is singularity-free. The goal of this note is to discuss the cosmological model and the cosmic coincidence problem for this oscillating universe. Concretely we will calculate the fraction of a period of the universe when the universe is in the coincidence state by generalizing the discussions in [@Sch; @CW; @Ave] to the cyclic universe model. Consider a cosmological model, in which all energy components consist of dust matter (including cold dark matter and baryon matter) with $w=0$ and phantom dark energy with a constant equation-of-state parameter $w<0$. In this note we will not consider possible interaction between dust matter and dark energy. In this case, dust matter has the vanishing equation-of-state parameter, $w=0$, its energy density satisfies $$\label{eq3} \rho_m = \rho_{m0}a_0^3/a^3,$$ where $a$ is the scale factor of the universe, $\rho_{m0}$ is a constant and $a_0$ is the present value of the scale factor. For a flat universe, one can set $a_0=1$, then $\rho_{m0}$ can be explained as the current energy density of dust matter. So from now on we set $a_0=1$. According to the continuity equation, $$\dot \rho +3 H(\rho +p)=0,$$ the energy density of phantom dark energy has the form $$\label{eq5} \rho_e=\rho_{e0}a^{-3(1+w)},$$ where $\rho_{e0}$ is the current dark energy density. Substituting (\[eq3\]) and (\[eq5\]) into (\[eq2\]), one has $$\label{eq6} H^2 =H_0^2 \left ( \Omega_{m0} a^{-3} +\Omega_{e0}a^{-3(1+w)} \right ) \frac{1-(\rho_{m0}a^{-3}+\rho_{e0}a^{-3(1+w)}) /2\sigma }{ 1-(\rho_{m0}+\rho_{e0})/2\sigma },$$ where $H_0$ is the current Hubble parameter, $\Omega_{m0}$ and $\Omega_{e0}$ are current fraction energy densities of dust matter and dark energy, respectively. Integrating (\[eq6\]) yields $$\label{eq7} t =H_0^{-1} \int da \ a ^{1/2}(1-\Omega_{e0}(1-a^{-3w}))^{-1/2} (1-\frac{\rho_{m0}a^{-3} +\rho_{e0}a^{-3(1+w)}}{2\sigma})^{-1/2},$$ where we have considered that current energy density of the universe is much less than the brane tension, that is, $\rho_{m0}+\rho_{e0} \ll \sigma$, and $\Omega_{m0}+\Omega_{e0} \approx 1$. Note that the total energy density of the universe is given by $$\rho = \rho_{m0} a^{-3} +\rho_{e0} a^{-3(1+w)}.$$ Clearly the energy density is dominated by the dust matter at the early-time expansion phase, and by the phantom dark energy at late-time expansion because $w<-1$. Therefore the two turning points, satisfying $\rho=2\sigma$, approximately are $$\label{eq9} a_{min} \approx \left(\frac{\rho_{m0}}{2\sigma}\right)^{1/3},\ \ \ a_{max} \approx \left(\frac{2\sigma}{\rho_{e0}}\right)^{1/(3|1+w|)}.$$ When the oscillating universe reaches the maximal scale factor $a_{max}$ from the minimal one $a_{min}$, it takes $$\label{eq10} T =H_0^{-1} \int^{a_{max}}_{a_{min}} da \ a ^{1/2}(\Omega_{m0}+ \Omega_{e0}a^{-3w})^{-1/2} (1-\frac{\rho_{m0}a^{-3} +\rho_{e0}a^{-3(1+w)}}{2\sigma})^{-1/2}.$$ Denote the current energy density of the universe by $\rho_0=\rho_{m0}+\rho_{e0}$, one has $\rho_0/\sigma \ll 1$, where $\rho_0$ is the current critical density given by $\rho_0=8\pi G H_0^2/3$. Further defining $s \equiv \rho_0/2\sigma$, we can rewrite (\[eq9\]) and (\[eq10\]) as $$a_{min} \approx (s\Omega_{m0})^{1/3},\ \ \ a_{max} \approx (s\Omega_{e0})^{-1/(3|1+w|)},$$ and $$\label{eq12} T =H_0^{-1} \int^{a_{max}}_{a_{min}} da \ a ^{1/2}(\Omega_{m0}+ \Omega_{e0}a^{-3w})^{-1/2} (1-s (\Omega_{m0}a^{-3} +\Omega_{e0}a^{-3(1+w)}))^{-1/2},$$ respectively. According to current observation data, $H_0= 72 km \cdot s^{-1} Mpc^{-1}$, which gives the critical energy density $\rho_0= 0.42\times 10^{-46}Gev^4$ and current Hubble age of the universe $H_0^{-1}=13.58 Gyr$. Note that the $\Lambda$CDM model tells us that the age of the universe is about $13.7Gyr$. In addition, we will take $\Omega_{m0}=0.27$ and $\Omega_{e0}=0.73$ in what follows [@WMAP]. In Fig. 1 we plot the ratio of the period of the oscillating universe to the current Hubble age $H_0^{-1}$ for the case of $\sigma= (10^3Gev)^4$. The ratio approximately varies from $3$ to $9$ as $w$ changes from $-1.4$ to $-1.1$. In particular, we find that the ratio is not sensitive to the tension $\sigma$ of the brane. For example, if $\sigma=(1Mev)^4$, the ratio for this case is almost indistinguishable to the case of $\sigma=(10^3Gev)^4$. The insensitivity can be understood as follows.  The integrand in (\[eq12\]) diverges at $a_{min}$ and $a_{max}$, and it decreases very quickly from infinity at $a_{min}$ to an almost vanishing value at some place. It remains almost unchanged until $a$ reaches a value near $a_0=1$, where it starts increasing to a finite value when $a$ gets around $a_0=1$, and then decreases again until $a$ arrives at a point near $a_{max}$, from which the integrand increases again very quickly to infinity at $a_{max}$. We plot a sketch of the integrand in Fig. 2. ![A sketch of the integrand in the integration (\[eq12\])](int.eps) We can separate the integration in (\[eq12\]) into three parts, $$T =T_1+T_2+T_3,$$ where $$\begin{aligned} \label{eq14} T_1 &=& H_0^{-1} \int^{a_{1}}_{a_{min}} da \ a ^{1/2}(\Omega_{m0}+ \Omega_{e0}a^{-3w})^{-1/2} (1-s (\Omega_{m0}a^{-3} +\Omega_{e0}a^{-3(1+w)}))^{-1/2} \nonumber \\ & \approx &H_0^{-1} \int^{a_{1}}_{a_{min}} da \frac{a ^{1/2}}{ \sqrt{\Omega_{m0}}\sqrt{ 1-s \Omega_{m0}a^{-3}}} \nonumber \\ &=& H_0^{-1} \frac{2}{3\sqrt{\Omega_{m0}}} \sqrt{a^3-\Omega_{mo}s} \left |^{a_1}_{a_{min}} \right. = H_0^{-1}(1.28 \sqrt{a_1^3-a_{min}^3}), \end{aligned}$$ $$T_2 =H_0^{-1} \int^{a_{2}}_{a_{1}} da \ a ^{1/2}(\Omega_{m0}+ \Omega_{e0}a^{-3w})^{-1/2} (1-s (\Omega_{m0}a^{-3} +\Omega_{e0}a^{-3(1+w)}))^{-1/2},$$ and $$\begin{aligned} \label{eq16} T_3 &=& H_0^{-1} \int^{a_{max}}_{a_{2}} da \ a ^{1/2}(\Omega_{m0}+ \Omega_{e0}a^{-3w})^{-1/2} (1-s (\Omega_{m0}a^{-3} +\Omega_{e0}a^{-3(1+w)}))^{-1/2} \nonumber \\ &\approx & H_0^{-1} \int^{a_{max}}_{a_{2}} da \ \frac{a ^{(1+3w)/2}}{\sqrt{\Omega_{e0}}\sqrt{1-s\Omega_{e0}a^{-3(1+w)}}} \nonumber \\ &=& H_0^{-1} \left( \frac{2 a^{3(1+w)/2}}{3(1+w)\sqrt{\Omega_{e0}}} \sqrt{1-s\Omega_{e0}a^{-3(1+w)}} \right) \left |^{a_{max}}_{a_{2}} \right. \nonumber \\ &=& H_0^{-1}\left (-0.78\frac{a_2^{3(1+w)/2)}}{1+w}\sqrt{1-\frac{a_2^{-3(1+w)}}{a_{max}^{-3(1+w)}}} \right) \end{aligned}$$ respectively. During $T_1$, the scale factor expands from $a_{min}$ to $a_1$, and the dust matter is dominant in the energy density of the universe, where the phantom dark energy can be neglected. On the other hand, during $T_3$ the scale factor varies from $a_2$ to $a_{max}$, and the phantom dark energy is dominant. During $T_2$, both terms, dust matter and dark energy, make contributions. We note that the contributions to $T$ of both terms, $T_1$ and $T_3$, are very small, compared to that of $T_2$. Take an example: if $\sigma= (10^3Gev)^4$ and $w=-1.4$, one has $s=2.1* 10^{-59}$, $a_{min} =1.78*10^{-20}$, and $a_{max}=0.81*10^{49.2}$. Even we take $a_1=10^{-3}$ and $a_2=10^4$, (\[eq14\]) and (\[eq16\]) then give us $T_1H_0 = 0.128$ and $T_3H_0=0.0078$, respectively. Take another extremal example: if $\sigma=(1Mev)^4$ and $w=-1.4$, one then has $s=2.1*10^{-35}$, $a_{min}=1.78*10^{-12}$ and $a_{max}=0.7*10^{29.2}$. If taking $a_1 = 10^{-3}$ and $a_2=10^4$, one still has $T_1H_0=0.128$ and $T_3H_0= 0.0078$ since $a_1\gg a_{min}$ and $a_2 \ll a_{max}$. Furthermore, it can be seen from (\[eq14\]) and (\[eq16\]) that a smaller $a_1$ leads to a smaller $T_1H_0$ and a larger $a_2$ gives a smaller $T_3H_0$. Fig. 1 shows that these small contributions $T_1H_0$ and $T_3H_0$ are indeed negligible. Further, if one defines the ratio of the dark energy density to that of dust matter as $$r \equiv \frac{\rho_{e}}{\rho_{m}}= \frac{\Omega_{e0}}{\Omega_{m0}}a ^{-3w},$$ the integration (\[eq7\]) can be rewritten as $$t= -\frac{1}{3w} \frac{\sqrt{\Omega_{m0}}}{H_0\Omega_{e0}} \int dr \left( \frac{\Omega_{e0}}{r\Omega_{m0}}\right)^{(1+2w)/2w} (1+r)^{-1/2} \left( 1-s \Omega_{m0}(1+r) \left( \frac{r \Omega_{m0}}{\Omega_{e0}}\right)^{1/w}\right)^{-1/2},$$ and the period $T$ in (\[eq10\]) can be expressed as $$\label{eq13} T = -\frac{1}{3w} \frac{\sqrt{\Omega_{m0}}}{H_0\Omega_{e0}} \int^{r_{max}}_{r_{min}} dr \left( \frac{\Omega_{e0}}{r\Omega_{m0}}\right)^{(1+2w)/2w} (1+r)^{-1/2} \left( 1-s \Omega_{m0}(1+r) \left( \frac{r \Omega_{m0}}{\Omega_{e0}}\right)^{1/w}\right)^{-1/2},$$ where $$r_{min} = \frac{\Omega_{e0}}{\Omega_{m0}}\left(s \Omega_{m0}\right)^{-w}, \ \ r_{max}= \frac{\Omega_{e0}}{\Omega_{m0}} \left ( s\Omega_{e0}\right) ^{-w/(1+w)}.$$ Following [@Sch], we define a scale $r_0$. If $r$ falls in the range $1/r_0 <r<r_0$, it is regarded that the universe is in the coincidence state. The duration $t_U$ of the universe in the coincidence state is then $$t_U =-\frac{1}{3w} \frac{\sqrt{\Omega_{m0}}}{H_0\Omega_{e0}} \int^{r_0}_{1/r_0} dr \left( \frac{\Omega_{e0}}{r\Omega_{m0}}\right)^{(1+2w)/2w} (1+r)^{-1/2} \left( 1-s \Omega_{m0}(1+r) \left( \frac{r \Omega_{m0}}{\Omega_{e0}}\right)^{1/w}\right)^{-1/2}.$$ Denote by $g$ the ratio of the duration $t_U$ of the universe in the coincidence state to the period $T$ of the oscillating universe, namely, $$\label{eq22} g \equiv \frac{t_U}{T}=\frac {\int^{r_0}_{1/r_0} dr \left( \frac{\Omega_{e0}}{r\Omega_{m0}}\right)^{(1+2w)/2w} (1+r)^{-1/2} \left( 1-s \Omega_{m0}(1+r) \left( \frac{r \Omega_{m0}}{\Omega_{e0}}\right)^{1/w}\right)^{-1/2}} { \int^{r_{max}}_{r_{min}} dr \left( \frac{\Omega_{e0}}{r\Omega_{m0}}\right)^{(1+2w)/2w} (1+r)^{-1/2} \left( 1-s \Omega_{m0}(1+r) \left( \frac{r \Omega_{m0}}{\Omega_{e0}}\right)^{1/w}\right)^{-1/2}}.$$ We plot in Fig. 3 the ratio $g$ for the case with brane tension $\sigma= (10^3Gev)^4$. Once again, the ratio is not sensitive to the tension of the brane. The result shown in Fig. 3 is almost the same as that showed in Fig. 1 of paper [@Sch], where the results hold for the case of usual Friedman equation in general relativity. In other words, the latter is for the case $\sigma \to \infty$ in (\[eq1\]) or (\[eq2\]). But note that the two cases are quite different for the evolution of the universe: in the current brane world scenario, the universe is an oscillating one without singularity, while for a phantom dominated universe in general relativity the universe begins with a big bang and ends with a big rip. This insensitivity can be seen from (\[eq22\]) that in the numerator because $s$ is a very tiny quantity so that the factor $(1-s\Omega_{m0}(1+r)(r\Omega_{m0}/\Omega_{e0})^{1/w}$ is negligible during the integration range $1/r_0 <r<r_0$. In the denominator, although the integrand diverges at $r_{min}$ and $r_{max}$, as we analyzed above, the contributions from these two points are small and negligible as well.  Note that for a given $w$ and $r_0$, the fraction of time $g$ that the oscillating universe spends in the so-called coincidence state is not small as expected. We can see from Fig. 3 that for a larger scale $r_0$, one has a larger $g$; For a fixed scale $r_0$, a larger $w$ gives a smaller $g$. This can be understood as follows. From Fig. 1 we note that for a larger $w$, one has a larger period $T$ of the oscillating universe so that the ratio $g$ gets smaller. Fig. 3 shows that it is not so strange that we just live in a period when the dust matter and dark energy densities are roughly comparable. In this way the coincidence problem can be significantly ameliorated in this oscillating universe model without singularity. In summary we have analyzed an oscillating universe model without singularity in brane world scenario. The early-time expansion phase is dominated by dust matter and the late-time expansion phase is dominated by phantom dark energy. We have found that the period of the oscillating universe is not so sensitive to the tension of brane, but is sensitive to the equation-of-state parameter $w$ of phantom dark energy. The ratio of the period to the current Hubble age of the universe approximately varies from $3$ to $9$ as $w$ changes from $-1.4$ to $-1.1$. Further we calculated the fraction of time $g$ that the oscillating universe spends in the coincidence state. The result shows that it is also not sensitive to the tension of the brane, but that the fraction of time of the universe in the coincidence state is comparable to the period of the oscillating universe. Acknowledgments {#acknowledgments .unnumbered} =============== The authors thank R. G. Cai for useful suggestions and discussions. [99]{} S. Perlmutter [*et al.*]{} \[Supernova Cosmology Project Collaboration\], Astrophys. J.  [**517**]{}, 565 (1999) \[arXiv:astro-ph/9812133\]. A. G. Riess [*et al.*]{} \[Supernova Search Team Collaboration\], Astron. J.  [**116**]{}, 1009 (1998) \[arXiv:astro-ph/9805201\]. P. J. E. Peebles and B. Ratra, Rev. Mod. Phys.  [**75**]{}, 559 (2003) \[arXiv:astro-ph/0207347\]; T. Padmanabhan, Phys. Rept.  [**380**]{}, 235 (2003) \[arXiv:hep-th/0212290\]; V. Sahni, arXiv:astro-ph/0403324. P. J. Steinhardt, L. M. Wang and I. Zlatev, Phys. Rev. D [**59**]{}, 123504 (1999) \[arXiv:astro-ph/9812313\]; I. Zlatev and P. J. Steinhardt, Phys. Lett. B [**459**]{}, 570 (1999) \[arXiv:astro-ph/9906481\]. C. Armendariz-Picon, V. Mukhanov and P. J. Steinhardt, Phys. Rev. Lett.  [**85**]{}, 4438 (2000) \[arXiv:astro-ph/0004134\]; C. Armendariz-Picon, V. Mukhanov and P. J. Steinhardt, Phys. Rev. D [**63**]{}, 103510 (2001) \[arXiv:astro-ph/0006373\]. T. Chiba, T. Okabe and M. Yamaguchi, Phys. Rev. D [**62**]{}, 023511 (2000) \[arXiv:astro-ph/9912463\]. H. Wei and R. G. Cai, arXiv:hep-th/0412045. R. A. Knop [*et al.*]{}, Astrophys. J.  [**598**]{}, 102 (2003) \[arXiv:astro-ph/0309368\]; A. G. Riess [*et al.*]{} \[Supernova Search Team Collaboration\], Astrophys. J.  [**607**]{}, 665 (2004) \[arXiv:astro-ph/0402512\]. A. Vikman, Phys. Rev. D [**71**]{}, 023515 (2005) \[arXiv:astro-ph/0407107\]. R. R. Caldwell, Phys. Lett. B [**545**]{}, 23 (2002) \[arXiv:astro-ph/9908168\]. B. McInnes, arXiv:astro-ph/0210321. R. J. Scherrer, arXiv:astro-ph/0410508. R. G. Cai and A. Wang, arXiv:hep-th/0411025. P. P. Avelino, arXiv:astro-ph/0411033. L. Randall and R. Sundrum, Phys. Rev. Lett.  [**83**]{}, 4690 (1999) \[arXiv:hep-th/9906064\]. P. Binetruy, C. Deffayet and D. Langlois, Nucl. Phys. B [**565**]{}, 269 (2000) \[arXiv:hep-th/9905012\]; P. Binetruy, C. Deffayet, U. Ellwanger and D. Langlois, Phys. Lett. B [**477**]{}, 285 (2000) \[arXiv:hep-th/9910219\]. R. Maartens, D. Wands, B. A. Bassett and I. Heard, Phys. Rev. D [**62**]{}, 041301 (2000) \[arXiv:hep-ph/9912464\]. Y. Shtanov and V. Sahni, Phys. Lett. B [**557**]{}, 1 (2003) \[arXiv:gr-qc/0208047\]. M. G. Brown, K. Freese and W. H. Kinney, arXiv:astro-ph/0405353. C. L. Bennett [*et al.*]{}, Astrophys. J. Suppl.  [**148**]{}, 1 (2003) \[arXiv:astro-ph/0302207\]; D. N. Spergel [*et al.*]{} \[WMAP Collaboration\], Astrophys. J. Suppl.  [**148**]{}, 175 (2003) \[arXiv:astro-ph/0302209\]. [^1]: Email address: yanggc@wznc.zj.cn [^2]: Email address: anzhong\_wang@baylor.edu

--- abstract: 'We investigate the role of the environment in processing molecular gas in radio galaxies (RGs). We observed five RGs at $z=0.4-2.6$ in dense Mpc-scale environment with the IRAM-30m telescope. We set four upper-limits and report a tentative CO(7$\rightarrow$6) detection for COSMOS-FRI 70 at $z=2.63$, which is the most distant brightest cluster galaxy (BCG) candidate detected in CO. We speculate that the cluster environment might have played a role in preventing the refueling via environmental mechanisms such as galaxy harassment, strangulation, ram-pressure, or tidal stripping. The RGs of this work are excellent targets for ALMA as well as next generation telescopes such as the [*James Webb Space Telescope.* ]{}' --- [**1. Introduction.**]{} Molecules in galaxies can help to trace star forming regions, even close to an active galactic nucleus (AGN, Omont 2007). Radio galaxies (RGs) are a precious tool to discover distant galaxy groups and (proto-)clusters at high-$z$, since they are often the brightest cluster galaxies (BCGs, Zirbel 1996). We consider two RGs at $z = 0.39$ and 0.61 within the DES SN deep fields (DES collaboration 2015) and three additional COSMOS-FRI RGs from the Chiaberge et al. (2009) sample, at $z=0.97$, 0.91, and 2.63. They have been selected since they show evidence of significant star formation, SFR$_{24\mu m}\sim(20-250)M_\odot$/yr, based on their emission at $\sim24\mu$m in the observer frame (WISE or Spitzer-MIPS). All five sources are found in (proto-)cluster candidates by using the Poisson Probability Method (PPM, Castignani et al. 2014ab) that searches for overdensities using photometric redshifts. [**2. Methods and Results.**]{} We observed the five targets with the IRAM-30m telescope targeting several CO(J$\rightarrow$J-1) lines, one for each source, at $\sim$(1.2-1.4) mm in the observer frame (Castignani et al. 2018). Data reduction and analysis were performed using the CLASS software of the GILDAS package.[^1] Our observations yielded four CO upper limits and one tentative CO(7-6) detection for COSMOS-FRI 70, which makes it the most distant BCG candidate detected in CO. There are in fact pieces of evidence that the radio source is [ the BCG of one of the most distant proto-clusters hosting a RG.]{} i) The stellar mass is exceptionally high ($\log(M_\star/M_\odot)\simeq11$, Baldi et al. 2013). ii) By applying Galfit (Peng et al. 2002) to the archival high-resolution HST ACS F814W (I-band) image a Sèrsic index $4.3\pm0.6$ is found, consistently with that of early type galaxies. iii) Color-color and color-magnitude plots further suggest that COSMOS-FRI 70 is indeed a [ star forming massive elliptical]{}, $\sim0.3$mag brighter than all photometrically selected (proto-)cluster members (Castignani et al. 2018). Our results are reported in Table \[tab:radio\_galaxies\_properties\_mol\_gas\] and Fig. \[fig:figure\]. [**3. Conclusions.**]{} All target RGs have molecular gas properties that are consistent with the predictions for main sequence (MS) field galaxies. However they also show that high-$z$ BCGs i) tend to be gas poor and ii) have a relatively short depletion time scale. We thus speculate that the cluster environment might have played a role in preventing the refueling via environmental mechanisms such as galaxy harassment, strangulation, ram-pressure, or tidal stripping. ------------------- -------------- ----------------------- -------------------------------- --------------------- ------------------------ -------------------------------- ------------------------ -- -- -- -- -- Galaxy ID $z_{spec}$ CO(J$\rightarrow$J-1) $S_{\rm CO(J\rightarrow J-1)}$ $M({\rm H_2})$ $\tau_{\rm dep}$ $\frac{M({\rm H_2})}{M_\star}$ $\tau_{\rm dep, MS}$ (Jy km s$^{-1}$) ($10^{10}~M_\odot$) ($10^9$ yr) ($10^8~M_\odot$) ($10^9$ yr) (1) (2) (3) (4) (5) (6) (7) (8) [DES-RG 399]{} [0.388439]{} 3$\rightarrow$2 $<1.5$ $<$1.0 $<$0.91 $<$0.16 $1.10^{+0.16}_{-0.14}$ [DES-RG 708]{} [0.60573]{} 3$\rightarrow$2 $<1.5$ $<$2.6 $<$0.36 $<$0.14 $1.12^{+0.21}_{-0.17}$ [COSMOS-FRI 16]{} [0.9687]{} 4$\rightarrow$3 $<5.5$ $<$18.8 $<$0.58 $<$2.29 $0.91^{+0.16}_{-0.14}$ [COSMOS-FRI 31]{} [0.9123]{} 4$\rightarrow$3 $<5.2$ $<$15.8 — $<$2.71 $0.90^{+0.15}_{-0.13}$ [COSMOS-FRI 70]{} [2.625]{} 7$\rightarrow$6 $0.69\pm0.31$ $5.0\pm2.2$ $0.20^{+0.13}_{-0.19}$ $0.22^{+0.15}_{-0.16}$ $0.66^{+0.17}_{-0.13}$ ------------------- -------------- ----------------------- -------------------------------- --------------------- ------------------------ -------------------------------- ------------------------ -- -- -- -- -- : (1) galaxy name; (2) spectroscopic redshift; (3-4) CO(J$\rightarrow$J-1) transition and flux; (5) molecular gas mass; (6) depletion time scale $\tau_{\rm dep}=M({\rm H_2})/{\rm SFR_{24\mu m}}$; (7) molecular gas to stellar mass ratio; (8) $\tau_{\rm dep}$ predicted for MS field galaxies by Tacconi et al. (2018).[]{data-label="tab:radio_galaxies_properties_mol_gas"} \ [2]{} Baldi, R. et al. 2013, ApJ, 762, 30 Castignani, G. et al. 2014a, ApJ, 792, 113 Castignani, G. et al. 2014b, ApJ, 792, 114 Castignani, G. et al. 2018, submitted to A&A Chiaberge, M. et al. 2009, ApJ, 696, 1103 DES Collaboration 2016, MNRAS, 460, 1270 Emonts, B. et al. 2013, MNRAS, 430, 3465 Omont, A., 2007, RPPh, 70, 1099O Peng, C. Y. et al. 2002, AJ, 124, 266 Tacconi, L. et al. 2018, ApJ, 853, 179 Webb, T. M. A. et al. 2017, ApJ, 844, 17 Zirbel, E. L. 1996, ApJ, 473, 713 [^1]: https://www.iram.fr/IRAMFR/GILDAS/

--- abstract: 'Avalanche behaviors, characterized by power-law statistics and structural relaxation that induces shear localization in amorphous plasticity, play an essential role in deciding the mechanical properties of amorphous metallic solids (i.e., metallic glasses). However, their interdependence is still not fully understood. To investigate the influence of structural relaxation on elementary avalanche behavior, we perform molecular-dynamics simulations for the shear deformation test of metallic glasses using two typical metallic-glass models comprising a less-relaxed (as-quenched) glass and a well-relaxed (well-aged) glass exhibiting a relatively homogeneous deformation and a shear-band-like heterogeneous deformation, respectively. The data on elementary avalanches obtained from both glass models follow the same power-law statistics with different maximum event sizes, and the well-relaxed glass shows shear localization. Evaluating the spatial correlation functions of the nonaffine squared displacements of atoms during each elementary avalanche event, we observe that the shapes of the elementary avalanche regions in the well-relaxed glasses tend to be anisotropic, whereas those in the less-relaxed glasses are relatively isotropic. Furthermore, we demonstrate that a temporal clustering in the direction of the avalanche propagation emerges, and a considerable correlation between the anisotropy and avalanche size exists in the well-relaxed glass model.' author: - 'Tomoaki Niiyama$^{1}$' - 'Masato Wakeda$^{2}$' - 'Tomotsugu Shimokawa$^{3}$' - 'Shigenobu Ogata$^{4, 5}$' bibliography: - '/Users/niyama/Documents/Articles.bib' title: Structural relaxation affecting shear transformation avalanches in metallic glasses --- \#1 Introduction {#sec:Intro} ============ Whereas metals usually form ordered (i.e., latticed) structures under moderate conditions, various glass states can be realized by quenching the multicomponent metals from their liquid state. These types of amorphous metals are called [*metallic glasses*]{} [@Masumoto1971MG; @Greer1995MG]. Such materials have excellent material properties, such as high strength, corrosion resistance, and soft magnetic properties [@Greer1995MG; @Masumoto1971MG; @Loffler2003ReviewMG; @Ashby2006MGforStructuralMater]. However, a high level of macroscopic brittleness and catastrophic failure of the metallic glasses caused by [*shear localization*]{} hinders their applicability as structural materials, whereas localized deformation (i.e., shear banding) induces ductility at microscopic scales [@Pampillo1972MGshearFracture; @Zhang2003MGshearFracture; @Ashby2006MGforStructuralMater; @Gludovatz2013MGfatigueFracture]. Thus, this localization of deformation is a major concern for the brittle failure of metallic glasses. Some experimental and numerical studies have reported that structural relaxation using specific thermal treatments determined whether the shear plastic deformation of metallic glasses was localized or homogeneous [@Shi2007MG-DisorderT; @Kumar2009MGembrittle; @Ogata2006localizationMG; @Zhang2015Processing-dep-MG; @Wakeda2015MGRejuv; @Miyazaki2016RejuveMG]. Studies on molecular dynamics (MD) simulations indicated that well-relaxed glasses (i.e., well-aged glasses) using thermal relaxation exhibit shear banding by localized deformation, whereas less-relaxed glasses (i.e., as-quenched glasses) exhibit homogeneous deformation [@Shi2007MG-DisorderT; @Zhang2015Processing-dep-MG; @Wakeda2015MGRejuv; @Miyazaki2016RejuveMG]. Thus, understanding the effects of structural relaxation by thermal treatments on the shear localization is expected to lead to an improvement of the ductility of metallic glasses. In the context of nonequilibrium physics, the avalanche behavior in plasticity (intermittent plasticity or [*avalanche plasticity*]{}) observed in various types of amorphous solids, such as glasses, granular materials, colloids, and metallic glasses has gathered considerable attention [@Dalton2001GranularSOC; @Maloney2004AmorphousAvalanches; @Salerno2012AmorphousAvalanches; @Hatano2015granular-avalache; @Sun2010MetallicGlassSOC; @Ren2012MG-Chaos2SOC; @Antonaglia2014BulkMetallicGlassSOC]. One of the significant characteristics of avalanche plasticity is its power-law statistics. The probability that a plastic event of size of $s$ occurs is proportional to an algebraic function, $P(s) \propto s^{-\alpha}$, where $\alpha$ is a constant. This statistical feature, following a power-law distribution, is a sign of nonequilibrium critical phenomena, including self-organized criticality [@Bak1987SOC; @SOC1998Jensen]. The same power-law behavior also emerges in crystalline solids through the collective motion of dislocations, and the behavior is thought to represent the intrinsic nature of plasticity in solids [@Miguel2001Intermittent-di; @Zaiser2006IntermittentPlasticityReview; @Csikor2007DislocationAvalanche; @Ispanovity2014NotDepinning; @Cui2017DDDSOCNano; @Papanikolaou2018AvalanchePlasticityOverview]. The plastic events obeying power-law statistics in amorphous solids correspond to avalanche-like collective motions of local atomistic rearrangements. The minimum unit of plastic deformation in amorphous solids is considered to be a set of atomic rearrangements in a local region known as a shear transformation zone (STZ) [@Argon1979STZ; @FalkLanger1998STZ]. The deformation of an STZ can activate the deformation of other STZs through the redistribution of the elastic energy stored in the STZ. The chain-reaction propagation of this type of deformation in STZs behaves like an avalanche (we refer to this as an elementary avalanche as described in Section  \[sec:avalanche-statistics\]). The plastic deformation of amorphous solids is a result of several shear transformation avalanches [@Boioli2017STZactivationVolume]. Thus, the localization of the deformation (i.e., shear banding in amorphous solids) can be considered a spatial concentration of shear transformation avalanches. Hence, avalanche plasticity is expected to be closely related to ductility in metallic glasses [@Sun2010MetallicGlassSOC]. Annealing’s influence of structural relaxation on avalanche plasticity (e.g., the difference of the avalanche behaviors between the localized deformation of well-relaxed glasses and homogeneous deformation of less-relaxed glasses) is still not well understood. Furthermore, it is not known how avalanches in well-relaxed glasses result in shear banding or how structural relaxation influences the shape of the avalanche. One may expect that less-relaxed glasses would not exhibit avalanche behaviors, because the excess free volume within their atomic configurations should prevent avalanche formation. Elucidating the relationship between the localization and behavior of the avalanche plasticity is expected to contribute to the improvement of the mechanical properties of metallic glasses. Furthermore, it is expected to provide an understanding of nonequilibrium critical behaviors of amorphous plasticity. In this study, we investigate the influence of structural relaxation on avalanche plasticity and the contribution of the avalanche to shear localization via MD simulations using well-relaxed and less-relaxed metallic glasses exhibiting localized and homogeneous deformations induced by specific thermal treatments. First, we confirm the localization and the avalanche statistics of plastic deformation in the two metallic glasses (Section  \[sec:avalanche-statistics\]). Next, we analyze the avalanche shapes in the two metallic glasses, (Section  \[sec:avalanche-geometry\] and Section  \[sec:spatio-temporal\]), by extracting individual avalanche events from our simulations and calculating spatial correlation functions of the nonaffine squared displacements [@FalkLanger1998STZ]. Finally, we discuss the evolution of the avalanche shape over time and the correlation of the shape with the magnitude of the avalanche events (Sections  \[sec:anisotropy-size\] and  \[sec:size-depend\]). Numerical method {#sec:method} ================ In the present study, the atomic structure of a less-relaxed glass model with homogeneous deformations was prepared by quenching a copper–zirconium (Cu–Zr) binary alloy from the liquid state, whereas a well-relaxed glass model with localized deformations was achieved using specific thermal annealing of the homogeneous glass, as described below. For these two typical glass configurations, we performed MD simulations of shear deformation with constant temperatures and strain rates. We selected the Cu–Zr system to perform the simulations, because this alloy exhibits excellent glass formability [@Matsubara2007Zr-basedMG]. To perform the MD simulations, we used the Lennard–Jones potential and parameters for the Cu–Zr mixtures, which were developed by Kobayashi [*et al.*]{} [@Kobayashi1980LJforCuZrAlloy]. The atomic radii of Cu and Zr at this potential are approximately 2.7 and 3.3 Å, respectively [@Kobayashi1980LJforCuZrAlloy]. The number of atoms in the simulations was $50,000$ with a $1:1$ ratio of Cu and Zr atoms. To obtain the atomic structure of the two different glasses, we applied a specific thermal loading with the following conditions used in a previous study to generate either homogeneous or localized deformations [@Wakeda2015MGRejuv]. First, the randomly packed configurations of the Cu–Zr atoms under the periodic boundary condition were heated to a temperature of $3,000$ K, greater than the melting temperature. Furthermore, the molten configurations were equilibrated for $100$ ps under an isothermal-isobaric (NPT) ensemble with zero normal stresses after performing equilibration for $100$ ps at the same temperature under the canonical ensemble (NVT ensemble). Quenching the equilibrated liquid to $0$ K with a cooling rate $10^{13}$ K/s resulted in the formation of a less-relaxed glass with homogeneous deformation, because the structure did not undergo any structural relaxation by annealing. Henceforth, we refer to the glass as the [*as-quenched model*]{}. The glass structure exhibiting localized deformation was obtained after additional thermal loading resulting in structural relaxation. After quenching the equilibrated liquid to $0$ K with a relatively slow cooling rate of $10^{12}$ K/s, we heated the quenched structure to $900$ K (slightly higher than the glass transition temperature [@Wakeda2015MGRejuv] $T_g = 898$ K), then we annealed it for $2$ ns under the NPT ensemble. Next, we quenched the well-annealed configurations to $0$ K at a rate of $3 \times 10^{11}$ K/s. This thermal annealing leads to a structural relaxation in the glass structure without recrystalization, whereas the annealing temperature was slightly higher than $T_g$. We confirmed that the radial distribution function of this well-annealed configuration did not show a significant difference with the as-quenched configuration. The structural relaxation resulting from this annealing process was established by the evaluation of the change in the atomistic volumes (as illustrated in the next paragraph) and the aging of the as-quenched and the well-annealed configuration [@Wakeda2015MGRejuv]. Henceforth, this well-relaxed glass configuration will be referred to as the [*well-aged model*]{}. The simulation cell volume of the as-quenched model at the initial state was $V_{AQ} = 893.97 \pm 0.02$ nm$^3$, whereas that of the well-aged model was $V_{WA} = 890.19 \pm 0.02$ nm$^3$. The variation of the number density was $(\rho_{WA}-\rho_{AQ})/\rho_{AQ} = (V_{AQ}-V_{WA})/V_{WA} \simeq 0.42 \%$, where $\rho_{AQ}$ and $\rho_{WA}$ were the number densities of the as-quenched and the well-aged model, respectively. This density variation is comparable to those reported in some experimental and numerical studies [@Gerling1982MGDensity; @Nagel1998FreeVolumeMG; @Chen1978MGStructRelax]. This variation indicates that the as-quenched model contained a larger atomic-free volume than the well-aged model because structural relaxation occurred during thermal annealing in the latter model. For these two typical glass configurations, we added simple shear deformation with an engineering strain rate, $\dot{\gamma} = \dot{\gamma}_{zx} = 10^7$ 1/s, for $100$ ns under the NPT ensemble condition at $10$ K using the Lee–Edwards periodic boundary condition [@Tuckerman2010MDsimulations] and zero normal stresses using the Parrinello–Rahman method [@Parrinello1980RahmanMethod], where $\gamma_{xy}$ and $\gamma_{yz}$ were fixed at zero. The simulation cell for the periodic boundary condition had a cubic shape with $9.62$ nm edges at the initial state. The above simulations were performed using LAMMPS [@Plimpton1995LAMMPS]. To remove thermal fluctuations from the original time series of the shear stress, $\sigma^*_{xz}(t)$, obtained from the simulations, we smoothed the time series using a Gaussian filter. Gaussian filtering is an averaging method involving a Gaussian weight. This is shown in the following equations. $$\begin{aligned} \sigma_{xz}(t) = \int^{\infty}_{-\infty} G(t'-t) \cdot \sigma^*_{xz}(t) \ {\text{d} {t'}},\end{aligned}$$ where $G(t)$ is the Gaussian weight, and $$\begin{aligned} G(t) = \frac{1}{\sqrt{2 \pi \delta^2}} \exp \left[ - t^2/2 \delta^2 \right],\end{aligned}$$ where $\delta$ represents the extent of the filter. In this study, we computed the sum over a discrete range from $-3 \delta$ to $+3 \delta$, as shown below, instead of calculating the above integral over an infinite range, because $\sigma^*_{xz}(t)$ is discrete time series. $$\begin{aligned} \sigma_{xz}(m \Delta t) = \sum^{m+3d}_{n=m-3 d} \frac{1}{\sqrt{2 \pi d^2}} \exp \left[ -\frac{(n-m)^2}{2 d^2} \right] \sigma^*_{xz}(n \Delta t),\end{aligned}$$ where $d = \delta/ \Delta t$, $m = t/ \Delta t$. Here, the standard deviation, $\delta$, and time segment, $\Delta t$, were chosen as $2$ ps and $4$ fs, respectively. To quantify the extent of local plastic deformations in the simulations, we employed the nonaffine squared displacement, $D^2_{\text{min}}$, developed by Falk and Langer [@FalkLanger1998STZ]. The quantity well-represents atomic displacements that cannot be represented by affine transformations (i.e., the atomic displacement by nonelastic deformation in amorphous solids). Avalanche statistics of as-quenched and well-aged glasses {#sec:avalanche-statistics} ========================================================= In Figure  \[fig:t-stress\](a), the evolution over time of the shear stress, $\sigma_{xz}$, obtained in our MD simulations for the as-quenched and well-aged models, are depicted by the red and blue lines, respectively. The stress increased almost monotonously until $\dot{\gamma} t \simeq 0.05$ ($t \simeq 5$ ns). Around $\dot{\gamma} t \simeq 0.07$, the well-aged model exhibited a significant overshoot and a sudden descent in shear stress compared to the as-quenched model. This yielding drop is a common feature of localized deformation of relaxed metallic glasses [@Shi2006MGstrainlocalize; @Shi2007MG-DisorderT; @Zhang2015Processing-dep-MG; @Wakeda2015MGRejuv]. After the drop, both models showed serrated stress fluctuations comprising numerous increases and sudden drops of the shear stress. Whereas the increasing shear stress was caused by elastic deformation caused by external shear deformation, the sudden stress drops were the result of plastic deformation (avalanche) events. This serrated behavior indicates the emergence of avalanche plasticity and is consistent with results obtained from previous experimental and numerical studies of amorphous and crystalline solids [@Maloney2004AmorphousAvalanches; @Sun2010MetallicGlassSOC; @Antonaglia2014BMGSOC; @Antonaglia2014BulkMetallicGlassSOC; @AnanthakrishnaPRE1999crossover; @Niiyama2016avalancheGB]. ![ (a) Stress-strain curves of the as-quenched model (red) and the well-aged model (blue). (b) Enlarged view of the stress drops corresponding to the $(n-1)$-th and the following $n$-th elementary avalanche events, indicated by the arrows. []{data-label="fig:t-stress"}](Fig-1-2.eps){width="9cm"} ![ Snapshots of the (a) as-quenched and (b) well-aged model at $\dot{\gamma} t = 0.4$. Atoms are colored according to their nonaffine squared displacements, $D^2_{\text{min}}$ (in $\AA^2$), where displacements were calculated in reference to the positions of the atoms at $\dot{\gamma} t = 0$. []{data-label="fig:snapshots-cum"}](Fig-snapshots-cum.eps){width="9cm"} Snapshots of the as-quenched and well-aged models at $\dot{\gamma} t = 0.4$ are shown in Figs. \[fig:snapshots-cum\](a) and (b), respectively, where the color of the atoms represents the magnitude of the nonaffine squared displacements, $D^2_{\text{min}}$, from the initial position of each atom [@FalkLanger1998STZ]. The magnitude of the displacement indicates the extent of plastic deformation caused by local atomic rearrangements. Visualization of the snapshots and calculation of the displacements were performed using the Open Visualization Tool (OVITO) [@Stukowski2010Ovito]. From the snapshots, shear localization occurred in the well-aged model, in contrast to the relatively homogeneous deformation pattern in the as-quenched model. Atoms exhibiting plastic deformation localized in two band-like regions along the horizontal and vertical directions in the snapshot of the well-aged model \[Fig. \[fig:snapshots-cum\](b)\]. The emergence of the vertical shear banding was caused by conjugate shear strain, in turn caused by the applied simple shear deformation (a similar behavior of supercooled liquids was discussed in Reference  [@Furukawa2009AnisotropicAmorphousDeform]). The above results reproducing a heterogeneous deformation with the large yielding drop in the thermal annealed metallic glass and homogeneous deformation in the non-annealed glass were consistent with those of a previous study [@Shi2007MG-DisorderT; @Zhang2015Processing-dep-MG; @Wakeda2015MGRejuv]. ![ Statistical distributions of the (a) stress drop, (b) duration, and (c) waiting time of plastic deformation (avalanche) events. The distributions obtained from the as-quenched and well-aged models are represented by red squares and blue circles, respectively.[]{data-label="fig:dist"}](Fig-dist.eps){width="9cm"} To verify whether our simulations accurately reproduced the avalanche plasticity, we calculated the statistical distributions of the stress drop, $\Delta\sigma$, duration, $T$, and waiting time, $\tau$, of each [*elementary plastic deformation event*]{}. The elementary plastic deformation event is defined as the plastic deformation caused during a monotonic decrease in the shear stress, $\sigma_{xz}(t)$. The period of the $n$-th deformation event ranges from the start time, $t^{(s)}_{n}$ (where the stress reaches the $n$-th local maximum) to the end time, $t^{(e)}_{n}$ (where the stress reaches the local minimum just after the maximum), as depicted in Fig. \[fig:t-stress\](b). By the definition, the stress drop, duration, and waiting time of the $n$-th elementary deformation event are determined by $\Delta\sigma_{n} = \sigma_{xz}(t_{n}^{(s)}) - \sigma_{xz}(t_{n}^{(e)})$, $T_{n} = t_{n}^{(e)} - t_{n}^{(s)}$, and $\tau_n = t_{n}^{(s)} - t_{n-1}^{(e)}$, respectively. The statistical distributions of the stress drop, duration, and waiting time, $P(\Delta\sigma)$, $P(T)$, and $P(\tau)$ are depicted in Figs. \[fig:dist\](a), (b), and (c), respectively. All distributions decayed algebraically over a range of one or more orders of magnitude (i.e., the stress fluctuation by plastic deformation follows power-law statistics). The exponents of the power-law distributions are not significantly out of the range of those reported in previous studies; the exponents of event size, duration, and waiting time have been estimated around $1$, $3$, and $1$, respectively, in proceding studies [@Maloney2004AmorphousAvalanches; @Sun2010MetallicGlassSOC; @Hatano2015granular-avalache; @Salerno2013inertiaGlassAvalanches; @Zhang2017scalingGlassAvalanches]. This indicates that our simulations successfully reproduced the avalanche behavior of amorphous solids, where each elementary plastic deformation corresponded to one avalanche of plastic deformation. Thus, we simply refer to elementary deformation events as [*elementary avalanche events*]{} or [*avalanche events*]{}. By comparing the statistics of the as-quenched and well-aged models, we can see that both distributions follow the same power-law distribution. However, there is a notable difference in the size distributions, $P(\Delta \sigma)$, where the maximum size of the stress drops in the well-aged model is approximately four times larger than that observed in the as-quenched model \[Fig. \[fig:dist\](a)\]. Thus, the well-aged glasses have the potential to produce larger avalanches than the as-quenched glasses, whereas plastic deformation of the well-aged model is limited to a narrow band region as depicted in Fig. \[fig:snapshots-cum\](b). Technique for evaluating the avalanche geometry {#sec:avalanche-geometry} =============================================== In this section, to clarify the spatial features of avalanche plasticity in the as-quenched and well-aged glasses, we introduce a spatial correlation function of the nonaffine squared displacements [@FalkLanger1998STZ] and demonstrate that this correlation function can quantify the geometry of the elementary avalanche events. The nonaffine squared displacements for the $n$-th avalanche event were calculated using the displacements of atoms at $t^{(e)}_n$ with reference to the positions of the atoms at $t^{(s)}_n$. We refer to the nonaffine squared displacement of the $i$-th atom at the $n$-th avalanche event as $D^2_{\text{min}}({\mbox{\boldmath $r$}}_i(t_n^{(e)}))$, where ${\mbox{\boldmath $r$}}_i(t_n^{(e)})$ is the position of the $i$-th atom at $t_n^{(e)}$. ![ Snapshots at the three typical elementary avalanche events that were associated with stress drops at (a) $\dot{\gamma} t = 0.0765$, (b) $0.0960$, and (c) $0.0988$, where atoms are colored according to their nonaffine squared displacements, $D^2_{\text{min}}({\mbox{\boldmath $r$}}_i(t_n^{(e)}))$ (in $\AA^2$), and only atoms with $D^2_{\text{min}} \ge 1~\AA^2$ are shown (see text). (d) Segmentary view of the stress-strain curves including the stress drops corresponding to the snapshots (a), (b), and (c) indicated by arrows, respectively. Enlarged stress-strain curve of the as-quenched model around the stress drop event at $\dot{\gamma} t = 0.0765$ is imposed. []{data-label="fig:snapshots"}](Fig-snapshots.eps){width="9cm"} Typical snapshots of atoms at several avalanche events are depicted in Fig. \[fig:snapshots\], where atoms with $D^2_{\text{min}} < 1\ \AA$ are not shown. It can be observed that atoms contributing to plastic deformation tend to localize in space. As observed in Fig. \[fig:snapshots\](a), the participating atoms clump spherically (the cluster consists of 333 atoms) after a small avalanche with a very small stress drop ($\Delta \sigma = 0.817 \times 10^{-3}$ GPa) occurs in the as-quenched model. This clumping shape indicates that the avalanche of atomistic rearrangements in the event developed in an [*isotropic*]{} manner. In a larger avalanche event that is observed in the as-quenched model \[Fig. \[fig:snapshots\](b)\], the avalanche shape is no longer spherical and does not show any discernible direction. Thus, it can be considered that this massive avalanche is also isotropic. This is quantitatively verified later. In contrast to the as-quenched model, participating atoms in a large avalanche event in the well-aged model clearly gather around a band-like region along the $z$-axis \[Fig. \[fig:snapshots\](c)\] (i.e., the avalanche developed [*anisotropically*]{}). Note that the highly deformed region (red in the snapshot) approximately corresponds to the shear-banding region depicted [in Fig. \[fig:snapshots-cum\](b).]{} Whereas the direct observation is useful to yield intuitive recognition of the geometrical nature of avalanches, the observation is qualitative and depends on the threshold value of $D_{\text{min}}^2$. Thus, the quantification of the extent of the propagated region in an avalanche event with no consideration of a threshold value is required. For this, we introduce a planar spatial correlation function, $C_{xy}, C_{yz}$, and $C_{zx}$, of $D^2_{\text{min}}({\mbox{\boldmath $r$}}_i(t_n^{(e)}))$ at an avalanche event described below. In this study, $C_{\alpha \beta}$ is introduced as a two-body correlation function that can be given as follows (its correlation is limited to a plane parallel to the $\alpha \beta$ plane): $$\begin{aligned} C_{\alpha \beta}(r_{\alpha \beta}) &= \frac{ {\left< \delta D({\mbox{\boldmath $r$}}_i) \cdot \delta D({\mbox{\boldmath $r$}}_j) \right>}_{ij}^{{\alpha \beta}} } { {\left< \delta D({\mbox{\boldmath $r$}}_i)^2 \right>}_i }, \label{eq:corr-func}\end{aligned}$$ where $(\alpha, \beta, \gamma)$ are the cyclic permutations of $(x, y, z)$, $r_{\alpha \beta}$ is the distance between two points on the $\alpha \beta$ plane, and $\delta D({\mbox{\boldmath $r$}}_i) \equiv D^2_{\text{min}}({\mbox{\boldmath $r$}}_i(t_n^{(e)})) - {\left< D^2_{\text{min}}({\mbox{\boldmath $r$}}_k(t_n^{(e)})) \right>}_k$. The bracket, ${\left< \right>}_k$, refers to the average of any one body scalar quantities $A({\mbox{\boldmath $r$}}_k)$ over all the atoms: ${\left< A({\mbox{\boldmath $r$}}_k) \right>}_k = \sum_{k=1}^N A({\mbox{\boldmath $r$}}_k)/N$, where $N$ is the total number of atoms in a system. The bracket, ${\left< \right>}^{{\alpha \beta}}_{ij}$, is the average of any two-body scalar quantities, $B({\mbox{\boldmath $r$}}_i, {\mbox{\boldmath $r$}}_j)$, over specific particle pairs, such that the distance between two atoms on the ${\alpha \beta}$ plane, $\sqrt{(\alpha_i-\alpha_j)^2 + (\beta_i-\beta_j)^2}$, is in a range from $r_{\alpha \beta}$ to $r_{\alpha \beta} + \Delta r$, and the distance along the $\gamma$ axis, $|\gamma_i - \gamma_j|$, is less than $\Delta \gamma$. We used a $\Delta r$ value of $0.01$ nm and selected a sufficiently small width of $0.8$ nm for $\Delta \gamma$. By employing a window function, the planar average, ${\left< \right>}^{{\alpha \beta}}_{ij}$, is explicitly described as $$\begin{aligned} {\left< B({\mbox{\boldmath $r$}}_i, {\mbox{\boldmath $r$}}_j) \right>}_{ij}^{{\alpha \beta}} &= \frac{ {\left< w_{\alpha \beta} ( |\gamma_i-\gamma_j|; \Delta \gamma ) \ w_{r}( r_{\alpha \beta} - |{\mbox{\boldmath $r$}}_{ij}|; \Delta r ) \ B({\mbox{\boldmath $r$}}_i, {\mbox{\boldmath $r$}}_j) \right>}_{ij} } {{\left< w_{\alpha \beta} ( |\gamma_i-\gamma_j|; \Delta \gamma ) \ w_{r}( r_{\alpha \beta} - |{\mbox{\boldmath $r$}}_{ij}|; \Delta r ) \right>}_{ij} }, \label{eq:ave-ab}\end{aligned}$$ where $w_r$ and $w_{\alpha \beta}$ are the rectangular window functions given as follows: $$\begin{aligned} w(x; \Delta x) = \begin{cases} 1 & ( 0 \le x < \Delta x)\\ 0 & ( \text{otherwise}), \end{cases}\end{aligned}$$ and the bracket, ${\left< \right>}_{ij}$, is the average over all particle pairs: $ {\left< B({\mbox{\boldmath $r$}}_i, {\mbox{\boldmath $r$}}_j) \right>}_{ij} = \frac{2}{N(N-1)} \sum_{i<j}^N B({\mbox{\boldmath $r$}}_i, {\mbox{\boldmath $r$}}_j)$. Note that if one omits $w_{\alpha \beta}$ from Eq.  (\[eq:ave-ab\]), $C_{\alpha \beta}$ becomes the conventional spatial correlation function of the distance between any two atoms. By integrating the correlation function, we can obtain the spatial correlation length of an avalanche event along the $\alpha \beta$ plane corresponding to the average linear size of one avalanche area projected onto the $\alpha \beta$ plane in the following manner: $$\begin{aligned} {\left< r \right>}_{\alpha \beta} = { \int r_{\alpha \beta} \ C_{\alpha \beta}(r_{\alpha \beta}) \ {\text{d} {r_{\alpha \beta}}} }\ {\LARGE /} { \int C_{\alpha \beta}(r_{\alpha \beta}) \ {\text{d} {r_{\alpha \beta}}} }.\end{aligned}$$ The functions, especially correlation lengths, ${\left< r \right>}_{xy}, {\left< r \right>}_{yz}$, and ${\left< r \right>}_{zx}$, indicate how far one avalanche spread parallel to $xy$, $yz$, and $zx$ planar directions, respectively. The ratio of these correlation lengths provides information regarding the geometry of one avalanche (or the aspect ratio of the avalanche shape). It should be noted that the spatial correlations introduced in this study were applied to individual avalanche events, whereas similar spatial correlations employed in previous studies were applied to the accumulation of some avalanche events [@Nicolas2014GlassSpatioCorr; @Shang2014MGstraincorr]. ![ Spatial correlation functions of the nonaffine squared displacement at $\dot{\gamma} t$ values of (a) $0.0765$, (b) $0.0960$, and (c) $0.0988$, where the functions at $\dot{\gamma} t = 0.0765$ and $0.0960$ are obtained from the as-quenched model and that at $\dot{\gamma} t = 0.0988$ is obtained from the well-aged model. The correlation functions along the $xy, yz$, and $zx$ planes are depicted by the red, blue, and green lines, respectively. []{data-label="fig:corr-func"}](Fig-corr-func.eps){width="9cm"} Here we illustrate how this spatial correlation function works to characterize the geometry of the elementary avalanche events using several examples in the as-quenched and well-aged model. The planar spatial correlation functions for typical avalanches are shown in Fig. \[fig:corr-func\], where the correlation functions, $C_{xy}(r_{xy}), C_{yz}(r_{yz})$, and $C_{zx}(r_{zx})$, are colored red, black, and green, respectively. The correlation functions depicted in Fig. \[fig:corr-func\](a) are for the event at $\dot{\gamma} t = 0.0765$ in the as-quenched model. The corresponding snapshot is shown in Fig. \[fig:snapshots\](a). All correlation functions in the figure decay following the exponential function, $C_{\alpha \beta}(r_{\alpha \beta}) \propto \exp[ - r_{\alpha \beta}/\bar{r}]$, and show the characteristic length, $\bar{r} \simeq 0.5$ nm, which is consistent with the average correlation lengths of the event: ${\left< r \right>}_{xy} = 0.679, {\left< r \right>}_{yz} = 0.725$, and ${\left< r \right>}_{zx} = 0.627$ nm. The small differences between $\bar{r}$ and ${\left< r \right>}_{\alpha \beta}$ are attributed to the excluded volume effect of atoms that results in a lack of the correlation in the regime, $0 \le r_{\alpha \beta} \lesssim 0.2$ nm. The result clearly indicates that the avalanche region for this event is almost isotropic, consistent with the direct observation of the event \[Fig. \[fig:snapshots\](a)\]. The correlation functions with larger characteristic lengths, as shown in Fig. \[fig:corr-func\](b), correspond to a larger avalanche event at $\dot{\gamma} t = 0.0960$ in the as-quenched model, where the corresponding snapshot is shown in Fig. \[fig:snapshots\](b). The correlation functions decay exponentially in accordance with various cut-offs, and the correlation lengths are ${\left< r \right>}_{xy} = 0.886$, ${\left< r \right>}_{yz} = 1.204$, and ${\left< r \right>}_{zx} = 0.891$ nm. Whereas these various correlation lengths imply the anisotropic geometry of the event, the difference observed is approximately 36 % at most. In contrast with the characteristics of the events in the as-quenched model, a more anisotropic feature can be found in the well-aged model. The spatial correlation functions of an avalanche at $\dot{\gamma}t = 0.0988$ in the well-aged model \[Fig. \[fig:corr-func\](c)\] show that $C_{yz}$ only shows a small decrease with an increasing $r_{yz}$. Actually, it follows an algebraic decay, whereas $C_{xy}$ and $C_{zx}$ decay exponentially. As a result, the correlation length parallel to the $yz$ plane is about twice larger than that to the $xy$ and $xz$ planes. ${\left< r \right>}_{xy}, {\left< r \right>}_{yz}$, and ${\left< r \right>}_{zx}$ are $1.052, 1.917$, and $0.978$ nm, respectively. The anisotropy in the estimated correlation lengths of the avalanche is consistent with the snapshot depicted in Fig. \[fig:snapshots\](c). These results verify that spatial correlation functions and correlation lengths can be used to characterize the geometric features of one avalanche event. Spatiotemporal features of anisotropic avalanches {#sec:spatio-temporal} ================================================= In this section, we compare overall trends in avalanche geometry in the as-quenched and well-aged models by estimating the aspect ratio of all avalanche geometries. ![ Relationship between the spatial correlation lengths of $D^2_{\text{min}}$ on the $xy$, $yz$, and $zx$ planes for (a)(c)(e) the as-quenched model and (b)(d)(f) the well-aged model. []{data-label="fig:ave-R-corr"}](Fig-r_ab.eps){width="9cm"} Figure \[fig:ave-R-corr\] shows the correlation lengths, ${\left< r \right>}_{xy}$, ${\left< r \right>}_{yz}$, and ${\left< r \right>}_{zx}$, calculated from all avalanche events as functions of each other’s correlation length. The data shown in Figs. \[fig:ave-R-corr\](b) and (d) deviate from the linear relation (indicated by the diagonal line) to the upper-left region of the graphs (i.e., dashed boxes). This deviation indicates that avalanches in the well-aged model tend to evolve parallel to the $xy$ or $yz$ planar directions rather than the $xz$ direction. These $xy$ and $yz$ directions correspond to the observed shear banding \[Fig. \[fig:snapshots-cum\](b)\]. In this discussion, we focus on ${\left< r \right>}_{xy}$ and ${\left< r \right>}_{yz}$, the correlation lengths parallel to these two preferable directions. Figs. \[fig:ave-R-corr\](e) and (f) show ${\left< r \right>}_{yz}$ values obtained in the as-quenched and well-aged models as a function of ${\left< r \right>}_{xy}$. The data for the well-aged model spread farther away from the diagonal line, compared to those of the as-quenched model at higher correlation lengths indicated by the dashed boxes. These data indicate that the avalanches in the well-aged model tend to evolve parallel to the two directions, whereas avalanche evolution in the as-quenched model is nearly isotropic. Thus, structural relaxation by thermal annealing in metallic glasses can enhance both the localization of plastic deformations and anisotropy of the elementary avalanche propagations. This anisotropy seems to depend on the size of the avalanche event. In Figs. \[fig:ave-R-corr\](b), (d), and (f), the plots significantly deviate from the diagonal trend line for larger correlation lengths, ${\left< r \right>}_{\alpha \beta} \gtrapprox 1$ nm. Hence, anisotropy is mainly observed in avalanche events with a large deformation area (as discussed in Section  \[sec:anisotropy-size\]). ![ Aspect ratios of the region over which the avalanche propagates during a plastic event as a function of strain; $R = {\left< r \right>}_{yz}/{\left< r \right>}_{xy}$ for the (a) as-quenched model and (b) well-aged model. The boxes indicated by dashed line and dash-dotted line are events during primary and secondary stage, respectively. []{data-label="fig:t-aspect-R"}](Fig-t-aspectR.eps){width="9cm"} In addition to the spatial correlations, we investigate the temporal clustering of the avalanche anisotropy. Figure \[fig:t-aspect-R\] shows the aspect ratio of the geometry of each avalanche event in the form of the ratio of correlation lengths, $R = {\left< r \right>}_{xy}/{\left< r \right>}_{yz}$, as a function of time (strain). An $R$ value of 1 for an avalanche signifies isotropic behavior, whereas $R > 1$ and $R<1$ indicate that the anisotropic avalanche propagations are biased toward the $xy$ and $yz$ planar directions, respectively. Note that, in this figure, the aspect ratio is plotted on a logarithmic scale to show the two directions equivalently. Figure \[fig:t-aspect-R\](a) shows that $R$ of the as-quenched model is evenly spread around unity, indicating quite isotropic behavior. This is consistent with the results shown in Figs. \[fig:ave-R-corr\](a), (c), and (e). In contrast, the aspect ratio of the well-aged model shown in Fig. \[fig:t-aspect-R\](b) tends to be higher than unity in the primary stage (enclosed by the dashed line), whereas in the secondary stage, the $R$ values tend to be lower than unity from around $\dot{\gamma}t = 0.32$ (enclosed by a dashed-dotted line). This indicates that the preferred direction of avalanche propagation in the well-aged model switched from the $yz$ to $xy$ direction at around $\dot{\gamma}t = 0.32$. The direction of avalanche propagation is not determined randomly but is clustered over time. This temporal clustering behavior of the avalanche directions in the well-aged model may indicate weakening by plastic deformation. The region where a previous avalanche occurred can be weakened, facilitating subsequent avalanches occurring in the same region and direction. This behavior implies that avalanche plasticity in amorphous solids has the memory of an area that was weakened by previous avalanches. To comprehensively quantify the tendencies in the anisotropic feature of the avalanche propagation, we calculated the sample deviation, $d$, of the aspect ratio from unity: $$\begin{aligned} d^2 = {\left< (R' - 1)^2 \right>}, \label{eq:deviation-R}\end{aligned}$$ where the bracket represents the sample average of all avalanche events. We calculate the aspect ratio, $R'$, as ${\left< r \right>}_{xy}/{\left< r \right>}_{yz}$ or ${\left< r \right>}_{yz}/{\left< r \right>}_{xy}$ in such a way that $R' \ge 1$. We do this to treat the deviations in the two directions equivalently. The deviations calculated from the results of the as-quenched model and the well-aged model are $d = 0.1554$ and $0.1890$, respectively. The deviation of the well-aged model increases by approximately $22$%. The result also indicates that the structural relaxation by thermal treatment enhances the anisotropic propagation of the elementary avalanches in metallic glasses. Correlation between anisotropy and size of an avalanche {#sec:anisotropy-size} ======================================================= ![ Relationship between the aspect ratio $R$ and the magnitude of an avalanche event $\Delta \sigma$ for the (a) as-quenched model and (b) well-aged model. []{data-label="fig:aspect-R-ds"}](Fig-rel-ds-aspectR.eps){width="9cm"} The relationship between the magnitude and anisotropy of an avalanche event is investigated. Fig. \[fig:aspect-R-ds\] shows the relationship between the aspect ratio, $R$, and the stress drop, $\Delta \sigma$, corresponding to the magnitude of each elementary avalanche event, where only the region of interest of $\Delta \sigma \ge 10^{-3}$ GPa is shown. It can be seen that $\Delta \sigma$ has a wide range of magnitudes. The plots of the as-quenched model \[Fig. \[fig:aspect-R-ds\](a)\] show no significant correlation between $\Delta \sigma$ and $R$, whereas there is a clear correlation in the regime $\Delta \sigma > 10^{-1}$ GPa for the well-aged model as shown by the bifurcated plots spreading along the diagonal lines in Fig. \[fig:aspect-R-ds\](b). Thus, larger avalanche events result in stronger anisotropy of the deformation region. This trend is consistent with anisotropic avalanches being mainly observed for large deformation regions, as discussed in Section \[sec:spatio-temporal\]. See also Fig. \[fig:ave-R-corr\](b), (d), and (f). Thus, elementary deformation avalanches in well-aged metallic glasses have a tendency to propagate anisotropically in large deformation areas. The observation that larger avalanches show stronger anisotropy provides interesting insights. An avalanche of plastic deformation of metallic glasses develops mainly via propagation of local atomistic rearrangements from a small starting region. Thus, small avalanches can be considered as the propagation of atomic rearrangements with their development halts at an early stage. In contrast, the development of large avalanches does not stop until a later stage. Thus, we can consider that a small isotropic avalanche will develop anisotropically along a favored direction from a certain point in time rather than a small anisotropic avalanche increases in size with no change in its aspect ratio. In other words, the propagation of a deformation avalanche in metallic glasses may bifurcate during the avalanche growth. The time evolution behavior of the avalanches requires more detailed analysis. Size dependence of avalanche anisotropy {#sec:size-depend} ======================================= ![ (Upper panels) Aspect ratios of the avalanche regions as a function of strain; $R = {\left< r \right>}_{yz}/{\left< r \right>}_{xy}$ for the larger (a) as-quenched model and (b) well-aged model. (Lower panels) Relationship between the aspect ratio $R$ and the magnitude of an avalanche event $\Delta \sigma$ for the larger (a) as-quenched model and (b) well-aged model. Snapshots of the larger as-quenched and larger well-aged model at $\dot{\gamma} t = 0.4$ are inserted in (c) and (d), respectively. Atoms are colored according to $D^2_{\text{min}}$ (in $\AA^2$), where the displacements were calculated in reference to the positions of the atoms at $\dot{\gamma} t = 0$. []{data-label="fig:N=40E+4"}](Fig-L-all.eps){width="9cm"} Whereas the avalanches in the well-aged model tend to develop anisotropically, the aspect ratios of these anisotropic avalanches are not as large as one might expect. The observed $R$ values remain in the range of $0.5$–$2$, as shown in Fig. \[fig:t-aspect-R\](b). These small values are caused by the finite size effect. Thus, the integration range for calculating ${\left< r \right>}_{\alpha \beta}$ is limited to half of the length of the simulation cell defined by the periodic boundary condition. This also explains why ${\left< r \right>}_{yz}$ was only $< 2$ nm, even when $C_{yz}$ decayed algebraically \[Fig. \[fig:ave-R-corr\](c)\]. In the present simulations, we used a periodic cell with sides of $\sim 9.5$ nm. Therefore, the correlation length, ${\left< r \right>}_{\alpha \beta}$, can be only $2$ nm at a maximum, even when $C_{\alpha \beta}$ does not decay at all, because the maximum integration range is about $4$ nm. Thus, MD simulations with larger systems are expected to result in enhanced anisotropic behavior. To confirm this, we calculated the aspect ratios for additional MD simulations employing larger models with $19.3$ nm edges in the initial state ($400,000$ atoms), performed under the same simulation conditions. The temporal evolution of the aspect ratios of the elementary avalanche regions and the relationship between aspect ratio $R$ and the event size, $\Delta \sigma$, of each avalanche are shown in Fig. \[fig:N=40E+4\]. Snapshots of the larger as-quenched and well-aged models at $\dot{\gamma} t = 0.4$ are inserted in Figs. \[fig:N=40E+4\](c) and (d), respectively. The snapshot shows that the larger well-aged model also exhibits shear localization. An intense concentration of plastic deformation appears in a horizontal band-like area in the larger well-aged model \[Fig. \[fig:N=40E+4\](d)\]. Whereas the larger as-quenched model also exhibits a certain level of localization \[Fig. \[fig:N=40E+4\](c)\], its deformation region spreads much wider than that of the well-aged model. This difference indicates that the structural relaxation caused by thermal annealing enhances the localization of deformation, even in larger models. A relatively strong bias in the aspect ratio of isotropic shapes in the larger well-aged model, compared to the one in the larger as-quenched model, is clearly visible in Figs. \[fig:N=40E+4\](b). The aspect ratios of some extreme events in the larger well-aged model drop to $1/3$, whereas a switch in the preferred avalanche direction is not observed in this case. This significant anisotropic trend is also verified by the deviation of aspect ratios defined by Eq.  (\[eq:deviation-R\]) in the well-aged model, as $d = 0.473$. The as-quenched model also exhibits some anisotropic avalanche propagations. However, the deviation is only $d = 0.259$. Therefore, the structural relaxation increases the deviation by approximately $83$%. As shown in Figs. \[fig:N=40E+4\](c) and (d), whereas the correlation between the magnitude and anisotropy of the avalanche events can be seen in the larger as-quenched model, and not just in the larger well-aged model, the latter is linked to a more definite correlation than the former. This correlation is more apparent in the larger well-aged model than in the smaller one. Therefore, it is confirmed that the influence of the structural relaxation on the anisotropy remains, even in the larger systems. Moreover, the relative influence in the well-aged model, compared to the as-quenched model, grows as the system size increases. Discussion {#sec:discussion} ========== The avalanche motion of deformation in amorphous solids can be interpreted as the chain reaction of local shear transformation. Atoms in a small local area (an STZ [@Argon1979STZ; @FalkLanger1998STZ]) can rearrange to release external shear stress. This rearrangement (i.e., the shear transformation of the local atomic configuration) causes stress concentration around the area, as described by the Eshelby inclusion theory, and can induce other local rearrangements through the stress concentration. The chain reaction of this shear transformation results in a shear transformation avalanche. Thus, the avalanche behavior illustrated in this study poses an important question regarding the manner in which thermal preparation enhances or suppresses the chain reaction of the shear transformation. The thermal annealing used in this study causes relaxation of the internal structural states characterized by various features, such as short and medium-range order, free volumes, or effective disorder temperature of glasses [@Sheng2006SRO-MRO-Glass; @Cohen1959MolecularTransportLiquid; @Shi2007MG-DisorderT; @Falk2011STZtheory]. The structural states and the resultant shear banding are well described by the effective temperature [@Shi2007MG-DisorderT; @Falk2011STZtheory], but we herein provide a speculative explanation of the avalanche features based on the atomistic free volumes [^1] because free volume is a convenient and intuitive concept in the discussion of atomic-scale dynamics. As discussed before, the as-quenched model exhibiting relatively small avalanches has more free volume than the well-aged model as mentioned in Sec. \[sec:method\], one can consider that the excess free volume in the as-quenched metallic glasses could absorb and thus prevent chain reactions in local shear transformations. Moreover, large free volumes might allow the surrounding regions to transform along with directions other than that of external shear direction because excess free volumes can provide additional space for surrounding atoms to move and rearrange themselves, independent of the external shear direction. Conversely, the well-aged glass with less free volume lacks adequate space to absorb the chain reaction. This may explain the as-quenched glass model neither showing large nor anisotropic avalanches [^2] . Although there seems to be a significant relationship between shear banding and the avalanche behaviors, further researches on this subject is essential so that a more comprehensive study based on evidences can be performed. The scope of the present study comprises, only of a comparison of shear-banding patterns \[Fig. \[fig:snapshots-cum\](b)\] and the snapshot of a large avalanche event emerging in the well-aged model \[Fig. \[fig:snapshots\](c)\]; the focal region of the avalanche largely overlaps with the shear banding region. This shows that relatively few extremely large and anisotropic (strip-shaped) avalanches, not many small and isotropic avalanches contribute to shear banding in metallic glasses. The influence of environmental temperature on avalanche behaviors, playing an essential role in amorphous plasticity [@Rodney2009distSTZact; @Cao2013Strain-T-dependAmorphous; @Fan2014STZactivation] is another subject of future research. It is expected that the thermal fluctuation by the temperature will provide both enhancement and suppression effects on the avalanche growth and anisotropy; the enhancement is due to the reduction of atomistic free volume caused by structural relaxation driven by the temperature, and the suppression effect is produced by the thermal activation of the STZs, which is supposed to be activated as part of an avalanche when the temperature was low, prior to the avalanche. The opposition of these two effects at high temperature could affect the avalnache behaviors. In this discussion, we assumed that the excess free volumes of metallic glasses leads to the ease of deformation in local area. This assumption is inferred from the fact that the as-quench glass with a large free volume started to deform at a relatively lower stress than the well-aged glass with a higher energy state [@Wakeda2015MGRejuv] as shown in Fig. \[fig:t-stress\]. (Note that a large free volume does not always lead to less strength and high energy state [@Miyazaki2016RejuveMG].) A more valid and comprehensive explanation for the avalanche behaviors with respect to other factors such as effective temperature is necessary and should be a subject of future investigation. Conclusion ========== Using three-dimensional molecular dynamics simulations of shear deformations in Cu–Zr metallic-glass models, we compared the avalanche plasticity of a well-relaxed metallic-glass model (i.e., well-aged model) with localized deformation produced by thermal annealing and a less-relaxed glass model (as-quenched model) that did not undergo structural relaxation by annealing, showing homogeneous deformation. We focused on analyzing the geometrical feature of elementary shear deformations (i.e., elementary shear transformation avalanches). The simulations showed that the statistics of stress drops, durations, and waiting times of elementary avalanche events followed power-law distributions for both models. The as-quenched model showed a homogeneous deformation pattern, whereas a heterogeneous pattern like a shear band was observed for the well-aged model, as shown in the previous numerical and experimental studies [@Shi2007MG-DisorderT; @Zhang2015Processing-dep-MG; @Wakeda2015MGRejuv]. We quantified the geometrical features of elementary shear transformation avalanches by introducing a planar spatial correlation function of the nonaffine squared displacements of all atoms in each avalanche event. The spatial correlations and their characteristic lengths parallel to the $xy, yz$, and $zx$ planes revealed that deformation regions caused by each avalanche event in the well-aged glass tended to be anisotropic, whereas those in the as-quenched glass were generally isotropic. The direction of anisotropic avalanche propagation in the well-aged glass was not random but showed temporal clustering. Moreover, we demonstrated that the well-aged glass had a significant correlation between the anisotropy and avalanche magnitude. The observed differences between the two glass models might be attributed to differences in the atomic-free volume, which can be removed by thermal structural relaxation. This research was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) KAKENHI Grant-in-Aid for Young Scientists (B) (No. 16K17764), Grant-in-Aid for Scientific Research (A) (No. 17H01238), Grant-in-Aid for Young Scientists (A) (No. 17H04949), Challenging Research (Pioneering) (No. 17K18827). This research was also supported by “Exploratory Challenge on Post-K computer” (Challenge of Basic Science — Exploring Extremes through Multi-Physics and Multi-Scale Simulations) and used computational resources of the K computer provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID: hp180224 and hp190194). [^1]: In this discussion, we consider the atomic free volume not as a concept based on the configurational entropy but as a primitive one, i.e., excess volume per atom derived from a densely packed structure [^2]: The term, “anisotropy,” used in this study, refers to avalanche propagation that is dependent on the external loading directions. Note that this behavior can be confirmed, even in the as-quenched model, as observed in Figs. \[fig:ave-R-corr\](a) and \[fig:N=40E+4\](a) although it is relatively weak. Thus, the anisotropy discussed in this study should be considered a relative property, depending on the extent of structural relaxation

--- abstract: 'We study phase diagrams of a class of doped quantum dimer models on the square lattice with ground-state wave functions whose amplitudes have the form of the Gibbs weights of a classical doped dimer model. In this dimer model, parallel neighboring dimers have attractive interactions, whereas neighboring holes either do not interact or have a repulsive interaction. We investigate the behavior of this system via analytic methods and by Monte Carlo simulations. At zero doping, we confirm the existence of a Kosterlitz-Thouless transition from a quantum critical phase to a columnar phase. At low hole densities we find a dimer-hole liquid phase and a columnar phase, separated by a phase boundary which is a line of critical points with varying exponents. We demonstrate that this line ends at a multicritical point where the transition becomes first order and the system phase separates. The first-order transition coexistence curve is shown to become unstable with respect to more complex inhomogeneous phases in the presence of direct hole-hole interactions. We also use a variational approach to determine the spectrum of low-lying density fluctuations in the dimer-hole fluid phase.' author: - Stefanos Papanikolaou - Erik Luijten - Eduardo Fradkin title: 'Quantum criticality, lines of fixed points, and phase separation in doped two-dimensional quantum dimer models' --- Introduction {#sec:intro} ============ The behavior of doped Mott insulators is a long-standing open and challenging problem in condensed-matter physics. Mott insulators are the parent states of all strongly correlated electronic systems and as such play a crucial role in our understanding of high-$T_c$ superconductors (HTSC) and many other systems. Their strongly correlated nature implies that their behavior cannot be understood in terms of weakly coupled models. Except for the very special case of one spatial dimension, the physics of doped Mott insulators is currently only understood at a qualitative level. The solution of this challenging problem remains one of the most important directions of research in condensed-matter physics. Quantum dimer models (QDM)[@rokhsar88] provide a simplified, and rather crude, description of the physics of a Mott insulator. They provide a correct description of the physics of Mott insulators in regimes in which the spin excitations have a large spin gap. QDMs were proposed originally within the context of the resonating-valence bond (RVB) mechanism of HTSC.[@anderson87; @kivelson87] These systems are of great interest as they can yield hints on the behavior of more realistic models of quantum frustration. The main idea behind the formulation of QDMs is that, if the spin gap is large, the spin degrees of freedom become confined in tightly bound singlet states which, in the extreme limit of a very large spin gap, extend only over distance scales of the order of nearest-neighbor sites of the lattice. Thus, in this extreme regime, the Hilbert space can be approximately identified (up to some important caveats[@rokhsar88]) with the coverings of the lattice by valence bonds or dimers. Surprisingly, even at the level of the oversimplified picture offered by QDMs, the physics of doped Mott insulators remains poorly understood. In this paper we explore the phase diagrams, and the properties of their phases, of QDMs generalized to include interactions between dimers (or valence bonds), and between dimers and doped (charged) holes. Basic aspects of the physics of these models are reviewed in Ref.  and references therein. Undoped QDMs have been studied more extensively and by now they are relatively well understood.[@read-sachdev91; @sachdev-read91; @moessner01a; @moessner02a; @fradkin04; @vishwanath04; @ardonne04] On bipartite lattices, their ground states show either long-range crystalline valence bond order of different sorts or are quantum critical,[@rokhsar88; @fradkin91; @moessner02a] while on non-bipartite lattices their ground states are typically disordered and are topological fluids.[@moessner01a] The more physically relevant *doped* QDMs, with a finite density of charge carriers (holes), are much less understood, although some properties are known.[@fradkin90a; @balents05b; @alet05; @syljuasen05; @alet06; @Castelnovo06; @Poilblanc06] In QDMs a spin-$\frac{1}{2}$ hole fractionalizes into a bosonic [*holon*]{}, an excitation that carries charge but no spin, and a [*spinon*]{}, a fermionic excitation that carries spin but no charge.[@kivelson87; @rokhsar88; @fradkin90a; @fradkin91] Holons can be regarded as sites that do not belong to any dimer, whereas spinon pairs are broken dimers. This form of electron fractionalization is observable in the spectrum of these systems only in the topological disordered (spin-liquid) phases of the undoped QDM. Otherwise, as in the case of the valence bond crystalline states which exhibit long range dimer order, spinons and holons are [*confined*]{} and do not exist as independent excitations.[@fradkin91] In this paper, we consider several interacting QDMs on a square lattice at finite hole doping, and discuss their possible phases and phase transitions as a function of hole density and strength of the interactions. At any finite amount of doping the system will have a finite density of holes, which are hard core charged bosons in this description. To simplify the problem, in this work we do not consider the physical effects of the charge-neutral fermionic spinons which in principle should also be present. Thus, at this level of approximation, all spin-carrying excitations are effectively projected out. The remaining degrees of freedom are thus dimers (“spin-singlet valence bonds”) and charged hard-core bosonic holes. Already this simplified picture of a strongly correlated system is very non-trivial. For a certain relation between its coupling constants, known as the Rokhsar-Kivelson (RK) condition,[@rokhsar88] QDM Hamiltonians, both with and without holes, can be written as a sum of projection operators. These RK Hamiltonians are manifestly positive definite operators. For this choice of couplings, the ground-state wave function is a zero-energy state which is known exactly. This [*RK wave function*]{} is a local function of the degrees of freedom of the QDM, the local dimer and hole densities. The quantum-mechanical amplitudes of the RK wave functions turn out to have the same form as the Gibbs weights of a two-dimensional (2D) *classical* dimer problem with a finite density of holes. For the generalized doped QDMs that we consider, the norm of the exact ground-state wave function is equal to the partition function of a system of interacting classical dimers at finite hole density. This mapping to a 2D classical statistical mechanical system, for which there is a wealth of available results and methods, makes this class of problems solvable.[@rokhsar88; @moessner01a; @ardonne04; @Castelnovo06] In this work we will investigate the behavior of doped QDMs which satisfy the RK conditions by studying the correlations encoded in their ground state wave functions. The phase diagrams of these systems turn out to be quite rich. As we shall see, these simple models can describe many aspects of the physics of interest in strongly correlated systems, including a dimer-hole liquid phase (a Bose-Einstein condensate of holes), valence-bond crystalline states, phase separation, and more general inhomogeneous phases. The undoped version of this system was studied in detail in Ref. , where a quantum phase transition was found that was argued to belong to the Kosterlitz-Thouless (KT) universality class, from a critical phase to a columnar state with long-range order. In this paper we confirm that this is indeed the case. At finite hole density, hitherto available results are limited to the form of the associated RK QDM Hamiltonian[@Castelnovo06; @Poilblanc06] and numerical results for small systems. In this work we employ analytic methods[@kadanoff77; @ginsparg88; @boyanovsky89a; @lecheminant02] combined with advanced classical Monte Carlo (MC) simulations[@krauth03; @liu04] to probe the correlations in the [*doped*]{} RK wave functions, investigate the phase diagram and its phase transitions. The methods used here can be readily generalized to the case of non-bipartite lattices, for which a number of important results have been published.[@moessner01a; @fendley02; @trousselet07] In Section \[sec:hamiltonians\], we describe the construction of two generalized quantum dimer RK Hamiltonians that we used in our study. A similar but independent construction has been presented by Castelnovo and coworkers[@Castelnovo06] and by Poilblanc and coworkers[@Poilblanc06]. The RK wave functions of these generalizations of the quantum dimer model have either a fixed number of holes or a variable number of holes and a fixed hole fugacity. The ground-state wave functions of both models at their associated RK points correspond to a canonical dimer-monomer system in the canonical and grand-canonical ensembles respectively. Near the end of the paper, in Section \[sec:hole-int\], we introduce a third Hamiltonian, with an associated RK wave function, to study the effects of hole interactions which compete with phase separation at the first-order transitions that we find for both models. In Sections \[sec:mean-field\] and \[sec:exact-critical-behavior\] we study the correlations and the phase diagram for the ground states encoded in these wave functions by means of an analysis of the equivalent classical statistical system of dimers and holes for the RK Hamiltonians of Section \[sec:hamiltonians\]. In Section \[sec:mean-field\] we summarize the results of a mean-field theory for both non-interacting and interacting classical dimer models at finite doping. The details of the mean-field theory are presented in Appendix \[app:mean-field-details\], where we compute the hole-hole correlation function and derive a qualitative phase diagram as a function of hole density and dimer interaction parameters. The main result of this simple mean-field theory is that the phase diagram at finite hole density contains two phases, a dimer-hole fluid and a columnar dimer solid. The columnar-liquid transition is continuous at weak coupling and turns first order at a tricritical point. Naturally, the critical and tricritical behavior are not correctly described by the mean-field theory, although the general topology of the phase diagram is correct and, remarkably, even the location of the tricritical point is consistent with what we find in the MC simulations of Section \[sec:MC\]. In Section \[sec:exact-critical-behavior\] we present a detailed analytic theory of the critical behavior of interacting classical dimers. Sections \[sec:zero-density\] and \[sec:finite-density\] focus on the field-theoretic Coulomb-gas approach for this model at zero and finite doping, respectively. We show that, up to a critical value of a parameter, the undoped RK wave function describes a critical system with continuously varying critical exponents, with a phase transition (belonging to the 2D KT universality class) to a state with long-range columnar order. At finite hole doping we find a hole-dimer liquid phase (with short-range correlations) and a stable phase with long-range columnar order. At low hole densities the phase boundary is a line of fixed points with varying exponents ending at a KT-type multicritical point where the transition becomes first order. We present a field-theoretical treatment of this tricritical point and a theory of the evolution of the behavior of the columnar and orientational order parameters and of their susceptibilities along the phase boundary. Past the tricritical point the system is found to exhibit a strong tendency to phase separation, which we verify in our numerical simulations (Section \[sec:MC\]). In Section \[sec:hole-int\] we consider the effects of direct hole-hole interactions near the first-order phase boundary, and discuss one of the many inhomogeneous phases which arise in this regime instead of phase separation. In Section \[sec:MC\] we confirm our analytic predictions via extensive classical MC simulations of the generalized RK wave functions. For the study of the line of critical points at low doping we employ the canonical generalized geometric cluster algorithm (GGCA), whereas the first-order transition is studied via grand-canonical Metropolis-type simulations. The GGCA algorithm enables us to study relatively large systems, up to $400 \times 400$, for a range of dopings, $0 \leq x \leq 0.06$, and to investigate the finite-size scaling behavior. The accessible range of system sizes should be compared to what can be reached for full quantum models, away from the RK condition, where available methods, such as exact diagonalization and Green function Monte Carlo, allow the study of only very small systems with few holes.[@syljuasen05; @Poilblanc06] In the undoped case we confirm the existence of a Kosterlitz-Thouless transition from a line of critical points to an ordered columnar state, as found in the work of Alet and coworkers.[@alet05; @alet06] We study the scaling behavior of the columnar and orientational order parameters and of their susceptibilities. We also use a mapping of the orientational order parameter of the interacting classical dimer model to the staggered polarization operator obtained by Baxter for the six-vertex model[@baxter73] to fit our MC data and find an accurate estimate of the KT transition coupling in the undoped case. At low doping, we study the transition from the dimer-hole fluid phase to the columnar state. We confirm that the scaling dimension of the columnar order parameter operator is equal to $1/8$, as predicted by our analytic results of Section \[sec:finite-density\]. We also present a typical set of data that demonstrates how the scaling dimension of the orientational order parameter varies, again in agreement with the analytic results of Section \[sec:finite-density\], and use these results to locate numerically the phase boundary. We then turn to the behavior at larger doping and stronger couplings where the transition becomes first order. We study this regime using grand-canonical Metropolis Monte Carlo simulations. We confirm the first-order nature of the phase transition by means of a careful analysis of the finite-size scaling behavior of the order parameters across the phase boundary and of their susceptibilities. We use these results to locate the phase boundary in the first-order regime as well. In Section \[sec:hole-int\], we use MC simulations to study the effects of a direct hole-hole repulsion which suppresses the effects of phase separation, leading instead to a complex phase diagram of inhomogeneous phases, of which we only study its most commensurate case. In Section \[sec:SMA\], we study the elementary quantum excitations of the doped QDMs satisfying the RK condition using the single-mode approximation. We only present the main results and have relegated the details to Appendix \[app:fks\]. We find that in the dimer-hole liquid phase, hole and dimer density fluctuations have quadratic dispersions $E(k)\sim k^2$. Thus this phase should be characterized as a Bose-Einstein condensate of bosonic charged particles (holes), but not really a superfluid, for reasons similar to those of Rokhsar and Kivelson.[@rokhsar88] We summarize our overall conclusions in Section \[sec:conclusions\]. While this manuscript was being completed (and refereed) a number of independent studies of aspects of this problem have been published.[@Castelnovo06; @Poilblanc06; @alet06] Our results agree with those in these references wherever they overlap, as noted throughout this paper. Quantum Hamiltonians for Interacting Dimers at Finite Hole Density {#sec:hamiltonians} ================================================================== The Hamiltonian of the quantum dimer model (QDM) can be written in the Rokhsar-Kivelson (RK) form[@rokhsar88] as the sum of a set of mutually non-commuting projection operators $Q_p$, $$H=\sum_{\{p\}} Q_p \;,$$ where $\{p\}$ denotes the set of all plaquettes of the square lattice. Each projection operator $Q_p$ acts on the states of the dimers and holes of a plaquette $p$ (or set of plaquettes surrounding $p$). In the simplest case[@rokhsar88] each $Q_p$ acts only on the states labeled by the dimer occupation numbers of the links of the plaquette $p$. In this case, the ground state is described by the short-range RVB wave function[@kivelson87] $$\ket{\textrm{RVB}}=\sum_{ \{ \mathcal{C} \} } \ket{\mathcal{C}} \;,$$ where $\{ \mathcal{C} \} $ is the set of (fully packed) dimer coverings of the 2D square lattice, and $\{ \ket{\mathcal{C}} \}$ is a complete set of orthonormal states. If one regards the dimers as spin singlet states (with the spins residing on the lattice sites) each configuration represents a set of spin singlets or valence-bonds.[@anderson87; @kivelson87] The dimer representation ignores the over-completeness of the valence bond singlet states.[@rokhsar88] This problem can be made parametrically small using a number of schemes, including large $N$ approximations[@read-sachdev89] and decorated generalizations of the spin-$1/2$ Hamiltonians.[@Raman05] It is possible and straightforward to generalize the QDM construction so as to include other types of interactions and coverings. In Ref.  it was shown how to extend this structure to smoothly interpolate between the square and the triangular lattices. It was also shown there that the same ideas can be used to construct a quantum generalization of the two-dimensional classical Baxter (or eight-vertex) model. In all of these cases, the RK form of this generalized quantum dimer model has an exact ground-state wave function whose amplitudes are equal to the statistical (Gibbs) weights of an associated two-dimensional classical statistical mechanical system on the same lattice. Thus, if the classical problem happens to be a classical critical system, the associated wave function now describes a 2D problem at a quantum critical point. In Ref.  such quantum critical points were dubbed “conformal quantum critical points” since the long-distance structure of their [*ground-state wave functions*]{} exhibits 2D conformal invariance. Here we are interested in a different generalization of the QDM in which we consider dimer coverings (although not necessarily fully packed) of the square lattice. We will also consider 2D Hamiltonians whose wave functions correspond to classical interacting 2D dimer problems with local weights. Similar but independent constructions have also been proposed.[@castelnovo05; @Castelnovo06; @Poilblanc06] Trying to be as physical and local as possible, we keep the quantum-resonance terms as simple as before (single plaquette moves), but the potential terms (which again have a central plaquette[@kivelson87]) now have fine-tuned couplings that depend on the nearby plaquettes. Explicitly, we have (cf. Fig. \[dimerconfig\]): ![(color online) Illustration to clarify the construction of the Hamiltonian . The central dimer pair is present in all configurations $C_i$ and $C_i'$, whereas the four surrounding dimers may or may not be present. The index $i$ enumerates the $2^4$ possible arrangements of the surrounding dimers. The configuration $C'$ differs from the corresponding $C$ by a single flip resonance of the central dimer pair.[]{data-label="dimerconfig"}](Ham_fig.eps){width="48.00000%"} $$\begin{aligned} H_{d} &=& t\sum_i \Bigg[- \Big|C_i\Big>\Big<C_i'\Big| - \Big|C_i'\Big>\Big<C_i\Big| \nonumber\\ &+& w^{R_{C_i'}-R_{C_i}}\Big|C_i\Big>\Big<C_i\Big|+w^{R_{C_i}-R_{C_i'}}\Big|C_i'\Big>\Big<C_i'\Big|\Bigg] \;, \nonumber \\ && \label{Ham}\end{aligned}$$ where $R_{C_i}$ and $R_{C_i'}$ denote the number of pairs of present dimers formed in configurations $C_i$ and $C_i'$ respectively. The Hamiltonian of Eq.  is designed in such a way that it annihilates any superposition of dimer-configuration states which have amplitudes that are of the form $w^{N_p}$, where $w$ is the parameter appearing in the Hamiltonian and $N_p$ is the number of pairs of neighboring dimers in the configuration.[@ardonne04] In this sense, the Hamiltonian is a sum of projection operators and consequently there is a unique ground state for each topological sector which must be composed of the superposition of these specially weighted configurations. The ground-state wave function, $\ket{G}$, the state annihilated by all the projection operators, for this system is: $$\ket{ G}=\frac{1}{\sqrt{Z(w^2)}}\sum_{\{\mathcal{C}\}} w^{\displaystyle{N_p[\mathcal{C}]}} \ket{\mathcal{C}} \;, \label{Gzw}$$ where $Z(w^2)$, the normalization of this state, $$\begin{aligned} Z(w^2) =\sum_{\{\mathcal{C}\}} w^{\displaystyle{2N_p[\mathcal{C}] } } \label{z1}\end{aligned}$$ has the form the partition function of classically interacting dimers with a coupling $u=-2\ln w$ between parallel neighboring dimers. In the following, we will assume an attractive coupling, $u<0$ or $w>1$. The case $u>0$ was studied for the fully-packed case in Ref. . There are two different ways in which we can add doping to our system, while still being able to determine the ground state. If we add the following fine-tuned hole-related terms to the initial Hamiltonian  $$\begin{aligned} H^{\rm hole}_{\rm canonical}= - t_{\rm hole}\sum_{<ijk>}\Bigg\{\Bigg[\Big|\raisebox{-0.5ex}{\includegraphics{Ham22.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics{Ham23.eps}}\Big| + \mathit{h.c.}\Bigg] \nonumber\\ -\Big|\raisebox{-0.5ex}{\includegraphics{Ham22.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics{Ham22.eps}}\Big| -\Big|\raisebox{-0.5ex}{\includegraphics{Ham23.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics{Ham23.eps}}\Big|\Bigg\} \label{Ham2}\end{aligned}$$ then the resulting ground state becomes $$\ket{ G_{N_h}}=\frac{1}{\sqrt{Z(w^2)}}\sum_{\{\mathcal{C}_{N_h}\}} w^{\displaystyle{N_p[\mathcal{C}_{N_h}]}}\ket{\mathcal{C}_{N_h}} \;, \label{Gzw2}$$ where the number of holes $N_h$ is now fixed at a specified value. The norm of this wave-function, $Z(w^2)$, is the *canonical* partition function for the set of dimer coverings with a fixed number of holes. On the other hand, if we add the following terms, which do not conserve the number of holes in the system, to the Hamiltonian  $$\begin{aligned} H^{\rm hole}_{\rm {grand-canonical}} = - \tilde t_{\rm hole}\sum_{links}\Bigg\{ \Bigg[\Big|\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham20.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham21.eps}}\Big| + h.c.\Bigg] \nonumber\\ - z^2\Big|\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham20.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham20.eps}}\Big|-z^{-2}\Big|\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham21.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham21.eps}}\Big| \Bigg\} \nonumber \\ && \label{Ham3}\end{aligned}$$ then the ground state of the system becomes $$\ket{ G}=\frac{1}{\sqrt{Z(w^2,z^2)}}\sum_{\{\mathcal{C}\}} w^{\displaystyle{N_p[\mathcal{C}]}} z^{\displaystyle{N_h[\mathcal{C}]}} \ket{\mathcal{C}} \;, \label{Gzw3}$$ with $$Z(w^2,z^2)=\sum_{\{\mathcal{C}\}} w^{2N_p[\mathcal{C}]} z^{2N_h[\mathcal{C}]} \;. \label{Zzw}$$ Equation  includes a natural, short-range repulsion between neighboring holes and an off-diagonal term which represents creation-annihilation of dimers. Furthermore, there is a dimer fugacity term, as in every perturbative derivation of a quantum dimer model.[@rokhsar88] We note that Eq.  has the same form as a grand partition function for dimer coverings of the square lattice. This partition function now depends not only on the interaction $u$ defined below Eq. , but also on the hole chemical potential $\mu/|u| = 2\ln z$. Since the canonical and grand-canonical ensembles become equivalent in the thermodynamic limit, the two ground-state wave functions and  must correspond to the same ground-state physics. Furthermore, it is clear that for $w=1$ the models are located at the usual RK point of the quantum dimer model on the square lattice. For a system with periodic boundary conditions, each configuration $\mathcal{C}$ contains only an *even* number of holes, with half of the holes on either sublattice. We remark that the fact that the exact ground-state wave function is a sum (as opposed to a product) of the ground states of sectors labeled by the number of holes on the lattice, is due to the resonance term that we have used to represent the motion of holes. In particular, we have assumed that a dimer can *break* into two holes which themselves repel each other. In the limit of very strong hole-hole repulsions, in strong-coupling perturbation theory, it is straightforward to recover a fixed-hole density sector with a single-hole resonance move involving three sites in any direction; the coupling strength in Eq.  then becomes $t_{\rm hole}\sim z^4\tilde t_{\rm hole}$, and thus, in the limit $z\rightarrow 0$, it reduces to an effective hopping amplitude for the holes. Mean-Field Results {#sec:mean-field} ================== To examine the physics described by the ground-state wave functions obtained in the previous section we start with a discussion of a mean-field theory of the phase diagram. We use the standard approach of regarding the probability densities, obtained by squaring the wave function, as the Gibbs weights of a classical two-dimensional system and focus on the interacting dimer model on the square lattice at finite hole density. Although mean-field theory is insufficient to describe two-dimensional critical systems, it is a useful tool to obtain qualitative features of the phase diagram as well as the behavior deep in the phases, away from criticality. The details of this theory are presented in Appendix \[app:mean-field-details\]. We begin by constructing a Grassmann representation of the partition function for an interacting dimer model using the standard methods introduced by Samuel.[@samuel80; @Samuel] The resulting theory involves Grassmann (anti-commuting) variables residing on the sites of the square lattice. The action of the Grassmann integral is non-trivial and is parametrized by the hole fugacity $z$ and a coupling between dimers $V=z^2 (e^u-1)$, where $u=-2\ln w$. Since the action of the resulting Grassmann path integral is not quadratic in the Grassmann variables, it cannot be reduced to the computation of a determinant. Thus, we use a standard mean-field approach which, in this case, involves the introduction of two Hubbard-Stratonovich (bosonic) fields $\phi_i$ and $\chi_{ij}$, defined on the sites and links of the square lattice respectively. Upon integrating out the Grassmann variables one obtains an effective theory for the fields $\phi_i$ and $\chi_{ij}$ which, as usual, is solved within a saddle-point expansion. The dimer $m_0$ and hole $n$ densities, as well as the columnar order parameter $m$, can be expressed straightforwardly in terms of the fields $\phi_i$ and $\chi_{ij}$. From this effective theory one can compute an effective potential $\Gamma$ and the configurations of the observables of interest, $n$, $m_0$ and $m$, as functions of $z$ and $V$, and determine the phase diagram. As a function of hole density (or hole fugacity) and $u$, we find that the phase diagram has two phases (shown qualitatively in Fig. \[fig:sketch\]), namely a dimer-hole liquid phase and a hole-poor phase with long-range columnar order. The nature of the transitions between these phases is incorrectly described by the mean-field theory, particularly at zero doping and near the tricritical point. The correct behavior is the subject of a detailed analysis in the subsequent sections. Nevertheless, the mean-field phase diagram correctly predicts that at low hole densities and moderate values of $u$ the transition between the dimer-hole liquid and the columnar solid phase is continuous, that for large $u$ the transition is first order, and that there is a tricritical point at $$u_{tr} \simeq 2.733, \qquad z_{tr} \simeq 0.075 \;.$$ Remarkably, the Monte Carlo simulations presented in Section \[sec:MC\] yield a tricritical point at a location quite consistent with these values. Another correct prediction of the mean-field theory is the behavior of the connected hole density correlation function deep in the dimer-hole liquid phase. This prediction, also discussed in detail in Appendix \[app:mean-field-details\], fits surprisingly well the Monte Carlo simulations of Krauth and Moessner,[@krauth03] performed for a system of classical non-interacting ($u=0$) dimers at finite hole density. The mean-field result is consistent with the simulations for a quite broad region of densities even quite close to the fully packed limit $z \to 0$ where, naturally, a wrong correlation length exponent is predicted. Phase Diagram and Correlations for Interacting Dimers and Holes {#sec:exact-critical-behavior} =============================================================== We now turn to a more precise analysis of the phase transitions of the interacting classical dimer models as a function of hole density on interaction parameter $u$. Here we take advantage of a wealth of information and methods from two-dimensional systems, exact solutions and conformal field theory, to analyse the behavior in detail and extract conclusions that will be quite useful for the analysis of the wave functions. We begin with a discussion of the undoped case, and then discuss the physics at finite hole density. Interacting Dimers at Zero Hole Density {#sec:zero-density} --------------------------------------- It is a well known fact that both classical and quantum two-dimensional dimer models can be represented in terms of height models. For the classical case, this mapping is well known[@blote82a; @blote82b; @henley97a; @ardonne04]. The mapping for the quantum case has also been discussed extensively[@fradkin90a; @levitov90; @zheng-sachdev89; @fradkin91; @ardonne04; @fradkin04]. In both cases the mapping relates each dimer configuration to a configuration of a set of integer-valued (height) variables, $h(\vec r)$, which reside on the sites of the dual lattice, a square lattice in the case of interest here. Thus, this mapping amounts to a duality transformation. An alternative picture follows from realizing that dimer configurations can be mapped onto the degenerate ground-state configurations of fully frustrated Ising models[@blote82a; @moessner03a]. In our case, the corresponding spin model is the fully frustrated Ising model on the square lattice (FFSI) at zero temperature. In the Ising model picture, each dimer is dual to an unsatisfied bond of the fully frustrated Ising model and classical dimer interactions correspond to second neighbor interactions in the square lattice FFSI[@fradkin81]. It is easy to see that holes correspond to unfrustrated plaquettes in the FFSI. In this work we use primarily the language of the height representation. To map the square-lattice classical dimer model onto a height model, we follow the prescription given in Ref. . One first assigns a height variable to each plaquette. In going around a vertex on the even sub-lattice clockwise, the height changes by $+3$ if a dimer is present on the link between the plaquettes, and by $-1$ if no dimer is present on that link. On the odd sub-lattice, the heights change by $-3$ and $+1$ respectively. The dimer constraint, that one lattice site belongs to one and only one dimer, implies (for the square lattice) that, for the fully packed case, there are four possible configurations of dimers for each lattice site. In the dual height model this is reproduced by the period four property $ h \equiv h+4$ of the allowed height configurations. It is easy to see that for the allowed configuration the average values that the height variables can take at a given site of the direct lattice (a vertex) are $\pm 3/2, \pm 1/2$. On the other hand, a uniform shift [*of all*]{} the heights by one unit, $h \to h+1$, leads to an equivalent state. This mapping works strictly speaking only for the fully packed case. Holes are sites that don’t belong to any dimer and thus represent violations of the full packing rule. They play the role of topological defects (“vortices”) in the (dual) height representation. The exact solution of the non-interacting fully packed dimer model on the square (and other) lattice has been known for a long time[@fisher63]. In particular the long-distance behavior of the dimer density correlation functions and the hole density correlation functions are known explicitly[@fisher63; @youngblood80]. These correlation functions obey power law behaviors and show that this is a critical system. Here we will use the standard approach to map the exact long distance behavior of two-dimensional critical systems to the behavior of the simplest critical system, the Gaussian or free boson model[@kadanoff77]. This approach is consistent for the free dimer model on an even lattice with periodic boundary conditions since its central charge (or conformal anomaly) is also $c=1$. We will use as reference states (“ideal states” in the terminology of Kondev and Henley[@kondev-henley96]) the four columnar states of the dimer coverings, which have the largest number of flippable plaquettes, and use them to define an effective field theory for this problem[@fradkin04]. We will assign a uniform value to the coarse-grained height field $h=0,1,2,3$ to each reference (columnar) state. Let $n_x(\vec r)$ and $n_y(\vec y)$ represent the coarse grained dimer densities of the horizontal link with endpoints at the pairs of lattice sites $\vec r$ and $\vec r + \vec e_x$, and vertical links with endpoints $\vec r$ and $\vec r + \vec e_y$ respectively. Here $\vec e_x$ and $\vec e_y$ are two lattice unit vectors along the $x$ and $y$ directions respectively, with a lattice spacing of $1$. We can now define the columnar local order parameter as the two component vector $$\begin{aligned} \mathcal{O}_x(x,y)&=&n_x(x,y)-n_x(x+1,y)\nonumber \\ \mathcal{O}_y(x,y)&=&n_y(x,y)-n_y(x,y+1)\end{aligned}$$ which clearly correspond to the $(\pi,0)$ and $(0,\pi)$ Fourier components of the dimer densities. This two-component order parameter takes four distinct values for each one of the columnar states, and changes sign under shifts by one lattice spacing in either direction. It also transforms as a vector under $90^{\circ}$ rotations. Thus it is the order parameter for columnar order. ### Effective Field Theory: the non-interacting case {#sec:dimer-qft} The fluctuations of the free field $h(\vec r)$ are described by a continuum Gaussian (free boson) model. We will find it simpler to work with the rescaled height field $\phi=\frac{\pi}{2} h$. For this field the periodicity condition $h \to h+4$ becomes $ \phi \to \phi + 2 \pi$. (For the rescaled field the ideal states are $\phi=0,\pi/2,\pi,3\pi/2$.) Thus, the allowed operators are $2\pi$ periodic functions of $\phi$, and are either derivatives of $\phi$, or the exponential (or [*charge*]{}) operators $\exp(\pm i \phi)$, $\exp(\pm 2 i \phi)$, $\exp(3 i \phi)$ and $\exp(\pm 4 i \phi)$, which are $2\pi$ periodic functions of $\phi$. The action $S$ for the rescaled field is $$S=\int d^2x \frac{K}{2} \left( \nabla \phi\right)^2 \label{eq:gaussian-action}$$ For the free dimer model the stiffness is $K=\frac{1}{4\pi}$ (see below). By matching the exact correlation functions of the free dimer model on the square lattice one readily finds the following operator identification of the coarse-grained dimer densities in terms of free field operators[@fradkin04] $$\begin{aligned} n_x-\frac{1}{4}&=&\frac{1}{2\pi}(-1)^{x+y}\partial_y \phi + \frac{1}{2}[(-1)^{x} e^{\displaystyle{i \phi }}+ {\rm c.c.}]\\ n_y-\frac{1}{4}&=& \frac{1}{2\pi}(-1)^{x+y+1}\partial_x \phi+ \frac{1}{2}[(-1)^{y} i \; e^{\displaystyle{i \phi}}+ {\rm c.c.}]\nonumber \\ && \label{densities}\end{aligned}$$ In Ref.  it was shown that this is an operator identity for the free dimer model on the square lattice in the sense the the asymptotic long-distance behavior of the dimer density correlation functions computed with this Gaussian model are the same as the exact long distance correlation functions for the free dimer problem on the square lattice[@fisher63; @youngblood80] provided the stiffness $K=\frac{1}{4\pi}$. Notice that, with this identification, when the field $\phi$ takes each of the values $0,\pi/2,\pi,3\pi/2$ (the “ideal states”) the density operators take four distinct values which reflect the broken symmetries of the four columnar states. From the operator identification of Eq.  the columnar order parameter is, up to a normalization constant, $$\mathcal{O}_x = \cos \phi , \qquad \mathcal{O}_y = \sin\phi$$ Due to the effects of dimer-dimer interactions the form of this effective action is $$S=\int d^2 x\; \frac{K}{2} \left(\nabla \phi\right)^2+\textrm{perturbations}$$ The effect of the interactions is a finite renormalization of the stiffness $K$ away from its free dimer value, $K=\frac{1}{4\pi}$. We saw above that, due to the dimer constraints, the allowed charge operators are $O_n(\vec r)=\exp(i n \phi(\vec r))$. We also saw that the columnar order parameter is proportional to the operator $\frac{1}{2}(O_1(\vec r)+ O_{-1})(\vec r)=\cos \phi(\vec r)$, and carries the unit of charge $n=1$. One can also define [*vortex*]{} or [*magnetic*]{} operators[@kadanoff77], and example of which is the hole. A vortex operator causes the field $\phi$ to wind by $2\pi m$, where $m$ is the vorticity (or [*magnetic charge*]{}). One can similarly define a general composite operator $O_{n,m}(\vec r)$ with $n$ units of (electric) charge and $m$ units of vorticity (or magnetic charge). Its scaling dimensions, $\Delta(n,m)$ are[@kadanoff77] $$\Delta_{n,m}(K)=\frac{n^2}{4\pi K}+\pi K m^2$$ We can now use these results to identify a few operators of interest and give their scaling dimensions. These results are summarized in Table \[table:0density\] 1. The columnar order parameter is the elementary charge operator $O_{\pm 1,0}$ and has no vorticity. On the (columnar) ideal states $0$, $\pi/2$, $\pi$, and $3\pi/2$ this operator takes the values $1$ ,$i$ ,$-1$, and $-i$ respectively. Its scaling dimension is $\Delta_{1,0}(K)=\frac{1}{4\pi K}$. At the free dimer point, $K=\frac{1}{4\pi}$, its scaling dimension is $\Delta_{1,0}(\frac{1}{4\pi})=1$. This is consistent with the exact results[@fisher63; @youngblood80] that the density correlation function falls off as $1/r^2$. The operator identification of Eq.  is based on these facts[@fradkin04]. 2. The operator $O_{\pm 2,0}=\exp(\pm 2i \phi)$ takes the values $1$, $-1$, $1$ and $-1$ on each of the ideal columnar states. It is clearly the order parameter for symmetry breaking by $90^\circ$ rotations: it is the order parameter for [*orientational*]{} symmetry. 3. The operator with the highest allowed electric charge is $O_{\pm4,0}=\exp(\pm4i\phi)$. Its scaling dimension is $\Delta_{4,0}(K)=\frac{4}{\pi K}$. At the free dimer point it has dimension $\Delta_{4,0}(\frac{1}{4\pi})=16$, and it is a strongly irrelevant operator. This operator arises naturally due to the fact that the microscopic heights $h$ take integer values, and hence height configurations which differ by an uniform integer shift are physically equivalent. This operator does not break any physical symmetry of the dimer model. 4. The hole operator is represented by the fundamental vortex operator $O_{0,\pm 1}$. A vortex with unit positive magnetic charge corresponds to a hole on one sublattice, and a vortex with unit negative magnetic charge to a hole on the other sublattice. The scaling dimension of the vortex (hole) operator is $\Delta_{0,1}(K)=\pi K$. At the free dimer value, the scaling dimension of the hole operator is $\Delta_{0,1}(\frac{1}{4\pi})=\frac{1}{4}$, which is consistent with the exact result that the hole-hole correlation function decays $1/\sqrt{r}$ at large distances[@fisher63]. 5. The operator which describes a pair of holes on nearby sites of the [*same*]{} sublattice is represented by the operators $O_{0,\pm 2}$ which carry two units of magnetic charge (vorticity). This operator creates (or destroys) a [ *diagonal dimer*]{} connecting nearby points on the same sublattice.[@read-sachdev91] In the $2+1$-dimensional quantum dimer model, this operator is useful to describe the possible pairing of holes. This operator has dimension $\Delta_{0,2}(K)=4\pi K$. Its scaling dimension at the free dimer point is $\Delta_{0,2}(\frac{1}{4\pi})=1$, and is relevant for $K<\frac{1}{2\pi}$. As noted in Refs. and , this operator maps the square lattice into a deformed triangular lattice. The irrelevancy of this operator for $K>\frac{1}{2\pi}$ implies that this line of fixed points also exists for a deformed triangular lattice, as discussed recently in Ref. . 6. The free dimer problem is a free fermion system, which can be solved by Pfaffian methods.[@fisher63; @samuel80; @fendley02] This is actually a theory with two free real (Majorana) fermions or, equivalently, one free complex (Dirac) fermion, at its (massless) fixed point, whose central charge is also $c=1$. The appropriate fermion operator is a composite operator of the order and disorder operators[@kadanoff77; @kadanoff-ceva71] which, in this case, is $O_{\frac{1}{2},1}$. At the free dimer point the fermion operator has scaling dimension $\Delta_{\frac{1}{2},1}(\frac{1}{4\pi})=\frac{1}{2}$ and (conformal) spin $nm=\frac{1}{2}$ (as it should for a free fermion). At particular values of $K$ it is also possible to define parafermion operators[@fradkin-kadanoff80; @zamolodchikov-fateev85] operators which obey fractional statistics. For instance, at the KT point, $K=\frac{2}{\pi}$ (see below), the operator $O_{1,\frac{1}{4}}$ has dimension $\frac{1}{4}$ and spin $\frac{1}{4}$, and it is a [*semion*]{}. In fact, and not surprisingly, a fermionic approach[@fendley02] can be used to map this critical line onto an Euclidean version of the Luttinger-Thirring model. The coarse-grained height model description we sue here corresponds to the bosonization approach of the fermionic version of this problem. [|C||C|C|C|C|C|C|]{} &[columnar]{}&[rotational]{}&&[hole]{}&[hole pair]{}\ &O\_[1,0]{} & O\_[2,0]{} & O\_[4,0]{} & O\_[0,1]{}& O\_[0,2]{}\ K=& 1 & 4 & 16 & & 1\ K=& & & 2 & 2 & 8\ \[table:qn\] ### Effective Field Theory: interactions and phase transition {#sec:KT} We now turn to the effects of dimer-dimer interactions. We recall that we are considering only interactions of a pair of dimers in a plaquette. The interaction energy is $$H_{\rm int}=-u \sum_{\vec r} \Big[ n_x(\vec r) n_x(\vec r+\vec e_y)+n_y(\vec r) n_x(\vec r+\vec e_x)\Big]$$ We wish to find the corresponding operator in terms of the coarse grained (rescaled) height field $\phi$. This can be done by using the operator product expansion (OPE) of the coarse-grained form of the dimer density operators, Eq. , in terms of the field $\phi$. We will also need the standard OPE of the (normal ordered) charge operators[@witten78; @kadanoff-brown79; @boyanovsky89a] $$\begin{aligned} &&:\cos (n \phi(x)) : \; :\cos (n \phi(y)):= \frac{1}{2} :\cos (2n\phi(x)):\nonumber \\ &&+\frac{1}{(\mu^2|x-y|^2)^{n^2}}\Big[1-n^2|x-y|^2 :\left(\nabla \phi \right)^2:+\ldots\Big] \nonumber \\ && \\ &&:\sin (n \phi(x)) : \; :\sin (n \phi(y)):= - \frac{1}{2} :\cos (2n\phi(x)):\nonumber \\ &&+\frac{1}{(\mu^2|x-y|^2)^{n^2}}\Big[1-n^2 |x-y|^2 : \left(\nabla \phi \right)^2:+\ldots \Big]\nonumber \\ && \label{ope1}\end{aligned}$$ where $\mu$ is a short-distance cutoff and the ellipsis represents the contributions of irrelevant operators. Using these results we find that the net effect of the interactions is to renormalize the stiffness $K$ upwards $$K=\frac{1}{4\pi}+\frac{1}{2}\left(1+\frac{1}{\pi^2}\right) u+ \mathcal{O}(u^2) \label{Kint}$$ where we have denoted $u>0$ for [*attractive*]{} interactions. This expression for the renormalization of $K$ is only accurate to linear order in the dimer-dimer interaction. Higher order renormalizations (in $u$) would result if the effects of irrelevant operators are also taken into account. The relation between $K$ and the microscopic model is non-universal and can only be determined either from an exact solution or from a numerical simulation. One can determine the function $K(u)$ from the Monte Carlo simulations we present elsewhere in this paper. What is important is that these non-universal effects affect only the relation between the coefficients of the effective theory and not the form of the effective theory itself. Thus, the effective action of the field theory for the interacting classical dimer model at zero hole density has the form $$S=\int d^2x \left[ \frac{K}{2} \left(\nabla \phi\right)^2+g \cos(4\phi)\right] \label{Seff-0density}$$ where we have included the effects of the charge $4$ perturbation, $\cos (4\phi)=\cos(2\pi h)$, which biases the coarse-grained height field to take integer values. For an [*anisotropic*]{} dimer-dimer interaction, which arises form a term which weights differently the interactions between parallel horizontal dimers from those of parallel vertical dimers, we would have also found a $\cos(2\phi)$ operator in addition to an anisotropy for the stiffness. Thus, an anisotropy in the dimer-dimer interaction is a relevant perturbation which couples to the [*orientational*]{} order parameter $\cos(2\phi)$. In Table \[table:0density\] we see that as the attractive interactions grow there will be a critical value of the interaction $u$ at which the stiffness $K(u_c)=\frac{2}{\pi}$. At this point, the $\cos(4\phi)$ operator has scaling dimension $\Delta_{2,0}=2$, where it becomes marginal. For $u>u_c$ ($K>K(u_c)$) this operator becomes relevant. We also see that at $K(u_c)=\frac{2}{\pi}$ the columnar order parameter, $\cos \phi$, has scaling dimension $1/8$ and it is the most relevant operator in this problem. Thus this is a phase transition from a [*critical phase*]{}, for $K<\frac{2}{\pi}$, without long-range order but with power law correlations, to a phase with long-range columnar order, for $K>\frac{2}{\pi}$, in which the columnar order parameter has a non-vanishing expectation value. This is a standard Kosterlitz-Thouless (KT) transition[@kosterlitz73; @kosterlitz74; @jose77] which is naturally described by the sine-Gordon field theory[@luther75; @wiegmann78; @amit80; @dennijs81] whose (Euclidean) action is given in Eq. . The only difference between this problem and the standard KT transition, [*e.g.*]{} the classical 2D XY model, is that the phase with a finite correlation length is ordered: it is a columnar state with a four-fold degenerate non-uniform state, whereas the finite-correlation length phase of the XY model (and of its dual surface roughening model) has a non-degenerate translation invariant state. In spite of these global differences, this phase transition is in the KT universality class. Thus the well known behavior of the correlation functions at the KT transition apply to this case as well [@alet05; @alet06; @Castelnovo06; @Poilblanc06]. In Section \[sec:MC\] we verify this behavior by a detailed Monte Carlo study of the columnar and orientational order parameters and of their associated susceptibilities for the interacting dimer model. Interacting Dimers at Finite Hole Density {#sec:finite-density} ----------------------------------------- We now consider the dimer model at finite hole density, away from the full packing condition. The classical partition function for this problem is given in Eq. , where the weights (fugacities) $z$ and $w$ are related to the coupling constant $u$ and the chemical potential $\mu$ as described earlier. Recall that the 2D classical partition function $Z(w,z)$ is the norm of the ground-state wave function $\ket{G}$, of Eq. , of the 2D doped quantum dimer model. In terms of a sum over configurations of electric charges $n$ and magnetic charges (vortices) $m$, the partition function $Z(w,z)$ is equivalent to that of a generalized (neutral) Coulomb gas (GCG) of electric and magnetic charges in two dimensions[@kadanoff78; @nienhuis87] $$\begin{aligned} Z(w,z)=\!\!\!\!\!\!{\sum_{\{ n(\vec r), m(\vec R)\} }}^{\!\!\!\!\!\!\!\! \prime} \; \exp\Bigg[ \frac{N^2}{4\pi K} \sum_{\vec r,\vec r^{\; \prime}} n(\vec r) \; \ln |\vec r-\vec r^{\;\prime}|\; n(\vec r^{\; \prime})\nonumber\\ +\pi K \sum_{\vec R,\vec R^{\; \prime}} m(\vec R) \; \ln |\vec R-\vec R^{\;\prime}| \; m(\vec R^{\; \prime})\Bigg] \nonumber \\ \times\; \exp\Bigg[\sum_{\vec r}\ln w \; n(\vec r)^2+\sum_{\vec R} \ln z\; m(\vec R)^2 \nonumber\\ -i \sum_{\vec r, \vec R} N \; n(\vec r) \; m(\vec R) \; \Theta(\vec r-\vec R)\Bigg]\;\;\;\;\; \label{GCG}\end{aligned}$$ where prime denotes that the sum is restricted to neutral configurations with vanishing total charge and vanishing total vorticity, [*i.e.*]{} $\sum_{\vec r} n(\vec r)=\sum_{\vec R} m(\vec R)=0$. Here $\Theta(\vec r - \vec R)$ is the angle between a vortex at $\vec R$, as seen from a charge at $\vec r$, measured with respect to the (arbitrary) $x$ axis. It is the Cauchy-Riemann dual of the logarithm, $$\begin{aligned} G(\vec r -\vec r^{\;\prime})&=&-\ln |\vec r-\vec r^{\;\prime}| \\ \Theta(\vec r-\vec r^{\;\prime})&=&-\tan^{-1}\left(\frac{y-y^\prime}{x-x^\prime}\right) \\ -\nabla^2 G(\vec r - \vec r^{\;\prime})&=&2\pi \; \delta^2(\vec r - \vec r^{\;\prime})\\ \partial_\mu G&=&\epsilon_{\mu \nu} \partial_\nu \Theta\end{aligned}$$ As usual, the logarithmic interaction is regularized so that it vanishes for $\vec r=\vec r^{\prime}$. The short distance behavior of the interactions is absorbed in the fugacities $z$ and $w$. For the case we are discussing here, the GCG of interest has $N=4$, and Eq.  is just the the Coulomb-gas form of the partition function of the $\mathbb{Z}_4$ model, and for the related 2D Ashkin-Teller model. This is a well understood system [@jose77; @wiegmann78; @boyanovsky89a; @lecheminant02] and in our case, it corresponds to the grand-partition function for the doped interacting dimer model on the square lattice at low hole densities. In the limit of low fugacities, $z\ll 1$ and $w\ll 1$, the GCG is equivalent to a (generalized) sine-Gordon field theory in two-dimensional Euclidean space-time, whose (Euclidean) Lagrangian is given by[@wiegmann78; @boyanovsky89a] $$\mathcal{L}_E=\frac{1}{2} \left(\partial_\mu \phi\right)^2-\frac{2z}{a^2} \cos\left(\frac{N}{\sqrt{K}}\; \phi\right)-\frac{2w}{a^2} \cos \left(2\pi \sqrt{K} \; \widetilde \phi \right) \label{eff-action-euclidean}$$ Here $\widetilde \phi$ is the dual field, defined by $$\epsilon_{\mu \nu} \partial_\nu \phi=i \partial_\mu \widetilde \phi$$ In Eq.  we can see by inspection that the operators $\cos(\frac{N}{\sqrt{K}}\; \phi) $ and $\cos(2\pi\sqrt{K}\; \widetilde \phi)$ can be identified respectively with the operators $\frac{1}{2}(O_{N,0}+O_{-N,0})$ and $\frac{1}{2}(O_{0,1}+O_{0,-1})$ discussed above. We will use this effective field theory to study the transition between the liquid and the ordered phases of the interacting dimer model. At $z=w=0$ this is the KT transition discussed above. For general $N$, both operators have the scaling dimension if $\frac{N}{\sqrt{K}}=2\pi \sqrt{K}$, [*i.e.*]{} the theory is self-dual, which happens for $K=\frac{N}{2\pi}$. For $N=4$, $K=\frac{2}{\pi}$, both operators have scaling dimension $2$ and both are marginal. This is the only case we will discuss here. (A detailed discussion of the more general case of $N>4$ was given by Lecheminant [*et al*]{}.[@lecheminant02]) For $N=4$ the Euclidean Lagrangian becomes[@lecheminant02] $$\mathcal{L}_E=\frac{1}{2} \left(\partial_\mu \phi\right)^2-\frac{2z}{a^2} \cos\left(\sqrt{8\pi } \; \phi\right) -\frac{2w}{a^2} \cos\left(\sqrt{8\pi} \; \widetilde \phi \right) \label{self-dual}$$ It turns out that this a problem which can be solved exactly.[@ginsparg88; @lecheminant02] The most direct way of doing this is to perform an analytic continuation from 2D Euclidean space to $1+1$-dimensional Minkowski space-time, [*i.e.*]{} to think of this problem as a $1+1$-dimensional quantum field theory. The [*Hamiltonian density*]{} of the equivalent $1+1$-dimensional field theory is $$\begin{aligned} \mathcal{H}&=&\frac{1}{2}(\partial_x \widetilde \phi)^2+\frac{1}{2}(\partial_x \phi)^2 \nonumber\\ &&-\frac{2z}{a^2}\cos\left(\sqrt{8\pi}\;\phi\right)-\frac{2w}{a^2}\cos\left(\sqrt{8\pi}\;\widetilde \phi \right)\nonumber \\ &=& \frac{1}{2}(\partial_x \widetilde \phi)^2+\frac{1}{2}(\partial_x \phi)^2 \nonumber\\ &&-2\frac{(z+w)}{a^2} \cos (\sqrt{8\pi} \phi_L)\; \cos (\sqrt{8\pi} \phi_L) \nonumber \\ &&+2\frac{(z-w)}{a^2} \sin (\sqrt{8\pi} \phi_L)\; \sin (\sqrt{8\pi} \phi_L) \label{1+1d-H}\end{aligned}$$ where we have used the fact that $\Pi$, the canonical momentum conjugate to the field $\phi$ is simply related to the dual field $\widetilde \phi$, $$\Pi=\frac{\delta \mathcal{L}_M}{\delta \phi}=\partial_t \phi=-\partial_x \widetilde \phi \label{Pi-dual}$$ and obey equal-time canonical commutation relations $$\left[\phi(x),\Pi(y)\right]=i \delta(x-y) \label{ccr}$$ In Eq.  we used the decomposition of the field $\phi$ and the dual field $\widetilde \phi$ into right and left moving fields $\phi_R$ and $\phi_L$, $$\begin{aligned} \phi&=&\phi_L+\phi_R \qquad \widetilde \phi=\phi_L-\phi_R \nonumber \\ \phi_L&=&\frac{1}{2}(\phi+\widetilde \phi ) \qquad \phi_R=\frac{1}{2} (\phi - \widetilde \phi ) \label{right-left}\end{aligned}$$ whose propagators are $$\begin{aligned} \langle \phi_L(z) \phi_L(w) \rangle=-\frac{1}{4\pi} \ln(z-w)\nonumber\\\langle \phi_R(\bar z) \phi_R(\bar w) \rangle=-\frac{1}{4\pi} \ln(\bar z-\bar w) \label{propagators-LR}\end{aligned}$$ where we have used the complex coordinates (not to be confused with the coupling constants!) $z=\tau+ix=i(t+x)$ and $\bar z=\tau-ix=i(t-x)$, in imaginary and real time respectively. It is easy to see[@ginsparg88; @yellow; @lecheminant02] that the following dimension $1$ chiral operators $$\begin{aligned} &J_L^z=\frac{i}{\sqrt{2\pi}} \partial_z \phi_L &&J_L^\pm=\frac{1}{2\pi}: e^{\displaystyle{\pm i\sqrt{8\pi} \phi_L}}: \nonumber\\ &J_R^z=\frac{-i}{\sqrt{2\pi}} \partial_{\bar z} \phi_R &&J_R^\pm=\frac{1}{2\pi}e^{\displaystyle{\mp i\sqrt{8\pi} \phi_R}} \label{su2currents}\end{aligned}$$ with $J_{L,R}^\pm=J_{L,R}^x\pm i J_{L,R}^y$, are the generators of an $su(2)_1$ Kac-Moody algebra given by the OPE[@yellow; @lecheminant02] $$\begin{aligned} J_L^a(z) J_L^b(w)&=& \frac{\delta_{ab}}{8\pi^2(z-w)^2} +i \frac{\epsilon_{abc}}{8\pi^2(z-w)} J_L^c(w)\nonumber \\ J_R^a(\bar z) J_R^b(\bar w)&=& \frac{\delta_{ab}}{8\pi^2(\bar z-\bar w)^2} +i \frac{\epsilon_{abc}}{8\pi^2(\bar z-\bar w)} J_R^c(\bar w)\nonumber \\ && \label{su1}\end{aligned}$$ where $a,b,c=x,y,z$ and $\epsilon_{abc}$ is the Levi-Civita tensor. We also note for future use the following operator identifications (still for $K=2/\pi$) $$\begin{aligned} e^{\displaystyle{i\sqrt{8\pi}\phi}} &\equiv& O_{4,0} \nonumber \\ e^{\displaystyle{i\sqrt{8\pi} \widetilde \phi}}&\equiv&O_{0,1} \nonumber \\ J_L^\pm \sim e^{\displaystyle{i\sqrt{8\pi}\phi_L}}&\equiv& O_{\pm 2,\pm \frac{1}{2}} \nonumber \\ J_R^\pm \sim e^{\displaystyle{i\sqrt{8\pi} \phi_L}}&\equiv&O_{\mp 2,\pm \frac{1}{2}} \label{op-id1}\end{aligned}$$ The Hamiltonian of Eq.  can be written in terms of these generators and has the Sugawara form[@yellow]: $$\begin{aligned} \mathcal{H}&=&\frac{2\pi}{3}\left(\vec J_L \cdot \vec J_L+\vec J_R \cdot \vec J_R\right)\nonumber\\ &&-8\pi^2(z+w)J_L^x J_R^x -8\pi^2(z-w) J_L^y J_R^y \label{1+1d-Hsu2}\end{aligned}$$ Thus, one recovers the old result[@luther75; @ginsparg88] that at the KT transition the system has an effective ([*dynamical*]{}) $SU(2)$ symmetry. Given the existence of an $su(2)_1$ symmetry one expects to find, in addition to the “spin one”(vector) representation (the $su(2)$ currents), operators associated with the spin $1/2$ representation of $su(2)_1$. In the Wess-Zumino-Witten (WZW) version of this theory[@witten84; @yellow], there is an operator with this property, the field $g(z,\bar z)$ of the WZW model. This is a $2 \times 2$ matrix-valued field with scaling dimension $1/2$. In our theory we also have an operator with scaling dimension $1/2$, the operator $O_{2,0} \sim \exp(i \sqrt{2\pi} \phi)$ which we saw was related to the order parameter for broken rotational symmetry; see Table \[table:0density\]. Thus, the operators of the spin $1/2$ (spinor)representation of $su(2)_1$ are identified with the operators $$g(z,\bar z) \sim \begin{pmatrix} e^{\displaystyle{-i \sqrt{2\pi} \phi}} & e^{\displaystyle{-i \sqrt{2\pi} \widetilde \phi }} \\ e^{\displaystyle{ i \sqrt{2\pi} \phi}} & e^{\displaystyle{ i \sqrt{2\pi} \widetilde \phi }} \end{pmatrix} \sim \begin{pmatrix} O_{-2,0} & O_{0,-\frac{1}{2} } \\ O_{2,0} & O_{0, \frac{1}{2} } \end{pmatrix} \label{spin1/2}$$ Following the approach of Refs.  we will perform a global $SU(2)$ rotation by $\pi/2$ about the $y$ axis which maps $J_{L,R}^x \to J_{L,R}^z$, $J_{L,R}^z \to -J_{L,R}^x$ and $J_{L,R}^y \to J_{L,R}^y$, after which the Hamiltonian becomes $$\begin{aligned} \mathcal{H}&=&\frac{2\pi}{3}\left(\vec J_L \cdot \vec J_L+\vec J_R \cdot \vec J_R\right)\nonumber\\ &&-8\pi^2(z+w)J_L^z J_R^z -8\pi^2(z-w) J_L^y J_R^y \label{1+1d-Hsu2-rotated}\end{aligned}$$ Once again, one can introduce a new bose field, which we will call $\Phi$, and its dual $\widetilde \Phi$, to use the representation of the $su(2)_1$ current algebra of the form of Eq. . Using that $\partial_z \Phi_L=-i \partial_x \Phi_L$ and $\partial_{\bar z} \Phi_R=i \partial_x \Phi_R$, we can rewrite the Hamiltonian as $$\begin{aligned} \mathcal{H}&=&\mathcal{H}_0+\mathcal{H}_{\rm pert} \nonumber \\ \mathcal{H}_0&=&\frac{1}{2} (\partial_x \Phi)^2+\frac{1}{2} (\partial_x \widetilde \Phi)^2- 4\pi(z+w) \partial_x \Phi_L \partial_x \Phi_R \nonumber \\ \mathcal{H}_{\rm pert}&=&-\frac{2(z-w)}{a^2} \sin(\sqrt{8\pi}\Phi_L) \sin(\sqrt{8\pi}\Phi_R) \label{Hrotated}\end{aligned}$$ Thus, along the phase boundary line $z=w$ the term $\mathcal{H}_{\rm pert}$ is absent and we see that the effective Hamiltonian $\mathcal{H}_0$ involves only marginal operators.[@jose77; @wiegmann78; @ginsparg88; @lecheminant02] Similarly, the operators in the spin-$1/2$ representation transform as an $su(2)$ spinor under a $\pi/2$ rotation about the $y$ axis in $su(2)_1$, leading to the identifications $$\begin{aligned} O_{-2,0}& \to &\;\;\; O_{0,\frac{1}{2}}=\;\;\; e^{\displaystyle{i\sqrt{2\pi} \widetilde \Phi}}\nonumber\\ O_{2,0} &\to& -O_{0,-\frac{1}{2}}=-e^{\displaystyle{-i\sqrt{2\pi} \widetilde \Phi}}\nonumber \\ O_{0,-\frac{1}{2}} &\to& -O_{2,0}=-e^{\displaystyle{i \sqrt{2\pi} \Phi}}\nonumber\\ O_{0,\frac{1}{2}} &\to& -O_{-2,0}=-e^{\displaystyle{-i\sqrt{2\pi} \Phi}} \label{gtransf}\end{aligned}$$ where the operators on the right hand side of Eq.  are vertex operators written in terms of the new bosons $\Phi$ and $\widetilde \Phi$. Notice that this is a duality transformation. However, there is an operator in this theory, namely $O_{\pm1,0}$, which does not have $su(2)_1$ quantum numbers. As a result it will have a quite different behavior along the phase boundary. In contrast, all the other operators in this theory carry $su(2)_1$ quantum numbers. (As remarked in Ref.  this is not truly an $su(2)_1$ theory, although it contains it, but rather connected with a $\mathbb{Z}_2$ orbifold as well.) It is straightforward to check that the operators $O_{\pm 1,0}$ do not have an OPE with the operators which do transform under $su(2)_1$ (or, rather, that the OPE involve only irrelevant operators.) Since the marginal operator which deforms the $su(2)_1$ theory along the phase boundary does transform under $su(2)_1$, the operator $O_{\pm 1,0}$ will not mix (in the sense of its OPE) with the marginal operator either. We will see that this implies that the dimension of this operator remains equal to $1/4$ along the entire line, a result that is also well known (see for instance Ref. .) In contrast, the operators of the spin-$1/2$ representation, Eq. , do transform under $su(2)_1$, a fact which is generated by their OPE’s with the $su(2)_1$ generators[@yellow] and we will now see that their scaling dimensions do change along the phase boundary line. In Section \[sec:MC\] we present evidence from Monte Carlo simulations in support of both statements. We can now use this approach to solve this problem exactly along the phase boundary. Formally the Hamiltonian $\mathcal{H}_0$ of Eq. , is equivalent to a spinless Luttinger model with attractive backscattering interactions[@emery79]. As in the case of the Luttinger model, the problem is solved by means of a Bogoliubov transformation of the right and left moving bosons. This procedure breaks the $su(2)_1$ symmetry explicitly. We introduce a new bose field $\chi$ and its dual field $\widetilde \chi$. The left and right moving components of these fields, $\chi_L$ and $\chi_R$, are linearly related to the left and right moving fields $\Phi_L$ and $\Phi_R$ by $$\begin{aligned} \chi_L&=&\frac{1}{2}\left(\sqrt{\kappa}+\frac{1}{\sqrt{\kappa}}\right) \Phi_L+ \frac{1}{2}\left(\frac{1}{\sqrt{\kappa}}-\sqrt{\kappa}\right) \Phi_R \nonumber\\ \chi_R&=&\frac{1}{2}\left(\frac{1}{\sqrt{\kappa}}-\sqrt{\kappa}\right) \Phi_L+ \frac{1}{2}\left(\sqrt{\kappa}+\frac{1}{\sqrt{\kappa}}\right) \Phi_R \nonumber \\ && \label{chi-phi}\end{aligned}$$ where $\kappa$ is given by $$\kappa=\sqrt{\frac{1+2\pi (z+w)}{1-2\pi(z+w)}} \label{kappa}$$ The inverse transformation of Eq. , which relates $\Phi_{L}$ and $\Phi_{R}$ to $\chi_{L}$ and $\chi_{R}$ has the same form and it is obtained simply by replacing $\kappa$ by $1/\kappa$. The Hamiltonian $\mathcal{H}_0$ in terms of the new fields becomes $$\mathcal{H}_0=\frac{v}{2} \left[\left(\partial_x \widetilde \chi \right)^2+\left(\partial_x \chi\right)^2\right] \label{H0}$$ where the “dimensionless velocity” $v$ is $$v=\sqrt{1-4\pi^2(z+w)^2} \label{velocity}$$ which can be absorbed in a suitable rescaling of the $x$ coordinate, $x \to x \sqrt{v}$. Notice that the parameter $\kappa$ plays the same role as the stiffness $K$ defined above, which governed the change of the scaling dimensions at zero doping. Similarly, $\kappa$ governs the change of the scaling dimensions along the $z=w$ phase boundary of the systems at finite hole density. Here too, the relationship between this stiffness $\kappa$ and the microscopic interactions is non-universal, and the validity of Eqs.  and  is restricted to the weak coupling regime in which this continuum theory holds. Notice that, since $z\geq 0$ and $w\geq 0$, we will always have $\kappa \geq 1$. This fact will play an important role below. We can now use these results to determine the scaling dimension of the perturbation $\mathcal{H}_{\rm pert}$ along the phase boundary. It is straightforward to write the perturbation $\mathcal{H}_{\rm pert}$ in terms of the new field $\chi$ and its dual $\widetilde \chi$: $$\begin{aligned} \mathcal{H}_{\rm pert} &=&-\frac{2(z-w)}{a^2} \sin(\sqrt{8\pi}\Phi_L) \sin(\sqrt{8\pi}\Phi_R) \nonumber \\ &=&\frac{(z-w)}{a^2} \left[\cos\left( \sqrt{8\pi\kappa} \; \chi \right)-\cos\left(\sqrt{\frac{8\pi}{\kappa}}\; \widetilde \chi\right) \right]\nonumber\\ && \label{Hpert}\end{aligned}$$ Critical behavior along the phase boundary {#sec:critical} ------------------------------------------ ### The correlation length exponent {#sec:correlation} From Eq.  we find that the operator which perturbs the line of fixed points along the phase boundary at finite density, $\mathcal{H}_{\rm perp}$, involves two operators whose scaling dimensions are $2\kappa>2$ and $\displaystyle{\frac{2}{\kappa}}<2$ respectively, since $\kappa >1$. Thus this operator becomes more relevant along the phase boundary, away from the KT point, which in this language has $\kappa=1$. In fact, if we neglect the effects of the irrelevant operator (which is a safe thing to do only away from the KT point since its only important effect is a finite renormalization of $\kappa$) we see that the effective theory in the vicinity of the phase boundary is a sine-Gordon theory for the dual field $\widetilde \chi$. Since the scaling dimension of the relevant operator is $2/\kappa$, it follows that, away from the KT point, the correlation length $\xi$ diverges as the phase boundary is approached as $$\xi \sim |z-w|^{-\nu}, \qquad \nu=\frac{1}{2-\displaystyle{\frac{2}{\kappa}}}=\frac{\kappa}{2(\kappa-1)} \label{xi-exponent}$$ Thus, the correlation length exponent decreases (from infinity!) along the phase boundary away from the KT point. It is apparent from the form of the perturbation that away from the KT point there is simple scaling, up to contributions of strictly irrelevant operators. On the other hand, as $\kappa \to 1$ the relevant operator becomes marginally relevant and the irrelevant operator becomes marginally irrelevant. Thus, as $\kappa \to 1$ we should expect logarithmic corrections to scaling, and a complex crossover near $\kappa=1$. ### The columnar order parameter and its susceptibility {#sec:columnar} On the other hand, we noted above that the dimension of the columnar order parameter operator, $\frac{1}{2}(O_{1,0}+O_{-1,0})$ remains fixed at the KT value of $\Delta_{1,0}=1/8$. Hence for this operator we find $\eta_{1,0}=1/4$. On the other hand from scaling we know that the susceptibility exponent obeys the scaling relation $\gamma_{1,0}=(2-\eta) \nu$, where $\nu$ is given by Eq. . Hence $$\gamma_{1,0}= \frac{7\kappa}{8(\kappa-1)} \label{gamma-10}$$ is the susceptibility exponent of the columnar order parameter, which also increases along the phase boundary, even though $\gamma_{1,0}/\nu=7/4$ along the whole phase boundary (provided the transition remains continuous!.) ### The orientational order parameter and its susceptibility {#sec:orientational} We can use the operator identifications to look at the behavior of the orientational order parameter which we saw above is the operator $O_{2,0}$ of the original version of the theory. We also saw that this operator is a component of the spin-$1/2$ representation of $su(2)_2$. We also found how it transforms. In particular, we have $$O_{\pm 2,0} \to \mp e^{\displaystyle{\mp i \sqrt{\displaystyle{\frac{2\pi}{\kappa}}}\; \widetilde\chi}} \label{O20-map}$$ Along the phase boundary the scaling dimension of this operator is $$\Delta_{2,0}=\frac{1}{2\kappa}< \frac{1}{2} \label{dim20}$$ which it is always relevant, and $$\eta_{2,0}=\frac{1}{\kappa} \label{eta20}$$ Using once again the scaling relation $\gamma_{2,0}=(2-\eta_{2,0})\nu$, we find that the susceptibility exponent for the orientational order parameter is $$\gamma_{2,0}=\frac{2\kappa-1}{2(\kappa-1)} \label{gamma20}$$ which also decreases along the phase boundary away from the KT point. Tricritical Point, First-Order Transition, and Phase Separation {#first-order} --------------------------------------------------------------- Let us now discuss how this critical line turns into a first-order transition at a multicritical point. In Sections \[sec:MC\] and \[sec:1storder\] we use Monte Carlo simulations to show that this is indeed what happens. In Section \[sec:finite-density\] we used mean-field methods which indicated that the transition eventually should become first order. For this to work we should be able to predict the existence of a tricritical point along the phase boundary at which the transition becomes first order. It turns out that this is the case and that the first-order transition is triggered by an effective attractive interaction between holes on the same sublattice, leading to phase separation. To see how this happens we need to discuss the effects of irrelevant operators along the phase boundary. As we stated above, their most important effect is a finite and non-universal renormalization of $\kappa$ away from the value given in Eq. . We have also focused on the role of the operators $O_{4,0}$ and $O_{0,1}$ as they are both marginal at the KT transition. However, we also saw that one combination of these two operators remains marginal along the phase boundary and its coupling constant determines the value of $\kappa$, through Eq. . On the other hand, the other combination is the sum of a relevant operator, $\cos(\sqrt{(8\pi/\kappa)} \widetilde \chi)$ with scaling dimension $2/\kappa$, and of an irrelevant operator, $\cos(\sqrt{8\pi \kappa} \chi)$ with scaling dimension $2\kappa$. The dimension of the irrelevant operator increases along the phase boundary (thus becoming more irrelevant) while the dimension of the relevant operator decreases as $\kappa$ increases (thus becoming more relevant.) One possible scenario for a first-order transition is found by noting that as $\kappa \to \infty$, the dimension of the relevant operator vanishes, and the “thermal eigenvalue” $\gamma_{0,1}/\nu \to 2$. Thus at this point, naturally provided this limit is accessible, the line of fixed points reaches a discontinuity fixed point[@nienhuis76] and the transition becomes first order. However, at this point the theory becomes pathological (as $\kappa \to \infty$, $v \to 0$) and one may suspect that other physical effects, contained in irrelevant operators, may intervene before this happens. We have so far neglected other operators which are even more irrelevant at the KT transition. For example, the operators $O_{8,0}$ and $O_{0,2}$, have dimension $8$ at the KT point. Recall that the operator $O_{0,2}$ represents pairs of holes on the [*same sublattice*]{}. Both of these operators are present in any lattice problem (such as the interacting dimer model) and play no significant role at the KT transition (beyond a non-universal but otherwise trivial shift of the critical coupling) and for this reason they were (correctly) neglected. However, along the phase boundary the scaling dimensions of these operators change. Using the OPE, it is easy to see that along the phase boundary both operators contain the operators (among others which are less important) $O_{0,2} \sim \cos(2\sqrt{(8\pi/\kappa)} \widetilde \chi)$ with scaling dimension $8/\kappa$, and $O_{8,0} \sim \cos(2\sqrt{8\pi \kappa} \chi)$ with scaling dimension $8\kappa$. Even though they are not explicitly present in our starting theory, these operators will be generated under renormalization and close enough to the KT point, $\kappa \gtrsim 1$, they both are and remain irrelevant. However, although $\kappa$ also changes in a non-universal manner, the dependence of the dimensions with $\kappa$ does not, as it follows from the structure of the theory. Thus, provided the dependence between $\kappa$ and the microscopic couplings allow it, it may be possible to reach a point along the phase boundary at which $\kappa=4$. This will happen at a critical value of the coupling constant $u$ and a critical value of the hole density $\rho$ (or, equivalently at a critical value of the hole fugacity $z$ (cf. Fig. \[fig:sketch\]) ). At this critical value of $\kappa$, the scaling dimension of the operators $O_{0,2}$ becomes equal to $2$, and together with the strictly marginal operator $O_{4,0}$, there are now two marginal operators at this point. Thus the system is at a [*tricritical*]{} point at this value of the parameters.[@Bruce75] Past this point, $O_{0,2}$ becomes marginally relevant along the phase boundary. In this regime, the effective field theory at the phase boundary is a sine-Gordon theory for the field $\tilde \chi$ with the marginally relevant operator $O_{0,2}$. Since the sine-Gordon theory in this regime is massive, it has a finite correlation length and since $O_{0,2}$ is marginally relevant the correlation length along the phase boundary, which has now become a coexistence curve, has an essential singularity as a function of the distance to the tricritical point, [*i.e.*]{} a KT-like transition. Thus, the transition becomes [*first order*]{} along the phase boundary past the tricritical point with a correlation length that scales like $\xi\sim e^{{\rm const}./\sqrt{s}}$, where $s$ is the distance to the tricritical point measured along the coexistence curve. In contrast, the correlation length across the phase boundary (below the tricritical point) exhibits conventional power-law scaling. Closely related scenarios for the existence of such tricritical points have been suggested in other systems, such as the extended Hubbard model in one-dimension[@fradkin-Hirsch83], the two-dimensional classical Ashkin-Teller model[@Grest81; @Fradkin84], and the dilute 4-state Potts model[@cardy80], which is a statistical system with very similar phase diagram. What happens as the tricritical point is reached, can be understood more physically by noting that at that point the operator $O_{0,2}$, which measures the probability amplitude for a pair of holes (on the same sublattice), becomes relevant. The relevance of $O_{0,2}$ indicates that holes on the same sublattice now have a strong effective attractive interaction, have a strong tendency to pairing and consequently phase separate. The effective field theory description given above corresponds to the grand-canonical picture, since the coupling constants are simple functions of the hole fugacity. On the other hand, in the canonical description, [*i.e.*]{} at fixed hole density $\rho$, the coexistence curve opens up into a two-phase region: there is phase separation between hole-poor regions with local columnar dimer order and hole-rich regions. The jump in the hole and dimer densities (as well as in the order parameters) across the first-order transition is governed by the correlation length at the coexistence curve. Thus, close to the tricritical point the jump in the densities ([*i.e.*]{} the width in density of the two-phase region) has the scaling form $\Delta \rho \sim \xi^{-2}$ and therefore vanishes with an essential singularity as the tricritical point is approached. Similar scaling behavior applies to the discontinuity of the columnar and orientational order parameters across the two-phase region. In the subsequent sections we will give further evidence for the nature of the phase transitions in this system, including the first-order transition, using Monte Carlo simulations in the canonical and grand-canonical ensemble. Monte Carlo Simulations {#sec:MC} ======================= We now employ Monte Carlo simulations to map out the phase diagram of the doped quantum dimer models at their generalized RK points. This approach is complementary to the analytic approach of Sections \[sec:mean-field\], \[sec:zero-density\] and \[sec:finite-density\], and of Appendix \[app:mean-field-details\]. We first introduce (subsection \[sec:methods\]) a canonical Monte Carlo algorithm for interacting dimers. In subsection \[sec:zero-doping\] we apply this method to the case of the interacting fully packed classical dimer model on the square lattice, a system that has been studied recently in some detail,[@alet05; @alet06; @Castelnovo06; @Poilblanc06] and study its phase transition. In subsections \[sec:low-doping\] and \[sec:1storder\] we consider the case of the doped dimer model at low doping and map out the critical line, verifying the theoretical scenario discussed in Section \[sec:finite-density\]. In subsection \[sec:1storder\] we combine the cluster algorithm with conventional grand-canonical moves that permit us to determine the first-order transition line as a function of dimer fugacity $z_d$ and interaction strength $u$. We also estimate the location of the multicritical point discussed in Appendix \[app:mean-field-details\] and Section \[sec:finite-density\], using data from both canonical and grand-canonical simulations. Algorithm for classical interacting dimers {#sec:methods} ------------------------------------------ At high dimer coverage (low doping), conventional Monte Carlo algorithms become very inefficient. On the other hand, in Ref. , it was demonstrated that a geometric cluster algorithm (GCA) can work efficiently for dimers that only have a repulsive hard-core interaction. We briefly summarize this algorithm here. The overlap of two hard-core dimer configurations generates a transition graph. This graph consists of disjoint subgraphs of dimers alternating between the two configurations. In the presence of holes, there are two possible types of graphs: an *open graph* which always terminates on a hole or a *closed loop*. Any Monte Carlo move corresponds to a transition graph of the initial and final configurations. In the geometric cluster algorithm, the two subgraphs are related by a global lattice symmetry. The algorithm obtains long transition graphs with minimal overhead: moves are never rejected, and each dimer encountered during the construction of the graph participates in the move. The construction proceeds as follows.[@krauth03] First, a “seed” dimer and a symmetry axis are chosen at random. The seed dimer is reflected with respect to the symmetry axis, and if it overlaps with other dimers these are reflected as well. This proceeds in an iterative fashion until there are no more dimer overlaps or, equivalently, when an open or closed graph has been formed. On the square lattice, the algorithm is ergodic if we allow both diagonal and horizontal-vertical axes passing through sites of the lattice. The first choice allows to change the numbers of horizontal and vertical dimers, whereas the second one permits to move through the different winding number sectors. Transition graphs generated by the algorithm are symmetric with respect to the symmetry axis, and cross it at most twice. We now extend this approach to dimers with additional interactions by exploiting the *generalized* geometric cluster algorithm proposed by Liu and Luijten.[@liu04; @liu05a] Now, in a single cluster move, multiple transition graphs and/or open graphs are formed simultaneously while retaining the rejection-free character of the algorithm. This is achieved by also reflecting dimers that do *not* overlap, with a probability that depends on the dimer-dimer coupling. When a dimer $i$, located at $\vec r^{\rm old}_i$, is reflected to a new position $\vec r^{\rm new}_i$, there are two classes of dimers that interact with dimer $i$: a) dimers which interact with it *before* it is reflected and b) dimers which interact with $i$ *after* it is reflected. Dimers $j$, located at positions $\vec r_j$, that belong to any of the two classes are included in the cluster (will be reflected with respect to the symmetry axis) with a probability $$p_{ij}=\max \left[1-e^{-\frac{\delta\mathcal{U}_{ij}}{k_BT}},0\right] \;,$$ where $\delta\mathcal{U}_{ij} =V(|\vec{r}^{\rm new}_i - \vec{r}_j|)-V(| \vec{r}^{\rm old}_i -\vec{r}_j|)$ and $V(r)$ represents the interaction between two dimers at a separation $r$. Thus the cluster addition probability for dimer $j$ depends *solely* on the energy difference corresponding to a change in relative position of $i$ and $j$. In the limit of a pure hard-core repulsion, this generalized geometric cluster algorithm reduces to the original GCA. The GGCA applies only to simulations in the canonical ensemble. To perform Monte Carlo simulations in the grand-canonical ensemble, we alternate the cluster moves with conventional grand-canonical Metropolis moves, consisting of insertion and deletion attempts of single dimers. Zero doping: Kosterlitz-Thouless transition to a columnar valence-bond crystal {#sec:zero-doping} ------------------------------------------------------------------------------ According to the theoretical study of Section \[sec:zero-density\] and also from the results of Refs. , we expect to find a Kosterlitz-Thouless transition at zero doping, as a function of the dimer interaction. To detect and locate this transition, we exploit the fact that there is an ordered phase in the large-$u$ region and define columnar and orientational order parameters, $$\begin{aligned} {C}({\bf r}) &\equiv & \sum_{i=x,y} \left[ n_i({\bf r})-n_i({\bf r}+{\bf e}_i)\right] \;, \\ {R}({\bf r}) &\equiv & n_x({\bf r})n_x({\bf r}+ {\bf e}_y)-n_y({\bf r})n_y({\bf r}+{\bf e}_x) \;, \label{orderpar}\end{aligned}$$ where $n_i ({\bf r})$ denotes the dimer density at ${\bf r}$. $\langle C({\bf r}) \rangle$ is non-vanishing only in a columnar-ordered phase, providing a signature of translational symmetry breaking (with a four-fold degeneracy), whereas $\langle R ({\bf r})\rangle $ measures the breaking of invariance under $\pi/2$ rotations. In terms of the most relevant operators of the effective theory of Section \[sec:zero-density\], using OPE, we make the following identifications: $$\begin{aligned} C &\sim&\frac{1}{2}(O_{1,0} + O_{-1,0}) \;, \\ R&\sim&\frac{1}{2}(O_{2,0}+O_{-2,0}) \;.\end{aligned}$$ Having a proper correspondence between the effective theory described in Section \[sec:zero-density\] and the microscopic order parameters , we may verify our predictions. Since the conventional fourth-moment ratio (directly related to the Binder cumulant[@Binder81]) of the order parameter generally does not show a well-defined crossing at a KT transition, we instead use a scaling function of the form of the spontaneous staggered polarization of the six-vertex model,[@baxter73] which maps on the same vertex operator as $R$ in the Coulomb-gas representation, and is in the same universality class. Keeping only the most relevant terms, this function has the form, $$\langle R(u)\rangle = \left(a\frac{1}{\sqrt{u-u_c}} + \cdots\right)e^{\left[-\frac{c}{\sqrt{u-u_c}} + d\sqrt{u-u_c} + \cdots \right]} \;. \label{baxterscaling}$$ From a careful nonlinear least-squares fit to the numerical data outside the finite-size regime (cf. Fig. \[fig:orderpar\]) we obtain $u_c=1.508 \pm 0.003$. ![(color online) The orientational order parameter $R$ in the undoped case: it vanishes for $u<u_c$ and has an essential singularity at $u_c$. The curves represent Monte Carlo data for different lattice sizes, interpolated via multiple histogram reweighting. Inset: data collapse, with a least-squares fit to the exact scaling function for the staggered polarization operator of the six-vertex model  related to $R$ by a universality mapping as discussed in the text.[]{data-label="fig:orderpar"}](PSB_und_orderpar.eps){width="45.00000%"} ![(color online) Fourth-order amplitude ratio of the columnar order parameter $C$ in the undoped case. The curves for all system sizes essentially coincide for the entire critical phase ($u<u_C$), as expected for a KT transition.[]{data-label="fig:CSB_cum_0"}](csb_cumul_1.0.eps){width="45.00000%"} ![(color online) Fourth-order amplitude ratio of the orientational order parameter $R$ in the undoped case. In contrast to the amplitude ratio of the columnar order parameter $C$ (Fig. \[fig:CSB\_cum\_0\]), this quantity exhibits a strong finite-size dependence, leading to an effective crossing point that can be used to locate the transition point.[]{data-label="fig:PSB_cum_0"}](psb_cumul_1.0.eps){width="45.00000%"} For completeness, we also investigate the behavior of the fourth-order amplitude ratios $Q_{M} = \langle{M}^2\rangle^2/\langle {M}\rangle^4$ with $M = C$, $R$. The behavior of $Q_C$, shown in Fig. \[fig:CSB\_cum\_0\], is similar to what is expected for the $XY$ model, namely a collapse of all curves in the critical low-$u$ phase and no well-defined crossing of curves for different system sizes. In contrast, $Q_R$ (Fig. \[fig:PSB\_cum\_0\]) is found to exhibit such strong finite-size effects in the critical phase that its behavior almost resembles that of a regular continuous phase transition. This anomalous behavior explains why the crossing point of the curves for different system sizes could be exploited to obtain an accurate estimate of the critical coupling[@alet05; @alet06]. In the figures presented in this section, the multiple histogram reweighting method[@ferrenberg89] has been used to interpolate all data obtained at different values for the coupling parameter. This allows us to accurately locate crossing points and extrema in the curves. Alet and coworkers[@alet05; @alet06] and Poilblanc and coworkers[@Poilblanc06] used transfer-matrix calculations and Monte Carlo simulations to study the critical behavior of the undoped system for $u>0$ (attractive dimer interactions), whereas Castelnovo and coworkers[@Castelnovo06] used transfer-matrix methods to study primarily the $u<0$ (“repulsive”) regime. In addition, in Ref.  the doped interacting dimer model was also briefly studied for low doping by means of numerical transfer-matrix techniques. All these results are consistent and complementary to those presented in the following subsection. Low doping: Line of fixed points {#sec:low-doping} -------------------------------- We now proceed to the low-doping regime. We perform simulations in the canonical ensemble, for couplings near the critical region and for hole densities $\rho_{h}=0.004, 0.01, 0.02, 0.04, 0.06$. Figure \[susc\] shows, for $\rho_{h}=0.06$, the susceptibilities of the columnar and orientational order parameters, $C$ and $R$, followed by the corresponding fourth-order amplitude ratios in Fig. \[cumul\]. According to the predictions of Section \[sec:finite-density\], we expect that the columnar order parameter $C$, which maps onto the $O_{1,0}$ effective operator in the scaling limit, will retain its scaling dimension $1/8$ along the critical line that emerges from the KT transition point for low hole doping and which constitutes the phase boundary between the dimer-hole liquid and the columnar solid phases. This constant value of the scaling dimension of $C$ is the most salient signature of this critical line: The scaling dimensions of all the other operators change continuously along the phase boundary. To test this prediction, we extract the anomalous dimensions $\eta_C$ and $\eta_R$ (which are equal to twice their scaling dimension) by means of finite-size scaling. The maximum of the susceptibility scales as $\chi^{\rm max}\sim L^{2-\eta}+\cdots$. Subleading scaling contributions are omitted in the fitting expression since the results for sufficiently large lattice sizes satisfy simple scaling \[cf. Figs. \[C-susc\] and \[R-susc\]\]. The correlation-length exponent $\nu$ can be extracted from the slope of the fourth-order amplitude ratios of both order parameters at the critical point. Here, instead, we obtain it from the scaling behavior of the location of susceptibility maximum, which scales as $u_{\chi^{\rm max}}=u_c + {\rm const.}\; L^{-1/\nu}+\cdots$ (cf. Figs.\[susc1pos\] and \[susc2pos\]). The critical coupling $u_c$, in turn, is obtained from the crossing points of the fourth-order amplitude ratio (Fig. \[cumul\]). By repeating this procedure for different hole densities, we find $\eta_C$ and $\eta_R$, as well as $\nu$ as a function of $\rho_h$. We note that for the *undoped* case the (logarithmic) finite-size corrections are so strong that the anomalous exponents are very difficult to determine. By including subleading corrections to the susceptibility expressions at the KT transition, we find strong indications that $\eta_{C}=1/4$ and $\eta_{R}=1$, satisfying the established theoretical description of this KT transition. The density dependence of the exponents is shown in Fig. \[exponents1\]. Clearly, they behave very differently: For the columnar parameter $C$, its anomalous dimension remains unchanged and equal to $1/4$, while for $R$ it decreases monotonically from $1$ to $1/4$ where the transition is expected to become first-order, according to the scenario presented in Section \[sec:finite-density\]. The results from our Monte Carlo simulations are thus consistent with the predictions we made in our theoretical analysis. The evolution of the correlation length exponent along the phase boundary is shown in Fig. \[exponents2\]. This exponent behaves *qualitatively* as predicted for finite doping, it exhibits a monotonically decreasing behavior along the line of fixed points. A direct *quantitative* comparison to the field-theoretical prediction is not possible, since the simulations are performed in the canonical ensemble. Even though the dimer density could be mapped to a *dimer* fugacity, the field theory assumes a fixed *hole* fugacity. In addition, the evolution of the exponent in Fig. \[exponents2\] is slower than predicted because in the simulations we do not approach the phase boundary perpendicularly in the simulations, which leads to an *effective* exponent $\nu$ that is smaller than the one computed from the scaling dimension of the relevant operator in the field theory. ![Phase diagram at low doping, with the critical line of fixed points separating the dimer-hole liquid phase \[$u<u_c(x)$\] from the columnar solid \[$u>u_c(x)$\].[]{data-label="rhojc"}](rho_vs_uc.eps){width="34.00000%"} In Fig. \[rhojc\] we summarize our results for the location of the phase boundary between the dimer-hole liquid phase and the columnar solid phase for hole densities $\rho_h \leq 0.06$. The behavior of the critical line for $\rho_h \to 0$ is consistent with its expected scaling behavior $\rho_h \propto \xi^{-2} (\rho_h=0)$ which is based on simple dimensional analysis, with the only length scale of the problem being the correlation length of the undoped problem. High doping: First Order Transition and Phase Separation {#sec:1storder} -------------------------------------------------------- Beyond the multicritical point predicted in Sections \[sec:mean-field\] and \[sec:finite-density\] (see also Appendix \[app:mean-field-details\]), we expect a first-order transition line in the $z_d$–$u$ phase diagram, where $z_d$ denotes the dimer fugacity. This line separates the crystalline from the liquid phase. According to our scenario, an entropic attraction between holes on the same sublattice becomes marginal and leads naturally to phase separation between a hole-rich liquid phase and a hole-poor columnar crystalline phase. Since the multicritical point is characterized by a marginally relevant operator, complicated crossover will be observed in the first-order transition region in the vicinity of this point.[@cardy80] In addition, close to the multicritical point, the first-order transition will be very weak, with a discontinuity that vanishes with an essential singularity as a function of the distance to the multicritical point along the phase coexistence curve. At this transition, all observables, such as the latent heat, should vanish in a similar way, making the numerical study of the transition close to the multicritical point particularly difficult. To confirm the existence of the discontinuous transition we perform grand-canonical Monte Carlo simulations with single-dimer insertions and deletions (alternated with canonical geometrical cluster moves to accelerate the relaxation of the configurations), for couplings $u= 3.0$, $3.5$, $4.0$, $5.0$, $6.0$, as a function of the dimer fugacity $z_d$. Figure \[dd\] shows the dimer density as a function of dimer chemical potential $\mu_d$. Although with increasing system size a jump in the dimer density develops, it does not become very pronounced. However, consideration of the heat capacity $C_V$ \[Fig. \[heatcap\]\] confirms the presence of a single phase transition at fixed coupling constant $u$, as $C_V$ exhibits a peak at a chemical potential that matches the location of the jump in $\rho_d$. We emphasize that this classical heat capacity is *not* the heat capacity of the $(2+1)$-dimensional QDM. Indeed, $C_V$ does not have any physical meaning in terms of the ground-state wave function that we are considering, because the ground-state energy cannot be changed through variation of the parameters $u$ or $\mu_d$. Another confirmation of the phase transition is obtained from the susceptibilities of the orientational \[Fig. \[susc1fo1\]\] and columnar \[Fig. \[susc2fo1\]\] order parameters, $\chi_R$ and $\chi_C$, respectively. Both quantities exhibit a peak at a chemical potential that approaches, with increasing system size, the location of the peak observed in $C_V$. To confirm the nature of the phase transition, we consider the scaling of the peaks in $C_V$, $\chi_R$, and $\chi_C$. For a first-order transition, these quantities should exhibit a $\delta$-function singularity in the thermodynamic limit or equivalently, for finite systems their peaks must scale with the lattice size in a finite system. We find that the heat-capacity peak, apart from a constant background, indeed scales with the lattice size $L^2$ for the range of system sizes (up to $L=140$) that we considered. This is supported by the behavior of the system-size dependent maxima in $\chi_R$ and $\chi_C$, see Figs.\[susc1fo\] and \[susc2fo\], which both scale as $L^2$, indicating that $\eta = 0$. In addition, all local order parameters must develop a discontinuity at the transition point as the system size increases. Whereas the jump in the dimer density \[Fig. \[dd\]\] is not very sharp, the jump in the columnar and orientational order parameters, $C$ and $R$, is quite pronounced already for the system sizes studied here, see Figs.\[op1\] and \[op2\]. Furthermore, the location the order-parameter jump provides a good indication of the transition point. The strongest evidence, however, is provided by the fourth-order amplitude ratio of the density, $Q_\rho$. At a first-order transition, this quantity displays a specific behavior, as discussed in Ref. . In particular, the positions of two minima observed in Fig. \[dencum\] approach, in the thermodynamic limit, the densities of the two coexisting phases. Outside the coexistence region, $Q_\rho$ approaches a limiting value $1/3$, characteristic of Gaussian fluctuations. This type of behavior is not found at a continuous phase transition, and should be considered as a strong indicator for the occurrence of a first-order transition. This is particularly important since the very weak nature of the first-order transition makes it impossible to unambiguously confirm the existence of a double peak in the histograms of the internal energy for the system sizes that we considered. Whereas the first-order transition becomes more pronounced at higher couplings, and it thus should become easier to distinguish the two peaks in the energy histogram, in practice those simulations are seriously hampered by the very large relaxation times encountered for large dimer interactions. By repeating the analysis presented here for different couplings, and estimating the coexistence chemical potential from the convergence point observed at the order-parameter discontinuity (cf. Fig. \[op\]), we derive the phase diagram in the $z_d$–$u$ plane, see Fig. \[zupd\]. Effects of repulsive hole-hole interactions near the first-order transition region {#sec:hole-int} ================================================================================== We now discuss briefly the effects of additional interactions near the coexistence curve. It is clear that additional interactions near the first-order transition line should stabilize more complex ordered inhomogeneous phases. The simplest interaction that competes with the tendency of holes to pair and phase separate from the crystal is a weak nearest-neighbor hole-hole repulsion $V_h$. The addition of such an interaction to the classical dimer model would lead to an additional energy cost for homogeneous and isotropic clusters of holes. Remarkably, for a range of dimer interactions $u$, this energy cost leads to the formation of commensurate hole stripes with period $3$, in a region in the phase diagram located between the dimer-columnar crystal and the hole-dimer liquid. For general values of $u$, $V_h$ and hole densities one expects a complex phase diagram, most likely similar to what is found in theories of commensurate-incommensurate transitions, which we do not explore here in detail but are discussed in Refs. . This phase can be thought of as the ground-state wave function of a quantum Hamiltonian constructed using the approach described in Section \[sec:hamiltonians\]. The quantum Hamiltonian that leads to the prescribed wave function includes a generalized form of the hole-related Hamiltonian of Eq. \[Ham2\], $$\begin{aligned} H =H_d + t_{\rm hole}\sum_i \Bigg[- \Big|C^{h}_i\Big>\Big<C^{h'}_i\Big| - \Big|C^{h'}_i\Big>\Big<C^{h}_i\Big| \nonumber\\ + y^{R_{C^{h'}_i}-R_{C^{h}_i}}\Big|C^{h}_i\Big>\Big<C^{h}_i\Big|+y^{R_{C^{h}_i}-R_{C^{h'}_i }}\Big|C^{h'}_i\Big>\Big<C^{h'}_i\Big|\Bigg] \;, \label{Hholeint}\end{aligned}$$ where $R_{C_{i}^h}$ and $R_{C_{i}^{h'}}$ denote the number of pairs of holes formed in the corresponding configurations $C_{i}^{h}$ and $C_{i}^{h'} $ (cf. Fig. \[addit\_int\]). More specifically, the ground-state wave function is $$\ket{ G^{int}_{N_h}}=\frac{1}{\sqrt{Z(w^2,y^2,N_{h})}}\sum_{\{\mathcal{C}_{N_h}\}} w^{\displaystyle{N^{d}_p[\mathcal{C}_{N_h}]}}y^{\displaystyle{N^{h}_p[\mathcal{C}_{N_h}]}}\ket{\mathcal{C}_{N_h}} \;. \label{Gzw2int}$$ This wave function has a counterpart in the grand-canonical ensemble, for which the exactly solvable quantum Hamiltonian is a generalization, in exactly the same way as above, of Eq. . ![(color online) A particular hopping process realized in the Hamiltonian Eq. . The potential terms that are present in the Hamiltonian depend on the number of additional pairs of holes that are formed after the hopping process. In the process shown above, this number is 1.[]{data-label="addit_int"}](Ham_fig_add_hole.eps){width="50.00000%"} ![(color online) The hole density structure factor $S(k)$ for dimer coupling $u=5.0$, hole repulsion $V_h=0.5$ and dimer chemical potential $\mu_d=-1.0$, for linear system size $L=128$. The four peaks at $(\pm 2\pi/3,0)$ and $(0,\pm 2\pi/3)$ correspond to the ordered configuration of hole-stripes shown in Fig. \[cstripe2\].[]{data-label="holecf"}](hole_corr_map.eps){width="48.00000%"} ![Snapshot of a part of a typical ordered configuration that appears at the couplings $u=5.0$, $V_h=0.5$, $\mu_d=-1.0$ and linear system size $L=128$. Holes prefer to form commensurate stripes of period $3$, so as to minimize the effect of the weak hole repulsions.[]{data-label="cstripe2"}](config2_stripes.eps){width="50.00000%"} By performing grand-canonical simulations for weak hole repulsions $V_h\le\frac{1}{10}u$ in the regime of strong dimer attractions, $u>4.0$, where the first-order transition is more pronounced, a hole-stripe phase was observed. In particular, for the couplings $u=5.0,V_h=0.5$ and a dimer chemical potential $-1.1<\mu_d<-0.8$ (between the liquid phase $\mu_d \lesssim -1.2$ and the columnar phase $\mu_d \gtrsim -0.8$), the hole density structure factor shows non-trivial peaks at $(k_x,k_y)=(\pm 2\pi/3,0),(0,\pm 2\pi/3)$, which sharpen as the lattice size increases (see Fig. \[holecf\]). A snapshot of the ordered phase (Fig. \[cstripe2\]) illustrates how the holes order in stripes with a period of three lattice spacings, whereas the dimers are still ordered in a columnar pattern. In this way, the holes minimize the effect of the hole-hole repulsions and the dimers simultaneously maximize the effect of the attractive dimer-dimer interactions. The same pattern is also found for $u=4.0,4.5,6.0$, in similar regimes for the hole-hole interaction. More generally, we expect that, as the liquid phase is approached in the regime of strong dimer couplings, a sequence of hole-commensurate phases will be stabilized, leading ultimately to incommensurate phases next to the liquid phase. The formation of this phase diagram is similar in spirit to the ones discussed in Refs. . Elementary quantum excitations of doped quantum Dimer Models {#sec:SMA} ============================================================= In the preceding two sections we discussed the properties of the ground-state wave functions and the behavior of equal-time correlation functions of several operators of physical interest. There we used extensively the connection that exists for these type of wave functions between the computation of equal-time correlators of local operators and computations of similar objects in the equivalent two-dimensional classical statistical mechanical system of interacting dimers and holes. In this section we will be interested in the spectrum of low lying excitations which is inherently a quantum mechanical property. Unfortunately, as it usually the case in QDMs,[@rokhsar88] all we know is the ground-state wave function. The low-lying excitation spectrum is not known exactly but it can be computed approximately using the variational principle. This is the single mode approximation (SMA), which is a useful tool for studying the excited states of many body systems.[@feynman72; @mahan90; @arovas91; @girvin85; @arovas88] It is particularly useful in the case of quantum dimer models at their RK points due to the fact that the exact ground-state wave function is known exactly. The computation of the low lying collective modes in the QDM was done by Rokhsar and Kivelson.[@rokhsar88] Alternatively, one can describe qualitatively the low-lying spectrum using the effective field theory of the quantum dimer models (and their generalizations) at criticality, the [*quantum Lifshitz model*]{} of Ref. . In this section we will consider only the SMA spectrum in the dimer-hole liquid phase. Similar calculations can be done in the phase with long-range columnar order. In the dimer-hole liquid phase the equal-time correlation function of the hole density operator, [*i.e.*]{} the one-body density matrix, approaches a constant at long distances. Thus the wave function for this phase exhibits a Bose condensate of holes. Since the holes are charged, this is a charge Bose condensate. To determine if it is a superfluid (or more precisely a superconductor) it is necessary to show that it has a finite superfluid stiffness, [*i.e.*]{} a critical velocity. This can be determined form the spectrum of density fluctuations and hence from the spectrum of collective modes. It will turn out that, in spite of the more correlated nature of the wave functions we consider here, the result will be similar to that of Rokhsar and Kivelson,[@rokhsar88] [*i.e.*]{} no superfluid stiffness. Given the more general structure (while still local) of the wave functions we study here, we conclude that wave functions associated with Hamiltonians satisfying the RK condition in general do not describe superfluid states. We begin by summarizing the SMA procedure, focusing on doped quantum dimer models. Firstly, one must know the exact ground state of the system $|0\rangle$ and the type of excitations which saturate the frequency sum rule. In our case, there are two candidates: the dimer density and the hole density excitations. Since it follows from a variational principle, the SMA provides a proof of existence only for gapless excitations but not for gapped ones. The energy of an excitation created by an operator with wave vector $\bf k$, which we will denote by $\hat\rho_{\bf k}$, acting on the ground state is bounded from above as follows[@feynman72; @mahan90; @arovas91] $$\begin{aligned} E_{\bf k} - E_0 \le \frac{f(\bf k)}{s(\bf k)}=\frac{\langle0|\left[\hat\rho(-\bf k),\left[\cal{H},\hat\rho(\bf \bf k)\right]\right]|0\rangle}{\langle0|\hat\rho(-\bf k)\hat\rho(\bf k)|0\rangle} \;,\end{aligned}$$ where $f(\bf k)$ is the “oscillator strength” and $s(\bf k)$ is the structure factor ([*i.e.*]{} the equal-time correlation function) for the operator $\hat \rho(\bf k)$. In the case of doped QDMs at their RK point we know their ground-state wave functions exactly and they have (by construction) zero ground-state energy, $E_0=0$. Thus the excitation spectrum must be positive. The system will have gapless excitations if $E_{\bf k}-E_0$ vanishes at some wave vector. It is worth noting that there are two distinct ways for the SMA bound to vanish close to some ${\bf k}={\bf k}_0$. One way is if $f(\bf k)$ vanishes at ${\bf k}_0$. This can happen only if the commutator $\left[\cal{H},\hat\rho({\bf k}_0)\right]$ vanishes. This means that $\rho({\bf k}_0)$ is a conserved quantity. The other way occurs when $s(\bf k)$ becomes infinite at ${\bf k}_0$. This is a signature of a nearby density-ordered state like the columnar dimer crystal we found in the phase diagram of the dimer models under study, shown in Fig. \[fig:sketch\]. In the following, we will use the following operator definitions. For dimers, $\hat\sigma_{\hat \alpha}^{d}({\bf r})$ denotes the dimer density operator and it takes the values $\pm1$ if a dimer is present or absent at the link which begins at the position ${\bf r}$ and has direction $\hat \alpha =\hat x,\hat y$. For holes, $\hat\sigma^{h}({\bf r})$ denotes the hole density operator and it takes the values $\pm1$ if a hole is present or absent at the position ${\bf r}$. Details of the derivations of the SMA oscillator strength functions for both models are given in Appendix \[app:fks\]. Here we just quote the main results. The fixed hole density model ----------------------------- ### Hole density excitations For the fixed hole density model the SMA oscillator strength function $f(\bf k)$ for hole density excitations is given by $$\begin{aligned} f({\bf k})=4t_{\rm hole}{\bf q}^2\end{aligned}$$ for ${\bf k}=(\pi,\pi)+\bf q$. From the results of the Section \[sec:mean-field\] and Appendix \[app:mean-field-details\], we may conclude that: For ${\bf k}=(\pi,\pi)+{\bf q}$, the structure factor near $(\pi,\pi)$ scales like $s^{(\pi,\pi)}({\bf k})\propto \frac{1}{q^2+\xi^{-2}}$ and thus, $s^{(\pi,\pi)}(0)$ is a constant. Thus $E_{\bf k}^{(\pi,\pi)}-E_0\propto {\bf q}^2$. For ${\bf k}=(0,0)+{\bf q}$, the results from the following section, which formally hold only for low density of dimers, lead to the conclusion that $s^{(0,0)}({\bf k})\propto \frac{1}{q^2-\xi^{-2}}$ and $s^{(0,0)}(0)$ is again a constant but has strong oscillatory behavior in real space. Thus, from the correlation function we computed in and Appendix \[app:mean-field-details\], we conclude that these excitations are also quadratic in the momentum ${\bf q}$,  $E_{\bf k}-E_{0}\le {\bf q}^2$. This result is consistent with the compressibility argument, given in Ref.  which would also give quadratic dispersion in this case (keeping in mind that in this case the compressibility is infinite when the system is doped (constant number of holes)). However, it is important to stress that the compressibility argument is a stronger condition because our calculation of the correlation function is legitimate for low dimer densities. ### Dimer density excitations For the fixed hole density model the SMA oscillator strength function $f(\bf k)$ for dimer density excitations is given by $$\begin{aligned} f({\bf k}) = f_{\rm dimer-flip}({\bf k}) + f_{\rm hole(1)}({\bf k}) \label{fkh}\end{aligned}$$ Close to the wave vector ${\bf Q}_0=(\pi,\pi)$ with ${\bf k}={\bf Q}_0 + {\bf q}$, both oscillator strengths $f_{\rm dimer-flip}({\bf k})$ and $f_{hole(1)}$ vanish quadratically (cf. Appendix B) and more specifically, $$\begin{aligned} f_{\rm dimer-flip}({\bf q}) = 8tq^2\\ f_{\rm hole(1)}({\bf q})=-t_{\rm hole}q^2 \label{fkh2}\end{aligned}$$ Given the fact that the dimer density structure factor is a constant at $Q_0=(\pi,\pi)$ and combining Eqs. and , we may conclude that there are gapless dimer density excitations at $Q_0=(\pi,\pi)$. In addition to these results, we may have additional branches of dimer density excitations, specially when there is a divergence of the structure factor at some wave vector ${\bf Q}_{0}$ which would would be an indication of dimer order at a nearby phase. In particular, this happens at the phase boundary between the hole-dimer liquid phase and the columnar-ordered crystalline phase. The fixed hole fugacity model ----------------------------- As above, we can study the hole density excitations of the second model, the grand-canonical model, in which the hole density is not fixed, but is determined by a parameter $z$ which plays the role of a hole fugacity in the wave function. By the fact that there is no conservation of the number of holes is explicitly broken, we can predict that the numerator $f({\bf k})$ will vanish only at ${\bf Q}_0 =(\pi,\pi)$. This happens because of the bipartite lattice symmetry that enforces the number holes on each sublattice to be equal. For the fixed hole fugacity model the SMA oscillator strength function $f(\bf k)$ for hole density excitations is $$\begin{aligned} f_{\rm hole(2)}({\bf k})=4t_{\rm pairing}(2+\cos({\bf k}_x)+\cos({\bf k}_y)) \label{pairing}\end{aligned}$$ The formula shows that the numerator $f_{\rm hole(2)}({\bf k})$ vanishes only at the wave vector ${\bf Q}_0=(\pi,\pi)$ quadratically, as expected. The dimer density excitations are not gapless (in the sense that the numerator does not vanish for any ${\bf Q}_0$) because the dimer-flip contribution vanishes at $(\pi,\pi),(0,\pi)$ and the dimer-breaking term gives a constant contribution. This is expected due to the violation of the dimer number conservation. It is important to stress that the behavior of the structure factor is exactly the same as before (in the fixed hole-density model). The reason is the equivalence of the canonical and grand-canonical ensemble in the thermodynamic limit for classical systems. It is certainly worth noting that, although the two ground-state wave functions of Eqs.  and  have the same ground-state physics, due to the essential equivalence of the classical canonical and grand-canonical ensembles in the thermodynamic limit, the nature of their quantum elementary excitations is drastically different. Conclusions {#sec:conclusions} =========== In this work we have constructed generalizations of the quantum dimer model to include the effects of dimer correlations as well as finite hole doping in the wave function. Throughout we considered the case of bosonic (charged) holes and neglected the fermionic spinons. We have constructed generalized RK Hamiltonians whose ground-state wave functions describe the effects of (attractive) dimer correlations and finite hole doping. We have discussed the rich phase diagram and critical behavior of three doped interacting quantum dimer models at their RK point using both analytic methods and numerical simulations. We have shown that the ground-state wave function embodies a complex phase diagram which consists of dimer-hole liquid and columnar phases separated by a critical line with varying exponents, ending at a multicritical point with a Kosterlitz-Thouless structure where the transition becomes first order. The critical behavior along the low doping section of the phase boundary was investigated in detail and the predictions of our scaling analysis were confirmed with large-scale Monte Carlo simulations. Monte Carlo simulations were also used to show that the transition between the dimer-hole liquid and the columnar solid does indeed become first order, and to estimate the location of the multicritical point and of the first-order phase boundary. In the high-doping regime, near the first-order transition line, additional repulsive interactions among holes were shown to generate, at the expense of the two-phase region, an even richer phase diagram with phases in which the dimer-hole system becomes inhomogeneous. In the regime of strong dimer coupling, $u=5=0.$, and weak hole interactions, $V_h=0.5$, we found a stripe phase with wave vectors $(2\pi/3,0)$ and $(0,2\pi/3)$. In general, we expect the two-phase region to be replaced by a complex phase diagram with a large number of commensurate and incommensurate phases. This physics is well known in the context of two-dimensional classical statistical mechanics of systems with competing interactions.[@pokrovsky-talapov; @bak82; @fisher-selke] However, it is interesting to see how it arises at the level of the exact ground-state wave function of models of strongly correlated systems, particularly given the current interest on this type of phenomena in high-temperature superconductors and related systems.[@zaanen89; @kivelson93; @Kivelson98; @emery99; @subir-rmp; @balents05b] In this paper we have also presented an analysis of the low-lying excitations of the quantum system and found that, much as in the case of the Rokhsar-Kivelson quantum dimer model, the doped system is a Bose-Einstein condensate but not a superfluid, since the superfluid stiffness vanishes even though both dimers and holes interact. It is apparent that in order to render the Bose-Einstein condensate a true superfluid it is necessary to violate the RK condition which forces the wave function to have a local structure. The effects of violations to the RK condition are poorly understood, and we have not investigated this important problem which is essential to determine the generic phase diagram of these models. We thank C. Castelnovo, C. Chamon, P. Fendley, S. Kivelson, M. Lawler, R. Moessner, C. Mudry, V. Pasquier, P. Pujol, K. S. Raman, S. Sondhi, and M. Troyer for many discussions. This work was supported in part by the National Science Foundation through grants NSF DMR 0442537 (EF), and CAREER Award NSF DMR 0346914 (EL), and by the U.S. Department of Energy, Division of Materials Sciences under Award DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign (EL and EF). The calculations presented here were in part performed at the Materials Computation Center of the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign (EL), the Turing XServe Cluster at the University of Illinois and at NCSA Teragrid cluster facilities under award PHY060022. Mean-Field Theory for dimers and holes {#app:mean-field-details} ====================================== In this Appendix we give details of the mean-field theory summarized in Section \[sec:mean-field\]. We will use Grassmann variable methods to write down the partition functions for classical interacting and doped dimers. We will use these methods to derive a simple mean-field theory for this system. While mean-field theory, as it is well known, fails to give the correct critical behavior in two-dimensional systems, it offers a good qualitative description of the phases and, surprisingly, even of the gross features of the phase diagram. In subsequent sections we will use more sophisticated analytic and numerical methods to study the phase transitions. Non-Interacting dimers at finite hole density ---------------------------------------------- The classical dimer-hole partition function can be formulated in terms of a Grassmann functional integral, according to the prescription of Ref. . The classical partition function of the dimer problem on any lattice which is defined by the connectivity matrix $M$ and with fugacity $z$ for the dimers and 1 for the holes is: $$\begin{aligned} \mathcal{Z}_{\rm dimer} &=& \int \mathcal{D}\eta\mathcal{D}\eta^\dagger e^{\sum_i \eta_i\eta^{\dagger}_i + \frac{z}{2}\sum_{ij}M_{ij}\eta_i\eta_{i}^\dagger\eta_j\eta_{j}^\dagger}\nonumber\\ &&\end{aligned}$$ For the square lattice, $M_{ij}$ is 1 if $i,j$ are nearest neighbor sites and zero otherwise. For the triangular lattice, $M_{ij}$ is 1 if $i,j$ are nearest neighbors and also next-nearest neighbor sites, but only along one diagonal, if $j = i + \hat{\bf{x}} +\hat{\bf{y}}$ or $j = i - \hat{\bf{x}} -\hat{\bf{y}}$. Also, the fugacity z ranges from 0 to $\infty$. When $z \rightarrow 0$, the system is filled with holes and there are very few dimers and when $z\rightarrow\infty$ the system approaches the close-packed limit with dimers. Both limits are worth of study due to the existence of an important theorem by Heilmann and Lieb[@lieb1970] which proves under rather general assumptions the absence of any phase transitions with doping in this model. Thus, the identification of the phase in the *few dimers* limit is enough to conclude about the phase that the system enters when doped. In this section we will study this problem in the limit in which the dimers are dilute. In this regime a simple mean-field theory of the Hartree type is expected to be accurate.[@Samuel] Such a crude approximation should break near criticality, [*i.e.*]{} near the close packing limit. To proceed, we apply a Hubbard-Stratonovich transformation to the above partition function: $$\begin{aligned} e^{\frac{z}{2}\sum_{ij}M_{ij}\eta_i\eta_{i}^\dagger\eta_j\eta_{j}^\dagger} &=& \mathcal{N}\int\mathcal{D}\phi e^{-\frac{1}{2z}\sum_{ij}\phi_iM^{-1}_{ij}\phi_j+\sum_i\eta_i\eta_{i}^\dagger\phi_i} \nonumber \\ &&\end{aligned}$$ where $\mathcal{N}$ is an irrelevant normalization constant. We may also add sources for the hole density operators $J\eta\eta^\dagger$. In this way, the classical dimer partition function may be written as follows: $$\begin{aligned} \mathcal{Z}_{\rm dimer} &=& \int\mathcal{D}\phi\mathcal{D}\eta\mathcal{D}\eta^\dagger e^{-\frac{1}{2z}\sum_{ij}\phi_iM^{-1}_{ij}\phi_j+\sum_i\eta_i\eta_{i}^\dagger(\phi_i + 1 + J_i)}\nonumber\\ &&\end{aligned}$$ where he have dropped the normalization constant $\mathcal{N}$, as we will do in what follows. (Note that what makes the above problem *unsolvable* is the term ‘1’ in the exponent!) Upon integrating out the Grassmann variables we find: $$\begin{aligned} \mathcal{Z}_{\rm dimer} &=& \int \mathcal{D}\phi \; e^{-\frac{1}{2z}\sum_{ij}(\phi_i-J_i)M^{-1}_{ij}(\phi_j-J_j)+\sum_i\ln(\phi_i + 1)} \nonumber \\ &&\end{aligned}$$ In the limit $z\rightarrow 0$ we may formulate a legitimate and well-defined mean-field theory. To this end we rewrite the partition function in the following way: $$\begin{aligned} \mathcal{Z}_{\rm dimer} &=&\int \mathcal{D}\phi \; e^{-\frac{1}{z}\left[\frac12\sum_{ij}(\phi_i-J_i)M^{-1}_{ij}(\phi_j-J_j)-z\sum_i\ln(\phi_i + 1)\right]}\nonumber\\ &=& \int \mathcal{D}\phi \; e^{-\frac{1}{z}\mathcal{S}(\phi)}\end{aligned}$$ where $$\begin{aligned} S(\phi) &=& \frac{1}{2}\sum_{ij}(\phi_i-J_i)M^{-1}_{ij}(\phi_j-J_j)-z\sum_i\ln(\phi_i + 1) \nonumber\\ &&\end{aligned}$$ is the effective action. As $z\rightarrow 0$, we have a theory which has a well-defined saddle point and the perturbation around this point will be in powers of $z$ which plays the role of an effective coupling constant. In this way, we have a very fast convergent expansion. The saddle-point is defined as follows: $$\begin{aligned} \frac{\delta\mathcal{S}}{\delta\phi_i}\bigg|_{\phi_i=\bar\phi} = 0\end{aligned}$$ and we take the following equation for $\bar\phi$ : $$\begin{aligned} \bar\phi_i = J_i + z\sum_{j}\frac{M_{ij}}{\bar\phi_j + 1}\end{aligned}$$ In this approximation, the density of holes is given by: $$\begin{aligned} \rho_i =\langle\eta_i\eta^{\dagger}_i\rangle &=& \frac{\partial \ln \mathcal{Z}_{dimer}}{\partial J_i}\\ &=& -\frac{1}{z}\frac{\partial \mathcal{S}}{\partial J_i}\\ &=& \frac{1}{\bar\phi_i + 1}\end{aligned}$$ So, the source $J_i$ in terms of the density of the holes $\rho_i$, is given by: $$\begin{aligned} J_i &=& \bar\phi_i - z\sum_{j}\frac{M_{ij}}{\bar\phi_j + 1} \\ &=& \frac{1}{\rho_i} - z\sum_j M_{ij}\rho_j - 1\end{aligned}$$ The Legendre transform of the effective action $\mathcal{S}$ is: $$\begin{aligned} \Gamma(\rho_i) &=&\frac{1}{z}\mathcal{S}(\bar\phi_j(\rho_i),J_j(\rho_i))+\sum_{i}J_i(\rho_j)\rho_i \\ &=& -\frac{z}{2}\sum_{ij}\rho_iM_{ij}\rho_j + \sum_i\ln(\rho_i)+\sum_i(1-\rho_i) \nonumber \\ && \label{LegendreTransform}\end{aligned}$$ We specialize now to the case of uniform hole density $\rho_i = \rho$ and also for the case of the square lattice where the number of nearest neighbors is $2D=4$ and assuming that the number of sites on the lattice is $N$. Then, $$\begin{aligned} \Gamma(\rho) &=& -\frac{z}{2} (4N\rho^2) + N\ln(\rho) + N(1-\rho) \end{aligned}$$ At this level of approximation (“Hartree”) the equation of state becomes $$\begin{aligned} J = - 4z\rho + \frac{1}{\rho} - 1 \label{eqnstate0}\end{aligned}$$ So, when the sources are set to zero, the density in terms of the fugacity, in the limit $z\rightarrow0$, is: $$\begin{aligned} \rho\; &=& \frac{2}{1+\sqrt{1+16z}}\end{aligned}$$ As a check of the approximation, we may expand the result in the region of $z\rightarrow0$, where the result should be $\rho\simeq1$ for the hole density: $$\begin{aligned} \rho \; = 1 - 4z +O(z^2)\end{aligned}$$ Or, equivalently, for $\rho \to 1^-$, $$\begin{aligned} z = \frac{1}{4\rho} \left(\frac{1}{\rho}-1\right) = \frac{1}{4}\left(1-\rho \right)+O((1-\rho)^2) \label{eqnstate2}\end{aligned}$$ The hole density-density correlation function can be obtained in the following way by using the Legendre transform: $$\begin{aligned} \mathcal{G}_{ij}&=&\langle\eta_i\eta^{\dagger}_i\eta_j\eta^{\dagger}_j\rangle - \langle\eta_i\eta^{\dagger}_i\rangle\langle\eta_j\eta^{\dagger}_j\rangle\nonumber \\ &=& \frac{\partial^2\ln Z_{dimer}}{\partial J_i\partial J_j} = \frac{\partial\rho_i}{\partial J_j}\end{aligned}$$ and also, $$\begin{aligned} \mathcal{G}_{ij}=\left[\frac{\partial^2 \Gamma}{\partial\rho_i\partial \rho_j}\right]^{-1} \label{invG}\end{aligned}$$ By using and , we have: $$\begin{aligned} \left[\mathcal{G}_{ij}\right]^{-1}=\frac{\partial^2 \Gamma}{\partial\rho_i\partial \rho_j} = - zM_{ij} -\delta_{ij}\frac{1}{\rho_{i}^2}\end{aligned}$$ Since $M_{ij}$ is a function of the distance between sites and vanishes except for nearest neighbors, its Fourier transform is: $$\begin{aligned} M(\vec q)&=& a^2\sum_{i}e^{-i{\bf q}\cdot({\bf r}_i - {\bf r}_j)}M({\bf r}_i - {\bf r}_j)\nonumber \\ &=& 2a^2\sum_{\alpha=1}^{2}\cos q_\alpha a\end{aligned}$$ Also, we set $a=1$ and finally the hole density connected correlation function is: $$\begin{aligned} \mathcal{G}(\vec q) &=& \frac{1}{-\frac{1}{\rho^2}-2z\sum_{\alpha=1}^{2}\cos q_\alpha }\end{aligned}$$ For momenta near ${\bf Q}=(\pi,\pi)$, ${\bf q} ={\bf Q}+{\bf p} $, with ${\bf p}$ small, it becomes $$\begin{aligned} \mathcal{G}({\bf p} + {\bf Q})&\simeq& \;-\;\frac{\rho \xi^{-2}}{\xi^{-2} + {\bf p}^2 }\end{aligned}$$ where $\xi$ is the correlation length $$\xi =\sqrt{\displaystyle{\frac{1-\rho}{4\rho}}} \label{eq:corr-length}$$ Finally, the connected hole density correlation function (the structure factor) in real space for $\rho \to 1$ is: $$\begin{aligned} \mathcal{G}(r) &= & \frac{(-1)^{r_x+r_y+1}}{\sqrt{\pi\xi r}} e^{-\xi r}\;\;\;\;\;\; \label{hhcfn}\end{aligned}$$ If we restore the units, the correlation length is $\xi = a\sqrt{\frac{1-\rho a^2}{4\rho a^2}}$. Surprisingly, given how crude this approximation is, the result of Eq.  is consistent with the numerical results provided by Krauth and Moessner[@krauth03] for much of the dimer density range they studied. Significant deviations are seen only upon approach of the close packing regime where the classical dimer model is critical and this mean-field calculation fails, [*e.g.*]{} the correlation length diverges as $\rho \to 0$ with exponent $1/2$, given by Eq. , (the mean-field value). The correct value of the exponent can be deduced from Table \[table:0density\] and it is $1/(2-1/4)=4/7$ ($1/4$ being the scaling dimension of the hole operator for the non-interacting case.) As we show in Section \[sec:zero-density\] (and in Table \[table:0density\]), the dimension of the hole operator grows from the value $1/4$ for free dimers to a value of $2$ at the (Kosterlitz-Thouless) transition to the columnar state, where it should exhibit an essential singularity due to the marginality of the hole operator. Adding interactions between dimers ---------------------------------- Clearly, the mean-field method for the non-interacting dimer-hole system can be easily extended to dimer-hole systems with local interactions. In the case of attractive interactions between parallel dimers, the partition function of the monomer-dimer system should be $$\begin{aligned} \mathcal{Z}_{\rm d-int} = \int \mathcal{D}\eta\mathcal{D}\eta^\dagger \exp\Bigg( \sum_i \eta_i\eta^{\dagger}_i + \frac{z}{2}\sum_{ij}M_{ij}\eta_i\eta_{i}^\dagger\eta_j\eta_{j}^\dagger\nonumber\\ +\frac{V}{4}\sum_{ijkl}\tilde M_{ijkl}\eta_i\eta_{i}^\dagger\eta_j\eta_{j}^\dagger\eta_k\eta_{k}^\dagger\eta_l\eta_{l}^\dagger\Bigg)\nonumber\\\end{aligned}$$ where $V=z^2(e^{u}-1)$ for an attractive interaction of strength $u>0$ between parallel neighboring dimers. $M_{ij}$ represents the coordination array of the lattice and it takes the value $1$ when $i$ is nearest neighbor to $j$ and otherwise is zero. In the same respect, $\tilde M_{ijkl}$ takes the value $1$ only when $i,j,k,l$ are arranged on a square plaquette and is zero otherwise. The sums run along all possible lattice sites for each index. We may perform two Hubbard-Stratonovich transformations by introducing the fields $\chi,\phi$ and then we have (again, dropping all normalization constants): $$\begin{aligned} \mathcal{Z}_{\rm d-int}&& =\int \mathcal{D}\eta\mathcal{D}\eta^\dagger \mathcal{D}\chi \exp\Bigg[\sum_i \eta_i\eta^{\dagger}_i -\frac{1}{V}\sum_{ijkl}\chi_{ij}(\tilde M_{ijkl})^{-1}\chi_{kl}+ \sum_{ij}\eta_i\eta_{i}^\dagger\eta_j\eta_{j}^\dagger(\chi_{ij}+\frac{z}{2}M_{ij})\Bigg]\nonumber \\ && =\int \mathcal{D}\eta\mathcal{D}\eta^\dagger \mathcal{D}\chi\mathcal{D}\phi \exp\Bigg[-\frac{1}{V}\sum_{ijkl}\chi_{ij}\tilde M_{ijkl})^{-1}\chi_{kl} -\frac{1}{4}\sum_{i}\phi_i(\chi_{ij}+\frac{z}{2}M_{ij})^{-1}\phi_j+ \sum_{ij}\eta_i\eta_{i}^\dagger(\phi_i+1)\Bigg]\;\;\;\;\; \nonumber \\ &&=\int \mathcal{D}\chi\mathcal{D}\phi \exp\Bigg[-\frac{1}{V}\sum_{ijkl}\chi_{ij}(\tilde M_{ijkl})^{-1}\chi_{kl} -\frac{1}{4}\sum_{ij}\phi_i(\chi_{ij}+\frac{z}{2}M_{ij})^{-1}\phi_j + \sum_{i}\ln(\phi_i+1)\Bigg]\end{aligned}$$ Just as we did above, we introduce a set of auxiliary sources $J_{ij}^\chi$ defined on links $ij$, and $J_i^\phi$ defined on sites $i$. The partition function now reads $$\begin{aligned} && \mathcal{Z}_{\rm d-int}[J^\chi,J^\phi] =\int \mathcal{D}\chi\mathcal{D}\phi \; e^{\displaystyle{-\frac{1}{Vz}\mathcal{S}(\phi,\chi;J^\phi,J^\chi)}}\end{aligned}$$ where the action now is: $$\begin{aligned} \mathcal{S}(\phi,\chi;J^\phi,J^\chi)&=&z\sum_{ijkl}(\chi_{ij}-J^{\chi}_{ij})(\tilde M_{ijkl})^{-1}(\chi_{kl}-J^{\chi}_{kl}) \nonumber \\ &&+\frac{Vz}{4}\sum_{ij}(\phi_i-J^{\phi}_i)(\chi_{ij}+\frac{z}{2}M_{ij})^{-1}(\phi_j-J^{\phi}_j) -Vz\sum_{i}\ln(\phi_i+1) \nonumber \\ && \label{act-int} \end{aligned}$$ We define the densities conjugate to the fields $\phi_i,\chi_{ij}$ as: $$\begin{aligned} n_i = \frac{\partial \ln \mathcal{Z}_{d-int}}{\partial J_i^\phi}=\frac{1}{2}\sum_{j}(\chi_{ij}+zM_{ij})^{-1}\phi_{j}, \qquad m_{ij}=\frac{\partial \ln \mathcal{Z}_{d-int}}{\partial J_{ij}^\chi}=\frac{2}{V}\sum_{kl}(\tilde M_{ijkl})^{-1}\chi_{kl}\end{aligned}$$ Solving for the fields and replacing in Eq., we have finally for the effective potential or equivalently the Gibbs free energy : $$\begin{aligned} \Gamma(z,V)= \frac{V}{4}\sum_{ijkl}m_{ij}\tilde M_{ijkl}m_{kl} &+& \sum_{ij}n_i\bigg(\frac{V}{2}\sum_{kl}\tilde M_{ijkl}m_{kl}+\frac{z}{2}M_{ij}\bigg)n_j \nonumber \\ &-& \sum_{i}\ln\Bigg[2\sum_{j}\Bigg(\frac{V}{2}\sum_{kl}\tilde M_{ijkl}m_{kl} + \frac{z}{2}M_{ij}\Bigg)n_j + 1 \Bigg] \nonumber \\ &&\end{aligned}$$ The ordered phase of dimers is expected to be a columnar one. So, for $m_{ij}=(-1)^{x_i}\delta_{i,j-\hat x}m + m_0$ and $n_j=n$, the free energy is: $$\begin{aligned} \frac{\Gamma(z,V)}{N}&=&Vm^2+2Vm_{0}^2+4Vm_{0}n^2+2zn^2 \nonumber \\ &&-\frac{1}{2}\ln\Bigg(1+8n\bigg(V(\frac{m}{2}+m_0)+\frac{z}{2}\bigg)\Bigg) -\frac{1}{2}\ln\Bigg(1+8n\bigg(V(-\frac{m}{2}+m_0)+\frac{z}{2}\bigg)\Bigg) \nonumber \\ && \label{gibbs}\end{aligned}$$ Now, we proceed by solving two of the three extremal (saddle-point) equations: $$\frac{\delta\Gamma}{\delta n}=0 \quad {\rm and} \quad \frac{\delta\Gamma}{\delta m_0}=0$$ which take the explicit form $$\begin{aligned} 8Vm_0n+4zn- \frac{4\bigg(V(\frac{m}{2}+m_0)+\frac{z}{2}\bigg)}{1+8n\bigg(V(\frac{m}{2}+m_0)+\frac{z}{2}\bigg)} - \frac{4\bigg(V(-\frac{m}{2}+m_0)+\frac{z}{2}\bigg)}{1+8n\bigg(V(-\frac{m}{2}+m_0)+\frac{z}{2}\bigg)}=0 \label{sp1} \\ 4Vm_{0}+4Vn^2-\frac{4nV}{1+8n\bigg(V(-\frac{m}{2}+m_0)+\frac{z}{2}\bigg) } -\frac{4nV}{1+8n\bigg(V(\frac{m}{2}+m_0)+\frac{z}{2}\bigg)}=0 \label{sp2}\end{aligned}$$ For $z=V=0$, the solution of the saddle-point equations is trivial $m_0=n=1$. For small $z$, we can solve Eqns. and recursively, expanding also in the small order parameter $m$ (this is correct close to a continuous phase transition or to a weakly first order transition): $$\begin{aligned} n&&\simeq1-4z-8V-\frac{1}{4(2V+z)}(\frac{16V^2m^2}{(1+8V+4z)^2}+ \frac{256V^4m^4}{(1+8V+4z)^4}+\frac{4096V^6m^6}{(1+8V+4z)^6})\nonumber \\ && \label{n}\\ m_0&&\simeq \frac{2}{1+8V+4z}-1+\frac{1}{4V}(\frac{128V^3m^2}{(1+8V+4z)^3}+ \frac{2048V^5m^4}{(1+8V+4z)^5}+\frac{32768V^7m^6}{(1+8V+4z)^7})\nonumber \\ && \label{m0}\end{aligned}$$ Replacing Eqs. , in the free energy , and also expanding in the small parameter $z$, we have: $$\begin{aligned} \frac{\Gamma(z,V)}{N}=C_0(z,V_r)+C_2(z,V_r)m^2+C_4(z,V_r)m^4 +C_6(z,V_r)m^6 + \cdots\end{aligned}$$ where $V_r\equiv\frac{V}{z^2}=e^u-1$ and $$\begin{aligned} C_0(z,V_r)&&=-2z+(8-2V_r)z^2 +\frac{32}{3}(-5+3V_r)z^3+O(z^4)\label{c0}\\ C_2(z,V_r)&&=V_rz^2(1+8V_rz^2-128V_rz^3 -256(-7+V_r)V_{r}z^4)+O(z^7)\label{c2}\\ C_4(z,V_r)&&=-32V_{r}^4z^7(1-2(19+V_r)z +4(184+V_r(4+V_r))z^2+ 8(1336+ V_r(-232+V_r(4+V_r)))z^{3}) + O(z^{11})\nonumber \\ &&\label{c4}\\ C_6(z,V_r)&&= 128V_{r}^6z^{10}(1-4(V_r+12)z +4(340+V_r(28+3V_r))z^2)+O(z^{13})\label{c6}\end{aligned}$$ In the limit of very low dimer density, $z\rightarrow 0$, from Eqs. -, there is a clear first-order phase transition from an empty lattice ($n=1,m=0$) to a columnar dimer crystal ($m\neq0$) because $C_2>0,C_4<0$ and $C_6>0$. In particular, in this limit, the first-order transition happens when: $$\begin{aligned} C_2 &=&\frac{C_{4}^2}{4C_6}\\ z^2V_r &\simeq&\frac{(32V_{r}^4z^7)^2}{512V_{r}^6z^{10}}\\ e^u &=& 1+\frac{1}{2z^2}\simeq \frac{1}{2}z^{-2}\Longrightarrow 2z^2e^u=1 \label{zrw0}\end{aligned}$$ The condition can be derived through a very simple argument: In the limit of very low dimer densities, the non-local effects which are related to the hard-core dimer constraint are negligible. If we consider just a single plaquette, then the contribution from four holes on this plaquette to the Gibbs weight of the partition function will be just unity but the contribution of two parallel dimers arranged either vertically or horizontally will be $2z^2e^u$. When $2z^2e^u=1$, the holes become unstable to the formation of pairs of parallel nearest neighboring dimers and the result is a columnar dimer crystal. When $C_2=C_4=0$ (whereas $C_6$ remains positive), there is a mean-field tricritical point where the transition ultimately becomes continuous. Using the approximate Eqs. -, we can have an estimate for the tricritical point. Solving the set of equations we arrive to the following estimate: $$\begin{aligned} z_{tr} &=&0.075\\ u_{tr} &=& 2.733\end{aligned}$$ Remarkably enough, these estimates are very close to the estimates from the grand-canonical simulations that are presented in Section \[sec:MC\]. However, we should be very cautious on taking these estimates too seriously since the terms in the expansion of the coefficients $C_0,C_2,C_4,C_6$ in powers of $z$, generically have alternating signs with increasingly large constant coefficients which suggests that this expansion is not convergent. In any case, mean-field approximations, such as the one presented here, fail in two dimensions. The actual multicritical point has a more complex analytic structure, akin to a Kosterlitz-Thouless transition, than suggested by these Landau-Ginzburg type arguments. On the other hand, the multicritical point can be approached from the high dimer density limit, where the transition is continuous and thus, an effective field theory description is possible. As we show in the following Sections \[sec:zero-density\],\[sec:finite-density\], this multicritical point is controlled by a marginally relevant operator and belongs to the Kosterlitz-Thouless universality class. Derivation of the SMA oscillator strength functions $f(\bf k)$. {#app:fks} =============================================================== In this Appendix we present the details of the calculations of the SMA oscillator strength functions $f(\bf k)$ discussed in Section \[sec:SMA\]. The fixed hole-density model {#sec:fixed-density} ---------------------------- ### Hole density excitations The only term of the Hamiltonian which does not commute with the hole density operator is the hopping term for the holes. This term, in terms of destruction-creation Pauli operators ${\sigma_{\hat\alpha}^{d}}^{\pm},{\sigma^{h}} ^{\pm}$ can be written as: $$\begin{aligned} \mathcal{T}_{\rm t-hole}= \hspace{6cm}\nonumber\\ -\tilde t_{\rm hole}\sum_{<ijk>}{\sigma^{h}}^-({\bf r}_i){\sigma^{h}}^+({\bf r}_k){\sigma^{d}_{\hat\alpha}}^+({\bf r}_{ij}){\sigma^{d}_{\hat\alpha}}^-({\bf r}_{jk})\end{aligned}$$ At this point, we are not interested for the possible orientations of the dimers with respect to the holes, as the hole density operator commutes with the dimer density operator. Now, we will repeatedly use that $$\begin{aligned} \left[{\sigma^{h}}^{\pm},\sigma^{h}\right] = \mp 2{\sigma^{h}}^{\pm}\end{aligned}$$ at the same position in real space. At any distance different from zero, the commutator vanishes. For a given hole at a position ${\bf R}$, there are four possible positions ${\bf R}'={\bf R} \pm \hat x \pm\hat y$ where it may move through the available hopping term. By counting contributions from all these terms for every site of the lattice, we exactly take into account all the terms of the Hamiltonian including Hermitean conjugates. If ${\bf R}$ is the initial hole position and ${\bf R}+{\bf r}_0$ the final one, then ${\bf r}_0$ has eight possible values: ${\bf r}_0=\pm \hat x \pm \hat y $. The first commutator can now be computed for any site ${\bf R}$(The operator $\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0)$ contains each of the above eight hopping terms). We have: $$\begin{aligned} \left[-t_{\rm hole}\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0),{\sigma^{h}}({\bf k})\right] &=& -t_{\rm hole}\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0)\left(e^{-i{\bf k}\cdot{\bf R}}-e^{-i{\bf k}\cdot({\bf R}+{\bf r}_0)}\right)\\ \left[{\sigma^{h}}(-{\bf k}),-t_{\rm hole}\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0)\right] &=& t_{\rm hole}\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0)\left(e^{i{\bf k}\cdot{\bf R}}-e^{i{\bf k}\cdot({\bf R}+{\bf r}_0)}\right)\\ \left[{\sigma^{h}}(-{\bf k}),\left[-t_{\rm hole}\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0),{\sigma^{h}}({\bf k})\right]\right] &=& 2t_{\rm hole}\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0)\left(1-\cos({\bf k}\cdot{\bf r}_0)\right)\;\;\;\end{aligned}$$ Thus, the oscillator strength $f({\bf k})$ can now be calculated: $$\begin{aligned} f({\bf k})&=& \sum_{{\bf R},{\bf r}_0}\langle\left[{\sigma^{h}}(-{\bf k}),\left[-t_{\rm hole}\mathcal{T}_{\rm hole}({\bf R},{\bf r}_0),{\sigma^{h}}({\bf k})\right]\right]\rangle 2t_{\rm hole}\sum_{{\bf r}_0}\left(1-\cos({\bf k}\cdot{\bf r}_0)\right)\end{aligned}$$ If we set ${\bf k}={\bf Q}_0+{\bf q}$ where ${\bf q}$ is assumed to be small, then: For ${\bf Q}_0=(0,0)$, the above expression can be expanded: $$\begin{aligned} f({\bf k})&=& 4t_{\rm hole}{\bf q}^2\end{aligned}$$ For ${\bf Q}_0=(\pi,\pi)$ we have: $$\begin{aligned} f({\bf k})&=& 2t_{\rm hole}\sum_{{\bf r}_0}\left[1-(-1)^{r_{0x}+r_{0y}}\cos({\bf q}\cdot{\bf r}_0)\right]\end{aligned}$$ $r_{0x}+r_{0y}$ can take only the values $0$ and $2$. Thus, $(-1)^{r_{0x}+r_{0y}}=1$ for all ${\bf r}_{0}$’s. In this way, if we expand around $(\pi,\pi)$, we have again: $$\begin{aligned} f({\bf k})=4t_{\rm hole}{\bf q}^2\end{aligned}$$ It is obvious that for any other ${\bf k}$’s, the oscillator strength $f({\bf k})$ cannot vanish. So, the upper bound to the excitation energy for the holes is: $$\begin{aligned} E_{\bf k} - E_0 \le \frac{f({\bf k})}{s({\bf k})}\end{aligned}$$ ### Dimer density excitations {#sec:dimer-density-excitations} In the same way as before, we may calculate the numerator for the case where we use the dimer density operator ${\sigma^{d}_{\hat\alpha}}$ instead of the hole density. Now, the operator does not commute with both the dimer-flip term and the hole-hopping terms. In the case of the dimer-flip term the commutator will give the following contribution [@rokhsar88; @moessner03c]:(the dimer-density operator is taken to be at $\hat \alpha =\hat x$ direction) $$\begin{aligned} f_{\rm dimer-flip}({\bf k})=8t\sum_{{\bf r}=\pm y}\left[1+\cos({\bf k}\cdot{\bf r}_0)\right]\end{aligned}$$ where $r_0=\pm\hat y$ for the case of a horizontal dimer. This term comes from the original quantum dimer model. As it was pointed out in Ref. , it vanishes quadratically at $Q_{0}=(\pi,\pi)$ and $Q_{1}=(0,\pi)$. In particular, at $Q_0=(\pi,\pi)$, with ${\bf k}={\bf Q}_0 +{\bf q}$, it vanishes as $$\begin{aligned} f_{\rm dimer-flip}({\bf q})=8tq^2 \label{fkh1}\end{aligned}$$ In the case of hole-hopping terms which mix horizontal and vertical dimers ${\bf r}_0'=-{\bf r}_0/2=(\pm\hat x\pm\hat y)/2$ (where ${\bf r}_0$ denotes the displacement vector for the hole in the considered move), we have: $$\begin{aligned} \left[-t_{\rm hole}\mathcal{T}_{\rm hole}^{(1)}({\bf R},{\bf r}_0),{\sigma^{d}_{\hat x}}({\bf k})\right]&=& -t_{\rm hole}\mathcal{T}_{\rm hole}^{(1)}({\bf R},{\bf r}_0)\left(e^{-i{\bf k}\cdot({\bf R}+{\bf r}_0/2)}-e^{-i{\bf k}\cdot{\bf R}}\right)\\ \left[{\sigma^{d}_{\hat x}}(-{\bf k}),\left[-t_{\rm hole}\mathcal{T}_{\rm hole}^{(1)}({\bf R},{\bf r}_0),{\sigma^{d}_{\hat x}}({\bf k})\right]\right] &=& 2t_{\rm hole}\mathcal{T}_{\rm hole}^{(1)}({\bf R},{\bf r}_0)(1-\cos({\bf k}\cdot{\bf r}_0/2))\end{aligned}$$ $$\begin{aligned} f_{\rm hole(1)}({\bf k})= 2t_{\rm hole}\sum_{{\bf r}_0}(1-\cos({\bf k}\cdot{\bf r}_0/2))\end{aligned}$$ The fixed hole fugacity model {#sec:fixed-fugacity} ----------------------------- Let’s follow the same strategy as before (${\bf r}_0=\hat x,\hat y$): $$\begin{aligned} \left[-\tilde t_{\rm hole}\mathcal{T}_{\rm hole}^{(2)}({\bf R},{\bf r}_0),{\sigma^{h}}({\bf k})\right]&=& \tilde t_{\rm hole}\mathcal{T}_{\rm hole}^{(2)}({\bf R},{\bf r}_0)\left(e^{-i{\bf k}\cdot{\bf R}}+e^{-i{\bf k}\cdot({\bf R}+{\bf r}_0)}\right)\end{aligned}$$ where $\mathcal{T}_{\rm hole}^{(2)}$ denotes the resonance part of the Hamiltonian in Eq. . We also have, $$\begin{aligned} \left[{\sigma^{h}}(-{\bf k}),\left[-\tilde t_{\rm hole}\mathcal{T}_{\rm hole}^{(2)}({\bf R},{\bf r}_0),{\sigma^{h}}({\bf k})\right]\right]= 2\tilde t_{\rm hole}\mathcal{T}_{\rm hole}^{(2)}({\bf R},{\bf r}_0)(1+\cos({\bf k}\cdot{\bf r}_0))\end{aligned}$$ Finally, after adding the contribution of the hermitean conjugate part of the Hamiltonian, we have: $$\begin{aligned} f_{\rm hole(2)}({\bf k})=4t_{\rm pairing}(2+\cos({\bf k}_x)+\cos({\bf k}_y))\end{aligned}$$ [86]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ****, (). , , , ****, (). , ** (, , ), . , ****, (). , ****, (). , ****, (). , , , ****, (). , , , , , ****, (). , , , ****, (). , , , ****, (). , ****, (). , , , , , ****, (). , , , , , , ****, (). , ****, (). , , , , , ****, (). , , , , ****, (). , , , , , , ****, (). , ****, (). , in **, edited by (, ). , ****, (). , , , ****, (). , ****, (). , ****, (). , , , ****, (). , , , (), . , ****, (). , ****, (). , , , ****, (). , , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , in **, edited by (, , ), . , , , ** (, , ). , ****, (). , , , ****, (). , in **, edited by , , (, , ), p. , . , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ** (, , ). , ** (, , ). , ** (), , , , ****, (). , , , ****, (). , ****, (). , in **, edited by , , , , (, , ), . , , , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, ().

--- abstract: 'Probability models are proposed for passage time data collected in experiments with a device designed to measure particle flow during aerial application of fertilizer. Maximum likelihood estimation of flow intensity is reviewed for the simple linear Boolean model, which arises with the assumption that each particle requires the same known passage time. M-estimation is developed for a generalization of the model in which passage times behave as a random sample from a distribution with a known mean. The generalized model improves fit in these experiments. An estimator of total particle flow is constructed by conditioning on lengths of multi-particle clumps.' address: - 'Department of Statistics, North Carolina State University, Raleigh, NC, USA' - 'Department of Agricultural and Biological Engineering, University of Illinois, Urbana, IL, USA' author: - 'Jason A. Osborne' - 'Tony E. Grift' bibliography: - 'refs-short.bib' title: 'M-estimation of Boolean models for particle flow experiments' --- \#1\#2 ( [c]{} \#1\ \#2 ) 3[M\_[3+]{}]{} ł\#1[\_[\#1]{}]{} \#1[\_[\#1]{}(t)]{} \#1[\_[\#1]{}(t;)]{} \#1[\_[\#1]{}]{} \#1[\_[ \#1 ]{}]{} Introduction ============ Measuring the outflow of granular particles from an airborne spreader during the aerial application of fertilizer or pesticide presents agricultural engineers with a difficult problem. The goal of uniform distribution over a targeted field requires knowledge about flow rate of the material as it is dropped from the aircraft. Windspeed, air speed, granule properties, humidity, and temperature have been identified ([@casady]) as factors which can lead to variability in these outflow rates and hence amounts of material that reach the target. Typically, applicators are calibrated annually so that they achieve an average target flow rate. In practice, pilots use a simple lever-operated gate to change the flow rate in order to account for extreme values of these factors. This adjustment is based on intuition, without any feedback from measurement of particle flow. One approach to providing the pilot with more information uses an optical sensor device ([@grift4]) which measures the velocity (in meters per second) and size of clumps of particles as they flow through the spreader duct. This device has two photo-sensitive arrays of optical sensors that receive a signal from a light source. As a particle passes an active area, it blocks this light thereby interrupting the signal received by the sensors. As long as all of the sensors in the array are receiving a high signal, the channel is classified as unoccupied and this is taken as an indication that there are no particles flowing through at that instant. If the signal to any one of sensors is interrupted, this is interpreted as the presence of at least one particle, constituting a clump, in flow. The two sensor arrays are $0.00078$ meters apart and it is possible to measure the time in seconds that it takes a clump to move from one array to the other, $\Delta t_f$. The total time that either array is blocked, $\Delta t_b$, is also measured, facilitating calculation of velocity in meters per second, $v=0.00078/\Delta t_f$ and clump length in meters, $ CL = v \Delta t_b.$ These observable clump lengths, either in terms of physical length in meters or time in seconds, are the basis for inference about particle flow in the system. Such a measurement device is called a type II counter ([@pyke1]). @grift2 and @grift3 carried out bench-scale experiments to evaluate the optical sensor device in situations designed to simulate the flow of fertilizer particles through an airborne spreader duct. In these experiments, a known number of spherical particles with a known mean diameter of $4.45mm$ was dropped from predetermined heights through a duct on which the sensor device was installed. The heights from which the particles were dropped was controlled at several values to simulate a range of particle velocities and flow rates. A histogram representing the distribution of particle clump lengths in units of time, obtained from one run of these experiments, is shown in Figure 1. The relative frequencies for clump lengths (in $msec$) are based on dropping 4000 spherical steel particles (actually BBs) from a fixed height. In this paper, simple linear Boolean models are used to describe the clump length data generated by the Grift experiments, thus providing a basis for inference about flow rates and total particle flow during a dispersal period. In particular, flow intensity is quantified by a single rate parameter in a simple Boolean model. Maximum-likelihood is reviewed in cases where particles require a fixed time for passage and an $M$-estimator is obtained in more general cases. Assessment of total particle flow utilizing this estimator is also developed. Section 2 introduces the Boolean models and establishes notation and terminology. Results for the clump length distribution derived in @hall1 are used to develop $M$-estimation of flow intensity and the $M$-estimator is compared with maximum likelihood and other moment estimators by simulation. In section 3, two estimators of total particle flow are proposed, including one obtained by derivation of the conditional expectation of the number of particles in a clump given clump length under the equal diameters model. Simulations are carried out to give an assessment of the performance of this estimator in the random passage times model. The methods are evaluated based on their performance with the experimental data in Section 4. Section 5 concludes. Estimation ========== To obtain a probability model for the clump-length data, particles are assumed to be identically spherical with a known diameter, $d_0$, and to arrive at the sensor according to a homogeneous Poisson process with unknown intensity $\lambda.$ Passage of particles is assumed to continue unabated upon arrival at the sensor. In one version of the model, the particles are travelling at a constant velocity, say $v_0$, and the [*segment length*]{} ([-@hall1]), or time required for any single particle to pass the sensor, is constant at $t_0=d_0/v_0$. In a second version, velocities or diameters are assumed to vary in such a way that segment lengths behave as a random sample from a population with a known mean $\mu$ and an unknown variance $\sigma^2$. The two models will be referred to as deterministic segment length (DSL) or random segment length (RSL) models, respectively. Suppose that particle flow is observed for $t$ time units. Let the number of particles arriving at the sensor in this time period be denoted by $A(t)$. Let $N(t)$ denote the number of complete particle clumps observed by time $t$. Let $Y_1,Y_2,\ldots,Y_{N(t)}$ denote the lengths of these clumps and $Z_1,Z_2,\ldots$ the spacings between them. Let the unobservable number of particles comprising clump $i$ be called the clump [*order*]{} and be denoted by $K_i$. Figure 2 illustrates the clumping process using an example with $A(t)=7$ particles arriving at a sensor at times $1.9, 5.9,6.8,7.5,11.6,12.8$ and $17.1 \ msec$ during an observation period of $t=20 \ msec$. If particles are assumed to have diameter $4.45\ mm$ and to be travelling at a constant velocity of $2.225 \ mm/msec$, the passage time required for each, or deterministic segment length, is $d_0=2 \ msec$, leading to four clumps of lengths $y_1=2,y_2=3.6,y_3=3.2,y_4=2 \ msec $ that exit the sensor at times $3.9,9.5,14.8$ and $19.1 \ msec$, respectively. Spacings between clumps would be of length $z_1=1.9,z_2=2.0, z_3=2.1$ and $z_4=2.3\ msec$ and the four clump orders would be $k_1=1,k_2=3,k_3=2,k_4=1$. The particle clumps constitute a coverage process on one dimension. [@hall1] describes the process as a simple linear Boolean model; simple because the clump-lengths are line segments and linear because the events occur in one dimension, the time line. Linear Boolean models also arise as linear transects from higher dimensional convex-grain Boolean models. In the language of queueing theory, the number of particles in a clump at the sensor at a given time forms an $M/D/\infty$ queue in the DSL model and an $M/G/\infty$ queue in the RSL model and clump-lengths are called [*busy periods*]{}. There is much literature on these models from queueing theory ([-@daley]). For statistical inference for the distribution of diameters or more complex quantities describing the grain process, or for Boolean models in higher dimensions, see [@molchanov]. [@handley2] derived a discrete approximation to the distribution of clump-length in the linear Boolean model and used it for likelihood inference. [@crespi] have employed the linear Boolean model for monitoring events of viral activity in human subjects. Likelihood ---------- Specification of the clump-length density, $f(y;\lambda)$ is difficult outside of the case where particle diameters are degenerate. In the DSL model, [@hall1] has shown that the density has point mass $\emla$ at $y=t_0$, and is otherwise given by $$\begin{aligned} %\lefteqn{f(y;\lambda,t_0)=\lambda e^{-\lambda t_0}}\\ \lefteqn{f(y;\lambda,t_0)=\lambda \frac{\emla}{1-\emla}}\\ & &\left[1+\sum_{j=1}^{s-1} \frac{(-1)^j}{j!}\{\lambda(y-(j+1)t_0)\}^{j-1} e^{-j\lambda t_0}\{\lambda(y-(j+1)t_0)+j\}\right]\end{aligned}$$ where $y>t_0$ and $s$ is the largest integer such that $t_0<y/s$. The continuous part of the density is uniform over $(t_0, 2t_0)$, and decreasing for $y>2t_0$. For small $\lambda$, $f(y;\lambda,t_0)$ can be approximated by the uniform distribution on $(t_0,2t_0)$, for large $\lambda$ it can be approximated by an exponential distribution. The fitted density $f(y;\lambda=0.40,t_0=2.00 msec)$ overlays the probability histogram of experimental clump-lengths in Figure 1. For RSL models, likelihood inference is difficult because of the complexity of the clump-length distributions ([-@handley1]). In the DSL model, an approximate likelihood function can be specified by ignoring the residual lifetime of the process. The residual lifetime is the duration of the last incomplete clump or spacing. A clump is a singleton if there are no arrivals within $t_0$ time units of the start of the clump, an event which occurs with probability $\emla$. Let $M_1=\#\{y_i:y_i=t_0\}$ denote the number of singleton clumps. By independence of clump-lengths, the approximate partial Boolean likelihood can be factored into components for singleton point masses and multi-particle clump-length densities: $$\begin{aligned} \tilde{\cal{L}}(\lambda;y_1,\ldots,y_{N(t)}) &=& \underbrace{e^{-m_1\lambda t_0}}_{\mbox{singletons}} %\underbrace{(1-e^{-a\lambda})^{N-M_1} \prod_{i:y_i>a} f(y_i;\lambda)}_{\mbox{complete multi-particle lengths.}} \underbrace{\prod_{i:y_i>t_0} f(y_i;\lambda,t_0)}_{\mbox{multi-particle lengths.}}\end{aligned}$$ Spacings $z_1,z_2,\ldots$ are not available for the experiments analyzed in section 4. For cases where the $z_i$ are available, an approximate complete Boolean likelihood may be obtained by multiplying the partial likelihood by the likelihood from an exponential random sample, $\lambda^N e^{-\lambda \sum z_i}$. For large $t$, the maximum likelihood estimator of $\lambda$ based on $\tilde{\cal{L}}$ is approximately normally distributed. However, the analytic expression for the Fisher information is unwieldy, particularly for large clump-lengths, where the degree of the polynomial components of the clump-length density is high. Alternatively, approximate confidence regions can be constructed from the likelihood ratio test statistic, which has an approximate $\chi^2$ distribution on 1 degree of freedom. In the RSL model, where segment lengths are distributed as a random sample from a known distribution with distribution function $G(x;\theta)$, the clump-length density and resulting likelihood are considerably more complex. Let $f_{RSL}(y;\lambda,\theta)$ denote the clump-length density, which depends on the unknown parameters, $\lambda$ and $\theta$. Ignoring the residual lifetime, the partial likelihood of the complete clumps is then $$\begin{aligned} %\tilde{\cal{L}}_{GLBM}(\lambda;y_1,\ldots,y_{N(t)},z_1,z_2,\ldots) &=& %\tilde{\cal{L}}_{G}(\lambda;y_1,\ldots,y_{N(t)},z_1,z_2,\ldots) &=& %\tilde{\cal{L}}_{G}(\lambda;y_1,\ldots,y_{N(t)}) &=& \tilde{\cal{L}}_{RSL}(\lambda,\theta;y_1,\ldots,y_{N(t)}) &=& \lambda^{N} e^{-\lambda \sum z_i} %\prod_{i=1}^{N(t)} g(y_i;\lambda). %\prod_{i=1}^{N(t)} f(y_i;\lambda). %\prod_{i=1}^{N(t)} f_G(y_i;\lambda). %\prod_{i=1}^{N(t)} f_{RSL}(y_i;\lambda). \prod_{i=1}^{N(t)} f_{RSL}(y_i;\lambda,\theta).\end{aligned}$$ [@hall1] shows that the Laplace transform $\gamma$ of $Y$ is $$\gamma(s) = 1+ \frac{s}{\lambda}- %\left(\lambda\int_0^\infty \exp\{-st - \lambda \int_0^t\{1-G(x)\}dx\}dt\right)^{-1}.$$ (\_0\^{-st - \_0\^t{1-G(x;)}dx}dt)\^[-1]{}.$$ [@stadje] obtains the clump-length distribution function $F_{RSL}(y)$ by inversion of $\gamma$, but it is an infinite sum of self-convolutions of a function that may involve an integral with no analytic solution, making inference based on $\tilde{\cal{L}}_{RSL}$ difficult. M-estimation ------------ An important issue in estimation of $\lambda$ is robustness under model misspecification. Inspection of the clump-lengths from the experimental data, such as the run depicted in Figure 1, reveals that the number of clumps with lengths slightly in excess of $t_0$ is greater than expected, so that the distribution between $t_0$ and $2t_0$ is not uniform. This can be caused by variability in diameter or velocity or by errors of measurement. A desirable property for any estimator is robustness to this departure from model assumptions. For mean segment length $\mu$, the mean clump-length is given by $$E(Y;\lambda) = \frac{e^{\lambda \mu}-1}{\lambda}$$ in either the DSL or RSL model, regardless of the distribution of segment lengths ([-@hall1]). For known $\mu$, consider the $M-$estimator $\tilde{\lambda}$ which satisfies $$\bar{y} = \frac{e^{\tilde\lambda \mu}-1}{\tilde\lambda}.$$ A solution exists by the mean value theorem with $E(Y;\lambda)$ increasing in $\lambda$. Though there is no analytic solution, the equation can be solved rapidly using any root-finding procedure, such as the [uniroot]{} function in the $R$ statistical software package ([@rdoc]). A starting point that works in simulations is given by $\tilde\lambda=(\bar{y}-\mu)/(2\mu^2)$, which is the solution obtained using a second order expansion of $e^{\tilde\lambda \mu}$ about 0. An interesting aspect of the sampling distribution of $\tilde\lambda$ is that it is negative whenever $\bar{y} < \mu$, an event whose probability is small as long as $\lambda \mu$ is not too small. This estimating equation for $\lambda$ can be written $$\sum_i \psi(y_i,\lambda) =0$$ where $\psi(y,\lambda)=y-\lambda^{-1}(e^{\lambda \delta}-1)$. Large-sample theory for $M$-estimators, (see, e.g. [-@boos]) can be used for inference about $\lambda$. For a random sample of $n$ clump-lengths $y_1,\ldots,y_n$, the asymptotic distribution of $\tilde\lambda$ is given by $$\sqrt{n}(\tilde\lambda-\lambda) \stackrel{\cal{L}}{\longrightarrow} N(0,C/B^2)$$ where $B$ and $C$ are functions of $\lambda$ defined by $$\begin{aligned} B(\lambda) & = & E(-\frac{\partial}{\partial \lambda}\psi(Y_1,\lambda))\\ C(\lambda) & = & E( \psi^2(Y_1,\lambda)).\end{aligned}$$ Since $\psi$ is linear in $Y$, the expectation operations are straightforward: $$\begin{aligned} %B(\lambda) & = & \frac{\ela(\lambda \mu_t - 1) + 1}{\lambda^2} \\ B(\lambda) & = & \frac{e^{\lambda\mu}(\lambda \mu - 1) + 1}{\lambda^2} \\ C(\lambda) & = & \Var(Y;\lambda).\end{aligned}$$ The variance of $Y$ depends on the distribution of segment lengths. In the DSL model with $t_0=\mu$, $$\Var(Y) = \lambda^{-2} (e^{2 \lambda \mu}-2\lambda \mu e^{\lambda \mu} - 1).$$ In the RSL model with segment lengths distributed according to the general distribution function $G(x)$, clump-lengths have variance $$\Var(Y) = 2 \lambda^{-1} e^{\lambda \mu}\int_0^\infty \left(\mbox{exp}\left[\lambda \int_t^\infty (1-G(x)) dx\right]-1\right)dt-\lambda^{-2}(e^{\lambda \mu}-1)^2$$ which can be estimated using the sample variance of clump-lengths, $s_y^2.$ Estimators for the variance of $\tilde\lambda$ are then given by $$\widehat{\Var}(\tilde\lambda) = n^{-1}\frac{\tilde\lambda^2(e^{2\tilde\lambda \mu}-2 \tilde\lambda \mu e^{\tilde\lambda \mu} -1)}{(e^{\tilde\lambda\mu}(\tilde\lambda \mu -1)+1)^2}$$ in the DSL model and $$\widehat{\Var}_{\mbox{G}}(\tilde\lambda) = n^{-1}\frac{\tilde\lambda^4 s_y^2}{(e^{\tilde\lambda \mu}(\tilde\lambda \mu -1)+1)^2}$$ in either the DSL or RSL model. In large samples, approximate confidence intervals for $\lambda$ can be constructed from these estimates along with the normal approximation for $\tilde\lambda$. Other estimators ---------------- For the DSL model with common deterministic passage time $t_0$, other method-of-moments (MOM) estimators can be constructed using only the clumpcount ($N(t)$) and singleton count ($M_1$) statistics. The sequence of i.i.d. sums $\{Z_i+Y_i\}$ is a renewal process. Elementary renewal theory ([-@cox]) yields that as $t \rightarrow \infty$, $$\frac{N(t)-t/\mu_R}{\sigma_R\sqrt{t/\mu_R^3}} \stackrel{\cal{L}}{\longrightarrow} N(0,1)$$ where $\mu_R$ and $\sigma_R^2$ denote the mean and variance of a randomly sampled renewal period. In DSL model with deterministic common passage time $t_0$, $$\begin{array}{ccccc} %\mu_R &=& E(Z+Y) &=&\lambda^{-1}\ela \\ %\mu_R &=& E(Z+Y) &=&\lambda^{-1}e^{\lambda \mu} \\ \mu_R &=& E(Z+Y) &=&\lambda^{-1}e^{\lambda t_0} \\ %\sigma^2 &=& \Var(Z+Y) &=&\lambda^{-2}(e^{2\lambda d}-2\lambda d \ela). %\sigma^2 &=& \Var(Z+Y) &=&\lambda^{-2}(e^{2\lambda \mu_t}-2\lambda \mu_t \ela). %\sigma^2 &=& \Var(Z+Y) &=&\lambda^{-2}(e^{2\lambda \mu}-2\lambda \mu e^{\lambda \mu}). \sigma_R^2 &=& \Var(Z+Y) &=&\lambda^{-2}(e^{2\lambda t_0}-2\lambda t_0 e^{\lambda t_0}). \end{array}$$ Moments for $N(t)$ are then $$\begin{aligned} %E[N(t)] & \approx & \lambda t e^{-\lambda d} \\ %E[N(t)] & \approx & \lambda t e^{-\lambda \mu_t} \\ %E[N(t)] & \approx & \lambda t e^{-\lambda \mu} \\ E[N(t)] & \approx & \lambda t e^{-\lambda t_0} \\ %\Var[N(t)] & \approx & \lambda t \left(e^{-\lambda d} - 2d\lambda e^{-2\lambda d}\right). %\Var[N(t)] & \approx & \lambda t \left(e^{-\lambda \mu_t} - 2\mu_t\lambda e^{-2\lambda \mu_t}\right). %\Var[N(t)] & \approx & \lambda t \left(e^{-\lambda \mu} - 2\mu\lambda e^{-2\lambda \mu}\right). \Var[N(t)] & \approx & \lambda t \left(e^{-\lambda t_0} - 2\lambda t_0 e^{-2\lambda t_0}\right).\end{aligned}$$ The probability that a randomly selected clump is a singleton is $e^{-\lambda t_0}$ so that $E(M_1)=\lambda t e^{-2\lambda t_0}$. A MOM estimator based on the observed number of singletons is then $$\tilde\lambda_{S}=-\frac{1}{t_0}\log\left(\frac{M_1}{N(t)}\right).$$ [@grift2] and [@grift3] base estimation of total mass flow on this estimator. Other estimators of $\lambda$ can be constructed by consideration of [*vacancy*]{}, $V\approx \sum Z_i$, or total time that that the sensor is unoccupied. [@hall1] develops asymptotic theory for a number of vacancy-based estimators. Measurements of $V$ were not available from the experiments discussed in section 4, and vacancy-based estimators are not considered further. Simulation ---------- Simulations were undertaken to provide some information about the performance of these estimators, with three goals in particular: a comparison of the efficiency of the moment estimator $\tilde\lambda$ relative to the MLE under the DSL model, an investigation of the robustness of the MLE under the RSL model and a comparison of coverage probabilities of confidence intervals resulting from the two variance estimates of the asymptotically normal $M$-estimator, $\tilde\lambda$. Particle arrivals were generated according to a Poisson process. Three cases with an increasing degree of clumping were simulated using flow intensities of $\lambda=0.1,0.2$ and $0.3$. Two times were considered for the length of the total observation period, $t=1000$ and $t=10000$. Preliminary experiments with particles far enough apart so that there was no clumping indicated that measured passage times were normally distributed. So, passage times for individual particles were generated from a normal distribution with a mean of $\mu=5$ with three different standard deviations, $\sigma=0,0.5,1$. The first of these standard deviations leads to the DSL model, the others to RSL models. The approximate mean clump counts for the DSL model were $E[K] \approx 1.6,2.7,4.5$ for the three flow rates, $\lambda=0.1,0.2,0.3$, respectively. The simulation experiment then had a crossed $3 \times 2 \times 3$ design, with $n=500$ independent datasets generated per combination of $\lambda,t$ and $\sigma$. Normal plots and Kolmogorov-Smirnov statistics did not indicate any obvious non-normality for either the MLE or $\tilde\lambda$. Table 1 summarizes the results of the simulation. The bias of the $M$-estimate relative to $\lambda$ and the efficiency relative to the MLE are given in the middle section. Though the bias of the MLE formulated under the DSL model dissipates with increasing $\lambda$ or $t$, it does not exhibit robustness to heterogenous segment lengths, in the sense that it has larger variance than the $M$-estimate. Empirical coverage probabilities for $95\%$ confidence intervals based on the LRT and those of the form $\tilde\lambda \pm 1.96 SE$ where $SE$ denotes the appropriate estimated asymptotic standard error from Section 2 are given in the right section of Table 1. For the shorter simulations $(t=1000)$, there is a tendency for coverage probabilities based on $\tilde\lambda$ to be low. For datasets with a larger number of clumps ($t=10000$), the nominal coverages for intervals based on $\tilde\lambda$ are reached. With $n=500$ simulations, the Monte Carlo standard error is such that any sample proportion less than 0.934 is significantly less than the nominal 0.95 with comparisonwise error rate 0.05. Additionally, the intervals around the $M$-estimate that use the standard error, $SE_G$, which is a function of the sample variance of the clump-lengths, appear to do better for the RSL models with large $N(t)$, particularly for the noisy segment length $\sigma=1$ case. The likelihood ratio interval gives coverages consistent with nominal levels in simulations with the DSL model, but breaks down under the RSL model where the likelihood is misspecified. In summary, the recommendation based on these simulations is that the $M$-estimator is reasonably efficient under the DSL model and robust to the conditions of the RSL model. Confidence intervals based on the standard error $SE_G$ meet nominal coverage probabilities in large samples under either model. ---------- ------- ----------- ------------------- ------- ------ ------- --------------------- ----------------------- Rel. Rel. $\sigma$ $t$ $\lambda$ $\overline{N(t)}$ Bias Eff. LRT $SE(\tilde\lambda)$ $SE_G(\tilde\lambda)$ 0 1000 0.1 60.1 0.01 0.89 0.966 0.950 0.944 0 1000 0.2 73.2 -0.01 0.95 0.956 0.956 0.934 0 1000 0.3 66.7 -0.01 0.98 0.948 0.946 0.944 0 10000 0.1 605.8 0.00 0.83 0.940 0.942 0.942 0 10000 0.2 734.5 0.00 0.88 0.960 0.956 0.950 0 10000 0.3 669.6 0.00 0.97 0.942 0.938 0.942 0.5 1000 0.1 60.2 0.56 8.3 0.128 0.922 0.916 0.5 1000 0.2 73.0 0.15 2.2 0.768 0.910 0.902 0.5 1000 0.3 66.3 0.05 1.1 0.940 0.952 0.950 0.5 10000 0.1 606.1 0.57 82.4 0.000 0.942 0.946 0.5 10000 0.2 736.1 0.15 17.2 0.002 0.946 0.954 0.5 10000 0.3 670.1 0.05 3.3 0.646 0.938 0.940 1 1000 0.1 60.5 0.56 7.3 0.136 0.914 0.940 1 1000 0.2 73.5 0.13 1.9 0.798 0.922 0.920 1 1000 0.3 66.9 0.03 0.98 0.952 0.944 0.942 1 10000 0.1 605.3 0.56 71.9 0.000 0.922 0.952 1 10000 0.2 735.7 0.14 13.6 0.016 0.924 0.946 1 10000 0.3 668.6 0.04 3.0 0.694 0.944 0.954 ---------- ------- ----------- ------------------- ------- ------ ------- --------------------- ----------------------- : Simulation: relative efficiency and coverage probability of $\lambda$ estimators Estimation of total particle flow ================================= In the case where either $\lambda$ is known or variance in its estimation is negligible, total particle flow may be estimated by $E[A(t)]=\lambda t$. When $t$ is not available, another estimator can be formed by substitution of $t \approx \sum Y_i + \sum E(Z_i)$ into the expression giving $\widehat{E[A(t)]} = \lambda\sum Y_i + N(t).$ In the DSL model, clump orders ($K_1,K_2,\ldots$) may be shown ([@Pippenger]) to be geometrically distributed. A clump is of order one ($K_i=1$) if there are no arrivals within $t_0$ time units of the start of the clump, which occurs with probability $\emla$. A clump is of order two if there is exactly 1 arrival within $t_0$ units and none in the next $t_0$ time units, an event which occurs with probability $(1-\emla)\emla$ and so on. $K_1,K_2,\ldots$ are then independent geometric random variables with support on positive integers: $$\Pr(K_i=k) = (1-\emla)^{k-1}\emla \ \ \ \mbox{ for } k=1,2,\ldots$$ with $E(K_i)=e^{\lambda t_0}$ and $\Var(K_i)=e^{2\lambda t_0}-e^{\lambda t_0}$. If the system is vacant when observation ends at time $t$, then total particle flow may be expressed as the sum of these clump orders: $ A(t)=K_1 + \cdots + K_{N(t)}$. If the system is occupied at time $t$, there is a partial clump that contributes a relatively small amount of particle flow for large $t$. Expressing total particle flow $A(t)$ as the sum of clump orders each with mean $e^{\lambda t_0}$ suggests the estimator $\hat{A}_1(t;\lambda)=N(t)e^{\lambda t_0}$. When evaluated at the $M$-estimator $\tilde\lambda$, with mean passage time $\mu=t_0$, the two estimators of total particle flow become equivalent: $\tilde{A}_0(\tilde\lambda) = N(t)e^{\tilde\lambda \mu} = \hat{A}_1(t;\tilde\lambda)$. More efficiency might be gained by conditioning on the clump lengths. The estimator $p(y)$ of an individual clump order which is a function of the clump length $y$ and minimizes the mean squared error $E[(K-p(y))^2]$, is the [*Bayes*]{} estimate, or $p(y)=E(K|Y=y)$. An estimate of mean total particle flow $E[A(t)]=E[\sum K_i]$ is then given by summing over clumps: $$\hat{A}_{B}(t;\lambda) = \sum_{i=1}^{N(t)} E(K_i|Y_i;\lambda).$$ Of course $E[K_i|Y_i=t_0;\lambda]=1.$ The approach used by @hall1 to derive the clump length density $f(y)$ in the DSL model may be extended to obtain the conditional mean of clump orders, $E(K|Y)$. Let the beginning of a clump be the origin and let $k$ denote an integer greater than unity. The joint event $K=k$ and $Y\in (y,y+dy)$ occurs if and only if there is a particle arrival at $(y-t_0,y-t_0+\Delta y)$, no arrival in $(y-t_0+\Delta y,y)$, exactly $k-2$ arrivals in $(0,y-t_0)$, and the nearest neighbor of each of these $k-2$ arrival times is not further than $t_0$ time units away. Since the first three of these conditions are independent and the fourth is conditionally independent of the first two given the third, the joint probability of these four events is the product $$\lambda \Delta y \emla \frac{(\lambda(y-t_0))^{k-2}}{(k-2)!}e^{-\lambda(y-t_0)} p_{k-2}\left(\frac{t_0}{y-t_0}\right)$$ where $p_n(u)$ denotes the chance that the largest division formed by a random sample of $n$ points taken from the unit interval does not exceed $u$. This probability is given by $$\begin{aligned} p_n(u) %& = & \sum_{j=0}^{n+1} (-1)^j \choose{n+1}{j} (1-ju)_+^n \\ & = & \sum_{j=0}^{[u^{-1}]} (-1)^j \choose{n+1}{j} (1-ju)^n \\ & = & 1 - (n+1)(1-u)^n + \choose{n+1}{2}(1-2u)^n - \ldots\end{aligned}$$ where $[\cdot]$ denotes the largest integer not exceeding the argument. Division by $f(y)$ and differentiation with respect to $y$ yields the conditional density $$\Pr(K=k|Y=y) = \frac{\lambda e^{-\lambda y}}{f(y)}\frac{(\lambda(y-t_0))^{k-2}}{(k-2)!} p_{k-2}\left(\frac{t_0}{y-t_0}\right).$$ If $s=[y/t_0]$, then summation over positive integers yields an exact expression for the conditional mean: $$\begin{aligned} E(K|Y=y) & = & \sum_{k=s+1}^\infty k \Pr(K=k|Y=y) \\ %& = & \frac{\lambda e^{-\lambda a}}{f(y)}\sum_{k=s+1}^\infty k \frac{\left(\lambda(y-d)\right)^{k-2}}{(k-2)!}p_{k-2}(\frac{d}{y-d}) \\ & = & \frac{\lambda e^{-\lambda t_0}}{f(y)}\sum_{k=s+1}^\infty k \frac{\left(\lambda(y-t_0)\right)^{k-2}}{(k-2)!}p_{k-2}(\frac{t_0}{y-t_0}) \\ %& = & \frac{\lambda e^{-\lambda a}}{f(y)}\sum_{k=s+1}^\infty k \frac{\left(\lambda(y-d)\right)^{k-2}}{(k-2)!} \sum_{j=0}^{s-1} (-1)^j \choose{k-1}{j}\left(1-\frac{jd}{y-d}\right)^{k-2}. & = & \frac{\lambda e^{-\lambda t_0}}{f(y)}\sum_{k=s+1}^\infty k \frac{\left(\lambda(y-t_0)\right)^{k-2}}{(k-2)!} \sum_{j=0}^{s-1} (-1)^j \choose{k-1}{j}\left(1-\frac{jt_0}{y-t_0}\right)^{k-2}.\end{aligned}$$ Inspection of $\Pr(K=k|Y=y;\lambda)$ reveals that for $t_0<y<2t_0$, $K$ has the translated Poisson distribution with mean and variance that are linear in $y$. For larger $y$, numerical evaluation of $E(K|Y=y)$ be difficult. Inspection of plots for larger $y$ and various values of $\lambda$ indicates that after a jump discontinuity of $\lambda t_0 \emla (1-\emla)^{-1}$ at $y=2t_0$, approximate linearity extends to $y>2t_0$. For cases where $N(t)$ is large and there is heavy clumping, $E(K|Y=y)$ can be approximated by linear interpolation to save computational effort. Simulation ---------- The performances of these estimators of mean total particle flow are compared using the simulated data from section 2. Error for either $\hat{A}_1$ or $\hat{A}_B$, as a percentage of the mean particle flow is assessed using the relative root mean squared error, RRMSE: $$RRMSE(\hat{A}(t)) = \frac{1}{\overline{A(t)}} \sqrt{500^{-1}\sum_i (\hat{A}_i(t)-A_i(t))^2}$$ where $i$ indexes the 500 simulated datasets. Table 2 summarizes relative bias and RRMSE of estimates obtained by substitution of the $M-$estimates $\tilde\lambda$ into the expressions $\hat{A}_1(t)=N(t)e^{\lambda \mu_t}$ and $\hat{A}_B(t;\lambda)$ for each simulated experimental condition. The estimation based on clumpwise estimated clump orders $\hat{A}_B$, is competitive under the DSL model ($\sigma_t=0$) for smaller sample sizes, ($t=1000$). It suffers from some positive bias in RSL models that appears to decrease as flow rate $\lambda$ increases, though it remains inferior to $\hat{A}_1$ despite smaller variance and higher correlation with $A(t)$. In the RSL model, many singleton clumps have clump lengths slightly in excess of the mean singleton passage time $\mu_t$ and so have estimated orders in excess of 1. This may lead to a positive bias for the clumpwise estimators which is particularly acute when support is high near $Y=\mu_t$. This theory is supported by the poor performance under light clumping, when $\lambda=0.1$ and density near $Y=\mu_t$ is highest among values of $\lambda$ considered in the simulation. ------------ ------- ----------- ------------- ------------- ------------- ------------- $\sigma_t$ $t$ $\lambda$ $\hat{A}_1$ $\hat{A}_B$ $\hat{A}_1$ $\hat{A}_B$ 0 1000 0.1 -0.008 -0.007 0.042 0.033 0 1000 0.2 -0.012 -0.011 0.048 0.045 0 1000 0.3 -0.014 -0.013 0.049 0.048 0 10000 0.1 -0.001 -0.001 0.014 0.01 0 10000 0.2 0.000 0.000 0.015 0.014 0 10000 0.3 -0.002 -0.002 0.016 0.015 0.5 1000 0.1 -0.003 0.178 0.048 0.185 0.5 1000 0.2 -0.008 0.061 0.051 0.076 0.5 1000 0.3 -0.011 0.013 0.054 0.052 0.5 10000 0.1 0.000 0.184 0.014 0.184 0.5 10000 0.2 -0.001 0.067 0.015 0.068 0.5 10000 0.3 -0.003 0.022 0.015 0.026 1 1000 0.1 -0.002 0.189 0.058 0.198 1 1000 0.2 -0.010 0.063 0.053 0.079 1 1000 0.3 -0.011 0.018 0.052 0.052 1 10000 0.1 0.001 0.192 0.018 0.193 1 10000 0.2 0.000 0.072 0.017 0.074 1 10000 0.3 -0.001 0.026 0.016 0.031 ------------ ------- ----------- ------------- ------------- ------------- ------------- : Error in estimation of total particle flow from simulations. In summary, for minimal relative error, these simulations suggest the use of the simple $\hat{A}_1(t)$ estimator, which is unbiased and involves less computation than the clumpwise estimator $\hat{A}_B(t)$. A slight loss of efficiency under the DSL model may be offset by the superior performance in the RSL model. Expressed relative to total particle flow, the root MSE was not larger than $5.8\%$ in any of the conditions simulated here. Experimental data ================= An optical sensor was used to measure clump lengths and clump velocities in experiments ([@grift2; @grift3]) in which a known number of spherical particles was dropped through a device simulating an aerial spreader duct. Various quantities of several kinds of particles (BBs, urea fertilizer) were dropped at several velocities. The data considered here include 10 runs with 4000 identical steel particles (BBs) dropped from each of two heights and 5 runs with 2000 BBs dropped from a fixed height. Mean ($\bar{y}$) and variance ($s_y^2$) of physical lengths (in $mm$) appear in Table 3 along with other statistics from the experiments. Division by mean velocity ($\bar{v}=2.23 mm/msec$) was used to transform the measurements to the time line (in $msec$) to obtain Figure 1. In general, velocity was reasonably constant within a run of the experiment. The data were imperfect and some outlier removal was undertaken. For example, the counter returned several clumps with negative velocities or negative physical lengths, or sometimes both. Additionally, each run contained a very small number of extremely short clumps, much less than the particle diameter, possibly due to matter other than the particles of interest blocking the sensor. The number of questionable clump measurements that were removed did not exceed $1\%$ for any of the 25 runs. Run $N$ $\bar{y}$ $s_y^2$ $\tilde\lambda(SE)$ $\hat{A}_1$ $\hat{A}_B$ ----- ------ ----------- --------- --------------------- ------------- ------------- -- 1 2958 5.22 3.07 0.070 (0.003) 4041 4921 2 2930 5.22 2.91 0.070 (0.003) 3997 4946 3 2891 5.26 3.39 0.073 (0.003) 4008 4874 4 2935 5.22 3.00 0.070 (0.003) 4000 4944 5 2990 5.16 2.88 0.065 (0.003) 3986 4941 6 2941 5.20 3.00 0.068 (0.003) 3984 4883 7 2983 5.15 2.84 0.064 (0.003) 3969 4900 8 2956 5.16 2.90 0.065 (0.003) 3952 4846 9 2894 5.24 3.12 0.071 (0.003) 3976 4831 10 2914 5.25 3.11 0.073 (0.003) 4025 4931 11 1821 6.76 11.77 0.176 (0.005) 3988 4299 12 1770 6.85 11.56 0.182 (0.005) 3976 4303 13 1805 6.80 12.30 0.179 (0.005) 4000 4321 14 1748 6.96 12.57 0.188 (0.005) 4038 4333 15 1800 6.85 12.13 0.182 (0.005) 4040 4340 16 1784 6.93 14.82 0.186 (0.005) 4089 4403 17 1772 6.93 12.56 0.187 (0.005) 4064 4341 18 1788 6.89 13.28 0.184 (0.005) 4052 4346 19 1812 6.78 12.01 0.178 (0.005) 3995 4317 20 1790 6.84 11.98 0.181 (0.005) 4005 4330 21 746 7.54 17.10 0.219 (0.008) 1981 2143 22 791 7.24 13.20 0.204 (0.007) 1959 2141 23 777 7.46 13.15 0.215 (0.007) 2024 2184 24 774 7.30 13.23 0.207 (0.007) 1941 2102 25 745 7.57 13.39 0.221 (0.007) 1989 2133 : Estimation from experiments with BBs In these experiments, total particle flow is fixed and total flow time varies with run and is not observed. The opposite is true for the application of mass flow measurement during aerial application of fertilizer particles. The theoretical results regarding inference for the random particle flow $A(t)$ for fixed $t$ do not necessarily hold under the conditions of the experiment, where $A(t)$ is fixed and $t$ varies and is not observed. However, Table 3 provides some indication that estimates for total particle flow, $A(t)$, have good empirical performance when it is treated as random, at least in these experiments. The observed value of the estimator $\hat{A}_1$ is given in the penultimate column. It appears to perform reasonably well under these conditions. The average of $\hat{A}_1$ over runs 1-20 is 9 and the root mean squared error from 4000 is 35.5, which is 0.9% of the target. There is some evidence of positive bias in the high intensity runs 11-20. A two-sided $t$-test of the hypothesis that $E[\hat{A}_1]=4000$ under the conditions of runs 11-20 yielded a $p$-value of $0.065$ on $df=9$. Higher flow rates lead to more clumps per particle, fewer singletons, and larger variance in estimation of clump order, either conditionally as in $\hat{A}_B$ or unconditionally, as in $\hat{A}_1$. The standard deviations of $\hat{A}_1$ under the light (runs 1-10) and heavy (runs 11-20) clumping conditions with 4000 BBs were $s_l=26.4$ and $s_h=37.1$, respectively. The estimates $\hat{A}_B$, which are based upon the DSL model, exhibit substantial positive bias, as they did for data simulated under the RSL model. The same is true for the MLE of $\lambda$. To assess the goodness of fit of the linear Boolean models, probability histograms of the clump length data were checked for agreement with the estimated density $f(y;\tilde\lambda,d)$. One such check appears in Figure 1, which exhibits reasonable fit except for slightly lowered mass at the mean segment length, $\mu=2 msec$ and slightly more observations just above the mean segment length than expected under uniformity of this part of the density. All of the other histograms exhibited the same three distinctive features of a spike near this fixed segment length, near uniformity between one and two of these lengths and a long right tail. Quantile plots and Kolmogorov-Smirnov goodness-of-fit tests, for estimates in runs 1-10 or runs 11-20 do not indicate any non-normality in the distribution of $N$, $\tilde\lambda$ or $\hat{A}_1$. Conclusion ========== Two versions of a simple linear Boolean model are proposed to describe passage times of clumps of particles in a type II counter system; one assumes deterministically equal passage times for all particles, while the other assumes these to be distributed about a known mean with unknown variance. An $M$-estimator of flow intensity is developed that is intuitively sensible, computationally feasible, and robust to conditions where either particle velocity and/or diameters have substantial variability or are being measured with error by the type II counter. For total mass flow, $A(t)$, two estimators are developed. The first is simply product of the number of clumps, $N(t)$ and the estimate of the mean number of particles per clump. The second more complex estimator is the clumpwise sum of conditional mean clump orders ($K$), given clumplengths ($Y$). In models where segment lengths were deterministically equal, the Bayes estimator exploiting the conditional mean clump order had relative root mean squared error not exceeding $5.0\%$, and always lower than that of the simpler estimator based only on the $M$-estimate of flow rate and the number of clumps. Under the most favorable conditions, with light particle flow and a long dispersal period, the relative error was as small as $1\%$. While the Bayes estimator did well in data simulated from the DSL model, it was outperformed by the simpler estimator in simulations where the segment lengths vary according to a normal distribution and in the bench-scale experiments. The relative root mean square error when using the estimator of based ranged between $1.4\%$ and $5.1\%$. The relative root mean square errors for the experimental data were $0.6\%$ and $1.1\%$, in the low and high intensity runs with 4000 BBs, respectively and $1.8\%$ in the runs with 2000 BBs.

--- abstract: 'By extending local $U(1)$ gauge symmetry to discontinuous case, we find that under one special discontinuous $U(1)$ gauge transformation the symmetric and antisymmetric wave functions can transform into each other in one dimensional quantum mechanics. The free spinless fermionic system and bosonic system with $\delta$-type vector gauge potential are proved to be equivalent. The relation also holds in higher space-time dimensions.' --- [**Boson-Fermion Duality from Discontinuous Gauge Symmetry**]{} Ji Xu and Shuai Zhao\ Can bosons and fermions transform into each other? It is possible in supersymmetry, a hot candidate for solving the hierarchy problem, which is still waiting to be examined by high energy experiment. However, in low dimensional non-relativistic systems, the equivalence of bosonic and fermionic systems are reported long time ago[@Girardeau:1960; @Girardeau:1965; @Mattis1965; @Coleman:1974bu]. The specifieness of $1+1$ dimensions leads to bosonization [@Tomonaga:1950zz; @Luttinger:1963zz], which has become a popular procedure in condensed matter physics. The relation of thermodynamic properties between $1D$ bosonic and fermionic system is also analyzed [@schmidt:1998; @crescimanno:2001]. Massless boson and fermion theories in $1+1$ dimensional Minkowski and curved space-time are proved to be equivalent [@freundlich:1972; @davies:1978]. Because the spin-statistics relation is based on relativity, non-relativistic system can escape from the spin-statistics relation, thus boson can be spinning and fermion can be spinless. The relation between boson and spinless fermion may shed light in the general properties of boson-fermion duality. In [@cheon:pla; @Cheon:1998iy] an one-dimensional fermionic many-body system is found to be equivalent to a system of bosonic particles interacting through $\delta$-type interaction. The 1+1 dimensional systems with derivative $\delta$-functions and momentum dependent interactions are also discussed [@BasuMallick:2009km; @Grosse:2004rp], and the roles of symmetry and supersymmetry played in point interactions have been realized [@Cheon:2000tq; @Fulop:1999pf; @Ohya:2014lsa; @Nagasawa:2002un]. Thanks to the gauge symmetry, one can tune the phase of a wave function by gauge transformations. For the difference between boson and fermion is from the different phase factor when exchanging two identity particles, it provides the probability of connecting bosons and fermions from gauge symmetry. It is usually considered that the gauge transformation should be continuous. However, the key idea of the local gauge symmetry is that the choice of phase at one space-time point should not be affected by another point. The phase of two points should be independent to each other even the two points are close enough. Thus the gauge transformation can be discontinuous. To understand this in another way, let’s consider two points $x$ and $y$ with small distance. In gauge theory, the gauge transformation of the two points are related by a Wilson line $U_1(x,y)$, which is shown in Fig.\[gauge\]. Generally the difference of the phase $\alpha(x)$ and $\alpha(y)$ is finite. When $x$ and $y$ is close to each other and keep the difference of the phase $\alpha(x)$ and $\alpha(y)$ as finite, we will get a discontinuous gauge transformation. One can also link $x$ and $y$ from another Wilson line $U_2(x,y)$ with finite length when $|x-y|\to 0$, $U^{\dag}_1$ and $U_2$ then form a Wilson loop. When gauge transformation along $U_1(x,y)$ is discontinuous, the gauge transformation along $U_2(x,y)$ can still be continuous, which means one can realize discontinuous transformation through a continuous one. ![Local gauge transformation at two points $x$ and $y$. Two Wilson line between $x$ and $y$ are presented. The solid line is “short” Wilson line while the dashed line denotes the “longer” one. When $|x-y|\to 0$, the gauge transformation along $U_1(x,y)$ will be discontinuous. []{data-label="gauge"}](gauge.eps){width="6cm"} In the present work we try to connect one-dimensional spinless fermionic system and bosonic system with discontinuous gauge transformation. We will show that free fermions can be mapped into bosons with a $\delta$-function type vector-potential, and vice versa, which reveal an interesting connection between boson-fermion duality and gauge symmetry. It is also possible to generalize the discussions on one-dimensional system to other space-time dimensions and symmetries. We start with one-body problem in one dimensional quantum mechanics. Consider a charged scalar particle in a magnetic field. Its motion is governed by Schrödinger equation with gauge potential $A_i=(A_0,A_1)$ $$\begin{aligned} \left[-\frac{1}{2}\left(\partial_x+ie A_1\right)^2+V(x)\right]\phi(x,t)=i\left(\frac{\partial }{\partial t}+i eA_0\right)\phi(x,t),\label{scheq}\end{aligned}$$ For simplicity we adopt the system of natural units and take the particle mass $m=1$, $\phi(x,t)$ is the wave function, $e$ is the electric charge of the particle. It is well known that Eq.(\[scheq\]) is invariant under $U(1)$ gauge transformation $$\begin{aligned} \phi (x,t)\to \phi'(x,t)&=\phi(x,t)e^{i\alpha(x,t)},\\ A_i(x,t)\to A'_i(x,t)&=A_i(x,t)-\frac{1}{e}\frac{\partial}{\partial x_i}\alpha(x,t),\end{aligned}$$ where $i=0,~1$ corresponding to $t,~x$ respectively. As discussed above, the choice of the phase at one point should not be affected by another, so it is reasonable that the phase function $\alpha(x,t)$ can be discontinuous. Here we let $\alpha(x,t)=\pi \theta(x)$, where $\theta(x)$ is the Heaviside step function, $\alpha$ is independent of $t$. Then we have the discontinuous gauge transformation $$\begin{aligned} \phi (x,t)\to \phi'(x,t)&=\phi(x,t)e^{i\pi\theta(x)}\label{gt},\\ A_0(x,t)\to A'_0(x,t)&=A_0(x,t),\\ A_1(x,t)\to A'_1(x,t)&=A_1(x,t)-\frac{\pi}{e} \delta(x),\end{aligned}$$ $\delta(x)$ is the Dirac delta function. The transformation in Eq.(\[gt\]) can be expressed as $$\begin{aligned} \phi'(x,t)=\left\{ \begin{aligned} -&\phi(x,t),~~~~&x>0;\\ &\phi(x,t),~~~~&x<0. \end{aligned} \right.\end{aligned}$$ Now we assume that $\phi(x,t)$ is an odd function of $x$, i.e., $\phi(-x,t)=-\phi(x,t)$, then for $\phi'(x,t)$ we have $\phi'(-x,t)=\phi'(x,t)$, which means that $\phi'(x,t)$ is an even function. So we have transform an odd wave function into an even wave function, at the cost of introducing a $\delta$-type potential in the gauge field. Even functions can also be transformed into odd functions under the same transformation. Before further discussions we should make a few remarks on the vector potential. In fact there is no magnetic field $B$ in $1+1$ dimensions, even if it exists, charged particle will not couple to the magnetic field. Further more, it is a little tricky to talk about “vector potential”, because $A$ is actually a scalar in $1+1$ dimensions [@elect]. In our case, we just take $A$ as an auxiliary vector potential, also for the sake of simplicity, we will set $A_i=0$ and only keep the $\delta$ term in the following discussions for $1+1$ dimensions. $A$ will make sense in higher dimensions. With these assumptions, after the gauge transformation, the new field $\phi'(x,t)$ then satisfies the Schrödinger equation $$\begin{aligned} -\frac{1}{2}\Big[\partial_x-i\pi\delta(x)\Big]^2\phi'(x,t)=i\frac{\partial}{\partial t}\phi'(x,t).\label{dteq}\end{aligned}$$ To see how anti-symmetric wave function transform into symmetric more clearly, we consider a class of equations which approach to Eq.(\[dteq\]) when $\epsilon\to 0$. We restrained the system with an infinite square potential well, thus one can study the stationary state problem $$\begin{aligned} -\frac{1}{2}\left[\left(\frac{\partial}{\partial x}-i \frac{\epsilon}{x^2+\epsilon^2}\right)^2+V(x)\right]\phi'(x)=E_n \phi'(x)\end{aligned}$$ with $$\begin{aligned} V(x)=\left\{ \begin{aligned} &0,~~~~~~&|x|<a;\\ &\infty,~~~~~~&|x|>a, \end{aligned} \right.\end{aligned}$$ where $E_n$ is the energy level. The eigenfunctions for different values of $\epsilon$ are shown in Fig.\[wave\]. When $\epsilon$ is large, the anti-symmetric imaginary part is dominant. When $\epsilon$ takes small value the symmetric real part become important. When $\epsilon\to 0$, $\epsilon/(x^2+\epsilon^2)\to \pi\delta(x)$, and the anti-symmetric imaginary part of wave function disappears, we finally arrive at a symmetric wave function. ![The wave functions of Eq.(\[gt\]) correspond to $n=2$ at $\epsilon=10$ (a), $\epsilon=1$ (b), $\epsilon=0.1$ (c) and $\epsilon=0.001$ (d). The solid lines denote the real parts while the dashed lines show the imaginary parts of the wave functions.[]{data-label="wave"}](wave.eps){width="15cm"} In [@cheon:pla][@Cheon:1998iy], a fermionic system in an $\varepsilon(x;c)$ potential is proved to be equivalent with a bosonic system in $ \delta(x;v)$ potential, with the coupling strength reversed, i.e., $v=1/c$, here $\varepsilon(x;c)\equiv\chi(x;-1,0,-1,-4c)$, and $\delta(x;v)\equiv \chi(x;-1,-v,-1,0)=v \delta(x)$. $\chi$ is defined by [@Cheon:1998iy] $$\begin{aligned} \chi(x;\alpha,\beta,\gamma,\delta)=\displaystyle{\lim_{a\to +0}}\left[\left(-\frac{1}{a}+\frac{\gamma-1}{\delta}\right)\delta(x+a)+\frac{1-\alpha\gamma}{\beta a^2} \delta(x)+\left(-\frac{1}{a}+\frac{\alpha-1}{\delta}\right)\delta(x-a)\right].\end{aligned}$$ In fact what we have discussed above is an extreme case of such a strong-weak duality. To see this clearly, Eq.(\[dteq\]) can be reexpressed as a Schrödinger equation with a $\delta^2(x)$ potential $$\begin{aligned} -\frac{1}{2}\frac{\partial^2}{\partial x^2}\phi'(x,t)-\pi^2\delta^2(x)\phi'(x,t)=i\frac{\partial }{\partial t}\phi'(x,t),\end{aligned}$$ because $\exp(i\pi\theta(x))=\exp(-i\pi\theta(x))$, the wave function satisfies the same equation when replacing $\pi$ with $-\pi$. When $c=a/2\to 0$, $\varepsilon(x;c)\to 0$, Schrödinger equation with $\varepsilon$ interaction then degenerates to a free particle equation; In another hand, $v=1/c\to \infty$, $v\delta (x)$ then becomes a $\delta$-interaction with infinite coupling strength, which can be regarded as a $\delta^2(x)$ interaction, because that when $x$ is close to $0$, $\delta^2(x)\sim \delta(0)\delta(x)$, which is a $\delta(x)$ interaction with an infinite coupling strength $\delta(0)$. Thus a fermionic system with zero coupling strength $\varepsilon$-interaction is equivalent to a bosonic system with infinite coupling strength $\delta$-interaction. Now we turn to the two-particle system with a wave function $\Psi(x_1,x_2,t)$, $x_i(i=1,~2)$ is the coordinate of the $i$-th particle. Introducing the discontinuous gauge transformation $$\begin{aligned} &&\Psi(x_1, x_2, t)\to \Psi'(x_1, x_2, t)=\Psi(x_1,x_2,t)e^{i\pi\theta(x_1-x_2)},\label{gt2}\end{aligned}$$ one can immediately get that $$\begin{aligned} \Psi'(x_1, x_2, t)=\left\{\begin{aligned} -&\Psi(x_1, x_2, t),~~~~~~&x_1>x_2;\\ &\Psi(x_1,x_2,t),~~~~~~&x_1<x_2. \end{aligned}\right.\end{aligned}$$ Notice that the sign before the $\delta$ function is opposite for the two particles. Now we assume that $\Psi(x_1,x_2,t)$ is anti-symmetric under the exchange of $x_1$ and $x_2$, i.e., a wave function for fermionic system, then $\Psi'(x_1,x_2,t)$ should not change when exchanging $x_1$ and $x_2$, which is a wave function of bosons. Thus we have transformed a wave function of fermion into bosons under a discontinuous gauge transformation, at the cost of introducing a $\delta$-type interaction in the derivative of wave function. The two-body Schrödinger equation then reads $$\begin{aligned} -\frac{1}{2}\left[\left(\frac{\partial}{\partial x_1}-i \pi\delta(x_1-x_2)\right)^2+\left(\frac{\partial}{\partial x_2}+i\pi\delta(x_1-x_2)\right)^2\right]\Psi'(x_1,x_2,t)=i\frac{\partial}{\partial t}\Psi'(x_1,x_2,t).\end{aligned}$$ Similarly one can consider $n$-particle system, $x_i$ is the coordinate of the $i$-th particle. Then we perform the gauge transformation on the wave function $\Psi(x_1,\cdots,x_i,\cdots,x_n)$: $$\begin{aligned} && \Psi(x_1,\cdots,x_i,\cdots,x_n)\to \Psi'(x_1,\cdots,x_n)=\Psi(x_1,\cdots,x_n)\exp{\left[i\pi\displaystyle{\sum_{1\leq i<j \leq n}\theta(x_i-x_j)}\right]}.\label{gtn}\end{aligned}$$ One can examine that $\Psi'$ will be multiplied by a $(-1)$ when exchanging the coordinates of any pair of particles. The wave function then satisfies the $n-$body Schrödinger equation $$\begin{aligned} -\frac{1}{2}\displaystyle{\sum_{i=1}^n}\left[\frac{\partial}{\partial x_i}-i\pi\displaystyle{\sum_{\substack{l,~k\\1\leq l<i<k\leq n}}}\left(\delta(x_i-x_k)-\delta(x_l-x_i)\right)\right]^2\Psi'(x_1,\cdots,x_n)=i\frac{\partial}{\partial t}\Psi'(x_1,\cdots,x_n).\end{aligned}$$ The $\delta$-function in the above equation can be understood as Pauli exclusion principle, which states that two or more identical fermions cannot occupy the same quantum state within a quantum system simultaneously. In one dimensional quantum mechanics, when the interaction term contains $V=\delta(x_i-x_j)$, particles can not have the same coordinates at the same time, because that when $x_i=x_j$, $V$ will be infinite. In one dimensional system the quantum states are characterized by the space-time coordinates, so Pauli exclusion principle states that two fermions can not have the same coordinates at the same time, which indicates that there is a $\delta(x_i-x_j)$ interaction in the equation of motion. The above discussions on one dimensional system can also be generalized into other space-time dimensions. Now we will show that for an one-component scalar field with $U(1)$ symmetry, one can also build up boson-fermion duality in $2+1$ dimensions. Consider a gauge potential $\vec{A}=(-\frac{\pi}{2e}\delta(x),-\frac{\pi}{2e}\delta(y))$ and a symmetric wave function $\phi(\vec{r})$. For any $\vec{r}$ and $-\vec{r}$, one can link these two points with product of two Wilson lines $U_{L_2}(\vec{r},0)$ and $U_{L_1}(0,-\vec{r}\,)$: $$\begin{aligned} U_{L_2}(\vec{r},0)U_{L_1}(0,-\vec{r}\,)&=\exp\left(ie\int_{L_1} \vec{A}\cdot d \vec{l}\right)\exp\left(ie\int_{L_2} \vec{A}\cdot d \vec{l}\right)\nonumber\\ &=\exp\left\{-i\frac{\pi}{2}\int^{0}_{r} ~dR \Big[\cos(\theta+\pi) ~\delta\left(R\cos(\theta+\pi)\right)+\sin(\theta+\pi) ~\delta\left(R\sin(\theta+\pi)\right)\Big]\right.\nonumber\\ &~~~~~~~~\,\,\,\,-i\left.\frac{\pi}{2}\int^{r}_{0} ~dR \Big[\cos\theta ~\delta(R\cos\theta)+ \sin\theta ~\delta(R\sin\theta)\Big] \right\}\nonumber\\ &=-1,\end{aligned}$$ where $L_1$ is a straight line from $-\vec{r}$ to $0$ and $L_2$ is a straight line from $0$ to $\vec{r}$, $r$ and $\theta$ is the polar radius and angular respectively. If $\phi(\vec{r})$ is an odd function, then $\phi'(\vec{r})$ will be an even function. Note that one can link $\vec{r}$ and $-\vec{r}$ by Wilson lines along other paths. However, every path links $\vec{r}$ and $-\vec{r}$ will pass $x$ and $y$ axes once. Every time passing through an axes will contribute a phase $\exp(i\pi/2)$, finally contributes a factor $(-1)$ to the relative phase of the two points. In 2+1 dimensions we can also consider two components vector wave function under $U(1)\times U(1)$ symmetry. Consider a wave function $\Phi(\vec{r},t)=(\phi_1(\vec{r},t),~\phi_2(\vec{r},t))$, where $\vec{r}=(x,y)$. We perform the gauge transformation $$\begin{aligned} \Phi(\vec{r},t)=(\phi_1(\vec{r},t),~\phi_2(\vec{r},t))&\to &\Phi'(\vec{r},t)= (e^{i\pi\theta(x)}\phi_1(\vec{r},t),~~e^{i\pi\theta(y)}\phi_2(\vec{r},t))\nonumber\\ &=&\left\{\begin{aligned} &(-\phi_1(\vec{r},t),-\phi_2(\vec{r},t)),~~~~~~&x>0,~y>0;\\ &(\phi_1(\vec{r},t),-\phi_2(\vec{r},t)),~~~~~~&x<0,~y>0;\\ &(\phi_1(\vec{r},t),\phi_2(\vec{r},t)),~~~~~~&x<0,~y<0;\\ &(-\phi_1(\vec{r},t),\phi_2(\vec{r},t)),~~~~~~&x>0,~y<0. \end{aligned}\right.\label{vec}\end{aligned}$$ If $\Phi(\vec{r},t)$ is an odd function with $\vec{r}$, i.e. $\Phi(-\vec{r},t)=\Phi(\vec{r},t)$, then $\Phi'(-\vec{r},t)=-\Phi'(\vec{r},t)$, which means that $\Phi'(\vec{r},t)$ is an even function. Fig.\[graph3d\] shows that an odd vector wave function transforms into an even vector wave function under Eq.(\[vec\]). ![The gauge transformation Eq.(\[vec\]). (a) shows an anti-symmetric vector wave function in 2 spatial dimensions. (b) is a symmetric vector wave function which is got by acting Eq.(\[vec\]) on (a).[]{data-label="graph3d"}](graph3d.eps){width="14cm"} The gauge transformation on gauge potential $A_i$ then becomes $$\begin{aligned} {A}_0(\vec{r},t)&\to A'_0(\vec{r},t),\\ {A}_x(\vec{r},t)&\to {A}'_x(\vec{r},t)={A}_x(\vec{r},t)-\frac{\pi}{e}\delta(x),\\ {A}_y(\vec{r},t)&\to {A}'_y(\vec{r},t)={A}_y(\vec{r},t)-\frac{\pi}{e}\delta(y).\end{aligned}$$ One can also construct many-body equations in 2+1 dimensions, just as what we have done for $1+1$ dimensional case. These discussions can also be generalized to higher space-time dimensions. The Aharonov-Bohm effect [@Aharonov:1959fk] states that outside the region where magnetic field $B$ is confined, the vector potential may still causes observable effects. The above discussions also reveal some novel properties of vector potential. If the vector potential is confined in a $n-1$ dimensional hyperplane in $n$ spatial dimensions, it can also have physical effects. For certain configurations they may affect the shape of wave functions. In addition, this work only consider scalar wave functions governed by Schrödinger equation. Similar discussions can be performed to Dirac equations as well. Further more, in this work we only consider $U(1)$ and $U(1)\times U(1)$ groups. It should be also interesting of generalizing this work to non-Abelian symmetries. We leave these for a further research. To summarize, we build up boson-fermion duality under the spirit of local gauge symmetry. The non-relativistic bosonic (fermionic) system can be mapped into a fermionic (bosonic) system with $\delta$-type gauge interactions. This duality may open a door to a deeper understanding of the relationship between exchange statistics and gauge symmetry. [**Acknowledgments**]{} We thank Dr. Jian-Ping Dai for valuable discussions. [99]{} M. Girardeau, J. Math. Phys. [**1**]{}, 516 (1960). M. Girardeau, Phys. Rev. B [**139**]{}, 500 (1965). D. C. Mattis and E. H. Lieb, J. Math. Phys. [**6**]{}, 304 (1965). S. R. Coleman, Phys. Rev. D [**11**]{}, 2088 (1975). S. Tomonaga, Prog. Theor. Phys.  [**5**]{}, 544 (1950). J. M. Luttinger, J. Math. Phys.  [**4**]{}, 1154 (1963). H. J. Schmidt, J. Schnack, Physica. A 260, 479 (1998). M. Crescimanno and A. S. Landsberg, Phys. Rev. A [**63**]{}, 035601 (2001). Y. Freundlich, Nucl. Phys. B [**36**]{}, 621 (1972). P. C. W. Davies, J. Phys. A [**11**]{}, 179 (1978). T. Cheon and T. Shigehara, Phys.  Lett. A [**243**]{}, 111 (1998). T. Cheon and T. Shigehara, Phys. Rev. Lett.  [**82**]{}, 2536 (1999). B. Basu-Mallick and T. Bhattacharyya, Mod. Phys. Lett. A [**25**]{}, 715 (2010). H. Grosse, E. Langmann and C. Paufler, J. Phys. A [**37**]{}, 4579 (2004) T. Cheon, T. Fulop and I. Tsutsui, Annals Phys.  [**294**]{}, 1 (2001). T. Fulop and I. Tsutsui, Phys. Lett. A [**264**]{}, 366 (2000). S. Ohya, J. Phys. Conf. Ser.  [**563**]{}, no. 1, 012021 (2014). T. Nagasawa, M. Sakamoto and K. Takenaga, Phys. Lett. B [**562**]{}, 358 (2003). H. J. W. Muller-Kirsten, [*Electrodynamics: An Introduction Including Quantum Effects*]{}, World Scientific, 2004. Y. Aharonov and D. Bohm, Phys. Rev.  [**115**]{}, 485 (1959).

--- abstract: 'The CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining all CM curves of genus $2$ defined over the *reflex field*. We solve the problem by showing that the list given in Bouyer–Streng [@bouyer Tables 1a, 1b, 2b, and 2c] is complete.' author: - 'P[i]{}nar K[i]{}l[i]{}çer[^1]  and Marco Streng[^2]' bibliography: - 'mybib.bib' title: | The CM class number one problem for\ curves of genus $2$ --- Introduction ============ By a *curve*, we always mean a projective smooth geometrically irreducible algebraic curve. A curve $C$ over a field $k$ of genus $g$ has *complex multiplication* (CM) if the endomorphism ring of its Jacobian over $\overline{k}$ is an order ${\mathcal{O}}$ in a *CM field* of that is, a totally imaginary quadratic extension of a totally real number field of We say that $C$ by ${\mathcal{O}}$. For example, an elliptic has CM if ${\text{\normalfont{End}}}(E_{\overline{k}})$ is an order in an imaginary quadratic field $K$. An elliptic curve $E$ with CM by an order ${\mathcal{O}}_{K}$ can be defined over ${\mathbb{Q}}$ if and only if the class group ${\text{\normalfont{Cl}}}_{K}:=I_{K}/P_{K}$ is trivial. The CM class number one problem for elliptic curves asks to determine all imaginary quadratic fields of class number one. This problem was solved by Baker [@baker] (1966) and Stark [@stark] (1967); the fields are $K\cong{\mathbb{Q}}(\sqrt{-d})$ where $d\in \{1,\ 2,\ 3,\ 7,\ 11,\ 19,\ 43,\ 67,\ 163\}$. We consider the analogous problem for curves of genus $2$. In , we recall the definition of the *CM class group* $I_{K^r}/I_0(\Phi^r)$ and we call its order the *CM class number*. If a curve $C$ has CM by a maximal and is defined over ${\mathbb{Q}}$, then the CM class group is trivial, see . In [@umeg], Murabayashi and Umegaki listed all quartic CM fields $K$ with curves having CM defined over ${\mathbb{Q}}$. This list contains only cyclic quartic CM fields, but not the generic dihedral quartic CM fields because curves cannot be defined over ${\mathbb{Q}}$ in the dihedral case, see Shimura [@goro1 Proposition 5.17]. The simplest examples that do include the dihedral case are those defined over the reflex field, equivalently, those of CM class number one (see [@bouyer]). We give the complete list of CM class number one fields for curves of \[thm: non-normallist\] There exist exactly $63$ isomorphism classes of non-normal quartic CM fields with CM class number one. The fields are listed in Theorem \[thm: non-normallist2\]. \[thm: cycliclist\] There exist exactly $20$ isomorphism classes of cyclic quartic CM fields with CM class number one. The fields are listed in Theorem \[thm: cycliclist2\]. The list in contains the list in [@umeg]. \[cor: curves\] There are exactly $125$ curves of genus $2$ defined over the reflex field with CM by ${\mathcal{O}}_K$ for some quartic non-biquadratic $K$. The fields are the fields in Theorems \[thm: non-normallist\] and \[thm: cycliclist\], and the curves are those of Bouyer–Streng [@bouyer Tables 1a, 1b, 2b, and 2c]. This follows from the list given by Bouyer–Streng in [@bouyer] and There are exactly $21$ absolutely simple curves of genus $2$ defined with CM. The fields and $19$ of the curves are given in van Wamelen [@wame]. The other two curves are $y^2 = x^6-4x^5+10x^3-6x-1$ and $y^2 = 4x^5+40x^4-40x^3+20x^2+20x+3$ given in Theorem 14 of Bisson–Streng [@bisson]. The $19$ curves given in [@bouyer] are the curves of genus $2$ defined with CM , see [@umeg] or Corollary \[cor: curves\]. In [@bisson], Bisson–Streng prove that there are only $2$ curves of genus $2$ defined with CM by a non-maximal order inside one of the fields of Theorem \[thm: cycliclist2\]. finishes the proof. There are finitely many curves of genus 2 with CM by an order ${\mathcal{O}}$ in a number field $K$ defined over the reflex field. The fields are those of the complete list of orders can be computed using the methods of [@bisson] and the curves using the methods of [@bouyer]. In Section \[section: CM\], we give some definitions and facts from CM theory, and state the CM class number one problem for curves of genus $g\leq 2$. In Section \[non-normal\], we prove The strategy is as follows. We first show that there are only finitely many such CM fields by bounding their discriminant. The bound will be too large for practical purposes, but by using ramification theory and $L$-functions, we improve the bound which we then use to enumerate the fields. Section \[cyclic\] proves Theorem \[thm: cycliclist\] using the same strategy as in Section \[non-normal\]. Acknowledgement {#acknowledgement .unnumbered} --------------- The authors would like to thank Maarten Derickx and Peter Stevenhagen for useful discussions. Complex Multiplication {#section: CM} ====================== We refer to Shimura [@goro3] and Lang [@lang] as references for this section. Let $K$ be a CM field of degree $2g$ and $N'$ be a number field that contains a subfield isomorphic (over ${\mathbb{Q}}$) to a normal closure of $K$. There exists an automorphism $\rho$ such that for every embedding $\tau: K \rightarrow {\mathbb{C}}$ we have $\overline{\cdot}\circ \tau = \tau \circ \rho$; we call it *complex conjugation* and denote it by $\overline{\cdot}$. If $\phi$ is an embedding of CM fields $K_1 \rightarrow K_2$, then we have $\overline{\cdot}\circ \phi = \phi \circ \overline{\cdot}$. We denote $\phi\circ \overline{\cdot}$ by $\overline{\phi}$. A *CM type* of $K$ with values in $N'$ is a set $\Phi$ of embeddings into $N'$ such that exactly one embedding of each of the $g$ complex conjugate pairs of embeddings $\phi, \ \overline{\phi}: K \rightarrow N'$ is in $\Phi$. Let $\Phi$ be a CM type of $K$ with values in $N'$. Let $L\supset K$ be a CM field such contains a subfield isomorphic to a normal closure of $L$. Then the CM type of $L$ *induced* by $\Phi$ is $\{\phi\in{\text{\normalfont{Hom}}}(L,N'):\ \phi|_{K}\in\Phi\}$. A CM type $\Phi$ is *primitive* if it is not induced from a CM type of a proper CM subfield. We say that CM types $\Phi_1$ and $\Phi_2$ of $K$ are *equivalent* if there is an automorphism $\sigma$ of $K$ such that $\Phi_1=\Phi_2\sigma$ holds. Let $N$ be a normal closure of $K$. From now on, we identify $N'$ with $N$ by making a choice of isomorphism and replacing $N'$ by a subfield. The *reflex field* of $(K,\ \Phi)$ is $$K^{r}={\mathbb{Q}}(\{\sum_{\phi\in\Phi}\phi(x)\ |\ x\in K\}) \subset N$$ and satisfies ${\text{\normalfont{Gal}}}(N/K^r)=\{\sigma\in{\text{\normalfont{Gal}}}(N/{\mathbb{Q}}):\sigma\Phi=\Phi\}$. For example, if a is Galois over ${\mathbb{Q}}$, then the reflex field $K^r$ of $K$ is a subfield of $K$. be the CM type of $N$ induced by $\Phi$. Then the reflex field $K^r$ is a CM field with CM type $\Phi^{r}=\{\sigma^{-1}|_{K^r}:\ \sigma\in\Phi_N\}.$ The pair $(K^r,\Phi^r)$ is called the *reflex* of $(K,\Phi)$. The *type norm* is the multiplicative map $$\begin{aligned} {\text{\normalfont{N}}}_{\Phi}:K&\rightarrow K^r,\\ x&\mapsto\prod_{\phi\in\Phi}\phi(x),\end{aligned}$$ satisfying ${\text{\normalfont{N}}}_{\Phi}\overline{{\text{\normalfont{N}}}_{\Phi}} = {\text{\normalfont{N}}}_{K/{\mathbb{Q}}}(x)\in{\mathbb{Q}}$. The type norm induces a homomorphism between the groups of fractional and $I_{K^r}$ by sending ${\mathfrak{b}}\in I_K$ to ${\mathfrak{b}}'\in I_{K^r}$ such that ${\mathfrak{b}}'{\mathcal{O}}_{N}=\prod\limits_{\phi\in\Phi}\phi({\mathfrak{b}}){\mathcal{O}}_{N}$ (Shimura [@goro3 Proposition 29]). An *abelian variety* $A$ over a field $k$ of characteristic $0$ of dimension $g$ has CM by an order ${\mathcal{O}}_K$ if $K$ has and there is an embedding $\theta:K\rightarrow {\text{\normalfont{End}}}_{\overline{k}}(A_{\overline{k}})\otimes{\mathbb{Q}}$. Let ${\text{\normalfont{Tgt}}}_{0}(A)$ be the tangent space of $A$ over $\overline{k}$ at $0$. Given $A$ with CM by ${\mathcal{O}}_K$, be the set of homomorphisms $K \rightarrow \overline{k}$ occurring in the representation of $K$ on ${\text{\normalfont{End}}}_{\overline{k}}({\text{\normalfont{Tgt}}}_{0}(A_{\overline{k}}))$. Then $\Phi$ is a CM type , and we say that $(A,\theta)$ is of type $(K,\Phi)$. A polarized abelian variety of type $(K,\Phi)$ is a triple $P=(A,C, \theta)$ formed by an abelian variety $(A,\theta)$ of type $(K,\Phi)$ and a polarization $C$ such that $\theta(K)$ is stable under the involution of ${\text{\normalfont{End}}}_{\overline{k}}(A)\otimes{\mathbb{Q}}$ determined by $C$. For more details see Shimura [@goro3 Chapter 14]. We say that an abelian variety is *absolutely simple* if it is not isogenous over $\overline{k}$ to product of abelian varieties of lower dimension. \[Shimura [@goro3], §8.2\]\[simple\] An *abelian variety of type* $(K,\Phi)$ is simple if and only if $\Phi$ is primitive. \[main theorem\] Let $(K,\Phi)$ be a primitive CM type and $(K^r,\Phi^r)$ be the reflex of $(K,\Phi)$. Let $P=(A,C, \theta)$ be a polarized abelian variety of type $(K,\Phi)$. Let $M$ be the field of moduli of $(A,C)$. Then the composite $M_{K}=K^r\cdot M$ is the unramified class field corresponding to the ideal group $$I_{0}(\Phi^{r}):=\{{\mathfrak{b}}\in I_{K^{r}}:\ {\text{\normalfont{N}}}_{\Phi^{r}}({\mathfrak{b}})=(\alpha),\ {\text{\normalfont{N}}}_{K^r/{\mathbb{Q}}}({\mathfrak{b}})=\alpha\overline{\alpha},\ \text{\normalfont{for some}} \ \alpha\in K^{\times} \}.$$ The *Jacobian* $J(C)$ of a curve $C/k$ of genus $g$ is an abelian variety of such that we have $J(C)({\overline{k}}) = {\text{\normalfont{Pic}}}^{0}(C_{\overline{k}})$; for details we refer to [@milnejac]. We say that a curve $C$ has CM by ${\mathcal{O}}_K$, if there is an embedding $\theta:K\rightarrow {\text{\normalfont{End}}}_{\overline{k}}(J(C)_{\overline{k}})\otimes{\mathbb{Q}}$. If a curve $C$ has CM by ${\mathcal{O}}_K$ and is defined over $K^r$, then the CM class group $\textbf{C}_{\Phi^r}:=I_{K^r}/I_{0}(\Phi^{r})$ is trivial. The CM class number one problem for CM fields of degree $2g$ asks to list all primitive CM types $(K,\Phi)$ degree $2g$ such that the CM class group $I_0(\Phi^r)$ is trivial. In the case $g=2$, these are exactly the endomorphism algebras of simple CM curves of genus $2$ defined over the reflex field. Theorems \[thm: non-normallist\] and \[thm: cycliclist\] solve this problem for non-biquadratic quartic CM fields. In the genus-2 case, the quartic CM field $K$ is either cyclic Galois, biquadratic Galois, or non-Galois with Galois group $D_4$ (Shimura [@goro3 Example 8.4(2)]). We restrict to the CM curves with simple Jacobian, which, by Theorem \[simple\], have primitive types. By [@goro3 Example 8.4(2)] their CM fields are not biquadratic. The non-normal quartic CM fields {#non-normal} ================================ This section, which is the largest in this paper, proves the main theorem in the case of non-normal fields. The case of cyclic fields is much easier and is Section \[cyclic\]. Suppose that $K/{\mathbb{Q}}$ is a non-normal quartic CM field with real quadratic The normal closure $N$ is a dihedral CM field of degree $8$ with Galois group $G:={\text{\normalfont{Gal}}}(N/{\mathbb{Q}})= \langle x,\ y:y^{4}=x^{2}=(xy)^{2}={\text{\normalfont{id}}}\rangle$. Complex conjugation $\overline{\cdot}$ is $y^2$ in this notation and the CM field $K$ is the subfield of $N$ fixed Let $\Phi$ be a CM type with values We can (and do) identify $N$ with a subfield of $N'$ in such a way that $\Phi=\{{\text{\normalfont{id}}}, y|_{K}\}$. Then the reflex field $K^r$ is the fixed field of $\langle xy\rangle$, which is a non-normal quartic CM field non-isomorphic to $K$ with reflex type $\Phi^{r}=\{{\text{\normalfont{id}}}, y^{3}|_{K^r}\}$, (see [@goro3 Examples 8.4., 2(C)]). Denote the quadratic subfield of $K^{r}$ by $F^r$. \[diagram\] (0, -1.2) node\[\][$N$]{}; (0, -3) node\[\][$N_+$]{}; (-1.5, -3) node\[\][$K$]{}; (-3, -3) node\[\][$K'$]{}; (1.5, -3) node\[\][$K^r$]{}; (3, -3) node\[\][$K'^r$]{}; (-2.25, -4.5) node\[\][$F$]{}; (2.25, -4.5) node\[\][$F^r$]{}; (0, -4.5) node\[\][$F_+$]{}; (0, -6) node\[\][${\mathbb{Q}}$]{}; (0, -4) – (0, -3.25); (0, -2.50) – (0, -1.5); (-0.50, -3.25) – (-2.0, -4.25); (0.50, -3.25) – (2.0, -4.25); (-2.5, -4.2) – (-3, -3.28); (2.5, -4.2) – (3, -3.28); (-2.25, -4.2) – (-1.5, -3.28); (2.25, -4.2) – (1.5, -3.28); (-2.75, -2.75) – (-0.25, -1.25); (2.75, -2.75) – (0.25, -1.25); (-1.25, -2.75) – (-0.15, -1.50); (1.25, -2.75) – (0.15, -1.50); (-2.25, -4.75) – (-0.50, -5.75); (2.25, -4.75) – (0.50, -5.75); (0, -4.75) – (0, -5.50); (0, -1.2) node\[\][$1$]{}; (0, -3) node\[\][$\langle y^2\rangle$]{}; (-1.5, -3) node\[\][$\langle x\rangle$]{}; (-3, -3) node\[\][$\langle xy^2\rangle$]{}; (1.5, -3) node\[\][$\langle xy\rangle$]{}; (3, -3) node\[\][$\langle xy^3\rangle$]{}; (-2.25, -4.5) node\[\][$\langle x,\ y^2\rangle$]{}; (2.25, -4.5) node\[\][$\langle xy,\ y^2\rangle$]{}; (0, -4.5) node\[\][$\langle y\rangle$]{}; (0, -6) node\[\][$G$]{}; (0, -4) – (0, -3.25); (0, -2.50) – (0, -1.5); (-0.50, -3.25) – (-2.0, -4.25); (0.50, -3.25) – (2.0, -4.25); (-2.5, -4.2) – (-3, -3.28); (2.5, -4.2) – (3, -3.28); (-2.25, -4.2) – (-1.5, -3.28); (2.25, -4.2) – (1.5, -3.28); (-2.75, -2.75) – (-0.25, -1.25); (2.75, -2.75) – (0.25, -1.25); (-1.25, -2.75) – (-0.15, -1.50); (1.25, -2.75) – (0.15, -1.50); (-2.25, -4.75) – (-0.50, -5.75); (2.25, -4.75) – (0.50, -5.75); (0, -4.75) – (0, -5.50); Let $N_+$ be the maximal totally real subfield of $N$, and let $F_+$ be the fixed field An effective bound for non-normal quartic CM fields with CM class number one {#finiteness} ---------------------------------------------------------------------------- In this section, we find an effective upper bound for the discriminant of the non-normal quartic CM fields with CM class number one. We first prove the following relation between the relative class number $h^{*}_K:=h_K/h_F$ and the number $t_K$ of ramified primes in $K/F$. \[prop2\] Let $K$ be a non-biquadratic quartic CM field with the real quadratic subfield $F$. Assuming $I_{0}(\Phi^{r})=I_{K^r}$, we have $h^{*}_{K}=2^{t_K-1}$, where $t_K$ is the number of ramified primes in $K/F$. Moreover, we have $h^{*}_{K^r}=2^{t_{K^r}-1}$, where $t_{K^r}$ is the number of ramified primes In the case where $K/{\mathbb{Q}}$ is cyclic quartic, this result is $(i)\Rightarrow (iii)$ of Proposition 4.5 in Murabayashi [@murab]. On the other hand, if $K$ is a non-normal quartic CM field, Louboutin proves $h^{*}_{K}\approx \sqrt{d_K/d_F}$ with an effective error bound, see Proposition \[louboutin lower bound\]. Putting this together with the result in Proposition \[prop2\] gives approximately $\sqrt{d_K/d_F}\leq 2^{t_K-1}$. As the left hand side grows more quickly than the right, this relation will give a bound on the discriminant, precisely see Proposition \[effectiveness\]. We start the proof of Proposition \[prop2\] with the following lemma. \[lem1\] Let $K$ be a CM field with real quadratic subfield $F$ and group of roots of unity $\mu_K=\{\pm1\}$. Let $H$ denote the group ${\text{\normalfont{Gal}}}(K/F)$. Put $I^{H}_{K}=\{{\mathfrak{b}}\in I_{K}\ |\ \overline{{\mathfrak{b}}}={\mathfrak{b}}\}$. Then we have $h_{K}^{*}=2^{t_{K}-1}[I_K:I^{H}_{K}P_K]$, where $P_K$ is the group of principal fractional ideals in $K$. We have the exact sequence $$\label{exact}1\rightarrow I_{F} \rightarrow I_{K}^{H} \rightarrow \bigoplus_{{\mathfrak{p}}\,\text{prime}\, \text{of}\, F}{\mathbb{Z}}/e_{K/F}({\mathfrak{p}}){\mathbb{Z}}\rightarrow 1$$ and $$\bigoplus\limits_{{\mathfrak{p}}\ \text{prime of}\ F}{\mathbb{Z}}/e_{K/F}({\mathfrak{p}}){\mathbb{Z}}\quad\cong\quad({\mathbb{Z}}/2{\mathbb{Z}})^{t_K}.$$ Define $P^{H}_K=P_K\cap I_K^{H}$. The map $\varphi:I_{K}^{H}\rightarrow \frac{I_{K}}{P_{K}}$ induces an isomorphism $I_{K}^{H}/P_{K}^{H}\cong \text{im}(\varphi)=I^{H}_{K}P_K/P_K$, so by (\[exact\]), we have $$h_F = [I_F:P_F] = \frac{[I_{K}^{H}:P_{K}^{H}][P_{K}^{H}:P_{F}]}{[I_{K}^{H}:I_{F}]}=2^{-t_K}[I^{H}_{K}P_K:P_K][P_{K}^{H}:P_{F}],$$ hence $$h^{*}_{K}:=\frac{h_{K}}{h_F}=2^{t_K}\frac{[I_K:I_{K}^{H}P_K]}{[P_{K}^{H}:P_{F}]}.$$ It now suffices to prove $[P_{K}^{H}:P_{F}]=2$, for which we claim that there is a surjective group homomorphism ${\phi}: P_{K}^{H}{\rightarrow} \mu_K = \{\pm1\}$ given by $ \phi((\alpha))={\alpha}/{\overline{\alpha}}$ with kernel $P_F$. *Proof of the Claim*. The map $\phi$ is well-defined because the image is independent of the choice of $\alpha$ as ${\mathcal{O}}^{*}_K={\mathcal{O}}^{*}_F$ (cf. [@loub5 Lemma 1]). It is clear that it is a homomorphism. If $\phi((\alpha))=1$, then ${\alpha}/{\overline{\alpha}}=1$, so ${\alpha}={\overline{\alpha}}$. This means ${\alpha}\in F$ and hence $(\alpha)\in P_F$. Therefore, the kernel is $P_F$. As $K=F(\sqrt{-\beta})$ with a totally positive element $\beta$ we have $\phi((\sqrt{-\beta}))=-1$, so $\phi$ is surjective. Hence $[P_{K}^{H}:P_{F}]=|{\text{\normalfont{Im}}}(\phi)|$ equals $2$. \[lemext\] Let $K$ be a non-normal quartic CM field with real quadratic Then we have $[I_K:I^{H}_{K}P_K]\leq[I_{K^r}:I_{0}(\Phi^r)]$. Moreover, we have $[I_{K^r}: I^{H'}_{K^r}P_{K^r}]\leq[I_{K^r}:I_{0}(\Phi^r)]$, where $H'={\text{\normalfont{Gal}}}(K^r/F^r)$. To prove the first assertion, we show that the kernel of the map ${\text{\normalfont{N}}}_{\Phi}:I_K\rightarrow I_{K^r}/I_{0}(\Phi^r)$ is contained in $I^{H}_{K}P_K$. For any ${\mathfrak{a}}\in I_K$, we can compute (see [@goro2 (3.1)]) $${\text{\normalfont{N}}}_{\Phi^{r}}{\text{\normalfont{N}}}_{\Phi}({\mathfrak{a}})={\text{\normalfont{N}}}_{K/{\mathbb{Q}}}({\mathfrak{a}})\frac{{\mathfrak{a}}}{\overline{{\mathfrak{a}}}}.$$ Suppose ${\text{\normalfont{N}}}_{\Phi}({\mathfrak{a}})\in I_{0}(\Phi^r)$. Then $N_{K/{\mathbb{Q}}}({\mathfrak{a}})\frac{{\mathfrak{a}}}{\overline{{\mathfrak{a}}}}=(\alpha)$, where $\alpha\in K^{\times}$ and $\alpha\overline{\alpha}=N_{K^r/{\mathbb{Q}}}(N_{\Phi}({\mathfrak{a}}))=N_{K/{\mathbb{Q}}}({\mathfrak{a}})^2\in{\mathbb{Q}}$. So $\frac{{\mathfrak{a}}}{\overline{{\mathfrak{a}}}}=(\beta)$, where $\beta={\text{\normalfont{N}}}_{K/{\mathbb{Q}}}({\mathfrak{a}})^{-1}\cdot\alpha$, and hence $\beta\overline{\beta}=1$. There is a $\gamma\in K^{\times}$ such that $\beta=\frac{\overline{\gamma}}{\gamma}$ (this is a special case of Hilbert’s Theorem 90, but can be seen directly by taking $\gamma = \overline{\epsilon} + \overline{\beta}\epsilon$ for any $\epsilon\in {K}$ with $\gamma\neq 0$.). Thus we have ${\mathfrak{a}}=\overline{\gamma{\mathfrak{a}}}\cdot(\frac{1}{\gamma})\in I^{H}_{K}P_{K}$ and therefore $[I_K:I^{H}_{K}P_K]\leq[I_{K^r}:I_{0}(\Phi^r)]$. Now we prove the second assertion. By [@goro2 (3.2)], we have $${\text{\normalfont{N}}}_{\Phi}{\text{\normalfont{N}}}_{\Phi^r}({\mathfrak{b}})={\text{\normalfont{N}}}_{K^r/{\mathbb{Q}}}({\mathfrak{b}})\frac{{\mathfrak{b}}}{\overline{{\mathfrak{b}}}}.$$ Suppose ${\mathfrak{b}}\in I_{0}(\Phi^r)$. Then $N_{K^r/{\mathbb{Q}}}({\mathfrak{b}})\frac{{\mathfrak{b}}}{\overline{{\mathfrak{b}}}}=(\alpha)$, where $\alpha\in {K^r}^{\times}$ and $\alpha\overline{\alpha}=N_{\Phi}(N_{K^r/{\mathbb{Q}}}({\mathfrak{b}}))$ $=N_{K^r/{\mathbb{Q}}}({\mathfrak{b}})^2\in{\mathbb{Q}}$. We finish the proof of ${\mathfrak{b}}\in I^{H'}_{K^r}P_{K^r}$ exactly as above. Now let us recall and prove Proposition \[prop2\]. [prop2]{} Let $K$ be a (non-biquadratic) quartic CM field with the real quadratic subfield $F$. Assuming $I_{0}(\Phi^{r})=I_{K^r}$, we have $h^{*}_{K}=2^{t_K-1}$, where $t_K$ is the number of ramified primes in $K/F$. Moreover, we have $h^{*}_{K^r}=2^{t_{K^r}-1}$, where $t_{K^r}$ is the number of ramified primes Since $\mu_K=\{\pm 1\}$, by Lemma \[lem1\], we have $h^{*}_K=2^{t_{K}-1}[I_K:I_{K}^{H}P_K]$. We showed in Lemma \[lemext\] that, under the assumption $I_{0}(\Phi^{r})=I_{K^{r}}$, the quotient $I_K/I_{K}^{H}P_K$ is trivial. Therefore, we have $h^{*}_{K}=2^{t_K-1}$. Similarly, by Lemma \[lem1\], we have $h^{*}_{K^r}=2^{t_{K^r}-1}[I_{K^r}:I_{K^r}^{H^r}P_{K^r}]$ and hence $h^{*}_{K^r}=2^{t_{K^r}-1}$ follows from Lemma \[lemext\]. The next step is to use the following bound from analytic number theory. Let $d_M$ denote the discriminant of a number field $M$. \[louboutin lower bound\] (Louboutin [@loub2], Remark 27 (1)) Let $N$ be the normal closure of a non-normal quartic CM field $K$ with Galois group $D_4$. Assume $d_{N}^{1/8}\geq222$. Then $$\label{rel.cl.2}\qquad \qquad \quad h^{*}_{K}\geq\frac{2\sqrt{d_{K}/d_{F}}}{\sqrt{e}\pi^{2}(\log(d_{K}/d_{F})+0.057)^{2}}. \qquad \qquad \qquad \qquad\, \qed$$ \[effectiveness\] Suppose $I_{0}(\Phi^{r})=I_{K^{r}}$. Then we have $d_K/d_F\leq2\cdot10^{15}$. Let $$f(D) = \frac{2\sqrt{D}}{\sqrt{e}\pi^2(\log(D)+0.057)^2}\quad \text{and}\quad g(t)=2^{-t+1}f(p_tp_{t+1}\Delta^2_t),$$ where $p_j$ is the $j$-th prime and $\Delta_k=\prod^{k}_{j=1}p_j$. Here, if $D = d_K/d_F$, then $f$ is the right hand side of the inequality (\[rel.cl.2\]) in Proposition \[louboutin lower bound\]. The quotient $d_{K}/d_{F}$ is divisible by the product of ramified primes so $d_{K}/d_{F}\geq\Delta_{t_K}$. On the other hand, the function $f$ is monotonically increasing for $D>52$, so if $t_K\geq4$ then $f(d_{K}/d_{F})\geq f(\Delta_{t_K})$. Therefore, by Proposition \[prop2\], we get that if $I_{0}(\Phi^{r})=I_{K^{r}}$, then $$\label{lower bd. with delta} {2^{t_K-1}}\geq f(d_{K}/d_{F})\geq f(\Delta_{t_K})$$ and hence $g(t)\leq 1$. The function $g$ is monotonically increasing for $t\geq 4$ and is greater than $1$ if $t_K>14$. Therefore, we get $t_K\leq 14$ and $h^{*}_{K} \leq 2^{13}$, hence $d_K/d_F< 2\cdot10^{15}$. The bound that we get in Proposition \[effectiveness\] is unfortunately too large to list all the fields. In the following section we study ramification of primes in $N/{\mathbb{Q}}$ and find a sharper upper bound for $d_{K^r}/d_{F^r}$, see Proposition \[t&lt;6\]. Almost all ramified primes are inert in $F$ and $F^r$ {#ramification} ----------------------------------------------------- In this section, under the assumption $I_{0}(\Phi^{r})=I_{K^{r}}$, we study the ramification behavior of primes in $N/{\mathbb{Q}}$, and prove that almost all ramified primes are inert in $F^r$. Thus $d_{K^r}/d_{F^r}$ grows as the square of the product of such ramified primes and we get a lower bound on $f(d_{K^r}/d_{F^r})$ of (\[lower bd. with delta\]) that grows even faster than what we have had before, so, in the following section \[betterupper\], we obtain a better upper bound . In this section, we prove the following theorem. \[Fr\] Let $K$ be a non-biquadratic quartic CM field of type $\Phi$ and $F$ be its quadratic subfield. Suppose $I_{0}(\Phi^{r})=I_{K^r}$. Then $F={\mathbb{Q}}(\sqrt{p})$ and $F^r={\mathbb{Q}}(\sqrt{q})$, are prime numbers with $q\not\equiv3\ ({\text{\normalfont{mod}}}\ 4)$ and $(p/q)=(q/p)=1$. Moreover, all the ramified primes (distinct from the ones lying above $p$ and $q$) in $K^r/F^r$ are inert in $F$ and $F^r$. We begin the proof with exploring the ramification behavior of primes in $N/{\mathbb{Q}}$, under the assumption $I_0(\Phi^r)=I_{K^r}$. ### Ramification of primes in $N/{\mathbb{Q}}$ \[inertia gp. not V4\] Let $M/L$ be a Galois extension of number fields and ${\mathfrak{q}}$ be a prime over an odd prime ideal ${\mathfrak{p}}$ (that is, the prime ${\mathfrak{p}}$ lies over an odd prime in ${\mathbb{Q}}$) of $L$. Then there is no surjective homomorphism from a subgroup of $I_{\mathfrak{q}}$ to a Klein four group $V_4$. For an odd prime ideal ${\mathfrak{p}}$ in $L$, suppose that there is a surjective homomorphism from a subgroup of $I_{\mathfrak{p}}$ to $V_4$. In other words, suppose a prime of $F$ over ${\mathfrak{p}}$ is totally ramified in a biquadratic intermediate extension $E/F$ of $M/L$. The biquadratic intermediate extension $E/F$ has three quadratic intermediate extensions $E_i = F(\sqrt{\alpha_i})$ for $i=1,\ 2,\ 3$. Without loss of generality, take ${\text{\normalfont{ord}}}_{\mathfrak{p}}(\alpha_i)\in\{0,1\}$ for each $i$. Note ${\mathcal{O}}_{E_i}$ contains ${\mathcal{O}}_F[\sqrt{\alpha_i}]$ of relative discriminant $4\alpha_i$ over ${\mathcal{O}}_F$. Since ${\mathfrak{p}}$ is odd, this implies that the relative discriminant $D_i$ of ${\mathcal{O}}_{E_i}$ has ${\text{\normalfont{ord}}}_{\mathfrak{p}}(D_i) = {\text{\normalfont{ord}}}_{\mathfrak{p}}(\alpha_i)$. At the same time, $E_3 = F(\sqrt{\alpha_1\alpha_2})$, so ${\mathfrak{p}}$ ramifies in $E_i$ for an even number of $i$’s. In particular, ${\mathfrak{p}}$ is not totally ramified in $E/F$. \[ram1\] Let $K$ be a non-biquadratic quartic CM field of type $\Phi$ with the real quadratic subfield $F$ and let $K^r$ be its reflex field with the quadratic subfield $F^r$. Then the following assertions hold. - If a prime $p$ is ramified in both $F$ and $F^r$, then it is totally ramified in $K/{\mathbb{Q}}$ and $K^r/{\mathbb{Q}}$. - If an odd prime $p$ is ramified in $F$ (in $F^r$, respectively) as well as in $F_+$, splits in $F^r$ (in $F$, respectively). Moreover, at least one of the primes above $p$ is ramified in $K^r/F^r$ (in $K/F$, respectively). The statements (i) and (ii) are clear from Table \[table\] on page . Alternatively, one can also prove the statements as follows: - Let ${\mathfrak{p}}_{N}$ be a prime of $N$ above $p$ that is ramified in both $F/{\mathbb{Q}}$ and $F^r/{\mathbb{Q}}$. Then the maximal unramified subextension of $N/{\mathbb{Q}}$ is contained in $F_+$. Therefore, the inertia group of ${\mathfrak{p}}_{N}$ contains ${\text{\normalfont{Gal}}}(N/F_+)=\langle y \rangle$. By computing ramification indices in the diagram of subfields one by one, we see that the prime $p$ is totally ramified in $K$ and $K^r$. - Let $p$ be an odd prime that is ramified in $F/{\mathbb{Q}}$ and $F_+/{\mathbb{Q}}$ and ${\mathfrak{p}}_{N}$ be a prime above $p$ in $N$. The inertia group of an odd prime cannot be a biquadratic group by Lemma \[inertia gp. not V4\], so ${\text{\normalfont{I}}}_{{\mathfrak{p}}_N}$ is a proper subgroup of ${\text{\normalfont{Gal}}}(N/F^r)$. Since ${\text{\normalfont{I}}}_{{\mathfrak{p}}_N}$ is a normal subgroup in ${\text{\normalfont{D}}}_{{\mathfrak{p}}_N}$, the group ${\text{\normalfont{D}}}_{{\mathfrak{p}}_N}$ cannot be the full Galois group ${\text{\normalfont{Gal}}}(N/{\mathbb{Q}})$. is a proper subgroup of ${\text{\normalfont{Gal}}}(N/F^r)$ and hence $p$ splits in $F^r$. Moreover, is ramified in $F$, hence in $K$, hence in $K^r$, at least one of the primes above $p$ in $F^r$ is ramified in $K^r$. Since $F$ and $F^r$ are symmetric in $N/{\mathbb{Q}}$, the same argument holds for $F^r$ as well. \[ram2\] Assuming $I_{0}(\Phi^{r})=I_{K^{r}}$, if $K^r$ has a prime ${\mathfrak{p}}$ of prime norm $p$ then $F={\mathbb{Q}}(\sqrt{p})$. By assumption, we have $${\text{\normalfont{N}}}_{\Phi^{r}}({\mathfrak{p}})=(\alpha)\ \text{such that}\ \alpha\overline{\alpha}={\text{\normalfont{N}}}_{K^r/{\mathbb{Q}}}({\mathfrak{p}})=p.$$ Since $\overline{{\mathfrak{p}}}={\mathfrak{p}}$, we have $(\alpha)=(\overline{\alpha})$, and so $\alpha=\epsilon\overline{\alpha}$ for a unit $\epsilon$ in ${\mathcal{O}}^{*}_{K}$ with absolute (hence a root of unity). Since $\mu_K=\{\pm1\}$, we get $\alpha^{2}=\pm p$. The case $\alpha^{2}=-p$ is not possible, since $K$ has no imaginary quadratic intermediate field. Hence we and so $\sqrt{p}\in F$. \[totram\]Suppose $I_{0}(\Phi^{r})=I_{K^{r}}$. If $p$ is totally ramified in $K^r/{\mathbb{Q}}$, or splits in $F^r/{\mathbb{Q}}$ and at least one of the primes over $p$ in $F^r$ ramifies in $K^r/F^r$, then $F={\mathbb{Q}}(\sqrt{p})$. \[propo1\] Suppose $I_{0}(\Phi^{r})=I_{K^r}$. Then $F={\mathbb{Q}}(\sqrt{p})$, where $p$ is a rational prime. Suppose that there is an odd prime $p$ that is ramified in $F$. Then $p$ is ramified either in $F$ and $F^r$ or in $F$ and $F_+$. If $p$ is ramified in both $F$ and $F^r$, then by Lemma \[ram1\]-(i), the prime $p$ is totally ramified in $K^r/{\mathbb{Q}}$. If $p$ is ramified in $F$ and $F_+$, then by Lemma \[ram1\]-(ii), the splits in $F^r$ and at least one of the primes over $p$ in $F^r$ ramifies in $K^r/F^r$. In both cases, by Corollary \[totram\], we have $F={\mathbb{Q}}(\sqrt{p})$. Therefore, if an odd prime $p$ is ramified in $F$, then we have $F={\mathbb{Q}}(\sqrt{p})$. If no odd prime ramifies in $F$ then the only prime that ramifies in $F$ is $2$, and so we \[ram3\] Suppose $I_{0}(\Phi^{r})=I_{K^{r}}$. Then the following assertions are true. - If a rational prime $l$ is unramified in both $F/{\mathbb{Q}}$ and $F^r/{\mathbb{Q}}$, but is ramified or $K^r/{\mathbb{Q}}$, then all primes above $l$ in $F$ and $F^r$ are ramified in $K/F$ and $K^{r}/F^{r}$ and $l$ is inert in $F^r$. - If $F={\mathbb{Q}}(\sqrt{p})$ with a prime number $p\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$, then $2$ is inert in $F^r$. <!-- --> - It follows from Table \[table\] for except the statement that $l$ is inert in $F^r$. Suppose that $l$ splits in $F^r$. Then by Corollary \[totram\], we have $\sqrt{l}\in F$, but we assumed that $l$ is unramified in $F$. Therefore, the prime $l$ is inert in $F^r$. - If $2$ is ramified in $F^r$, then by Lemma \[ram1\]-(i), the prime $2$ is totally ramified in $K$ and $K^r$. If $2$ splits in $F^r$, then at least one of the primes in $F^r$ above $2$ ramifies in $K^r$ since $2$ ramifies in $K/{\mathbb{Q}}$. Therefore, by Corollary \[totram\], in both cases we have $F={\mathbb{Q}}(\sqrt{2})$, contradiction. Hence the prime $2$ is inert in $F^r$. ### Equality of $t_K$ and $t_{K^r}$ In the previous section, we proved that the primes that are unramified in $F$ , but are ramified in $K^r/F^r$ are inert in $F^r$. Thus these primes contribute to the number of ramified primes $t_{K^r}$ in $K^r/F^r$ with one prime, on the other hand they contribute to $t_K$ with at least one prime and exactly two if the prime splits in $F^r/{\mathbb{Q}}$. So if we could prove $t_K=t_{K^r}$, then that would approximately say that all such primes are inert in both $F$ and $F^r$. \[rel.cl.eq.\] (Shimura, [@goro2 Proposition A.7.]) Let the notation be as above. Then, we have $h^{*}_K = h^{*}_{K^r}$. The idea of the proof is to first show $$\label{zeta}\zeta_K(s)/\zeta_F(s)=\zeta_{K^r}(s)/\zeta_{F^r}(s)$$ and then use the class number formula. Louboutin [@loub5 Theorem A] shows (\[zeta\]) by writing the Dedekind zeta functions of $K$, $K^r$, $F$ and $F^r$ as a product of Artin $L$-functions and finding relations between these combinations of $L$-functions (see, [@loub5 Theorem A]). We can also get this equality by comparing the local factors of the Euler products of the Dedekind $\zeta$-functions of the fields. By using Table \[table\], we see that each ramified prime in $N/{\mathbb{Q}}$ has the same factors in the Euler products of the quotients of the Dedekind $\zeta$-functions on both sides of (\[zeta\]). As an example, we take a rational with ramification type (6) in Table \[table\], where the local factors for $p$ of the Dedekind $\zeta$-functions are as follows: $$\begin{aligned} \zeta_K(s)_{p}&= \frac{1}{1-{\text{\normalfont{N}}}{{\mathfrak{p}}_{K,1}}^{-s}}\cdot\frac{1}{1-{\text{\normalfont{N}}}{{\mathfrak{p}}_{K,y}}^{-s}}=\frac{1}{1-(p^2)^{-s}}\cdot \frac{1}{1-{p}^{-s}},\\ \zeta_F(s)_{p}&= \frac{1}{1-{\text{\normalfont{N}}}{{\mathfrak{p}}_{F,1}}^{-s}}\cdot \frac{1}{1-{\text{\normalfont{N}}}{{\mathfrak{p}}_{F,y}}^{-s}}=\bigg ( \frac{1}{1-{p}^{-s}}\bigg )^2,\\ \zeta_{K^r}(s)_{p}&= \frac{1}{1-{\text{\normalfont{N}}}{{\mathfrak{p}}_{K^r,1}}^{-s}}=\frac{1}{1-(p^2)^{-s}},\\ \zeta_{F^r}(s)_{p}&= \frac{1}{1-{\text{\normalfont{N}}}{{\mathfrak{p}}_{K^r,1}}^{-s}}= \frac{1}{1-{p}^{-s}}.\end{aligned}$$ So for such a prime, we get $$\zeta_K(s)_{p}/\zeta_F(s)_{p} = \frac{1}{1+{p}^{-s}} = \zeta_{K^r}(s)_{p}/\zeta_{F^r}(s)_{p}.$$Similarly, by using Table 3.5.1 in [@goren], we can get this equality for the unramified primes as well. The analytic class number formula at $s = 0$ (see, [@wash Chapter 4]) says that the Dedekind zeta function $\zeta_M(s)$ of an algebraic number field $M$ has a zero at $s=0$ and the derivative of $\zeta_M(s)$ at $s=0$ has the value $$-\frac{h_M\cdot R_M}{ w_M},$$ where $h_M$ is the class number; $R_M$ is the regulator; and $w_M$ is the order of the group of roots of unity $\mu_M$. Since $w_K=2=w_F$ and $R_K=2R_F$ (see Washington, [@wash Proposition 4.16]), the analytic class number formula at $s = 0$ gives $$\begin{aligned} \lim_{s\rightarrow 0} \frac{ \zeta_K(s)}{\zeta_F(s)} =2h^*_{K}.\end{aligned}$$ Therefore, the result follows by the identity (\[zeta\]). \[equal\] Assuming $I_{0}(\Phi^{r})=I_{K^{r}}$, we have $t:=t_{K}=t_{K^r}$. By Proposition \[prop2\], we have $h^{*}_{K}=2^{t_K-1}$ and $h^{*}_{K^r}=2^{t_{K^r}-1}$. Then by Proposition \[rel.cl.eq.\], we get $t_{K}=t_{K^r}$. ### Proof of Proposition \[Fr\] [Fr]{} Let $K$ be a non-biquadratic quartic CM field of type $\Phi$ and $F$ be its quadratic subfield. Suppose $I_{0}(\Phi^{r})=I_{K^r}$. Then $F={\mathbb{Q}}(\sqrt{p})$ and $F^r={\mathbb{Q}}(\sqrt{q})$, are prime numbers with $q\not\equiv3\ ({\text{\normalfont{mod}}}\ 4)$ and $(p/q)=(q/p)=1$. Moreover, all the ramified primes (distinct from the ones lying above $p$ and $q$) in $K^r/F^r$ are inert in $F$ and $F^r$. We first prove that if a prime $l$ ramifies in both $F$ and $F^r$, then it is equal where $F={\mathbb{Q}}(\sqrt{p})$. Indeed, by Lemma \[ram1\]-(i), the prime $l$ is totally ramified in $K^r/{\mathbb{Q}}$ and hence by Corollary \[totram\], we get $F={\mathbb{Q}}(\sqrt{l})$, so $l=p$. Now we see that there are four types of prime numbers that ramify in $N/{\mathbb{Q}}$: - The prime $p$, which is ramified in $F$ and possibly in $F^r$. - The primes that are unramified in $F$, but ramified in $F^r$, say $q_1,\ \cdots,\ q_s$. - The primes that are unramified in $F$ and $F^r$, but ramified in $K$, say $r_1,\ \cdots,\ r_m$. - If $p\equiv\, 3\ ({\text{\normalfont{mod}}}\, 4)$, then $2\neq p$ is ramified in $F$ and is inert in $F^r$ by Let $i_2=1$ if $p \equiv\, 3\ ({\text{\normalfont{mod}}}\, 4)$, and $i_2=0$ if $p \not\equiv\, 3\ ({\text{\normalfont{mod}}}\, 4)$. We will compute the contribution of each ramification type to the number of ramified primes $t_K$ in $K/F$ and $t_{K^r}$ in $K^r/F^r$. Let $f_{p}$ and $f^{r}_{p}$ be the contributions of the primes over $p$ to $t_K$ and $t_{K^r}$, respectively. We claim $t_K\geq f_{p}+ s+ m + i_2$ with equality only if all primes of type (III) are inert in $F$ and $t_{K^r}=f^{r}_{p}+m+i_2$. Proof of the claim: By including Lemma \[ram2\], we see that for $i=1,\cdots,s$ *exactly* one of the primes above $q_i$ in $F$ ramifies in $K/F$ and the unique prime in $F^r$ does not ramify in $K^r/F^r$. By , we see that for $j=1,\cdots, m$ the prime $r_j$ is inert in $F^r$ so contributes with *exactly* one prime to $t_{K^r}$, and with *at least* one prime to $t_K$ and with exactly one if and only if $r_j$ is inert in $F/{\mathbb{Q}}$. If $p\equiv3\ ({\text{\normalfont{mod}}}\ 4)$, then by Lemma \[ram3\]-(ii), the prime $2$ is inert in $F^r$. As furthermore $2$ is ramified in $F$ and $F\not\cong{\mathbb{Q}}(\sqrt{2})$, the prime $2$ has the decomposition (14) in Table \[table\], so it contributes *exactly* with one prime to So we get $t_K\geq f_{p}+ s+ m + i_2$ with equality if and only if all primes of type (III) are inert in $F$ and $t_{K^r}=f^{r}_{p}+m+i_2$, which proves the claim. We observe that $s>0$ holds. Indeed, if $s=0$, then all primes that ramify in $F^r$ also ramify in $F$. Hence $d_{F^r}$ divides $d_F$, which is equal to $p$ if $p\equiv 1\ ({\text{\normalfont{mod}}}\ 4)$ and $4p$ otherwise. So $F^r\cong F$, a contradiction. If $p$ ramifies in both $F$ and $F^r$, then by Lemma \[ram1\]-(i), we have $f_p=f^{r}_{p}=1$. The same is true if $p$ is of type (14) of Table \[table\]. By Corollary \[equal\], we have $t_K=t_{K^r}$, so in this case $m+i_2 \geq s+ m + i_2$, so $s=0$, a contradiction. Therefore, the is not ramified in $F^r$ and is not of type (14), leaving only the possibility $(q/p)=1$. By Table \[table\], we see that $f^{r}_{p}-f_{p}=1$. Hence $t_K=t_{K^r}$ implies that all primes of type (III) are inert in $F$ In particular, since $p$ is unramified in $F^r$, we get $F^r={\mathbb{Q}}(\sqrt{q})$ for a prime $q\not\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$. Moreover, Table \[table\] implies $(p/q)=1$. \[key\] -------- --------------------------- --------------------------- --------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------- ------------------------------------------------- ----------------------------------------------------- ---------------------------------------------------------------------------------- ---------------------------------------------- ----------------- Case ${\text{\normalfont{I}}}$ ${\text{\normalfont{D}}}$ [decomp. of $p$ in $N$]{} [decomp. of $p$ in $K$]{} [decomp. of $p$ in $F$]{} [decomp. of $p$ in $F_+$]{} [decomp. of $p$ in $F^r$]{} [decomp. of $p$ in $K^r$]{} $N_{\Phi^r}({\mathfrak{p}}_{K^{r},1})$ $\sqrt{p}\in F$ (1)\* $\langle y^2\rangle$ $\langle y^2\rangle$ ${\mathfrak{p}}^{2}_{N,1} {\mathfrak{p}}^{2}_{N,x} {\mathfrak{p}}^{2}_{N,y} {\mathfrak{p}}^{2}_{N,xy}$ ${\mathfrak{p}}^{2}_{K,1} {\mathfrak{p}}^{2}_{K,y}$ ${\mathfrak{p}}_{F,1} {\mathfrak{p}}_{F,y}$ ${\mathfrak{p}}_{F_+,1} {\mathfrak{p}}_{F_+,y}$ ${\mathfrak{p}}_{F^{r},1} {\mathfrak{p}}_{F^{r},y}$ ${\mathfrak{p}}^{2}_{K^{r},1} {\mathfrak{p}}^{2}_{K^{r},y}$ ${\mathfrak{p}}_{K,1} {\mathfrak{p}}_{K,y}$ (2) $\langle y^2\rangle$ $\langle y\rangle$ ${\mathfrak{p}}^{2}_{N,1} {\mathfrak{p}}^{2}_{N,y}$ ${\mathfrak{p}}^{2}_{K,1}$ ${\mathfrak{p}}_{F,1}$ ${\mathfrak{p}}_{F_+,1} {\mathfrak{p}}_{F_+,y}$ ${\mathfrak{p}}_{F^{r},1}$ ${\mathfrak{p}}^{2}_{K^{r},1}$ $p$ (3) $\langle y^2\rangle$ $\langle x,\ y^2\rangle$ ${\mathfrak{p}}^{2}_{N,1} {\mathfrak{p}}^{2}_{N,y}$ ${\mathfrak{p}}^{2}_{K,1} {\mathfrak{p}}^{2}_{K,y}$ ${\mathfrak{p}}_{F,1} {\mathfrak{p}}_{F, y}$ ${\mathfrak{p}}_{F_+,1}$ ${\mathfrak{p}}_{F^{r},1}$ ${\mathfrak{p}}^{2}_{K^{r},1}$ $p$ (4)\* $\langle y^2\rangle$ $\langle xy,\ y^2\rangle$ ${\mathfrak{p}}^{2}_{N,1} {\mathfrak{p}}^{2}_{N,y}$ ${\mathfrak{p}}^{2}_{K,1}$ ${\mathfrak{p}}_{F,1}$ ${\mathfrak{p}}_{F_+,1}$ ${\mathfrak{p}}_{F^{r},1} {\mathfrak{p}}_{F^{r},y}$ ${\mathfrak{p}}^{2}_{K^{r},1} {\mathfrak{p}}^{2}_{K^{r},y}$ ${\mathfrak{p}}_{K,1}$ (a) $\langle x\rangle$ $\langle x\rangle$ ${\mathfrak{p}}^{2}_{N,1} {\mathfrak{p}}^{2}_{N,y}{\mathfrak{p}}^{2}_{N,y^2}{\mathfrak{p}}^{2}_{N,y^3}$ ${\mathfrak{p}}_{K,1}{\mathfrak{p}}^{2}_{K,y}{\mathfrak{p}}_{K,y^2}$ ${\mathfrak{p}}_{F,1}{\mathfrak{p}}_{F,y}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{2}_{K^{r},1}{\mathfrak{p}}^{2}_{K^{r},y^2}$ ${\mathfrak{p}}_{K,1}{\mathfrak{p}}_{K,y}$ (b) $\langle xy^2\rangle$ $\langle xy^2\rangle$ ${\mathfrak{p}}^{2}_{N,1}{\mathfrak{p}}^{2}_{N,y}{\mathfrak{p}}^{2}_{N,y^2}{\mathfrak{p}}^{2}_{N,y^3}$ ${\mathfrak{p}}^{2}_{K,1}{\mathfrak{p}}_{K,y}{\mathfrak{p}}_{K,y^3}$ ${\mathfrak{p}}_{F,1}{\mathfrak{p}}_{F,y}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{2}_{K^{r},1}{\mathfrak{p}}^{2}_{K^{r},y}$ ${\mathfrak{p}}_{K,1}{\mathfrak{p}}_{K,y^3}$ (a) $\langle x\rangle$ $\langle x,\ y^2\rangle$ ${\mathfrak{p}}^{2}_{N,1}{\mathfrak{p}}^{2}_{N,y}$ ${\mathfrak{p}}_{K,1}{\mathfrak{p}}^{2}_{K,y}$ ${\mathfrak{p}}_{F,1}{\mathfrak{p}}_{F,y}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{2}_{K^{r},1}$ $p$ (b) $\langle xy^2\rangle$ $\langle x, y^2\rangle$ ${\mathfrak{p}}^{2}_{N,1}{\mathfrak{p}}^{2}_{N,y}$ ${\mathfrak{p}}^{2}_{K,1}{\mathfrak{p}}_{K,y}$ ${\mathfrak{p}}_{F,1}{\mathfrak{p}}_{F,y}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{2}_{K^{r},1}$ $p$ (a) $\langle xy\rangle$ $\langle xy\rangle$ ${\mathfrak{p}}^{2}_{N,1}{\mathfrak{p}}^{2}_{N,y}{\mathfrak{p}}^{2}_{N,y^2}{\mathfrak{p}}^{2}_{N,y^3}$ ${\mathfrak{p}}^{2}_{K,1}{\mathfrak{p}}^{2}_{K,y^3}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}_{F^{r},1}{\mathfrak{p}}_{F^{r},y}$ ${\mathfrak{p}}^{2}_{K^{r},1}{\mathfrak{p}}_{K^{r},y}{\mathfrak{p}}_{K^{r},y^3}$ ${\mathfrak{p}}_{K,1}{\mathfrak{p}}_{K,y^3}$ (b) $\langle xy^3\rangle$ $\langle xy^3\rangle$ ${\mathfrak{p}}^{2}_{N,1}{\mathfrak{p}}^{2}_{N,y}{\mathfrak{p}}^{2}_{N,y^2}{\mathfrak{p}}^{2}_{N,y^3}$ ${\mathfrak{p}}^{2}_{K,1}{\mathfrak{p}}^{2}_{K,y}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}_{F^{r},1}{\mathfrak{p}}_{F^{r},y}$ ${\mathfrak{p}}_{K^{r},1}{\mathfrak{p}}^{2}_{K^{r},y}{\mathfrak{p}}_{K^{r},y^2}$ ${\mathfrak{p}}^{2}_{K,1}$ (a) $\langle xy\rangle$ $\langle xy, y^2\rangle$ ${\mathfrak{p}}^{2}_{N,1}{\mathfrak{p}}^{2}_{N,y}$ ${\mathfrak{p}}^{2}_{K,1}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}_{F^{r},1}{\mathfrak{p}}_{F^{r},y}$ ${\mathfrak{p}}^{2}_{K^{r},1}{\mathfrak{p}}_{K^{r},y}$ ${\mathfrak{p}}_{K,1}$ (b) $\langle xy^3\rangle$ $\langle xy, y^2\rangle$ ${\mathfrak{p}}^{2}_{N,1}{\mathfrak{p}}^{2}_{N,y}$ ${\mathfrak{p}}^{2}_{K,1}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}_{F^{r},1}{\mathfrak{p}}_{F^{r},y}$ ${\mathfrak{p}}_{K^{r},1}{\mathfrak{p}}^{2}_{K^{r},y}$ $p$ (9) $\langle y\rangle$ $\langle y\rangle$ ${\mathfrak{p}}^{4}_{N,1}{\mathfrak{p}}^{4}_{N,y}$ ${\mathfrak{p}}^{4}_{K,1}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}_{F_+,1}{\mathfrak{p}}_{F_+,y}$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{4}_{K^{r},1}$ ${\mathfrak{p}}^{2}_{K,1}$ (10) $\langle y\rangle$ $G$ ${\mathfrak{p}}^{4}_{N,1}$ ${\mathfrak{p}}^{4}_{K,1}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}_{F_+,1}$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{4}_{K^{r},1}$ ${\mathfrak{p}}^{2}_{K,1}$ (11)\* $\langle x, y^2\rangle$ $\langle x, y^2\rangle$ ${\mathfrak{p}}^{4}_{N,1}{\mathfrak{p}}^{4}_{N,y}$ ${\mathfrak{p}}^{2}_{K,1}{\mathfrak{p}}^{2}_{K,y}$ ${\mathfrak{p}}_{F,1}{\mathfrak{p}}_{F,y}$ ${\mathfrak{p}}^{2}_{F_+,1}$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{4}_{K^{r},1}$ ${\mathfrak{p}}_{K,1}{\mathfrak{p}}_{K,y}$ (12)\* $\langle x, y^2\rangle$ $G$ ${\mathfrak{p}}^{4}_{N,1}$ ${\mathfrak{p}}^{2}_{K,1}$ ${\mathfrak{p}}_{F,1}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{4}_{K^{r},1}$ ${\mathfrak{p}}_{K,1}$ (13) $\langle xy, y^2\rangle$ $\langle xy, y^2\rangle$ ${\mathfrak{p}}^{4}_{N,1}{\mathfrak{p}}^{4}_{N,y}$ ${\mathfrak{p}}^{4}_{K,1}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}^{2}_{F_+,1}$ ${\mathfrak{p}}_{F^{r},1}{\mathfrak{p}}_{F^{r},y}$ ${\mathfrak{p}}^{2}_{K^{r},1}{\mathfrak{p}}^{2}_{K^{r},y}$ ${\mathfrak{p}}^{2}_{K,1}$ (14) $\langle xy,y^2\rangle$ $G$ ${\mathfrak{p}}^{4}_{N,1}$ ${\mathfrak{p}}^{4}_{K,1}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}_{F_+,1}^2$ ${\mathfrak{p}}_{F^{r},1}$ ${\mathfrak{p}}^{2}_{K^{r},1}$ $p$ (15) $G$ $G$ ${\mathfrak{p}}^{8}_{N,1}$ ${\mathfrak{p}}^{4}_{K,1}$ ${\mathfrak{p}}^{2}_{F,1}$ ${\mathfrak{p}}^{2}_{F_+,1}$ ${\mathfrak{p}}^{2}_{F^{r},1}$ ${\mathfrak{p}}^{4}_{K^{r},1}$ ${\mathfrak{p}}^{2}_{K,1}$ -------- --------------------------- --------------------------- --------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------- ------------------------------------------------- ----------------------------------------------------- ---------------------------------------------------------------------------------- ---------------------------------------------- ----------------- : This table lists all $19$ pairs $({\text{\normalfont{I}}},{\text{\normalfont{D}}})$ where $1\neq {\text{\normalfont{I}}}\triangleleft {\text{\normalfont{D}}}\leq D_4 = \langle x,\ y \rangle$ and ${\text{\normalfont{D}}}/{\text{\normalfont{I}}}$ is cyclic, partitioned into $15$ conjugacy classes (1) – (15). In particular, it contains all possible inertia and decomposition groups of ramified primes of $N$. This table is a corrected subset of [@goren Table 3.5.1]. We restricted to ${\text{\normalfont{I}}}\neq 1$, added the case 8-(b), which is missing in [@goren Table 3.5.1], and corrected the type norm column of some cases. The cases (11) – (15) can only occur for the prime $2$, see Lemma \[inertia gp. not V4\]. If there is a checkmark in the last column, then by , such splitting implies $\sqrt{p}\in F$ (i.e., $F={\mathbb{Q}}(\sqrt{p})$) under the assumption $I_{0}(\Phi^{r})=I_{K^{r}}$. The cases do not occur under the assumption $I_{0}(\Phi^{r})=I_{K^{r}}$ because $p$ is not ramified in $F$ in these cases, but on the other hand $\sqrt{p}\in F$ by Lemma \[ram2\]. \[table:nonlin\] A sharper bound for $d_{K^r}/d_{F^r}$ {#betterupper} ------------------------------------- \[t&lt;6\] Suppose $I_{0}(\Phi^{r})=I_{K^{r}}$ and $d_{N}^{1/8}\geq222$. Then we have $h_{K^r}^{*}\leq2^{5}$ and $d_{K^r}/d_{F^r}\leq3\cdot10^{10}$. Under the assumption $I_{0}(\Phi^{r})=I_{K^{r}}$, in Propositions \[propo1\] and \[Fr\] we proved $F={\mathbb{Q}}(\sqrt{p})$ and $F^r={\mathbb{Q}}(\sqrt{q})$, where $p$ and $q$ are prime numbers. Additionally, we proved that at least one of the ramified primes above $p$ in $F^r$ is ramified in $K^r/F^r$, and the other ramified primes in $K^r/F^r$ are inert in $F^r$, say $r_1,\cdots,r_{t-1}$. Therefore, we have $d_{K^r}/d_{F^r}\geq pqr^2_1\cdots r^2_{t-1}$. Let $$f(D) = \frac{2\sqrt{D}}{\sqrt{e}\pi^2(\log(D)+0.057)^2}\quad \text{and}\quad g(t)=2^{-t+1}f(p_tp_{t+1}\Delta^2_t),$$ where $p_j$ is the $j$-th prime and $\Delta_k=\prod^{k}_{j=1}p_j$. If $D = d_{K^r}/d_{F^r}$, then $h_{K^r} \geq f(D)$ by Proposition \[louboutin lower bound\]. Recall that, by the proof of Proposition \[effectiveness\], the function $f$ is monotonically increasing for $D>52$. Therefore, if $t>3$, then $f(d_{K^r}/d_{F^r})>f(p_tp_{t+1}\Delta^2_t)$. So in that case by Proposition \[prop2\] and Corollary \[equal\], we have $h^{*}_{K^r}=2^{t-1}$, hence we get $g(t)\leq1$. Further, the function $g$ is monotonically increasing for $t\geq4$ and is greater than $1$ for $t=7$. So we get $t\leq 6$. Enumerating the fields {#main} ---------------------- To specify quartic CM fields, we use the following notation of the ECHIDNA database [@echidna]. Given a quartic CM field $K$, be the discriminant of the real quadratic subfield $F$. Write $K=F(\sqrt{\alpha})$ where $\alpha$ is a totally negative element and take $\alpha$ such that $A:=-{\text{\normalfont{Tr}}}_{F/{\mathbb{Q}}}(\alpha)>0$ is minimal and let $B:={\text{\normalfont{N}}}_{F/{\mathbb{Q}}}(\alpha)$. We choose $\alpha$ with minimal $B$ if there is more than one $B$ with the same $A$. We use the triple $[D,A,B]$ to uniquely represent the isomorphism class of the CM field $K\cong{\mathbb{Q}}[X]/(X^4 + AX^2 + B)$. \[thm: non-normallist2\] There exist exactly $63$ isomorphism classes of non-normal quartic CM fields $K$ that have a CM type $\Phi$ satisfying $I_{0}(\Phi^{r})=I_{K^{r}}$, where $K^r$ is the reflex field of $\Phi$. The fields are given by $K\cong{\mathbb{Q}}[X]/(X^4 + AX^2 + B)\supset {\mathbb{Q}}(\sqrt{D})$ where $[D,A,B]$ ranges over $$\begin{aligned} \nonumber &[5,13,41],\ [5,17,61],\ [5,21,109],\ [5,26,149],\ [5,34,269],\ [5,41,389],\\ \nonumber & [8,10,17],\ [8,18,73],\ [8,22,89],\ [8,34,281],\ [8,38,233],\ [13,9,17],\\ \nonumber & [13,18,29],\ [13,29,181],\ [13,41,157],\ [17,5,2],\ [17,15,52],\ [17,46,257],\\ \nonumber & [17,47,548],\ [29,9,13],\ [29,26,53],\ [41,11,20],\ [53,13,29],\ [61,9,5],\\ \nonumber & [73,9,2],\ [73,47,388],\ [89,11,8],\ [97,94,657],\ [109,17,45],\\ \nonumber & [137,35,272],\ [149,13,5],\ [157,25,117],\ [181,41,13],\ [233,19,32],\ [269,17,5],\\ \nonumber & [281,17,2],\ [389,37,245] \end{aligned}$$ with class number $1$; $$\begin{aligned} \nonumber & [5,11,29],\ [5,33,261],\ [5,66,909],\ [8,50,425],\ [8,66,1017],\ [17,25,50],\\ \nonumber & [29,7,5],\ [29,21,45],\ [101,33,45],\ [113,33,18],\ [8,14,41],\ [8,26,137],\\ \nonumber & [12,8,13],\ [12,10,13],\ [12,14,37],\ [12,26,61],\ [12,26,157],\ [44,8,5],\\ \nonumber & [44,14,5],\ [76,18,5],\ [172,34,117],\ [236,32,20] \end{aligned}$$ with class number $2$;   $[257,23,68]$\ with class number $3$;   $[8,30,153],\ [12,50,325],\ [44,42,45]$\ with class number $4$. We start the proof by combining the ramification results into the following explicit form \[construct.1\] Suppose $I_{0}(\Phi^{r})=I_{K^{r}}$. Then there exist prime numbers $p$, $q$, and $s_1< \cdots<s_{u}$ with $u\in\{t_{K^r}-1,\ t_{K^r}-2\}$ such that all of the following hold. We have $F={\mathbb{Q}}(\sqrt{p})$ and $F^r={\mathbb{Q}}(\sqrt{q})$ with $q\not\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$ and $(p/q)=(q/p)=1$. There exists a prime ${\mathfrak{p}}$ lying above $p$ in $F^r$ that ramifies in $K^r$, an odd $j>0$ in ${\mathbb{Z}}$ and a totally positive generator $\pi$ of ${\mathfrak{p}}^j$. Moreover, for exactly one such ${\mathfrak{p}}$ and each such $\pi$ and $j$, we have $K^r\cong{\mathbb{Q}}(\sqrt{-\pi s_1\cdots s_{u}})$. By Proposition \[Fr\], we have $F={\mathbb{Q}}(\sqrt{p})$ and $F^r={\mathbb{Q}}(\sqrt{q})$, and $q$ are prime numbers with $q\not\equiv3\ ({\text{\normalfont{mod}}}\ 4)$ and $(p/q)=(q/p)=1$. There exists a totally positive element $\beta$ in ${F^r}^*$ such that $K^r=F^r(\sqrt{-\beta})$, is uniquely defined (without loss of generality, we can take $\beta$ in ${\mathcal{O}}_{F^r}$). Since ${\mathcal{O}}_{K^r} \supset {\mathcal{O}}_{F^r}[\sqrt{-\beta}] \supset {\mathcal{O}}_{F^r}$, the quotient of the discriminant ideals $\Delta({\mathcal{O}}_{K^r}/{\mathcal{O}}_{F^r})/\Delta({\mathcal{O}}_{F^r}[\sqrt{-\beta}]/{\mathcal{O}}_{F^r})$ is a square ideal in ${\mathcal{O}}_{F^r}$, see Cohen [@cohen pp.79], where $\Delta({\mathcal{O}}_{F^r}[\sqrt{-\beta}]/{\mathcal{O}}_{F^r})=(-4\beta)$. As $\beta$ is unique up to squares, and we can $\beta'\in\beta(F^*)^2$ for each prime ${\mathfrak{l}}$ of ${\mathcal{O}}_{F^r}$, we get $$\label{order} {\text{\normalfont{ord}}}_{{\mathfrak{l}}}((\beta))\equiv \begin{cases} 1\ ({\text{\normalfont{mod}}}\ 2)&\quad \text{if ${\mathfrak{l}}$ is ramified in $K^r/F^r $ and ${\mathfrak{l}}{\hspace{-4pt}\not|\hspace{2pt}}2$,}\\ 0\ ({\text{\normalfont{mod}}}\ 2) &\quad \text{if ${\mathfrak{l}}$ is not ramified in $K^r/F^r $,}\\ 0\ \text{or}\ 1\ ({\text{\normalfont{mod}}}\ 2)&\quad \text{if ${\mathfrak{l}}$ is ramified in $K^r/F^r $ and ${\mathfrak{l}}|2$.} \end{cases}$$ Let ${\mathfrak{l}}_1,\cdots,{\mathfrak{l}}_{t_{K^r}}\subseteq {\mathcal{O}}_{F^r}$ be the primes over the prime numbers $l_1, \cdots, l_{t_{K^r}}$, respectively, that ramify in $K^r/F^r$. Let $n_i>0$ be minimal such that ${{\mathfrak{l}}_i}^{n_i}$ is generated by a totally positive $\lambda_i\in{\mathcal{O}}_{F^r}$. Since $F^r={\mathbb{Q}}(\sqrt{q})$ with prime $q\not\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$, genus theory implies that ${\text{\normalfont{Cl}}}_{F^r}={\text{\normalfont{Cl}}}^+_{F^r}$ has odd order, and so $n_i$ is odd. Let $$\alpha = \prod_{i=1}^{t_{K^r}}\lambda_i^{({\text{\normalfont{ord}}}_{{\mathfrak{l}}_i}((\beta))\ {\text{\normalfont{mod}}}\ 2)}.$$ By proving the following two claims we finish the proof. Claim 1. We have $\alpha/\beta\in ({F^r}^*)^2$. Claim 2. We have $\alpha = \pi s_1\cdots s_u$ with $\pi$, $s_i$ and $u$ as in the statement. *Proof of Claim 1.* We first prove that $({\alpha}/{\beta}) = {(\alpha)}/{(\beta)}$ is a square ideal. Let ${\mathfrak{l}}$ be any prime of $F^r$. If ${\mathfrak{l}}$ is unramified in $K^r/F^r$, then by (\[order\]), we have ${\text{\normalfont{ord}}}_{{\mathfrak{l}}}((\beta)) \equiv 0\ ({\text{\normalfont{mod}}}\ 2)$ so ${\text{\normalfont{ord}}}_{{\mathfrak{l}}}((\alpha)) = 0$. If ${\mathfrak{l}}$ is ramified in $K^r/F^r$, then there such that ${\mathfrak{l}}={\mathfrak{l}}_i$, so ${\text{\normalfont{ord}}}_{{\mathfrak{l}}}((\alpha))\equiv{\text{\normalfont{ord}}}_{{\mathfrak{l}}_i}((\beta))\cdot{\text{\normalfont{ord}}}_{{\mathfrak{l}}_i}((\lambda_i)) \equiv {\text{\normalfont{ord}}}_{{\mathfrak{l}}_i}((\beta))\ ({\text{\normalfont{mod}}}\ 2)$ as $n_i = {\text{\normalfont{ord}}}_{{\mathfrak{l}}_i}((\lambda_i))$ is odd. Therefore, the ideal $(\frac{\alpha}{\beta})$ is a square of an ideal ${\mathfrak{a}}$ in ${\mathcal{O}}_{F^r}$. Thus ${\mathfrak{a}}^2$ is generated by the totally positive $\alpha/\beta$. So the ideal class $[{\mathfrak{a}}]$ is $2$-torsion in ${\text{\normalfont{Cl}}}^+_{F^r}$, which has an odd order, so there is a totally positive element $\mu\in F^r$ that generates ${\mathfrak{a}}$. So $\alpha/\beta = \mu^2\cdot v$ for some $v\in ({\mathcal{O}}^{*}_{F^r})^+$. Since ${\text{\normalfont{Cl}}}_{F^r}={\text{\normalfont{Cl}}}^{+}_{F^r}$, the norm of the fundamental unit $\epsilon$ is negative. Therefore, a unit is totally positive if and only if it is a square . Hence, we have $\alpha/\beta\in ({F^r}^*)^2$. *Proof of Claim 2.* For any given $i$, if $l_i$ is inert in $F^r/{\mathbb{Q}}$, then $n_i=1$ and $\lambda_i=l_i\in{\mathbb{Z}}_{>0}$ is prime. If $l_i$ is not inert in $F^r/{\mathbb{Q}}$ then $l_i\in \{p,\ q\}$, by If $l_i=q$, then ${\mathfrak{l}}_i$ is not ramified in $K^r/F^r$ otherwise by Corollary \[totram\] we get $\sqrt{q}\in F$. So $l_i = p$. Let $\{s_1,\cdots,s_u\} = \{l_i\ :\ l_i\ \text{is inert in}\ F^r/{\mathbb{Q}}\ \text{and ramified in}\ K^r/F^r\ \text{and}\ {\text{\normalfont{ord}}}_{{\mathfrak{l}}_i}((\beta))\equiv 1\ ({\text{\normalfont{mod}}}\ 2)\}$. Then $u\in\{t_{K^r}-1,\ t_{K^r}-2,\ t_{K^r}-3\}$ by (\[order\]). Let $p{\mathcal{O}}_{F^r}={\mathfrak{p}}{\mathfrak{p}}'$. Then we have $\alpha=\pi^{a}\pi'^{a'}\prod_{i=1}^{u} s_{i}^{(1\ {\text{\normalfont{mod}}}\ 2)}$, where $\pi$ and $\pi'$ are totally positive generators of ${\mathfrak{p}}^j$ and ${\mathfrak{p}}'^{j}$ for odd $j$. Here, we have $\prod_{i=1}^{u} s_{i}^{(1\ {\text{\normalfont{mod}}}\ 2)}\in{\mathbb{Z}}$ and $a,\ a'\in\{0,\ 1\}$. If $a = a'$, then $\alpha\in{\mathbb{Z}}$, which leads to contradiction since $K^r$ is non-biquadratic. So for a unique ${\mathfrak{p}}$, we can take $a_1=1$ and $a_2=0$. In particular, we have $u\in\{t_{K^r}-1,\ t_{K^r}-2\}$. Combining Lemma \[construct.1\] and the bound on the discriminant in Proposition \[t&lt;6\], we now have a good way of listing the fields. Next, we need a fast way of eliminating fields from our list if they have CM class number $>1$. The following lemma is a special case of Theorem D in Louboutin [@loub5]. \[quick\] Let $K$ be a non-biquadratic quartic CM field with real quadratic subfield $F$. Let $d_K$ and $d_F$ be the absolute values of the discriminants of $K$ and $F$. Then assuming $I_{0}(\Phi^r)=I_{K^r}$, if a rational prime $l$ totally splits in $K^r/{\mathbb{Q}}$, then $l\geq \frac{\sqrt{d_K/{d_F}^2}}{4}$. Let $l$ be a prime that totally splits in $K^r/{\mathbb{Q}}$. Let $\mathfrak{l}_{K^r}$ be a prime ideal in $K^r$ above $l$. By the assumption $I_{0}(\Phi^r)=I_{K^r}$, there exists $\tau\in K^{\times}$ such that ${\text{\normalfont{N}}}_{\Phi^r}(\mathfrak{l}_{K^r})=(\tau)$ and $\tau\overline{\tau}=l$. Here $\tau\neq\overline{\tau}$, since $\sqrt{l}\not\in K$. Then since ${\mathcal{O}}_{K}\supset{\mathcal{O}}_{F}[\tau]$ and $\Delta({\mathcal{O}}_{F}[\tau]/{\mathcal{O}}_{F}) = (\tau-\overline{\tau})^2$, we have $d_{K}/d^{2}_{F} = {\text{\normalfont{N}}}_{F/{\mathbb{Q}}} (d_{K/F}) = {\text{\normalfont{N}}}_{F/{\mathbb{Q}}}(\Delta({\mathcal{O}}_{K}/{\mathcal{O}}_{F})) \leq {\text{\normalfont{N}}}_{F/{\mathbb{Q}}}((\tau-\overline{\tau})^2)$. Moreover, since $\tau\overline{\tau}=l$, we have $(\tau-\overline{\tau})^2\leq (2\sqrt{l})^2$ for all embeddings of $F$ into ${\mathbb{R}}$, hence $d_{K}/d^{2}_{F} \leq {\text{\normalfont{N}}}_{F/{\mathbb{Q}}}((\tau-\overline{\tau})^2)\leq 16l^2$. Every prime $s_i$ as in Lemma \[construct.1\] divides $\Delta({K^r}/{F^r})$ and so $s_i^2|d_{K^r}$, hence $s_i^4|d_N$. The primes $p$ and $q$ are ramified in $F$ and $F^r$, so $p^4$ and $q^4$ divide the of the normal closure $N$ of degree $8$. Hence $d_N\geq p^{4}q^{4}s^{4}_1\cdots s^{4}_{t-1}$. \[algo2\] **Output:** $[D,\ A,\ B]$ representations of all non-normal quartic CM fields $K$ satisfying $I_{0}(\Phi^r)=I_{K^r}$. - Find all square-free integers smaller than $3\cdot10^{10}$ having at most $8$ prime divisors and find all square-free integers smaller than $222^2$. - Order the prime factors of each of these square-free integers as tuples of primes $(p,q,s_1,\cdots,s_{u})$ with $s_{1}<\cdots <s_u$ in $(u+1)(u+2)$-ways, then take only the tuples satisfying $q\not\equiv3\ ({\text{\normalfont{mod}}}\ 4)$, $(p/q)=(q/p)=1$ and $(p/s_i)=(q/s_i)=-1$ for - For each $(p,q,s_1,\cdots,s_{u})$, let $F^{r}={\mathbb{Q}}(\sqrt{q})$, write $p{\mathcal{O}}_{F^r}={\mathfrak{p}}{\mathfrak{p}}'$, and take $\alpha=\pi\cdot s_1\cdots s_u\in F^{r}$, where $\pi$ is a totally positive generator of ${\mathfrak{p}}^{j}$ for an odd $j\in{\mathbb{Z}}_{>0}$. Construct $K^{r}=F^{r}(\sqrt{-\alpha})$. - Eliminate the fields $K^r$ that have totally split primes in $K^r$ below the bound ${\sqrt{d_{K}/d^{2}_F}}/{4}$. (In this step we eliminate most of the CM fields.) - For each $\mathfrak{q}$ with norm $Q$ below the bound $12\log(|d_{K^r}|)^2$, check whether it is in $I_{0}(\Phi^r)$ as follows. List all quartic Weil $Q$-polynomials, that is, monic integer polynomials of degree $4$ such that all roots in ${\mathbb{C}}$ have absolute value $\sqrt{Q}$. For each, take its roots in $K$ and check whether ${\text{\normalfont{N}}}_{\Phi^r}(\mathfrak{q})$ is generated by such a root. If not, then $\mathfrak{q}$ is not in $I_0(\Phi^r)$, so we throw away the field. - For each $K^r$, compute the class group of $K^r$ and test $I_{0}(\Phi^r)/P_{K^r}=I_{K^r}/P_{K^r}$. - Find $[D,\ A,\ B]$ representations for the reflex fields $K$ of the remaining pairs $(K^r,\Phi^r)$. Note that Step $4$ and Step $5$ of the algorithm above do not affect the validity of the algorithm by Lemma \[quick\]. These two steps are only to speed up the computation. Suppose that a non-normal quartic CM field $K$ satisfies $I_{0}(\Phi^r)=I_{K^r}$. Then by Lemma \[construct.1\], we have $F={\mathbb{Q}}(\sqrt{p})$ and $F^r={\mathbb{Q}}(\sqrt{q})$, where $p$ and $q$ are prime numbers with $q\not\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$ and $(p/q)=(q/p)=1$. Also by Lemma \[construct.1\], there exist a prime ${\mathfrak{p}}$ lying above $p$ in $F^r$ that ramifies in $K^r$ and a totally positive element $\alpha = \pi s_1\cdots s_{u}$, is a totally positive generator of ${\mathfrak{p}}^{j}$ for an odd $j\in{\mathbb{Z}}_{>0}$ such that $K^r=F^{r}(\sqrt{-\alpha})$. By the ramified primes in $K^r/F^r$ that are distinct from ${\mathfrak{p}}$ are inert and $F^r$. As $s_1, \cdots,\ s_{u}$ are such primes, we have $(p/s_i)=-1$ and $(q/s_i)=-1$. By we have either $h^{*}_{K^r}=2^{t_{K^r}-1}\leq2^5$ and $d_{K^r}/d_{F^r}\leq3\cdot10^{10}$ or $d_N<222^8$. Therefore, the CM field $K$ is listed. We implemented the algorithm in SAGE [@sage; @pari; @streng] and obtained the list of the fields in Theorem \[thm: non-normallist2\]. The implementation is available online at [@kilicer_code]. This proves Theorems \[thm: non-normallist2\] and \[thm: non-normallist\]. This computation takes few days on a computer. There are no fields eliminated in Step 6, because they turned out to be already eliminated in Step 5. The cyclic quartic CM fields {#cyclic} ============================ Murabayashi and Umegaki determined the *cyclic* CM fields whose ring of integers are isomorphic to the endomorphism rings of the (simple) Jacobians of curves defined over ${\mathbb{Q}}$. Such fields have CM class number one, however there are more examples, for example, the fields in Table 1b of [@bouyer] have CM class number one, but the curves corresponding to these fields have not a model over ${\mathbb{Q}}$. We apply the strategy in the previous section to cyclic quartic CM fields and list all cyclic quartic CM fields with CM class number one. Murabayashi [@murab Proposition 4.5] proves that the relative class group of cyclic quartic CM fields with CM class number one is $2^{t_K-1}$, where $t_K$ is the number of ramified primes in $K/F$. This result also follows from Proposition \[prop2\] in Section \[finiteness\]. Suppose that $K/{\mathbb{Q}}$ is a cyclic quartic CM field with ${\text{\normalfont{Gal}}}(K/{\mathbb{Q}})=\langle\sigma\rangle$. Since $K/{\mathbb{Q}}$ is normal, we consider CM types with values in $K$. The CM type, up to equivalence, is $\Phi=\{{\text{\normalfont{id}}}, \sigma\}$, which is primitive. The reflex field $K^r$ is $K$ and the reflex type of $\Phi$ is the CM type $\{{\text{\normalfont{id}}},\sigma^{3}\}$ (Example 8.4(1) of [@goro3]). In this notation complex is $\sigma^2$. Suppose $K\cong{\mathbb{Q}}(\zeta_5)$, where $\zeta_m$ denotes a primitive $m$-th of unity. Then the class group of $K$ is trivial, so the equality $I_{0}(\Phi^{r})=I_{K}$ holds. Hence $K={\mathbb{Q}}(\zeta_5)$ will occur in the list of cyclic quartic fields satisfying $I_{0}(\Phi^{r})=I_{K}$. From now on, suppose $K\not\cong{\mathbb{Q}}(\zeta_5)$. \[ramification1\](Murabayashi [@murab], Lemma 4.2) If $I_{0}(\Phi^{r})=I_{K}$, then there is exactly one totally ramified prime in $K/{\mathbb{Q}}$ (i.e., $F={\mathbb{Q}}(\sqrt{p})$ with prime $p\not \equiv 3\ ({\text{\normalfont{mod}}}\ 4)$) and the other ramified primes of $K/{\mathbb{Q}}$ are inert in $F/{\mathbb{Q}}$. \[cyc2\] Suppose $I_{0}(\Phi^{r})=I_{K}$. The relative class number $h^{*}_K$ equals $1$ if and only if $K/F$ has exactly one ramified prime. This ramified prime is $\sqrt{p}$ when $F={\mathbb{Q}}(\sqrt{p})$. Now, we determine such CM fields by using a lower bound on their relative class numbers from analytic number theory. (Louboutin [@loub1], Theorem 5)\[lower1\] Let $K$ be a cyclic quartic CM field of and discriminant $d_K$. Then we have $$\label{eq:cyclic}\qquad \qquad \qquad h^{*}_{K}\geq \frac{2}{3e \pi^2}\left(1-\frac{4\pi e^{1/2}}{d_{K}^{1/4}}\right)\frac{f_{K}}{(\log(f_{K})+0.05)^2}. \qquad \qquad\, \qed$$ \[lemrel\] Let $K$ be a cyclic quartic CM field satisfying $I_{0}(\Phi^r) = I_{K}$. Then we have $h^{*}_K\leq2^5$ and $f_K<2.1\cdot10^5$. Lemma \[ramification1\] implies, under the assumption, that there is exactly one totally ramified prime in $K/{\mathbb{Q}}$ and the other ramified primes of $K/{\mathbb{Q}}$ are inert in $F/{\mathbb{Q}}$. Let $\Delta_t$ be the product of the first $t$-primes. The ramified primes in $K/{\mathbb{Q}}$ divide the conductor $f_K$, so we have $f_K>\Delta_{t_K}$. Further, by Proposition 11.9 and 11.10 in Chapter VII [@neukrich], we have $d_K=f^2_K\cdot d_F$ so $d_K>\Delta_{t_K}^2$. The right hand side of (\[eq:cyclic\]) is monotonically increasing with $f_K>2$. Further, by Proposition \[prop2\], we have $h^*_K = 2^{t_K-1}$, so, by dividing the both side of (\[eq:cyclic\]) by $2^{t_K-1}$, we obtain $$\label{eq: cycliceqn} 1\geq \frac{2}{3e \pi^2}\left(1-\frac{4\pi e^{1/2}}{\Delta_{t_K}^{1/2}}\right)\frac{\Delta_{t_K}}{2^{t_K}(\log(\Delta_{t_K})+0.05)^2}.$$ The right hand side of (\[eq: cycliceqn\]) is monotonically increasing with $t_K\geq 2$, and if $t_K=7$, then the right hand side is greater than $1$. Hence $t\leq 6$. So we get $h^{*}_{K}\leq 2^5$, and therefore, we get $f_K< 2.1\cdot10^5$. \[thm: cycliclist2\] There exist exactly $20$ isomorphism classes of cyclic quartic with CM class number one. The fields are given by $K\cong{\mathbb{Q}}[X]/(X^4 + AX^2 + B)\supset {\mathbb{Q}}(\sqrt{D})$ where $[D,A,B]$ ranges over $$[5,5,5],\ [8,4,2],\ [13,13,13],\ [29, 29, 29],\ [37,37,333],\ [53, 53, 53],\ [61, 61, 549]$$ with class number $1$; $$[5, 65, 845],\ [5, 85, 1445],\ [5, 10, 20],\ [8, 12, 18],$$$$[8, 20, 50],\ [13, 65, 325],\ [13, 26, 52],\ [17, 119, 3332]$$ with class number $2$; $$[5, 30, 180],\ [5, 35, 245],\ [5, 15, 45],\ [5, 105, 2205],\ [17, 255, 15300]$$ with class number $4$. We start proving with the following lemma. \[construct.2\] If a cyclic quartic CM field $K$ satisfies $I_{0}(\Phi^r)=I_{K}$, then there exist prime numbers $p$, $s_1, \cdots, s_u$ $\in{\mathbb{Z}}$ such that $F={\mathbb{Q}}(\sqrt{p})$ with $p\not\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$ and $(p/s_i)=-1$ for all $i$, and we have $K^r\cong{\mathbb{Q}}(\sqrt{-\epsilon s_1\cdots s_{u}\sqrt{p}})$ with $u\in\{t_K-1,\ t_K-2\}$ for every $\epsilon\in{\mathcal{O}}^{*}_{F}$ with $\epsilon\sqrt{p}>>0$. By Proposition \[ramification1\], we have $F={\mathbb{Q}}(\sqrt{p})$, where $p$ is prime $p\not\equiv3\ ({\text{\normalfont{mod}}}\ 4)$. If there are $t_K$ ramified primes in $K/F$, the ones that are distinct from the one are inert in $F/{\mathbb{Q}}$, by Proposition \[ramification1\], denote them by $s_1, \cdots, s_{t_K}$. There exists a totally positive element $\beta$ in ${F}^*$ (without loss and generality, we can take $\beta$ in ${\mathcal{O}}_{F}$) such that $K=F(\sqrt{-\beta})$, where $\beta$ is uniquely defined up to $({F}^*)^2$. As in the proof of Lemma \[construct.1\] in the previous section, we will define a totally positive element $\alpha\in F^*$ with respect to the ramified primes in $K/F$ and show that $\alpha$ and $\beta$ differ by a factor in $(F^*)^2$. Let $\epsilon\in{\mathcal{O}}^{*}_{F}$ such that $\epsilon\sqrt{p}>>0$. This element exists since $p\not\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$. is unique up to squares and we can take ${\mathfrak{l}}$-minimal $\beta'\in\beta(F^*)^2$ for each prime ${\mathfrak{l}}$ then we get the cases in (\[order\]) for ${\text{\normalfont{ord}}}_{{\mathfrak{l}}}((\beta))$. If $p\neq 2$ and the prime $(2)$ in ${\mathcal{O}}_F$ is ramified in $K/F$ with ${\text{\normalfont{ord}}}_{(2)}((\beta))\equiv 0\ ({\text{\normalfont{mod}}}\ 2)$, then take $\alpha:= \epsilon s_1\cdots s_{u}\sqrt{p}$ with $u=t_K-2$. If $p = 2$ and ${\text{\normalfont{ord}}}_{(\sqrt{2})}((\beta))\equiv 0\ ({\text{\normalfont{mod}}}\ 2)$, then take $\alpha:= s_1\cdots s_{u}$ with $u=t_K-1$. For all other cases in (\[order\]), take $\alpha:= \epsilon s_1\cdots s_{u}\sqrt{p}$ with $u=t_K-1$. By construction of $\alpha$, for all ideals ${\mathfrak{l}}\subset {\mathcal{O}}_F$ we have ${\text{\normalfont{ord}}}_{{\mathfrak{l}}}((\alpha/\beta))\equiv 0\ ({\text{\normalfont{mod}}}\ 2)$. So $(\alpha/\beta)={\mathfrak{a}}^2$ for a fractional ${\mathcal{O}}_F$-ideal ${\mathfrak{a}}$. The ideal ${\mathfrak{a}}$ is a $2$-torsion element Since $F={\mathbb{Q}}(\sqrt{p})$ with $p\not\equiv 3\ ({\text{\normalfont{mod}}}\ 4)$, genus theory implies that ${\text{\normalfont{Cl}}}_F={\text{\normalfont{Cl}}}^{+}_F$ has odd order. Therefore, there is a totally positive element $\mu$ that generates ${\mathfrak{a}}$. So $\alpha/\beta=\mu^2\cdot v$ for some $v\in {\mathcal{O}}^{+}_F$. Since ${\text{\normalfont{Cl}}}_F={\text{\normalfont{Cl}}}^{+}_F$, the fundamental unit has negative norm, and so $ {\mathcal{O}}^{+}_F=({\mathcal{O}}_F)^2$. Hence, $\alpha/\beta\in (F^{*})^2$. In the case $p = 2$ and ${\text{\normalfont{ord}}}_{(\sqrt{2})}((\beta))\equiv 0\ ({\text{\normalfont{mod}}}\ 2)$, we get $K=F(\sqrt{- s_1\cdots s_u})$, which is a biquadratic extension of ${\mathbb{Q}}$. Therefore, we get $K = {\mathbb{Q}}(\sqrt{-\epsilon s_1\cdots s_{u}\sqrt{p}})$ with $u\in\{t_{K-1},\ t_{K-2}\}$. \[algo1\] **Output:** $[D,\ A,\ B]$ representations of all cyclic quartic CM satisfying $I_{0}(\Phi^r)=I_{K}$. - Find all square-free integers less than $2.1\cdot10^5$ and having at most $6$ prime divisors. - Order the prime factors of each of these square-free integers as tuples of primes $(p,s_1,\cdots, s_{u})$ with $s_1<\cdots <s_u$ in $(u+1)$-ways, then take only the tuples satisfying $p\not\equiv3\ ({\text{\normalfont{mod}}}\ 4)$ and $(p/s_i)=-1$ for all $i$. - For each $(p,s_1,\cdots, s_{u})$, let $F={\mathbb{Q}}(\sqrt{p})$ and take a totally positive element $\alpha = \epsilon s_1 \cdots s_u\sqrt{p}$, where $\epsilon$ is a fundamental unit in $F$ such that $\epsilon\sqrt{p}>>0$. Construct $K=F(\sqrt{-\alpha})$. - Eliminate the fields $K$ that have totally split primes in $K$ below the bound ${\sqrt{d_{K}/d^{2}_F}}/{4}$. (In this step we eliminate most of the CM fields.) - For each $\mathfrak{q}$ with norm $Q$ below the bound $12\log(|d_{K^r}|)^2$, check whether it is in $I_{0}(\Phi^r)$ as follows. List all quartic Weil $Q$-polynomials, that is, monic integer polynomials of degree $4$ such that all roots in ${\mathbb{C}}$ have absolute value $\sqrt{Q}$. For each, take its roots in $K$ and check whether ${\text{\normalfont{N}}}_{\Phi^r}(\mathfrak{q})$ is generated by such a root. If not, then $\mathfrak{q}$ is not in $I_0(\Phi^r)$, so we throw away the field. - For each $K$ compute the class group of the fields $K$ and test $I_{0}(\Phi^r)/P_{K}=I_{K}/P_{K}$. - Find $[D,\ A,\ B]$ representations for the quartic CM class number one fields $K$. The idea of the proof of this algorithm is exactly as the proof of In this algorithm, Step 1 follows from Proposition \[lemrel\]; Step 2 and 3 follow from Lemma \[construct.2\]; Step 4 follows from Lemma \[quick\]. We implemented the algorithm in SAGE [@sage; @pari; @streng] and obtained the list of the fields in Theorem \[thm: cycliclist2\]. The implementation is available online at [@kilicer_code]. This proves Theorems \[thm: cycliclist\] and \[thm: cycliclist2\]. This computation takes few hours on a computer. [^1]: [Leiden University]{}, <http://pub.math.leidenuniv.nl/~kilicerp/>, <pinarkilicer@gmail.com> [^2]: Leiden University, <http://pub.math.leidenuniv.nl/~strengtc/>, [marco.streng@gmail.com](marco.streng@gmail.com)

--- abstract: 'In this paper, we propose `FedGP`, a framework for privacy-preserving data release in the federated learning setting. We use generative adversarial networks, generator components of which are trained by `FedAvg` algorithm, to draw privacy-preserving artificial data samples and empirically assess the risk of information disclosure. Our experiments show that `FedGP` is able to generate labelled data of high quality to successfully train and validate supervised models. Finally, we demonstrate that our approach significantly reduces vulnerability of such models to model inversion attacks.' author: - 'Sarit Kraus Department of Computer Science, Bar-Ilan University, Israel pcchair@ijcai19.org' - Aleksei Triastcyn - | Boi Faltings Artificial Intelligence Lab\ Ecole Polytechnique Fédérale de Lausanne\ Lausanne, Switzerland {aleksei.triastcyn, boi.faltings}@epfl.ch, bibliography: - 'fml\_2019.bib' title: Federated Generative Privacy --- Introduction {#sec:introduction} ============ The rise of data analytics and machine learning (ML) presents countless opportunities for companies, governments and individuals to benefit from the accumulated data. At the same time, their ability to capture fine levels of detail potentially compromises privacy of data providers. Recent research [@fredrikson2015model; @shokri2017membership; @hitaj2017deep] suggests that even in a black-box setting it is possible to argue about the presence of individual examples in the training set or recover certain features of these examples. Among methods that tackle privacy issues of machine learning is the recent concept of *federated learning* (FL) [@mcmahan2016communication]. In the FL setting, a central entity (*server*) wants to train a model on user data without actually copying these data from user devices. Instead, users (*clients*) update models locally, and the *server* aggregates these models. One popular approach is the federated averaging, `FedAvg` [@mcmahan2016communication], where *clients* do local on-device gradient descent using their data, then send these updates to the *server* where they get averaged. Privacy can further be enhanced by using secure multi-party computation (MPC) [@yao1982protocols] to allow the server access only average updates of a big group of users and not individual ones. Despite many advantages, federated learning does have a number of challenges. First, the result of FL is a single trained model (therefore, we will refer to it as a *model release* method), which does not provide much flexibility in the future. For instance, it would significantly reduce possibilities for further aggregation from different sources, e.g. different hospitals trying to combine federated models trained on their patients data. Second, this solution requires data to be labelled at the source, which is not always possible, because user may be unqualified to label their data or unwilling to do so. A good example is again a medical application where users are unqualified to diagnose themselves but at the same time would want to keep their condition private. Third, it does not provide provable privacy guarantees, and there is no reason to believe that the aforementioned attacks do not work against it. Some papers propose to augment FL with differential privacy (DP) to alleviate this issue [@mcmahan2017learning; @geyer2017differentially]. While these approaches perform well in ML tasks and provide theoretical privacy guarantees, they are often restrictive (e.g. many DP methods for ML assume, implicitly or explicitly, access to public data of similar nature or abundant amounts of data, which is not always realistic). ![Architecture of our solution for two clients. Sensitive data is used to train a GAN (local critic and federated generator) to produce a private artificial dataset, which can be used by any ML model.[]{data-label="fig:architecture"}](architecture_fedgp){width="\linewidth"} In our work, we address these problems by proposing to combine the strengths of federated learning and recent advancements in generative models to perform privacy-preserving *data release*, which has many immediate advantages. First, the released data could be used to train any ML model (we refer to it as the *downstream task* or the *downstream model*) without additional assumptions. Second, data from different sources could be easily pooled, providing possibilities for hierarchical aggregation and building stronger models. Third, labelling and verification can be done later down the pipeline, relieving some trust and expertise requirements on users. Fourth, released data could be traded on data markets[^1], where anonymisation and protection of sensitive information is one of the biggest obstacles. Finally, data publishing would facilitate transparency and reproducibility of research studies. The main idea of our approach, named `FedGP`, for *federated generative privacy*, is to train generative adversarial networks (GANs) [@goodfellow2014generative] on clients to produce artificial data that can replace clients real data. Since some clients may have insufficient data to train a GAN locally, we instead train a federated GAN model. First of all, user data still remain on their devices. Second, the federated GAN will produce samples from the common cross-user distribution and not from a specific single user, which adds to overall privacy. Third, it allows releasing entire datasets, thereby possessing all the benefits of private *data release* as opposed to *model release*. Figure \[fig:architecture\] depicts the schematics of our approach for two clients. To estimate potential privacy risks, we use our *post hoc* privacy analysis framework [@triastcyn2019generating] designed specifically for private data release using GANs. Our contributions in this paper are the following: - on the one hand, we extend our approach for private data release to the federated setting, broadening its applicability and enhancing privacy; - on the other hand, we modify the federated learning protocol to allow a range of benefits mentioned above; - we demonstrate that downstream models trained on artificial data achieve high learning performance while maintaining good average-case privacy and being resilient to model inversion attacks. The rest of the paper is structured as follows. In Section \[sec:related\_work\], we give an overview of related work. Section \[sec:preliminaries\] contains some preliminaries. In Section \[sec:approach\], we describe our approach and privacy estimation framework. Experimental results are presented in Section \[sec:evaluation\], and Section \[sec:conclusion\] concludes the paper. Related Work {#sec:related_work} ============ In recent years, as machine learning applications become a commonplace, a body of work on security of these methods grows at a rapid pace. Several important vulnerabilities and corresponding attacks on ML models have been discovered, raising the need of devising suitable defences. Among the attacks that compromise privacy of training data, model inversion [@fredrikson2015model] and membership inference [@shokri2017membership] received high attention. Model inversion [@fredrikson2015model] is based on observing the output probabilities of the target model for a given class and performing gradient descent on an input reconstruction. Membership inference [@shokri2017membership] assumes an attacker with access to similar data, which is used to train a “shadow” model, mimicking the target, and an attack model. The latter predicts if a certain example has already been seen during training based on its output probabilities. Note that both attacks can be performed in a black-box setting, without access to the model internal parameters. To protect privacy while still benefiting from the use of statistics and ML, many techniques have been developed over the years, including $k$-anonymity [@sweeney2002], $l$-diversity [@machanavajjhala2007], $t$-closeness [@li2007t], and differential privacy (DP) [@dwork2006]. Most of the ML-specific literature in the area concentrates on the task of privacy-preserving model release. One take on the problem is to distribute training and use disjoint datasets. For example, [@shokri2015privacy] propose to train a model in a distributed manner by communicating sanitised updates from participants to a central authority. Such a method, however, yields high privacy losses [@abadi2016deep; @papernot2016semi]. An alternative technique suggested by [@papernot2016semi], also uses disjoint training sets and builds an ensemble of independently trained teacher models to transfer knowledge to a student model by labelling public data. This result has been extended in [@papernot2018scalable] to achieve state-of-the-art image classification results in a private setting (with single-digit DP bounds). A different approach is taken by [@abadi2016deep]. They suggest using differentially private stochastic gradient descent (DP-SGD) to train deep learning models in a private manner. This approach achieves high accuracy while maintaining low DP bounds, but may also require pre-training on public data. A more recent line of research focuses on private data release and providing privacy via generating synthetic data [@bindschaedler2017plausible; @huang2017context; @beaulieu2017privacy]. In this scenario, DP is hard to guarantee, and thus, such models either relax the DP requirements or remain limited to simple data. In [@bindschaedler2017plausible], authors use a graphical probabilistic model to learn an underlying data distribution and transform real data points (seeds) into synthetic data points, which are then filtered by a privacy test based on a *plausible deniability* criterion. This procedure would be rather expensive for complex data, such as images. @fioretto2019privacy  employ dicision trees for a hybrid model/data release solution and guarantee stronger $\varepsilon$-differential privacy, but like the previous approach, it would be difficult to adapt to more complex data. Alternatively, @huang2017context  introduce the notion of *generative adversarial privacy* and use GANs to obfuscate real data points w.r.t. pre-defined private attributes, enabling privacy for more realistic datasets. Finally, a natural approach to try is training GANs using DP-SGD [@beaulieu2017privacy; @xie2018differentially; @zhang2018differentially]. However, it proved extremely difficult to stabilise training with the necessary amount of noise, which scales as $\sqrt{m}$ w.r.t. the number of model parameters $m$. It makes these methods inapplicable to more complex datasets without resorting to unrealistic (at least for some areas) assumptions, like access to public data from the same distribution. On the other end of spectrum, @mcmahan2016communication  proposed federated learning as one possible solution to privacy issues (among other problems, such as scalability and communication costs). In this setting, privacy is enforced by keeping data on user devices and only submitting model updates to the server. It can be augmented by MPC [@bonawitz2017practical] to prevent the server from accessing individual updates and by DP [@mcmahan2017learning; @geyer2017differentially] to provide rigorous theoretical guarantees. Preliminaries {#sec:preliminaries} ============= This section provides necessary definitions and background. Let us commence with approximate differential privacy. A randomised function (mechanism) $\mathcal{M}: \mathcal{D} \rightarrow \mathcal{R}$ with domain $\mathcal{D}$ and range $\mathcal{R}$ satisfies $(\varepsilon, \delta)$-differential privacy if for any two adjacent inputs $d, d' \in \mathcal{D}$ and for any outcome $o \in \mathcal{R}$ the following holds: $$\begin{aligned} \Pr\left[\mathcal{M}(d)=o\right] \leq e^\varepsilon \Pr\left[\mathcal{M}(d') = o\right] + \delta.\end{aligned}$$ Privacy loss of a randomised mechanism $\mathcal{M}: \mathcal{D} \rightarrow \mathcal{R}$ for inputs $d, d' \in \mathcal{D}$ and outcome $o \in \mathcal{R}$ takes the following form: $$\begin{aligned} L_{(\mathcal{M}(d) \| \mathcal{M}(d'))} = \log\frac{\Pr\left[\mathcal{M}(d) = o \right]}{\Pr\left[\mathcal{M}(d') = o \right]}.\end{aligned}$$ The Gaussian noise mechanism achieving $(\varepsilon, \delta)$-DP, for a function $f: \mathcal{D} \rightarrow \mathbb{R}^m$, is defined as $$\begin{aligned} \mathcal{M}(d) = f(d) + \mathcal{N}(0, \sigma^2),\end{aligned}$$ where $\sigma > C \sqrt{2\log\frac{1.25}{\delta}} / \varepsilon$ and $C$ is the L2-sensitivity of $f$. For more details on differential privacy and the Gaussian mechanism, we refer the reader to [@dwork2014algorithmic]. In our privacy estimation framework, we also use some classical notions from probability and information theory. The Kullback–Leibler (KL) divergence between two continuous probability distributions $P$ and $Q$ with corresponding densities $p$, $q$ is given by: $$\begin{aligned} D_{KL}(P \| Q) = \int_{-\infty}^{+\infty} p(x) \log\frac{p(x)}{q(x)} dx.\end{aligned}$$ Note that KL divergence between the distributions of $\mathcal{M}(d)$ and $\mathcal{M}(d')$ is nothing but the expectation of the privacy loss random variable $\mathbb{E}[L_{(\mathcal{M}(d) \| \mathcal{M}(d'))}]$. Finally, we use the Bayesian perspective on estimating mean from the data to get sharper bounds on expected privacy loss compared to the original work [@triastcyn2019generating]. More specifically, we use the following proposition. \[prop:bayesian\] Let $[l_1, l_2, \ldots, l_n]$ be a random vector drawn from the distribution $p(L)$ with the same mean and variance, and let $\overline{L}$ and $S$ be the sample mean and the sample standard deviation of the random variable $L$. Then, $$\begin{aligned} \Pr\left(\mathbb{E}[L] > \overline{L} + \frac{F_{n-1}^{-1}(1 - \gamma)}{\sqrt{n-1}} S \right) \leq \gamma,\end{aligned}$$ where $F_{n-1}^{-1}(1 - \gamma)$ is the inverse CDF of the Student’s $t$-distribution with $n-1$ degrees of freedom at $1 - \gamma$. The proof of this proposition can be obtained by using the maximum entropy principle with a flat (uninformative) prior to get the marginal distribution of the sample mean $\overline{L}$, and observing that the random variable $\frac{\mathbb{E}[L] - \overline{L}}{S/\sqrt{n-1}}$ follows the Student’s $t$-distribution with $n - 1$ degrees of freedom [@oliphant2006bayesian]. Federated Generative Privacy {#sec:approach} ============================ In this section, we describe our algorithm, what privacy it can provide and how to evaluate it, and discuss current limitations. Method Description ------------------ In order to keep participants data private while still maintaining flexibility in downstream tasks, our algorithm produces a federated generative model. This model can output artificial data, not belonging to any real user in particular, but coming from the common cross-user data distribution. Let $\{u_1, u_2, \ldots, u_n\}$ be a set of *clients* holding private datasets $\{d_1, d_2, \ldots, d_n\}$. Before starting the training protocol, the *server* is providing each *client* with generator $G_i^0$ and critic $C_i^0$ models, and *clients* initialise their models randomly. Like in a normal FL setting, the training process afterwords consists of communication rounds. In each round $t$, *clients* update their respective models performing one or more passes through their data and submit generator updates $\triangle G_i^t$ to the *server* through MPC while keeping $C_i^t$ private. In the beginning of the next round, the *server* provides an updated common generator $G^t$ to all *clients*. This approach has a number of important advantages: - Data do not physically leave user devices. - Only generators (that do not come directly into contact with data) are shared, and critics remain private. - Using artificial data in downstream tasks adds another layer of protection and limits the information leakage to artificial samples. This is esprecially useful given that ML models can be attacked to extract training data [@fredrikson2015model], sometimes even when protected by DP [@hitaj2017deep]. What remains to assess is how much information would an attacker gain about original data. We do so by employing a notion introduced in an earlier work [@triastcyn2019generating] that we name *Differential Average-Case Privacy*. It is important to clarify why we do not use the standard DP to provide stronger theoretical guarantees: we found it extremely difficult to train GANs with the amount of noise required for meaningful DP guarantees. Despite a number of attempts [@beaulieu2017privacy; @xie2018differentially; @zhang2018differentially], we are not aware of any technically sound solution that would generalise beyond very simple datasets. Differential Average-Case Privacy {#sec:dap} --------------------------------- Our framework builds upon ideas of *empirical DP* (EDP) [@abowd2013differential; @schneider2015new] and *on-average KL privacy* [@wang2016average]. The first can be viewed as a measure of sensitivity on posterior distributions of outcomes [@charest2017meaning] (in our case, generated data distributions), while the second relaxes DP notion to the case of an average user. More specifically, we say the mechanism $\mathcal{M}$ is $(\mu, \gamma)$-DAP if for two neighbouring datasets $D, D'$, where data come from an observed distribution, it holds that $$\begin{aligned} \label{eq:dap} \Pr(\mathbb{E}[|L_{(\mathcal{M}(D) \| \mathcal{M}(D'))}|] > \mu) \leq \gamma.\end{aligned}$$ For the sake of example, let each data point in $D, D'$ represent a single user. Then, $(0.01, 0.001)$-DAP could be interpreted as follows: with probability $0.999$, a typical user submitting their data will change outcome probabilities of the private algorithm on average by $1\%$[^2]. Generative Differential Average-Case Privacy {#sec:gdap} -------------------------------------------- In the case of generative models, and in particular GANs, we don’t have access to exact posterior distributions, a straightforward EDP procedure in our scenario would be the following: *(1)* train GAN on the original dataset $D$; *(2)* remove a random sample from $D$; *(3)* re-train GAN on the updated set; *(4)* estimate probabilities of all outcomes and the maximum privacy loss value; *(5)* repeat *(1)–(4)* sufficiently many times to approximate $\varepsilon$, $\delta$. If the generative model is simple, this procedure can be used without modification. Otherwise, for models like GANs, it becomes prohibitively expensive due to repetitive re-training (steps *(1)–(3)*). Another obstacle is estimating the maximum privacy loss value (step *(4)*). To overcome these two issues, we propose the following. First, to avoid re-training, we imitate the removal of examples directly on the generated set $\widetilde{D}$. We define a similarity metric $sim(x, y)$ between two data points $x$ and $y$ that reflects important characteristics of data (see Section \[sec:evaluation\] for details). For every randomly selected real example $i$, we remove $k$ nearest artificial neighbours to simulate absence of this example in the training set and obtain $\widetilde{D}^{-i}$. Our intuition behind this operation is the following. Removing a real example would result in a lower probability density in the corresponding region of space. If this change is picked up by a GAN, which we assume is properly trained (e.g. there is no mode collapse), the density of this region in the generated examples space should also decrease. The number of neighbours $k$ is defined by the ratio of artificial and real examples, to keep density normalised. Second, we relax the worst-case privacy loss bound in step *(4)* by the expected-case bound, in the same manner as on-average KL privacy. This relaxation allows us to use a high-dimensional KL divergence estimator [@perez2008kullback] to obtain the expected privacy loss for every pair of adjacent datasets ($\widetilde{D}$ and $\widetilde{D}^{-i}$). There are two major advantages of this estimator: it converges almost surely to the true value of KL divergence; and it does not require intermediate density estimates to converge to the true probability measures. Also since this estimator uses nearest neighbours to approximate KL divergence, our heuristic described above is naturally linked to the estimation method. Finally, having obtained sufficiently many sample pairs $(\widetilde{D}, \widetilde{D}^{-i})$, we use Proposition \[prop:bayesian\] to determine DAP parameters $\mu$ and $\gamma$. This is an improvement over original DAP, because this way we can get much sharper bounds on expected privacy loss. Limitations {#sec:limitations} ----------- Our approach has a number of limitations that should be taken into consideration. First of all, existing limitations of GANs (or generative models in general), such as training instability or mode collapse, will apply to this method. Hence, at the current state of the field, our approach may be difficult to adapt to inputs other than image data. Yet, there is still a number of privacy-sensitive applications, e.g. medical imaging or facial analysis, that could benefit from our technique. And as generative methods progress, new uses will be possible. Second, since critics remain private and do not leave user devices their performance can be hampered by a small number of training examples. Nevertheless, we observe that even in the setting where some users have smaller datasets overall discriminative ability of all critics is sufficient to train good generators. Lastly, our empirical privacy guarantee is not as strong as the traditional DP and has certain limitations [@charest2017meaning]. However, due to the lack of DP-achieving training methods for GANs it is still beneficial to have an idea about expected privacy loss rather than not having any guarantee. Evaluation {#sec:evaluation} ========== [**Setting**]{} [**Dataset**]{} [**Baseline**]{} [**CentGP**]{} [**FedGP**]{} ----------------- ----------------- ------------------ ---------------- --------------- MNIST (500) $98.10\%$ $97.35\%$ $79.45\%$ MNIST (1000) $98.55\%$ $97.39\%$ $93.38\%$ MNIST (2000) $98.92\%$ $97.41\%$ $96.23\%$ MNIST (500) $97.31\%$ $83.26\%$ MNIST (1000) $98.78\%$ — $95.89\%$ MNIST (2000) $98.76\%$ $96.88\%$ : Accuracy of student models trained on artificial samples of FedGP compared to non-private centralised baseline and CentGP. In parenthesis we specify the average number of data points per client. []{data-label="tab:accuracy"} In this section, we describe the experimental setup and implementation, and evaluate our method on MNIST [@lecun1998gradient] and CelebA [@liu2015faceattributes] datasets. Experimental Setting -------------------- We evaluate two major aspects of our method. First, we show that training ML models on data created by the common generator achieves high accuracy on MNIST (Section \[sec:learning\]). Second, we estimate expected privacy loss of the federated GAN and evaluate the effectiveness of artificial data against model inversion attacks on CelebA face attributes (Section \[sec:privacy\]). Learning performance experiments are set up as follows: 1. Train the federated generative model (*teacher*) on the original data distributed across a number of users. 2. Generate an artificial dataset by the obtained model and use it to train ML models (*students*). 3. Evaluate students on a held-out test set. We choose two commonly used image datasets, MNIST and CelebA. MNIST is a handwritten digit recognition dataset consisting of 60000 training examples and 10000 test examples, each example is a 28x28 size greyscale image. CelebA is a facial attributes dataset with 202599 images, each of which we crop to 128x128 and then downscale to 48x48. In our experiments, we use Python and Pytorch framework.[^3] For implementation details of GANs and privacy evaluation, please refer to [@triastcyn2019generating]. To train the federated generator we use FedAvg algorithm [@mcmahan2016communication]. As a *sim* function introduced in Section \[sec:gdap\] we use the distance between InceptionV3 [@szegedy2016rethinking] feature vectors. [**Setting**]{} [**Dataset**]{}   $\mu$ $\gamma$ ----------------- ----------------- --- ---------- ---------- MNIST (500)   $0.0117$ MNIST (1000)   $0.0069$ MNIST (2000)   $0.0021$ CelebA   $0.0009$ non-i.i.d. MNIST (500)   $0.0090$ MNIST (1000)   $0.0044$ MNIST (2000)   $0.0020$ : Average-case privacy parameters: expected privacy loss bound $\mu$ and probability $\gamma$ of exceeding it.[]{data-label="tab:privacy"} [**Setting**]{} [**Dataset**]{} $\mu_c$ $\mu_b$ $\gamma$ ----------------- ----------------- ---------- ---------- ---------- MNIST (500) $3.6858$ $0.0117$ MNIST (1000) $1.8856$ $0.0069$ MNIST (2000) $0.6469$ $0.0021$ CelebA $0.2703$ $0.0009$ non-i.i.d. MNIST (500) $2.8393$ $0.0090$ MNIST (1000) $1.3777$ $0.0044$ MNIST (2000) $0.6911$ $0.0020$ : Empirical privacy parameters: expected privacy loss bound $\mu$ and probability $\gamma$ of exceeding it.[]{data-label="tab:privacy"} Learning Performance {#sec:learning} -------------------- First, we evaluate the generalisation ability of the student model trained on artificial data. More specifically, we train a student model on generated data and report test classification accuracy on a held-out real set. We compare learning performance with the baseline centralised model trained on original data, as well as the same model trained on artificial samples obtained from the centrally trained GAN (`CentGP`). Since critics stay private and can only access data of a single user, the size of each individual dataset has significant effect. Therefore, in our experiment we vary sizes of user datasets and observe its influence on training. In each experiment, we specify an average number of points per user, while the actual number is drawn from the uniform distribution with this mean, with some clients getting as few as 100 data points. We also study two settings: i.i.d. and non-i.i.d data. In the first setting, distribution of classes for each client is identical to the overall distribution. In the second, every client gets samples of 2 random classes, imitating the situation when a single user observes only a part of overall data distribution. Details of the experiment can be found in Table \[tab:accuracy\]. We observe that training on artificial data from the federated GAN allows to achieve $96.9\%$ accuracy on MNIST with the baseline of $98.8\%$. We can also see how accuracy grows with the average user dataset size. A less expected observation is that non-i.i.d. setting is actually beneficial for `FedGP`. A possible reason is that training critics with little data becomes easier when this data is less diverse (i.e. the number of different classes is smaller). Comparing to the centralised generative privacy model `CentGP`, we can also see that `FedGP` is more affected by sharding of data on user devices than by overall data size, suggesting that further research in training federated generative models is necessary. Privacy Analysis {#sec:privacy} ---------------- Using the privacy estimation framework (see Sections \[sec:dap\] and \[sec:gdap\]), we fix the probability $\gamma$ of exceeding the expected privacy loss bound $\mu$ in all experiments to $10^{-15}$ and compute the corresponding $\mu$ for each dataset and two settings. Table \[tab:privacy\] summarises the bounds we obtain. As anticipated, the privacy guarantee improves with the growing number of data points, because the influence of each individual example diminishes. Moreover, the average privacy loss $\mu$, expectedly, is significantly smaller than the typical worst-case DP loss $\varepsilon$ in similar settings. To put it in perspective, the average change in outcome probabilities estimated by DAP is ${\sim}1\%$ even in more difficult settings, while the state-of-the-art DP method would place the worst-case change at hundreds or even thousands percent without giving much information about a typical case. ![Results of the model inversion attack. Top to bottom: real target images, reconstructions from the non-private model, reconstructions from the model trained by `FedGP`.[]{data-label="fig:reconstruction"}](celeba_reconstruction_crop){width="\linewidth"} [****]{} [**Baseline**]{} [**FedGP**]{} ------------- ------------------ --------------- Detection $25.5\%$ $1.2\%$ Recognition $2.8\%$ $0.1\%$ : Face detection and recognition rates (pairs with distances below $0.99$) for images recovered by model inversion attack from the non-private baseline and the model trained by `FedGP`.[]{data-label="tab:face_recognition"} On top of estimating expected privacy loss bounds, we test `FedGP`’s resistance to the *model inversion attack* [@fredrikson2015model]. More specifically, we run the attack on two student models: trained on original data samples and on artificial samples correspondingly. Note that we also experimented with another well-known attack on machine learning models, the membership inference [@shokri2017membership]. However, we did not include it in the final evaluation, because of the poor attacker’s performance in our setting (nearly random guess accuracy for given datasets and models even on the non-private baseline). Moreover, we only consider passive adversaries and we leave evaluation with active adversaries, e.g. [@hitaj2017deep], for future work. In order to run the attack, we train a student model (a simple multi-layer perceptron with two hidden layers of 1000 and 300 neurons) in two settings: the real data and the artificial data generated by the federated GAN. As facial recognition is a more privacy-sensitive application, and provides a better visualisation of the attack, we pick the CelebA attribute prediction task to run this experiment. We analyse real and reconstructed image pairs using OpenFace [@amos2016openface] (see Table \[tab:face\_recognition\]). It confirms our theory that artificial samples would shield real data in case of the downstream model attack. In the images reconstructed from a non-private model, faces were detected $25.5\%$ of the time and recognised in $2.8\%$ of cases. For our method, detection succeeded only in $1.2\%$ of faces and the recognition rate was $0.1\%$, well within the state-of-the-art error margin for face recognition. Figure \[fig:reconstruction\] shows results of the model inversion attack. The top row presents the real target images. The following rows depict reconstructed images from the non-private model and the model trained on the federated GAN samples. One can observe a clear information loss in reconstructed images going from the non-private to the `FedGP`-trained model. Despite failing to conceal general shapes in training images (i.e. faces), our method seems to achieve a trade-off, hiding most of the specific features, while the non-private model reveals important facial features, such as skin and hair colour, expression, etc. The obtained reconstructions are either very noisy or converge to some average feature-less faces. Conclusions {#sec:conclusion} =========== We study the intersection of federated learning and private data release using GANs. Combined these methods enable important advantages and applications for both fields, such as higher flexibility, reduced trust and expertise requirements on users, hierarchical data pooling, and data trading. The choice of GANs as a generative model ensures scalability and makes the technique suitable for real-world data with complex structure. In our experiments, we show that student models trained on artificial data can achieve high accuracy on classification tasks. Moreover, models can also be validated on artificial data. Importantly, unlike many prior approaches, our method does not assume access to similar publicly available data. We estimate and bound the expected privacy loss of an average client by using differential average-case privacy thus enhancing privacy of traditional federated learning. We find that, in most scenarios, the presence or absence of a single data point would not change the outcome probabilities by more than $1\%$ on average. Additionally, we evaluate the provided protection by running the model inversion attack and showing that training with the federated GAN reduces information leakage (e.g. face detection in recovered images drops from $25.5\%$ to $1.2\%$). Considering the importance of the privacy research, the lack of good solutions for private data publishing, and the rising popularity of federated learning, there is a lot of potential for future work. In particular, a major direction of advancing current research would be achieving differential privacy guarantees for generative models while still preserving high utility of generated data. A step in another direction would be to improve our empirical privacy concept, e.g. by bounding maximum privacy loss rather than average, or finding a more principled way of sampling from outcome distributions. [^1]: <https://www.datamakespossible.com/value-of-data-2018/dawn-of-data-marketplace> [^2]: Because $e^{0.01} \approx 1.01.$ [^3]: <http://pytorch.org>

--- abstract: | In this report I give the short historical review some of the first steps that were done to the invention of SUSY in Kharkov team headed by D.Volkov. This paper is dedicated to the memory of Prof. Yu. Gol’fand, whose ideas of SUSY inspired the most active developments in High Energy Physics over thirty years. author: - 'V. Akulov [^1]' title: 'Non-linear way to Supersymmetry and N-extended SUSY [^2] ' --- [*”Geometry of space is associated with mathematical group”*]{} Felix Klein , ”Erlagen Program” 1872 This year, the science community celebrates the 30th anniversary of SUSY. I have been asked to give a short historical introduction to the first steps in this direction that were done by our group headed by Prof. Dmitry Volkov. Rochester’s High Energy Conference was held in Kiev (Ukraine, Soviet Union) in 1970. Prof. Yu. Golfand announced 2 reports for this conference, but the Org. Committee gave him time only for one. Prof. Yu. Gol’fand preferred to discuss the problem of vacuum in QED, because the problem of Superalgebra Poincaré, which was obtained already, seemed to him very complicated for a first discussion. But the abstract of this report, with Superalgebra Poincaré, was published in the Rotaprint edition of the Proceeding [@/1/]. It was the first publication about the superalgebra Poincaré. Unfortunately, in the final version of Proceedings did not appear this abstracts. But the first attempts of the introduction of superalgebras in physics was given in paper by G.Stavraki [@/2/] in 1966 and H.Miyazava [@/3/] in 1968. The mathematical background for supersymmetry was constructed in 1970. Felix Berezin(Moscow) and Gregory Kats (Kiev) [@/4/] published the paper about the groups with a commuting and anticommuting parameters in the Russian journal ”Mathematicheskiy Zbornik” – that is how the supergroups appeared in mathematics. Before that Prof. D. Volkov investigated the connection between spin and statistics and rediscovered (after M.Green,1953) [@/5/] the parastatistics in 1959. Later he considered fermionic Regge trajectory. The success of the application of Goldstone’s Theorem [@/6/] to $\pi$-meson physics stimulated the desire for the generalization of this theorem to fermionic case. Another hint was linked with the incorrect Heisenberg’s idea about the neutrino as a Goldstone particle connected with broken discrete symmetry – P-parity. I was a post-graduated student at that time and Prof. D. Volkov proposed this theme for my Ph.D. Thesis in 1971. Prof. D. Volkov headed the Research Laboratory in Theoretical Physics Division, which was headed by academician A. Akhieser, in the Ukrainian Physics and Technology Institute (UPhTI) in Kharkov. Volkov’s laboratory included also the post-graduated students – V. Tkach, V. Soroka, L. Gendenshtein, A. Zheltukhin, V. Gershun, A. Pashnev and later D. Sorokin and I. Bandos, A. Gumenchuk, A. Nurmagombetov. The main direction of research activity was connected with the application of Group theory to the Particle Physics. E. Cartan’s book ”Geometry of Lie Groups and symmetric spaces” was a main textbook for us during period. Only the one who had finished studing of that textbook, could participate in further activity. D.Volkov understood that we needed much greatermathematical background to achieve this goal. As he said: ”I’m afraid to rediscover what is already dicovered”. For using Klein & Cartan’s approach, we were needed the new sort of group – the group with anticommuting and commuting parameters. We participated in the seminars of Mathematical Physics, headed by Prof. V. Marchenko in the Mathematics Department of Kharkov State University. D. V. Volkov discussed with V. Marchenko our problem and one of the participant, V.Golodets, told us that he has read the paper of F. Berezin and G. Kats dealing with new kind of group. We first attempted the use of the exponential representation for the Poincaré Supergroup, but this approach was more complicated. However the last part of the paper by Berezin-Kats contained an example of the matrix realization of supergroup on the graded Pauli matrices SL$\left( 1|1\right) =\sigma_{0},\sigma_{+}, \sigma_{-}$ with the generalized commutator – the supercommutator $\left[ \sigma_{\pm}, \sigma_0\right] =0$, $\left\{\sigma_{+}, \sigma_{-}\right\} =\sigma_0$ Note,that a superalgebra contained the unit matrix in contrary to the standard Lee algebra and satisfied generalized the Jakobi’s identity. Then we constructed a matrix representation for a supergroup $3\times 3$ using graded $\lambda$-matrices by Gell-Man: $$\lambda ==\left| \begin{array}{ccc} \lambda _{11} & \lambda _{12} & \lambda _{13} \\ \lambda _{21} & \lambda _{22} & \lambda _{23} \\ \lambda _{31} & \lambda _{32} & \lambda _{33} \end{array} \right|$$ where only $\lambda_{13}, \lambda_{23}, \lambda_{31}, \lambda_{32}$ have a grassmannian parity. That was an examples of graded $SU\left( 2\right|1)$. Then, using the well-known representation for the Poincaré group like an upper triangular matrices $$\begin{array}{l} P= \left( \begin{array}{cc} \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} & iT \\ 0 & \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \end{array} \right) \times \left( \begin{array}{cc} L & 0 \\ 0 & L^{\left( +-1\right) } \end{array} \right) \end{array}$$ where each of the block is two by two matrix, and $L=L\left( l\right)$ is a matrix of the group $SL\left( 2,C\right)$, which corresponds to the Lorentz group, and $T=T\left( x\right)$ is a Hermitian matrix corresponding to the group of translations. We constructed an extended Poincaré supergroup, insert to the center of matrices new blocks $$\begin{array}{l} G = KH = \left( \begin{array}{ccc} 1 & \Xi & 1/2\Xi \Xi ^{+}+iT \\ 0 & 1 & \Xi ^{+} \\ 0 & 0 & 1 \end{array} \right) \times\left( \begin{array}{ccc} L & 0 & 0 \\ 0 & U & 0 \\ 0 & 0 & L^{+-1} \end{array} \right) =\\ \\ \phantom{G= KH } = \left( \begin{array}{ccc} L & \Xi U & \left( 1/2\Xi \Xi ^{+}+iT\right) L^{+-1} \\ 0 & U & \Xi ^{+}L^{+-1} \\ 0 & 0 & L^{+-1} \end{array} \right) \end{array}$$ In the matrices (2), the block $U=U\left(u\right)$ corresponds to the representation of a unitary group $N\times N$ unitary matrices with parameters $u$. In the simplest case, $U$ is the one-dimensional identity matrix. The block $\Xi=\Xi \left( \xi \right)$ is determined by two indexes: one of them carried by spinor index, corresponding to the group $SL\left( 2,C\right)$ and another carried by unitary index. This block is an arbitrary rectangular matrix with two rows and $N$ columns. The block $\Xi^{+}\left(\xi\right)$ is the Hermits conjugate of the block $\Xi\left(\xi\right)$. It is assumed that the matrices $\Xi$ and $\Xi^{+}$ are linear with respect to the group spinor parameters $\xi$ and $\xi^{*}$, so that these anticommute with each other and commute with another parameters. In the definitions (1) and (2) we have written down representations of the matrices as products of two factors, each of them determines a certain group. The principal advantage gained by representing the group as product is that when one acts on the left on the products (1) and (2) with matrices of the group the parameters in the left-hand factors are transformed solely through themselves and correspond to certain homogeneous spaces.For the Poincaré group (1), the homogeneous space defined in this manner is ordinary 4-dimensional Minkovsky’s space-time. For the Poincaré supergroup (2), the homogeneous space or corresponding coset contains ordinary space-time as well as additional anticommuting spinor degrees of freedom – this space was later called ”superspace” by A. Salam and J. Strathdee. A transformation of the group parameters corresponding to the matrix product (2) $$G\left(g^{^{\prime \prime}}\left(g^{^{\prime }},g\right)\right) = G\left(g^{^{\prime}}\right) G\left(g\right)$$ expressed in terms of the parameters in the definition of the individual blocks, had the form $$L(l^{^{\prime \prime }}) = L(l^{\prime })L(l)$$ $$U(u^{\prime \prime}) = U(u^{\prime})U(u)$$ $$\Xi(\xi ^{\prime \prime}) = L(l^{\prime})\Xi(\xi)U^{-1}(u^{\prime })+\Xi \left( \xi ^{\prime }\right)$$ $$\begin{array}{l} T(x^{\prime \prime }) = L(l^{\prime })T(x)L^{+-1}(l^{\prime }) + T(x^{\prime }) + \\ \phantom{T(x^{\prime \prime }) = } 1/2i\left[\Xi(\xi^{\prime})U(u^{\prime })\Xi (\xi )L^{+-1}(l^{\prime }) - \right. \\ \phantom{T(x^{\prime \prime }) = asdf} \left. L(l^{\prime })\Xi \left( \xi \right) U^{-1}(u^{\prime })\Xi ^{+}(\xi ^{\prime })\right] \end{array}$$ The transformation (4) and (5) correspond to the ordinary transformations of the Lorentz group and the Unitary group. The structure of the first term in the transformation (6) is due to the circumstances already noted that the matrix $\Xi$ has one spinor and one unitary index. The second term corresponds to translations in the spinor space. In the transformation (7), the first two terms describe a transformation of translations in the Poincaré group. The last term in (7) establishes the relationship between translations in the ordinary space and spinor space. Note that the transformations (6) and (7) do not contain the parameters $l^{\prime\prime}, u^{\prime\prime}, l, u$. This means that the variables $x$ and $\xi$ do indeed form a homogeneous space under left shifts in formula (3).In order to distinguish group parameters from the coordinates of the homogeneous space, we shall henceforth denote the latter by variables $x$ for translations in the Poincaré group and $\theta$ for spinor translations. Expanding the matrices $\Xi$ and $T$ with respect to a complete system of matrices, we obtain: $$\Xi\left(\theta\right) = Q_\alpha^k\theta_k^\alpha ,$$ $$\Xi^{+}\left(\theta^{*}\right) = Q_{k\alpha *}\theta^{k\alpha *},$$ $$T(x) = \widetilde{\sigma}_\mu x^\mu ,$$ where $\left(Q_\alpha^k\right)_b^a = \delta_\alpha^a\delta_b^k$, and $\widetilde{\sigma}_\mu$ are the relativistic Pauli matrices. We defined a transformation of the coordinates of the homogeneous space under transformations corresponding to the parameters $\xi$. Replacing $\xi, \xi^{\prime}, \xi^{\prime\prime}, x, x^{\prime\prime}$ by $\theta, \xi, \theta^{\prime }, x, x^{\prime}$ we obtained $$\theta _\alpha ^{i\prime }=\theta _\alpha ^i+\xi _\alpha ^i,$$ $$x^{\mu\prime} = x+1/2i\left(\xi^{*}\sigma^\mu\theta - \theta^{*}\sigma^\mu\xi\right)$$ Under transformations of the Poincaré group, the $x^\mu$ transform as ordinary coordinates. The transformation of these quantities under the Lorentz group and the unitary group is determined by their indices. Note that the spinors $\theta$ transform under all transformations of $G$ only among themselves. The matrices $Q_\alpha^i, Q_{i\alpha^{*}}^{+}$ and $P^\mu = \widetilde{\sigma}^\mu$ together with the generators of the Lorentz group and the unitary group form the complete set of the generators of $G$ in this representation. The commutation relations for these generators can be readily found from the definitions and have the form $$\left\{Q_\alpha^i, Q_{k\beta^{*}}^{+}\right\} = 1/2\delta_k^i\sigma_{\alpha\beta ^{*}}^\mu\widetilde{\sigma}_\mu.$$ $$\left\{ Q_\alpha ^i,Q_\beta ^k\right\} = \left\{ Q_{i\alpha^{*}}^{+},Q_{k\beta ^{*}}^{+}\right\} =0.$$ $$\left[\widetilde{\sigma}_\mu, Q_\alpha^i\right] = \left[\widetilde{\sigma}_\mu, Q_{k\beta^{*}}^{+}\right] = \left[\widetilde{\sigma}_\mu, \widetilde{\sigma}_\mu\right] = 0.$$ The commutation relations of the operators $Q, Q^{+}$ and $P_\mu = \widetilde{\sigma}_\mu$ with the generators $L_{\mu\nu}$ of the Lorentz group and $I_{ik}$ of the unitary group are uniquely determined by their Lorentz and unitary indexes. At the same time, operator $\widetilde{\sigma}_\mu$, which is associated with translations in the Poincaré group, satisfies the commutation relations for the energy-momentum operator. The commutation relations for the operators $Q$ and $Q^{+}$ are determined by anticommutators, since the corresponding group parameters are anticommuting. Thus, basing on the extended Poincaré supergroup, we constructed a nontrivial unification of the space-time symmetries like the Lorentz group or the Poincaré group with the internal symmetries $U\left(n\right)$. This way we bypassed the ”no-go” theorem of Coleman-Mandula (The restriction to Lie group has no a priori grounds). The above coset $\widetilde{P/H}$ included the grassmannian spinor coordinates with unitary index along with the conventional 4-dimensional Minkovski space and was later called ”Extended Superspace” by A.Salam and J.Strathdee. Our next step was the construction of the action integral invariant against such supergroup. We introduced the Cartan-Maurer differential 1-form, which are the coefficients of the supergroup generators in the expression: $G^{-1}dG = H^{-1}K^{-1}dK$, $H+H^{-1}dH = \omega^\mu P_\mu +\omega_\alpha^iQ_i^\alpha + \omega^{i\alpha *}Q_{i\alpha *}+...$, where the ellipses correspond to omitted terms with generators unitary group and Lorentz group. Expanding the product $$K^{-1}dK= \left( \begin{array}{ccc} 0 & d\Xi & 1/2\left( d\Xi\Xi^{*} - \Xi d\Xi^{*}\right) + idT \\ 0 & 0 & d\Xi ^{*} \\ 0 & 0 & 0 \end{array} \right)$$ with respect to the group generators,we obtained the following expressions for the Cartan-Maurer forms: $$\omega_\alpha^i = d\theta_\alpha^i,$$ $$\omega^{i\alpha ^{*}} = d\theta^{i\alpha ^{*}},$$ $$\omega^\mu = dx^\mu - 1/2i\left(\theta^{*}\sigma^\mu d\theta - d\theta^{*}\sigma^\mu\theta\right)$$ It is readily seen by direct calculations that under the transformations of super Poincaré group this forms are indeed invariant. To construct an invariant action integral it is sufficient to consider combinations of this forms that are invariant under transformations of the Lorentz group and the unitary group can be represented under the condition that $\theta$ is a function of $x$ and $\omega \left( d\right) = \omega_\mu \left( \theta, \partial \theta /\partial x\right) dx^\mu$ in the form: $$S=\int L\left( \theta ,\partial \theta /\partial x\right) d^4x$$ The presence of the volume element $d^4x$ in the expression for action imposes important restrictions on the structure of the accepted combinations of the forms $\omega$. To restrict the number of possible invariants, we used an additional requirement that the degree of the derivatives $\partial \theta /\partial x$ in the action be minimal, which corresponds to allowing in the $S$-matrix for only the lowest powers of the momenta of the Goldstone fermions. Among the invariant outer products we had the unique product,which contained only the differential forms $\omega^\mu$: $$S = \int \omega^\mu \wedge \omega^\nu \wedge \omega^\rho \wedge \omega^\lambda \epsilon_{\mu \nu \rho \lambda } = \int d^4x\det \left| \omega_\nu ^\mu \right|$$ that would describe a spontaneous broken supersymmetry. It had the form of the Born-Infeld action in the absence of the gauge $F$-field – as was noted by R. Kallosh in 1997, from the modern perspective one can say that it was one of the first version of $D-3$ branes [@/7/]. During the summer of 1972 we finished this work and sent the short version to JETP Lett. and Phys.Lett.B. [@/8/]. The latter was a great problem for our Institute, because all papers intended for publication abroad had to be cleared in Moscow. This would normally take 3 month or more. Only after the positive decision from Moscow could we send our paper abroad. In the autumn of 1972 we attended the International seminar on the ”$\mu-e$ problem” in Moscow. Prof. D. Volkov wanted to attract an attention to new kind of Goldstone particle and to announced on the possible universal neutrino interaction (Of course, we did not assume that neutrino is realistic Goldstone particle, because we used the $U(1)$ subgroup, but not $SU(2)$). Prof. E. Fradkin invited him to give a two hour talk at the Theoretical Division of FIAN. At that time Prof. V. Ogievetsky (who was very close to SUSY idea – he tried to consider the Rarita-Schwinger’s field as the gauge field) told us about the paper by Yu. Golfand - E. Likhtman in JETP Lett., that contained a similar algebra. Yu. Golfand (who worked in FIAN at that time) was absent during Volkov’s report, neither did E.Likhtman attend it. We returned back to Kharkov and read the Golfand-Likhtman’s paper, that contained the super Poincaré algebra and the action of an Abelian gauge model with linear realized supersymmetry. We added the reference about their paper and send our detailed report to the Russian journal ”Theoreticheskaya and Mathematicheskaya Fizika” and published in Kiev preprint [@/9/]. Further development had to connect with the generalization to the local extended Poincaré supergroup. We announced that the local version would contain also the Rarita-Schwinger field with spin 3/2 like superpartner of graviton. I had to finish my Ph.D.thesis at this time. D.Volkov and V.Soroka constructed the local version, using the supermatrix approach (2). Gauge fields was introduced and first version of supergravity appeared in [@/10/]. But the breakthrough in SUSY was began following the paper by J.Wess and B.Zumino [@/11/]. In conclusion I would like to thank the Org. Committee and special to Misha Shifman for the invitation and the support. [99]{} Y. Golfand, E. Likhtman Proceed. HEP (Rochester) Conference, Kiev, 1970 Rotaprint version (editor L.Enkovskiy); JETP LETT. v.13, p.323, 1971 G. Stavraki Proceeding of HEP School, Yalta, (Ukraine) 1966 , p.356 H. Miyazava Phys. Rev. v.170, p.1596, 1968 F. A. Berezin, G. I. Kats ”Matematicheskiy sbornik”, 1970, v.83, p.343 D. Volkov JETP, v.36, p.1560, 1959 S. Weinberg Phys. Rev. Lett. v.18, p.507, 1967 R. Kallosh Lect. Notes in Phys., v.509, p.49, 1997 D. Volkov, V. Akulov JETP Lett. v.16, p.621, 1972; Phys. Lett. 46B, p.109, 1973 V. Akulov, D. Volkov Preprint ITPh-73-51, Kiev, 1973; Theoreticheskaya and Mathematicheskaya Fizika, v.18, p.39, 1974 D. Volkov, V. Soroka JETP Lett. v.18, p.529, 1973; Theor. & Math. Phys. v.3, p.291, 1974 J. Wess, B. Zumino Nucl. Phys. B70, p.39, 1974 [^1]: Physics Department, Baruch College of the City University of New York New York, NY 10010 , USA E-mail: akulov@gursey.baruch.cuny.edu [^2]: Invited talk, will be published in Proceed. of SUSY-30, Supl. of NP B

--- abstract: | Let $\{X(t)= (X_1(t),X_2(t))^T,\ t \in \mathbb{R}^N\}$ be an $\mathbb{R}^2$-valued continuous locally stationary Gaussian random field with $\E[X(t)]=\mathbf{0}$. For any compact sets $A_1, A_2 \subset \R^N$, precise asymptotic behavior of the excursion probability $$\mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg),\ \ \text{ as }\ u \rightarrow \infty$$ is investigated by applying the double sum method. The explicit results depend not only on the smoothness parameters of the coordinate fields $X_1$ and $X_2$, but also on their maximum correlation $\rho$. address: - 'Department of Statistics, University of Nebraska-Lincoln, 340 Hardin Hall North Wing, Lincoln, NE 68583-0963 ' - 'Department of Statistics and Probability, Michigan State University, 619 Red Cedar Road, C413 Wells Hall, East Lansing, MI 48824-1027 ' author: - - title: 'Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields[^1]' --- Introduction {#Introduction} ============ For a real-valued Gaussian random field $X = \{X(t),$ $ t\in T \}$, where $T$ is the parameter set, defined on probability space $(\Omega, \mathcal{F}, \P)$, the excursion probability $\mathbb{P}\{\sup_{t\in T}X(t)>u\}$ has been studied extensively. Extending the seminal work of [@Pickands_1969], [@Piterbarg_1996] developed a systematic theory on asymptotics of the aforementioned excursion probability for a broad class of Gaussian random fields. Their method, which is called the double sum method, has been further extended by [@Chan_Lai_2006] to non-Gaussian random fields and, recently, by [@DHJ_14] to a non-stationary Gaussian random field $\{X(s, t), (s, t)\in \R^2\}$ whose variance function attains its maximum on a finite number of disjoint line segments. For smooth Gaussian random fields, more accurate approximation results have been established by using integral and differential-geometric methods (see, e.g., [@Adler_2000], [@Adler_Taylor_2007], [@Azais_Wschebor_2009] and the references therein). For Gaussian and asymptotically Gaussian random fields, the change of measure method was developed by [@Nardi_Siegmund_Yakir_2008] and [@Yakir_2013]. Many of the results in the aforementioned references have found important applications in statistics and other scientific areas. We refer to [@Adler_Taylor_Worsley_2012] and [@Yakir_2013] for further information. However, only a few authors have studied the excursion probability of multivariate random fields. [@Piterbarg_Stamatovich_2005] and [@Debicki_Kosinski_2010] established large deviation results for the excursion probability in multivariate case. [@Anshin_2006] obtained precise asymptotics for a special class of nonstationary bivariate Gaussian processes, under quite restrictive conditions. [@Hashorva_Ji_2014] recently derived precise asymptotics for the excursion probability of a bivariate fractional Brownian motion with constant cross correlation. The last two papers only consider multivariate processes on the real line $\mathbb{R}$ with specific cross dependence structures. [@Cheng_Xiao_2014] established a precise approximation to the excursion probability by using the mean Euler characteristics of the excursion set for a broad class of smooth bivariate Gaussian random fields on $\mathbb{R}^N$. In the present paper we investigate asymptotics of the excursion probability of non-smooth bivariate Gaussian random fields on $\mathbb{R}^N$, where the methods are totally different from the smooth case. Our work is also motivated by the recent increasing interest in using multivariate random fields for modeling multivariate measurements obtained at spatial locations (see, e.g., [@Gelfand_Diggle_Fuentes_Guttorp_2010], [@Wackernagel_2003]). Several classes of multivariate spatial models have been introduced by [@Gneiting_Kleiber_Schlather2010], [@Apanasovich_Genton_Sun_2012] and [@Kleiber_Nychka_2012]. We will show in Section 2 that the main results of this paper are applicable to bivariate Gaussian random fields with Matérn cross-covariances introduced by [@Gneiting_Kleiber_Schlather2010]. Furthermore, we expect that the excursion probabilities considered in this paper will have interesting statistical applications. Let $\{X(t),t \in \mathbb{R}^N\}$ be an $\mathbb{R}^2$-valued (not-necessarily stationary) Gaussian random field with $\E[X(t)]=\mathbf{0}$. We write $X(t)\triangleq(X_1(t),X_2(t))^T$ and define $$r_{ij}(s,t):= \E[X_i(s)X_j(t)],\ i,j=1,2.$$ Let $|t|:=\sqrt{\sum_{j=1}^Nt_j^2}$ be the $l^2$-norm of a vector $t\in \mathbb{R}^N$. Throughout this paper, we impose the following assumptions. - $r_{ii}(s,t)=1-c_i|t-s|^{\alpha_i}+o(|t-s|^{\alpha_i})$, where $\alpha_i\in (0,2)$ and $ c_i>0$ ($i=1,2$) are constants. - $|r_{ii}(s,t)|<1$ for all $|t-s|>0$, $i=1,2$. - $r_{12}(s,t)=r_{21}(s,t):=r(|t-s|)$. Namely, the cross correlation is isotropic. - The function $r(\cdot): [0,\infty)\rightarrow \mathbb{R}$ attains maximum only at zero with $r(0)=\rho\in (0,1)$, i.e., $|r(t)|<\rho$ for all $t>0$. Moreover, we assume $r'(0)=0, r''(0) < 0$ and there exists $\eta>0$, for any $s\in [0,\eta]$, $r''(s)$ exists and continuous. The cross correlation defined here is meaningful and common in spatial statistics where it is usually assumed that the correlation decreases as the distance between two observations increases (see, e.g., [@Gelfand_Diggle_Fuentes_Guttorp_2010], [@Gneiting_Kleiber_Schlather2010]). We only assume that the cross correlation is twice continuously differentiable around the area where the maximum correlation is attained, which is a weaker assumption than that in [@Cheng_Xiao_2014] who considered smooth bivariate Gaussian fields. For any compact sets $A_1, A_2 \subset \R^N$, we investigate the asymptotic behavior of the following excursion probability $$\label{bivariate excursion probablity} \mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u, \, \max_{t\in A_2} X_2(t)>u\bigg),\ \ \text{ as }\ u \rightarrow \infty.$$ The main results of this paper are Theorems 2.1 and 2.2 below, which demonstrate that the excursion probability (\[bivariate excursion probablity\]) depends not only on the smoothness parameters of the coordinate fields $X_1$ and $X_2$, but also on their maximum correlation $\rho$. The proofs of our Theorems 2.1 and 2.2 will be based on the double sum method. Compared with the earlier works of [@Ladneva_Piterbarg_2000], [@Anshin_2006] and [@Hashorva_Ji_2014], the main difficulty in the present paper is that the correlation function of $X_1$ and $X_2$ attains its maximum over the set $D:=\{(s,s): \, s\in A_1 \cap A_2\}$ which may have different geometric configurations. Several non-trivial modifications for carrying out the arguments in the double sum method have to be made. This paper raises several open questions. First, the cases of $\alpha_1= 2$ or $\alpha_2= 2$ have not been considered in this paper. The main difficulty is that, when $\alpha_1 = 2$, the sample functions of $X_1$ may either be differentiable or non-differentiable. In view of the method in this paper, the proof of Lemma \[uniformly conditional convergence\] on the uniform convergence of finite dimensional distributions for bivariate process breaks down when $\alpha_1= 2$ or $\alpha_2= 2$. Studying the asymptotics of (\[bivariate excursion probablity\]) when $\alpha_1= 2$ or/and $\alpha_2= 2$ requires different methods for dealing with differentiable or non-differentiable cases. When both $X_1$ and $X_2$ have twice continuously differentiable sample functions, this problem has been studied by [@Cheng_Xiao_2014]. The authors plan to study the remaining cases in their future work. Second, it would be interesting to study the excursion probabilities when $\{X(t),\,t \in \mathbb{R}^N\}$ is anisotropic or non-stationary, or taking values in $\R^d$ with $d \ge 3$. In the last problem, the covariance and cross-covariance structures become more complicated. We expect that the pairwise maximum cross correlations and the size (e.g., the Lebesgue measure) of the set where all the pairwise cross correlations attain their maximum values (if not empty) will play an important role. The rest of the paper is organized as follows. Section \[sec\_main results and discussion\] states the main theorems with some discussions and provides an application of the main theorems to the bivariate Gaussian fields with Matérn cross-covariances introduced by [@Gneiting_Kleiber_Schlather2010]. We state the key lemmas and provide proofs of our main theorems in Section \[sec\_proof of main results\]. The proofs of the lemmas are given in Section \[sec\_proof of lemmas\]. We end the introduction with some notation. For any $t\in \mathbb{R}^N$, $|t|$ denotes its $l^2$-norm. An integer vector $\mathbf{k}\in \mathbb{Z}^N$ is written as $\mathbf{k}=(k_1,...,k_N)$. For $\mathbf{k}\in \mathbb{Z}^N$ and $T \in \mathbb{R}_+= [0, \infty)$, we define the cube $[\mathbf{k}T,(\mathbf{k}+1)T]:= \prod_{i=1}^N [k_iT,(k_i+1)T]$. For any integer $n$, $mes_n(\cdot)$ denotes the $n$-dimensional Lebesgue measure. An unspecified positive and finite constant will be denoted by $C_0$. More specific constants are numbered by $C_1,C_2, \ldots.$ Main Results and Discussions {#sec_main results and discussion} ============================ We recall the Pickands constant first (see, [@Pickands_1969; @Piterbarg_1996; @Dieker_Yakir_2014]). Let $\chi = \{\chi(t),t\in \mathbb{R}^N\}$ be a (rescaled) fractional Brownian motion with Hurst index $\alpha/2 \in (0, 1)$, which is a centered Gaussian field with covariance function $\mathbb{E}[\chi(t)\chi(s)]=|t|^{\alpha}+|s|^{\alpha}-|t-s|^{\alpha}$. As in [@Ladneva_Piterbarg_2000] and [@Anshin_2006], we define for any compact sets $\mathbb{S}, \mathbb{T} \subset \mathbb{R}^N$, $$\label{H_alpha(S,T)} H_\alpha(\mathbb{S},\mathbb{T}):=\int_0^\infty e^x\cdot\mathbb{P}\Big(\sup_{s\in \mathbb{S}} \big(\chi(s)-|s|^\alpha \big)>x,\, \sup_{t\in \mathbb{T}} \big(\chi(t)-|t|^\alpha\big)>x\Big)\,dx.$$ Let $H_\alpha(\mathbb{T})=H_\alpha(\mathbb{T},\mathbb{T})$. Then, the Pickands constant is defined as $$\label{Pickands constant} H_\alpha:=\lim_{T\rightarrow \infty} \frac{H_\alpha([0,T]^N)}{T^N},$$ which is positive and finite (cf. [@Piterbarg_1996]). Before moving to the tail probability of extremes of a bivariate Gaussian random field, let us consider the tail probability of a standard bivariate Gaussian vector $(\xi,\eta)$ with correlation $\rho$. It is known that (see, e.g., [@Ladneva_Piterbarg_2000]) $$\begin{aligned} \mathbb{P}(\xi>u,\eta>u)=&\Psi(u,\rho)(1+o(1)),\ \text{as}\ u\rightarrow \infty,\end{aligned}$$ where $$\begin{aligned} \Psi(u,\rho):=\frac{(1+\rho)^2}{2\pi u^2 \sqrt{1-\rho^2}}\exp\left(-\frac{u^2}{1+\rho}\right).\end{aligned}$$ The exponential part of the tail probability above is determined by the correlation $\rho$. As shown by Theorems \[theorem\_Jordan measurable sets\] and \[theorem: Jordan measurable sets with mes zero\] below, similar phenomenon also happens for the tail probability of double extremes of $\{X(t),t \in \R^N\}$, where the exponential part is determined by the maximum cross correlation of the coordinate fields $X_1$ and $X_2$. We will study double extremes of $X$ on the domain $A_1\times A_2$ where $A_1,A_2$ are bounded Jordan measurable sets in $\mathbb{R}^N$. That is, the boundaries of $A_1$ and $A_2$ have $N$-dimensional Lebesgue measure $0$ (see, e.g., [@Piterbarg_1996], p.$105$). We only consider the case when $A_1\cap A_2\neq \emptyset$, in which the maximum cross correlation $\rho$ can be attained. If $mes_N(A_1\cap A_2)\neq 0$, we have the following theorem. \[theorem\_Jordan measurable sets\] Let $\{X(t),t \in \R^N\}$ be a bivariate Gaussian random field that satisfies the assumptions in Section \[Introduction\]. If $mes_N(A_1\cap A_2)\neq 0$, then as $u \rightarrow \infty$, $$\label{double extremes asymptotics_Jordan measurable sets} \begin{split} &\mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg) \\ &=(2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}c_1^{\frac{N}{\alpha_1}} c_2^{\frac{N}{\alpha_2}} mes_N(A_1\cap A_2) H_{\alpha_1}H_{\alpha_2} \\ &\qquad \times (1+\rho)^{-N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\, u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)(1+o(1)). \end{split}$$ If $mes_N(A_1\cap A_2)=0$, the above theorem is not informative. We have not been able to obtain a general explicit formula. Instead, we consider the special cases $$\label{Eq:intM} A_1=A_{1,M}\times \prod_{j=M+1}^N[S_j,T_j]\ \hbox{ and }\ A_2=A_{2,M}\times \prod_{M+1}^N[T_j,R_j],$$ where $A_{1,M}$ and $A_{2,M}$ are $M$ dimensional Jordan sets with $mes_M(A_{1,M}\cap A_{2,M}) \neq 0$ and $S_j\leq T_j\leq R_j,\, j=M+1, \ldots, N, \, 0\leq M\leq N-1$. For simplicity of notation, let $mes_0(\cdot)\equiv 1$. Our next theorem shows that the excursion probability is smaller than that in (\[double extremes asymptotics\_Jordan measurable sets\]) by a factor of $u^{M-N}$. \[theorem: Jordan measurable sets with mes zero\] Let $\{X(t),t \in \R^N\}$ be a bivariate Gaussian random field that satisfies the assumptions in Section \[Introduction\], and let $A_1,A_2$ be as in (\[Eq:intM\]) such that $mes_M(A_{1,M}\cap A_{2,M})>0$. Then as $u \to \infty$, $$\label{double extremes asymptotics_Jordan measurable sets_mes zero} \begin{split} &\mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg) \\ &=(2\pi)^{\frac{M}{2}}(-r''(0))^{-\frac{2N-M}{2}}c_1^{\frac{N}{\alpha_1}}c_2^{\frac{N} {\alpha_2}} H_{\alpha_1}H_{\alpha_2} mes_M(A_{1,M}\cap A_{2,M})\\ &\qquad \times (1+\rho)^{2N-M-\frac{2N}{\alpha_1}-\frac{2N}{\alpha_2}}\, u^{M+N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-2)}\Psi(u,\rho)(1+o(1)). \end{split}$$ The following are some additional remarks about Theorems \[theorem\_Jordan measurable sets\] and \[theorem: Jordan measurable sets with mes zero\]. - The excursion probability in (\[bivariate excursion probablity\]) depends on the region where the maximum cross correlation is attained. In our setting, the maximum cross correlation $\rho$ is attained on $D:=\{(s,s)\ |\ s\in A_1\cap A_2\}$. - For Theorem \[theorem: Jordan measurable sets with mes zero\], let us consider the extreme case when $M=0$, i.e., $A_1\cap A_2=\{(T_1,...,T_N)\}$. The exponential part still reaches $-\frac{u^2}{1+\rho}$, although the maximum cross correlation $\rho$ is attained at a single point. - To compare our results with [@Anshin_2006], we consider a centered Gaussian process $\{X(t)=(X_1(t),X_2(t)),t \in \mathbb{R}\}$ and $A_1=A_2=[0,T]$. In our setting, the cross correlation attains its maximum on the line $D=\{(s,s)\ |\ s\in [0,T]\}$, while in [@Anshin_2006] it only attains at a unique point in $[0,T]\times[0,T]$ because of the assumption $\mathbf{C2}$. This is the reason why the power of $u$ in our settings is $\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-3$ instead of $\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-4$ in [@Anshin_2006]. - Even though Theorem \[theorem: Jordan measurable sets with mes zero\] only deals with a special case of $A_1$, $A_2$ with $mes_N(A_1\cap A_2)=0$, its method of proof can be applied to more general cases provided some information on $A_1$ and $A_2$ is specified. The key step is to reevaluate the infinite series in Lemma \[lem\_Riemann Sum-2\]. We end this section with an application of Theorems \[theorem\_Jordan measurable sets\] and \[theorem: Jordan measurable sets with mes zero\] to bivariate Gaussian random fields with the Matérn correlation functions introduced by [@Gneiting_Kleiber_Schlather2010]. The Matérn correlation function $M(h|\nu,a)$, where $a>0,\nu>0$ are scale and smoothness parameters, is widely used to model covariance structures in spatial statistics. It is defined as $$M(h|\nu,a):=\frac{2^{1-\nu}}{\Gamma(\nu)}(a|h|)^\nu K_\nu(a|h|),$$ where $K_\nu$ is a modified Bessel function of the second kind. In [@Gneiting_Kleiber_Schlather2010], the authors introduce the full bivariate Matérn field $X(s)=(X_1(s),X_2(s))$, i.e., an $\mathbb{R}^2$-valued Gaussian random field on $\mathbb{R}^N$ with zero mean and matrix-valued covariance functions: $$C(h)=\left(\begin{array}{ll} C_{11}(h)&C_{12}(h)\\ C_{21}(h)&C_{22}(h) \end{array} \right),$$ where $C_{ij}(h):=\E[X_i(s+h)X_j(s)]$ are specified by $$\begin{aligned} C_{11}(h)&=&\sigma_1^2M(h|\nu_1,a_1),\\ C_{22}(h)&=&\sigma_2^2M(h|\nu_2,a_2),\\ C_{12}(h)&=&C_{21}(h)=\rho\sigma_1\sigma_2M(h|\nu_{12},a_{12}).\label{Matern Cross Covariance}\end{aligned}$$ According to [@Gneiting_Kleiber_Schlather2010], the above model is valid if and only if $$\label{validy cond} \begin{split} \rho^2 &\leq \frac{\Gamma(\nu_1+N/2)\Gamma(\nu_2+N/2)}{\Gamma(\nu_1)\Gamma(\nu_2)} \frac{\Gamma(\nu_{12})^2}{\Gamma(\nu_{12}+N/2)^2} \frac{a_1^{2\nu_1}a_2^{2\nu_2}}{a_{12}^{4\nu_{12}}}\\ & \qquad \ \ \times \inf_{t\geq 0}\frac{(a_{12}^2+t^2)^{2\nu_{12}+N}}{(a_1^2+t^2)^{\nu_1+N/2}(a_2^2+t^2)^{\nu_2+N/2}}. \end{split}$$ Especially, when $a_1=a_2=a_{12}$, condition (\[validy cond\]) is reduced to $$\rho^2\leq \frac{\Gamma(\nu_1+N/2)\Gamma(\nu_2+N/2)}{\Gamma(\nu_1)\Gamma(\nu_2)} \frac{\Gamma(\nu_{12})^2}{\Gamma(\nu_{12}+N/2)^2},$$ in which case the choice of $\rho$ is fairly flexible. Here we focus on a standardized bivariate Matérn field, that is, we assume $\sigma_1=\sigma_2=1$, $a_1=a_2=a_{12}=1$ and $\rho>0$. Moreover, we assume $\nu_1,\nu_2\in (0,1)$ and $\nu_{12}> 1$. In this case, the bivariate Matérn field $\{X(t), \, t \in \R^N\}$ satisfies the assumptions in Section \[Introduction\]. Indeed, Assumption i) in Section \[Introduction\] is satisfied since $$\begin{aligned} M(h|\nu_i,a)=1-c_i|t|^{2\nu_i}+o(|t|^{2\nu_i}),\end{aligned}$$ where $c_i=\frac{\Gamma(1-\nu_i)}{2^{2\nu_i}\Gamma(1+\nu_i)},\ i=1,2$ (see, e.g., [@Stein_1999Interpo], p. 32). Assumption ii) holds immediately if we use the following integral representation of $M(h|\nu,a)$ (see, e.g., [@Abramowitz_Stegun_1972], Section $9.6$) $$\begin{aligned} \label{integral form for Matern correlation} M(h|\nu,a)=\frac{2\Gamma(\nu+1/2)}{\sqrt{\pi}\Gamma(\nu)} \int_0^\infty \frac{\cos(a|h|r)}{(1+r^2)^{\nu+1/2}}\, dr.\end{aligned}$$ Assumption iii) holds by the definition of cross correlation in . For Assumption iv), we only need to check the smoothness of $M(h|\nu,a)$. By another integral representation of $M(h|\nu,a)$ (see, e.g., [@Abramowitz_Stegun_1972], Section $9.6$), i.e., $$\begin{aligned} M(h|\nu,a)=\frac{2^{1-2\nu}(a|h|)^{2\nu}}{\Gamma(\nu+1/2)\Gamma(\nu)}\int_1^\infty e^{-a|h|r}(r^2-1)^{\nu-1/2}\,dr,\end{aligned}$$ one can verify that $M(h|\nu,a)$ is infinitely differentiable when $|h|\neq 0$. Meanwhile, $M''(0|\nu,a)$ exists and is continuous when $\nu>1$ which can be proven by taking twice derivatives to the integral representation in w.r.t. $|h|$. So Assumption iv) holds. Applying Theorem \[theorem\_Jordan measurable sets\] to the double excursion probability of $X(s)$ over $[0,1]^N$, we have $$\begin{aligned} \label{double extremes for matern field} &&\mathbb{P}\Big(\max_{s\in [0,1]^N} X_1(s)>u,\, \max_{t\in [0,1]^N} X_2(t)>u\Big)\nonumber\\ &&=(2\pi)^{\frac{N}{2}}(-C_{12}''(0))^{-\frac{N}{2}}c_1^{\frac{N}{2\nu_1}}c_2^{\frac{N}{2\nu_2}}(1+\rho)^{-N(\frac{1} {\nu_1}+\frac{1}{\nu_2}-1)} H_{2\nu_1}H_{2\nu_2} \nonumber\\ &&\quad \times u^{N(\frac{1}{\nu_1}+\frac{1}{\nu_2}-1)}\Psi(u,\rho)(1+o(1)),\ \ \text{ as }\, u \rightarrow \infty.\end{aligned}$$ Secondly, when the two measurements are observed on two regions which only share part of boundaries, we use Theorem \[theorem: Jordan measurable sets with mes zero\] to obtain the excursion probability. For example, if $X_1(s)$ are observed on the region $[0,1]^N$ and $X_2(s)$ on $[0,1]^{N-1}\times [1,2]$, then as $u\rightarrow \infty$, $$\begin{aligned} \label{double extremes for matern field 2} &&\mathbb{P}\bigg(\max_{s\in [0,1]^N} X_1(s)>u,\, \max_{t\in [0,1]^{N-1}\times [1,2]} X_2(t)>u\bigg)\nonumber\\ &&=(2\pi)^{\frac{N-1}{2}}(-C_{12}''(0))^{-\frac{N+1}{2}}c_1^{\frac{N}{2\nu_1}}c_2^{\frac{N}{2\nu_2}} (1+\rho)^{1-N(\frac{1}{\nu_1}+\frac{1}{\nu_2}-1)} H_{2\nu_1}H_{2\nu_2} \nonumber\\ &&\qquad \times u^{N(\frac{1}{\nu_1}+\frac{1}{\nu_2}-1)-1}\Psi(u,\rho)(1+o(1)).\end{aligned}$$ Proofs of the main results {#sec_proof of main results} ========================== The proofs of Theorems \[theorem\_Jordan measurable sets\] and \[theorem: Jordan measurable sets with mes zero\] are based on the double sum method ([@Piterbarg_1996]) and the work of [@Ladneva_Piterbarg_2000]. Since the latter deals with the tail probability $\mathbb{P}(\max_{t\in[T_1,T_2]}X(t)>u,\, \max_{t\in[T_3,T_4]}X(t)>u)$ of a univariate Gaussian process $\{X(t),t\in \mathbb{R}\}$, their method is not sufficient for carrying out the double sum method for a bivariate random field. Lemmas \[main lemma\] and \[double double local extremes lemma\] below extend Lemma $1$ and Lemma $9$ in [@Ladneva_Piterbarg_2000] to the bivariate random field $\{(X_1(t), X_2(t)),\, t \in \mathbb{R}^N\}$. Moreover, we have strengthened the conclusions by showing that the convergence is uniform in certain sense. This will be useful for dealing with sums of local approximations around the regions where the maximum cross correlation is attained. The details will be illustrated in the proof of Theorem \[theorem\_Jordan measurable sets\] (see, e.g., , ). In the following lemmas, $\{X(t), \, t \in \R^N\}$ is a bivariate Gaussian random field as defined in Section \[Introduction\]. \[main lemma\] Let $s_u$ and $t_u$ be two $\mathbb{R}^N$-valued functions of $u$ and let $\tau_u:=t_u-s_u$. For any compact sets $\mathbb{S}$ and $\mathbb{T}$ in $\mathbb{R}^N$, we have $$\label{local double extremes asymptotics} \begin{split} &\mathbb{P}\bigg(\max_{s\in s_u+u^{-2/\alpha_1}\mathbb{S}} X_1(s)>u, \max_{t\in t_u+u^{-2/\alpha_2}\mathbb{T}} X_2(t)>u\bigg)\\ &=\frac{(1+\rho)^2}{2\pi \sqrt{1-\rho^2}}H_{\alpha_1}\left(\frac{c_1^{1/\alpha_1}\mathbb{S}}{(1+\rho)^{\frac{2}{\alpha_1}}}\right)H_{\alpha_2}\left(\frac{c_2^{1/\alpha_2}\mathbb{T}}{(1+\rho)^{\frac{2}{\alpha_2}}}\right)\\ & \qquad \times u^{-2} \exp\left(-\frac{u^2}{1+r(|\tau_u|)}\right)\, (1+o(1)), \end{split}$$ where $o(1)\rightarrow 0$ uniformly w.r.t. $\tau_u$ satisfying $|\tau_u|\leq C_0\sqrt{\log u}/u$ as $u \rightarrow \infty$. \[double double local extremes lemma\] Let $s_u,\, t_u$ and $\tau_u$ be the same as in Lemma \[main lemma\]. For all $T>0$, $\mathbf{m},\mathbf{n}\in \mathbb{Z}^N$, we have $$\begin{aligned} \label{double double local extremes asymptotics} &\mathbb{P}\bigg(\max_{s\in s_u+u^{-2/\alpha_1}[0,T]^N} X_1(s)>u, \, \max_{t\in t_u+u^{-2/\alpha_2}[0,T]^N} X_2(t)>u, \nonumber\\ &\qquad \ \max_{s\in s_u+u^{-2/\alpha_1}[\mathbf{m}T,(\mathbf{m}+1)T]}X_1(s)>u,\, \max_{t\in t_u+u^{-2/\alpha_2}[\mathbf{n}T,(\mathbf{n}+1)T]} X_2(t)>u\bigg)\nonumber\\ &=\frac{(1+\rho)^2}{2\pi \sqrt{1-\rho^2}\, u^{2}}\, e^{-\frac{u^2}{1+r(|\tau_u|)}}\,H_{\alpha_1}\bigg(\frac{c_1^{1/\alpha_1}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_1}}},\frac{c_1^{1/\alpha_1}[\mathbf{m}T,(\mathbf{m}+1)T]}{(1+\rho)^{\frac{2}{\alpha_1}}}\bigg)\nonumber\\ &\qquad \qquad \times\, H_{\alpha_2}\bigg(\frac{c_2^{1/\alpha_2}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_2}}},\frac{c_2^{1/\alpha_2}[\mathbf{n}T,(\mathbf{n}+1)T]}{(1+\rho)^{\frac{2}{\alpha_2}}}\bigg)\,\big(1+o(1)\big),\end{aligned}$$ where $H_\alpha(\cdot,\cdot)$ is defined in and $o(1)\rightarrow 0$ uniformly for all $s_u$ and $t_u$ that satisfy $|\tau_u|\leq C_0\sqrt{\log u}/u$ as $u \rightarrow \infty$. Proofs of Lemmas \[main lemma\] and \[double double local extremes lemma\] will be given in Section 4. Now we proceed to prove our main theorems. Let $\Pi=A_1\times A_2,\ \delta(u)=C\sqrt{\log u}/u$, where $C$ is a constant whose value will be determined later. Let $$\begin{aligned} \label{domain D} \mathcal{D}= \big\{(s,t)\in \Pi: |t-s|\leq \delta(u) \big\}.\end{aligned}$$ Since $$\begin{aligned} &\mathbb{P}\bigg(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\bigg) \leq \mathbb{P}\bigg(\max_{s\in A_1} X_1(s)>u,\, \max_{t\in A_2} X_2(t)>u\bigg)\nonumber\\ & \leq \mathbb{P}\bigg(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\bigg)%\nonumber\\ +\mathbb{P}\bigg(\bigcup_{(s,t)\in\Pi\setminus \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\bigg),\nonumber\end{aligned}$$ it is sufficient to prove that, by choosing appropriate constant $C$, we have $$\label{around maximum} \begin{split} &\mathbb{P}\bigg(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\bigg) \\ &=(2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}c_1^{\frac{N}{\alpha_1}}c_2^{\frac{N}{\alpha_2}}(1+\rho)^{-N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)} mes_N(A_1\cap A_2) \\ &\qquad \times H_{\alpha_1}H_{\alpha_2} \, u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)(1+o(1)),\ \ \text{ as } u \rightarrow \infty \end{split}$$ and $$\begin{aligned} \label{off maximum} \lim_{u \rightarrow \infty}\frac{\mathbb{P}\left(\bigcup_{(s,t)\in\Pi\setminus \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\right)}{\mathbb{P}\left(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\right)}=0.\end{aligned}$$ We prove first. For any fixed $T>0$ and $i=1,2$, let $d_i(u)=Tu^{-\frac{2}{\alpha_i}}$ and, for any $\mathbf{k} =(k_1,\ldots,k_N)\in \mathbb{Z}^N$, define $$\begin{aligned} \Delta^{(i)}_\mathbf{k} \triangleq \prod_{j=1}^N[k_jd_i(u),(k_j+1)d_i(u)] = [\mathbf{k}d_i(u),(\mathbf{k}+1)d_i(u)].\end{aligned}$$ Let $$\label{mathcal C} \mathcal{C}=\{(\mathbf{k} ,\mathbf{l}): \Delta^{(1)}_\mathbf{k}\times \Delta^{(2)}_\mathbf{l} \cap \mathcal{D}\neq \emptyset \}\ \text{ and }\ \mathcal{C}^\circ =\{(\mathbf{k},\mathbf{l}):\Delta^{(1)}_\mathbf{k}\times \Delta^{(2)}_\mathbf{l} \subseteq \mathcal{D} \}.$$ It is easy to see that $$\bigcup_{(\mathbf{k},\mathbf{l})\in \mathcal{C}^\circ}\Delta^{(1)}_\mathbf{k}\times \Delta^{(2)}_\mathbf{l}\subseteq \mathcal{D} \subseteq \bigcup_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}\Delta^{(1)}_\mathbf{k}\times \Delta^{(2)}_\mathbf{l}.$$ Thus the LHS of is bounded above by $$\label{upper bound around maximum} \begin{split} &\mathbb{P}\bigg(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u,X_2(t)>u)\}\bigg)\\ &\leq \sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}\mathbb{P}\bigg(\max_{s\in \Delta^{(1)}_\mathbf{k}} X_1(s)>u, \max_{t\in \Delta^{(2)}_\mathbf{l} } X_2(t)>u\bigg)\\ &= \sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}\mathbb{P}\bigg(\max_{s\in \mathbf{k}d_1(u)+\Delta^{(1)}_0} X_1(s)>u, \max_{t\in \mathbf{l}d_2(u)+\Delta^{(2)}_0 } X_2(t)>u\bigg). \end{split}$$ Let $$\begin{split} \tau_{\mathbf{k}\mathbf{l}}&:=\mathbf{l}d_2(u)-\mathbf{k}d_1(u)\\ &\ =(l_1d_2(u)-k_1d_1(u),...,l_Nd_2(u)-k_Nd_1(u)). \label{def_tau_kl} \end{split}$$ For $(\mathbf{k},\mathbf{l})\in \mathcal{C}$, $|\tau_{\mathbf{k}\mathbf{l}}|\leq \delta(u) +\sqrt{N}(d_1(u)+d_2(u))\leq 2\delta(u)$ for all $u$ large enough, since $d_1(u)=o(\delta(u))$ and $d_2(u)=o(\delta(u))$, as $u\rightarrow \infty$. By applying Lemma \[main lemma\] to the RHS of , we obtain $$\label{upper bound around maximum 2} \begin{split} &\mathbb{P}\bigg(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u,X_2(t)>u)\}\bigg)\\ &\leq \frac{(1+\rho)^2(1+\gamma(u))}{2\pi \sqrt{1-\rho^2}\, u^{2}}H_{\alpha_1}\left(\frac{c_1^{1/\alpha_1}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_1}}}\right)H_{\alpha_2}\left(\frac{c_2^{1/\alpha_2}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_2}}}\right)\\ &\qquad \ \times \sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}\exp\left(-\frac{u^2}{1+r(|\tau_{\mathbf{k}\mathbf{l}}|)}\right)\\ &=H_{\alpha_1}\left(\frac{c_1^{1/\alpha_1}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_1}}}\right)H_{\alpha_2}\left(\frac{c_2^{1/\alpha_2}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_2}}}\right) \Psi(u,\rho)(1+\gamma(u))\\ &\ \ \qquad \times \sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} \exp\left\{-u^2\left(\frac{1}{1+r(|\tau_{\mathbf{k}\mathbf{l}}|)}-\frac{1}{1+\rho}\right)\right\}, \end{split}$$ where the global error function $\gamma(u)\to 0$, as $u\rightarrow \infty$. The uniform convergence of in Lemma \[main lemma\] guarantees that the local error term $o(1)$ for each pair $(\mathbf{k},\mathbf{l})\in \mathcal{C}$ is uniformly bounded by $\gamma(u)$. The series in the last equality of is dealt by the following key lemma, which gives the power of the threshold $u$ in . \[lem\_Riemann Sum\] Recall the set $\mathcal{C}$ defined in . Let $$\label{Eq:h} h(u):=\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} \exp\left\{-u^2\left(\frac{1} {1+r(|\tau_{\mathbf{k}\mathbf{l}}|)}-\frac{1}{1+\rho}\right)\right\}.$$ Then, under the assumptions of Theorem \[theorem\_Jordan measurable sets\], we have $$\begin{aligned} \label{h(u)} h(u)&=(2\pi)^{N/2}(-r''(0))^{-N/2}(1+\rho)^NT^{-2N}mes_N(A_1\cap A_2)\nonumber\\ &\qquad \times u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}(1+o(1)),\ \ \text{ as }\ u\rightarrow \infty.\end{aligned}$$ Moreover, if we replace $\mathcal{C}$ in (\[Eq:h\]) by $\mathcal{C}^\circ$ defined in , then (\[h(u)\]) still holds. We defer the proof of Lemma \[lem\_Riemann Sum\] to Section \[sec\_proof of lemmas\] and continue with the proof of Theorem \[theorem\_Jordan measurable sets\]. Applying (\[h(u)\]) to , we obtain $$\label{upper bound around maximum 5} \begin{split} &\mathbb{P}\bigg(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\bigg)\\ &\leq (2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}(1+\rho)^NT^{-2N}mes_N(A_1\cap A_2) H_{\alpha_1}\left(\frac{c_1^{1/\alpha_1}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_1}}}\right)\\ &\ \ \ \times H_{\alpha_2}\left(\frac{c_2^{1/\alpha_2}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_2}}}\right)\, u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)(1+\gamma_1(u)), \end{split}$$ where $\gamma_1(u)\to 0$, as $u\rightarrow \infty$. Hence, $$\label{upper bound around maximum 3} \begin{split} &\limsup_{u\rightarrow \infty}\frac{\mathbb{P}\left(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\right)}{u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)}\\ &\leq (2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}(1+\rho)^N mes_N(A_1\cap A_2)\\ &\qquad \times T^{-2N}H_{\alpha_1}\left(\frac{c_1^{1/\alpha_1}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_1}}}\right)H_{\alpha_2}\left(\frac{c_2^{1/\alpha_2}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_2}}}\right). \end{split}$$ The above inequality holds for every $T>0$. Therefore, letting $T \rightarrow \infty$, we have $$\begin{aligned} \label{upper bound around maximum 4} &\limsup_{u\rightarrow \infty}\frac{\mathbb{P}\left(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u)\}\right)}{u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)} \leq (2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}} \nonumber\\ &\quad\qquad \times c_1^{\frac{N}{\alpha_1}}c_2^{\frac{N}{\alpha_2}}(1+\rho)^{-N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}mes_N(A_1\cap A_2)H_{\alpha_1}H_{\alpha_2}.\end{aligned}$$ On the other hand, the lower bound for LHS of can be derived as follows. Let $$\mathcal{B}=\{(\mathbf{k},\mathbf{l},\mathbf{k}',\mathbf{l}'):\, (\mathbf{k},\mathbf{l})\neq(\mathbf{k}',\mathbf{l}'), (\mathbf{k},\mathbf{l}),(\mathbf{k}',\mathbf{l}')\in \mathcal{C}\}.$$ By Bonferroni’s inequality and symmetric property of $\mathcal{B}$, the LHS of is bounded below by $$\begin{aligned} \label{lower bound around maximum} &\mathbb{P}\bigg(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u, X_2(t)>u\}\bigg)\nonumber\\ &\geq\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}^\circ}\mathbb{P}\bigg(\max_{s\in \Delta^{(1)}_\mathbf{k}} X_1(s)>u,\, \max_{t\in \Delta^{(2)}_\mathbf{l} } X_2(t)>u\bigg)\nonumber\\ & \qquad -\frac{1}{2}\sum_{(\mathbf{k},\mathbf{l},\mathbf{k}',\mathbf{l}')\in \mathcal{B}}\mathbb{P}\bigg(\max_{s\in \Delta^{(1)}_\mathbf{k}} X_1(s)>u, \max_{t\in \Delta^{(2)}_\mathbf{l} } X_2(t)>u, \\ &\qquad \qquad \qquad \max_{s\in \Delta^{(1)}_{\mathbf{k}'}} X_1(s)>u, \, \max_{t\in \Delta^{(2)}_{\mathbf{l}'} } X_2(t)>u\bigg)\nonumber\\ &\triangleq\Sigma_1-\Sigma_2. \nonumber\end{aligned}$$ Since $\mathcal{C}^\circ$ and ${\mathcal{C}}$ are almost the same, a similar argument as in $\sim$ shows that $\Sigma_1$ is bounded below by $$\begin{aligned} \Sigma_1&\geq(2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}(1+\rho)^N mes_N(A_1\cap A_2)T^{-2N}H_{\alpha_1}\left(\frac{c_1^{1/\alpha_1}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_1}}}\right)\nonumber\\ &\qquad \ \times H_{\alpha_2}\left(\frac{c_2^{1/\alpha_2}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha_2}}}\right)\, u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)(1-\gamma_2(u)),\end{aligned}$$ where $\gamma_2(u)\to 0$, as $u\rightarrow \infty$. Hence, letting $T\rightarrow \infty$, we have $$\label{lower bound around maximum 1} \begin{split} &\liminf_{u\rightarrow \infty}\frac{\Sigma_1}{u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)}\ \geq (2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}c_1^{\frac{N}{\alpha_1}}c_2^{\frac{N}{\alpha_2}}\\ &\qquad \qquad \times (1+\rho)^{-N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}mes_N(A_1\cap A_2)H_{\alpha_1}H_{\alpha_2}. \end{split}$$ Next, we consider $\Sigma_2$ in (\[lower bound around maximum\]). To simplify the notation, we let $$\begin{aligned} \label{I(k,l,k',l') def} I(\mathbf{k},\mathbf{l},\mathbf{k}',\mathbf{l}') &:= \mathbb{P}\bigg(\max_{s\in \Delta^{(1)}_\mathbf{k}} X_1(s)>u,\, \max_{t\in \Delta^{(2)}_\mathbf{l} } X_2(t)>u,\nonumber\\ & \qquad \qquad \qquad \qquad \max_{s\in \Delta^{(1)}_{\mathbf{k}'}} X_1(s)>u, \max_{t\in \Delta^{(2)}_{\mathbf{l}'} } X_2(t)>u\bigg).\nonumber\end{aligned}$$ For $\mathbf{m}=(m_1, \ldots,m_N)\in \mathbb{Z}^N$, let $$\label{def_mathcal_H(m)} \mathcal{H}_{\alpha,c}(\mathbf{m})\triangleq H_\alpha\left(\frac{c^{1/\alpha}[0,T]^N}{(1+\rho)^{\frac{2}{\alpha}}}, \frac{c^{1/\alpha}[\mathbf{m}T,(\mathbf{m}+1)T]}{(1+\rho)^{\frac{2}{\alpha}}}\right).$$ Rewriting $\Sigma_2$ and applying Lemma \[double double local extremes lemma\], we obtain $$\begin{aligned} \label{lower bound around maximum 2} %\Sigma_2=&\frac{1}{2}\sum_{(\mathbf{k},\mathbf{l},\mathbf{k}',\mathbf{l}')\in \mathcal{B}} I(\mathbf{k},\mathbf{l},\mathbf{k}',\mathbf{l}')\nonumber\\ \Sigma_2&=\frac{1}{2}\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}\bigg(\sum_{\substack{(\mathbf{k}',\mathbf{l}')\in \mathcal{C}\\\mathbf{k}'=\mathbf{k},\mathbf{l}'\neq \mathbf{l}}} +\sum_{\substack{(\mathbf{k}',\mathbf{l}')\in \mathcal{C}\\ \mathbf{k}'\neq \mathbf{k},\mathbf{l}'=\mathbf{l}}} +\sum_{\substack{(\mathbf{k}',\mathbf{l}')\in \mathcal{C}\\\mathbf{k}'\neq \mathbf{k},\mathbf{l}'\neq \mathbf{l}}}\bigg)I(\mathbf{k},\mathbf{l},\mathbf{k}',\mathbf{l}') \nonumber\\ =& \frac{(1+\rho)^2(1+\gamma_3(u))}{4\pi \sqrt{1-\rho^2}\,u^2}\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} e^{-\frac{u^2}{1+r(|\tau_{\mathbf{k}\mathbf{l}}|)}}\, \bigg(\mathcal{H}_{\alpha_1,c_1}(\mathbf{0})\sum_{\substack{(\mathbf{k}',\mathbf{l}')\in \mathcal{C}\\\mathbf{k}'=\mathbf{k},\mathbf{l}'\neq \mathbf{l}}}\mathcal{H}_{\alpha_2,c_2}(\mathbf{l}'-\mathbf{l})\nonumber\\ &+\mathcal{H}_{\alpha_2,c_2}(\mathbf{0})\sum_{\substack{(\mathbf{k}',\mathbf{l}')\in \mathcal{C}\\\mathbf{k}'\neq \mathbf{k},\mathbf{l}'=\mathbf{l}}}\mathcal{H}_{\alpha_1,c_1}(\mathbf{k}'-\mathbf{k})+ \sum_{\substack{(\mathbf{k}',\mathbf{l}')\in \mathcal{C}\\\mathbf{k}'\neq \mathbf{k},\mathbf{l}'\neq \mathbf{l}}}\mathcal{H}_{\alpha_1,c_1}(\mathbf{k}'-\mathbf{k})\mathcal{H}_{\alpha_2,c_2}(\mathbf{l}'-\mathbf{l})\bigg)\nonumber\\ \leq &\frac{(1+\rho)^2(1+\gamma_3(u))}{4\pi \sqrt{1-\rho^2}u^2}\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}e^{-\frac{u^2}{1+r(|\tau_{\mathbf{k}\mathbf{l}}|)}} \bigg(\mathcal{H}_{\alpha_1,c_1}(\mathbf{0})\sum_{\mathbf{n}\neq \mathbf{0}}\mathcal{H}_{\alpha_2,c_2}(\mathbf{n})\nonumber\\ &+\mathcal{H}_{\alpha_2,c_2}(\mathbf{0})\sum_{\mathbf{m}\neq \mathbf{0}}\mathcal{H}_{\alpha_1,c_1}(\mathbf{m})+\sum_{\mathbf{m}\neq \mathbf{0},\mathbf{n}\neq \mathbf{0}}\mathcal{H}_{\alpha_1,c_1}(\mathbf{m})\mathcal{H}_{\alpha_2,c_2}(\mathbf{n})\bigg),\end{aligned}$$ where $\gamma_3(u)\to 0$, as $u\rightarrow \infty$. According to the uniform convergence of , the local error term $o(1)$ for each pair $(\mathbf{k'},\mathbf{l'})\in \mathcal{C}$ is bounded above by $\gamma_3(u)$ . To estimate $\mathcal{H}_{\alpha,c}(\cdot)$, we make use of the following lemma, whose proof is again postponed to Section \[sec\_proof of lemmas\]. \[lem\_mathcal\_H(m)\] Recall $\mathcal{H}_{\alpha,c}(\cdot)$ defined in . Let $i_0= {\rm argmax}_{1\leq i \leq N}|m_i|$. Then there exist positive constants $C_1$ and $T_0$ such that for all $T \ge T_0$, $$\begin{aligned} \label{mathcal_H(0) bounds} &\mathcal{H}_{\alpha,c}(\mathbf{0})\leq C_1 T^N;\\ \label{mathcal_H(1) bounds} &\mathcal{H}_{\alpha,c}(\mathbf{m})\leq C_1 T^{N-\frac{1}{2}},\ \text{when}\ |m_{i_0}|=1;\\ \label{mathcal_H(m) bounds} &\mathcal{H}_{\alpha,c}(\mathbf{m})\leq C_1 T^{2N}e^{-\frac{c}{8(1+\rho)^2}(|m_{i_0}|-1)^\alpha T^\alpha}, \ \text{when}\ |m_{i_0}|\geq 2.\end{aligned}$$ Consequently, $$\label{mathcal_H(m) summation bounds} \sum_{\mathbf{m}\in \mathbb{Z}^N\setminus\{\mathbf{0}\}}\mathcal{H}_{\alpha,c}(\mathbf{m})\leq C_1 T^{N-\frac{1}{2}}.$$ Applying Lemmas \[lem\_Riemann Sum\] and \[lem\_mathcal\_H(m)\] to the RHS of , we obtain $$\begin{aligned} \Sigma_2&\leq \frac{C_0(1+\rho)^2(1+\gamma_3(u))}{4\pi \sqrt{1-\rho^2}\, u^2}T^{2N-\frac{1}{2}} \sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}\exp\left(-\frac{u^2}{1+r(|\tau_{\mathbf{k}\mathbf{l}}|)}\right)\nonumber\\ &\leq C_0 (2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}(1+\rho)^N mes_N(A_1\cap A_2)T^{-\frac{1}{2}} \nonumber\\ &\qquad \qquad \times u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)(1+\gamma_4(u)),\end{aligned}$$ where $\gamma_4(u)\to 0$, as $u\rightarrow \infty$. By letting $u\rightarrow \infty$ and $T\rightarrow\infty$ successively, we have $$\begin{aligned} \label{lower bound around maximum 22} &\limsup_{u\rightarrow \infty}\frac{\Sigma_2}{u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)}=0.\end{aligned}$$ By combining , and , we have $$\begin{aligned} \label{lower bound around maximum 3} &\liminf_{u\rightarrow \infty}\frac{\mathbb{P}\left(\bigcup_{(s,t)\in \mathcal{D}} \{X_1(s)>u,X_2(t)>u)\}\right)}{u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)}\nonumber\\ &\geq \liminf_{u\rightarrow \infty}\frac{\Sigma_1}{u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)}-\limsup_{u\rightarrow \infty}\frac{\Sigma_2}{u^{N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}\Psi(u,\rho)}\\ &\geq (2\pi)^{\frac{N}{2}}(-r''(0))^{-\frac{N}{2}}c_1^{\frac{N}{\alpha_1}}c_2^{\frac{N}{\alpha_2}}(1+\rho)^{-N(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1)}mes_N(A_1\cap A_2)H_{\alpha_1}H_{\alpha_2}. \nonumber\end{aligned}$$ It is now clear that follows from and . Now we prove . Define $$\label{Y(s,t)} Y(s,t):=X_1(s)+X_2(t),\ \text{for}\ (s,t)\in \Pi\setminus \mathcal{D}.$$ For $x=(s_1,t_1), y=(s_2,t_2) \in \Pi\setminus \mathcal{D} $, let $|x-y|=\sqrt{|s_1-s_2|^2+|t_1-t_2|^2}$. Then we can verify that $$\label{global holder} \mathbb{E}|Y(x)-Y(y)|^2\leq C_0 |x-y|^{\min(\alpha_1,\alpha_2)}, \ \forall x,y\in \Pi\setminus \mathcal{D}.$$ By applying Theorem $8.1$ in [@Piterbarg_1996], we obtain that the numerator of is bounded above by $$\begin{aligned} \label{off maximum 1} &&\mathbb{P}\bigg(\bigcup_{(s,t)\in\Pi\setminus \mathcal{D}} \{X_1(s)>u,X_2(t)>u)\}\bigg)\leq \mathbb{P}\left(\max_{(s,t)\in\Pi\setminus \mathcal{D}}Y(s,t)>2u\right)\nonumber\\ &&\leq C_0u^{-1+\frac{2N}{\min(\alpha_1,\alpha_2)}}\exp\left(-\frac{u^2}{1+\max_{(s,t)\in\Pi\setminus \mathcal{D}}r(|t-s|)}\right).\end{aligned}$$ Since $r(|t-s|)=\rho+\frac{1}{2}r''(0)|t-s|^2(1+o(1))$ and $r(\cdot)$ attains maximum only at zero, we have $$\begin{aligned} \label{off maximum 2} \max_{(s,t)\in\Pi\setminus \mathcal{D}}r(|t-s|) \leq \rho-\frac{1}{3}(-r''(0))\delta^2(u)\end{aligned}$$ for $ u$ large enough. So is at most $$\label{off maximum 3} \begin{split} &C_0u^{-1+\frac{2N}{\min(\alpha_1,\alpha_2)}}\exp\left(-\frac{u^2}{1+\rho-\frac{1}{3}(-r''(0))\delta^2(u)}\right) \\ &\leq C_0u^{-1+\frac{2N}{\min(\alpha_1,\alpha_2)}}\exp\left(-\frac{u^2}{1+\rho}\right)\exp\left(-\frac{\frac{1}{3}(-r''(0))\delta^2(u)u^2}{(1+\rho)^2}\right) \\ &=\frac{2\pi\sqrt{1-\rho^2}C_0}{(1+\rho)^2}u^{1+\frac{2N}{\min(\alpha_1,\alpha_2)}-\frac{-r''(0)}{3(1+\rho)^2}C^2}\Psi(u,\rho), \end{split}$$ where the inequality holds since $\frac{1}{x-y}\geq \frac{1}{x}+\frac{y}{x^2}, \forall x>y$. Compare with , it is easy to see holds if and only if $$1+\frac{2N}{\min(\alpha_1,\alpha_2)}-\frac{-r''(0)}{3(1+\rho)^2}C^2<N\Big(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-1\Big)$$ Hence, by choosing the constant $C$ satisfying $$C>\left[\frac{3(1+\rho)^2}{-r''(0)}\left(N\Big(\frac{2}{\min(\alpha_1,\alpha_2)}+1-\frac{2}{\alpha_1}-\frac{2}{\alpha_2}\Big)+1\right)_{+}\right]^\frac{1}{2},$$ we conclude . From the proof of Theorem \[theorem\_Jordan measurable sets\], we see that the exponential decaying rate of the excursion probability is only determined by the region where the maximum cross correlation is attained. In the case of $mes_N(A_1\cap A_2)=0$ but $ A_1\cap A_2 \neq \emptyset$, the exponential part, $e^{-\frac{u^2}{1+\rho}}$, remains the same. Yet, the dimension reduction of $A_1\cap A_2$ does affect the polynomial power of the excursion probability, which is determined by the quantity $$h(u)=\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} \exp\left\{-u^2\left(\frac{1}{1+r(|\tau_{\mathbf{k}\mathbf{l}}|)}-\frac{1}{1+\rho}\right)\right\}$$ in Lemma \[lem\_Riemann Sum\]. Under the assumptions of Theorem \[theorem: Jordan measurable sets with mes zero\], the set $\mathcal{C}$ and the behavior of $h(u)$ change. We will make use of the following lemma which plays the role of Lemma \[lem\_Riemann Sum\]. \[lem\_Riemann Sum-2\] Under the assumptions of Theorem \[theorem: Jordan measurable sets with mes zero\], we have $$\begin{aligned} \label{h(u)_mes zero} h(u)&=(2\pi)^{M/2}(-r''(0))^{M/2-N}(1+\rho)^{2N-M}T^{-2N}mes_M(A_{1,M}\cap A_{2,M})\nonumber\\ &\times u^{M+N\big(\frac{2}{\alpha_1}+\frac{2}{\alpha_2}-2\big)}(1+o(1)),\ \text{as}\ u\rightarrow \infty.\end{aligned}$$ Moreover, if we replace $\mathcal{C}$ with $\mathcal{C}^\circ$ defined in , then the above statement still holds. The rest of the proof of Theorem \[theorem: Jordan measurable sets with mes zero\] is the same as that of Theorem \[theorem\_Jordan measurable sets\] and it is omitted here. Proof of Lemmas {#sec_proof of lemmas} =============== For proving Lemma \[main lemma\], we will make use of the following \[uniformly conditional convergence\] Let $s_u$ and $t_u$ be two $\mathbb{R}^N$-valued functions of $u$ and let $\tau_u:=t_u-s_u$. For any compact rectangles $\mathbb{S}$ and $\mathbb{T}$ in $\mathbb{R}^N$, define $$\begin{aligned} \label{tangent field} \xi_u(s)&:=u(X_1(s_u+u^{-2/\alpha_1}s)-u)+x,\quad \forall \, s\in \mathbb{S},\nonumber\\ \eta_u(t)&:=u(X_2(t_u+u^{-2/\alpha_2}t)-u)+y,\quad \forall \, t\in \mathbb{T}\end{aligned}$$ and for any $t \in \mathbb{R}^N,$ let $$\label{bivariate FBM} \xi(t):=\sqrt{c_1}\chi_1(t)-\frac{c_1|t|^{\alpha_1}}{1+\rho},\ \ \ \eta(t):=\sqrt{c_2}\chi_2(t)-\frac{c_2|t|^{\alpha_2}}{1+\rho},$$ where $\chi_1(t),\chi_2(t)$ are two independent fractional Brownian motions with indices $\alpha_1/2$ and $\alpha_2/2$, respectively. Then, the finite dimensional distributions (abbr. f.d.d.) of $(\xi_u(\cdot),\eta_u(\cdot))$, given $X_1(s_u)=u-\frac{x}{u},\, X_2(t_u)=u-\frac{y}{u}$, converge uniformly to the f.d.d. of $(\xi(\cdot),\eta(\cdot))$ for all $s_u$ and $t_u$ that satisfy $|\tau_u|\leq C_0\sqrt{\log u}/u$. Furthermore, as $u \rightarrow \infty$, $$\begin{aligned} \label{cond maximum unif convergence} &\mathbb{P}\bigg(\max_{s\in \mathbb{S}}\xi_u(s)>x,\max_{t\in \mathbb{T}}\eta_u(t)>y\ \Big |\ X_1(s_u) =u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\bigg)\nonumber\\ &\rightarrow \mathbb{P}\bigg(\max_{s\in \mathbb{S}}\xi(s)>x,\max_{t\in \mathbb{T}}\eta(t)>y\bigg),\end{aligned}$$ where the convergence is uniform for all $s_u$ and $t_u$ that satisfy $|\tau_u|\leq C_0\sqrt{\log u}/u$. First, we prove the uniform convergence of finite dimensional distributions. Given $X_1(s_u)=u-\frac{x}{u},\, X_2(t_u)=u-\frac{y}{u}$, the distribution of the bivariate random field $(\xi_u(\cdot),\eta_u(\cdot))$ is still Gaussian. Thanks to the following lemma (whose proof will be given at the end of this section), it suffices to prove the uniform convergence of conditional mean and conditional variance. \[lem\_uniform convergence of f.d.d.\] Let $X(u,\tau_u)=(X_1(u,\tau_u),\ldots,X_n(u,\tau_u))^T$ be a Gaussian random vector with mean $\mu(u,\tau_u)=(\mu_1(u,\tau_u),...,\mu_n(u,\tau_u)^T$ and covariance matrix $\Sigma(u,\tau_u)$ with entries $\sigma_{ij}(u,\tau_u) ={\rm Cov}(X_i(u,$ $\tau_u),X_j(u,\tau_u)),\ i,j=1,2,\dots,n$. Similarly, let $X=(X_1,\ldots,X_n)^T$ be a Gaussian random vector with mean $\mu=(\mu_1,...,\mu_n)$ and covariance matrix $\Sigma=(\sigma_{ij})_{i,j=1,...,n}$. Assume that $\Sigma$ is non-singular. Let $F_u(\cdot)$ and $F(\cdot)$ be the distribution functions of $X(u,\tau_u)$ and $X$ respectively. If $$\begin{aligned} \label{mean_var_unif_conver} &\lim_{u\rightarrow \infty} \max_{\tau_u}|\mu_j(u,\tau_u)-\mu_j|=0,\nonumber\\ &\lim_{u\rightarrow \infty} \max_{\tau_u}|\sigma_{ij}(u,\tau_u)-\sigma_{ij}|=0,\ \ i,j=1,2,\dots, n,\end{aligned}$$ then for any $x\in \mathbb{R}^N$, $$\begin{aligned} \lim_{u\rightarrow \infty} \max_{\tau_u}|F_u(x)-F(x)|=0.\end{aligned}$$ We continue with the proof of Lemma \[uniformly conditional convergence\] and postpone the proof of Lemma \[lem\_uniform convergence of f.d.d.\] to the end of this section. Recall that, for two random vectors $X,Y\in \mathbb{R}^m$, their covariance is defined as ${\rm Cov}(X,Y):=\mathbb{E}[(X-\mathbb{E}X)(Y-\mathbb{E}Y)^T]$ and the variance matrix of $X$ is defined as ${\rm Var}(X):={\rm Cov}(X,X)$. The conditional mean of $(\xi_u(t),\eta_u(t))^T$ given $X_1(s_u)=u-\frac{x}{u},\, X_2(t_u)=u-\frac{y}{u}$, is $$\begin{aligned} \label{conditional mean} &\mathbb{E}\left( \begin{array}{l} \xi_u(t)\\ \eta_u(t) \end{array} \left | \begin{array}{l} X_1(s_u)=u-\frac{x}{u}\\ X_2(t_u)=u-\frac{y}{u} \end{array} \right. \right) = \mathbb{E}\bigg( \begin{array}{l} \xi_u(t)\\ \eta_u(t) \end{array} \bigg) \nonumber\\ & \qquad +{\rm Cov}\left(\left( \begin{array}{l} \xi_u(t)\\ \eta_u(t) \end{array} \right), \left( \begin{array}{ll} X_1(s_u)\\ X_2(t_u) \end{array} \right) \right) \left({\rm Var}\left( \begin{array}{ll} X_1(s_u)\\ X_2(t_u) \end{array} \right) \right)^{-1}\left( \begin{array}{l} u-\frac{x}{u}\\ u-\frac{y}{u} \end{array} \right)\nonumber\\ &=\left( \begin{array}{l} -u^2+x\\ -u^2+y \end{array} \right)+\frac{u}{1-r^2(|\tau_u|)}\left( \begin{array}{ll} r_{11}(s_u+u^{-2/\alpha_1}t,s_u)&r(|\tau_u-u^{-2/\alpha_1}t|)\\ r(|\tau_u+u^{-2/\alpha_2}t|)& r_{22}(t_u+u^{-2/\alpha_2}t,t_u) \end{array} \right)\nonumber\\ &\qquad \times \left( \begin{array}{cc} 1&-r(|\tau_u|)\\ -r(|\tau_u|)&1 \end{array} \right)\left( \begin{array}{l} u-\frac{x}{u}\\ u-\frac{y}{u} \end{array} \right)\nonumber\\ & \triangleq \left( \begin{array}{l} a_1(u)\\ a_2(u) \end{array} \right),\end{aligned}$$ where $$\begin{aligned} a_1(u)=& -\frac{u^2(1-r_{11}(s_u+u^{-2/\alpha_1}t,s_u))-u^2(r(|\tau_u-u^{-2/\alpha_1}t|)-r(|\tau_u|))}{1+r(|\tau_u|)}\nonumber\\ &+\frac{(x-yr(|\tau_u|))(1-r_{11}(s_u+u^{-2/\alpha_1}t,s_u))}{1-r^2(|\tau_u|)}\nonumber\\ &+\frac{(y-xr(|\tau_u|))(r(|\tau_u|)-r(|\tau_u-u^{-2/\alpha_1}t|))}{1-r^2(|\tau_u|)}\end{aligned}$$ and $$\begin{aligned} a_2(u)=&-\frac{u^2(1-r_{22}(t_u+u^{-2/\alpha_2}t,t_u))-u^2(r(|\tau_u+u^{-2/\alpha_2}t|)-r(|\tau_u|))}{1+r(|\tau_u|)}\nonumber\\ &+\frac{(y-xr(|\tau_u|))(1-r_{22}(t_u+u^{-2/\alpha_1}t,t_u))}{1-r^2(|\tau_u|)}\nonumber\\ &+\frac{(x-yr(|\tau_u|))(r(|\tau_u|)-r(|\tau_u+u^{-2/\alpha_2}t|))}{1-r^2(|\tau_u|)}.\end{aligned}$$ Applying the mean value theorem twice, we see that for $u$ large enough, $$\begin{aligned} \label{cross corr diff} &&|r(|\tau_u+u^{-2/\alpha}t|)-r(|\tau_u|)|\leq |u^{-2/\alpha}t|\cdot \max_{\substack{s\ \text{is between}\\ |\tau_u|\ \text{and}\ |\tau_u+u^{-2/\alpha}t|}}|r'(s)|\nonumber\\ &&\leq |u^{-2/\alpha}t|\cdot \max_{|s|\leq 2C_0\sqrt{\log u}/u}|r'(s)| \nonumber\\ &&\leq|u^{-2/\alpha}t|\cdot \max_{|s|\leq 2C_0\sqrt{\log u}/u}\left(|s|\cdot\max_{ |t|\leq |s|}|r''(t)|\right)\nonumber\\ &&\leq2 C_0|t|\sqrt{\log u}\cdot u^{-1-2/\alpha}\cdot \max_{|t| \leq 2 C_0\sqrt{\log u}/u}|r''(t)|\nonumber\\ &&\leq 4 C_0|r''(0)||t|\sqrt{\log u}\cdot u^{-1-2/\alpha},\end{aligned}$$ where the second inequality holds because of $u^{-2/\alpha}=o(\sqrt{\log u}/u)$, as $u \rightarrow \infty$ and the last inequality holds since $r''(\cdot)$ is continuous in a neighborhood of zero. Thus implies that, as $u \rightarrow \infty$, $$\label{cross corr diff 2} u^2|r(|\tau_u+u^{-2/\alpha}t|)-r(|\tau_u|)|\leq 4 C_0|r''(0)||t|\sqrt{\log u}\cdot u^{1-2/\alpha}\rightarrow 0,$$ where the convergence is uniform for all $s_u$ and $t_u$ that satisfy $|\tau_u|\leq C_0\sqrt{\log u}/u$. We also notice that for $ i=1,2$ and all $s\in \mathbb{R}^N$, $$\label{auto corr diff} 1-r_{ii}(s+u^{-2/\alpha}t,s)=c_iu^{-2}|t|^{\alpha_i}+o(u^{-2}), \, \text{as}\ u\rightarrow \infty.$$ By , , and , we conclude that, as $u \rightarrow \infty,$ $$\begin{aligned} \label{conditional mean 2} &\mathbb{E}\left( \begin{array}{l} \xi_u(t)\\ \eta_u(t) \end{array} \left | \begin{array}{l} X_1(s_u)=u-\frac{x}{u}\\ X_2(t_u)=u-\frac{y}{u} \end{array} \right. \right)\rightarrow\left( \begin{array}{l} -\frac{c_1|t|^{\alpha_1}}{1+\rho}\\ -\frac{c_2|t|^{\alpha_2}}{1+\rho} \end{array} \right),\end{aligned}$$ where the convergence is uniform w.r.t. $s_u$ and $t_u$ satisfying $|\tau_u|\leq C_0\sqrt{\log u}/u$. Next, we consider the conditional covariance matrix of $(\xi_u(t)-\xi_u(s),\eta_u(t)-\eta_u(s))^T$. $$\begin{aligned} \label{conditional variance} &{\rm Var}\left(\left( \begin{array}{l} \xi_u(t)-\xi_u(s)\\ \eta_u(t)-\eta_u(s) \end{array} \right)\left | \begin{array}{l} X_1(s_u)\\ X_2(t_u) \end{array} \right. \right)\nonumber\\ &={\rm Var}\left( \begin{array}{l} \xi_u(t)-\xi_u(s)\\ \eta_u(t)-\eta_u(s) \end{array} \right) - {\rm Cov}\left(\left( \begin{array}{l} \xi_u(t)-\xi_u(s)\\ \eta_u(t)-\eta_u(s) \end{array} \right), \left( \begin{array}{ll} X_1(s_u)\\ X_2(t_u) \end{array} \right) \right)\nonumber\\ &\qquad \times{\rm Var}\left( \begin{array}{l} X_1(s_u)\\ X_2(t_u) \end{array} \right)^{-1} {\rm Cov}\left(\left( \begin{array}{l} \xi_u(t)-\xi_u(s)\\ \eta_u(t)-\eta_u(s) \end{array} \right), \left( \begin{array}{ll} X_1(s_u)\\ X_2(t_u) \end{array} \right) \right)^T.\end{aligned}$$ Let $h_u(t,s):=r(|\tau_u+u^{-2/\alpha_2}t-u^{-2/\alpha_1}s|)$. Applying and , we obtain $$\begin{aligned} \label{conditional variance 1} &{\rm Var}\left( \begin{array}{l} \xi_u(t)-\xi_u(s)\\ \eta_u(t)-\eta_u(s) \end{array} \right)\nonumber\\ &=\left( \begin{array}{ll} 2u^2(1-r_{11}(s_u+ &u^2(h_u(t,t)-h_u(s,t)\\ \quad u^{-2/\alpha_1}s,\, s_u+u^{-2/\alpha_1}t))&\quad -h_u(t,s)+h_u(s,s))\\ \vspace{-2mm}\\ u^2(h_u(t,t)-h_u(s,t)&2u^2(1-r_{22}(t_u+\\ \quad-h_u(t,s)+h_u(s,s))&\quad u^{-2/\alpha_2}s,\, t_u+u^{-2/\alpha_2}t)) \end{array} \right)\nonumber\\ &=\left( \begin{array}{cc} 2c_1|t-s|^{\alpha_1}(1+o(1))&o(1)\\ o(1)&2c_2|t-s|^{\alpha_2}(1+o(1)) \end{array} \right),\end{aligned}$$ where $o(1)$ converges to zero uniformly w.r.t. $\tau_u$ satisfying $|\tau_u|\leq C_0\sqrt{\log u}/u$, as $u \rightarrow \infty$. Also, we have $$\begin{aligned} \label{conditional variance 2} &{\rm Cov}\left[\left( \begin{array}{l} \xi_u(t)-\xi_u(s)\\ \eta_u(t)-\eta_u(s) \end{array}\right), \left( \begin{array}{l} X_1(s_u)\nonumber\\ X_2(t_u) \end{array} \right)\right]\nonumber\\ &=\left( \begin{array}{ll} u(r_{11}(s_u+u^{-2/\alpha_1}t,s_u)&u(r(|\tau_u-u^{-2/\alpha_1}t|)\\ \quad -r_{11}(s_u+u^{-2/\alpha_1}s,s_u))&\quad -r(|\tau_u-u^{-2/\alpha_1}s|))\\ \vspace{-2mm}\\ u(r(|\tau_u+u^{-2/\alpha_2}t|)&u(r_{22}(t_u+u^{-2/\alpha_2}t,t_u)\\ \quad-r(|\tau_u+u^{-2/\alpha_2}s|))&\quad -r_{22}(t_u+u^{-2/\alpha_2}s,t_u)) \end{array} \right)\nonumber\\ &=\left( \begin{array}{ll} o(1)&o(1)\\ o(1)&o(1) \end{array} \right),\end{aligned}$$ $\text{as}\ u\rightarrow \infty,$ and $$\begin{aligned} \label{conditional variance 3} &{\rm Var}\left( \begin{array}{l} X_1(s_u)\\ X_2(t_u) \end{array} \right)^{-1}=\frac{1}{1-r^2(|\tau_u|)} \left( \begin{array}{ll} 1&-r(|\tau_u|)\\ -r(|\tau_u|)&1 \end{array} \right).\end{aligned}$$ By – , we conclude that as $u\rightarrow \infty$, $$\begin{aligned} \label{conditional variance 4} {\rm Var}\left(\left( \begin{array}{l} \xi_u(t)-\xi_u(s)\\ \eta_u(t)-\eta_u(s) \end{array} \right)\bigg | \begin{array}{l} X_1(s_u)\\ X_2(t_u) \end{array} \right)\rightarrow\left( \begin{array}{cc} 2c_1|t-s|^{\alpha_1}&0\\ 0&2c_2|t-s|^{\alpha_2} \end{array} \right),\end{aligned}$$ where the convergence is uniform w.r.t. $\tau_u$ satisfying $|\tau_u|\leq C_0\sqrt{\log u}/u$. Hence, the uniform convergence of f.d.d. in Lemma \[uniformly conditional convergence\] follows from , and Lemma \[lem\_uniform convergence of f.d.d.\]. Now we prove the second part of Lemma \[uniformly conditional convergence\]. The continuous mapping theorem (see, e.g., [@Billingsley_1968], p. $30$) can be used to prove holds when $s_u$ and $t_u$ are fixed. Since we need to prove uniform convergence w.r.t. $s_u$ and $t_u$, we use a discretization method instead. Let $$\begin{aligned} \label{f(u,x,y)} &f(u,x,y):=\mathbb{P}\bigg(\max_{s\in \mathbb{S}}\xi_u(s)>x,\, \max_{t\in \mathbb{T}}\eta_u(t)>y\ \Big |\nonumber\\ &\qquad \qquad \qquad \qquad X_1(s_u)=u-\frac{x}{u}, X_2(t_u)=u-\frac{y}{u}\bigg)\end{aligned}$$ and $$\begin{aligned} \label{f(x,y)} &f(x,y) :=\mathbb{P}\Big(\max_{s\in \mathbb{S}}\xi(s)>x,\, \max_{t\in\mathbb{T}}\eta(t)>y\Big).\end{aligned}$$ Without loss of generality, suppose that $\mathbb{S}=[a,b]^N$ and $\mathbb{T}=[c,d]^N$, where $a<b,c<d$. For any $\delta\in (0,1),$ let $m=\left\lfloor \frac{b-a}{\delta}\right\rfloor$, $n=\left\lfloor \frac{d-c}{\delta}\right\rfloor$ and let $$\begin{aligned} \mathcal{S}_m&:= \big\{s_{\mathbf{k}}\ |\ s_{\mathbf{k}}=(x_{k_1},\,..., \,x_{k_N}), \ \mathbf{k}=(k_1,...,k_N)\in \{0,1,...,m+1\}^N\big\},\\ \mathcal{T}_n&:= \big\{t_\mathbf{l}\ |\ t_\mathbf{l}=(y_{l_1},\,...,\,y_{l_N}), \ \mathbf{l}=(l_1,...,l_N)\in \{0,1,...,n+1\}^N \big\},\end{aligned}$$ where $x_i, y_i$ are defined as $$\begin{aligned} a&=x_0<x_1<\cdots<x_m\leq x_{m+1}=b,\ x_i=a+i\delta, \ i=0,1,\ldots,m,\nonumber\\ c&=y_0<y_1<\cdots<y_n\leq y_{n+1}=d, \ \ y_i=c+i\delta, \ i=0,1,\ldots,n.\end{aligned}$$ Then $[a,b]^N\times[c,d]^N$ can be divided into $\delta$-cubes with vertices in $\mathcal{S}_m\times \mathcal{T}_n$. The function $f(u,x,y)$ in is bounded below by $$\begin{aligned} \label{lower bound of cmuc} &f_{m,n}(u,x,y):=\mathbb{P}\bigg(\max_{s\in \mathcal{S}_m}\xi_u(s)>x,\, \max_{t\in \mathcal{T}_n}\eta_u(t)>y\ \Big | \nonumber\\ &\qquad\qquad\qquad\qquad \qquad \ X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\bigg)\end{aligned}$$ and is bounded above by $g_{m,n}(u,x,y)$ which is defined as $$\begin{aligned} \label{upper bound of cmuc} &\mathbb{P}\bigg(\max_{s\in \mathcal{S}_m}\xi_u(s)>x-\epsilon,\, \max_{t\in \mathcal{T}_n}\eta_u(t)>y-\epsilon\, \Big |\, X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\bigg)\nonumber\\ &+\mathbb{P}\bigg(\max_{s\in \mathbb{S}}\xi_u(s)> x,\, \max_{s\in \mathcal{S}_m}\xi_u(s)\leq x-\epsilon\, \Big|\, X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\bigg)\nonumber\\ &+\mathbb{P}\bigg(\max_{t\in \mathbb{T}}\eta_u(t)>y,\, \max_{t\in \mathcal{T}_n}\eta_u(t)\leq y-\epsilon\, \Big|\, X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\bigg)\nonumber\\ & \triangleq f_{m,n}(u,x-\epsilon,y-\epsilon)+s_{m,n}(u,x,y)+t_{m,n}(u,x,y),\end{aligned}$$ where $\epsilon>0$ is any small constant. Let $$\begin{aligned} \label{lower bound of cmuc 1} &f_{m,n}(x,y) :=\mathbb{P}\bigg(\max_{s\in \mathcal{S}_m}\xi(s)>x,\, \max_{t\in \mathcal{T}_n}\eta(t)>y\bigg).\end{aligned}$$ Since the finite dimensional distributions of $(\xi_u(\cdot), \eta_u(\cdot))$ converge uniformly to those of $(\xi(\cdot),\eta(\cdot))$, we have $$\label{lower bound of cmuc 3} \lim_{u\rightarrow \infty} \max_{|\tau_u|\leq C_0\sqrt{\log u}/u}|f_{m,n}(u,x,y)-f_{m,n}(x,y)|=0.$$ The continuity of the trajectory of $(\xi(\cdot),\eta(\cdot))$ yields$$\label{lower bound of cmuc 4} \lim_{\substack {m\rightarrow \infty \\ n\rightarrow \infty }}f_{m,n}(x,y)=f(x,y).$$ By and , we conclude $$\label{lower bound of cmuc 5} \lim_{\substack {m\rightarrow \infty \\ n\rightarrow \infty }}\lim_{u\rightarrow \infty} \max_{|\tau_u|\leq C_0\sqrt{\log u}/u}|f_{m,n}(u,x,y)-f(x,y)|=0.$$ Let us consider the conditional probability $s_{m,n}(u,x,y)$ in . $$\begin{aligned} \label{upper bound of cmuc 1} &s_{m,n}(u,x,y)\nonumber\\ &\leq \mathbb{P}\bigg(\max_{|s-t|\leq \delta}|\xi_u(s)-\xi_u(t)|>\epsilon\, \left| \, X_1(s_u)=u-\frac{x}{u},\,X_2(t_u)=u-\frac{y}{u}\right.\bigg)\nonumber\\ &\leq \frac{1}{\epsilon}\mathbb{E}\left(\max_{|s-t|\leq \delta}|\xi_u(s)-\xi_u(t)|\, \left | \, X_1(s_u)=u-\frac{x}{u},\, X_2(t_u)=u-\frac{y}{u}\right.\right)\nonumber\\ &=\frac{1}{\epsilon}\mathbb{E}_{\mathbb{P}_u}\bigg(\max_{|s-t|\leq \delta}|x(s)-x(t)|\bigg),\end{aligned}$$ where $\mathbb{P}_u$ is the probability measure on $(C(\mathbb{S}),\mathcal{B}(C(\mathbb{S}))$ defined as $$\begin{aligned} &\mathbb{P}_u(A):=\mathbb{P}\bigg(\xi_u(\cdot)\in A\ \left |\ X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\right.\bigg),\end{aligned}$$ for all $A \in \mathcal{B}(C(\mathbb{S}))$ and $x(\cdot)$ is the coordinate random element on $(C(\mathbb{S}), \mathcal{B}(C(\mathbb{S})),$ $\mathbb{P}_u)$, i.e., $x(t,\omega)=\omega (t), \ \forall \omega \in C(\mathbb{S})$ and $t \in \mathbb{S}$. Consider the canonical metric $$\begin{aligned} d_u(s,t):&= \left[\mathbb{E}_{\mathbb{P}_u} \big(|x(s)-x(t)|^2\big)\right]^{1/2}\nonumber\\ &=\left[\mathbb{E}\left(|\xi_u(s)-\xi_u(t)|^2\ \left | \ X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\right.\right)\right]^{1/2}. \nonumber\end{aligned}$$ By , for $u$ large enough and all $s_u$, $t_u$ such that $|\tau_u|\leq C_0\sqrt{\log u}/u$, we have $$\label{Eq:canmetric} d_u(s,t)\leq 2\sqrt{c_1}|s-t|^{\alpha_1/2},$$ which implies $\forall s\in \mathbb S$, $$\begin{aligned} \big\{t\in \mathbb S\ |\ |t-s|\leq (\epsilon/2\sqrt{c_1})^{\frac{2}{\alpha_1}}\big\}\subseteq \{t\in \mathbb S\ |\ d_u(s,t)\leq \epsilon\}.\end{aligned}$$ Hence $$\label{Eq:N} N_{d_u}(\mathbb{S},\epsilon)\leq C_0 \epsilon^{-2N/\alpha_1},$$ where $N_{d_u}(\mathbb{S},\epsilon)$ denotes the minimum number of $d_u$-balls with radius $\epsilon$ that are needed to cover $\mathbb{S}$. By Dudley’s Theorem (see, e.g., Theorem 1.3.3 in [@Adler_Taylor_2007]) and (\[Eq:canmetric\]), we have $$\label{upper bound of cmuc 2} \mathbb{E}_{\mathbb{P}_u} \bigg(\max_{|s-t|\leq \delta}|x(s)-x(t)|\bigg) \leq K\int_0^{2\sqrt{c_1}\delta^{\alpha_1/2}} \sqrt{\log N_{d_u}(\mathbb{S},\epsilon)}\, d\epsilon,$$ where $K< \infty$ is a constant (which does not depend on $\delta$) and, thanks to (\[Eq:N\]), the last integral goes to 0 as $\delta \rightarrow 0$ (or, equivalently, as $m\rightarrow \infty, n\rightarrow \infty$). By and , we conclude that $$\label{upper bound of cmuc 4} \lim_{\substack {m\rightarrow \infty \\ n\rightarrow \infty }}\limsup_{u\rightarrow \infty} \max_{|\tau_u|\leq C_0\sqrt{\log u}|/u}|s_{m,n}(u,x,y)|=0.$$ A similar argument shows that $$\label{upper bound of cmuc 5} \lim_{\substack {m\rightarrow \infty \\ n\rightarrow \infty }}\limsup_{u\rightarrow \infty} \max_{|\tau_u|\leq C_0\sqrt{\log u}|/u}|t_{m,n}(u,x,y)|=0.$$ Since $$\begin{aligned} \label{limit of f(u,x,y)} &|f(u,x,y)-f(x,y)|\leq |f_{m,n}(u,x,y)-f(x,y)|+|g_{m,n}(u,x,y)-f(x,y)|\nonumber\\ &\leq |f_{m,n}(u,x,y)-f(x,y)|+|f_{m,n}(u,x-\epsilon,y-\epsilon)-f(x-\epsilon,y-\epsilon)|\nonumber\\ &\qquad + |f(x-\epsilon,y-\epsilon)-f(x,y)|+|s_{m,n}(u,x,y)|+|t_{m,n}(u,x,y)|,\end{aligned}$$ we combine , and to obtain $$\begin{aligned} \label{limit of f(u,x,y) 1} &\limsup_{u\rightarrow \infty} \max_{|\tau_u|\leq C_0\sqrt{\log u}|/u}|f(u,x,y)-f(x,y)| \leq |f(x-\epsilon,y-\epsilon)-f(x,y)|\nonumber\\ &\qquad + \lim_{\substack {m\rightarrow \infty \\ n\rightarrow \infty }}\limsup_{u\rightarrow \infty} \max_{|\tau_u|\leq C_0\sqrt{\log u}/u} \Big(|f_{m,n}(u,x,y)-f(x,y)| +|s_{m,n}(u,x,y)|\nonumber\\ &\qquad+|t_{m,n}(u,x,y)| + |f_{m,n}(u,x-\epsilon,y-\epsilon)-f(x-\epsilon,y-\epsilon)|\Big)\nonumber\\ &= |f(x-\epsilon,y-\epsilon)-f(x,y)|.\nonumber\end{aligned}$$ Since the last term $\rightarrow 0$ as $\epsilon \downarrow 0$, we have completed the proof of the second part of the lemma. Now we are ready to prove the main lemmas in Section 3. Let $\phi(a,b)$ be the density of $(X_1(s_u),X_2(t_u))^T$, i.e., $$\phi(a,b)=\frac{1}{2\pi\sqrt{1-r^2(|\tau_u|)}}\exp\left\{-\frac{1}{2}\frac{a^2-2r(|\tau_u|)ab+b^2}{1-r^2(|\tau_u|)}\right\}.$$ By conditioning and a change of variables, the LHS of becomes $$\begin{aligned} \label{main lemma 1} &\mathbb{P}\bigg(\max_{s\in s_u+u^{-2/\alpha_1}\mathbb{S}} X_1(s)>u, \max_{t\in t_u+u^{-2/\alpha_2}\mathbb{T}} X_2(t)>u\bigg)\nonumber\\ &=\int_{\mathbb{R}^2} \mathbb{P}\bigg(\max_{s\in s_u+u^{-2/\alpha_1}\mathbb{S}} X_1(s)>u, \max_{t\in t_u+u^{-2/\alpha_2}\mathbb{T}} X_2(t)>u\ \left |\ X_1(s_u)=u-\frac{x}{u},\right.\nonumber\\ &\qquad \qquad \ \ X_2(t_u)=u-\frac{y}{u}\bigg)\phi\Big(u-\frac{x}{u},u-\frac{y}{u}\Big)u^{-2}dxdy\nonumber\\ &=\frac{1}{2\pi\sqrt{1-r^2(|\tau_u|)}u^2}\exp\left(-\frac{u^2}{1+r(|\tau_u|)}\right)\int_{\mathbb{R}^2} f(u,x,y)\tilde\phi(u,x,y)dxdy,\end{aligned}$$ where $f(u,x,y)$ is defined in with $\xi_u(\cdot),\eta_u(\cdot)$ in , and where $$\begin{aligned} &\tilde \phi(u,x,y)\nonumber\\ :=&\exp\bigg\{-\frac{1}{2(1-r^2(|\tau_u|))}\Big(\frac{x^2+y^2}{u^2}-2(1-r(|\tau_u|))(x+y)-2r(|\tau_u|)\frac{xy}{u^2}\Big)\bigg\}.\end{aligned}$$ Since $\max_{|\tau_u|\leq C_0\sqrt{\log u}/u}|r(|\tau_u|)-\rho|\rightarrow 0$ as $u\rightarrow \infty$, it is easy to check that $$\label{lim of tilde phi} \max_{|\tau_u|\leq C_0\sqrt{\log u}/u}\left|\tilde{\phi}(u,x,y)-e^{\frac{x+y}{1+\rho}}\right|\rightarrow 0, \ \text{ as }\ u\rightarrow \infty.$$ Recall $H_\alpha(\cdot)$ in and $f(x,y)$ in . Since $\xi(\cdot)$, $\eta(\cdot)$ are independent, and $$\begin{aligned} \{\xi(t), t \in \R^N\} \stackrel{d}{=} \bigg\{(1+\rho)\Big[\chi_1\Big(\Big(\frac{\sqrt{c_1}}{1+\rho}\Big)^{\frac{2}{\alpha_1}}t\Big)- \left|\Big(\frac{\sqrt{c_1}}{1+\rho}\Big)^{\frac{2}{\alpha_1}}t\right |^{\alpha_1}\Big], t\in \R^N\bigg\}, \\ \{ \eta(t), t \in \R^N\} \stackrel{d}{= } \bigg\{(1+\rho)\Big[\chi_2\Big(\Big(\frac{\sqrt{c_2}}{1+\rho}\Big)^{\frac{2}{\alpha_2}}t\Big)- \left|\Big(\frac{\sqrt{c_2}}{1+\rho}\Big)^{\frac{2}{\alpha_2}}t\right |^{\alpha_2}\Big], t\in \R^N\bigg\},\end{aligned}$$ where $\stackrel{d}{=}$ means equality of all finite dimensional distributions, we have $$\begin{aligned} \label{main lemma 4} &\int_{\mathbb{R}^2} f(x,y)e^{\frac{x+y}{1+\rho}}dxdy\nonumber\\ &=\int_{\mathbb{R}}e^{\frac{x}{1+\rho}}\mathbb{P}\Big(\max_{s\in \mathbb{S}}\xi(s)>x\Big)dx \int_{\mathbb{R}}e^{\frac{y}{1+\rho}}\mathbb{P}\Big(\max_{t\in \mathbb{T}}\eta(t)>y\Big)dy\nonumber\\ &=(1+\rho)^2H_{\alpha_1}\bigg(\frac{c_1^{1/\alpha_1}\mathbb{S}}{(1+\rho)^{\frac{2}{\alpha_1}}}\bigg) H_{\alpha_2}\bigg(\frac{c_2^{1/\alpha_2}\mathbb{T}}{(1+\rho)^{\frac{2}{\alpha_2}}}\bigg).\end{aligned}$$ By and , to conclude the lemma, it suffices to prove $$\label{main lemma 2} \lim_{u\rightarrow \infty}\int_{\mathbb{R}^2}\max_{|\tau_u|\leq C_0\sqrt{\log u}/u}\left|f(u,x,y) \tilde \phi(u,x,y) -f(x,y)e^{\frac{x+y}{1+\rho}}\right|\,dxdy=0.$$ Firstly, applying Lemma \[uniformly conditional convergence\] together with , we have $$\begin{aligned} \label{main lemma 3} &\max_{|\tau_u|\leq C_0\sqrt{\log u}/u}\left|f(u,x,y)\tilde \phi(u,x,y)-f(x,y)e^{\frac{x+y}{1+\rho}}\right| \rightarrow 0, \ \text{as}\ u\rightarrow \infty.\end{aligned}$$ Secondly, as in [@Ladneva_Piterbarg_2000], we can find an integrable dominating function $g\in L({\mathbb{R}^2})$ such that for $u$ large enough, $$\max_{|\tau_u|\leq C_0\sqrt{\log u}/u}\left|f(u,x,y)\tilde \phi(u,x,y)-f(x,y)e^{\frac{x+y}{1+\rho}}\right|\leq g(x,y).$$ Therefore, follows from the dominated convergence theorem. This finishes the proof. Specifically, - Case $1$: For the quadrant $\{(x,y)\ |\ x<0,y<0\}$, we have $$\tilde \phi(u,x,y)\leq e^{\frac{x+y}{1+r(|\tau_u|)}}\leq e^{\frac{x+y}{1+\rho}}, \ \forall |\tau_u| \leq C\sqrt{\log u}/u,$$ and $\max_{|\tau_u| \leq C\sqrt{\log u}/u}|f(u,x,y)|\leq 1,\ |f(x,y)|\leq 1$. So we can choose $g(x,y)=2e^{\frac{x+y}{1+\rho}}.$\ - Case $2$: For the quadrant $\{(x,y)\ |\ x>0,y<0\}$, when $u$ is large, we have $$\tilde \phi(u,x,y)\leq \exp\bigg(\frac{y}{1+\rho}+\frac{x}{0.9+\rho}\bigg), \ \forall |\tau_u| \leq C\sqrt{\log u}/u.$$ By Borel inequality, there exists $K,\epsilon>0$, $$\begin{aligned} &f(u,x,y)\leq \mathbb{P}\bigg(\max_{s\in \mathbb{S}}\xi_u(s)>x\ \left |\ X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\right.\bigg)\leq Ke^{-\epsilon x^2},\end{aligned}$$ which also implies $f(x,y)\leq Ke^{-\epsilon x^2}$. So we can choose $g(x,y)=2Ke^{-\epsilon x^2}\exp\big(\frac{y}{1+\rho}+\frac{x}{0.9+\rho}\big)$.\ - Case $3$: For the quadrant $\{(x,y)\ |\ x<0,y>0\}$, using the very similar argument as Case $2$, we choose $g(x,y)=2Ke^{-\epsilon y^2}\exp\big(\frac{x}{1+\rho}+\frac{y}{0.9+\rho}\big)$.\ - Case $4$: For the quadrant $\{(x,y)\ |\ x>0,y>0\}$, when $u$ is large, we have $$\tilde \phi(u,x,y)\leq e^{\frac{x+y}{0.9+\rho}}, \ \forall |\tau_u| \leq C\sqrt{\log u}/u.$$ By Borel inequality, there exists $K,\epsilon>0$, $$\begin{aligned} &f(u,x,y)\leq \mathbb{P}\bigg(\max_{(s,t)\in \mathbb{S}\times \mathbb{T}}\xi_u(s)+\eta_u(t)>x+y\ \left |\ X_1(s_u)=u-\frac{x}{u},X_2(t_u)=u-\frac{y}{u}\right.\bigg)\nonumber\\ &\leq Ke^{-\epsilon (x+y)^2},\end{aligned}$$ which also implies $f(x,y)\leq Ke^{-\epsilon (x+y)^2}$. So we can choose $g(x,y)=2Ke^{-\epsilon (x+y)^2+\frac{x+y}{0.9+\rho}}$. We first claim that for any compact sets $\mathbb{S}$ and $\mathbb{T}$, the identity $$\label{relation of H(S,T) and H(S)} H_\alpha(\mathbb{S})+H_\alpha(\mathbb{T})-H_\alpha(\mathbb{S}\cup \mathbb{T})=H_\alpha(\mathbb{S},\mathbb{T})$$ holds. Indeed, if we let $X=\sup_{t\in \mathbb{S}}(\chi(t)-|t|^\alpha)$ and $Y=\sup_{t\in \mathbb{T}}(\chi(t)-|t|^\alpha)$, then $$\begin{aligned} &H_\alpha(\mathbb{S})+H_\alpha(\mathbb{T})-H_\alpha(\mathbb{S}\cup \mathbb{T})=\mathbb{E}\big(e^X\big) +\mathbb{E}\big(e^Y\big)-\mathbb{E}\big(e^{\max(X,Y)}\big)\\ &=\mathbb{E}\big(e^X1_{\{X<Y\}}\big)+\mathbb{E}\big(e^Y1_{\{X\geq Y\}}\big)=\mathbb{E}\big(e^{\min(X,Y)}\big)=H_\alpha(\mathbb{S},\mathbb{T}).\end{aligned}$$ Now let $\mathbb{T}_1=[0,T]^N$, $\mathbb{T}_2=[\mathbf{m}T,(\mathbf{m}+1)T]$ and $\mathbb{T}_3=[\mathbf{n}T,(\mathbf{n}+1)T]$. Consider the events $$\begin{aligned} &A=\left\{\max_{s\in s_u+u^{-2/\alpha_1}\mathbb{T}_1} X_1(s)>u\right\},\ \ B=\left\{\max_{s\in s_u+u^{-2/\alpha_1}\mathbb{T}_2} X_1(s)>u\right\},\\ &C=\left\{\max_{t\in t_u+u^{-2/\alpha_2}\mathbb{T}_1} X_2(t)>u\right\},\ \ D=\left\{\max_{t\in t_u+u^{-2/\alpha_2}\mathbb{T}_3} X_2(t)>u\right\}.\end{aligned}$$ It is easy to check that the LHS of is equal to $$\begin{aligned} \label{double double local extremes asymptotics 1} &\mathbb{P}(A\cap B\cap C\cap D)\nonumber\\ &=[\mathbb{P}(A\cap C)+\mathbb{P}( B\cap C)-\mathbb{P}((A\cup B)\cap C)]\nonumber\\ &+[\mathbb{P}(A\cap D)+\mathbb{P}( B\cap D)-\mathbb{P}((A\cup B)\cap D)]\nonumber\\ &-[\mathbb{P}(A\cap (C\cup D))+\mathbb{P}( B\cap (C\cup D))-\mathbb{P}((A\cup B)\cap (C\cup D))].\end{aligned}$$ Let $R(u)=\frac{(1+\rho)^2}{2\pi \sqrt{1-\rho^2}}u^{-2}\exp\left(-\frac{u^2}{1+r(|\tau_u|)}\right)$ and $q_{\alpha,c}=\frac{(1+\rho)^{2/\alpha}}{c^{1/\alpha}}$. By Lemma \[main lemma\], we have $$\begin{aligned} &\mathbb{P}(A\cap C)=R(u)H_{\alpha_1}\left(\frac{\mathbb{T}_1}{q_{\alpha_1,c_1}}\right)H_{\alpha_2}\left(\frac{\mathbb{T}_1}{q_{\alpha_2,c_2}}\right)(1+\gamma_1(u)),\\ &\mathbb{P}(B\cap C)=R(u)H_{\alpha_1}\left(\frac{\mathbb{T}_2}{q_{\alpha_1,c_1}}\right)H_{\alpha_2}\left(\frac{\mathbb{T}_1}{q_{\alpha_2,c_2}}\right)(1+\gamma_2(u)),\\ &\mathbb{P}((A\cup B)\cap C)=R(u)H_{\alpha_1}\left(\frac{\mathbb{T}_1\cup \mathbb{T}_2}{q_{\alpha_1,c_1}}\right)H_{\alpha_2}\left(\frac{\mathbb{T}_1}{q_{\alpha_2,c_2}}\right)(1+\gamma_3(u)),\end{aligned}$$ where, for $i = 1, 2, 3$, $\gamma_i(u)\rightarrow 0$ uniformly w.r.t. $\tau_u$ satisfying $|\tau_u|\leq C_0\sqrt{\log u}/u$, as $u\rightarrow \infty$. These, together with (\[relation of H(S,T) and H(S)\]), imply $$\begin{aligned} \label{double double local extremes asymptotics 2} &\mathbb{P}(A\cap C)+\mathbb{P}( B\cap C)-\mathbb{P}((A\cup B)\cap C)\nonumber\\ &=R(u)H_{\alpha_2}\left(\frac{\mathbb{T}_1}{q_{\alpha_2,c_2}}\right)H_{\alpha_1}\left(\frac{\mathbb{T}_1}{q_{\alpha_1,c_1}}, \frac{\mathbb{T}_2}{q_{\alpha_1,c_1}}\right)(1+o(1)).\end{aligned}$$ Similarly, we have $$\begin{aligned} \label{double double local extremes asymptotics 3} &\mathbb{P}(A\cap D)+\mathbb{P}( B\cap D)-\mathbb{P}((A\cup B)\cap D)\nonumber\\ &=R(u)H_{\alpha_2}\left(\frac{ \mathbb{T}_3 }{q_{\alpha_2,c_2}}\right)H_{\alpha_1} \left(\frac{\mathbb{T}_1}{q_{\alpha_1,c_1}},\frac{\mathbb{T}_2}{q_{\alpha_1,c_1}}\right)(1+o(1))\end{aligned}$$ and $$\begin{aligned} \label{double double local extremes asymptotics 4} &\mathbb{P}(A\cap (C\cup D))+\mathbb{P}( B\cap (C\cup D))-\mathbb{P}((A\cup B)\cap (C\cup D))\nonumber\\ &=R(u)H_{\alpha_2}\left(\frac{\mathbb{T}_1\cup \mathbb{T}_3 }{q_{\alpha_2,c_2}}\right)H_{\alpha_1}\left(\frac{\mathbb{T}_1}{q_{\alpha_1,c_1}},\frac{\mathbb{T}_2}{q_{\alpha_1,c_1}}\right)(1+o(1)).\end{aligned}$$ By – , we have $$\begin{aligned} &\mathbb{P}(A\cap B\cap C\cap D)\nonumber\\ &=R(u)H_{\alpha_1}\left(\frac{\mathbb{T}_1}{q_{\alpha_1,c_1}},\frac{\mathbb{T}_2}{q_{\alpha_1,c_1}}\right) H_{\alpha_2,c_2}\left(\frac{\mathbb{T}_1}{q_{\alpha_2,c_2}},\frac{\mathbb{T}_3}{q_{\alpha_2,c_2}}\right)(1+o(1)),\end{aligned}$$ which concludes the lemma. \[proof of lemma\_Riemann Sum\] Let $f(|t|)=\frac{1}{1+r(|t|)}$. Recall $\tau_{\mathbf{k}\mathbf{l}}$ defined in and $|\tau_{\mathbf{k}\mathbf{l}}|\leq 2\delta(u)$, when $u$ is large. By Taylor’s expansion, $$\begin{aligned} f(|\tau_{\mathbf{k}\mathbf{l}}|)=f(0)+\frac{1}{2}f''(0)|\tau_{\mathbf{k}\mathbf{l}}|^2(1+\gamma_{\mathbf{k}\mathbf{l}}(u)),\end{aligned}$$ where $f(0)=\frac{1}{1+ \rho}$, $f''(0)=\frac{-r''(0)}{(1+\rho)^2}$ and, as $u\rightarrow \infty$, $\gamma_{\mathbf{k}\mathbf{l}}(u)$ converges to zero uniformly w.r.t. all $(\mathbf{k},\mathbf{l})\in \mathcal{C}$. Therefore, for any $\epsilon >0$, we have $$\label{bounds for h(u)} \sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} e^{-\frac{1}{2}f''(0)(1+\epsilon)u^2|\tau_{\mathbf{k}\mathbf{l}}|^2}\leq h(u)\leq \sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} e^{-\frac{1}{2}f''(0)(1-\epsilon)u^2|\tau_{\mathbf{k}\mathbf{l}}|^2}$$ when $u$ is large enough. For $ a>0$, let $$h(u,a):=\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} e^{-au^2|\tau_{\mathbf{k}\mathbf{l}}|^2}.$$ In order to prove (\[h(u)\]), it suffices to prove that $$\label{h(u,a)} \lim_{u\rightarrow \infty}u^Nd_1^N(u)d_2^N(u)h(u,a)=\left(\frac{\pi}{a}\right)^\frac{N}{2}mes_N(A_1\cap A_2).$$ To this end, we write $$\begin{aligned} \label{intuition} &u^Nd_1^N(u)d_2^N(u)h(u,a)\nonumber\\ &=\frac{1}{u^N}\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} e^{-a\sum_{j=1}^N({l_jud_2(u)-k_jud_1(u)})^2}\cdot(ud_1(u))^N(ud_2(u))^N.\end{aligned}$$ Let $$\begin{aligned} \label{bounds for h(u,a)} &p(u):=\frac{1}{u^N}\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} \min_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}e^{-a|t-s|^2}\cdot(ud_1(u))^N(ud_2(u))^N,\nonumber\\ &q(u):=\frac{1}{u^N}\sum_{(\mathbf{k},\mathbf{l})\in \mathcal{C}} \max_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}e^{-a|t-s|^2}\cdot(ud_1(u))^N(ud_2(u)^N.\end{aligned}$$ It follows from (\[intuition\]) that $$\label{Eq:pq1} p(u)\leq u^Nd_1^N(u)d_2^N(u)h(u,a)\leq q(u),$$ and $$\label{Eq:pq2} p(u)\leq \frac{1}{u^N}\int_{\substack{s \in uA_1,t\in uA_2\\ |t-s|\leq C\sqrt{\log u}}}e^{-a|t-s|^2}dtds\leq q(u).$$ Observe that $$\begin{aligned} \label{Eq:pq3} &\frac{1}{u^N}\iint_{\substack{s\in uA_1,t\in uA_2\\ |t-s|\leq C\sqrt{\log u}}}e^{-a|t-s|^2}dtds =\frac{1}{u^N}\iint_{\substack{y \in uA_1,x+y\in uA_2\\ |x|\leq C\sqrt{\log u}}}e^{-a|x|^2}dxdy\nonumber\\ &=\frac{1}{u^N}\int_{|x|\leq C\sqrt{\log u}}e^{-a|x|^2}dx\int_{\mathbb{R}^N}1_{\{y\in uA_1\cap (uA_2-x)\}}dy\nonumber\\ &=\int_{|x|\leq C\sqrt{\log u}}e^{-a|x|^2}dx\int_{\mathbb{R}^N}1_{\{z\in A_1\cap (A_2-x/u)\}}dz\nonumber\\ &\rightarrow \ mes_N(A_1\cap A_2)\int_{\mathbb{R}^N} e^{-a|x|^2}dx =\left(\frac{\pi}{a}\right)^\frac{N}{2}mes_N(A_1\cap A_2),\end{aligned}$$ as $u\rightarrow \infty$, where the convergence holds by the dominated convergence theorem. Indeed, $\int_{\mathbb{R}^N}1_{\{z\in A_1\cap (A_2-x/u)\}}dz$ is bounded by $\max_{|\epsilon|<1} mes_N(A_1\cap (A_2-\epsilon))$ uniformly for $|x|\leq C\sqrt{\log u}$ when $u$ is large enough. It follows from (\[Eq:pq1\])–(\[Eq:pq1\]) that, for concluding , it remains to verify $$\label{D(u) def} D(u):=q(u)-p(u)\rightarrow 0, \ \text{as}\ u\rightarrow \infty.$$ Define $$\begin{aligned} \mathcal{\hat D}:=\Big\{(s,t)\in A_1\times A_2:\, |t-s|\leq \delta(u)+\sqrt{N}d_1(u)+\sqrt{N}d_2(u)\Big\}.\end{aligned}$$ By the definition of $\mathcal{C}$ in , we see that $ \mathcal{D}\subseteq \bigcup_{(\mathbf{k},\mathbf{l})\in \mathcal{C}}\Delta_{\mathbf{k}}^{(1)}\times \Delta_{\mathbf{l}}^{(2)}\subseteq \mathcal{\hat D}. $ Since $d_1(u)=o(\delta(u))$ and $d_2(u)=o(\delta(u))$ as $u\rightarrow \infty$, the set $\mathcal{\hat D}$ is a subset of $\mathcal{\tilde D}:=\{(s,t)\in A_1\times A_2:\, |t-s|\leq 2\delta(u)\}$ when $u$ is large. Write $D(u)$ in (\[D(u) def\]) as a sum over $(\mathbf{k},\mathbf{l})\in \mathcal{C}$. To estimate the cardinality of $ \mathcal{C}$, we notice that $$\begin{aligned} \label{upper bound of mes(tilde D)} mes_{2N}(\mathcal{\tilde D})&= \iint_{s\in A_1, t\in A_2}1_{\{|t-s|\leq 2\delta(u)\}}dsdt \\ &=\int_{|x|\leq 2\delta(u)}\int_{y \in A_1\cap (A_2-x)}dydx \leq K\,\delta(u)^N,\end{aligned}$$ for all $u$ large enough, where $K=2^{N+1}\pi^{N/2}\Gamma^{-1}(N/2)\max_{| \epsilon| \leq 1} mes_N(A_1\cap (A_2-\epsilon))$. Hence, for large $u$, the number of summands in (\[intuition\]) is bounded by $$\label{number of sums in D(u)} \#\{(\mathbf{k},\mathbf{l})\ |\ (\mathbf{k},\mathbf{l})\in \mathcal{C}\}\leq \frac{mes_{2N}(\mathcal{\tilde D})} {mes_{2N}(\Delta_{\mathbf{k}}^{(1)}\times \Delta_{\mathbf{l}}^{(2)})}\leq \frac{K\,\delta(u)^N} {d_1^N(u)d_2^N(u)}.$$ Next, by applying the inequality $e^{-x}-e^{-y}\leq y-x$ for $y\geq x>0$ to each summand in $D(u)$, we obtain $$\begin{aligned} \label{bound for element in D(u)_0} &\max_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}e^{-a|t-s|^2}-\min_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}e^{-a|t-s|^2}\nonumber\\ %&=\exp\bigg(-a\min_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}|t-s|^2\bigg)-\exp\bigg(-a\max_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}|t-s|^2\bigg)\nonumber\\ &\leq a\left(\max_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}|t-s|^2-\min_{(s,t)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}}|t-s|^2\right)\nonumber\\ &=a \max \big(|t-s|+|t_1-s_1|)(|t-s|-|t_1-s_1|\big),\end{aligned}$$ where the last maximum is taken over $(s,t,s_1,t_1) \in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}\times u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}$ Since $|t-s|\leq 2\delta(u)$ for all $(t,s)\in u\Delta_\mathbf{k}^{(1)}\times u\Delta_\mathbf{l}^{(2)}$ when $u$ is large, the inequality $\big||t-s|-|t_1-s_1|\big|\leq |t-t_1|+|s-s_1|$ implies that is at most $$\begin{aligned} \label{bound for element in D(u)} 4a\sqrt{N}u^2\delta(u)\big(d_1(u)+d_2(u)\big)\end{aligned}$$ when $u$ is large enough. By and , we can verify that $$\begin{aligned} %\label{D(u) bound} D(u)&\leq \frac{1}{u^N}\frac{K(\delta(u))^N}{d_1^N(u)d_2^N(u)}4a\sqrt{N}u^2\delta(u)\big(d_1(u)+d_2(u)\big)\big(ud_1(u)\big)^N\big(ud_2(u)\big)^N\nonumber\\ &\leq C_0 (\log u)^{\frac{N+1}{2}}\big(u^{1-\frac{2}{\alpha_1}}+u^{1-\frac{2}{\alpha_2}}\big)\, \rightarrow 0,\ \text{ as }\ u\rightarrow \infty.\end{aligned}$$ Therefore holds. Similarly, we can check that the same statement holds while changing the set $\mathcal{C}$ to $\mathcal{C}^\circ$. Inequality holds immediately by Lemma $6.2$ in [@Piterbarg_1996]. Hence we only consider the case when $\mathbf{m}\neq \mathbf{0}$. Suppose that $\{X(t),t\in \mathbb{R}^N\}$ is a real valued continuous Gaussian process with $\E[X(t)]=0$ and covariance function $r(t)$ satisfying $r(t)=1-|t|^\alpha+o(|t|^\alpha)$ for a constant $\alpha \in (0,2)$. Applying Lemma $6.1$ in [@Piterbarg_1996], we see that for any $S>0$, $$\begin{aligned} \label{mathcal_H_alpha(m) with extreme prob} &\mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}[0,S]^N}X(t)>u,\, \max_{t\in u^{-2/\alpha}[\mathbf{m}S,(\mathbf{m}+1)S]}X(t)>u\bigg)\nonumber\\ &=\mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}[0,S]^N}X(t)>u\bigg)+\mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}[\mathbf{m}S,(\mathbf{m}+1)S]}X(t)>u\bigg)\nonumber\\ &\qquad \ \ -\mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}([0,S]^N\cup [\mathbf{m}S,(\mathbf{m}+1)S])}X(t)>u)\bigg)\nonumber\\ &=\Big(H_\alpha([0,S]^N)+H_\alpha([\mathbf{m}S,(\mathbf{m}+1)S])-H_\alpha([0,S]^N\cup [\mathbf{m}S,(\mathbf{m}+1)S])\Big)\nonumber\\ &\qquad \ \ \times \frac{1}{\sqrt{2\pi}u}e^{-\frac{1}{2}u^2}(1+o(1))\nonumber\\ &=H_\alpha([0,S]^N,[\mathbf{m}S,(\mathbf{m}+1)S])\frac{1}{\sqrt{2\pi}u}e^{-\frac{1}{2}u^2}(1+o(1)),\ \text{ as }\ u\rightarrow \infty,\end{aligned}$$ where the last equality holds thanks to . On the other hand, by applying Lemma $6.3$ in [@Piterbarg_1996] and the inequality $\inf_{s\in [0,1]^N,t\in [\mathbf{m},\mathbf{m}+1]}|s-t|\geq |m_{i_0}|-1$ (recall that $i_0$ is defined in Lemma \[lem\_mathcal\_H(m)\]), we have $$\begin{aligned} \label{Lemma 6.3 in Piterbarg} &\mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}[0,S]^N}X(t)>u,\, \max_{t\in u^{-2/\alpha}[\mathbf{m}S,(\mathbf{m}+1)S]}X(t)>u\bigg)\nonumber\\ &\leq C_0 S^{2N}\frac{1}{\sqrt{2\pi}u}e^{-\frac{1}{2}u^2}\exp\left(-\frac{1}{8}(|m_{i_0}|-1)^\alpha S^\alpha\right)\end{aligned}$$ for all $u$ large enough. It follows from and that $$H_\alpha([0,S]^N,[\mathbf{m}S,(\mathbf{m}+1)S])\leq C_0S^{2N}\exp\left(-\frac{1}{8}(|m_{i_0}|-1)^\alpha S^\alpha\right),$$ which implies by letting $S=\frac{c^{1/\alpha}T}{(1+\rho)^{2/\alpha}}$. When $|m_{i_0}|=1$, the above upper bound is not sharp. Instead, we derive in Lemma \[lem\_mathcal\_H(m)\] as follows. For concreteness, suppose that $i_0=N$ and $m_N=1$. By applying Lemmas $6.1$ - $6.3$ in [@Piterbarg_1996], we have $$\begin{aligned} &\mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}[0,S]^N}X(t)>u,\, \max_{t\in u^{-2/\alpha}[\mathbf{m}S,(\mathbf{m}+1)S]}X(t)>u\bigg)\nonumber\\ &\leq \mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}(\prod_{j=1}^{N-1}[m_jS,(m_j+1)S]\times[S,S+\sqrt{S}])}X(t)>u\bigg)\nonumber\\ &+\mathbb{P}\bigg(\max_{t\in u^{-2/\alpha}[0,S]^N}X(t)>u,\, \max_{t\in u^{-2/\alpha}(\prod_{j=1}^{N-1}[m_jS,(m_j+1)S]\times[S+\sqrt{S},2S+\sqrt{S}])}X(t)>u\bigg)\nonumber\\ &\leq C_0 S^{N-\frac{1}{2}}\frac{1}{\sqrt{2\pi}u}e^{-\frac{1}{2}u^2}+C_0 S^{2N}\frac{1}{\sqrt{2\pi}u}e^{-\frac{1}{2}u^2}e^{-\frac{1}{8}S^{\alpha/2}}\nonumber\\ &\leq C_0 S^{N-\frac{1}{2}}\frac{1}{\sqrt{2\pi}u}e^{-\frac{1}{2}u^2} \end{aligned}$$ for $u$ and $S$ large. Hence, when $|m_{i_0}|=1$, we have $$H_\alpha([0,S]^N,[\mathbf{m}S,(\mathbf{m}+1)S])\leq C_0 S^{N-\frac{1}{2}}$$ for large $S$. This implies by letting $S=\frac{c^{1/\alpha}T}{(1+\rho)^{2/\alpha}}$. Notice that $$\#\{m\in \mathbb{Z}^N\ |\ \max_{1\leq i\leq N}|m_i|=k\}=(2k+1)^N-(2k-1)^N,\ k=1,2,....$$ By , and the fact that $\int_T^\infty x^{N-1} e^{-a x^\alpha}dx \sim \frac {1} {a\alpha} T^{N-\alpha} e^{-aT^\alpha}$ as $T \to \infty$, we have $$\begin{aligned} &\sum_{\mathbf{m}\neq \mathbf{0}}\mathcal{H}_{\alpha,c}(\mathbf{m})=\sum_{k=1}^\infty\sum_{|m_{i_0}|=k}\mathcal{H}_{\alpha,c}(\mathbf{m})\\&\leq C_0 (3^N-1)T^{N-\frac{1}{2}}+C_0 \sum_{k=2}^\infty [(2k+1)^N-(2k-1)^N]T^{2N}e^{-\frac{c}{8(1+\rho)^2}(k-1)^\alpha T^\alpha}\\ &\leq C_0 (3^N-1)T^{N-\frac{1}{2}}+C_0 T^{2N}\int_{1}^\infty x^{N-1}e^{-\frac{c}{8(1+\rho)^2}x^\alpha T^\alpha}dx \leq C_0 T^{N-\frac{1}{2}}\end{aligned}$$ for $T$ large enough. This completes the proof of Lemma \[lem\_mathcal\_H(m)\]. \[Proof of Lemma \[lem\_Riemann Sum-2\]\] The proof is similar to that of Lemma \[lem\_Riemann Sum\]. Indeed, we only need to modify and in the proof of Lemma \[lem\_Riemann Sum\]. For any $ y=(y_1,...,y_N)\in \mathbb{R}^N$ and $1\leq i\leq j\leq N$, let $y_{i:j} =(y_i,...,y_j)$. On one hand, with a different scaling, $h(u,a)$ in has the following asymptotics: $$\begin{aligned} \label{intuition_2} &u^{2N-M}d_1^N(u)d_2^N(u)h(u,a) \approx\frac{1}{u^M}\iint_{\substack{y \in uA_1,x+y\in uA_2\\ |x|\leq C\sqrt{\log u}}}e^{-a|x|^2}dxdy\nonumber\\ &= \frac{1}{u^M}\int_{|x|\leq C\sqrt{\log u}}e^{-a|x|^2}\bigg(\int_{\mathbb{R}^M}1_{\{y_{1:M}\in uA_{1,M}\cap (uA_{2,M}-x_{1:M})\}}dy_{1:M}\nonumber\\ &\qquad \times\prod_{j=M+1}^N\int_{\mathbb{R}}1_{\{y_j\in [uS_j,uT_j]\cap[uT_j-x_j,uR_j-x_j]\}}dy_j\bigg)\, dx\nonumber\\ &=\int_{|x|\leq C\sqrt{\log u}}e^{-a|x|^2}\prod_{j=M+1}^N x_j 1_{\big\{x_j>0\big\}}\bigg(\int_{\mathbb{R}^M} 1_{\big\{z_{1:M}\in A_{1,M}\cap (A_{2,M}-x_{1:M}/u)\big\}}dz_{1:M}\bigg)\,dx\nonumber\\ & \rightarrow \ mes_M(A_{1,M}\cap A_{2,M})\int_{\mathbb{R}^M} e^{-a|x_{1:M}|^2}dx_{1:M}\prod_{j=M+1}^N \int_0^\infty x_je^{-ax_j^2}dx_j\nonumber\\ &=2^{M-N}\pi^{M/2}a^{M/2-N}mes_M(A_{1,M}\cap A_{2,M}),\end{aligned}$$ as $u\rightarrow \infty$. On the other hand, when $u$ is large enough, $mes_{2N}( \mathcal{\tilde D})$ defined in can be bounded above by $$\begin{aligned} \label{upper bound of mes(tilde D)_2} &mes_{2N}( \mathcal{\tilde D})=\iint_{s\in A_1, t\in A_2}1_{\{|t-s|\leq 2\delta(u)\}}dsdt\nonumber\\ &=\int_{|x|\leq 2\delta(u)}\Big(\int_{y_{1:M} \in A_{1,M}\cap (A_{2,M}-x_{1:M})}dy_{1:M}\Big)\prod_{j=M+1}^Nx_j1_{\{x_j>0\}}dx\nonumber\\ &=\delta(u)^{2N-M}\int_{|z|\leq 2}\Big(\int_{y_{1:M} \in A_{1,M}\cap (A_{2,M}-z_{1:M}\delta(u))}dy_{1:M}\Big)\prod_{j=M+1}^Nz_j1_{\{z_j>0\}}dz\nonumber\\ &\leq K\, \delta(u)^{2N-M},\end{aligned}$$ where $K=\max_{| \epsilon| \leq 1} mes_M(A_{1,M}\cap (A_{2,M}-\epsilon))\int_{|z|\leq 2}\prod_{j=M+1}^Nz_j1_{\{z_j>0\}}dz$. By and , can be obtained through the same argument in the proof of Lemma \[lem\_Riemann Sum\]. We omit the details. We end this section with the proof of Lemma \[lem\_uniform convergence of f.d.d.\]. Let $f_{u,\tau_u}(\cdot)$ and $f(\cdot)$ be the density function of $X(u,\tau_u)$ and $X$, respectively. It suffices to prove that for all $ x\in \mathbb{R}^N$, $$\begin{aligned} \label{uniform covergence of F_u} \int_{\{y\leq x\}}f(y)\max_{\tau_u}\bigg|\frac{f_{u,\tau_u(y)}}{f(y)}-1\bigg|dy\rightarrow 0, \ \text{as}\ u\rightarrow \infty,\end{aligned}$$ where $\{y\leq x\}= \prod_{i=1}^N(-\infty, x_i]$. First, we will find an upper bound for $\max_{\tau_u}|f_{u,\tau_u}(y)/f(y)-1|$. For any $ \epsilon>0$, define $$\begin{aligned} &\Gamma(u,\tau_u)=(\gamma_{ij}(u,\tau_u))_{i,j=1,...n}:=\frac{1}{\epsilon}(\Sigma(u,\tau_u)- \Sigma)\\ &e(u,\tau_u)=(e_{i}(u,\tau_u))_{i=1,...n}:=\frac{1}{\epsilon}(\mu(u,\tau_u)-\mu).\end{aligned}$$ By Assumption , there exists a constant $U>0$ such that for all $u>U$, $$\begin{aligned} \max_{\tau_u}|\mu_j(u,\tau_u)-\mu_j|<\epsilon,\ \ \ \max_{\tau_u}|\sigma_{ij}(u,\tau_u)-\sigma_{ij}|<\epsilon, \ \ i,j=1,\dots, n,\end{aligned}$$ which implies $|\gamma_{ij}(u,\tau_u)|\leq 1$ and $|e_{i}(u,\tau_u)|\leq 1$ for $u>U$. Let $\Sigma^{-1}=(v_{ij})_{i,j=1,...,n}$ be the inverse of $\Sigma$. When $\epsilon$ is small, the determinant of $\Sigma(u,\tau_u)$ satisfies $$\begin{aligned} |\Sigma(u,\tau_u)|=|\Sigma +\epsilon \Gamma(u,\tau_u)|=|\Sigma|(1+\epsilon \hbox{tr}(\Sigma^{-1} \Gamma(u,\tau_u))+O(\epsilon^2)),\end{aligned}$$ where $O(\epsilon^2)/\epsilon^2$ is uniformly bounded w.r.t. $\tau_u$ for large $u$ (see, e.g., [@Magnus_Heudecker_2007], p. $169$). Hence, when $\epsilon$ is small enough, we have $$\begin{aligned} \label{ratio of two var det} \left|\frac{|\Sigma(u,\tau_u)|}{|\Sigma|}-1\right|\leq 2\epsilon |\hbox{tr}(\Sigma^{-1}\Gamma(u,\tau_u))|\leq 2\epsilon \sum_{i,j}|v_{ij}|.\end{aligned}$$ Since $|\gamma_{ij}(u,\tau_u)|\leq 1,\ \forall i,j=1,...,n,\ \forall \tau_u$ for large $u$, as $\epsilon \rightarrow 0$, the inverse of $\Sigma(u,\tau_u)$ can be written as $$\begin{aligned} \Sigma(u,\tau_u)^{-1}=\Sigma^{-1}-\epsilon \Sigma^{-1}\Gamma(u,\tau_u)\Sigma^{-1}+O(\epsilon^2),\end{aligned}$$ where $O(\epsilon^2)/\epsilon^2$ is a matrix whose entries are uniformly bounded and independent of $\tau_u$ for large $u$ (see, e.g., [@Meyer_2000], p. $618$). Hence, $$\begin{aligned} %\label{diff of two exp power} d_{u,\tau_u}(y):=&-\frac{1}{2}\Big[(y-\mu(u,\tau_u))^T\Sigma^{-1}(u,\tau_u)(y-\mu(u,\tau_u))- (y-\mu)^T\Sigma^{-1}(y-\mu)\Big]\nonumber\\ = &-\frac{1}{2}(y-\mu)^T\big(-\epsilon \Sigma^{-1}\Gamma(u,\tau_u)\Sigma^{-1}+O(\epsilon^2)\big)(y-\mu)\nonumber\\ &+\epsilon e^T(u,\tau_u)\big(\Sigma^{-1}-\epsilon \Sigma^{-1}\Gamma(u,\tau_u)\Sigma^{-1}+O(\epsilon^2)\big)(y-\mu)\nonumber\\ &-\frac{1}{2}\epsilon^2e^T(u,\tau_u)\big(\Sigma^{-1}-\epsilon \Sigma^{-1}\Gamma(u,\tau_u)\Sigma^{-1}+O(\epsilon^2)\big)e(u,\tau_u). \end{aligned}$$ Since $|\gamma_{ij}(u,\tau_u)|$ and $|e_{i}(u,\tau_u)|$ are uniformly bounded by $1$ w.r.t. $\tau_u$ for all $u>U$, we derive that for any $y\in \mathbb{R}^N$, $$\begin{aligned} \label{limit of d_u,tau_u(y)} \max_{\tau_u}|d_{u,\tau_u}(y)|\rightarrow 0,\ \text{as}\ u\rightarrow \infty.\end{aligned}$$ By and , for $y\in \mathbb{R}^N$, $$\begin{aligned} \label{pointwise conv of integrand} \max_{\tau_u}\left|\frac{f_{u,\tau_u(y)}}{f(y)}-1\right|=\max_{\tau_u}\Big | e^{d_{u,\tau_u}(y)} \frac{|\Sigma(u,\tau_u)|^{-1/2}}{|\Sigma|^{-1/2}}-1\Big|\rightarrow 0,\ \text{as} \ u\rightarrow \infty.\end{aligned}$$ If we could further find an integrable function $g(y)$ on $\mathbb{R}^N$, $$\begin{aligned} \label{dominating function g} f(y)\max_{\tau_u}\left|\frac{f_{u,\tau_u}(y)}{f(y)}-1\right|\leq g(y),\end{aligned}$$ then holds by the dominated convergence theorem. Given a constant $C_0$, let $ A_I:=\{(a_{ij})_{i,j=1}^n\in \mathbb{R}^{N\times N}\ |\ \max_{i,j} |a_{i,j}|\leq C_0\},\ b_I:=\{(b_i)_{i=1}^n\in \mathbb{R}^N\ | \ \max_{i} |b_i|\leq C_0\} $. Then there exist constants $C_2,C_3$, such that $$\begin{aligned} |x^TAx|\leq C_2 x^Tx,\ \ \ |b^Tx|\leq C_3+x^Tx, \ \ \forall x\in \mathbb{R}^N, \forall A\in A_I, \forall b\in b_I.\end{aligned}$$ Hence, there exists a constant $C_4>0$ such that $$\begin{aligned} \label{ratio of two exp} |d_{u,\tau_u}(y)|\leq& C_4\epsilon(y-\mu)^T(y-\mu)+ C_4\epsilon ,\end{aligned}$$ By and , for small $\epsilon$ and large $u$, there exists a constant $K$ such that $$\begin{aligned} &\max_{\tau_u}\left|\frac{f_{u,\tau_u(y)}}{f(y)}-1\right| \leq Ke^{C_4\epsilon(y-\mu)^T(y-\mu)}+1\end{aligned}$$ On the other hand, for all $y\in \mathbb{R}^N$, $$\begin{aligned} f(y)\leq (2\pi)^{-n/2}|\Sigma|^{-1/2}e^{-\frac{\lambda}{2}(y-\mu)^T(y-\mu)},\end{aligned}$$ where $\lambda$ is the minimum eigenvalue of $\Sigma^{-1}$. If we choose $\epsilon<\frac{\lambda}{2C_4}$ and define $$\begin{aligned} g(y):=(2\pi)^{-n/2}|\Sigma|^{-1/2}e^{-\frac{\lambda}{2}(y-\mu)^T(y-\mu)}(Ke^{C_4\epsilon(y-\mu)^T(y-\mu)}+1),\end{aligned}$$ then holds and hence we have completed the proof. [20]{} (). . . (). . . (). . . , (). . . (). . . , (). . . (). . (). . . (). . . (). . . , . , . , , (). . . , (). . . , , , (). . . , (). . . (). . . (). . . (). . . . (). . . (). . . , (). . . (). . . (). . . (). . . (). . . (). 2nd ed. . (). . . [^1]: Research supported in part by NSF grants DMS-1307470 and DMS-1309856.

--- abstract: 'We describe results from a fully self–consistent three dimensional hydrodynamical simulation of the formation of one of the first stars in the Universe. Dark matter dominated pre-galactic objects form because of gravitational instability from small initidal density perturbations. As they assemble via hierarchical merging, primordial gas cools through ro-vibrational lines of hydrogen molecules and sinks to the center of the dark matter potential well. The high redshift analog of a molecular cloud is formed. When the dense, central parts of the cold gas cloud become self-gravitating, a dense core of $\sim 100\Ms$ undergoes rapid contraction. At densities $n>10^9 \ccc$ a $1\Ms$ proto-stellar core becomes fully molecular due to three–body formation. Contrary to analytical expectations this process does not lead to renewed fragmentation and only one star is formed. The calculation is stopped when optical depth effects become important, leaving the final mass of the fully formed star somewhat uncertain. At this stage the protostar is acreting material very rapidly ($\sim 10^{-2}\Ms \yr^{-1}$). Radiative feedback from the star will not only halt its growth but also inhibit the formation of other stars in the same pre–galactic object (at least until the first star ends its life, presumably as a supernova). We conclude that at most one massive ($M\gg1\Ms$) metal free star forms per pre–galactic halo, consistent with recent abundance measurements of metal poor galactic halo stars.' author: - Tom Abel - 'Greg L. Bryan' - 'Michael L. Norman' title: The Formation of the First Star in the Universe --- \#1[[*\[\#1\]*]{}]{} \#1[10\^[\#1]{}]{} \#1[n\_[ \#1]{}]{} \#1[k\_[[\#1]{}]{}]{} \#1[$^{\ref{#1}}$]{} \#1[*\#1* ]{} \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in Motivation ========== Chemical elements heavier than Lithium are synthesized in stars. Such “metals” are observed at times when the Universe was only $\lsim 10$% of its current age in the inter–galactic medium (IGM) as absorption lines in quasar spectra (see Ellison et al. 2000, and references therein). Hence, these heavy elements not only had to be synthesized but also released and distributed in the IGM within the first billion years. Only supernovae of sufficiently short lived massive stars are known to provide such an enrichment mechanism. This leads to the prediction that [*the first generation of cosmic structures formed massive stars (although not necessarily only massive stars).*]{} In the past 30 years it has been argued that the first cosmological objects form globular clusters (), super–massive black holes (), or even low mass stars (). This disagreement of theoretical studies might at first seem surprising. However, the first objects form via the gravitational collapse of a thermally unstable reactive medium, inhibiting conclusive analytical calculations. The problem is particularly acute because the evolution of all other cosmological objects (and in particular the larger galaxies that follow) will depend on the evolution of the first stars. Nevertheless, in comparison to present day star formation, the physics of the formation of the first star in the universe is rather simple. In particular: - the chemical and radiative of processes in the primordial gas are readily understood. - strong magnetic fields are not expected to exist at early times. - by definition no other stars exist to influence the environment through radiation, winds, supernovae, etc. - the emerging standard model for structure formation provides appropriate initial conditions. In previous work we have presented three–dimensional cosmological simulations of the formation of the first objects in the universe (, ) including first applications of adaptive mesh refinement (AMR) cosmological hydrodynamical simulations to first structure formation (, , ABN hereafter) . In these studies we achieved a dynamic range of up to $2\tento{5}$ and could follow in detail the formation of the first dense cooling region far within a pre–galactic object that formed self–consistently from linear density fluctuation in a cold dark matter cosmology. Here we report results from simulations that extend our previous work by another 5 orders of magnitude in dynamic range. For the first time it is possible to bridge the wide range between cosmological and stellar scale. Simulation Setup and Numerical Issues {#sec:sim} ===================================== We employ an Eulerian structured adaptive mesh refinement cosmological hydrodynamical code developed by Bryan and Norman (, ). The hydrodynamical equations are solved with the second order accurate piecewise parabolic method (; ) where a Riemann solver ensures accurate shock capturing with a minimum of numerical viscosity. We use initial conditions appropriate for a spatially flat Cold Dark Matter cosmology with 6% of the matter density contributed by baryons, zero cosmological constant, and a Hubble constant of 50 km/s/Mpc (). The power spectrum of initial density fluctuations in the dark matter and the gas are taken from the computation by the publicly available Boltzmann code CMBFAST () at redshift 100 (assuming an Harrison–Zel’dovich scale–invariant initial spectrum). We set up a three dimensional volume with 128 comoving kpc on a side and solve the cosmological hydrodynamics equations assuming periodic boundary conditions. This small volume is adequate for our purpose, because we are interested in the evolution of the first pre–galactic object within which a star may be formed by a redshift of $z\sim 20$. First we identify the Lagrangian volume of the first proto–galactic halo with a mass of $\sim 10^6\Ms$ in a low resolution pure N–body simulation. Then we generate new initial conditions with four initial static grids that cover this Langrangian region with progressively finer resolution. With a $64^3$ top grid and a refinement factor of 2 this specifies the initial conditions in the region of interest equivalent to a $512^3$ uni–grid calculation. For the adopted cosmology this gives a mass resolution of $1.1\Ms$ for the dark matter (DM, hereafter) and $0.07\Ms$ for the gas. The small DM masses ensure that the cosmological Jeans mass is resolved by at least ten thousand particles at all times. Smaller scale structures in the dark matter will not be able to influence the baryons because of their shallow potential wells. The theoretical expectation holds, because the simulations of ABN which had 8 times poorer DM resolution led to identical results on large scales as the simulation presented here. During the evolution, refined grids are introduced with twice the spatial resolution of the parent (coarser) grid. These child (finer) meshes are added whenever one of three refinement criteria are met. Two Langrangian criteria ensure that the grid is refined whenever the gas (DM) density exceeds 4.6 (9.2) its initial density. Additionally, the local Jeans length is always covered by at least 64 grid cells [^1] (4 cells per Jeans length would be sufficient, ). We have also carried out the simulations with identical initial conditions but varying the refinement criteria. In one series of runs we varied the number of mesh points per Jeans length. Runs with 4, 16, and 64 zones per Jeans length are indistinguishable in all mass weighted radial profiles of physical quantities. No change in the angular momentum profiles could be found, suggesting negligible numerical viscosity effects on angular momentum transport. A further refinement criterion that ensured the local cooling time scale to be longer than the local Courant time also gave identical results. This latter test checked that any thermally unstable region was identified. The simulation follows the non–equilibrium chemistry of the dominant nine species species (H, H$^+$, H$^-$, e$^-$, He, He$^+$, He$^{++}$, H$_2$, and H$_2^+$) in primordial gas. Furthermore, the radiative losses from atomic and molecular line cooling, Compton cooling and heating of free electrons by the cosmic background radiation are appropriately treated in the optically thin limit (, ). To extend our previous the studies to higher densities three essential modifications to the code were made . First we implemented the three–body molecular hydrogen formation process in the chemical rate equations. For temperatures below 300 K we fit to the data of Orel () to get $k_{3b} = 1.3\tento{-32} (T/300\K)^{-0.38}\cm^6\s^{-1}$. Above 300 K we then match it continuously to a powerlaw () $k_{3b} = 1.3\tento{-32} (T/300\K)^{-1}\cm^6\s^{-1}$. Secondly, we introduce a variable adiabatic index for the gas (). The dissipative component (baryons) may collapse to much higher densities than the collisionless component (DM). The discrete sampling of the DM potential by particles can then become inadequate and result in artificial heating of the baryons (cooling for the DM) once the gas density becomes much larger than the local DM density. To avoid this, we smooth the DM particles with a Gaussian of width 0.05  for grids with cells smaller than this length. At this scale, the enclosed gas mass substantially exceeds the enclosed DM mass. The standard message passing library (MPI) was used to implement domain decomposition on the individual levels of the grid hierarchy as a parallelization strategy. The code was run in parallel on 16 processors of the SGI Origin2000 supercomputer at the National Center for Supercomputing Applications at the University of Illinois at Urbana Champaign. We stop the simulation at a time when the molecular cooling lines reach an optical depth of ten at line center because our numerical method cannot treat the difficult problem of time–dependent radiative line transfer in multi–dimensions. At this time the code utilizes above 5500 grids on 27 refinement levels with $1.8\tento{7}\approx 260^3$ computational grid cells. An average grid therefore contains $\sim 15^3$ cells. Results ======= \[colorplate\] Characteristic mass scales -------------------------- Our simulations (Fig. \[colorplate\], Fig. \[5panel\]), identify at least four characterisic mass scales. From the outside going in, one observes infall and accretion onto the pre–galactic halo with a total mass of $7\tento{5}\Ms$, consistent with previous studies (, , , ABN, and for discussion and references). At a mass scale of about 4000 solar mass ($r\sim 10\pc$) rapid cooling and infall is observed. This is accompanyed by the first of three valleys in the radial velocity distribution (Fig. \[5panel\]E). The temperature drops and the molecular hydrogen fraction increases. It is here, at number densities of $\sim 10 \ccc$, that the high redshift analog of a molecular cloud is formed. Although the molecular mass fraction is not even 0.1% it is sufficient to cool the gas rapidly down to $\sim 200\K$. The gas cannot cool below this temperature because of the sharp decrease in the cooling rate below $\sim 200K$. At redshift 19 (Fig. \[5panel\]), there are only two mass scales; however, as time passes the central density grows and eventually passes $10^4 \ccc$, at which point the ro-vibrational levels of  are populated at their equilibrium values and the cooling time becomes independent of density (instead of inversely proportional to it). This reduced cooling efficiency leads to an increase in the temperature (Fig. \[5panel\]D). As the temperature rises, the cooling rate again increases (it is 1000 times higher at 800 K than at 200 K), and the inflow velocities slowly climb. In order to better understand what happens next, we examine the stability of an isothermal gas sphere. The critical mass for gravitational collapse given an external pressure $P_{ext}$ (BE mass hereafter) is given by Ebert () and Bonnor () as: $$\begin{aligned} M_{BE} = 1.18\Ms \frac{c_s^4}{G^{3/2}} P_{ext}^{-1/2}; \ \ \ c_s^2 =\frac{dP}{d\rho} = \frac{\gamma k_B T}{\mu m_H}.\end{aligned}$$ Here $P_{ext}$ is the external pressure and $G$, $k_B$, and $c_s$ the gravitational constant, the Boltzmann constant and the sound speed, respectively. We can estimate this critical mass locally if we set the external pressure to be the local pressure to find $M_{BE}\approx 20\Ms T^{3/2}n^{-1/2}\mu^{-2}\gamma^2$ where $\mu \approx 1.22$ is the mean mass per particle in units of the proton mass. Using an adiabatic index $\gamma=5/3$, we plot the ratio of the enclosed gas mass to this modified BE mass in Figure \[BEmass\]. Our modeling shows (Fig. \[BEmass\]), that by the fourth considered output time, the central 100 exceeds the BE mass at that radius, indicating unstable collapse. This is the third mass scale and corresponds to the second local minimum in the radial velocity curves (Fig. \[5panel\]E). The inflow velocity is $1\kms$ is still subsonic. Although this mass scale is unstable, it does not represent the smallest scale of collapse in our simulation. This is due to the increasing molecular hydrogen fraction. When the gas density becomes sufficiently large ($\sim 10^{10} \ccc$), three-body molecular hydrogen formation becomes important. This rapidly increases the molecular fraction (Fig. \[5panel\]C) and hence the cooling rate. The increased cooling leads to lower temperatures and even stronger inflow and. At a mass scale of $\sim 1 \Ms$, not only is the gas nearly completely molecular, but the radial inflow has become supersonic (Fig. \[5panel\]E). When the  mass fraction approaches unity, the increase in the cooling rate saturates, and the gas goes through a radiative shock. This marks the first appearance of the proto–stellar accretion shock at a radius of about 20 astronomical units from its center. Chemo–Thermal Instability ------------------------- When the cooling time becomes independent of density the classical criterion for fragmentation $t_{cool}< t_{dyn} \propto n^{-1/2}$ () cannot be satisfied at high densities. However, in principal the medium may still be subject to thermal instability. The instability criterion is $$\begin{aligned} \rho \left(\frac{\partial L}{\partial \rho}\right)_{T=const.} - T \left(\frac{\partial L}{\partial T}\right)_{\rho=const.} + L(\rho,T) > 0, \label{eq:ic}\end{aligned}$$ where $L$ denotes the cooling losses per second of a fluid parcel and $T$ and $\rho$ are the gas temperature and mass density, respectively. At densities above the critical densities of molecular hydrogen the cooling time is independent of density, i.e. $\partial L/\partial \rho = \Lambda(T)$ where $\Lambda(T)$ is the high density cooling function (e.g. ). Fitting the cooling function with a power-law locally around a temperature $T_0$ so that $\Lambda(T)\propto (T/T_0)^\alpha$ one finds $\partial L/\partial T = \rho \alpha \Lambda(T)/T$. Hence, under these circumstances the medium is thermally stable if $\alpha > 2$. Because, $\alpha > 4$ for the densities and temperatures of interest, we conclude that the medium is thermally stable. The above analysis neglects the heating from contraction, but this only strengthens the conclusion. If heating balances cooling one can neglect the $+L(\rho,T)$ term in equation (\[eq:ic\]) and find the medium to be thermally stable for $\alpha > 1$. \[BEmass\] However, here we neglected the chemical processes. The detailed analysis for the case when chemical processes occur on the collapse time–scale is well known (). This can be applied to primordial star formation () including the three–body formation of molecular hydrogen () which drives a chemo–thermal instability. Evaluating all the terms in this modified instability criterion (, equation 36) one finds the simple result that for molecular mass fractions $f<6/(2\alpha + 1)$ the medium is expected to be chemo–thermally unstable. These large molecular fractions illustrate that the strong density dependence of the three body   formation dominates the instability. Examining the three dimensional temperature and  density field we clearly see this chemo–thermal instability at work. Cooler regions have larger  fractions. However, no corresponding large density inhomogeneities are found and fragmentation does not occur. This happens because of the short sound crossing times in the collapsing core. When the  formation time–scale becomes shorter than the cooling time the instability originates. However, as long as the sound crossing time is much shorter than the chemical and cooling time scales the cooler parts are efficiently mixed with the warmer material. This holds in our simulation until the final output where for the first time the   formation time scale becomes shorter than the sound–crossing time. However, at this point the proto–stellar core is fully molecular and stable against the chemo–thermal instability. Consequently no large density contrasts are formed. Because at these high densities the optical depth of the cooling radiation becomes larger than unity the instability will be suppressed even further. Angular momentum {#sec:ang_mom} ---------------- Interestingly, rotational support does not halt the collapse. This is for two reaons. The first is shown in panel A of Fig. \[angtrans\], which plots the specific angular momentum against enclosed mass for the same seven output times discussed earlier. Concentrating on the first output (Fig. \[angtrans\]), we see that the central gas begins the collapse with a specific angular momentum only $\sim 0.1$% as large as the mean value. This type of angular momentum profile is typical of halos produced by gravitational collapse (e.g. ), and means that the protostellar gas starts out without much angular momentum to lose. As a graphic example of this, consider the central one solar mass of the collapsing region. It has only an order of magnitude less angular momentum at densities $n\gsim 10^{13}\cm^{-3}$ than it had at $n\gsim 10^{6}\cm^{-3}$ although it collapsed by over a factor 100 in radius. The remaining output times (Fig. \[angtrans\]) indicate that there is some angular momentum transport within the central $100 \Ms$ (since L plotted as a function of enclosed mass should stay constant as long as there is no shell crossing). In panel C, we divide $L$ by $r$ to get a typical rotational velocity and in panels B and D compare this velocity to the Keplerian rotational velocity and the local sound speed, respectively. We find that the typical rotational speed is a factor two to three below that required for rotational support. Furthermore, we see that this azimuthal speed never significantly exceeds the sound speed, although for most the mass below $100 \Ms$ it is comparable in value. We interpret this as evidence that it is shock waves during the turbulent collapse that are responsible for much of the transported angular momentum. A collapsing turbulent medium is different from a disk in Keplarian rotation. At any radius there will be both low and high angular momentum material, and pressure forces or shock waves can redistribute the angular momentum between fluid elements. Lower angular momentum material will selectively sink inwards, displacing higher angular momentum gas. This hydrodynamic transport of angular momentum will be suppressed in situation where the collapse proceeds on the dynamical time rather on the longer cooling time as in the presented case. This difference in cooling time and the widely different initial conditions may explain why this mechanism has not been observed in simulations of present day star formation (e.g. , and references therein). However, such situations may also arise in the late stages of the formation of present day stars and in scenarios for the formation of super–massive black holes. To ensure that the angular momentum transport is not due to numerical shear viscosity () we have carried out the resolution study discussed above. We have varied the effective spatial resolution by a factor 16 and found identical results. Furthermore, we have run the adaptive mesh refinement code with two different implementations of the hydrodynamics solver. The resolution study and the results presented here were carried out with a direct piecewise parabolic method adopted for cosmology (; ). We ran another simulation with the lower order ZEUS hydrodynamics () and still found no relevant differences. These tests are not strict proof that the encountered angular momentum transport is not caused by numerical effects; however, they are reassuring. Magnetic Fields? ---------------- The strength of magnetic fields generated around the epoch of recombination is minute. In contrast, phase transitions at the qantum–chromo–dynamic (QCD) and electro–weak scales may form even dynamically important fields. While there is a plethora of such scenarios for primordial magnetic field generation in the early universe they are not considered to be an integral part of our standard picture of structure formation. This is because not even the order of these phase transitions is known (), and references therein). Unfortunately, strong primordial small-scale ($\ll 1$ comoving ) magnetic fields are poorly constrained observationally (). The critical magnetic field for support of a cloud () allows a rough estimate up to which primordial magnetic field strengths we may expect our simulation results to hold. For this we also assume a flux frozen flow with no additional amplification of the magnetic field other than the contraction ($B\propto \rho^{2/3}$). For a comoving B field of $\gsim 3\tento{-11}\G$ on scales $\lsim 100\kpc$ the critical field needed for support may be reached during the collapse possibly modifying the mass scales found in our purely hydrodynamic simulations. However, the ionized fraction drops rapidly during the collapse because of the absence of cosmic rays ionizations. Consequently ambipolar diffusion should be much more effective in the formation of the first stars even if such strong primordial magnetic fields were present. Discussion ========== Previously we discussed the formation of the pre–galactic object and the primordial “molecular cloud” that hosts the formation of the first star in the simulated patch of the universe (). These simulations had a dynamic range of $\sim 10^5$ and identified a $\sim 100\Ms$ core within the primordial “molecular cloud” undergoing renewed gravitational collapse. The fate of this core was unclear because there was the potential caveat that three body  formation could have caused fragmentation. Indeed this further fragmentation had been suggest by analytic work () and single zone models (). The three dimensional simulations described here were designed to be able to test whether the three body process will lead to a break up of the core. [*No fragmentation due to three body  formation is found.*]{} This is to a large part because of the slow quasi–hydrostatic contraction found in ABN which allows sub–sonic damping of density perturbations and yields a smooth distribution at the time when three body  formation becomes important. Instead of fragmentation a single fully molecular proto–star of $\sim 1\Ms$ is formed at the center of the $\sim 100\Ms$ core. However, even with extraordinary resolution, the [*final*]{} mass of the first star remains unclear. Whether all the available cooled material of the surroundings will accrete onto the proto–star or feedback from the forming star will limit the further accretion and hence its own growth is difficult to compute in detail. Within $10^4\yr$ about $70\Ms$ may be accreted assuming that angular momentum will not slow the collapse (Fig. \[acrete\]). The maximum of the accretion time of $\sim 5\tento{6}\yr$ is at $\sim 600 \Ms$. However, stars larger than $100\Ms$ will explode within $\sim 2\Myr$. Therefore, it seems unlikely (even in the absence of angular momentum) that there would be sufficient time to accrete such large masses. A one solar mass proto–star will evolve too slowly to halt substantial accretion. From the accretion time profile (Fig. \[acrete\]) one may argue that a more realistic minimum mass limit of the first star should be $\gsim 30 \Ms$ because this amount would be accreted within a few thousand years. This is a very short time compared to expected proto–stellar evolution times. However, some properties of the primordial gas may make it easier to halt the accretion. One possibility is the destruction of the cooling agent, molecular hydrogen, without which the acreting material may reach hydrostatic equilibrium. This may or may not be sufficient to halt the accretion. One may also imagine that the central material heats up to $10^4$ K, allowing Lyman-$\alpha$ cooling from neutral hydrogen. That cooling region may expand rapidly as the internal pressure drops because of infall, possibly allowing the envelope to accrete even without molecular hydrogen as cooling agent. Additionally, radiation pressure from ionizing photons as well as atomic hydrogen Lyman series photons may become significant and eventually reverse the flow. The mechanisms discussed by Haehnelt (1993) on galactic scales will play an important role for the continued accretion onto the proto–star. This is an interplay of many complex physical processes because one has a hot ionized Strömgren sphere through which cool and dense material is trying to accrete. In such a situation one expects a Raleigh–Taylor type instability that is modified via the geometry of the radiation field. At the final output time presented here there are $\sim 4\tento{57}$ hydrogen molecules in the entire protogalaxy. Also the  formation time scale is long because there are no dust grains and the free electrons (needed as a catalyst) have almost fully recombined. Hence, as soon as the the first UV photons of Lyman Werner band frequencies are produced there will be a rapidly expanding photo–dissociating region (PDR) inhibiting further cooling within it. This photo-dissociation will prevent further fragmentation at the molecular cloud scale. I.e. no other star can be formed within the same halo before the first star dies in a supernova. The latter, however, may have sufficient energy to unbind the entire gas content of the small pre–galactic object it formed in (). This may have interesting feedback consequences for the dispersal of metals, entropy and magnetic field into the intergalactic medium (, ). Smoothed particle hydrodynamics (SPH, e.g. ), used extensively in cosmological hydrodynamics, has been employed () to follow the collapse of solid body rotating uniform spheres. The assumption of coherent rotation causes these clouds to collapse into a disk which developes filamentary structures which eventually fragment to form dense clumps of masses between $100$ and $1000$ solar masses. It has been argued that these clumps will continue to accrete and merge and eventually form very massive stars. These SPH simulation have unrealistic initial conditions and much less resolution then our calculations. However, they also show that many details of the collapse forming a primordial star are determined by the properties of the hydrogen molecule. We have also simulated different initial density fields for a Lambda CDM cosmology. There we have focused on halos with different clustering environments. Although we have not followed the collapse in these halos to proto-stellar densities, we have found no qualitative differences in the “primordial molecular cloud” formation process as discussed in ABN. Also other AMR simulations () give consistent results on scales larger than $1\pc$. In all cases a cooling flow forms the primordial molecular cloud at the center of the dark matter halo. We conclude that the molecular cloud formation process seems to be independent of the halo clustering properties and the adopted CDM type cosmology. Also the mass scales for the core and the proto–star are determined by the local Bonnor–Ebert mass. Consequently, we expect the key results discussed here to be insensitive to variations in cosmology or halo clustering. Conclusion ========== The picture arising from these numerical simulations has some very interesting implications. It is possible that all metal free stars are massive and form in isolation. Their supernovae may provide the metals seen in even the lowest column density quasar absorption lines (, and references therein). Massive primordial stars offer a natural explanation for the absence of purely metal free low mass stars in the Milky Way. The consequences for the formation of galaxies may be even more profound in that the supernovae provide metals, entropy, and magnetic fields and may even alter the initial power spectrum of density fluctuations of the baryons. Interestingly, it has been recently argued, from abundance patterns, that in low metallicity galactic halo stars seem to have been enriched by only one population of massive stars (). These results, if confirmed, would represent strong support for the picture arising from our ab initio simulations of first structure formation. To end on a speculative note there is suggestive evidence that links gamma ray bursts to sites of massive star formation (e.g. ). It would be very fortunate if a significant fraction of the massive stars naturally formed in the simulations would cause gamma ray bursts (e.g. ). Such high redshift bursts would open a remarkably bright window for the study of the otherwise dark (faint) Ages. 1. \[PD68\] Peebles, P. J. E. & Dicke, R. H. 1968, ApJ [**154**]{}, 891 2. \[H69\] Hirasawa, T. 1969, Prog. Theor. Phys. [**42**]{}, 523 3. \[PSS83\] Palla, F., Salpeter, E.E., Stahler, S.W. 1983, ApJ [**271**]{}, 632 4. \[A95\] Abel, T. 1995, thesis, University of Regensburg. 5. \[AANZ\] Abel, T., Anninos, P., Norman, M.L., Zhang, Y. 1998, ApJ [**508**]{}, 518. 6. \[ABN99\] Abel, T., Bryan, G. L., & Norman, M. L. 1999, in “Evolution of Large Scale Structure: From Recombination to Garching”, eds. Banday, T., Sheth, R. K. and Costa, L. N. 7. \[ABN00\] ABN: Abel, T., Bryan, G.L., Norman, M.L. 2000, ApJ [**540**]{}, 39 8. \[cosmo\] Friedmann models with a cosmological constant which currently seem to fit various observational test better differ from the standard CDM model considered here only slightly at the high redshifts modeled. 9. \[BN97\] Bryan, G.L., Norman, M.L. 1997, in [*Computational Astrophysics*]{}, eds. D.A. Clarke and M. Fall, ASP Conference \#123 10. \[BN99\] Bryan, G.L., Norman, M.L. 1999, in [*Workshop on Structured Adaptive Mesh Refinement Grid Methods*]{}, IMA Volumes in Mathematics No. 117, ed. N. Chrisochoides, p. 165 11. \[WC84\] Woodward, P. R., & Colella, P. 1984, J. Comput. Phys. [ **54**]{}, 115 12. \[Betal95\] Bryan, G.L., Cen, R., Norman, M.L., Ostriker, J.P. & Stone, J.M. 1994, ApJ [**428**]{}, 405 13. \[SZ96\] Seljak, U., & Zaldarriaga, M. 1996, ApJ [**469**]{}, 437 14. \[TKMH97\] Truelove, J. K., Klein, R. I., McKee, C. F., Holliman, J. H., Howell, L. H. & Greenough, J. A. 1997, ApJL [**489**]{}, L179 15. \[AAZN97\] Abel, T., Anninos, P., Zhang, Y., Norman, M.L. 1997, NewA [**2**]{}, 181 16. \[AZAN97\] Anninos, P., Zhang, Y., Abel, T., and Norman, M.L. 1997, NewA [**2**]{}, 209. 17. \[Orel\] Orel, A.E. 1987, J.Chem.Phys. [**87**]{}, 314 18. \[ON98\] Omukai, K. & Nishi, R. 1998, ApJ [**508**]{}, 141 19. \[Tegmark97\] Tegmark, M., Silk, J., Rees, M.J., Blanchard, A., Abel, T., Palla, F. 1997, ApJ [**474**]{}, 1. 20. \[BCL99\] Bromm, V., Coppi, P. S., & Larson, R. B. 1999, ApJL [ **527**]{}, L5 21. \[Ebert\] Ebert, R. 1955, Zs. Ap. 217 22. \[Bonnor\] Bonnor, W. B. 1956, MNRAS [**116**]{}, 351 23. \[Field65\] Field, G. B. 1965, ApJ [**142**]{}, 531 24. \[GP98\] Galli, D. & Palla, F. 1998, A&A [**335**]{}, 403 25. \[SY77\] Sabano, Y. & Yoshi, Y. 1977, PASJ [**29**]{}, 207 26. \[Silk83\] Silk, J. 1983, MNRAS [**205**]{}, 705 27. \[QZ88\] Quinn, P.J. & Zurek, W.H. 1988, ApJ [**331**]{}, 1 28. \[BB93\] Burkert, A. & Bodenheimer, P. 1993, MNRAS [**264**]{}, 798 29. \[NWB80\] Norman, M.L., Wilson, J.R. & Barton, R.T. 1980, ApJ [ **239**]{}, 968 30. \[SN92\] Stone, J.M. & Norman, M.L. 1992, ApJS [**80**]{}, 791 31. \[SOJ97\] Sigl, G., Olinto, A. V. & Jedamzik, K. 1997, Phys.Rev.D [**55**]{}, 4582 32. \[BFS97\] Barrow, J. D., Ferreira, P. G. and Silk, J. 1997, Physical Review Letters [**78**]{}, 3610 33. \[MS76\] Mouschovias, T. Ch., Spitzer, L. 1976, ApJ [**210**]{}, 326 34. \[Haehnelt\] Haehnelt, M.G. 1995, MNRAS [**273**]{}, 249 35. \[MF99\] Mac Low, M. & Ferrara, A. 1999, ApJ [**513**]{}, 142 36. \[F98\] Ferrara, A. 1998, ApJL [**499**]{}, L17 37. \[CB00\] Cen, R. & Bryan, G.L. 2000, ApJL [**546**]{}, L81 38. \[M92\] Monaghan, J. J. 1992, ARAA [**30**]{}, 543 39. \[MBA01\] Machacek, M.E., Bryan, G.L., & Abel, T 2001, ApJ [**548**]{}, 509 40. \[ESSP00\] Ellison, S. L., Songaila, A., Schaye, J. & Pettini, M. 2000, , 1175 41. \[WQ00\] Wasserburg, G. J. & Qian, Y. 2000, ApJL [**529**]{}, L21 42. \[R99\] Reichart, D. E. 1999, ApJL [**521**]{}, L111 43. \[CL00\] Ciardi, B. & Loeb, A. 2000, ApJ [**540**]{}, 687 44. T.A. happily acknowledges stimulating and insightful discussions with Martin Rees and Richard Larson. GLB was supported through Hubble Fellowship grant HF-0110401-98A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS5-26555. [^1]: The Jeans mass which is the relevant mass scale for collapse and fragmentation is thus resolved by at least $4\pi 32^3/3\approx 1.4\tento{5}$ cells.

--- abstract: 'In a network, a local disturbance can propagate and eventually cause a substantial part of the system to fail, in cascade events that are easy to conceptualize but extraordinarily difficult to predict. Here, we develop a statistical framework that can predict cascade size distributions by incorporating two ingredients only: the vulnerability of individual components and the co-susceptibility of groups of components (i.e., their tendency to fail together). Using cascades in power grids as a representative example, we show that correlations between component failures define structured and often surprisingly large [*groups*]{} of co-susceptible components. Aside from their implications for blackout studies, these results provide insights and a new modeling framework for understanding cascades in financial systems, food webs, and complex networks in general.' author: - Yang Yang - Takashi Nishikawa - 'Adilson E. Motter' title: 'Vulnerability and co-susceptibility determine the size of network cascades' --- The stability of complex networks is largely determined by their ability to operate close to equilibrium—a condition that can be compromised by relatively small perturbations that can lead to large cascades of failures. Cascades are responsible for a range of network phenomena, from power blackouts [@hines2009large] and air traffic delay propagation [@fleurquin2013] to secondary species extinctions [@sahasrabudhe2011rescuing; @eco:11] and large social riots [@watts2002; @brummitt2015]. Evident in numerous previous modeling efforts [@bak_1987; @kinney_2005; @buldyrev_2010; @watts2002; @goh_2003; @motter_2004; @dobson2007complex; @brummitt2015; @sahasrabudhe2011rescuing; @eco:11] is that dependence between components is the building block of these self-amplifying processes and can lead to correlations among eventual failures in a cascade. A central metric characterizing a cascade is its size. While the suitability of a size measure depends on the context and purpose, a convenient measure is the number of network components (nodes or links) participating in the cascade (e.g., failed power lines, delayed airplanes, extinct species). Since there are many known and unknown factors that can affect the details of cascade dynamics, the main focus in the literature has been on characterizing the statistics of cascade sizes rather than the size of individual events. This leads to a fundamental question: what determines the distribution of cascade sizes? ![ Example of co-susceptibility of cascading failures in a power grid. (a) Color-coded network of positive correlations between failures of transmission lines (blue dots) in the Texas network. In the gray background network, the nodes represent (all) transmission lines, while each link represents a direct physical connection through a common substation. (b) Network in (a) after a correlation-based repositioning of the nodes.[]{data-label="fig1"}](fig1.pdf "fig:"){width="3.5in"}\ In this Letter, we show that cascading failures (and hence their size distributions) are often determined primarily by two key properties associated with failures of the system components: the [*vulnerability*]{}, or the failure probability of each component, and the [*co-susceptibility*]{}, or the tendency of a group of components to fail together. The latter is intimately related to pairwise correlations between failures, as we will see below. We provide a concrete algorithm for identifying groups of co-susceptible components for any given network. We demonstrate this using the representative example of cascades of overload failures in power grids (Fig. \[fig1\]). Based on our findings, we develop the [*co-susceptibility model*]{}—a statistical modeling framework capable of accurately predicting the distribution of cascade sizes, depending solely on the vulnerability and co-susceptibility of component failures. We consider a system of $n$ components subject to cascading failures, in which a set of initial component failures can induce a sequence of failures in other components. Here we assume that the initial failures and the propagation of failures can be modeled as stochastic and deterministic processes, respectively (although the framework also applies if the propagation or both are stochastic). Thus, the cascade size $N$, defined here as the total number of components that fail after the initial failures, is a random variable that can be expressed as $$\label{eqn2} N = \sum_{\ell=1}^n F_\ell,$$ where $F_\ell$ is a binary random variable representing the failure status of component $\ell$ (i.e., $F_\ell = 1$ if component $\ell$ fails during the cascade, and $F_\ell = 0$ otherwise). While the $n$ components may be connected by physical links, a component may fail as the cascade propagates even if none of its immediate neighbors have failed [@brummitt2015; @Witthaut:2015; @dobson_2016]. For example, in the case of cascading failures of transmission lines in a power grid, the failure of one line can cause a reconfiguration of power flows across the network that leads to the overloading and subsequent failure of other lines away from the previous failures [@dobson_2016; @anghel2007stochastic]. A concrete example network we analyze throughout this Letter using the general setup above is the Texas power grid, for which we have 24 snapshots, representing on- and off-peak power demand in each season of three consecutive years. Each snapshot comprises the topology of the transmission grid, the capacity threshold of each line, the power demand of each load node, and the power supply of each generator node (extracted from the data reported to FERC [@FERC]). For each snapshot we use a physical cascade model to generate $K=5{,}000$ cascade events. In this model (which is a variant of that in Ref. [@anghel2007stochastic] with the power re-balancing scheme from Ref. [@Hines2011]), an initial perturbation to the system (under a given condition) is modeled by the removal of a set of randomly selected lines. A cascade following the initial failures is then modeled as an iterative process. In each step, power flow is redistributed according to Kirchhoff’s law and might therefore cause some lines to be overloaded and removed (i.e., to fail) due to overheating. The temperature of the transmission lines is described by a continuous evolution model and the overheating threshold for line removal is determined by the capacity of the line [@anghel2007stochastic]. When a failure causes part of the grid to be disconnected, we re-balance power supply and demand under the constraints of limited generator capacity [@Hines2011]. A cascade stops when no more overloading occurs, and we define the size $N$ of the cascade as the total number of removed lines (excluding the initial failures). This model [@source-code], accounting for several physical properties of failure propagation, sits relatively high in the hierarchy of existing power-grid cascade models [@opaModel; @henneaux2016; @dobson2012_vulnerability; @hines_2015], which ranges from the most detailed engineering models to simplest graphical or stochastic models. The model has also been validated against historical data [@Yang:2016]. In general, mutual dependence among the variables $F_\ell$ may be necessary to explain the distribution of the cascade size $N$. We define the [*vulnerability*]{} $p_\ell \equiv \langle F_\ell \rangle$ of component $\ell$ to be the probability that this component fails in a cascade event (including events with $N=0$). If the random variables $F_\ell$ are uncorrelated (and thus have zero covariance), then $N$ would follow Poisson’s binomial distribution [@Wang:1993], with average $\tilde{\mu}=\sum_\ell p_\ell$ and variance $\tilde{\sigma}^2=\sum_\ell p_\ell(1-p_\ell)$. However, the actual variance $\sigma^2$ of $N$ observed in the cascade-event data is significantly larger than the corresponding value $\tilde{\sigma}^2$ under the no-correlation assumption for all $24$ snapshots of the Texas power grid (with the relative difference, $\bar{\sigma}^2\equiv(\sigma^2 - \tilde{\sigma}^2)/\tilde{\sigma}^2$, ranging from around $0.18$ to nearly $39$). Thus, the mutual dependence must contribute to determining the distribution of $N$ in these examples. Part of this dependence is captured by the correlation matrix $C$, whose elements are the pairwise Pearson correlation coefficients among the failure status variables $F_\ell$. When the correlation matrix is estimated from cascade-event data, it has noise due to finite sample size, which we filter out using the following procedure. First, we standardize $F_\ell$ by subtracting the average and dividing it by the standard deviation. According to random matrix theory, the probability density of eigenvalues of the correlation matrix computed from $K$ samples of $T$ independent random variables follow the Marchenko-Pastur distribution [@mehta2004random], $\rho(\lambda) = K\sqrt{(\lambda_{+} - \lambda)(\lambda - \lambda_{-})}/(2 \pi \lambda T)$, where $\lambda_{\pm} = \bigl[1 \pm \sqrt{T/K}\,\bigr]^2$. Since those eigenvalues falling between $\lambda_{-}$ and $\lambda_{+}$ can be considered contributions from the noise, the sample correlation matrix $\widehat{C}$ can be decomposed as $\widehat{C} = \widehat{C}^{(\text{ran})}+\widehat{C}^{(\text{sig})}$, where $\widehat{C}^{(\text{ran})}$ and $\widehat{C}^{(\text{sig})}$ are its random and significant parts, respectively, which can be determined from the eigenvalues and the associated eigenvectors [@MacMahon:2015]. In the network visualization of Fig. \[fig1\](a), we show the correlation coefficients $\widehat{C}_{\ell\ell'}^{\text{(sig)}}$ between components $\ell$ and $\ell'$ estimated from the cascade-event data for the Texas grid under the 2011 summer on-peak condition. Note that we compute correlation only between those components that fail more than once in the cascade events. As this example illustrates, we observe no apparent structure in a typical network visualization of these correlations. However, as shown in Fig. \[fig1\](b), after repositioning the nodes based on correlation strength, we can identify clusters of positively and strongly correlated components—those that tend to fail together in a cascade. To more precisely capture this tendency of simultaneous failures, we define a notion of [*co-susceptibility*]{}: a given subset of $m$ components $\mathcal{I}\equiv\{\ell_1,\ldots,\ell_m\}$ is said to be co-susceptible if $$\label{eqn:co-sus} \gamma_\mathcal{I} \equiv \frac{\langle N_\mathcal{I} \vert N_\mathcal{I}\neq0 \rangle - \bar{n}_\mathcal{I}}{m - \bar{n}_\mathcal{I}} > \gamma_\text{th},$$ where $N_\mathcal{I}\equiv\sum_{j=1}^m F_{\ell_j}$ is the number of failures in a cascade event among the $m$ components, $\langle N_\mathcal{I} \vert N_\mathcal{I}\neq0 \rangle$ denotes the average number of failures among these components given that at least one of them fails, $\bar{n}_\mathcal{I}\equiv \sum_{j=1}^m p_{\ell_j}/\bigl[1-\prod_{k=1}^m(1-p_{\ell_k})\bigr] \ge 1$ is the value $\langle N_\mathcal{I} \vert N_\mathcal{I}\neq0 \rangle$ would take if $F_{\ell_1},\ldots,F_{\ell_m}$ were independent. Here we set the threshold in Eq.  to be $\gamma_\text{th}=\sigma_{N_\mathcal{I}}/(m - \bar{n}_\mathcal{I})$, where $\sigma_{N_\mathcal{I}}^2 \equiv \sum_{j=1}^m p_{\ell_j}(1-p_{\ell_j})/\bigl[1-\prod_{k=1}^m(1-p_{\ell_k})\bigr] - \bar{n}_\mathcal{I}^2 \prod_{k=1}^m(1-p_{\ell_k})$ is the variance of $N_\mathcal{I}$ given $N_\mathcal{I}\neq0$ for statistically independent $F_{\ell_1},\ldots,F_{\ell_m}$. By definition, the co-susceptibility measure $\gamma_\mathcal{I}$ equals zero if $F_{\ell_1},\ldots,F_{\ell_m}$ are independent. It satisfies $-(\bar{n}_\mathcal{I}-1)/(m-\bar{n}_\mathcal{I}) \le \gamma_\mathcal{I} \le 1$, where the (negative) lower bound is achieved if multiple failures never occur and the upper bound is achieved if all $m$ components fail whenever one of them fails. Thus, a set of co-susceptible components are characterized by significantly larger number of simultaneous failures among these components, relative to the expected number for statistically independent failures. While $\gamma_\mathcal{I}$ can be computed for a given set of components, identifying sets of co-susceptible components in a given network from Eq.  becomes infeasible quickly as $n$ increases due to combinatorial explosion. ![ Two-stage algorithm for identifying sets of co-susceptible components. (a) First stage, in which we partition the reduced unweighted correlation graph into multiple cliques $\{Q_k\}$, and index the cliques in descending order of size. (b) Second stage, in which we find the co-susceptible groups $\{R_k\}$ by recursively agglomerating cliques that have enough connections. \[fig2\]](fig2.pdf "fig:"){width="8.5cm"}\ Here we propose an efficient two-stage algorithm for identifying co-susceptible components [@source-code]. The algorithm is based on partitioning and agglomerating the vertices of the auxiliary graph $G_0$ in which vertices represent the components that fail more than once in the cascade-event data, and (unweighted) edges represent the dichotomized correlation between these components. Here we use $\widehat{C}^{(\text{sig})}_{\ell \ell'}>0.4$ as the criteria for having an edge between vertices $\ell$ and $\ell'$ in $G_0$. In the first stage \[illustrated in Fig. \[fig2\](a)\], $G_0$ is divided into non-overlapping cliques—subgraphs within which any two vertices are directly connected—using the following iterative process. In each step $k=1,2,\ldots$, we identify a clique of the largest possible size (i.e., the number of vertices it contains), denote this clique as $Q_k$, remove $Q_k$ from the graph $G_{k-1}$, and then denote the remaining graph by $G_k$. Repeating this step for each $k$ until $G_k$ is empty, we obtain a sequence $Q_1, Q_2, \ldots, Q_m$ of non-overlapping cliques in $G_0$, indexed in the order of non-increasing size. In the second stage, we agglomerate these cliques, as illustrated in Fig. \[fig2\](b). Initially, we set $R_k = Q_k$ for each $k$. Then, for each $k=2,3,\ldots,m$, we either move all the vertices in $R_k$ to the largest group among $R_1,\cdots, R_{k-1}$ for which at least $80\%$ of all the possible edges between that group and $R_k$ actually exist, or we keep $R_k$ unchanged if no group satisfies this criterion. Among the resulting groups, we denote those groups whose size is at least three by $R_1,R_2,\ldots,R_{m'}$, $m'\le m$. A key advantage of our method over applying community-detection algorithms [@MacMahon:2015] is that the edge density threshold above can be optimized for the accuracy of cascade size prediction. ![ Correlations between transmission line failures in the Texas power grid. (a–d) Estimated correlation matrix $\widehat{C}^{\text{(sig)}}$ \[(a),(b)\] and vulnerabilities $p_\ell$ \[(c),(d)\] under the 2011 summer on-peak condition. In (a) and (c), the transmission lines are indexed so that $p_\ell$ is decreasing in $\ell$. In (b) and (d), they are indexed so that the sets of co-susceptible lines appear as diagonal blocks. (e) Sizes of the groups of co-susceptible transmission lines, indicated by the lengths of the individual segments of the red portion of the top bar (on-peak snapshot) and of the blue portion of the bottom bar (off-peak snapshot) for each season. The gray portion of each bar corresponds to groups of fewer than three lines among groups $R_k$. (f) Relative difference $\bar{\sigma}^2$ between the variance of the cascade size and its counterpart under the no-correlation assumption. \[fig:correlation\]](fig3.pdf){width="\columnwidth"} We test the effectiveness of our general algorithm on the Texas power grid. As Figs. \[fig:correlation\](a) and \[fig:correlation\](b) show, the block-diagonal structure of $\widehat{C}^{\text{(sig)}}$ indicating high correlation within each group and low correlation between different groups becomes evident when the components are reindexed according to the identified groups. We note, however, that individual component vulnerabilities do not necessarily correlate with the co-susceptibility group structure \[see Fig. \[fig:correlation\](d), in comparison with Fig. \[fig:correlation\](c)\]. We find that the sizes of the groups of co-susceptible components vary significantly across the $24$ snapshots of the Texas power grid, as shown in Fig. \[fig:correlation\](e). The degree of co-susceptibility, as measured by the total number of co-susceptible components, is generally lower under an off-peak condition than the on-peak counterpart \[Fig. \[fig:correlation\](e)\]. This is consistent with the smaller deviation from the no-correlation assumption observed in Fig. \[fig:correlation\](f), where this deviation is measured by the relative difference in the variance, $\bar{\sigma}^2$ (defined above). Since high correlation within a group of components implies a high probability that many of them fail simultaneously, the groups identified by our algorithm tend to have high values of $\gamma_\mathcal{I}$. Indeed, our calculation shows that Eq.  is satisfied for all the $171$ co-susceptible groups found in the $24$ snapshots. ![ Validation of the co-susceptibility model. (a) Cumulative distributions $S_N(x)$ and $S_{\cal N}(x)$ of cascade sizes $N$ and ${\cal N}$ from the cascade-event data and from the co-susceptibility model, respectively, under the 2011 summer on-peak condition. Inset: Binned probabilities from the two distributions plotted against each other. The shaded area indicates the 95% confidence interval. (b) Distance between two distributions as a function of the total load in megawatts (MW) for all $24$ snapshots. []{data-label="fig:model-validation"}](fig4.pdf){width="\columnwidth"} Given the groups of components generated through our algorithm, the [*co-susceptibility model*]{} is defined as the set of binary random variables ${\cal F}_\ell$ (different from $F_\ell$) following the dichotomized correlated Gaussian distribution [@emrich1991method; @macke2009generating] whose marginal probabilities (i.e., the probabilities that ${\cal F}_\ell=1$) equal the estimates of $p_\ell$ from the cascade-event data and whose correlation matrix ${\cal C}$ is given by $${\cal C}_{\ell\ell'} = \begin{cases} \widehat{C}^{(\text{sig})}_{\ell\ell'} & \text{if $\ell,\ell'\in R_k$ for some $k\le m'$,}\\ 0 & \text{otherwise.} \end{cases}$$ We are thus approximating the correlation matrix $C$ by the block diagonal matrix ${\cal C}$, where the blocks correspond to the sets of co-susceptible components. In terms of the correlation network, this corresponds to using only those links within the same group of co-susceptible components for predicting the distribution of cascade sizes. Since individual groups are assumed to be uncorrelated, this can be interpreted as model dimensionality reduction, in which the dimension reduces from $n$ to the size of the largest group. We sample ${\cal F}_\ell$ using the code provided in Ref. [@macke2009generating]. In this implementation, the computational time for sampling scales with the number of variables with an exponent of $3$, so factors of $2.0$ to $15.2$ in dimensionality reduction observed for the Texas power grid correspond to a reduction of computational time by factors of more than $8$ to more than $3{,}500$. We now validate the co-susceptibility model for the Texas grid. We estimate the cumulative distribution function $S_{\cal N}(x)$ of cascade size, ${\cal N}\equiv \sum_\ell {\cal F}_\ell$, using $3{,}000$ samples generated from the model. As shown in Fig. \[fig:model-validation\](a), this function matches well with the cumulative distribution function $S_N(x)$ of cascades size $N$ computed directly from the cascade-event data. This is validated more quantitatively in the inset; the (binned) probability $p_{\cal N}(x)$ that $x \le {\cal N} \le x+\Delta x$ for the co-susceptibility model is plotted against the corresponding probability $p_N(x)$ for the cascade-event data, using a bin size of $\Delta x = N_{\max}/20$, where $N_{\max}$ denotes the maximum cascade size observed in the cascade-event data. The majority of the points lie within the 95% confidence interval for $p_{\cal N}(x)$, computed using the estimated $p_N(x)$. To validate the co-susceptibility model across all 24 snapshots, we use the Kolmogorov-Smirnov (KS) test [@massey1951kolmogorov]. Specifically, for each snapshot we test the hypothesis that the samples of ${\cal N}$ and the corresponding samples of $N$ are from the same distribution. Figure \[fig:model-validation\](b) shows the measure of distance between two distributions, $\sup_x |S_N(x)-S_{\cal N}(x)|$, which underlies the KS test, as a function of the total amount of electrical load in the system. We find that the null hypothesis cannot be rejected at the 5% significance level for most of the cases we consider \[$21/24=87.5$%, blue dots in Fig. \[fig:model-validation\](b)\]; it can be rejected in only three cases (red triangles, above the threshold distance indicated by the dashed line), all corresponding to high stress (i.e., high load) conditions. We also see that more stressed systems are associated with larger distances between the distributions, and a higher likelihood of being able to reject the null hypothesis. We believe this is mainly due to higher-order correlations not captured by $p_\ell$ and $C$. The identification of co-susceptibility as a key ingredient in determining cascade sizes leads to two new questions: (1) What gives rise to co-susceptibility? (2) How to identify the co-susceptible groups? While the first question opens an avenue for future research, the second question is addressed by the algorithm developed here (for which we provide a ready-to-use software [@source-code]). The co-susceptibility model is general and can be used for cascades of any type (of failures, information, or any other spreadable attribute) for which information is available on the correlation matrix and the individual “failure” probabilities. Such information can be empirical, as in the financial data studied in Ref. [@Plerou:2002], or generated from first-principle models, as in the power-grid example used here. Our approach accounts for correlations (a strength shared by some other approaches, such as the one based on branching processes [@dobson2012]), and does so from the fresh, network-based perspective of co-susceptibility. Finally, since co-susceptibility is often a nonlocal effect, our results suggest that we may need nonlocal strategies for reducing the risk of cascading failures, which bears implications for future research. This work was supported by ARPA-E Award No. DE-AR0000702 . P. Hines, J. Apt, and S. Talukdar, Large blackouts in North America: Historical trends and policy implications, Energ. Policy **37**, 5249 (2009). P. Fleurquin, J. J. Ramasco, and V. M. Eguiluz, Systemic delay propagation in the US airport network, Sci. Rep. **3**, 1159 (2013). J. A. Estes, J. Terborgh, J. S. Brashares, M. E. Power, J. Berger, W. J. Bond, and D. A. Wardle, Trophic downgrading of planet Earth, Science **333**, 301 (2011). S. Sahasrabudhe and A. E. Motter, Rescuing ecosystems from extinction cascades through compensatory perturbations, Nature Commun. **2**, 170 (2011). D. J. Watts, A simple model of global cascades on random networks, Proc. Natl. Acad. Sci. USA **99**, 5766 (2002). C. D. Brummitt, G. Barnett, and R. M. D’Souza, Coupled catastrophes: sudden shifts cascade and hop among interdependent systems, J. Roy. Soc. Interface **12**, 20150712 (2015). R. Kinney, P. Crucitti, R. Albert, and V. Latora, Modeling cascading failures in the North American power grid, Euro. Phys. J. B **46**, 101 (2005). S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, Catastrophic cascade of failures in interdependent networks, Nature **464**, 1025 (2010). K. I. Goh, D. S. Lee, B. Kahng, and D. Kim, Sandpile on scale-free networks, Phys. Rev. Lett. **91**, 148701 (2003). A. E. Motter, Cascade control and defense in complex networks, Phys. Rev. Lett. **93**, 098701 (2004). I. Dobson, B. A. Carreras, V. E. Lynch, and D. E. Newman, Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization, Chaos **17**, 026103 (2007). P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: An explanation of the $1/f$ noise, Phys. Rev. Lett. **59**, 381 (1987). D. Witthaut and M. Timme, Nonlocal effects and countermeasures in cascading failures, Phys. Rev. E **92**, 032809 (2015). I. Dobson, B. A. Carreras, D. E. Newman, and J. M. Reynolds-Barredo, Obtaining statistics of cascading line outages spreading in an electric transmission network from standard utility data, IEEE T. Power Syst. **99**, 1 (2016). M. Anghel, K. A. Werley, and A. E. Motter, Stochastic model for power grid dynamics, [*Proc. 40th Int. Conf. Syst. Sci.*]{} HICSS’07, Big Island, HI, USA, Vol. 1, 113 (2007). The data for the snapshots are obtained from Federal Energy Regulatory Commission (FERC) Form 715. P. Hines, E. Cotilla-Sanchez, and S. Blumsack, Topological models and critical slowing down: Two approaches to power system blackout risk analysis, [*Proc. 44th Int. Conf. Syst. Sci.*]{} HICSS’11, Kauai, HI, USA, 1 (2011). Source code is available for download at <https://github.com/yangyangangela/determine_cascade_sizes> B. A. Carreras, D. E. Newman, I. Dobson, and N. S. Degala, Validating OPA with WECC data, [*Proc. 46th Int. Conf. Syst. Sci.*]{} HICSS’13, Maui, HI, USA, 2197 (2013). P. Henneaux, P. E. Labeau, J. C. Maun, and L. Haarla, A two-level probabilistic risk assessment of cascading outages, IEEE T. Power Syst. **31**, 2393 (2016). P. D. Hines, I. Dobson, and P. Rezaei, Cascading power outages propagate locally in an influence graph that is not the actual grid topology, arXiv:1508.01775 (2016). B. A. Carreras, D. E. Newman, and I. Dobson, Determining the vulnerabilities of the power transmission system, [*Proc. 45th Int. Conf. Syst. Sci.*]{} HICSS’12, Maui, HI, USA, 2044 (2012). Y. Yang, T. Nishikawa, and A. E. Motter (to be published). Y. H. Wang, On the number of successes in independent trials, Stat. Sinica. [**3**]{}, 295 (1993). M. L. Mehta, [*Random Matrices*]{}, 3rd edn. (Academic Press, 2004). M. MacMahon and D. Garlaschelli, Community Detection for Correlation Matrices, Phys. Rev. X [**5**]{}, 021006 (2015). L. J. Emrich and M. R. Piedmonte, A method for generating high-dimensional multivariate binary variates, Amer. Statist. **45**, 302 (1991). J. H. Macke, P. Berens, A. S. Ecker, A. S. Tolias, and M. Bethge, Generating spike trains with specified correlation coefficients, Neural Comp. **21**, 397 (2009). F. J. Massey, Jr., The Kolmogorov-Smirnov test for goodness of fit, J. Amer. Statist. Assoc. **46**, 68 (1951). V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr, and H. E. Stanley, Random matrix approach to cross correlations in financial data, Phys. Rev. E [**65**]{}, 066126 (2002). I. Dobson, Estimating the propagation and extent of cascading line outages from utility data with a branching process, IEEE T. Power Syst. **27**, 2146 (2012).

--- abstract: 'Diversity plays a vital role in many text generating applications. In recent years, Conditional Variational Auto Encoders (CVAE) have shown promising performances for this task. However, they often encounter the so called KL-Vanishing problem. Previous works mitigated such problem by heuristic methods such as strengthening the encoder or weakening the decoder while optimizing the CVAE objective function. Nevertheless, the optimizing direction of these methods are implicit and it is hard to find an appropriate degree to which these methods should be applied. In this paper, we propose an explicit optimizing objective to complement the CVAE to directly pull away from KL-vanishing. In fact, this objective term guides the encoder towards the “best encoder" of the decoder to enhance the expressiveness. A labeling network is introduced to estimate the “best encoder". It provides a continuous label in the latent space of CVAE to help build a close connection between latent variables and targets. The whole proposed method is named Self Labeling CVAE (SLCVAE). To accelerate the research of diverse text generation, we also propose a large native *one-to-many* dataset. Extensive experiments are conducted on two tasks, which show that our method largely improves the generating diversity while achieving comparable accuracy compared with state-of-art algorithms.' author: - | Yuchi Zhang [^1], Yongliang Wang, Liping Zhang, Zhiqiang Zhang, Kun Gai\ Alibaba Group\ Beijing, China, 100102\ bibliography: - 'Bibliography-File.bib' title: | Improve Diverse Text Generation by Self Labeling\ Conditional Variational Auto Encoder --- Introduction ============ Text generating techniques are widely used in various tasks, such as dialogue generation [@serban2016building; @li2016deep; @zhao2017learning], image caption [@chen2015mind; @vinyals2015show; @xu2015show] and question-answer systems [@beamon2017justifying; @oh2017non], etc. Recently, encoder-decoder models such as SEQ2SEQ[@sutskever2014sequence] have been increasingly adopted in text generating tasks. Encoder-decoder models extract a semantic representation from the input and generate sentences coherent to the input according to this representation. They perform well in tasks which require accuracy and relativeness. However, applications such as dialogue systems further require results with diversity besides accuracy. Conventional encoder-decoder models are not good at handling such situations due to its deterministic nature. In open domain conversation systems, given the dialogue history, there may exist various kinds of responses which are grammatically correct and semantically meaningful. The dialogue bots should be able to model these multiple responses for the same input in training and give diverse answers like humans in predicting. Another application takes place in e-commerce recommendation systems. For a given item, multiple selling points and descriptions are needed for personalized recommendations. Fig. \[fig:one-to-many-example\] shows an example. With plenty of recommendation texts, different sentences can be selected to display to different users to meet their preferences or to the same user at various situations so that he/she does not feel monotonous. We summarize the above applications as the “one source, multiple targets” problems. As conventional encoder-decoder models encode same input patterns to same unique representative vectors without any variation, their ability of generating different sentences from one input are limited. Researchers have made efforts to improve the encoder-decoder models for more diverse generations. In the early periods, methods are proposed to interfere the inference stage of a well-trained encoder-decoder model to encourage abundant outputs [@li2016diversity; @vijayakumar2016diverse]. The drawback of such methods is that they do not optimize the encoder-decoder models to fit multi-target data and the quality of their generating results is limited by the trade-off between accuracy and diversity. Recently, variational encoder-decoders have shown great potentials in solving the “one source, multiple targets" problems [@bowman2016generating; @zhao2017learning; @shen2018improving]. These methods introduced an intermediate latent variable and assume that each configuration of the latent variable corresponds to a feasible response. Thus diverse responses can be generated by sampling the variable. However, both VAE and CVAE have encountered the KL-vanishing problem that the decoder tends to model the targets without making use of the latent variables. In this paper, we point out that during optimizing the objective of CVAE, the encoder is gradually pulled to a prior distribution and losing discriminative ability of the targets, while the decoder tends to fit the data even without the help of encoders. Thus KL-vanishing is rooted in the objective of CVAE. Current approaches, either weakening the decoder or strengthening the encoder to make compensation to the objective implicitly in advance, only mitigate this problem and are hard to determine how weak/strong should decoder/encoder be. Orthogonal to these efforts, we propose an explicit optimization objective for the encoder to move towards better expressiveness to fit current decoder. With this novel objective, the latent variable distribution from the encoder has the potential to be appropriately flexible in correspondence with decoder, which naturally coordinates the representative abilities of the encoder and decoder and enhances the utilization of latent variable by decoders. Specifically, an additional module called “labeling network” is used to estimate the “best encoder” for the current decoder. Then a loss which measures the difference between the latent variable of CVAE and predicted variable from labeling network is added to the original objective function of the CVAE. Since this loss pulls the encoder towards the “best encoder” approximated by the labeling network and in the meanwhile original CVAE pulls encoder to the prior, an equilibrium will be reached where KL-vanishing can be avoided. Additionally, the labeling network introduces a continuous label for each target which essentially reflects the structural constraints of the latent space. Therefore, it guarantees each $z$ in the latent space corresponds to a unique target, thus improves the coverage of the generations in target space. We alternatively train the “labeling network" and the CVAE structure, and call this model Self Labeling Conditional Variational Auto Encoder (SLCVAE). ![Example of recommendation texts of a dress. Some of them emphasize on the linen material while some others emphasize on the color etc.[]{data-label="fig:one-to-many-example"}](case){width="0.95\linewidth"} In summary our main contributions are: Firstly, we point out that the current CVAE objective function tends to encounter the *KL-Vanishing problem* due to the lack of explicit constraints on the connection between the latent variable and targets. Secondly, we propose the self labeling mechanism which connects the decoder with latent variable by a novel explicit optimization objective. With this objective, the encoder is pulled towards the “best encoder" defined by current decoder and the prior distribution simultaneously, which leads to equilibrium at which the encoder distribution is close to the prior and also remains the expressiveness. Thus the KL-vanishing problem can be significantly relieved. Further, extensive experiments demonstrates that our method called SLCVAE owns better ability to model multiple targets and improves the diversity of text generation without losing accuracy. Thirdly, a large scale dataset called EGOODS which contains native one-to-many text data of high quality is constructed to accelerate the research of diverse text generation. {width="0.95\linewidth"} \[fig:slcvae\] Related Work ============ In this section, we review the development of both the encoder-decoder models and VAE/CVAE based models for text generation. Encoder-decoder model --------------------- Encoder-decoder models are commonly adopted in NLP as they are able to fit complex data by end-to-end training. The presentation of SEQ2SEQ[@sutskever2014sequence] structure revolutionarily augmented the quality of Machine Translation (MT). And researchers soon introduce such structure into text generating systems[@serban2016building; @vinyals2015neural]. However, the purpose of the SEQ2SEQ models is to best fit the target sequence given the source sequence. Therefore two problems might probably happen when a SEQ2SEQ model is used for generating texts. One is that SEQ2SEQ often ends up with dull and generic responses. Such situation often takes place at conversation systems because safe and meaningless responses such as “I don’t know" or “I’m okay" have frequently appearance and then captured by the decoder. The other is the lack of ability of fitting multiple probable outputs,for the representative vector the decoder used to generate output is fixed and only depends on the inputs. These problems not only reduce the precision, but also limit the diversity of text generation. To tackle the above problems. Different ways of solutions have been proposed. [@li2016diversity] pointed out that the mutual information of sources and targets should be augmented during the decoding procedure. They proposed the MMI-antiLM algorithm which adds a language model penalty to unconditional high frequent responses. Their algorithm successfully solved the generic and dull response problem, is not applicable to handle the multi-target, since it just consider one target at a time. Beam Search (BS) methods for n-best outputs during the decoding procedure are commonly used in MT and could be introduced to text generation. However, as the greedy strategy in BS makes it tend to generate outputs with same prefixes, sentences generated respective to one source still look similar. [@vijayakumar2016diverse] modifies the strategy used in BS to be subject to a diverse behavior goal by reinforcement learning. However such method might reduce the coherence of the outputs. The above methods are based on improved strategies during the inference stage of a encoder-decoder model. But the encoder-decoder model itself is not substantially made better. Their strategies are actually a trade-off between diversity and coherence and thus are restricted. Additional information could still be used such as topics, speakers’ characteristics in a dialogue session or linguistic prior knowledges. These methods, however, are not applicable for common as extra inputs are required for their unique applications. VAE and CVAE ------------ Variational Auto Encoder [@rezende2014stochastic; @kingma2013auto] is a popular generative model. It makes use of a latent variable $z$ sampled from a prior distribution to generate data $x$. The logarithm likelihood of the data $x$ is optimized by maximizing the evidence lower bound (ELBO): $${\log{p(x)}}\geq{\mathbb{E}_{q(z|x)}[\log{p(x|z)}]-KL(q(z|x)||p(z))}\label{vae_elbo}$$ while both $q(z|x)$ and $p(z|x)$ are parameterized as encoder $q_{\phi}(z|x)$ and decoder $p_{\theta}(z|x)$. It is obvious that VAE encodes the input $x$ into a probability distribution rather than a fixed vector so that different $z$ could be chosen from the distribution to obtain different outputs $x$. The VAE model could be modified to be conditioned on a certain attribute $c$ such as dialogue contexts to generate outputs given a source pattern. And such modification leads the original VAE to conditional VAE called CVAE [@yan2016attribute2image; @sohn2015learning]. Needless to say, the output of CVAE now depends both on $z$ and $c$ and the ELBO becomes: $$\begin{gathered} {\log{p(x|c)}}\nonumber \\ \geq{\mathbb{E}_{q(z|x,c)}[\log{p(x|z,c)}]-KL(q(z|x,c)||p(z|c))}\label{cvae_elbo}\end{gathered}$$ VAE and CVAE seem to show great potential to generate diverse outputs, as the latent variable $z$ from a distribution could be modulated to help model different patterns. Some image generating tasks adopt VAE or CVAE and achieve good generative quality. In spite of this, difficulties are encountered while researchers attempted to generate texts via such structures [@bowman2016generating]. Directly optimized with Equation \[vae\_elbo\] or Equation \[cvae\_elbo\] will lead to the *KL-Vanishing problem* which is also called as the *posterior collapse problem*. For the VAE model, the encoder $q_{\phi}(z|x)$ perfectly fits the prior distribution $p(z)$ while ignores the inputs and the decoder generates outputs without referring to $z$. Same problems take place all conditioned on $c$ while CVAE is used. [@bowman2016generating] presented the *KL annealing* method (KLA) and *word-dropout* operation (WD) in VAE training to mitigate the *KL-Vanishing problem* in their sentence generating system. And [@zhao2017learning] introduced an additional *bag-of-word loss* which takes the latent variable as input and predicts the words which will appear in the target so that the connection of the latent variable and the outputs are augmented. They applied their *bag-of-word* loss in a CVAE structure for one-to-many text generation tasks and get excellent results in open domain dialogue generation of discourse-level diversity. The Proposed Method =================== Analysis of the KL-Vanishing Problem ------------------------------------ Considering VAE’s objective function Equation. \[vae\_elbo\], two facts are observed: (1) The second term $KL(q(z|x)||p(z))$ reaches its global minimum of 0 when $q(z|x)=p(z)$. (2) According to Jensen’s Inequality, $E_{p(z)}[\log{p(x|z)}]\leq\log\sum_{z}[p(x|z)p(z)]=\log{p(x)}$, and the equal sign of the inequation holds when and only when $x$ is independent of $z$. As a consequence, when $x$ and $z$ are independent, the ELBO objective degenerates to original $\log p(x)$ objective under which the decoder learns a plain language model. Thus the encoder $q(z|x) = p(z)$ and decoder $p(x|z) = p(x)$ which fits the dataset as a plain language model constitute a trivial solution of the objective of VAE. At this time, the KL divergence term in ELBO becomes 0, and we call this phenomenon *KL-Vanishing*. Although the *KL-Vanishing* point is a solution of Equation. \[vae\_elbo\], it is not actually we want. When *KL-Vanishing* takes place, Equation. \[vae\_elbo\] degenerates to $\log{p(x)}$. When the decoder is modeled by an RNN structure for text generation, it easily converges to a average language model of target sentences without regarding to $z$ under the degenerated objective. As a result, $z$ losses its ability of affecting $x$ and the decoder fits the average behavior of all $x$. \[fig:latentspace\] Previous works tried to avoid the *KL-Vanishing problem* by either breaking fact (1) or fact (2). Some of them put efforts into preventing $q(z|x)$ from collapsing to $p(z)$ by slowing down the optimizing procedure of the KL term. Others attempt to force the decoder to depends on the $z$ by weakening the decoder or making encoder more complicated. However, such methods fall into a dilemma: On the one hand, the objective function is optimized to reach its maximal. On the other hand, the optimal point should be avoided to prevent the occurrence of *KL-Vanishing*. These methods tried to find a good balance between the two contradictory facts without an explicit objective. Nevertheless, it is hard to find an appropriate trade-off point. Self Labeling CVAE ------------------ Our goal is to generate diverse $x$ using different $z$. From the perspective of the decoder, two conditions should be satisfied: First, each $z$ should correspond to a unique $x$ through the decoder. Second, $z$ should obey the prior distribution $p(z)$. Maximizing the ELBO objective encourages the latter by pulling encoder’s output distribution of $z$ to $p(z)$. However, with the $p(z|x)$ moving towards $p(z)$ during the optimizing procedure, $z$ losses the discriminative information and the decoder tends to fit the data even without the help of encoders. As a consequence, the first condition is violated and multiple $z$s will collapse to a same averaged output distribution $p(x)$, as is shown in Fig. \[fig:latentspace\](b). Thus we propose to strengthen the connection between the latent $z$ and target $x$ via maintaining the expressiveness of the encoder. As illustrated in Fig. \[fig:latentspace\](a), considering that an expressive $z$ has the ability to recover a unique target through the decoder, the decoder itself can then be used to find the most expressive $z'$ given a certain target $x$. This in concept equivalents to find the inverse image of $x$ of the decoder. So the inverse image $z'$ of $x$ can be regarded as the effectiveness label $x$ in the continuous latent space. And if $z'$ has been obtained, then we are reasonably motivated to pull the encoder distribution $p(z|x)$ to close $z'$ to maintain the expressiveness of the encoder. However, finding the inverse image of the decoder exactly is not an easy task. To overcome this, we introduce an extra network to estimate the inverse image of $x$ of the decoder, i.e. the effectiveness label. Thus we call this network the *labeling network* and denote the output of it as $z_{label}$. The *labeling network* can also be considered as an approximation to the ideal encoder for current decoder. Specifically, the *labeling network* shares the same network structure with the original encoder of VAE, but it only outputs the variable $z_{label}$ rather than the reparameterized distribution. As the output of the VAE encoder is a distribution $q(z|x)$, we put the expressive constraint on the expectation of the $L_2$ distance ${||z-z_{label}||}^2$ between encoded latent variable and $z_{label}$ over the encoder distribution $q(z|x)$. Thus an expressiveness objective function is defined as follows: $$\mathcal{L}_\mathsf{exp} = \mathbb{E}_{q(z|x)}[{||z-z_{label}||}^2]\label{eq:vae_pre}$$ which is minimized to encourage the encoder to be more expressive. By using $g(x)$ to denote the labeling network i.e. $z_{label}=g(x)$, and adding $\mathcal{L}_\mathsf{exp}$ as an additional term to the VAE’s objective function, we get the total objective function in following: $$\begin{gathered} \mathcal{L}_\mathsf{SLVAE} = -\mathbb{E}_{q(z|x)}[\log{p(x|z)}]+KL(q(z|x)||p(z))\nonumber \\ +\lambda\mathbb{E}_{q(z|x)}[{||z-g(x)||}^2]\label{eq:slvae}\end{gathered}$$ From this formulation, we can see that, $q(z|x)$ is not only pulling to $p(z)$ like before, but also pulling to the estimated “best encoder” for the decoder. The hyper-parameter $\lambda$ is used to control the importance of the expressiveness objective. The “best encoder” can expand a comprehensive coverage of the target space through the current decoder. Thus they will reach an equilibrium at which the $p(z|x)$ is close to $p(z)$ and also remains the expressiveness. As we use a labeling network to estimate the most expressive latent label given the decoder and strengthen the connection between the latent $z$ and target $x$ through the decoder itself, we call this method Self Labeling VAE (SLVAE). When it comes to CVAE, things remain the same except that everything is conditioned on $c$. And the final total objective function is: $$\begin{gathered} \mathcal{L}_\mathsf{SLCVAE} = -\mathbb{E}_{q(z|x,c)}[\log{p(x|z,c)}]+KL(q(z|x,c)||p(z|c))\nonumber \\ +\lambda\mathbb{E}_{q(z|x,c)}[{||z-g(x,c)||}^2]\label{eq:slcvae}\end{gathered}$$ Similarly, we call this model SLCVAE. At last, learning the *labeling network* $g(x, c)$ is somewhat straightforward. As we discussed before, $g(x,c)$ should be the “best encoder" for the decoder to recover $x$. Thus we should optimize $g(x,c)$ by maximizing the following objective function: $$\begin{gathered} \log{p(x|z_{label},c)}=\log{p(x|g(x,c),c)}\label{eq:labeling}\end{gathered}$$ with the decoder fixed. An alternative training schedule of the VAE/CVAE network and the labeling network is applied. Fig. \[fig:slcvae\] shows the overview of the whole proposed method. The CVAE part of our structure is a conventional CVAE with reparameterization trick[@kingma2013auto] by introducing a posterior and a prior network as described in [@zhao2017learning] in detail. The structure of the labeling network is similar to the encoder of the CVAE which consists of a target encoder and a context encoder which embed target and source sentences to representative vectors. The only difference is that the labeling encodes its inputs into a fixed $z_{label}$ rather than a distribution. Training Process ---------------- To optimize Equation. \[eq:slcvae\] and Equation. \[eq:labeling\], we parameterize all the three modules: the encoder $q_{\phi}(z|x, c)$ and decoder $p_{\theta}(x|z, c)$ of the CVAE, and the labeling network $g_{\gamma}(x, c)$. An alternative training schedule is used with two phases: the CVAE phase and the Labeling phase. In the CVAE phase, we minimize the loss function of the SLCVAE: $$\begin{gathered} \min_{\phi,\theta,\beta}\mathcal{L}_\mathsf{SLCVAE}=\min_{\phi,\theta,\beta}[-\mathbb{E}_{q_{\phi}(z|x,c)}[p_{\theta}(x|z,c)]\nonumber \\ +KL(q_{\phi}(z|x,c)||p_{\beta}(z|c)) \\ +\lambda\mathbb{E}_{q_{\phi}(z|x,c)}[{||z-g_{\gamma}(x,c)||^2}]]\nonumber\label{eq:lslcvae}\end{gathered}$$ where $\beta$ are parameters of the prior network. In this phase, the labelling network $g_{\gamma}(x,c)$ is fixed to provide a $z_{label}$ corresponding to each $x$. In the Labeling phase, we minimize the loss function of the labeling network: $$\begin{gathered} \min_{\gamma}\mathcal{L}_\mathsf{label}=\min_{\gamma}[-p_{\theta}(x|g_{\gamma}(x,c), c)]\label{eq:llabel}\end{gathered}$$ The decoder is fixed at this time to get the good expressive label for current decoder. See Algorithm \[alg:slcvae\] for the whole training procedure. The Adam[@kingma2015adam] optimizer is adopted to update parameters. Initialize ${\phi}, {\theta}, {\beta}, {\gamma}$ randomly $c, x \gets$ sample a mini-batch from dataset Sample latent $z \sim q_{\phi}(z|x, c)$ Calculate label $z_{label} \gets g_{\gamma}(x,c)$ Calculate the gradients: $\nabla_{\phi,\theta,\beta}\mathcal{L}_\mathsf{SLCVAE}$ Update CVAE parameters $\theta, \phi, \beta$ by Adam $c, x \gets$ sample a mini-batch from dataset Calculate the gradients: $\nabla_{\gamma}\mathcal{L}_\mathsf{label}$ Update labeling network parameter $\gamma$ by Adam The EGOODS Dataset ================== The text generating problem defined with “one source, multiple targets” is an active research topic and plays important roles in many tasks. However, there still lacks real one-to-many datasets to improve and evaluate the algorithms for this problem. Most current datasets come from dialogue system are essentially one-to-one corpora. Although there may exist various underlying responses for a certain question, these datasets only contain one answer for each dialogue context due to data source limitations. To fulfill the gap between the demand and status quo for one-to-many dataset, we collect a large scale item description corpus from a Chinese e-commerce website to construct the native one-to-many dataset. In this corpus, each item has one description provided by their sellers and multiple recommendation sentences written by third-party who is payed to make these sentences more attractive to customers. The descriptions provided by sellers are usually texts stacking many keywords of the item properties. In the contrary, the recommendation sentences are written according to item descriptions but read more smoothly. For the text generation task, we naturally use the seller’s descriptions as the source to generate multiple recommendation sentences mimicking humans. Since this corpus originates from a real business, texts are of high quality and coherent with sources. Thus it gives rise to a very large and native one-to-many dataset, which is called EGOODS. After simple cleaning and formatting, EGOODS dataset contains 3001140 source and target pairs from 789582 items in total. So each source item description has 3.8 target recommendation sentences on average. The dataset is split into training/validation/testing parts with respect to items, each of which contains 2961317/19536/20287 pairs. Experiments =========== Experimental Setups ------------------- ### Datasets Our experiments are conducted on two text generating tasks: open-domain dialogue generation and recommendation sentence generation. We evaluate the performance of our algorithm and compare it with several strong baselines on the two tasks. For the first task, the public dialogue dataset Daily Dialog (DD) [@li2017dailydialog] is used. DD dataset is collected from different websites under 10 topics. It contains 13118 multi-turn dialogue sessions in English, and is split into training/validation/testing set of 11118/1000/1000 sessions. The average number of turns per session is 8.85 and the average number of tokens per utterance is 13.85. To avoid too long dialogue contexts, we first split long sessions into multiple short full speaker turns containing no more than 6 utterances with an utterance level sliding window. As a result, the final DD dataset contains 39567/3681/3471 full speaker turns in training/validation/testing set. For each full speaker turn, we use all utterances but the last one as the dialogue context to predict the last one. Need to note that though there may exist various responses for a question, DD dataset essentially only contains one-to-one data. To better model and evaluate the diversity, the newly constructed one-to-many dataset EGOODS is adopted in the second task. ### Baselines We compared our SLCVAE to 4 strong baselines: SEQ2SEQ [@sutskever2014sequence], MMI-AntiLM [@li2016diversity], CVAE and CVAE with *bag-of-word loss* (CVAE+BOW) [@zhao2017learning]. Several training skills, such as *KL-annealing*(KLA) and *word dropout*(WD) [@bowman2016generating], are used in combination with baselines and our method to improve the performance. All methods are required to generate 10 responses for each given input. Note that although the SEQ2SEQ model uses deterministic encoding vectors, the widely adopted beam search strategy can be applied during inference procedure to generate 10-best decoding results which corresponds to 10 responses (denoted as SEQ2SEQ-BS). MMI-AntiLM method follows this idea and put MMI prior onto the beam search strategy. We tried different beam size from 10 to 100 and find that beam size set to 10 gives the best result for all dataset. It is worth noting that beam search is applied only to above two methods and will bring unfair advantages due to the exploration of a much larger search space. All other methods including ours use the greedy strategy during decoding to be consistent with previous work. We also tried another simple strategy that adds a random noise drawn from a gaussian distribution to the encoded vector to bring variability to SEQ2SEQ. We denote this method as SEQ2SEQ + noise, and use it as extra baselines with various standard deviations. ### Training The whole structure of SLCVAE is implemented with the famous open source library PyTorch[@paszke2017automatic]. In all experiments, English letters are all transformed to the lower case first. Encoders are bidirectional RNNs [@schuster1997bidirectional] with Gated Recurrent Units (GRU) [@chung2014empirical] and the decoders are unidirectional RNN with GRUs throughout all experiments. All RNNs have two layers. Since EGOODS dataset is much larger than DD, the network capacity increases accordingly. Specifically, for DD and EGOODS dataset respectively, the word embedding sizes are set to 32 and 128, and the hidden dimensions of RNN are also set as the same. In all VAE-based methods, the latent variable dimensions are set to 8 and 16 for two datasets separately. The Adam optimizer with a learning rate of 0.0001 is used to train all models with batch sizes of 64 and 128 for two datasets. Training skills of KLA and WD are also used to get further better performance. We also conduct an annealing strategy that the weight of our labeling error is increased over time synchronously with KLA, as the soft label provided by the labeling network is not that good in the early stages. We tune several hyper-parameters on the validation sets and measure the performances on the test sets for all baselines and our proposed method. Results ------- ### Evaluation Metrics In “one source, multiple targets” setting, for an input $c$, given $N$ hypothesis responses $h_i$ generated by a model and $M_c$ reference responses $r_j$, accuracy and diversity are two sides of the generations we need to concern. Automatic quantitative measures for these purposes are still an open research challenge [@liunot; @tong2018one]. [@zhao2017learning] proposed BLEU-precision and BLEU-recall metrics for discourse-level accuracy and diversity respectively as following: $$\begin{split} \mathsf{precision(c)} &= \frac{\sum_{i=1}^{N}\max_{j \in [1,M_c]}d(r_j, h_i)}{N} \\ \mathsf{recall(c)} &= \frac{\sum_{j=1}^{M_c}\max_{i \in [1,N]}d(r_j, h_i)}{M_c} \end{split}$$ BLEU-1, BLEU-2 and BLEU-3 are adopted and their average is calculated as the metrics. However, BLEU-recall is defined based on lexical similarity, which might penalize a reasonable but not same prediction. Following [@li2016diversity], we also use the number of *distinct* *n-gram* to measure the word-level diversity. The *distinct* is normalized to \[0, 1\] by dividing the total number of generated tokens. In summary, BLEU-precision is reported as the accuracy measure, and BLEU-recall, *distinct-1* and *distinct-2* are reported as diversity measures. {width="0.95\linewidth"} \[fig:show\_case\] We also conduct human evaluations on the EGOODS dataset. 7 human experts are employed to measure the fluency of generated sentences, coherence of each sentence to source and diversity. For fluency and coherence, experts are asked to vote to each sentence. Sentences which yield more than 4 votes are good sentences. The ratio of good sentences are reported. For diversity, 5 level of diverse scores are introduced. The higher the score, the more diverse the sentence is. The final diversity score of each sentence is the average score of all experts. ### Automatic Quantitative Measurement on Daily Dialog Table. \[tab:dd\_result\] shows the evaluation results of all methods on Daily Dialog dataset. Results for SEQ2SEQ-noise with different standard deviations are also listed. Training skills of KLA and WD are used for all CVAE based methods. We can see that our proposed method outperforms all baselines in terms of all the 4 metrics on this task. And it is worth noting that our method obtains much higher diversity measures no matter in discourse-level or word-level than all others. In the meanwhile, the accuracy metrics BLEU-precision of our method remains slightly better than the best baseline. This confirms our insight of the generating process that our labeling objective can lead to an equilibrium at which the KL-vanishing problem is significantly relieved and so result in better diversity. Remind that Daily Dialog is actually a *one-to-one* dataset. The better performance in diversity on DD demonstrates that our model can better exploit such training data without explicit *one-to-many* annotations. Further, the results show that CVAE based models beat conventional encoder-decoder methods in almost all metrics. This is consistent with those in previous work like [@zhao2017learning], and confirms the advantage of latent variable methods for generation tasks over encoder-decoder models with multi-decoding strategy. In addition, we find that with growing noise, the accuracy of SEQ2SEQ+noise decreases while the diversity increases significantly. Especially when the noise is small (e.g. 0.2), the diversity has obvious gains with only a slightly sacrifice on accuracy. Methods BLEU-prec BLEU-recall *distinct-1* *distinct-2* -------------------- ----------- ------------- -------------- -------------- SEQ2SEQ+BS 0.164 0.282 0.002 0.007 SEQ2SEQ+noise(0.2) 0.163 0.288 0.003 0.014 SEQ2SEQ+noise(0.5) 0.157 0.312 0.005 0.032 SEQ2SEQ+noise(0.8) 0.153 0.320 0.007 0.065 MMI-AntiLM 0.153 0.275 0.002 0.012 KLA+WD 0.212 0.345 0.010 0.041 KLA+WD+BOW 0.210 0.344 0.013 0.066 KLA+WD+SL **0.214** **0.354** **0.014** **0.078** : Results on Daily Dialog (DD). The bottom 3 lines are CVAE based methods. []{data-label="tab:dd_result"} ### Automatic Quantitative Measurement on EGOODS To better study current methods on the “one source, multiple targets” problem, experiments are conducted on our newly collected native *one-to-many* dataset EGOODS. Performances of different methods are shown in Table. \[tab:egoods\_result\]. First of all, our method achieves comparable accuracy with baselines and best diversity among all methods with an only exception of SEQ2SEQ+noise(0.8) on *distinct-2*. This demonstrates the effectiveness of SLCVAE on the *one-to-many* data. Although SEQ2SEQ+noise(0.8) gets the best *distinct-2*, its precision is sacrificed significantly due to the nosie, which means the results tend to be meaningless. In detail, our method harvests the much better gains on word-level diversity while is only slightly better than CVAE on BLEU-recall. We explain this in two folds: First, strong baselines can benefit from the large scale and *one-to-many* nature of EGOODS to better fit the multiple targets. Another reason is that automatically evaluating the quality of generated texts is very challenging. BLEU-recall only measures the coverage of hypothesis for the annotated targets, and could not judge good algorithms precisely when the annotations are limited. In such situation, *distinct* measures the vocabulary a model actually uses and demonstrates its absolute lexical diversity. Example results will show this in next subsection. Furthermore, we observed that SEQ2SEQ+BS obtains the best BLEU-precision among all methods on EGOODS, but it performs much worse on Daily Dialog. Meanwhile, the BLEU-recall gap between SEQ2SEQ+BS and the best result on EGOODS is obviously small than that on DD. We point out that our dataset especially designed for “one source, multiple targets” problem significantly improves the generation quality of SEQ2SEQ methods. Methods BLEU-prec BLEU-recall *distinct-1* *distinct-2* -------------------- ----------- ------------- -------------- -------------- SEQ2SEQ+BS **0.379** 0.388 0.0012 0.0042 SEQ2SEQ+noise(0.2) **0.379** 0.386 0.0021 0.0146 SEQ2SEQ+noise(0.5) 0.367 0.402 0.0029 0.0162 SEQ2SEQ+noise(0.8) 0.347 0.395 0.003 **0.0420** MMI-AntiLM 0.356 0.374 0.0021 0.0146 KLA+WD 0.373 **0.405** 0.0039 0.0216 KLA+WD+BOW 0.374 0.404 0.0039 0.0231 KLA+WD+SL 0.373 **0.405** **0.0049** 0.0270 : Results on EGOODS. The bottom 3 lines are CVAE based methods.[]{data-label="tab:egoods_result"} ### Human Evaluation on EGOODS Human evaluation results on EGOODS are shown in Table \[tab:human\_eval\]. Such results show that our method achieves comparable fluency and coherence as baseline methods, but our diversity is much higher than other models. Methods Fluency(%) Coherence(%) Diversity ------------ ------------ -------------- ----------- SEQ2SEQ+BS **96** 65 1.55 KLA+WD 87 64 3.12 KLA+WD+BOW 83 **66** 3.18 KLA+WD+SL 91 **66** **3.32** : Human evaluation results.[]{data-label="tab:human_eval"} Although the SEQ2SEQ+BS method achieves the best fluency, it sacrifices too much diversity, which means the result is monotonous and dull. ### Text Generating Examples To give an intuitive impression about generations, Fig. \[fig:show\_case\] shows an example of generated texts for EGOODS, and more results can be found in the supplementary material. 10 results are generated separately by SEQ2SEQ+BS, CVAE and our method SLCVAE. The results from all three methods are of good fluency and coherent to the input. But obviously SEQ2SEQ+BS fails to show different expressions thus gets poor diversity. Both CVAE model and our method tend to show stronger abilities in generating diversely than SEQ2SEQ+BS, since we can see that the generated results have better coverages for the references. Nevertheless, notice that the SLCVAE has a larger vocabulary and uses richer expressions that CAVE+BOW which is not reflected by BLEU-recall metrics. This finding is consistent with the quantitative experiment results we have discussed above. Conclusion ---------- “One source, multiple targets” is a common text generation task. Recently CVAE based methods shows great potentials for this task. However CVAE working with RNNs tends to run into the KL-vanishing problem that the RNN ends up with a trivial language model independent of the latent variable. In this paper, we analyze the objective of CVAE and give an intuitive explanation of the cause of KL-vanishing. Then we propose the self labeling mechanism which connects the decoder with latent variable by an explicit optimization objective. It leads the encoder to reach an equilibrium at which the decoder can take full advantage of the latent variable. Experiments show that SLCAVE largely improves the generating diversity. [^1]: Correspondence to Yuchi Zhang(yuchi.zyc@alibaba-inc.com)

--- abstract: 'We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than $10^3$. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.' author: - | Céline [Guervilly]{}$^{a,b}$,Philippe [Cardin]{}$^{a}$, Nathanaël [Schaeffer]{}$^{a}$\ [$^{a}$ ISTerre, Université de Grenoble 1/CNRS, F-38041, Grenoble, France]{}\ [$^{b}$ Department of Applied Mathematics and Statistics, Baskin School of Engineering,]{}\ [University of California, Santa Cruz, CA 95064, USA]{} bibliography: - 'ref.bib' title: 'A dynamo driven by zonal jets at the upper surface: Applications to giant planets' --- Introduction ============ The zonal ([*i.e.*]{} axisymmetric and azimuthally directed) jet streams visible at the surface of the giant planets are a persistent feature of the fluid dynamics of these planets (figure \[fig:prof\_surf\]). The gas giants (Jupiter and Saturn) display a strong eastward equatorial jet, extending to latitudes $\pm 20^\circ$ with a peak velocity exceeding $100$ m/s on Jupiter [@Por03], and to latitudes $\pm 30^\circ$ with a peak velocity exceeding $400$ m/s on Saturn [@San00]. At higher latitudes, alternating prograde (eastward) and retrograde (westward) jets of smaller amplitude are observed extending all the way to the poles. These profiles are fairly symmetric with respect to the equator. On the ice giants (Uranus and Neptune) the picture is rather different. A very intense retrograde equatorial current is present with maximum velocity of $100$ m/s on Uranus [@Sro05] and $400$ m/s on Neptune [@Sro01]. At higher latitudes, a single prograde jet of large amplitude is present in each hemisphere. Several decades of observations show that these zonal flows remain approximately steady [@Por03]. ![Zonal velocity measured at the surface in the planet’s mean rotating frame for each of the four giants by tracking cloud features in the outer weather layer. Profiles adapted from @Por03, @San00, @Sro01 and @Sro05.[]{data-label="fig:prof_surf"}](fig1.eps){width="80.00000%"} The origin of these zonal flows and the associated question of the depth to which they extend into the planets’ interiors have been areas of active research in rotating fluid dynamics for several decades ([*e.g.*]{} [@Jon09], and references therein; see also the review by [@Vas05]). In particular, several models have been proposed to explain the zonal wind pattern of Jupiter, and can be categorized into two main classes: weather layer models and deep convective layer models. The former assume that the zonal flows are produced in a shallow stably stratified region near cloud level. These models are able to reproduce the high latitude structures with alternating eastward and westward jets and a strong equatorial current [[*e.g.*]{} @Wil78; @Cho96]. These models tend to produce a retrograde equatorial jet [@Yan03], so they provide a plausible explanation for the retrograde equatorial flow of the ice giants but not for the prograde flow observed on gas giants. A parametrized forcing such as a strong equatorially-localized baroclinicity is required to force a shallow system to produce a prograde equatorial jet [@Wil03]. The second class of models is deep convection models which simulate most or all of the whole $10^4$km-thick molecular hydrogen layer [@Bus76; @Chr01b; @Chr02; @Man96]. The presence of deep convection is inferred from the observation that the atmospheres of the major planets emit more energy by long-wave radiation than they absorb from the Sun. Consequently their atmospheres must receive additional heat supplied by the interior of the planet. Recent numerical models using either a Boussinesq approximation [@Hei05] or an anelastic approximation [@Jon09] and low Ekman numbers (i.e. strong rotational effect compared with viscous dissipation) display alternating zonal jets at high latitudes. A strong eastward equatorial jet is a robust feature of these models where the Coriolis force dominates buoyancy, in good agreement with the gas giant observations. Interestingly, deep convection models suggest that the zonal velocity generated by non-linear interactions of convective motions ([*i.e.*]{} the motions directly forced by buoyancy) is roughly geostrophic, that is, invariant along the direction of the rotation axis. This feature is also present in strongly compressible models provided that the Ekman number is small enough, despite the increase of density with depth yielding ageostrophic convective motions [@Jon09; @Kas09]. When the convection is more vigorous such that the buoyancy force overcomes the Coriolis force, 3D turbulence homogenizes angular momentum; a retrograde jet forms in the equatorial region and a single strong prograde jet forms in the polar region, in good agreement with the ice giant observations [@Aur07]. ![Spectra of the magnetic field squared amplitude at the planetary radius for degrees $l$ and order $m$ up to $3$ obtained from inversion models of the magnetic measurements. The squared amplitude for a given degree $l$ is using a Schmidt normalisation for the spherical harmonics. The squared amplitude for a given mode $m$ is . $g_l^m$ and $h_l^m$ are the Gauss coefficients in gauss. After @Yu10 (Model Galileo 15), @Bur09 (Cassini measurements), @Con91 (model O$_8$) and @Her09 (AH$_5$ model from magnetic observations and auroral data).[]{data-label="fig:mag_spec"}](fig2.eps){width="80.00000%"} Another feature of the giant planets is their strong magnetic fields (figure \[fig:mag\_spec\]). The observed magnetic fields for gas and ice giants differ drastically [see for instance the recent review by @Rus10]. Jupiter and Saturn have a main axial dipole component (corresponding to $l=1$, $m=0$ in figure \[fig:mag\_spec\]), a feature shared with the Earth for instance [@Yu10; @Bur09]. Neptune and Uranus, on the other hand, have strong non-axial multipolar components (corresponding to $l=2,3$ in figure \[fig:mag\_spec\]) compared with the axial dipole component [@Con91; @Her09]. The magnetic field is generated in the deep, electrically conducting regions of the planets’ interiors: a metallic hydrogen layer for Jupiter and Saturn [@Nel99; @Gui05 and references therein] and an electrolyte layer composed of water, methane and ammonia [@Hub91; @Nel97] or superionic water [@Red11] for Uranus and Neptune. Numerical models of convective dynamos in rapidly rotating spherical shells typically produce axial dipolar dominated magnetic fields for moderate Rayleigh numbers and moderate Ekman numbers [[*e.g.*]{} @Ols99; @Aub04; @Chr07]. To explain the unusual large scale non-dipolar magnetic fields of Uranus and Neptune, models using peculiar parameter regimes or different convective region geometries have been proposed. The latter models show that a numerical dynamo operating in a thin shell surrounding a stably-stratified fluid interior produces magnetic field morphologies similar to those of Uranus and Neptune [@Hub95; @Holme96; @Sta06]. @Gom07 obtain weakly dipolar and strongly tilted dynamo magnetic fields when high magnetic diffusivities are used (or equivalently small electrical conductivity). Their results show that these peculiar fields are stable in the presence of strong zonal circulation and when the flow has a dominant effect over the magnetic fields. This feature is also emphasized by @Aub04 who find stable equatorial dipole solutions with a weak magnetic field strength and low Elsasser number (measure of the relative importance of the Lorentz and Coriolis forces) for moderately low Ekman numbers. They argue that the magnetic field geometry of the equatorial dipole solution is incompatible with the columnar convective motions and thus this morphology is stable only when Lorentz forces are weak. Although scaling laws derived from numerical simulations of dynamos driven by basal heating convection predict dipolar magnetic field in planetary parameter regimes [@Ols06], recent numerical simulations using more realistic parameter values (lower Ekman numbers) have not produced large scale magnetic fields so far, and require larger magnetic Reynolds numbers (measure of magnetic induction versus magnetic diffusion) [@Kag08]. Moreover, convection in the interior of Jupiter is often thought to be driven by secular cooling [@Ste03]. Numerical dynamos driven by secular cooling typically produce weak dipole or multipolar magnetic field for larger forcing [@Kut00; @Ols06] depending on boundary conditions [@Hor10]. Therefore the question of the generation of large scale magnetic field by turbulent convective motions in the planetary parameter regime remains open. The dichotomies observed in the magnetic fields and in the zonal wind profiles of the giant planets are rather striking. Up to now no study has tried to relate them directly, probably because the former is a feature of the deep interior whereas the latter is a characteristic of the surface. However, if some mechanism is able to transport angular momentum from the surface down to the deep, fully conducting region then the zonal motions may influence the generation of the magnetic field. In the non-magnetic deep convection models [@Hei05; @Jon09], zonal motions extend geostrophically throughout the electrically insulating molecular hydrogen layer down to the bottom of the model. On the other hand, due to the possible rapid increase of electrical conductivity with depth in the outer region, @Liu08 argued that the ohmic dissipation produced by geostrophic zonal motions shearing dipolar magnetic field lines would exceed the luminosity measured at the surface of Jupiter if the vertical extent of this geostrophic zonal motions exceeds 4% of the planet radius. However, the argument of @Liu08 is purely kinematic, that is the action of the magnetic forces on the flow and the feedback on the magnetic field are ignored. In a self-consistent magnetohydrodynamic model, the zonal flow would adjust toward a non-geostrophic state due to the action of magnetic forces if the electrical conductivity of the fluid is significant (@Gla08, see also the non-linear numerical simulations of convectively-driven dynamos of @Aub05). In this case, angular momentum may be transported along the magnetic field lines leading to a dynamical state close to the Ferraro state. This state minimizes the ohmic dissipation produced by the shearing of the poloidal magnetic field by the zonal flow as the poloidal magnetic field lines are aligned with angular velocity contours. Both scenarios, either geostrophic zonal balance or Ferraro state, imply the existence of multiple zonal jets of significant amplitude at the top of the fully conducting region beneath. The plausibility of each scenario depends on the radial profile of electrical conductivity, which is currently not well constrained within the giant planets [@Nel99]. The idea of the work presented in this paper is that these zonal jets may exert, by viscous or electromagnetic coupling, an external forcing at the top of the deeper conducting envelope. From previous studies [@Sch06; @Gue10] we know that the viscous coupling between a differentially rotating boundary and a low-viscosity electrically conducting fluid can generate a self-sustained magnetic field in different geometries. Zonal motions can be subject to barotropic shear instabilities which have a lengthscale independent of the viscosity, unlike convective instabilities. These instabilities are able to generate large scale magnetic fields, and so they are an interesting source of dynamo action under planetary interior conditions. In order to test the plausibility of a dynamo driven by this source in isolation, we use an incompressible 3D numerical dynamo model with a zonal velocity profile imposed at the top of a spherical shell containing a conducting fluid. We use a dynamical approach, that is non-linear interactions between the flow and the magnetic field are taken into account; therefore the fluid flow is free to adopt a three-dimensional structure as long as it satisfies the imposed viscous boundary conditions. The dynamics of the deep conducting region is usually assumed to be slower than the dynamics of the outer molecular hydrogen region due to magnetic braking, even if uncertainties remain in the electrical conductivity. The model presented in this paper assumes an idealized one-way coupling between the outer and deep regions. A more realistic model would need to account for the back reaction of the deep layer onto the outer layer; a study of the consistent dynamical interaction of the two layers is beyond the scope of this paper. For studies of more realistic coupling, see promising recent numerical models of self-consistent convectively-driven dynamos in spherical shells including radially variable electrical conductivity of @Hei11 and @Sta10. In these models, slow convective motions in the interior dynamo region coexist with strong zonal flow near the outer surface. Differential rotation in the interior is only partially inhibited by the strong magnetic field. In order to assess the role of the zonal wind profile on the topology of the sustained magnetic field, we use both Jupiter-like and Neptune-like zonal wind profiles. In the giant planets, as in rocky planets, it is usually assumed that the dynamo mechanism is driven by convective motions. The giant planets display a strong surface heat flux (with the exception of Uranus) meaning that heat transfer is efficient in the interior of the planet and thus mostly due to convection [@Gui07 and references therein]. Here we want to assess the efficiency of zonal velocity forcing alone, so we do not model convective motions. The first goal of this work is to quantify what amplitude of the zonal wind *inside* the conducting layer is needed to trigger the dynamo instability, so we do not model the exact or realistic coupling between the molecular hydrogen upper layer and the deep, electrically conducting region. Our second goal is to test to what extent the pattern of the zonal flow imposed at the top of the conducting layer influences the topology of the self-sustained magnetic field. We first describe the model and the numerical method used (section \[sec:model\_jupiter\]). Then we present numerical results from simulations in the non-magnetic case (section \[sec:hydro\_jupiter\]) followed by results from dynamo simulations (section \[sec:dynamo\_jupiter\]). The application of our results to planetary conditions is discussed in section \[sec:Summary\]. Model {#sec:model_jupiter} ===== We model the deep conducting layer of the giant planets as a thick spherical shell. At the top of the conducting layer we impose an axisymmetric azimuthal velocity to represent the zonal flow generated in the overlying envelope. The shell rotates around the $z$-axis at the imposed rotation rate $\Omega$. The aspect ratio is $\gamma=r_i/r_o$ where $r_i$ is the inner sphere radius, corresponding to a rocky core, and $r_o$ the outer sphere radius, corresponding to the top of the fully conducting region. The fluid is assumed incompressible with constant density $\rho$ and constant temperature, that is, no convective motions are computed. The assumption of incompressibility is made for simplicity, although the pressure scale height at the depths of the conducting layer is roughly $8000$km [@Gui04], that is, about $1/5$ of the thickness of the layer. The effects of compressibility may well play a role in the dynamics of the conducting regions [see for instance @Evo04]. For simplicity we model the angular momentum coupling with the external zonal flow as a rigid boundary condition for the velocity at the outer boundary, rather than as a shear stress condition. The flow is driven through a boundary forcing rather than a volume forcing to avoid directly imposing bidimensionality to the velocity field. As we are interested in the bulk magnetohydrodynamical process, the exact nature of the coupling (electromagnetic or viscous, shear stress or rigid) with the upper molecular hydrogen layer is not crucial for our study. We discuss the implication of the choice of the rigid boundary condition in section \[sec:hydro\_jupiter\]. The radial profile of electrical conductivity is not well constrained in the gas giants. In particular the existence of a first order or continuous transition between the molecular and metallic hydrogen phase is still an open question, although high-pressure experiments are in favor of a continuous transition [@Nel99]. We choose to model the outer boundary as electrically insulating to simplify the coupling between the layers. The conductivity is assumed constant throughout the whole modeled conducting layer. As we do not model the molecular hydrogen layer, we assume zonal geostrophic balance within this envelope for simplicity. The amplitude of the zonal motions at the outer boundary of our model is therefore the same as the surface winds. This idealized representation of the dynamics of the molecular hydrogen layer would be altered if the magnetic forces upset the zonal geostrophic balance. Depending on the magnitude and radial profile of the electrical conductivity, the amplitude of the zonal motions might be reduced, and the zonal flow contours would tend to align with the magnetic field lines, although we do not expect the characteristics of the zonal jets (narrow or wide, relative amplitude of the peaks) to be altered very much. We use two different synthetic azimuthal velocity profiles for the boundary forcing imposed at the top: a multiple jet profile for the gas giants with a profile based on Jupiter’s surface zonal winds (hereafter profile J) and a 3-band profile based on Neptune’s surface zonal winds (profile N). For Jupiter, we use the profile given in @Wic02b $$\begin{aligned} {\mathbf{U}}=U(s){\mathbf{e_{\phi}}}= U_0 \frac{s}{r_0 \cos(n_0\pi)} \cos{\left(}n_0 \pi \frac{s-r_0}{r_s-r_0} {\right)}{\mathbf{e_{\phi}}}, \label{eq:profileJ}\end{aligned}$$ where $s=r\sin\theta$, $r_s$ is the surface radius of the planet and . $n_0$ controls the numbers of jets. The profile at the radius $r_s$ best matches the observed profile at the surface for $n_0=4$ (figure \[fig:prof\_model\]). The profile $U(r_o,\theta)$ is used to drive the flow at the top of our simulated metallic hydrogen layer (figure \[fig:prof\_model\]). The ratio $\gamma_s=r_s/r_o$ determines the $U$ profile at $r_o$. We choose $\gamma_s=r_s/r_o=1/0.8=1.25$ following @Gui94. ![Zonal velocity profile imposed at the surface of model J (left) and model N (right) (solid lines). Both profiles are obtained by assuming that the zonal velocities are geostrophic for $r_s>r>r_o$ and using the profile represented by a dashed line at the surface of the planet ($r=r_s$): model J, profile (\[eq:profileJ\]) with $n_0=4$, $\gamma_s=r_s/r_o=1.25$ and $U_0=100$; model N: polynomial fit of order $10$ in latitude of the zonal wind profile measured at the surface of Neptune (figure \[fig:prof\_surf\]) with $\gamma_s=1/0.85=1.18$. For comparison the zonal wind profile measured at the surface of Jupiter is plotted in gray.[]{data-label="fig:prof_model"}](fig3.eps){width="80.00000%"} For the Neptune-like profile, we use the zonal velocity profile measured at the surface of Neptune, approximated by a polynomial of order $10$ in latitude. We project this surface velocity profile geostrophically down to $r_o$ using $\gamma_s=1/0.85=1.18$ [@Hub91] (figure \[fig:prof\_model\]). The existence of a rocky core at the centre of the giant planets is uncertain and depends on the poorly constrained composition of the planet. Estimates for the core mass are $0-14 m_{\oplus}$ for Jupiter (total mass $318 m_{\oplus}$), $6-17 m_{\oplus}$ for Saturn (total mass $95 m_{\oplus}$) and $0-4 m_{\oplus}$ for Uranus and Neptune (total mass $15 m_{\oplus}$ and $17 m_{\oplus}$ respectively) where $m_{\oplus}$ denotes the mass of the Earth [@Gui05]. If present, the rocky cores are therefore believed to be small. Following the interior model of Jupiter proposed by @Gui94 we use an aspect ratio $r_i/r_o=0.2$ for all the simulations performed. The inner core is assumed to be electrically conducting, with the same conductivity as the fluid in the conducting layer. We did not carry out simulations with an insulating core as the effect of the conductivity of the inner core on the dynamo mechanism is believed to be small [@Wic02]. The velocity boundary condition is no-slip at the inner boundary. The velocity ${\mathbf{u}}$ is scaled by $U_0$, the absolute value of the azimuthal velocity imposed at the equator of the outer sphere. The lengthscale is the radius of the outer sphere $r_o$. The magnetic field ${\mathbf{B}}$ is scaled by $\sqrt{\rho \mu_0 r_o \Omega U_0}$ where $\rho$ is the fluid density and $\mu_0$ is the vacuum magnetic permeability.\ We numerically solve the momentum equation for an incompressible fluid, $$\begin{aligned} Re {\frac{\partial {\mathbf{u}}}{\partial t}}+ Re {\left(}{\mathbf{u}}\cdot\boldsymbol{{\nabla}} {\right)}{\mathbf{u}}+ \frac{2}{E} \mathbf{e_z}\times {\mathbf{u}}= -\boldsymbol{{\nabla}}p +\boldsymbol{{\nabla}^2}{\mathbf{u}}+ \frac{1}{E} {\left(}\boldsymbol{{\nabla}} \times {\mathbf{B}}{\right)}\times {\mathbf{B}}, \label{eq:NS_Jupit}\end{aligned}$$ the continuity equation, $$\begin{aligned} \boldsymbol{{\nabla}} \cdot {\mathbf{u}}= 0 , \label{eq:divu_Jupit}\end{aligned}$$ and the magnetic induction equation, $$\begin{aligned} {\frac{\partial {\mathbf{B}}}{\partial t}} = \boldsymbol{{\nabla}} \times {\left(}{\mathbf{u}}\times {\mathbf{B}}{\right)}+ \frac{1}{Re Pm} \boldsymbol{{\nabla}^2}{\mathbf{B}}, \label{eq:ind_Jupit}\end{aligned}$$ $$\begin{aligned} \boldsymbol{{\nabla}} \cdot {\mathbf{B}}=0 ,\end{aligned}$$ where $p$ is the dimensionless pressure, which includes the centrifugal potential. The Reynolds number $Re=r_o U_0 / \nu$ parametrizes the mechanical forcing exerted on the system by controlling the amplitude of the zonal velocity. The magnetic Prandtl number $Pm=\nu / \eta$ measures the ratio of viscous to magnetic diffusivities. The magnetic Reynolds number $Rm$ is defined as $Rm=Re Pm$. The Ekman number $E=\nu / (\Omega r_o^2)$ measures the importance of the viscous term over the Coriolis force. The Rossby number $Ro=Re E=U_0/(\Omega r_o)$ is the ratio of inertial force to Coriolis force. Note that in our definition the Rossby number refers to the amplitude of the prescribed zonal jets at the surface, and not to the local flow velocity. The results presented in this paper were obtained with the PARODY code, a fully three-dimensional and non-linear code. The code was derived from @Dor97 by J. Aubert, P. Cardin, E. Dormy in the dynamo benchmark [@Chr01], and parallelised and optimised by J. Aubert and E. Dormy. The velocity and magnetic fields are decomposed into poloidal and toroidal scalars and expanded in spherical harmonic functions in the angular coordinates with $l$ representing the latitudinal degree and $m$ the azimuthal order. A finite difference scheme is used on an irregular radial grid (finer near the boundaries to resolve the boundary layers). A Crank-Nicolson scheme is implemented for the time integration of the diffusion terms and an Adams-Bashforth procedure is used for the other terms. Dynamics without the magnetic field {#sec:hydro_jupiter} =================================== For a rapidly rotating system in which the Coriolis force exactly balances the pressure force, the Proudman-Taylor constraint states that the flow is $z$-invariant and follows geostrophic contours. For an incompressible fluid in a bounded container, these geostrophic contours correspond to surfaces of equal height. In a sphere the only geostrophic motions are azimuthal and axisymmetric. In the giant planets’ conducting envelopes, the Ekman number is about $10^{-16}$ and the Rossby number is much smaller than $1$ [@Gui04]. In the absence of a magnetic field, we expect the Proudman-Taylor constraint to hold for large scale motions. As we want to reach the dynamical regime in which the flow is strongly geostrophic, the use of small Ekman and Rossby numbers is required. We carried out simulations for $10^{-5}>E>10^{-6}$ for model J and $10^{-5}>E>5\times10^{-6}$ for model N. The Rossby numbers are always smaller than $0.1$. For the profile J, in cases of low Ekman numbers ($E\le 2\times 10^{-6}$), we imposed longitudinal symmetry by calculating only the harmonics of a chosen order $m_s$. The required resolution for $E=10^{-6}$ is $500$ points on the radial grid and $l=580$ spherical harmonics degrees. Axisymmetric flow ----------------- When the imposed boundary forcing is small enough, [*i.e.*]{} when the Rossby number $Ro$ is less than a critical value $Ro_c$, the flow is axisymmetric and predominantly azimuthal (figure \[fig:Uaxi\_jupiter\]). The zonal jets imposed at the outer boundary extend into the volume along lines parallel to the axis of rotation. The use of no-slip boundary conditions yields a differential rotation between the boundary and the bulk of the fluid. This differential rotation is accommodated across viscous Ekman boundary layers, which scale as $(E/\cos \theta)^{1/2}$, where $\theta$ is the colatitude. By Ekman pumping, viscous forces within the Ekman layers drive axial motions of order $E^{1/2}$ within the bulk of the fluid (figure \[fig:Uaxi\_jupiter\]). These meridional circulations advect angular momentum from the boundary layer into the bulk of the fluid and cause the jets to propagate faster than by pure viscous diffusion. At low latitudes, the Ekman layer is thicker so the Ekman pumping is stronger, yielding to a more efficient driving of the zonal motions in the bulk by the outer boundary layer. For model J (figure \[fig:profilJ\_axi\_eq\]), the zonal velocity in the bulk relative to that imposed at the outer boundary is noticeably weaker for the inner jets than for the outer jets. When $E$ decreases this effect is less marked, and in the $E\to 0$ limit we expect the basic zonal velocity to be perfectly geostrophic in the whole volume. The comparison between the zonal velocity just below the Ekman layer and in the equatorial plane (figure \[fig:Profil\_axi\_eq\]) shows that the zonal velocity is geostrophic in the bulk of the fluid (outside of the boundary layers). For model N (figure \[fig:profilN\_axi\_eq\]), the zonal jets are wider, so the zonal flow already displays a strong geostrophic structure at $E=10^{-5}$. Note that the azimuthal velocity has to match the no-slip boundary condition at the inner core, and so an internal Stewartson layer forms on the axial cylinder tangent to the inner core [@Ste66]. Non-axisymmetric motions ------------------------ ### Model J #### Rossby wave at the onset When the boundary forcing (measured by $Ro$) becomes greater than a critical value $Ro_c$, the axisymmetric basic flow becomes unstable to a non-axisymmetric shear instability. The saturated instability takes the form of an azimuthal necklace of cyclonic and anticyclonic vortices aligned with the axis of rotation, is nearly $z$-independent and drifts eastward (figure \[fig:W\_Rec\_J\]). Close to the threshold, the radial extension of the pattern is large and occupies almost half of the gap. The pattern drifts with the same speed over its whole radial extension, even though the advection by the zonal flow velocity varies with $s$, implying that it is a single wave. @Wic02b studied the linear stability of the imposed zonal flow (\[eq:profileJ\]) in a spherical shell modeling the insulating molecular hydrogen layer of Jupiter (aspect ratio 0.8). For $E=10^{-4}$ they found nearly bidimensional instabilities that they described as drifting columns aligned with the rotation axis and similar to convective solutions. Although they do not identify these instabilities as waves, their characteristics are very similar to the ones obtained with our non-linear model. ![Non-zonal axial vorticity in the equatorial plane (right) and in a meridional slice (left) for model J at $E=4\times 10^{-6}$ and $Ro=1.01 Ro_c$ (blue: negative and red: positive). The black curve represents the zonal velocity in the equatorial plane.[]{data-label="fig:W_Rec_J"}](fig6_color.eps){width="40.00000%"} The nearly $z$-invariant structure and the prograde drift are two characteristics of Rossby waves propagating in a spherical container. The dispersion relation for the Rossby wave given by a local linear analysis is [[*e.g.*]{} @Fin08] $$\begin{aligned} \omega_{rw}(s)= -2 \Omega \beta \frac{m/s}{k_s^2+(m/s)^2} , \label{eq:w_RW}\end{aligned}$$ where $\beta=h^{-1}(dh/ds)=-s/(r_o^2-s^2)$ is related to the slope of the upper boundary of the spherical container of height $h$. $k_s$ and $m/s$ are the radial and azimuthal wavenumbers respectively. The theoretical Rossby wave frequency $\omega_{rw}$ can be calculated at a given radius assuming $k_s\approx m/s$ and using the wavenumber $m$ obtained from the numerical simulation. For different $E$, the frequency $\omega$ of the propagating wave observed in our numerical simulations always falls in the range $\omega_{rw}(s_1)<\omega<\omega_{rw}(s_2)$ where $s_1=0.56$ ($s_2=0.87$) is the smallest (resp. largest) radius where a significant vorticity associated with the presence of the wave can be seen in the numerical calculations. This strongly indicates that the shear instability occurs as a Rossby wave. The velocity of the zonal flow $U$ enters the dispersion relation of the Rossby wave through a Doppler shift $$\begin{aligned} \omega(s)=\omega_{rw}(s)+U(s)\frac{m}{s} .\end{aligned}$$ As reported earlier, $\omega(s)$ is constant in our numerical calculations so $\omega_{rw}(s)$ must adapt in the $s$-direction for the wave to be coherent. In a prograde jet $U>0$, $\omega_{rw}$ must decrease, which requires a local increase in $k_s$ in equation (\[eq:w\_RW\]) and so a local decrease in the radial lengthscale, which can be observed in figure \[fig:W\_Rec\_J\]. For small enough Ekman number (in practice $E < 5 \times 10^{-6}$), the critical wavenumber $m_c$ of the Rossby mode is independent of $E$. The radial lengthscale is determined by the width of the jet and the vortices are roughly circular in the equatorial plane (figure \[fig:W\_Rec\_J\]) suggesting that $m_c$ is controlled by the width of the jets. In a local approximation that neglects the curvature terms, a criterion of instability of barotropic shear flows has been derived by @Ing82 for an anelastic model in a full rotating sphere and by @Kuo49 for thin stably stratified “weather” layers. Using an inviscid Boussinesq model and for barotropic instability of a zonal flow $U$ in a sphere, this necessary condition implies a change of sign of a quantity $\Delta$ at some radius: $$\begin{aligned} \Delta = 2 \beta - Ro \frac{d \zeta}{ds}, \label{eq:crit_SC}\end{aligned}$$ where $\zeta$ is the vorticity of the zonal flow, $$\begin{aligned} \zeta = \frac{dU}{ds}+\frac{U}{s}.\end{aligned}$$ Note that the curvature terms have been taken into account here. In a sphere, $\beta$ is negative. Consequently, the zonal velocity profile is more prone to instability where the gradient of zonal vorticity is maximum and negative. Then for a profile $U$ of sinusoidal form, the first shear instability occurs at the maximum of the prograde jets, and thus, perhaps surprisingly, at a null value of the zonal velocity shear $dU/ds$. Note that our numerical simulations show instabilities with a large radial extent and with maximum amplitude located in a retrograde zonal jet (see figure \[fig:W\_Rec\_J\]), even though the local instability criterion predicts an onset in a prograde jet. This observation emphasizes that the local criterion does not predict the location of global saturated modes. The theoretical critical Rossby number obtained from applying the criterion (\[eq:crit\_SC\]) to the profile (\[eq:profileJ\]) imposed at the top of model is $Ro_c^{th}=0.0011$. The threshold of the first instability of the axisymmetric flow, denoted $Ro_c^{nlin}$, obtained with the numerical simulations are shown in figure \[fig:Roc\_E\_J\]. Despite the decrease of $Ro_c^{nlin}$ with the Ekman number, $Ro_c^{nlin}$ is still about four times larger than $Ro_c^{th}$ for $E=10^{-6}$ because the amplitude of the zonal flow within the bulk is reduced by viscous boundary layers in the numerical simulations. Due to computational limitations, it is not possible for us to carry out simulations at smaller $E$ with a fully non-linear code and prove the existence of an asymptotic regime for the inviscid instability threshold. For this purpose we used a dedicated linear code described in \[app:codeXSHELL\]. The linear code calculates linear perturbation solutions to the momentum equation using the geostrophic profile $U$ as the basic flow in the bulk of the fluid. The computational time is greatly reduced by the linear approach but is restricted to an analysis of the instability threshold. The growing solutions obtained with the linear code exhibit very similar features to the Rossby waves in the non-linear simulations (frequency, bidimensional structure, radial extent, location of the maximum amplitude in a retrograde jet). In figure \[fig:Roc\_E\_J\] the threshold $Ro_c^{lin}$ obtained with the linear code approaches asymptotically the value given by the local theory. For the same Ekman number, $Ro_c^{lin}$ is smaller than $Ro_c^{nlin}$ since the geostrophic zonal flow $U$ is used in the linear code, that is the jets in the bulk have greater amplitude than in the non-linear code. From our linear computations we conclude that the theoretical criterion (\[eq:crit\_SC\]) is relevant to explain the onset of instability obtained numerically. More details about the onset of the hydrodynamic instability can be found in @Gue10_thesis. ![ Critical Rossby number obtained from fully non-linear numerical simulations for model J ($Ro_c^{nlin}$, circles) compared to the theoretical Rossby number obtained with the local instability criterion (\[eq:crit\_SC\]) using the geostrophic profile (\[eq:profileJ\]) $U(s,\theta=\pi/2)$ ($Ro_c^{th}$, black line). The critical Rossby number obtained from the linear numerical calculation is also shown ($Ro_c^{lin}$, crosses).[]{data-label="fig:Roc_E_J"}](fig7.eps){width="\textwidth"} The characteristic time of the Rossby wave is $\tau_{rw}=1/\omega$. At the instability threshold, the numerical simulations give $\tau_{rw}\approx 18 \Omega^{-1}$ for $E< 5\times 10^{-6}$. The timescale of the zonal jets is $\tau_{zj}=r_o/U_0=\Omega^{-1}/Ro$. For $Ro=0.01$, we have $\tau_{zj}>\tau_{rw}$: the Rossby wave propagation is faster than the advection of the fluid by the zonal flow. The turnover time of a fluid particle trapped in a Rossby wave is $\tau_{to}=l/V_s$ where $l$ is the typical radial displacement of the particle and $V_s$ the typical cylindrical radial velocity of the particle. At $Ro=1.01Ro_c$, $V_s$ is typically $10^{-2} U_0$. In a rough approximation we use $l=\delta$, where $\delta$ is the width of the jets, $\delta \approx 0.1r_o$ for the profile J. Then we obtain $\tau_{to} \approx 0.1 r_o/(10^{-2} U_0) \approx 10 Ro^{-1} \Omega^{-1} \approx 10^{3}\Omega^{-1}$: the turnover time of the particle is much longer than the timescale of the wave. Consequently the particle oscillates rapidly as the wave propagates and is slowly advected by the zonal flow. In practice the radial displacement $l$ is typically smaller than $\delta$ and so the turnover time is slightly overestimated here. #### Supercritical regime When the Rossby number is increased in the supercritical regime, other prograde jets will eventually become unstable. A second Rossby wave appears in the weakly supercritical regime, at $Ro=1.06 Ro_c$ for $E=5\times 10^{-6}$, with a maximum velocity located in the retrograde zonal jet at larger radius than the first wave maxima ([*i.e.*]{} the wave appearing for $Ro=Ro_c$) (figure \[fig:U\_E5e6R3600\]). To fill the larger circumference at larger radius the instability has a slightly larger wave number, $m=22$ instead of $21$, while the radial width of the jet is comparable. The second wave propagates faster, in agreement with the Rossby wave dispersion relation (\[eq:w\_RW\]). Barotropic instabilities tend to broaden and weaken narrow jets by redistributing potential vorticity [see for instance @Ped87]. The smoothing of the jets saturates the amplitude of the Rossby waves. For this slightly supercritical regime the zonal flow profile is only weakly modified. Upon further increasing the forcing ($Ro=2.94 Ro_c$), several Rossby waves of different wavenumbers superpose and interact (figure \[fig:U\_E5e6R10000\]). The structure of the waves and the jets is still mainly bidimensional except in the viscous boundary layers. The typical cylindrical radial velocity is $V_s \approx 0.1 U_0$ and the Rossby number is $0.05$ so the turnover time is about $20\Omega^{-1}$ assuming that the radial displacement $l=\delta$, about the same order of magnitude as the timescale of the zonal jets. In figure \[fig:U\_s\_satur\] the time-averaged zonal flow in the equatorial plane is plotted for different $Ro$ up to $Ro=5.88 Ro_c$. As the forcing is increased, the Rossby waves gradually reduce the jet strength and broaden the jet width. For $Ro=2.94 Ro_c$, the retrograde jet at $s=0.81$ has been mostly destroyed leading to the widening of the zonal jet width. We note that the zonal flow becomes mostly westward for the strongest forcings. The amplitude of the zonal flow located at $s>0.9$ is hardly affected because the threshold to destabilise the outermost jets is high due to the large slope (related to $\beta$ in equation (\[eq:crit\_SC\])). For $Ro<2.35 Ro_c$, the amplitude of the non-axisymmetric velocity, relative to $U_0$, increases with the forcing (figure \[fig:U\_Roc\_satur\]). After reaching a maximum, at $Ro=2.35 Ro_c$, the amplitude of the non-axisymmetric flow decreases relative to $U_0$. The “efficiency” of the forcing to drive the non-zonal velocity is reduced as the Rossby waves smooth the gradient of vorticity and so affect their excitation mechanism. The back reaction on the forcing velocity in the upper molecular hydrogen layer is not taken into account in our model although it might significantly affect the zonal profile in the upper layer in the case of strong forcing. ### Model N The shear instability takes the form of an $m=2$ oscillation in the azimuthal direction (figure \[fig:W\_Rec\_N\]). It is a single wave propagating eastward with the same frequency over the shell, and is nearly $z$-invariant. The maxima of the non-zonal vorticity are located on each side of the prograde jet. The characteristics of this wave are similar to the Rossby wave obtained with model J. The frequency of this wave is in agreement with the frequency of a theoretical Rossby wave of wavenumber $m=2$ propagating at a radius $s=0.53$ (assuming that $k_s\approx m/s$ in the dispersion relation (\[eq:w\_RW\])). For $E=10^{-5}$ and $E=5\times 10^{-6}$, the critical Rossby numbers obtained with the non-linear numerical simulations are respectively $Ro_c^{nlin}=0.0335$ and $Ro_c^{nlin}=0.0325$. Using the instability criterion (\[eq:crit\_SC\]) with the profile imposed at the surface we obtain a critical Rossby number of $0.026$ in good agreement with the non-linear numerical results when the Ekman number decreases. ![Non-zonal axial vorticity in the equatorial plane (right) and in a meridional slice (left) for model N at $E=5\times 10^{-6}$ and $Ro=1.01Ro_c$ (blue: negative and red: positive). The black curve represents the zonal velocity in the equatorial plane.[]{data-label="fig:W_Rec_N"}](fig10_color.eps){width=".4\textwidth"} Magnetic field generation {#sec:dynamo_jupiter} ========================= The non-axisymmetric motions are of prime importance for the dynamo mechanism because a purely toroidal flow cannot generate a self-sustained magnetic field. We note that some axisymmetric poloidal flow is present when $Ro<Ro_c$ as a weak meridional circulation is created by the Ekman pumping. However these axisymmetric motions are weak at small Ekman numbers so we do not expect to find dynamos when the zonal flow is stable, that is when $Ro<Ro_c$, in the asymptotic inviscid regime. Indeed we did not find dynamos when $Ro<Ro_c$ (up to $Pm=10$). The non-axisymmetry associated with the hydrodynamic shear instability is a crucial element for the dynamo process: the stable zonal flow cannot sustain a magnetic field by itself. This is in agreement with the results obtained by @Gue10 with dynamos generated by spherical Couette flows (differential rotation between two concentric spheres). Characteristics of the magnetic field for model J ------------------------------------------------- We have performed dynamo simulations for $Ro=1.17-1.76Ro_c$ and $E=5\times 10^{-6}$. We find that the dynamo threshold occurs at a rather high value of the magnetic Prandtl number, . The critical magnetic Reynolds number (defined via the maximum forcing velocity) required for dynamo action is $Rm_c\approx20,000$ (see section \[sec:Summary\] for an estimate of the critical magnetic Reynolds number defined via the local velocity). For a given forcing, we have performed calculations just above the critical magnetic Prandtl number, $Pm_c$, and up to $2Pm_c$. The main features of the self-sustained magnetic field can be observed in figures \[fig:B\_ProfJ\] and \[fig:specB\_ProfJ\]. The magnetic field displays a dipolar symmetry, [*i.e.*]{} antisymmetry with respect to the equatorial plane, $$\begin{aligned} (B_r,B_{\theta},B_{\phi})(r,\pi-\theta,\phi)=(-B_r,B_{\theta},-B_{\phi})(r,\theta,\phi) . \label{eq:sym_dip}\end{aligned}$$ The magnetic field is predominantly toroidal and axisymmetric (corresponding to the mode $m=0$ in figure \[fig:spec\_fluid\_J\]). The toroidal magnetic field does not emerge from the conducting region as the outer region is electrically insulating. The strongest poloidal component is the axial dipole within the conducting region and outside of the outer sphere (corresponding to the harmonic $(l,m)=(1,0)$ in figure \[fig:specB\_ProfJ\]). Within the bulk of the flow, the axisymmetric poloidal magnetic field lines are mostly significantly bent where the Rossby wave causes a strong magnetic induction (figure \[fig:Baxi\_ProfJ\]). A magnetic field at the scale of the Rossby wave is produced in this region as can be observed on the spectra of magnetic energy (figure \[fig:spec\_fluid\_J\]) with significant peaks at $m=22$ in the poloidal and toroidal magnetic energies and at $l=23$ in the poloidal magnetic energy ($l-m$ is odd to preserve the dipolar symmetry). Close to the outer boundary, the axisymmetric poloidal magnetic field lines converge and diverge locally (figure \[fig:Baxi\_ProfJ\]). This is due to the induction of axisymmetric magnetic field by the secondary meridional circulation produced by Ekman pumping. This effect is very localized and generates a magnetic field of small latitudinal scale that decreases rapidly with radius. The spectrum and map of the radial magnetic field at the surface of our modeled planet (at radius $r_s=1.25r_o$) (Figs. \[fig:Bmap\_rs\_J\] and \[fig:spec\_rs\_J\]) show that the magnetic field is strongly dominated by the axial dipole. The magnetic field generated at the scale of the Rossby wave ($m=22$) is still visible in the spectrum of the magnetic field but its amplitude is weak at this radius: about four orders of magnitude smaller than the amplitude of the axisymmetric mode (note that the spectrum in figure \[fig:spec\_rs\_J\] represents the squared amplitude of the field). In all the simulations performed, no inversion of polarity of the axial dipole has been observed. The tilt of the dipole is rather weak, at most $2^{\circ}$ from the rotation axis. We found a secular variation of the dipole axis of about $1^{\circ}$ every $1000$ rotation periods or alternatively $0.001$ global magnetic diffusion time. Just above the dynamo threshold ($Pm_c<Pm\leqslant 2Pm_c$), the magnetic field is weak: the magnetic energy contained within the fluid conducting region is only about $5$% of the kinetic energy. The magnetic field does not strongly act back on the flow, except to produce its own saturation. A comparison between the zonal flow in the non-magnetic case and in the presence of the dynamo magnetic field does not reveal significant differences. The magnetic field lines of the poloidal field are almost aligned with the rotation axis and the flow structure (see figure \[fig:Baxi\_ProfJ\]) so the flow disruption due to Lorentz forces is weak. Characteristics of the magnetic field for model N {#sec:B_profN} ------------------------------------------------- We performed simulations at $Ro=1.05-1.5 Ro_c$ and $E=10^{-5}$. We find the dynamo threshold at $Pm_c\approx1$, that is, the critical magnetic Reynolds number is $Rm_c\approx4000$. The main features of the self-sustained magnetic field can be observed in figures \[fig:B\_ProfN\] and \[fig:specB\_ProfN\]. The self-sustained magnetic field displays an equatorial symmetry, [*i.e.*]{}$$\begin{aligned} (B_r,B_{\theta},B_{\phi})(r,\pi-\theta,\phi)=(B_r,-B_{\theta},B_{\phi})(r,\theta,\phi) . \label{eq:sym_quad}\end{aligned}$$ Within the fluid conducting region, the axisymmetric toroidal field is the strongest component whereas the poloidal field is dominated by the $m=2$ mode, not the axisymmetric $m=0$ mode. The $m=2$ mode corresponds to the magnetic field generated at the scale of the Rossby wave (figure \[fig:spec\_fluid\_N\]). The axisymmetric poloidal field is multipolar, mainly composed by the $(l,m)=(2,0)$ (axial quadrupole) and $(l,m)=(4,0)$ modes. At the surface of the planet (figure \[fig:spec\_rs\_N\]), the magnetic field appears to be mainly axisymmetric (with the $l=2$ and $l=4$ harmonics degrees dominant). The amplitude of the $m=2$ structure is weak, about two orders of magnitude smaller than the $m=0$ mode but still visible at high latitudes on the map of the radial field at the surface (figure \[fig:Bmap\_rs\_N\]). Due to the equatorial symmetry of the field, the magnetic field lines in the equatorial plane are roughly perpendicular to the cylindrical structure of the flow whereas they are nearly aligned at higher latitudes (figure \[fig:Baxi\_ProfN\]). As a result magnetic braking acting on the flow is stronger in the equatorial region than at high latitude regions. In the simulation performed here ($Pm_c<Pm\leqslant 2Pm_c$), the magnetic energy is weak compared to the kinetic energy (about $5$%) so the feedback of the magnetic field on the flow remains weak. For a stronger magnetic field (at larger magnetic Reynolds numbers), we expect that the flow disruption would become important. As a result the equatorially symmetric solution may become unstable and the magnetic field may switch to an axial dipolar symmetry. This is the result obtained by @Aub04 in convectively-driven dynamos: they found equatorial dipolar magnetic fields for Rayleigh numbers close to the convection onset; these solutions become unstable as the convective forcing is increased and an axial dipolar configuration is preferred. In summary, the flows driven by the profiles J and N produce very different poloidal magnetic fields: mainly a strongly axisymmetric dipole for the profile J and a weak multipolar axisymmetric field dominated by the magnetic field induced at the scale of the Rossby waves for the profile N. In both cases the magnetic field within the conducting region is mainly an axisymmetric toroidal field. The different magnetic field morphology is quite surprising given that the flows are quite similar: strong zonal flows and propagating Rossby waves. In the next section we review the dynamo mechanism that has been proposed to operate for similar flows and suggest the key difference between profiles J and N that determines the topology of their self-sustained magnetic fields. Dynamo mechanism ---------------- Using a quasi-geostrophic flow and a kinematic approach (no Lorentz force in the momentum equation), @Sch06 [hereafter SC06] obtain numerical dynamos generated by an unstable axisymmetric shear layer (Stewartson layer): for a strong enough forcing, the Stewartson layer is unstable to non-axisymmetric shear instabilities, which appear in the form of Rossby waves (of wavenumber about $10$ for the Ekman numbers and Rossby numbers they investigated). The self-sustained magnetic field has a strong axisymmetric toroidal component and a mostly axisymmetric poloidal component. SC06 show that the time dependence of the flow is a key ingredient for the dynamo effect: time-stepping the magnetic induction equation using a steady flow taken either from a snapshot or a time-average leads to the decay of the magnetic field. They characterize the dynamo process as an $\alpha \omega$ mechanism. In mean field theory, the $\alpha$ effect parameterizes the generation of an axisymmetric poloidal magnetic field from the correlation of small scale magnetic field and velocity. The $\alpha$ effect usually requires that the flow possess some helicity, the correlation between fluid velocity and vorticity, $H={\mathbf{u}}\cdot \boldsymbol{\omega}$ [[*e.g.*]{} @Mof78]. Flows displaying a columnar structure aligned with the axis of rotation, such as Rossby waves or convection columns [@Ols99], typically possess strong mean helicity. As these columns are essentially bidimensional vortical structures, the helicity is mainly produced by the term $u_z \omega_z$. In nearly $z$-invariant flow, the axial ($z$) velocity is mostly due to two terms: the slope effect and the Ekman pumping. The slope effect comes from the combination of mass conservation and impenetrable boundaries: a (cylindrical) radial velocity $u_s$ creates an axial velocity $u_z\sim z \beta u_s$ with $\beta=h^{-1} (dh/ds)$. In the limit of rapid rotation in a spherical container, this contribution is much larger (of order $1$) than the Ekman pumping (). However, the axial velocity produced by the slope effect is phase shifted by $\pi/2$ with $\omega_z$, and so does not allow the production of mean helicity. On the contrary, axial velocity produced by Ekman pumping is in phase with the axial vorticity and a dynamo mechanism based on the Ekman pumping associated to an azimuthal necklace of axial vortices is plausible [@Bus75]. In a numerical experiment at small Ekman numbers ($E=\mathcal{O}(10^{-8})$), SC06 artificially remove the Ekman pumping and observe dynamo action with nearly the same threshold showing that the Ekman pumping is unimportant in their dynamo mechanism. The crucial importance of the time dependence of the flow and the negligible contribution of the Ekman pumping lead SC06 to consider the involvement of the Rossby waves in the dynamo process. They conjecture that the propagation of the Rossby waves yields a proper phase shift between the non-axisymmetric magnetic field and velocity field in order to produce the axisymmetric poloidal magnetic field. @Ava09 have calculated the $\alpha$ tensor, describing the generation of a large scale magnetic field by correlation of small scale velocity and magnetic field, with a flow geometry corresponding to Rossby waves. In the absence of Ekman pumping, they show that the diagonal components of the $\alpha$ tensor, which are the relevant coefficients for the $\alpha$ effect, are non-zero if and only if the flow pattern is drifting relative to the mean flow. @Til08 explains that the time dependence of a velocity field can lead to dynamo action even when any particular snapshot of the velocity field cannot because the linear operator associated with the induction equation is non-normal. In particular, he shows that the simple time dependence of a propagating wave is enough for dynamo action. Several numerical studies report the importance of the time dependence of the velocity field, mainly of oscillating nature [@Reu09; @Gub08]. The idea that the propagation of Rossby waves may maintain a dynamo action is very appealing as their presence is ubiquitous in rotating fluid dynamics. A system in which no wave propagation occurs, and which is unable to produce $u_z$ by another mechanism, such as buoyancy, will rely on Ekman pumping to create axial velocity with the proper phase shift. However, in the limit of small Ekman number, the Ekman pumping vanishes and the dynamo threshold should become infinitely high. The dynamo mechanism relying on the propagation of Rossby waves is robust in the limit of small Ekman number as the presence of these waves does not rely on the action of viscosity. Due to the close resemblance of the flow (zonal motions and propagating Rossby wave) in our 3D numerical model and the kinematic quasi-geostrophic model of SC06, we now try to establish if the dynamo mechanism evoked in SC06 is at work in our 3D model. To formalize their idea, let us first consider a simple theoretical model. The velocity field is composed by a zonal flow, $\overline{U}(s) {\mathbf{e_{\phi}}}$, and the small scale velocity of a Rossby wave ${\mathbf{u}}^m$ with $$\begin{aligned} {\mathbf{u}}^m(s,\phi,z,t)=(u_s^m (s,z){\mathbf{e_s}}+u_{\phi}^m (s,z){\mathbf{e_{\phi}}} + u_z^m (s,z){\mathbf{e_z}}) e^{i (m\phi-\omega t)}\end{aligned}$$ where $u_s^m$, $u_{\phi}^m$ and $u_z^m$ are complex and $\omega$ is the frequency of the wave. The magnetic field is composed of an axisymmetric magnetic field $\overline{{\mathbf{B}}}$, and a magnetic field perturbation induced at the scale of the Rossby wave ${\mathbf{b}}^m$ with $$\begin{aligned} {\mathbf{b}}^m(s,\phi,z,t)=(b_s^m (s,z){\mathbf{e_s}}+b_{\phi}^m (s,z){\mathbf{e_{\phi}}} + b_z^m(s,z) {\mathbf{e_z}}) e^{i (m\phi-\omega t) +\lambda t}\end{aligned}$$ where $b_s^m$, $b_{\phi}^m$ and $b_z^m$ are complex and $\lambda$ is the growth rate of the magnetic field. The equations for the evolution of the poloidal components of $\overline{{\mathbf{B}}}$ in cylindrical coordinates $\overline{B_s}$ and $\overline{B_z}$ are $$\begin{aligned} \frac{\partial \overline{B_s}}{\partial t} &=& - \frac{\partial}{\partial z}\left( \overline{u_z^m b_s^m - u_s^m b_z^m} \right) + \eta \left( {\nabla}^2 \overline{B_s} - \frac{\overline{B_s}}{s^2} \right), \label{eq:B_s} \\ \frac{\partial \overline{B_z}}{\partial t} &=& \frac{1}{s}\frac{\partial}{\partial s}s \left( \overline{u_z^m b_s^m - u_s^m b_z^m} \right) + \eta {\nabla}^2 \overline{B_z}, \label{eq:B_z}\end{aligned}$$ where the overbar denotes an azimuthal average. It is immediately apparent that if $u_s^m$ ($u_z^m$) is out of phase by $\pi/2$ with $b_z^m$ (resp. $b_s^m$), then $\overline{B_{s}}$ and $\overline{B_{z}}$ will be decaying in time. If we suppose that $\overline{B_{\phi}}\gg \overline{B_s}, \overline{B_z}$ the equations for $b_s^m$ and $b_z^m$ are $$\begin{aligned} (\lambda - i c \frac{m}{s}) b_s^m &=& \frac{i m}{s} u_s^m \overline{B_{\phi}} + \eta \left({\nabla}^2 b_s^m - \frac{2}{s^2}\frac{\partial b_{\phi}^m}{\partial \phi} - \frac{b_s^m}{s^2} \right), \label{eq:b_s_m} \\ (\lambda - i c \frac{m}{s}) b_z^m &=& \frac{i m}{s} u_z^m \overline{B_{\phi}} + \eta {\nabla}^2 b_z^m. \label{eq:b_z_m}\end{aligned}$$ where $c=(\omega/(m/s) -\overline{U})$ is the phase speed of the wave relative to the mean flow $\overline{U}$. In the case of marginal stability ($\lambda=0$), if we neglect the magnetic diffusivity $\eta$ then we obtain that $b_s^m$ ($b_z^m$) is in phase with $u_s^m$ ($u_z^m$ resp.). Moreover if the axial velocity is mainly due to the slope effect then $u_z^m=z\beta u_s^m$ and so according to the equations (\[eq:b\_s\_m\])-(\[eq:b\_z\_m\]) $b_z^m \approx z \beta b_s^m$. This implies that the first term of the right hand side of equations (\[eq:B\_s\])-(\[eq:B\_z\]) is almost zero and thus $\overline{B_s}$ and $\overline{B_z}$ are decaying. Consequently magnetic diffusivity at the scale of $b_s^m$ and $b_z^m$ must play a role in the generation of the axisymmetric poloidal magnetic field by introducing a short phase lag between the velocity and magnetic modes. This phase lag depends on the spatial structures of $b_s^m$ and $b_z^m$, and hence the terms $\overline{u_s^m b_z^m}$ and $\overline{u_z^m b_s^m}$ do not cancel out. Note that the importance of magnetic diffusivity is well established in the $\alpha$ effect [@Rob07]. On the other hand, if the wave is not propagating, $c=0$, then $$\begin{aligned} - \frac{i m}{s} u_s^m \overline{B_{\phi}} &=& \eta \left({\nabla}^2 b_s^m - \frac{2}{s^2}\frac{\partial b_{\phi}^m}{\partial \phi} - \frac{b_s^m}{s^2} \right), \label{eq:b_s_m_c0} \\ - \frac{i m}{s} u_z^m \overline{B_{\phi}} &=& \eta {\nabla}^2 b_z^m. \label{eq:b_z_m_c0}\end{aligned}$$ In this case the magnetic field perturbations $b_s^m$ and $b_z^m$ are out of phase with $u_s^m$ and $u_z^m$ (as $u_s^m$ and $u_z^m$ are correlated by the slope effect) and so the averaged products $\overline{u_z^m b_s^m}$ and $\overline{u_s^m b_z^m}$ are zero. We can conclude that in order for this simple model to work as a mean-field dynamo (i) the wave must propagate and (ii) the magnetic diffusivity must act on the magnetic field generated at the scale of the waves. As the Rossby wave propagates, the location of the induction of the magnetic field perturbation is forced to drift with the same rate, but with a phase-shift. The phase-shift between the magnetic field perturbation and the Rossby wave depends on both the phase speed $c$ and the magnetic diffusivity $\eta$. The argument above implies that this phase-shift is essential for the dynamo mechanism. Using any particular snapshot of the velocity field for time stepping the magnetic induction in our numerical simulations with models J or N leads to the decay of the magnetic field. The failure of dynamo in the kinematic numerical experiment with both models is readily explained by our simple theoretical model. In figure \[fig:mecanisme\_alpha\], we plot the non-axisymmetric components of the velocity, $u_z^m$ and $u_s^m$ and magnetic field, $b_s^m$ and $b_z^m$ obtained in the numerical simulations for model J and model N in a plane located just above the equatorial plane ($b_s^m$ and $b_z^m$ are zero in the equatorial plane by dipolar symmetry in model J). The correlation of $u_z^m$ with $u_s^m$ confirms that $u_z^m$ is mainly produced by the slope effect for both models. For model J (figure \[fig:bsus\_ProfJ\]), we observe that $u_z^m$ and $b_s^m$ are in phase so $\overline{u_z^m b_s^m}$ has a significant amplitude. However, $b_z^m$ is out of phase with $u_s^m$, which means that $\overline{u_z^m b_s^m} \gg \overline{u_s^m b_z^m}$. This may be an effect of the magnetic diffusivity as $b_s^{m}$ and $b_z^{m}$ have different spatial structures, or due to radial derivatives of $\overline{B_s}$ and $\overline{B_z}$ that we neglect in equation (\[eq:b\_z\_m\]). Consequently $\overline{u_z^m b_s^m}$ mainly contributes to the generation of strong $\overline{B_s}$ and $\overline{B_z}$. For model N (figure \[fig:bsus\_ProfN\]) strong positive (negative) crescent-shaped patches of $b_s^m$ and $b_z^m$ are visible in the cyclonic (resp. anticyclonic) vortices, out of phase by $\pi/2$ with $u_s^m$ and $u_z^m$. Consequently these crescent-shaped structures of $b_s^m$ and $b_z^m$ do not contribute to the terms $\overline{u_z^m b_s^m}$ and $\overline{u_s^m b_z^m}$. The presence of these maxima of $b_s^m$ and $b_z^m$ are not explained by the theoretical model (equations (\[eq:b\_s\_m\]) and (\[eq:b\_z\_m\])) likely because of the neglect of the axial and radial derivatives of $\overline{B_s}$ and $\overline{B_z}$, which are important in this region (see figure \[fig:Baxi\_ProfN\]). Round-shaped lobes of $b_s^m$ and $b_z^m$ of weaker amplitude (located in the middle of the gap) are observed in phase with $u_s^m$ and $u_z^m$. Consequently these round-shaped structures of $b_s^m$ and $b_z^m$ contribute to the terms $\overline{u_z^m b_s^m}$ and $\overline{u_s^m b_z^m}$. Unlike model J (where $\overline{u_z^m b_s^m} \gg \overline{u_s^m b_z^m}$), $\overline{u_z^m b_s^m} \sim \overline{u_s^m b_z^m}$ so only a weak axisymmetric multipolar magnetic field is maintained in this case. At the surface of the planet this axisymmetric field is the dominant component but in comparison with the strongly axisymmetric dipolar field produced in model J, the field is of small amplitude: the amplitude of the axisymmetric radial field is about $10^{-3}\sqrt{\rho \mu_0}U_0$ for model J at $Rm=1.17Rm_c$ ($Ro=1.17Ro_c$ and $Pm\approx Pm_c$) (figure \[fig:spec\_rs\_J\]) while it is only $10^{-5}\sqrt{\rho \mu_0}U_0$ for model N at $Rm_c=2.4Rm_c$ ($Ro=1.20 Ro_c$ and $Pm=2 Pm_c$) (figure \[fig:spec\_rs\_N\]). The main difference between the Rossby waves in models J and N is their size. The phase speed of the Rossby wave, $c\approx \Omega \beta/(m/s)^2$, is about 100 times larger for a $m=2$ wave than a $m=22$ wave, for a fixed radius $s$ and rotation rate $\Omega$. On the other hand, the magnetic diffusion acts more rapidly on small scale structures. The typical propagation timescale for a Rossby wave of size $d$ is $\tau_{rw}=1/(\Omega \beta d)$ assuming that the radial and azimuthal lengthscales of the wave are similar. The magnetic diffusion timescale at the scale of the vortex $d$ is $\tau_{\eta}=d^2/\eta$. The ratio of the two timescales is $$\begin{aligned} \frac{\tau_{\eta}}{\tau_{rw}} = \frac{d^3 \Omega \beta}{\eta} .\end{aligned}$$ The dependence to the third power of the size, $d\propto 1/m$, shows that the magnetic diffusion timescale relative to the propagation timescale is about three orders of magnitude smaller for an $m=22$ mode than an $m=2$ mode for the same parameter values. For the simulation presented for model N, the ratio $\tau_{\eta}/\tau_{rw}$ is about $10^5$. For model J the ratio $\tau_{\eta}/\tau_{rw}$ is about $500$ so the propagation of the Rossby wave is still much more rapid than the magnetic diffusion. For both models, we found that the values of the small scale magnetic field in phase with the velocity is of the same order of magnitude. The velocity field of the vortices is also about the same order of magnitude for the two models. The difference between the two models is that, in model N, the magnetic diffusion acts too slowly on the $m=2$ magnetic structures compared to the wave propagation to produce a significant enough phase lag between $b_s^m$ ($b_z^m$) and $u_z^m$ ($u_s^m$). Consequently, the term $\overline{u_s^m b_z^m}-\overline{u_z^m b_s^m}$ is weak and leads to little generation of axisymmetric poloidal magnetic field. The last stage of the dynamo mechanism is the generation of the axisymmetric toroidal field. It can either be produced from the correlation of small scale velocity and magnetic field (as an $\alpha$ effect) or an $\omega$ effect, that is the shearing of the axisymmetric poloidal magnetic field by the mean zonal flow $\overline{U}$. SC06 find that the $\omega$ effect from the Stewartson layer is dominant in their numerical model. The zonal shear produced in the Stewartson layer is stronger than the shear we obtained with the profiles J and N, so it is not clear that the $\omega$ effect is important in our model *prima facie*. In $\alpha^2$ dynamos, both toroidal and poloidal components are typically of similar magnitudes [@Ols99]. Here, the strong toroidal magnetic field suggests that the $\omega$ effect is more important. To confirm this, we plot in figure \[fig:mecanisme\_omega\] the term responsible for the $\omega$ effect in the azimuthal component of the magnetic induction equation [@Gub87], $r \overline{B_r} \partial_r (r^{-1} \overline{U}) + r^{-1} \sin \theta \overline{B_{\theta}} \partial_{\theta} (\sin \theta^{-1} \overline{U})$. For model J, as we expect, this term is most significant in the region where the poloidal magnetic field lines are bent and misaligned with the zonal flow structure (see figure \[fig:Baxi\_ProfJ\]). The correlation of sign and location of the maxima of the $\omega$ effect in the bulk of the fluid with the axisymmetric azimuthal field indicates that it is mainly generated by the $\omega$ effect. Note that some $\omega$ effect is also present close to the outer boundary, where the poloidal magnetic field lines converge and diverge locally due to induction by the Ekman pumping. However no particularly strong axisymmetric azimuthal magnetic field is produced in this region (figure \[fig:Baxi\_ProfJ\]) so this small scale field diffuses probably very rapidly. For model N the outer part of the jet ($s>0.5$) is retrograde and creates a negative $\omega$ effect whose sign and location correlate with the axisymmetric azimuthal field, implying that the main dynamo process in the outer region is indeed the $\omega$ effect. However, $\overline{B_{\phi}}$ and the $\omega$ effect are anti-correlated in the inner region ($s<0.5$) so another dynamo process such as a correlation of small scale velocity and magnetic field must be at work there. We have not yet addressed the question of the selection of the axial dipolar symmetry or the axial quadrupolar symmetry. In kinematic dynamo calculations, @Gub00 show that minor changes in the flow can select very different eigenvectors. For a self-consistent system the selection rules are thus very subtle. As discussed in section \[sec:B\_profN\], @Aub04 found that axial quadrupolar symmetry is incompatible with the vertical structures of cyclones and anticyclones in convectively-driven dynamos, and so these solutions are unstable for strong convective flows. In our simulations of model N, this conclusion suggests that the axial quadrupolar symmetry would be unstable for larger magnetic Reynolds numbers, and an axial dipolar field would be preferred. The selection of a given symmetry does not modify our argument that the wavenumber of the Rossby mode determines the amplitude of the axisymmetric magnetic field since no particular latitudinal symmetry is assumed. In this study, it appears that the dynamo mechanism relies on a subtle balance between the Rossby wave propagation and the magnetic diffusion and therefore is closely related to the size of the Rossby waves. The dynamo field produced with this mechanism requires high magnetic Reynolds numbers ($Rm_c\approx20,000$ for model J and $Rm_c\approx 4000$ for model N). However, in the limit of small Ekman number, this dynamo mechanism is expected to keep a finite value of the critical magnetic Reynolds number [@Sch06], whereas for dynamos that rely on Ekman pumping the critical magnetic Reynolds number becomes infinitely high. Summary and discussion {#sec:Summary} ====================== We have numerically studied the dynamics of zonal flows driven by differential rotation imposed at the top of a conducting layer and how they sustain a magnetic field. Hydrodynamical instability -------------------------- In our hydrodynamical simulations, we found that the destabilisation of the zonal flow takes the form of a global (large radial extension) Rossby mode, even though the instability threshold is governed by a local criterion. The wavenumber depends on the width of the jets, and is independent of the viscosity and rotation rate provided that the former is sufficiently small. In the supercritical regime, several Rossby waves appear and saturate the amplitude of the zonal flow in the bulk of the fluid. They produce a widening of the jets and a strong damping of their amplitude, even for relatively small supercritical forcing ($Ro=2.94Ro_c$). Constraints on the dynamo mechanism ----------------------------------- In the limit of small Ekman number, we find that the Rossby wave appears for $Ro_c\approx0.001$ for a Jupiter-like zonal wind profile (model J) and $Ro_c\approx0.02$ for a Neptune-like profile (model N). In our numerical calculations, non-axisymmetric motions are necessary for dynamo action to occur. As the viscosity is large in the numerical simulations compared to the planetary values, the Reynolds number is much smaller in the simulations. To reach a sufficiently high magnetic Reynolds number, the magnetic Prandtl number is of order $1$, much larger than the expected planetary values. The critical magnetic Reynolds number $Rm_c$ is about $20,000$ for model J and $4000$ for model N. To make this dynamo mechanism work, two constraints must be satisfied: (i) $Ro>Ro_c$ and (ii) $Rm>Rm_c$. Equivalently this gives constraints on the amplitude of the zonal motions at the top of the conducting region, $U_0>Ro_c\Omega r_o$, and on the electrical conductivity within the conducting region, $\sigma>Rm_c/(U_0 r_o \mu_0)$. The extrapolation of the constraint (i) to the giant planets is straightforward as the hydrodynamical instability threshold is independent of the Ekman number, which is of order $10^{-15}-10^{-16}$ for Jupiter [@Gui04] and Neptune [@Ste83]. For Jupiter ($r_o \approx 56,000$ km and $\Omega=1.8\times10^{-4}$s$^{-1}$), the equatorial velocity at the top of the conducting region, $U_0$, must be larger than $10$ m/s to have $Ro>Ro_c=0.001$. For Neptune ($r_o \approx 21,000$ km and $\Omega=1.08\times10^{-4}$s$^{-1}$), $U_0$ must be larger than $45$ m/s to have $Ro>Ro_c=0.02$. For both cases, this constraint is quite strong as it only allows for a factor 10 decrease of the amplitude of the zonal wind between the surface of the planet and the top of the deep conducting region, independently of the location of the top of this region. The extrapolation of the constraint (ii) to the giant planets requires knowing how the critical Reynolds number scales with the Ekman number. When varying the Ekman number from $10^{-6}$ down to $10^{-8}$, @Sch06 found that $Rm_c$ remains constant (of the order of $10^4$ in their simulations, close to the values found in our study). Based on their results, we assume that $Rm_c$ is of the same order of magnitude when the Ekman number is close to the planetary values. For Jupiter, we obtain that the electrical conductivity should be larger than $30$S/m to have $Rm>Rm_c=20,000$ (using $U_0=10$ m/s). For Neptune, the electrical conductivity should be larger than $10$S/m to have $Rm>Rm_c=4000$ (using $U_0=45$ m/s). This constraint on the conductivity is less restrictive than the constraint on the amplitude of the zonal motions and should be satisfied in the deep conducting layer of Jupiter [@Nel99] and Neptune [@Nel97]. We conclude that the differential rotation imposed by the zonal winds at the top of the conducting regions is a plausible candidate to drive the dynamo mechanism in the giant planets although a strong constraint on the amplitude of the zonal jet applies. Given the assumptions used in our model, such as incompressibility, constant conductivity, unrealistically large viscosity and viscous coupling between electrically insulating and conducting regions, this conclusion remains tentative. However, the robust nature of Rossby waves in the asymptotic limit of small Ekman numbers makes this dynamo mechanism appealing for planetary physical conditions. Generation of the axisymmetric field and width of the jets ---------------------------------------------------------- With a simple theoretical model, we show that the production of the axisymmetric field depends on the propagation of the Rossby waves and on the magnetic diffusion acting at the scale of the vortices. This model is in agreement with our numerical results: the magnetic diffusion rate of the $m=2$ magnetic structures induced by the Rossby waves in model N is nearly negligible compared to the propagation rate of the wave: as a result a weak axisymmetric poloidal magnetic field is generated; the magnetic diffusion acting on the $m=22$ magnetic structures is not negligible compared to the propagation rate of the small size ($m=22$) Rossby wave of model J: a dominant axisymmetric poloidal magnetic field is therefore generated. Consequently, in this model, the width of the zonal jets has an important influence on the generation of the axisymmetric magnetic field by controlling the size of the Rossby waves. Our results suggest that the difference in the magnetic fields and the surface zonal winds may be related if a (hydrodynamic or magnetohydrodynamic) mechanism can transport angular momentum between the surface and the deep, electrically conducting region. The critical magnetic Reynolds number of this dynamo mechanism is large. However, in order to compare with other dynamos, a more significant number may be the critical local magnetic Reynolds number associated with magnetic induction by the Rossby wave velocity $Rm_c^l=V_s d/\eta$ where $V_s$ is the typical non-axisymmetric radial velocity and $d$ is the lengthscale of the Rossby mode. For the dynamo obtained in model J ($Rm_c= 20,000$), $V_s\approx0.1U_0$ and $m=22$ so we find $Rm_c^l \approx 570$. For the dynamo obtained in model N ($Rm_c= 4000$), $V_s\approx0.01U_0$ and $m=2$ so $Rm_c^l \approx 130$. Thus $Rm_c^l$ is roughly $2-10$ times larger than the magnetic Reynolds number needed for dynamo action with a convective forcing [@Chr06]. Magnetic field at the planets’ surfaces --------------------------------------- In our numerical model, we obtain a peak at small azimuthal scale in the magnetic field spectrum correlated with the width of the hydrodynamically unstable zonal jets. This is a testable prediction as the magnetic measurements of the forthcoming *Juno* mission (arrival at Jupiter in 2016) are expected to be of extraordinary quality due to the absence of a crustal magnetic field on Jupiter. For model J, we obtain a secular variation of the dipole tilt of about $1^{\circ}$ in $1000$ rotation periods or equivalently $0.001$ global magnetic diffusion time. The dipole is strongly axisymmetric with a tilt that does not exceed $2^{\circ}$. On Jupiter, the dipole axis tilt measured with the *Pioneer* and *Voyager* data compared with the *Galileo* measurements is larger (about $10^\circ$) and displays a secular variation of about $0.5^{\circ}$ in 20 years [@Rus01]. The strong axisymmetry of the dipolar field of model J is in better agreement with the magnetic field of Saturn with a dipole tilt less than $1^{\circ}$[@Rus10]. Convective motions within the conducting region ----------------------------------------------- In this work we have not taken into account the convective motions within the deep conducting region. @Wic02b studied the linear stability of an imposed zonal flow in a spherical shell modeling the molecular hydrogen layer of Jupiter. They found that the critical Rossby number of the shear instability onset is almost independent of the Rayleigh number, which measures the strength of the convection. They concluded that the shear instability is only weakly modified by the presence of convection. On the other hand, they showed that the convection onset is strongly influenced by the presence of the zonal circulation, with the convection either enhanced or damped depending on the direction of the shear. However, their study is linear, and so the results cannot be extrapolated beyond the weakly non-linear regime of convection. Whether or not our results apply in the presence of convection is a subject for future studies. In the presence of convection (which produces strong zonal motions and Rossby waves), and even for a convectively-driven dynamo [see for instance @Aub05; @Gro01], the mechanism described here may still impose a similar relationship between the magnetic field morphology and the zonal wind profile. Acknowledgments {#acknowledgments .unnumbered} =============== Financial support was provided by the Programme National de Planétologie of CNRS/INSU. C.G. was supported by a research studentship from Université Joseph-Fourier Grenoble and by the Center for Momentum Transport and Flow Organization sponsored by the US Department of Energy - Office of Fusion Energy Sciences. The computations presented in this article were performed at the Service Commun de Calcul Intensif de l’Observatoire de Grenoble (SCCI) and at the Centre Informatique National de l’Enseignement Supérieur (CINES). We thank Jonathan Aurnou, Toby Wood and the geodynamo group in Grenoble for useful discussions. The manuscript was substantially improved due to helpful suggestions by two anonymous referees. This is a preprint of an article whose final and definitive form has been published in Icarus. Icarus is available online at: <http://www.journals.elsevier.com/icarus>. Linear code used to compute the hydrodynamical instability threshold {#app:codeXSHELL} ==================================================================== In order to study the linear stability threshold at very low Ekman numbers, we designed a linear code derived from @Gil11. This three-dimensional spherical code uses second order finite differences in radius and pseudo-spectral spherical harmonic expansion. The linear perturbation $\mathbf{u}$ of the imposed background flow $\mathbf{U}$ is time-stepped from a random initial field with the following equation: $$\begin{aligned} \left( \frac{\partial}{\partial t} - \mathbf{\nabla}^2 \right) \mathbf{u} = -\left( \frac{2}{E} \mathbf{e_z} + \mathbf{\nabla} \times \mathbf{U} \right) \times \mathbf{u} + \mathbf{U} \times \mathbf{\nabla} \times \mathbf{u} -\mathbf{\nabla} p, \label{eq:codelin}\end{aligned}$$ together with the continuity equation $\mathbf{\nabla} \cdot \mathbf{u} = 0$, which allows us to eliminate the pressure term by using a poloidal-toroidal decomposition. The left hand side of equation \[eq:codelin\] is treated with a semi-implicit Crank-Nicolson scheme, whereas the right hand side is treated as an explicit Adams-Bashforth term. Thanks to the cylindrical symmetry of the base flow $\mathbf{U}$, all azimuthal modes $m$ of the perturbation $\mathbf{u}$ are independent, and we can compute them separately. The coupling with the background flow and the Coriolis force are handled in physical space, but a very fast implementation of the spherical harmonic transform (SHTns library) makes the code quite efficient. In order to determine the stability threshold at $E=10^{-7}$, we used $350$ points in the radial direction, and spherical harmonics truncated at . We use no-slip boundary conditions.

--- abstract: | Food webs can be regarded as energy transporting networks in which the weight of each edge denotes the energy flux between two species. By investigating 21 empirical weighted food webs as energy flow networks, we found several ubiquitous scaling behaviors. Two random variables $A_i$ and $C_i$ defined for each vertex $i$, representing the total flux (also called vertex intensity) and total indirect effect or energy store of $i$, were found to follow power law distributions with the exponents $\alpha\approx 1.32$ and $\beta\approx 1.33$, respectively. Another scaling behavior is the power law relationship, $C_i\sim A_i^\eta$, where $\eta\approx 1.02$. This is known as the allometric scaling power law relationship because $A_i$ can be treated as metabolism and $C_i$ as the body mass of the sub-network rooted from the vertex $i$, according to the algorithm presented in this paper. Finally, a simple relationship among these power law exponents, $\eta=(\alpha-1)/(\beta-1)$, was mathematically derived and tested by the empirical food webs. address: 'Department of Systems Science, School of Management, Beijing Normal University, Beijing, 100875' author: - 'Jiang Zhang, Liangpeng Guo' bibliography: - 'ecology.bib' title: Scaling Behaviors of Weighted Food Webs as Energy Transportation Networks --- Power Law ,Allometric Scaling ,Energy Flow Network Introduction ============ Scientists look for universal patterns of complex systems because such invariant features may help to unveil the principles of system organization[@Waldrop1992]. Complex network studies can not only provide a unique viewpoint of nature and society but also reveal ubiquitous patterns, e.g., small world and scale free, characteristic of various complex systems[@watts1998; @AlbertBarabasi2002]. However, ecological studies have shown that binary food webs, which depict trophic interactions in ecosystems, refuse to become part of the small world and scale free networks family [@Montoya2002; @Dunne2002]. Although some common features, including “two degrees separation”, which means the very small average distance, are shared among food webs [@Williams2002], other meaningful attributes such as degree distribution and clustering coefficient change with the size and complexity (connectance) of the network [@Dunne2002]. Weight information of complex networks such as air traffic network or metabolism networks, etc., can reveal more unique patterns and features that are never found in binary relationships[@Barrat2004; @Almaas2004; @Montis2007]. Food web weights have two different, yet correlated, meanings in ecology. One is the strength of the trophic interaction[@Emmerson2004; @Berlow2004]; the other is the amount of energy flow. They are correlated because interaction strength is the per capita measure of energy flow. Interaction strength-based weighted food webs exhibit new features such as a relationship between predator-prey body size ratio and interaction strength [@McCann1998; @Wootton2002; @Berlow2004]. Additionally, weights of food webs can also be denoted as the total amount of energy flow between two species when the whole system is in the steady state. Studies of energy flow networks in ecosystems have a long history [@odum1988; @Finn1976; @Szyrmer1987; @Higashi1986; @Baird1989; @higashi1993; @Patten1981; @Patten1982]. Many systematic indicators were designed to depict the macro-state of energy flow in ecosystem [@Fath1999; @Fath2001], of which some can not only reflect the direct energy flows between species but also indirect effects and inter-dependence of species[@Fath1999; @Finn1976]. Although some important discoveries were made about the general structure and function of ecological networks[@Fath1998; @Hannon1973; @levine1980; @Hannon1986; @Patten1985; @Ulanowicz1986; @Ulanowicz1997b], few focus on power law distributions and relations[@Ulanowicz1990]. Allometric scaling is an important universal pattern of flow systems. [@kleiber1932] found that the metabolism and body size of all species follow a ubiquitous power law relationship, with an exponent around 3/4. [@west1997] and [@banavar1999] explained this pattern as an emergent property of nutrient and energy transportation networks. This recognition encouraged people to realize that allometric scaling may be a universal feature for all transportation systems. [@garlaschelli2003] extended Banavar’s approach to binary food webs and found a similar allometric scaling power law relationship. Although Garlaschelli’s method as an algorithm had been applied to various networks, including the worldwide trade network[@Duan2007] and tree of life[@Herrada2008], it had several shortcomings. The first step of his algorithm is to obtain a spanning tree by cutting many edges in the original network so that a certain amount of information is lost[@garlaschelli2003]. [@allesina2005] improved this method by reducing the original network to a directed acyclic graph. Although less information is lost, cutting edges is still unavoidable. The second shortcoming of Garlaschelli’s approach and Allesina’s improvement is they can be applied to binary networks, but not weighted ones. This paper will combine the successful approaches in complex weighted networks and earlier studies on ecological flow networks to reveal the underlying heterogeneities and universal scaling behaviors of food webs. The study is organized as follows. In section \[sec.methods\], the basic ideas and steps for obtaining $A_i$ and $C_i$ are introduced. Afterwards, we apply these tools on 21 empirical food webs with energy flow information. Section \[sec.powerlawdisai\] and \[sec.powerlawdisci\] study the power law distributions of $A_i$ and $C_i$. We extend Garlarschelli’s approach to weighted food webs without cutting edges. The allometric scaling power law relationship between $A_i$ and $C_i$ is shown in section \[sec.allometricscaling\]. A simple mathematical relationship among scaling exponents of power law distributions and power law relation is derived and tested using empirical food webs in section \[sec.exponentrelation\]. Finally, the ecological meaning of $A_i$ and $C_i$, the distributions of flux matrix and fundamental matrix, consideration of node information, etc., are discussed in section \[sec.discussion\]. A simple example of our approach, a comparison to the existing methods, and the theorem regarding power law exponents are presented in the Appendix. Methods {#sec.methods} ======= In this section, we outline the basic idea and mathematical definition of our method. One simple example showing how the approach works will be discussed in the \[sec.example\] in detail. \[sec.representation\] Flux Matrix ----------- An ecological energy flow network is a weighted directed graph that represents relationships of ecological energy transfer. For a given graph, a matrix called flux matrix in this paper can be defined as representing the energy flux between species. $$\label{eqn.Fluxmatrix} F_{(N+2)\times (N+2)}=\{f_{ij}\}_{(N+2)\times (N+2)}$$ where $f_{ij}$ is the energy flux from species $i$ to $j$. Two special vertices represent the environment: vertex $0$ and vertex $N+1$. Vertex $0$ denotes the source of energy flow, whereas vertex $N+1$ represents the sink. We expect that the dissipative and exported energy will flow to vertex $N+1$. Therefore, there are in total $(N+2)\times (N+2)$ entries in the flux matrix. Fundamental Matrix ------------------ Suppose that the flow network is balanced, meaning that the total influx equals the efflux for each vertex $i\in [1,N]$. We can then define an $N\times N$ matrix $M$ from $F$ follows, $$\label{defm} m_{ij}={f_{ij} /(\sum_{k=1}^{N+1}{f_{ik}})}, \forall i,j \in [1,N]$$ and the fundamental matrix can be derived as $$\label{eqn.utilizationmatrix}U = I+M+M^2+ \cdots = \sum_{i=0}^{\infty}{M^i}=(I-M)^{-1}$$ where, $I$ is the unity matrix. Any element $u_{ij}$ in $U$ matrix denotes the influence $i$ to $j$ along all possible pathways. $U$ matrix was first introduced in economic input-output analyses[@leontief1951; @Leontief1966] to indicate the direct and indirect effects of good flows in various economic sectors. [@Hannon1973] was the first to apply this matrix to ecology[@Fath1999; @ulanowicz2004]. Given the flux matrix and fundamental matrix, two vertex-related variables, $A_i$ and $C_i$, which will later be shown to follow power law distributions, are defined. $A_i$ ----- We can calculate the total flux through any given vertex $i$ according to $F$. This value is also called node intensity in complex weighted network studies[@Almaas2004]. Because the network is balanced, we need only calculate the efflux of each node as $A_i$, $$\label{eqn.ai} A_i=\sum_{j=1}^{N+1}{f_{ij}}, \forall i \in [1,N]$$ $C_i$ ----- Another vertex-related index called $C_i$ can be defined to reflect the total indirect effects or the total energy store of the sub-network rooted from vertex $i$. $$\label{eqn.ci} C_i=\sum_{k=1}^N{\sum_{j=1}^N{(f_{0j}{u_{ji}}/ u_{ii})u_{ik}}}$$ We will provide an explanation of the indicator $C_i$ in \[sec.example\] by a simple example. $A_i$ is the total flow-through or intensity of vertex $i$. $C_i$ is the total influence of vertex $i$ on all vertices in the whole network. Suppose that of the many particles flowing in the network [@higashi1993], those passing vertex $i$ will be colored red. $C_i$ would then be the total number of red particles flowing in the network. Actually, these two variables are extended from the approach of [@garlaschelli2003] to calculate the allometric scaling of food webs (see \[sec.comparison\]). Balancing the Network --------------------- Sometimes the empirical network is not strictly balanced. To facilitate our algorithm, we can balance them artificially. Suppose $\sum_{j=0}^{N}{f_{ji}}\neq \sum_{j=1}^{N+1}{f_{ij}}$ for vertex $i$. We can add an edge with the weight $|f'_{ij}|$, $f'_{ij}=\sum_{j=0}^{N}{f_{ji}}-\sum_{j=1}^{N+1}{f_{ij}}$ to connect the vertex $i$ to $N+1$ or $0$. If $f'_{ij}>0$, the direction of this artificial edge is from $i$ to $N+1$. If $f'_{ij}<0$, the direction is from $0$ to $i$. Normally, the artificial edges have very small weights because most empirical food webs are almost balanced already. Power Laws ---------- After calculating the indicators of $A_i$ and $C_i$ for each vertex $i$, we will show that they follow the power law distributions in the high tails, which means that, $$\label{eqn.powerlawai} P(A_i>x)\thicksim x^{1-\alpha}$$ and $$\label{eqn.powerlawci} P(C_i>y)\thicksim y^{1-\beta}$$ for given $A_i$ and $C_i$ which are larger than given thresholds $x_0,y_0$ [@Clauset2007], and where $\thicksim $ represents “proportional to.” The cumulative probability distribution curves will be shown and the power law exponents $\alpha,\beta$ calculated in the next section. Furthermore, we will show that $A_i$ and $C_i$ satisfy a power law relationship, $$\label{eqn.allometricscaling} C_i\thicksim A_i^{\eta}$$ This relationship is also called the allometric scaling law because $A_i$ represents metabolism and $C_i$ is the equivalent body mass of the sub-system rooted from vertex $i$ (see \[sec.example\]). Results ======= Dataset ------- Twenty one food webs containing energy flow information from different habitats were studied(Table \[tab.foodwebdata\]). These food webs were obtained from an online database, and most are from published papers. In Table \[tab.foodwebdata\], we list the name and the number of nodes ($|N|$) and edges ($|E|$) in each web. The number of nodes does not include the “respiration” node, and the number of edges only counts the energy flows between species, and does not include the edges from (to) “input” and “output.” The weights of edges in these food webs are energy flows whose values vary across a large range because the units and time scales of the measurements are very different. [($|N|$ stands for the number of vertices of the network and $|E|$ is the number of edges. The webs are sorted by their number of edges.)]{} \[tab.foodwebdata\] Food web Abbre. $|N|$ $|E|$ -------------------------------------- ------------ ------- ------- -- -- Florida Bay, Dry Season BayDry 127 1969 Florida Bay, Wet Season BayWet 127 1938 Florida Bay Florida 127 1938 Mangrove Estuary, Dry Season MangDry 96 1339 Everglades Graminoid Marshes Everglades 68 793 Everglades Graminoids, Dry Season GramDry 68 793 Everglades Graminoids, Wet Season GramWet 68 793 Cypress, Dry Season CypDry 70 554 Cypress, Wet Season CypWet 70 545 Mondego Estuary - Zostrea site Mondego 45 348 St. Marks River (Florida) StMarks 53 270 Lake Michigan Michigan 38 172 Narragansett Bay Narragan 34 158 Upper Chesapeake Bay in Summer ChesUpper 36 158 Middle Chesapeake Bay in Summer ChesMiddle 36 149 Chesapeake Bay Mesohaline Net Chesapeake 38 122 Lower Chesapeake Bay in Summer ChesLower 36 115 Crystal River Creek (Control) CrystalC 23 81 Crystal River Creek (Delta Temp) CrystalD 23 60 Charca de Maspalomas Maspalomas 23 55 Rhode River Watershed - Water Budget Rhode 19 35 : Empirical Food Webs and Their Topological Properties After studying the scaling laws of these food webs, we divided the results into two main sections. First, the power law distributions that reflect the heterogeneity of $A_i$ and $C_i$ are shown. Second, the allometric scaling power law relationship that depicts the self-similar structures of energy flows is discussed. Power Law Distributions of $A_i$ {#sec.powerlawdisai} -------------------------------- We calculated the random variables $A_i$ for each of the 21 empirical food webs. Four of them are selected to plot in Figure \[fig.aidistribution\]. ![Cumulative probability of the variable $A_i$ for four selected food webs with the best fitted lines. The fitted lines start from $x_0$, and their slopes are listed in Table \[tab.aidistribution\][]{data-label="fig.aidistribution"}](Ai-result4.eps) $A_i$s of real food webs decay as power law in the high tail; therefore, the fitted lines start from given lower bounds $x_0$ [@Clauset2007]. These figures show that $A_i$ follows the power law distribution. There are obvious cutoffs in the tails that may be attributed to sampling effects[@Newman2005]. According to equation \[eqn.powerlawai\], the cumulative probability decays as $x$ slowly. The probability of finding a higher $A_i$ value (in the tail of the curve) is very small. Therefore, the number of samples in this interval becomes very few and statistical fluctuations are unavoidably large as a fraction of sample number. This phenomenon is obvious in other fields such as income[@Clementi2006], personal donations[@ChenWang2009] and the number of species per genus of mammals[@Newman2005]. The scaling exponent $\alpha$ for each food web was estimated according to the maximum likelihood approach [@Clauset2007]. The exponents and relative errors of power law fittings for all 21 food webs are listed in Table \[tab.aidistribution\]. [ ($\alpha$ is the power law distribution exponent of equation \[eqn.powerlawai\]; $x_0$ is the smallest value of $A_i$ that follows power law, $x_{max}$ is the largest $A_i$; $D$ is the KS statistic; $\sigma$ is the quantile of 95% confidence interval; and the number of samples is the total number of nodes following the power law distribution. The webs are sorted by their number of edges)]{} \[tab.aidistribution\] Food web $\alpha$ $x_0/x_{max}$ $D$ $\sigma$ No. of samples ------------ ---------- --------------- ------ ---------- ---------------- Baydry 1.25 8.45e-006 0.09 0.13 102 Baywet 1.23 4.24e-006 0.08 0.13 105 Florida 1.23 4.24e-006 0.08 0.13 105 Mangdry 1.24 1.55e-006 0.06 0.16 75 Everglades 1.23 4.20e-006 0.12 0.20 44 Gramdry 1.25 9.55e-006 0.11 0.21 41 Gramwet 1.23 4.20e-006 0.12 0.20 44 Cypdry 1.28 5.95e-005 0.10 0.20 44 Cypwet 1.23 2.06e-005 0.10 0.21 42 Mondego 1.33 3.87e-004 0.11 0.26 26 StMarks 1.35 9.36e-004 0.17 0.21 43 Michigan 1.32 1.80e-003 0.18 0.31 18 Narragan 1.26 1.10e-004 0.16 0.23 33 ChesUpper 1.28 9.54e-004 0.21 0.23 32 ChesMiddle 1.29 9.47e-004 0.17 0.24 30 Chesapeake 1.41 8.25e-003 0.22 0.29 21 ChesLower 1.77 5.79e-002 0.20 0.33 16 CrystalC 1.27 1.04e-004 0.13 0.32 17 CrystalD 1.25 3.10e-005 0.12 0.30 19 Maspalomas 1.54 2.22e-002 0.18 0.29 20 Rhode 1.56 1.35e-002 0.17 0.34 15 : Parameters of the power law distributions for $A_i$ In Table \[tab.aidistribution\], $x_0$s represent the lower bounds of the power law distributions. We normalized $x_0$ by dividing the maximum $A_i$ of each food web to avoid the large range variance of $x_0$ among different webs because their units and the measurement time scales are very different. $D$ is the statistic of the KS test[@Rousseau2000; @Goldstein2004]. Its value reflects the maximum distance between the cumulative probability of real data and the fitted model. Therefore, the smaller $D$ values indicate the better power law fitting. $\sigma$ is the quantile of the 95% confidence interval for different numbers of samples[@Noether1967], and is only a reference for $D$. If $D$ is smaller than $\sigma$, then we should accept the power-law hypothesis[@Noether1967]. From Table \[tab.aidistribution\], we know that all food webs pass the KS test. In the last column, “No. of samples” means the number of sample points that are larger than $x_0$ and follow the power law distribution. By comparing different rows, we know that the food webs with more edges can be better described by power laws because their $D$s are smaller. Further, the scaling exponent $\alpha$ and $x_0/x_{min}$ increase as the scale of the network decreases. All $\alpha$ values fall into the interval $[1.23,1.77]$, with an average of $1.32$. The power law distributions of $A_i$s reflect the heterogeneities of energy flux. Few nodes possess high $A_i$ values, while most nodes only share a small fraction of the energy flux. The exponent of power law reflects the degree of heterogeneity of the whole network. Therefore, larger food webs are more heterogeneous than smaller ones because their exponents are lower (Table \[tab.aidistribution\]). Although the power law distribution of $A_i$ can not give us concrete information about each vertex [@Fath1999; @Patten1981; @Patten1982], it helps us to understand the network as a whole. Power Law Distributions of $C_i$ {#sec.powerlawdisci} -------------------------------- The same approach can be applied to $C_i$s. The distributions of $C_i$s for four selected food webs are shown in Figure \[fig.cidistribution\]. ![Cumulative probability of the variable $C_i$s for four selected food webs with the best fitted lines. The slopes of the fitted lines are listed in Table \[tab.cidistribution\][]{data-label="fig.cidistribution"}](Ci-result4.eps) The curves of $C_i$ distributions are very similar to the curves in Figure \[fig.aidistribution\]. The estimated exponents and the KS test parameters are listed in Table \[tab.cidistribution\]. [ ($\beta$ is the power law distribution exponent of equation \[eqn.powerlawci\]; $y_0$ is the smallest value of $C_i$ that follows the power law, $y_{max}$ is the largest $C_i$; $D$ is the KS statistic; $\sigma$ is the quantile of 95% confidence interval; the number of samples is the total number of nodes following the power law distribution. The webs are sorted by their number of edges)]{} \[tab.cidistribution\] Food web $\beta$ $y_0/y_{max}$ $D$ $\sigma$ No. of samples ------------ --------- --------------- ------ ---------- ---------------- Baydry 1.23 2.97e-006 0.08 0.13 107 Baywet 1.23 2.71e-006 0.08 0.14 101 Florida 1.23 2.71e-006 0.08 0.14 101 Mangdry 1.25 2.03e-006 0.07 0.16 68 Everglades 1.23 2.16e-006 0.10 0.20 44 Gramdry 1.25 4.57e-006 0.10 0.21 42 Gramwet 1.23 2.16e-006 0.10 0.20 44 Cypdry 1.28 4.88e-005 0.10 0.20 44 Cypwet 1.25 3.13e-005 0.11 0.21 39 Mondego 1.29 9.61e-005 0.12 0.24 30 StMarks 1.42 1.27e-003 0.16 0.21 40 Michigan 1.30 7.22e-004 0.17 0.31 18 Narragan 1.24 1.22e-004 0.17 0.23 33 ChesUpper 1.28 6.03e-004 0.20 0.23 32 ChesMiddle 1.26 3.88e-004 0.19 0.24 30 Chesapeake 1.61 2.41e-002 0.18 0.33 16 ChesLower 1.84 5.02e-002 0.22 0.35 14 CrystalC 1.28 1.04e-004 0.13 0.32 17 CrystalD 1.23 1.35e-005 0.12 0.29 20 Maspalomas 1.56 1.63e-002 0.19 0.29 20 Rhode 1.52 8.93e-003 0.23 0.32 17 : Parameters of the power law distributions for $C_i$ Comparing Table \[tab.cidistribution\] with Table \[tab.aidistribution\], the exponents of $C_i$ distributions are slightly higher than those of $A_i$ distributions. The exponent $\beta$ and KS test statistic $D$ decrease with the scale of the network. The average exponent of these food webs is $1.33$, with all $C_i$ values falling into the interval $[1.23, 1.84]$. We have studied the heterogeneity of $C_i$s node by node. The nodes with high $A_i$ values always have high $C_i$ values, indicating a possible positive correlation between $A_i$ and $C_i$. Allometric Scaling Relations {#sec.allometricscaling} ---------------------------- The similarity between Figure \[fig.cidistribution\] and Figure \[fig.aidistribution\] shows that there must be some connections between $A_i$ and $C_i$. Allometric scaling of these flow networks revealed that the relationship between $A_i$ and $C_i$ is actually a power law. ![Allometric Scaling relationship between $A_i$s and $C_i$s of four selected food webs with the best fitted lines[]{data-label="fig.allometricscaling"}](allometricScalings.eps) As shown in Figure \[fig.allometricscaling\], the sample points aggregate around their fitted lines very well. This relationship is ubiquitous for all 21 food webs as shown in Table \[tab.allometricscaling\]. [(The second column lists $\eta$s of each food web with the errors; The webs are sorted by their number of edges)\ ]{} \[tab.allometricscaling\] Food web $\eta$ $R^2$ ------------ --------------- -------- -- Baydry 1.01$\pm$0.01 0.9946 Baywet 1.02$\pm$0.01 0.9946 Florida 1.02$\pm$0.01 0.9946 Mangdry 1.01$\pm$0.01 0.9967 Everglades 1.02$\pm$0.01 0.9992 Gramdry 1.03$\pm$0.01 0.9990 Gramwet 1.02$\pm$0.01 0.9992 Cypdry 1.00$\pm$0.02 0.9957 Cypwet 1.00$\pm$0.01 0.9970 Mondego 1.01$\pm$0.01 0.9989 StMarks 1.03$\pm$0.04 0.9784 Michigan 1.01$\pm$0.01 0.9986 Narragan 1.01$\pm$0.04 0.9910 ChesUpper 1.05$\pm$0.02 0.9966 ChesMiddle 1.04$\pm$0.02 0.9959 Chesapeake 0.99$\pm$0.02 0.9966 ChesLower 1.05$\pm$0.02 0.9974 CrystalC 0.89$\pm$0.23 0.7706 CrystalD 0.90$\pm$0.23 0.7722 Maspalomas 0.96$\pm$0.09 0.9656 Rhode 0.83$\pm$0.17 0.8658 : Allometric scaling of Empirical Food Webs We used the minimum square error method to find the best-fitted line (Table \[tab.allometricscaling\]). $R^2$s were larger than $0.9$ for all food webs except CrystalC, CrystalD and Rhode, whose scales are very small ($|N|<23$). The $R^2$s and exponents decrease with the scale of the network because the statistical significance decreases as the number of samples declines. All exponents $\eta$ fall into the interval $[0.83,1.05]$. The mean value of $\eta$s for these food webs, except CrystalC,CrystalD and Rhode, is $1.02$. We also show the $A_i$ and $C_i$ values of root nodes for all food webs, and fit them with a line on the log-log plot (Figure \[fig.rootpower\]). These power law relations reflect the self-similar nature of the weighted food webs. ![$A_i$, $C_i$ plot for root nodes of 21 food webs. The slope $\eta$ (the power law exponent) is 1.02$\pm$0.02[]{data-label="fig.rootpower"}](root.eps){width="12cm"} Relationship of Scaling Exponents {#sec.exponentrelation} --------------------------------- As we have shown, $A_i$ and $C_i$ are random variables following power law distributions with scaling exponents $\alpha$ and $\beta$, respectively. They also follow a power law relationship with the scaling exponent $\eta$. Is there any universal relationship among $\alpha,\beta$ and $\eta$? Actually, for any random variables following power law distributions and power law relations, we can prove a mathematical theorem (see \[sec.theorem\]). According to this theorem, the power law exponents $\alpha,\beta,\eta$ of the food webs should also satisfy equation \[eqn.exponentsrelation\]. We tested this hypothesis by calculating $(\alpha-1)/(\beta-1)-\eta$ for each of the 21 empirical food webs to obtain Figure \[fig.exponents\]. ![The differences between $\eta$ and ${(\alpha-1)}/{(\beta-1)}$ are calculated for all food webs to test the exponents relationship (equation \[eqn.exponentsrelation\]). The food webs are sorted according to their number of edges[]{data-label="fig.exponents"}](exponents.eps){width="12cm"} From Figure \[fig.exponents\], we know that most exponents of food webs satisfy this relation, and that the errors become larger as the scale of the network decreases. As the scaling behaviors we are studying are statistical properties, the significance of these regularities will increase with the number of samples. Therefore, scaling behaviors are more obvious and accurate for large scale networks because larger webs have more sample points. Discussion {#sec.discussion} ========== Ecological Meaning of Power Law Exponents ----------------------------------------- As demonstrated above, food webs as energy transportation networks always follow power law distributions and relations. Three important exponents ($\alpha$,$\beta$ and $\eta$) are derived from these power law regularities. The question of whether these exponents carry ecological meaning naturally follows, and at first, the three exponents all reflect integral properties of whole networks. $\alpha$ describes the heterogeneity of first passage energy flux distributions among vertices. The heterogeneity decreases with $\alpha$. Therefore the distributions of energy flows are more uneven in large food webs than the small ones. From table \[tab.aidistribution\], we also know that all $\alpha$ values fall into the interval $[1.23,1.77]$. According to the features of power law distributions, the means and variances of power law random variables with exponents smaller than 2 are divergent[@Newman2005]. Therefore, energy flux on food webs has no characteristic value. It is meaningless to find a specific species with the average energy flux as the representation of other species[@Newman2005]. The allometric scaling relation describes the self-similarity of flow networks. [@garlaschelli2003; @allesina2005] pointed out that allometric scaling exponents describe the transportation efficiency of binary food webs because $C_i$ is treated as the cost of transportation. The range of these exponents is between 1 (most inefficient network) and 2 (most efficient network). However, we believe that the exponent $\eta$ discussed in this paper does not describe the efficiency of the whole network. As pointed out in section \[sec.methods\] and \[sec.example\], $C_i$ can be understood as the energy store by the system, rooted from $i$ but not the cost of the transportation. Thus, the food web with higher $\eta$ can store more energy with the same consumption of metabolites ($A_i$). Therefore, we believe that the food webs with higher $\eta$ are more capable of storing energy by means of cycling the flows in the network. In Table \[tab.allometricscaling\], we see that the networks with larger scales have larger $\eta$ values. Consequently, food webs can increase their ability to store energy by increasing their complexity. Further, the range of exponents $\eta$ is not simply $[1,2]$ (see Table \[tab.allometricscaling\]). As [@garlaschelli2003; @allesina2005; @banavar1999] pointed out, the range $[1,2]$ is only suitable for spanning trees or directed acyclic graphs of the original binary food webs. However, our method considers more ingredients, including the energy flux as the weight of edges, the loop structures of energy flows, and the heterogenous energy dissipation of each node, than the mere topology of the food webs with homogenous nodes. That is the reason why the exponents are out of the range $[1,2]$. The exponent $\beta$ also describes the heterogeneity of indirect effects. It is determined by exponents $\alpha$ and $\beta$ via the theorem mentioned in section \[sec.exponentrelation\]. Flux Matrices and Fundamental Matrices -------------------------------------- As discussed in section \[sec.methods\], $A_i$ and $C_i$ are defined according to the flux matrix and fundamental matrix. Therefore, the scaling behaviors of the food webs are determined by the matrices. The properties of these matrices may help us to understand the origin of the scaling behaviors. The elements in flux matrices also follow power law distributions, with an average exponent 1.46 (see Table \[tab.matricesdistribution\]). We hypothesis that this power law determines the power law distribution of $A_i$. [($\alpha$ is the power law distribution exponent; $F_0$ is the smallest value of $f_{ij}$ that follows the power law; $F_{max}$ is the largest $f_{ij}$; $D$ is the KS statistic; $\sigma$ is the quantile of 95% confidence interval; number of samples is the total number of edges following the power law distribution. The webs are sorted by their number of edges)]{} \[tab.matricesdistribution\] Food web $\alpha$ $F_0/F_{max}$ $D$ $\sigma$ No. of Samples ------------ ---------- --------------- ------ ---------- ---------------- -- Baydry 1.33 3.82e-006 0.03 0.05 761 Baywet 1.33 4.91e-006 0.02 0.06 608 Florida 1.33 4.91e-006 0.02 0.06 608 Mangdry 1.32 2.81e-007 0.04 0.05 614 Everglades 1.37 2.98e-006 0.06 0.09 222 Gramdry 1.39 3.32e-006 0.06 0.10 194 Gramwet 1.37 2.98e-006 0.06 0.09 222 Cypdry 1.34 5.93e-005 0.05 0.10 203 Cypwet 1.29 1.15e-005 0.06 0.09 240 Mondego 1.40 4.80e-005 0.05 0.12 123 StMarks 1.84 1.29e-002 0.10 0.19 53 Michigan 1.43 5.88e-004 0.11 0.17 62 Narragan 1.70 1.43e-002 0.09 0.23 34 ChesUpper 1.32 1.95e-004 0.09 0.11 151 ChesMiddle 1.31 1.28e-004 0.11 0.12 134 Chesapeake 1.64 2.05e-002 0.17 0.24 30 ChesLower 1.65 1.17e-002 0.16 0.18 55 CrystalC 1.37 1.01e-004 0.08 0.24 30 CrystalD 1.29 6.56e-005 0.09 0.27 24 Maspalomas 1.71 2.10e-002 0.13 0.19 53 Rhode 1.84 2.65e-002 0.13 0.29 20 : Parameters of the power law distributions for $f_{ij}$ However, unlike other variables, the distributions of fundamental matrices are not power laws but rather more like log-normal because the tails of the curves decline quickly (see Figure \[fig.utilizationdistribution\]). We also studied all of the fundamental matrices of 21 empirical food webs, and noted that very few could pass the KS test. We presume that the calculation of the fundamental matrix in equation \[eqn.utilizationmatrix\] needs infinite operations on $F$ matrix (see section \[sec.methods\]). As a result, the noise in $F$ matrices is enlarged and accumulated in the tails of the distribution curves of $U$ matrix. However, the means by which the non-power law distribution of fundamental matrices determines power law distributions of $C_i$ and the allometric scaling power law relationship is an interesting problem for future studies. ![Cumulative distribution of fundamental matrices of four selected food webs[]{data-label="fig.utilizationdistribution"}](U-result4.eps) Information on Nodes -------------------- One of the weak points of our study on allometric scaling power law relationships is that the exponents are very close to 1. However, in this case, “allometry” just means the non-linear relationship between two variables, so, the relationship between $A_i$ and $C_i$ cannot be rigorously defined as an allometric scaling relationship. Because the calculations of fundamental matrices and $C_i$s are always based on linear algebra, the results are close to linear relationship. One possible way to mend this weak point is to further consider the information available about nodes. Indeed, much information on nodes, i.e., each species in the web, is ignored in this work. The biomass as the weight of each node is available for many food webs. According to the definition, $C_i$ is simply the energy store of the sub-system rooted from the vertex $i$. Therefore, biomass information should be included in $C_i$ because a large part of energy will flow into the species node stored as biomass. It is possible that a new approach of calculating $C_i$ including the biomass information for all species may break the linear relationship between $A_i$ and $C_i$. Another important node characteristic is the body size of a given species. The metabolic theory predicts that species body size of the species can not only determine metabolism, life span, and birth rate, etc. [@brown2004], but also play an important role in energy flows and food webs [@Cohen2003]. An integrated theory of weighted food webs based on energy flow networks should contain body size data. Concluding Remarks ================== This paper presents a new approach to reveal the scaling natures of weighted food webs as energy flow networks based on flux and fundamental matrices. The $A_i$,$C_i$ distributions and the relationship between them always follow power laws. The power law exponents $\alpha$,$\beta$ and $\eta$ satisfy a relationship, $\eta=(\alpha-1)/(\beta-1)$ as proved by the theorem \[thm.exponent\]. Power law exponents consistently change with network scales. We note that the allometric scaling exponent does not reflect the transportation efficiency of networks but rather the capability of storing energy, which is very different from previous studies. We also investigated the distributions of flux matrices and fundamental matrices, and suggested that biomass information should be incorporated into future studies. #### Acknowledgement Thanks for the support of National Natural Science Foundation of China(No.70601002 and No.70771011). We acknowledge Clauset, A. for providing the source code of power law fitting and KS test on his web site; and also the Pajek web site to provide food web data online. We also acknowledge three anonymous reviewers for advices. A Simple Example {#sec.example} ================ To understand the method introduced in the section \[sec.methods\], let’s look at a simple example (Figure \[fig.example\]).  [**(a)** is the original flow network; **(b)** is the balanced network from (a), the dashed arrows are artificial edges pointing to the sink node; **(c)** is the Markov chain calculated according to (b).]{}\[fig.example\] As shown in Figure \[fig.example\], the balanced flow network and the derived matrix $M$ can be obtained step by step according to the method described in section \[sec.methods\]. The fundamental matrix $U$ can then be calculated for this simple network. $$\label{eqn.exampleu}U = I+M+M^2+ \cdots=\left( \begin{array}{ccccc} 1 & 3/ 5 & 7/ 20 & 1/ 5 & 2/ 5\\ 0 & 42/ 41 & 7/82 & 14/41 & 28/41\\ 0 & 12/41 & 42/41 & 4/41 & 8/41\\ 0 & 3/164 & 21/328 & 165/164 & 21/41\\ 0 & 3/82 & 21/164 & 1/82 & 42/41 \end{array} \right)$$ Any entry $m_{ij}$ in the matrix $M$ is merely the probability of one particle flowing from vertex $i$ to vertex $j$[@Barber1978]. Furthermore, any entry $(i,j)$ in $M\cdot M$ represents the probability of a particle flowing from $i$ to $j$ along any path in $2$ steps. $M\cdot M\cdot M$ represents the probabilities after $3$ steps, etc. [@higashi1993]. Thus, the matrix $U$ simply takes in consideration all transfers of particles along all possible paths. Now, we will show how to calculate $A_i$ and $C_i$ for vertex 2. According to the definition, $A_i$ is the total flow-through of vertex $i$, so $A_2 = \sum_{j=1}^{6} {f_{2,j}}=60$. Suppose many particles are flowing in the network. They will be colored red once they flow through vertex $2$. These particles will keep their color and flow around the whole network along all possible pathways. Hence, the total number of red particles in the network is just $C_2$, which is computed as, $$\label{eqn.exampleC} C_2=G_2 \sum_{k=1}^5{ u_{2k}}=((100\times 3/5 + 0)/ (42/41))\sum_{k=1}^5{u_{2k}} =125$$ where the term $G_2=\sum_{j=1}^5{f_{0j}{u_{j2}}/ {u_{22}}}$ in equation \[eqn.exampleC\] is the total number of new particles which are colored red by vertex 2 in each time step. $\sum_{j=1}^N{f_{0j}{u_{j2}}}$ is the number of particles that flow into the system from the environment $0$ to the vertex $i$ along all possible pathways at each time step, with $f_{0j}=(100,0,0,0,0)$ in this example. By dividing by the term $u_{ii}$ to derive $G_i$ one avoids double counting the red particles[@higashi1993]. Thus, $C_i=G_i\sum_{k=1}^N{u_{ik}}$ is the total number of particles that have been colored red and flow to other nodes along all possible pathways at each time step. If we treat the red particles flowing in the network as a metabolic sub-system, we can calculate its allometric scaling relationship as Garlaschelli has done for binary food webs [@garlaschelli2003]. Thus, $A_i$ is the metabolism and $C_i$ is the energy store or body mass of the sub-system. Indeed, Garlaschelli’s approach can be recovered by our method, as shown in the next section. Comparisons to Existing Approaches {#sec.comparison} ================================== In this section, we will compare our approach to [@garlaschelli2003]’s method and [@allesina2005]’s method.  [ is a hypothetical food web (The letter in each vertex is its index). The black vertex is the root; [**(**b)]{} is a spanning tree of the original network (a). $A_i$ and $C_i$ are denoted inside and beside vertex $i$; [**(**c)]{} is the implicated flow network of (b), the numbers beside edges are flux. The dashed lines are additional edges; [**(**d)]{} is a directed acyclic graph of the original network, the numbers are $A_i$s and $C_i$s calculated by the method of [@allesina2005]; [**(**e)]{} is a constructed flow network according to (d); [**(**f)]{} is the network with numbers of $A_i$s and $C_i$s calculated by our method according to the flow structure of (e).]{}\[fig.hfoodweb\] Figure \[fig.hfoodweb\](a),(b) shows how Garlaschelli’s approach can be applied to a hypothetical food web to calculate $A_i$ and $C_i$ for each vertex. At first, a spanning tree is constructed from the original food web (Figure \[fig.hfoodweb\] (a)) by cutting edges. That way, each sub-tree rooted from any vertex can be viewed as a sub-system of the spanning tree. For example, the sub-tree with three vertices (b,f,g) rooted from the vertex b is a sub-system of the spanning tree. $A_i$ is the total number of vertices involved in this sub-tree and $C_i$ is the summation of $A_i$s for each vertex in this sub-tree. Therefore, in this example, $A_b$ is 3 and $C_b$ is 6. Finally, the universal allometric scaling relationship of $A_i$s and $C_i$s, with an exponent around $1.3$, was found for all food webs, according to [@garlaschelli2003]. Garlaschelli’s method was inspired by [@banavar1999]’s model to explain the Kleiber’s law (See Figure \[fig.hfoodweb\](c)). The spanning tree is simply Banavar’s optimal transportation network. Thus, energy flows into the whole system from the root along the links of the network to all nodes. Suppose that each node would consume 1 unit of energy in each time step. A flux with 1 unit representing the energy consumption by each node should then be added to the original spanning tree. In Figure \[fig.hfoodweb\](c), the energy dissipation by each node is added as a dotted line. As a result, $A_i$ of each node is just the total influx of this node. $C_i$ is the total flux (the total number of red particles colored by $i$) of the sub-tree rooted from $i$. Essentially, calculation of allometric scalings using Garlaschelli’s approach is based on this weighted flow network model. Therefore, our algorithm can derive the exact same values of $A_i$ and $C_i$ for the flow network (Figure \[fig.hfoodweb\](c)). [@allesina2005] extended Garlaschelli’s method. First, the original network is converted to a directed acyclic graph (DAG) as shown in Figure \[fig.hfoodweb\](d), then the sub-network originated from vertex $i$ is identified as the set of vertices that have at least one path from $i$. Therefore, vertices b,c,d,f,g,h belong to the sub-network rooted from b because they are all connected with b. $A_i$ is the number of vertices in the sub-network, and $C_i$ is the summation of all $A_i$s in this sub-network as shown in Figure \[fig.hfoodweb\](d). Because Allesina and Bodini’s approach is not based on weighted flows, it cannot be covered by our approach. However, we can construct a balanced flow network according to the original network as shown in Figure \[fig.hfoodweb\](e). The information of edges is added. Our approach can be applied to this flow network to calculate $A_i$ and $C_i$ values (Figure \[fig.hfoodweb\](f)). Comparing Figure \[fig.hfoodweb\](d) to (f), we find their $A_i$s are almost the same, except vertices a and h. To balance the network, more flows are added in vertex h and a, so their $A_i$ values are larger than those in Figure \[fig.hfoodweb\](d). This modification can not only influence $A_i$ values but also $C_i$s. The vertices in networks \[fig.hfoodweb\](d) and (f) therefore have different $C_i$ values. Although our approach requires weight information, it can be extended to more general flow networks, even those with cycles and loops. Also, as demonstrated by our approach, $C_i$ merely means the energy stored in the sub-system rooted from vertex $i$, which provides much clearer and more significant ecological meaning than previous works. A Theorem about Power Law Exponents {#sec.theorem} =================================== \[thm.exponent\] Suppose $X$ and $Y$ are two random variables following power law distributions. Their density functions, $p(x)\thicksim x^{-\alpha}$ and $p(y)\thicksim y^{-\beta}$ hold for any positive $x>x_0,y>y_0$, where $x_0$ and $y_0$ are lower bounds of $X$ and $Y$. Additionally, $X$ and $Y$ satisfy a power law relation, $Y\thicksim X^{\eta}$, then the exponents $\alpha,\beta,\eta$ have following relationship: $$\label{eqn.exponentsrelation} \eta={{(\alpha-1)}/{(\beta-1)}}$$ Because $X$ and $Y$ follow power law distributions, $p(x)=cx^{-\alpha}$, where $c$ is a constant that satisfies the normalization condition. And $Y= k X^{\eta}$, where $k$ is a constant, so, for any $y>k x_0^{\eta}$, $$\label{eqn.proof1} P\{Y>y\}=P\{kX^{\eta}>y\}=P\{X>({y/k})^{1/\eta}\}=\int_{({y/k})^{1/\eta}}^{+\infty}cx^{-\alpha}dx$$ Let $t=kx^\eta$, so $x=(t/k)^{1/\eta}$, $dx={(1/ \eta)}k^{-1/\eta}t^{(1/\eta)-1}dt$. Take it into equation \[eqn.proof1\], $$\label{eqn.proof2} P\{Y>y\}=\int_{y}^{+\infty}(ck^{(\alpha-1)/\eta}/ \eta) t^{{(1-\alpha)/\eta} -1}dt=\int_{y}^{+\infty} k' t^{{(1-\alpha)/ \eta}-1}dt$$ where $k'=ck^{(\alpha-1)/\eta}/ \eta$ is a constant. Because $Y$ follows power law distribution, $$\label{eqn.proof3} P\{Y>y\}=\int_{y}^{+\infty}c'y^{-\beta}dy$$ Compare equation \[eqn.proof2\] with equation \[eqn.proof3\], we know that, $$-\beta={(1-\alpha)/\eta} -1$$ Finally, $$\eta={(\alpha-1)/(\beta-1)}$$

--- abstract: 'Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials, and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi type identities, and associated Fomin-Greene operators.' address: - 'Kazakh-British Technical University, 59 Tole bi st. Almaty, Kazakhstan' - 'Department of Mathematics, UCLA, Los Angeles, CA 90095 ' author: - Damir Yeliussizov title: Duality and deformations of stable Grothendieck polynomials --- Introduction ============ Stable Grothendieck polynomials are certain symmetric power series first studied by Fomin and Kirillov [@fk; @fk1]. These functions arise as a stable limit of Grothendieck polynomials introduced by Lascoux and Schützenberger [@ls]. The stable Grothendieck polynomial $G_{\lambda}(x_1, x_2, \ldots)$ (corresponding to a Grassmannian permutation) can be viewed as a deformation and K-theory analog of the Schur function $s_{\lambda}(x_1, x_2, \ldots)$ [(while the Grothendieck polynomial is an analog of Schubert polynomial).]{} As a symmetric function, $G_{\lambda}$ has many similarities with $s_{\lambda}$. For example, it can be defined by the following ‘bi-alternant’ formula [@in; @kir1; @ms] $$G_{\lambda}(x_1, \ldots, x_n) = \frac{\det\left[x_i^{\lambda_j + n - j}(1 - x_i)^{j - 1} \right]_{1 \le i,j \le n}}{\prod_{1 \le i < j \le n}(x_i - x_j)}$$ and has a combinatorial presentation given by the generating series $$G_{\lambda}(x_1, x_2, \ldots) = \sum_{T} (-1)^{|T| - |\lambda|} \prod_{i \ge 1} x_i^{\# i\text{'s in }T},$$ where the sum runs over [*set-valued tableaux*]{} of shape $\lambda$; a generalization of semi-standard Young tableaux (SSYT), where boxes may contain sets of integers [@buch]. Let $\Lambda$ be the ring of symmetric functions in infinitely many variables $x = (x_1, x_2, \ldots)$. Denote by $\hat\Lambda$ the completion of $\Lambda$ which includes infinite linear combinations of the basis elements (in some distinguished basis of $\Lambda$, e.g. Schur functions). Note that $G_{\lambda} \in \hat\Lambda$, for instance $G_{(1)} = e_1 - e_2 + e_3 - \cdots$, where $e_k$ is the $k$th elementary symmetric function. It is remarkable that the product of stable Grothendieck polynomials has a finite decomposition $$\label{prodg} G_{\lambda} G_{\mu} = \sum_{\nu} (-1)^{|\nu| - |\lambda| - |\mu|}c_{\lambda \mu}^{\nu}{G}_{\nu}, \quad c_{\lambda \mu}^{\nu} \in \mathbb{Z}_{\ge 0}, \quad |\nu| \ge |\lambda| + |\mu|.$$ This result was proved by Buch [@buch] and he described the coefficients $c_{\lambda \mu}^{\nu}$ combinatorially using set-valued tableaux, generalizing the Littlewood-Richardson rule for Schur functions. Buch studied the Grothendieck ring of the Grassmannian $\mathrm{Gr}(k, \mathbb{C}^n)$ of $k$-planes in $\mathbb{C}^n$ as the quotient ring $\Gamma/\left\langle G_{\lambda}, \lambda \not\subseteq (n-k)^k \right\rangle$, where $\Gamma = \bigoplus_{\lambda} \mathbb{Z} \cdot G_{\lambda}$ is a ring with a basis of Grothendieck polynomials ($(n-k)^k$ is the partition of rectangular shape $k \times (n - k)$). By the fundamental duality isomorphism of the Grassmannian $\mathrm{Gr}(k, \mathbb{C}^n) \cong \mathrm{Gr}(n-k, \mathbb{C}^n)$, structure constants have the symmetry $c^{\nu}_{\lambda \mu} = c^{\nu'}_{\lambda' \mu'}$ where $\lambda'$ denotes the conjugate of $\lambda$. Therefore the involutive linear map $\tau : \hat\Lambda \to \hat\Lambda$ (or $\Gamma \to \Gamma$; the completion $\hat\Gamma$ of $\Gamma$ coincides with $\hat\Lambda$) defined on bases by setting $\tau(G_{\lambda}) = G_{\lambda'}$, is a ring homomorphism. The standard involutive ring automorphism $\omega$ (which maps $e_k$ to $h_k$) extended on $\hat\Lambda$ by mapping the Schur bases $s_{\lambda}$ to $s_{\lambda'}$, does not lead to self-duality for Grothendieck basis, $\omega(G_{\lambda}) = J_{\lambda} \ne G_{\lambda'}$ [@lp]. So another family $\{J_{\lambda}\}$ has the same structure constants. We first ask the following question. Is there a family $\{\widetilde{G}_{\lambda}\}$ which has the same structure constants as $\{{G}_{\lambda}\}$ and is self-dual under the standard involution $\omega$, $\omega(\widetilde{G}_{\lambda}) = \widetilde{G}_{\lambda'}$? Our aim is to deform stable Grothendieck polynomials to adjust it to the canonical involution $\omega$. We will give a concrete construction of these symmetric functions and state the following basic result. \[t1\] There is an automorphism $\phi : \hat\Lambda \to \hat\Lambda$ satisfying $\omega = \phi \tau \phi^{-1}$. Hence $\omega\phi(G_{\lambda}) = \phi(G_{\lambda'})$ and the symmetric function $ \widetilde{G}_{\lambda} := \phi(G_{\lambda})\in \hat\Lambda$ has the same ring structure and satisfies the duality $\omega(\widetilde{G}_{\lambda}) = \widetilde{G}_{\lambda'}$. As we will see, $\phi$ is given explicitly by the substitution $x \mapsto 2x/(2 + x)$. There is a comultiplication $\Delta : \Gamma \to \Gamma \otimes \Gamma$ given by $$\Delta(G_{\nu}) = \sum_{\lambda, \mu} (-1)^{|\lambda| + |\mu| - |\nu|}d^{\nu}_{\lambda \mu} G_{\lambda} \otimes G_{\mu}, \qquad d^{\nu}_{\lambda \mu} \in \mathbb{Z}_{\ge 0}.$$ Both product and coproduct $\Delta$ make $\Gamma$ a commutative and cocommutative bialgebra [@buch]. The completion $\hat\Gamma$ of $\Gamma$ is a Hopf algebra [@lp] with the antipode given by $S(G_{\lambda}(x)) = \omega(G_{\lambda}(-x))$ [@patrias]. The coproduct of $\Gamma$ is compatible with the dual family for $G_{\lambda}$ via the Hall inner product. These dual stable Grothendieck polynomials $g_{\lambda} \in \Lambda$ were explicitly described by Lam and Pylyavskyy [@lp] using reverse plane partitions. We have $$g_{\lambda} g_{\mu} = \sum_{\nu} (-1)^{|\lambda| + |\mu| - |\nu|}d^{\nu}_{\lambda \mu} g_{\nu}.$$ The constants $d^{\nu}_{\lambda \mu}$ are also symmetric up to diagram transpositions, $d^{\nu}_{\lambda \mu} = d^{\nu'}_{\lambda' \mu'}$ and similarly $\omega(g_{\lambda}) \ne g_{\lambda'}$. The dual polynomials $\{ g_{\lambda}\}$ is a (non-homogeneous) $\mathbb{Z}$-basis of $\Lambda$ and the linear map $\bar\tau : \Lambda \to \Lambda$ given by $\bar\tau(g_{\lambda}) = g_{\lambda'}$ is a ring homomorphism. Similarly, there is an automorphism $\bar\phi : \Lambda \to \Lambda$ satisfying $\omega = \bar\phi \bar\tau \bar\phi^{-1}$ and we introduce the polynomials $\widetilde{g}_{\lambda} \in \Lambda$ dual to $\widetilde{G}_{\lambda}$ via the Hall inner product, so they also satisfy the duality $\omega(\widetilde{g}_{\lambda}) = \widetilde{g}_{\lambda'}.$ Once we specify the elements $\widetilde{g}_{(1)}, \widetilde{g}_{(2)}, \ldots$ as (free) generators of (the polynomial ring) $\Lambda$, they characterize the automorphism $\bar\phi$ (and hence the dual family of the $G$-functions). The functions $\widetilde{G}_{\lambda}$ can be extended to $\widetilde{G}_{w}$ for any permutation $w \in S_n$, similarly as stable Grothendieck polynomials $G_{w}$ arise. Here we have the duality $\omega(\widetilde{G}_{w}) = \widetilde{G}_{w^{-1}},$ as well as $\omega(\widetilde{G}_{w}) = \widetilde{G}_{w_0w w_0},$ where $w_0 \in S_n$ is the longest permutation. More generally, the focus of this paper is on two-parameter versions of stable Grothendieck polynomials, the dual symmetric functions $G^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots),$ $g^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots)$. We call them the [*canonical stable Grothendieck functions*]{} and the [*dual canonical stable Grothendieck polynomials*]{}. In special cases they correspond to: - Schur functions $s_{\lambda} = G^{(0,0)}_{\lambda} = g^{(0,0)}_{\lambda}$; the case $\alpha + \beta = 0$ corresponds to certain deformed Schur functions; - stable Grothendieck polynomials $G_{\lambda} = G^{(0,-1)}_{\lambda}$ and its dual $g_{\lambda} = g^{(0,1)}_{\lambda}$; - weak stable Grothendieck polynomials $J_{\lambda} = \omega(G_{\lambda}) = G^{(-1,0)}_{\lambda'}$, and its dual $j_{\lambda} = \omega(g_{\lambda}) = g^{(1,0)}_{\lambda'}$; - and the functions discussed above: $\widetilde{G}_{\lambda} = G^{(-1/2, -1/2)}_{\lambda}$, $\widetilde{g}_{\lambda} = g^{(1/2, 1/2)}_{\lambda}$. The functions $G^{(\alpha, \beta)}_{\lambda} \in \hat\Lambda$, $g^{(\alpha, \beta)}_{\lambda} \in \Lambda$ are non-homogeneous symmetric, $$G^{(\alpha, \beta)}_{\lambda} = s_{\lambda} + \text{ higher degree terms}, \qquad g^{(\alpha, \beta)}_{\lambda} = s_{\lambda} + \text{ lower degree terms}.$$ The families $\{g^{(\alpha, \beta)}_{\lambda} \},$ $\{G^{(-\alpha, -\beta)}_{\lambda} \}$ are dual via the Hall inner product $\langle g^{(\alpha, \beta)}_{\lambda}, G^{(-\alpha, -\beta)}_{\mu}\rangle = \delta_{\lambda, \mu}$, and the involution acts on them as follows: $$\omega(G^{(\alpha, \beta)}_{\lambda}) = G^{(\beta, \alpha)}_{\lambda'},\qquad \omega(g^{(\alpha, \beta)}_{\lambda}) = g^{(\beta, \alpha)}_{\lambda'}.$$ Multiplication of the $G^{(\alpha, \beta)}_{\lambda}$ is governed by the same structure constants (up to scaling) as for the polynomials $G_{\lambda}$, we have $$G^{(\alpha, \beta)}_{\lambda} G^{(\alpha, \beta)}_{\mu} = \sum_{\nu} (\alpha + \beta)^{|\nu| - |\lambda| - |\mu|} c^{\nu}_{\lambda\mu} G^{(\alpha, \beta)}_{\nu}$$ and the comultiplication $\Delta$ is given by $$\Delta(G^{(\alpha, \beta)}_{\nu}) = \sum_{\lambda, \mu} (\alpha + \beta)^{|\lambda| + |\mu| -|\nu|} d^{\nu}_{\lambda \mu} G^{(\alpha, \beta)}_{\lambda} \otimes G^{(\alpha, \beta)}_{\mu}.$$ There are dual Hopf algebra structures with these dual bases parametrized by $\alpha, \beta$ and similar properties. In some sense, the first new parameter $\alpha$ in $G^{(\alpha, \beta)}_{\lambda}$ uncovers duality (under the involution $\omega$) of the $\beta$-Grothendieck polynomials $G^{(0, \beta)}_{\lambda}$. As we will see, construction of the canonical version $G^{(\alpha, \beta)}_{\lambda}$ corresponds to an appropriate substitution of variables. For the dual polynomials $g^{(\alpha, \beta)}_{\lambda}$, description appear to be more complicated, especially its combinatorial presentation. Combining the ‘unifying’ and duality (conjugation) properties described above, the reason for calling these symmetric functions as [*canonical*]{} is also the following. In the specialization $(\alpha, \beta) = (q, q^{-1})$, the elements $\{g^{(\alpha, \beta)}_{\rho} : \rho \text{ is a single row or column} \}$ (under the involution $\omega$) admit a similar characterization as the Kazhdan-Lusztig canonical bases. The elements $\{g^{(\alpha, \beta)}_{(k)} : k\in \mathbb{Z}_{> 0} \}$ then characterize the dual functions $g^{(\alpha, \beta)}_{\lambda}$ as generators for $\Lambda$. Let us summarize some further results about the functions $G^{(\alpha, \beta)}_{\lambda}$, $g^{(\alpha, \beta)}_{\lambda}$. Combinatorial formulas. They are based on several new types of tableaux: - [*Hook-valued tableaux for $G^{(\alpha, \beta)}_{\lambda}$.*]{} Each box of such tableau contains a semistandard [*hook*]{}, the hooks then ‘weakly increase’ in rows and ‘striclty increase’ in columns (Section \[hvt\]). This presentation combines set-valued tableaux [@buch] (sets are single column hooks) and weak set-valued (or multiset-valued) tableaux given in [@lp] for description of $J_{\lambda} = \omega(G_{\lambda})$ (multisets are single row hooks). - [*Rim border tableaux for $g^{(\alpha, \beta)}_{\lambda}$.*]{} These tableaux are constructed via a special decomposition of reverse plane partitions (RPP) into [*rim hooks*]{} on borders of the same entries (Section \[rrt\]). Here we also describe equivalent objects, called [*lattice forests on RPP*]{}. Similarly, for $\alpha \beta = 0$ combinatorics of $g^{(\alpha, \beta)}_{\lambda}$ corresponds to the known combinatorial presentations of the dual stable Grothendieck polynomials $g_{\lambda}$ and $j_{\lambda}=\omega(g_{\lambda})$ given in [@lp]. For $\alpha = 0$, we have generating series running over RPP with a special (column) weight and for $\beta = 0$ they run over SSYT with a special (row) weight. Combinatorial formulas are accompanied with various Pieri type formulas (Section \[spieri\]) for multiplying $G^{(\alpha, \beta)}_{\lambda}, g^{(\alpha, \beta)}_{\lambda}$ on the functions $e_{k},$ $h_k,$ $G^{(\alpha, \beta)}_{(k)},$ $g^{(\alpha, \beta)}_{(k)},$ $G^{(\alpha, \beta)}_{(1^k)},$ $g^{(\alpha, \beta)}_{(1^k)}.$ We prove the duality $\omega(G^{(\alpha, \beta)}_{\lambda}) = G^{(\beta, \alpha)}_{\lambda'}$ using the method of Fomin and Greene [@fg] on noncommutative Schur functions. Our approach is based on Schur operators (Section \[dfg\]), which also gives a way to define the functions indexed by skew shapes. In section \[schur\] we present Schur expansions and related combinatorics. In particular, we show that $g^{(\alpha, \beta)}_{\lambda}$ are Schur-positive (i.e. their transition coefficients are polynomials in $\alpha, \beta$ with positive integer coefficients). We give combinatorial and determinantal formulas for connection constants. Jacobi-Trudi type determinantal identities (Section \[jt\]). To obtain them, we use information given in Schur expansions. Using determinantal formulas for connection constants we obtain new determinantal identities via the Cauchy-Binet formula. In essence, this method gives combinatorial proofs of corresponding identities, which we discuss in detail for $\alpha = 0$. In section \[ggw\] we extend $G^{(\alpha, \beta)}_{\lambda}$ to the functions $G^{(\alpha, \beta)}_{w}$ indexed by permutations $w \in S_n$. In section \[skl\] we put the canonical stable Grothendieck polynomials $g^{(\alpha, \beta)}_{\lambda}, G^{(\alpha, \beta)}_{\lambda}$ (and more generally bases of the ring of symmetric functions) in context of the Kazhdan-Lusztig theory of canonical bases, we give corresponding characterization and discuss some related problems. Acknowledgements {#acknowledgements .unnumbered} ---------------- I am grateful to Pavlo Pylyavskyy for stimulation of this project, many helpful discussions and insightful suggestions. Initial stages of this work began while I was visiting the IMA (Institute for Mathematics and its Applications), and the major part was completed during my visit in the Department of Mathematics at MIT. I am thankful to Richard Stanley and Alexander Postnikov for their hospitality at MIT and for helpful conversations. I am grateful to Askar Dzhumadil’daev for his support and helpful discussions, parts of this work were reported in his seminar. I also thank Sergey Fomin and Victor Reiner for their comments, and the referees for helpful suggestions. Preliminaries and background on Grothendieck polynomials {#prelim} ======================================================== Symmetric functions ------------------- We assume some familiarity with basic theory, e.g. [@macdonald; @ec2]. Let $\Lambda$ be the ring of symmetric functions in infinitely many variables $x = (x_1, x_2, \ldots)$. The elementary symmetric function $e_k$ and complete homogeneous symmetric function $h_k$ are given by $$e_k = \sum_{i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}, \qquad h_k = \sum_{i_1 \le \cdots \le i_k} x_{i_1} \cdots x_{i_k}.$$ The ring $\Lambda$ is a polynomial ring generated by $e_1, e_2, \ldots$ (or $h_1, h_2, \ldots$) and the standard involutive ring automorphism $\omega : \Lambda \to \Lambda$ maps $e_k$ to $h_k$. [^1] Classical bases of $\Lambda$ are indexed by partitions. A [*partition*]{} is a sequence $\lambda = (\lambda_1 \ge \lambda_2 \ge \ldots)$ of nonnegative integers with only finitely many nonzero terms. The weight of a partition $\lambda$ is the sum $|\lambda| = \lambda_1 + \lambda_2 + \cdots.$ Any partition $\lambda$ can be viewed as a [*Young diagram*]{} which contains $\lambda_1$ boxes in the first row, $\lambda_2$ boxes in the second row and so on; equivalently it is the set $\{(i, j) : 1 \le i \le \ell, 1 \le j \le \lambda_i \}$, where $\ell = \ell(\lambda)$ is the number of nonzero parts of $\lambda$. (We use English notation for Young diagrams.) The partition $\lambda'$ with transposed Young diagram, is the [*conjugate*]{} of $\lambda$. We consider $\Lambda$ over $\mathbb{Z}$, e.g. with the Schur basis $\Lambda = \bigoplus_{\lambda} \mathbb{Z} \cdot s_{\lambda}$. We have $\omega(s_{\lambda}) = s_{\lambda'}$ and $\Lambda$ is equipped with the (standard) [*Hall inner product*]{} $\langle\cdot, \cdot \rangle : \Lambda \times \Lambda \to \mathbb{Z}$ which makes Schur functions an orthonormal basis, $\langle s_{\lambda}, s_{\mu}\rangle = \delta_{\lambda \mu},$ where $\delta$ is the Kronecker symbol. Denote by $\hat\Lambda$ the completion of $\Lambda$ which consists of symmetric power series (of unbounded degree). For the basis of Schur functions $s_{\lambda}$ each element $f \in \hat\Lambda$ can uniquely be written as (possibly an infinite sum) $ f = \sum_{\lambda} a_{\lambda} s_{\lambda}, a_{\lambda} \in \mathbb{Z}. $ The Hall inner product $\langle \cdot, \cdot \rangle$ and the ring automorphism $\omega$ extend as follows: $\langle \cdot, \cdot \rangle : \Lambda \times \hat\Lambda \to \mathbb{Z}$ by $\langle s_{\lambda}, s_{\mu} \rangle = \delta_{\lambda, \mu}$; and $\omega : \hat\Lambda \to \hat\Lambda$ by $\omega(s_{\lambda}) = s_{\lambda'}$. Grothendieck polynomials ------------------------ For a generic parameter $\beta$ (usually $\beta = \pm 1$), the [*stable Grothendieck polynomial*]{} $G^{\beta}_{\lambda}(x_1, \ldots, x_n)$ is a symmetric polynomial which can be defined as (see [@in; @kir1; @ms]) $$G^\beta_{\lambda}(x_1, \ldots, x_n) = \frac{\det\left[x_i^{\lambda_j + n - j}(1 - \beta x_i)^{j - 1} \right]_{1 \le i,j \le n}}{\prod_{1 \le i < j \le n}(x_i - x_j)}.$$ Combinatorially it is presented as the generating power series $$\label{svt} G^{\beta}_{\lambda}(x_1, x_2, \ldots) = \sum_{T} \beta^{|T| - |\lambda|} \prod_{i \ge 1} x_i^{\# \text{ of $i$'s in }T},$$ where the sum runs over [*set-valued tableaux*]{} [@buch], that are fillings of the boxes of the Young diagram of $\lambda$ with nonempty [*sets*]{} of positive integers such that if we replace each set by any of its elements the resulting tableau is always a [*semi-standard Young tableau*]{} (SSYT), i.e. the numbers weakly increase from left to right in each row and strictly increase from top to bottom in each column. In other words, a maximal element in each box of a set-valued tableau is less or equal than any element in a box to the right (in the same row) and strictly less than any element in boxes below (in the same column). Obviously, $ s_{\lambda} = G^{\beta}_{\lambda}$ for $\beta = 0$ and we set $G_{\lambda} = G^{\beta}_{\lambda}$ for $\beta = -1$. Note that $$\beta^{|\mu|} G^{\beta}_{\lambda} = G^{1}_{\lambda}(\beta x_1, \beta x_2, \ldots), \quad (-\beta)^{|\mu|} G^{\beta}_{\lambda} = G_{\lambda}(-\beta x_1, -\beta x_2, \ldots).$$ Originally, the functions $G_{\lambda}$ arose as a stable limit of (more general) Grothendieck polynomials indexed by permutations where the partition $\lambda$ corresponds to a Grassmannian permutation [@fk]. We touch this setting in more detail in section \[ggw\] when we consider $(\alpha, \beta)$-deformations of Grothendieck polynomials indexed by permutations. Buch [@buch] proved that the product of stable Grothendieck polynomials has a finite decomposition $$G_{\lambda} G_{\mu} = \sum_{\nu} (-1)^{|\nu| - |\lambda| - |\mu|} c^{\nu}_{\lambda \mu} G_{\nu}, \qquad c^{\nu}_{\lambda \mu}\in \mathbb{Z}_{\ge 0}$$ and described a combinatorial Littlewood-Richardson rule for $c^{\nu}_{\lambda \mu}$ using set-valued tableaux. He related the commutative ring $\Gamma = \bigoplus_{\lambda} \mathbb{Z} \cdot G_{\lambda}$ to the K-theory $K^{\circ} \mathrm{Gr}(k, \mathbb{C}^n)$ of the Grassmannian $\mathrm{Gr}(k, \mathbb{C}^n)$ of $k$-planes in $\mathbb{C}^n$. The Grothendieck group $K^{\circ} \mathrm{Gr}(k, \mathbb{C}^n)$ has a basis $[\mathcal{O}_{\lambda}]$ that are classes of the structure sheaves of the Schubert varieties in $\mathrm{Gr}(k, \mathbb{C}^n)$ indexed by partitions $\lambda$ whose diagram fit in the rectangle $k \times (n - k) = (n-k)^k$. Then, the quotient ring $\Gamma / \langle G_{\lambda}, \lambda \not\subseteq (n-k)^k \rangle$ is isomorphic to $K^{\circ} \mathrm{Gr}(k, \mathbb{C}^n)$ via the map $G_{\lambda} \to [\mathcal{O}_{\lambda}]$; i.e. the structure constants in the Grothendieck ring with the basis $[\mathcal{O}_{\lambda}]$ are the same: $$[\mathcal{O}_{\lambda}] \cdot [\mathcal{O}_{\mu}] = \sum_{\nu} (-1)^{|\nu| - |\lambda| - |\mu|} c^{\nu}_{\lambda \mu} [\mathcal{O}_{\nu}].$$ The basis $\{g_{\lambda} \}$ of $\Lambda$ dual to $\{G_{\lambda} \}$ via the Hall inner product was studied by Lam and Pylyavskyy [@lp]. They gave its formula as the generating series $$g_{\lambda}(x_1, x_2, \ldots) = \sum_{T} \prod_{i \ge 1} x_i^{\# \{\text{columns of $T$ containing }i\} },$$ where the sum runs over [*reverse plane partitions*]{}, i.e. entries weakly increase in rows and columns, of shape $\lambda$. This basis of [*dual stable Grothendieck polynomials*]{} agrees with the coproduct $\Delta : \Gamma \to \Gamma \otimes \Gamma$ of $G_{\lambda}$ and addresses K-homology of Grassmannians. The functions $G_{\lambda}, g_{\lambda}$ are not self-dual under the standard involution $\omega$. This map produces here other symmetric functions $J_{\lambda} = \omega(G_{\lambda}), j_{\lambda} = \omega(g_{\lambda})$. Combinatorially, $J_{\lambda}$ is described using [*weak set-valued tableaux*]{} (boxes may now contain multisets) and $j_{\lambda}$ using [*valued-set tableaux*]{} (which are SSYT with a special weighted decomposition) [@lp]. The canonical stable Grothendieck functions $G^{(\alpha, \beta)}_{\lambda}$ =========================================================================== Consider now the ring of symmetric functions $\Lambda$ and its completion $\hat\Lambda$ over $\mathbb{Z}[\alpha, \beta]$ for two generic parameters $\alpha, \beta$. Let $\hat\Lambda_{n}$ be a subring of $\hat\Lambda$ under specialization $x_{n+i} = 0$ for all $i \ge 1$, i.e. with finitely many variables $x_1, \ldots, x_n$. Let $\lambda$ be a partition. Define the [*canonical stable Grothendieck function*]{} $G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n)$ by the formula $$\label{eq1} G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n) = \frac{\displaystyle \det \left[\frac{x_i^{\lambda_j + n - j} (1 + \beta x_i)^{j - 1}}{(1-\alpha x_i)^{\lambda_j}} \right]_{1 \le i,j \le n} } {\displaystyle \det \left[{x_i^{n - j} } \right]_{1 \le i,j \le n} }.$$ Here $$\det \left[{x_i^{n - j} } \right]_{1 \le i,j \le n} = \prod_{1 \le i < j \le n} (x_i - x_j)$$ is the Vandermonde determinant. Since numerator and denominator of the expression above are both skew-symmetric, $G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n) \in \hat\Lambda_n$ is a well-defined symmetric power series in $x_1, \ldots, x_n$. Note that $G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n) = 0,$ if $\ell(\lambda) > n$. It is easy to see that stability property $\hat\Lambda_{n+1} \to \hat\Lambda_{n}$ holds: $$G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n, 0) = G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n).$$ Therefore we have an extended symmetric power series $G^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots) \in \hat\Lambda$ in infinitely many variables. From definition we obtain that $$G^{(\alpha, \beta)}_{\lambda} = s_{\lambda} + \text{higher degree terms},$$ the family $\{G^{(\alpha, \beta)}_{\lambda}\}$ forms a linearly independent set of elements of $\hat\Lambda$ and every element of $\hat\Lambda$ can uniquely be written as an infinite linear combination of these functions. We can compute (e.g. by induction on $n \to \infty$) that $$\begin{aligned} 1 + (\alpha + \beta)G_{(1)}^{(\alpha, \beta)}(x_1, x_2, \ldots) &= \prod_{i \ge 1}\frac{1+\beta x_i}{1 - \alpha x_i}, \label{ex1}\\ G_{(1)}^{(\alpha, \beta)}(x_1, x_2, \ldots) &= \sum_{i, j \ge 0} \alpha^i \beta^j s_{(i | j)},\end{aligned}$$ where $(i | j)$ denotes the hook shape partition $(i+1, 1^{j})$. To obtain both formulas from the example, one can e.g. prove by induction on $n \to \infty$ and using for $\lambda = (1,0,\ldots)$ that $$1 + (\alpha + \beta)G^{(\alpha, \beta)}_{(1)}(x_1, \ldots, x_{n}, x) = \frac{1 + \beta x}{1 - \alpha x} \left( 1 + (\alpha + \beta)G^{(\alpha, \beta)}_{(1)}(x_1, \ldots, x_{n}) \right).$$ As we see, the function $G^{(\alpha, \beta)}_{\lambda}$ is a certain deformation of Schur and stable Grothendieck polynomials. In special cases it corresponds to - Schur polynomials $G^{(0, 0)}_{\lambda} = s_{\lambda}$; - stable Grothendieck polynomials $G^{(0, -1)}_{\lambda} = G_{\lambda}$, $G^{(0, \beta)}_{\lambda} = G^{\beta}_{\lambda}$. Pivotal properties of this function are the following. \[omega1\] The functions $G^{(\alpha, \beta)}_{\lambda}$ satisfy: - self-duality $\omega(G^{(\alpha, \beta)}_{\lambda}) = G^{(\beta, \alpha)}_{\lambda'}$ - product has the finite decomposition $$G^{(\alpha, \beta)}_{\lambda} G^{(\alpha, \beta)}_{\mu} = \sum_{\nu} (\alpha + \beta)^{|\nu| - |\lambda| - |\mu|} c^{\nu}_{\lambda, \mu} G^{(\alpha, \beta)}_{\nu}.$$ The duality (i) will be proved later (in Section \[dfg\]) using the method of Fomin and Greene [@fg] on noncommutative Schur functions. Specializing $\alpha + \beta = -1$ in (ii), the multiplicative structure constants of $G^{(\alpha, \beta)}_{\lambda}$ coincide with those of $G^{}_{\lambda}$. This property (ii) will be clear from Proposition \[prop1\] below. Basic properties of $G^{(\alpha, \beta)}_{\lambda}$ --------------------------------------------------- \[prop1\] The following formulas hold $$\begin{aligned} G^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots) &= G^{(0,\alpha+ \beta)}_{\lambda} \left( \frac{x_1}{1 - \alpha x_1}, \frac{x_2}{1 - \alpha x_2}, \ldots \right),\label{e1} \\ G^{(\alpha, \beta)}_{\lambda}\left( \frac{x_1}{1 - \beta x_1}, \frac{x_2}{1 - \beta x_2}, \ldots \right) &= G^{(\alpha+ \beta, 0)}_{\lambda}(x_1, x_2, \ldots). \label{e2}\end{aligned}$$ It is straightforward to verify these identities for a finite variable functions by the determinantal formula , and then extend it to infinitely many variables. Note that as a relation between the stable Grothendieck polynomials $G_{\lambda}$ and the weak stable Grothendieck polynomials $J_{\lambda}$, similar formulas were given in [@pp]. For $\alpha + \beta = 0$ we have $$G^{(\alpha, -\alpha)}_{\lambda}(x_1, x_2, \ldots) = s_{\lambda} \left( \frac{x_1}{1 - \alpha x_1}, \frac{x_2}{1 - \alpha x_2}, \ldots \right),$$ which multiply by the usual Littlewood-Richardson rule. Consider a special case $(\alpha, \beta) \to (\beta/2, \beta/2).$ We know that the structure constants $c^{\nu}_{\lambda \mu}$ satisfy the symmetry $c^{\nu}_{\lambda \mu} = c^{\nu'}_{\lambda' \mu'}$, and thus the linear map $\tau : \hat\Lambda \to \hat\Lambda$ given by $\tau(G^{\beta}_{\lambda}) = G^{\beta}_{\lambda'}$ is a ring homomorphism. The [function]{} $\widetilde{G}^\beta_{\lambda} := G^{(\beta/2, \beta/2)}_{\lambda}$ is self-dual under the standard involution, $\omega(\widetilde{G}^\beta_{\lambda}) = \widetilde{G}^\beta_{\lambda'}.$ Hence combining this with Proposition \[prop1\], the map $\phi : \hat\Lambda \to \hat\Lambda$ which sends $G^{\beta}_{\lambda} \mapsto \widetilde{G}^\beta_{\lambda}$ is an automorphism given explicitly by the substitution of variables $x_i \mapsto \frac{x_i}{1 - \frac{\beta}{2} x_i}$ and we have $\omega = \phi \tau \phi^{-1}$, thus confirming Theorem \[t1\] for $\beta = -1$. The symmetry of $c^{\nu}_{\lambda \mu}$ also implies that there is another automorphism which maps $G^{(\alpha, \beta)}_{\lambda} \mapsto G^{(\alpha, \beta)}_{\lambda'}$, and it coincides with the standard involution $\omega$ for $\alpha = \beta$. The elements $G^{(\alpha, \beta)}_{(k)}, G^{(\alpha, \beta)}_{(1^k)}$ --------------------------------------------------------------------- Define the generating series $$\begin{aligned} H^{(\alpha, \beta)}(x;t) &:= 1 + (\alpha + \beta + t) \sum_{k \ge 1} G^{(\alpha, \beta)}_{(k)} t^{k - 1},\\ E^{(\alpha, \beta)}(x;t) &:= 1 + (\alpha + \beta + t) \sum_{k \ge 1} G^{(\alpha, \beta)}_{(1^k)} t^{k - 1}.\end{aligned}$$ We have $$\begin{aligned} H^{(\alpha, \beta)}(x;t) &= \prod_{j \ge 1} \frac{1 + \beta x_j}{1 - (\alpha + t) x_j}, \qquad E^{(\alpha, \beta)}(x;t) =\prod_{j \ge 1} \frac{1 + (\beta + t) x_j}{1 - \alpha x_j}.\end{aligned}$$ It is known that (e.g. [@lenart]) $$\begin{aligned} G^\beta_{(k)} = \sum_{i \ge 0} \beta^i s_{(k-1 | i)}, \qquad G^\beta_{(1^k)} = \sum_{i \ge 0} \beta^i \binom{i + k - 1}{i} e_{i + k}.\end{aligned}$$ The needed identities can be derived by standard manipulations and the substitutions $\beta \to \alpha + \beta$, $x_j \to \frac{x_j}{1 - \alpha x_j}$. We now give some variations on these series. First note that using equation we obtain $$\begin{aligned} \sum_{k \ge 1} G^{(\alpha, \beta)}_{(k)} t^{k - 1} &= G^{(\alpha + t, \beta)}_{(1)},\qquad \sum_{k \ge 1} G^{(\alpha, \beta)}_{(1^k)} t^{k - 1} = G^{(\alpha, \beta + t)}_{(1)}.\end{aligned}$$ Let us define $$\begin{aligned} h^{(\alpha, \beta)}_0 &:= 1 + (\alpha + \beta) G^{(\alpha, \beta)}_{(1)}, \quad h^{(\alpha, \beta)}_k := G^{(\alpha, \beta)}_{(k)} + (\alpha + \beta) G^{(\alpha, \beta)}_{(k+1)}, \quad k > 0,\label{a13}\\ e^{(\alpha, \beta)}_0 &:= 1 + (\alpha + \beta) G^{(\alpha, \beta)}_{(1)}, \quad e^{(\alpha, \beta)}_k := G^{(\alpha, \beta)}_{(1^k)} + (\alpha + \beta) G^{(\alpha, \beta)}_{(1^{k+1})}, \quad k > 0,\label{b13}\end{aligned}$$ or in other words, $$\begin{aligned} H^{(\alpha, \beta)}(x;t) &= \sum_{k \ge 0} h^{(\alpha, \beta)}_k t^k,\qquad E^{(\alpha, \beta)}(x;t) = \sum_{k \ge 0} e^{(\alpha, \beta)}_k t^k.\end{aligned}$$ It is easy to show that $$\begin{aligned} \label{14} {h^{(\alpha, \beta)}_{k}}/{h^{(\alpha, \beta)}_{0}} = h_k\left(\frac{x}{1 - \alpha x} \right), \qquad {e^{(\alpha, \beta)}_{k}}/{e^{(\alpha, \beta)}_{0}} = e_k\left(\frac{x}{1 + \beta x} \right).\end{aligned}$$ We have $$E^{(\alpha, \beta)}(t) H^{(-\beta, -\alpha)}(-t) = 1,$$ which implies the following relation between the elements $e^{(\alpha, \beta)}_{k}, h^{(-\beta, -\alpha)}_{\ell}$: $$\begin{aligned} \sum_{i} (-1)^i e^{(\alpha, \beta)}_{i} h^{(-\beta, -\alpha)}_{n - i} = \delta_{n,0}.\end{aligned}$$ In particular, $$\begin{aligned} e^{(\alpha, \beta)}_{0} h^{(-\beta, -\alpha)}_{0} = \left(1 + (\alpha + \beta) G^{(\alpha, \beta)}_{(1)} \right) \left(1 - (\alpha + \beta) G^{(-\beta, -\alpha)}_{(1)} \right) = 1.\end{aligned}$$ Each of the following four families is algebraically independent:\ $\{G^{(\alpha, \beta)}_{(k)} | k \in \mathbb{Z}_{> 0}\}$, $\{G^{(\alpha, \beta)}_{(1^k)} | k \in \mathbb{Z}_{> 0} \}$, and for $\alpha + \beta \ne 0$, $\{h^{(\alpha, \beta)}_{k} | k \in \mathbb{Z}_{\ge 0}\}$, $\{e^{(\alpha, \beta)}_{k} | k \in \mathbb{Z}_{\ge 0}\}$. If there is a relation between the elements $h^{(\alpha, \beta)}_{k}$ (or $e^{(\alpha, \beta)}_{k}$), then by there is a relation between the elements $h_i$ (or $e_i$) which is not true. Similarly, if there is a relation between $G^{(\alpha, \beta)}_{(k)}$ (or $G^{(\alpha, \beta)}_{(1^k)}$), then by definitions , there is a relation between $h^{(\alpha, \beta)}_{i}$ (or $e^{(\alpha, \beta)}_{i}$). Hook-valued tableaux {#hvt} ==================== In this section we describe the functions $G^{(\alpha, \beta)}_{\lambda}$ combinatorially using [*hook-valued tableaux*]{}. A Young diagram is called a [*hook*]{} if it has the form $(a | b) = (a+1, 1^b)$ for some $a, b \ge 0$. In this case, $a$ is called the [*arm*]{} of the hook and $b$ is called the [*leg*]{} of the hook. We now present a generalization of SSYT where boxes contain tableaux of hook shapes. Let $\max(T)$ (resp. $\min(T)$) of a tableau $T$ be the maximal (resp. minimal) number contained in $T$. For two arbitrary tableaux $T_1, T_2$ define the relations $T_1 \le T_2$ (resp. $T_1 < T_2$) if $\max(T_1) \le \min(T_2)$ (resp. $\max(T_1) < \min(T_2)$). So with these orders we can say that a sequence of nonempty tableaux is (weakly) increasing. A [*hook-valued tableau*]{} of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with SSYTs (instead of numbers) satisfying the following properties: - each box contains one SSYT of hook shape; - the hooks inside boxes weakly increase from left to right in rows and strictly increase from top to bottom in columns (with the orders defined above). An example of such tableau is given in Figure \[fig1\]. Let $a(T)$ and $b(T)$ be the sums of all arms and legs, respectively, of hooks in $T$. The [*weight*]{} of $T$ is then defined as $w_T(\alpha, \beta) = \alpha^{a(T)} \beta^{b(T)}$ and the monomial $x^T = \prod_i x_i^{a_i}$ where $a_i$ is the total number of occurrences of $i$ in $T$. Let $HT(\lambda)$ be the set of hook-valued tableaux of shape $\lambda$. This setting generalizes (and combines) the notions of set-valued tableaux [@buch] (set is a single column hook) and weak set-valued tableaux [@lp] (multiset is a single row hook), see Figure \[fig1x\]. \[hook\]The following formula holds $$G^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots) = \sum_{T \in HT(\lambda)} w_T(\alpha, \beta) x^{T}.$$ We use set-valued tableaux formula for $G^{\alpha + \beta}_{\lambda} = G^{(0, \alpha + \beta)}_{\lambda}$ and via the relation $$G^{(0, \alpha + \beta)}_{\lambda}\left(\frac{x_1}{1 - \alpha x_1}, \ldots \right) = G_{\lambda}^{(\alpha, \beta)}(x_1, \ldots)$$ from Proposition \[prop1\] obtain hook-valued interpretation. We have $$G^{(0, \alpha + \beta)}_{\lambda}\left(\frac{x_1}{1 - \alpha x_1}, \ldots \right) = \sum_{T \in SVT(\lambda)} (\alpha + \beta)^{|T| - |\lambda|} \left(\frac{x}{1 - \alpha x}\right)^T,$$ where $SVT(\lambda)$ is the set of set-valued tableaux of shape $\lambda$. Let $T \in SVT(\lambda)$. Imagine a set in a box of $T$ as a single column, where any element starting from the second row has the weight $(\alpha + \beta)$. Each element $i \in T$ has an expanded contribution $x_i/(1 - \alpha x_i) = x_i + \alpha x_i^2 + \alpha^2 x_i^3 + \cdots$ to the function and we may rewrite the last sum as $$G^{(\alpha, \beta)}_{\lambda} = G^{(0, \alpha + \beta)}_{\lambda}\left(\frac{x_1}{1 - \alpha x_1}, \ldots \right) = \sum_{T \in MSVT(\lambda)} (\alpha + \beta)^{|T| - |\lambda|} w_T(\alpha) x^T,$$ where $MSVT(\lambda)$ is the set of [*multiset-valued tableaux*]{} (sets are now replaced by multisets) and $w_T(\alpha) = \alpha^{a_T}$ where $a_T$ is the total number of [*extra*]{} copies of elements in all boxes of $T$. We now establish a weight-preserving bijection with the hook-valued tableaux. Suppose we have a multiset-valued tableau with weights $(\alpha + \beta)$ for each (non-first) distinct elements in its box and each element $i$ has $a_i$ copies contributing the weight $\alpha^{a_i - 1}$. To create a hook, for each element $i$ we do the following: - if $i$ is the first element in its box, put copies of $i$ in the first row (with weights $\alpha$); - otherwise there are two options: - put $i$ in the first column with the weight $\beta$ [and]{} put its extra copies to the first row with weights $\alpha$; [or]{} - put $i$ with its copies in the first row with weights $\alpha$. It is easy to see that the procedure is reversible. Notice that the hooks inside the boxes created by the rules (i), (ii) (a), (b) weakly increase in rows and strictly increase in columns, and hence we obtain a hook-valued tableaux. By this combinatorial formula we can extend the Grothendieck functions to any skew-shape. However the way we define $G^{(\alpha, \beta)}_{\lambda/\mu}$ in the next section is based on noncommutative operators and it differs from this combinatorial formula by having different boundary conditions, we allow to put tableau elements in a boundary of $\mu$ outside of the skew shape $\lambda/\mu$. Duality for $G^{(\alpha, \beta)}_{\lambda}$ {#dfg} =========================================== In this section we prove the duality $\omega(G^{(\alpha, \beta)}_{\lambda}) = G^{(\beta, \alpha)}_{\lambda'}.$ We describe noncommutative operators for canonical stable Grothendieck functions based on Schur operators. With this approach we then define the functions $G^{(\alpha, \beta)}_{\lambda/\mu}$ indexed by skew shapes. Noncommutative Schur functions ------------------------------ We use the theory of noncommutative Schur functions developed by Fomin and Greene [@fg]. Refer to [@bf] for a more general context and review of the method. For a given partition $\mu$ consider the free $\mathbb{Z}[\alpha, \beta]$-module $\mathbb{Z}_\mu = \bigoplus_{\mu \subset \lambda} \mathbb{Z}[\alpha, \beta] \cdot \lambda$ of all partitions that contain $\mu$. Given a set $u = (u_1, \ldots, u_N)$ of linear operators $u_i : \mathbb{Z}_{\mu} \to \mathbb{Z}_{\mu}$, consider the (noncommutative) ring $K\langle u_1, \ldots, u_N\rangle$ (over $K=\mathbb{Z}[\alpha, \beta]$ or $\mathbb{Z}$) generated by the variables $u$. The elementary symmetric functions $e_k(u)$ and the complete homogeneous symmetric functions $h_k(u)$ ($k \ge 0$) on $u$ are defined as follows $$e_k(u) := \sum_{N \ge i_1 > \ldots > i_k \ge 1} u_{i_1} \ldots u_{i_k}, \qquad h_k(u) := \sum_{1 \le i_1 \le \ldots \le i_k \le N} u_{i_1} \ldots u_{i_k}.$$ Then the noncommutative Schur functions $s_{\lambda}(u)$ can be defined via the Jacobi-Trudi identity[^2] $s_{\lambda} = \det[e_{\lambda'_i - i + j}]$, $$s_{\lambda}(u) := \sum_{\sigma\in S_{\ell = \ell(\lambda')}} {\mathrm{sgn}}(\sigma) e_{\lambda'_1 + \sigma(1) - 1}(u) \ldots e_{\lambda'_\ell + \sigma(\ell) - \ell}(u).$$ Denote $[a,b] = ab - ba$ the commutator. Let now $u = (u_1, \ldots, u_N )$ be a set of linear operators $u_i : \mathbb{Z}_{\mu} \to \mathbb{Z}_{\mu}$ satisfying the following commutation relations[^3]: $$\label{com1} [u_j, u_i] = 0, \quad |i- j| \ge 2;$$ $$\label{comx} [u_{i+1} u_i, u_i + u_{i+1}] = 0.$$ If these relations are satisfied, it is known that the noncommutative Schur function $s_{\lambda}(u)$ behaves like the usual Schur function [@fg]. In particular, the following properties hold. The noncommutative versions of symmetric functions commute: $$[e_{i}(u), e_{j}(u)] = [h_{i}(u), h_j(u)] = [s_{\lambda}(u), s_{\mu}(u)] = 0, \quad \forall\ i, j, \lambda, \mu$$ as well as the series $A(x), B(x)$ defined (for a single variable $x$ commuting with the $u$) by $$A(x) := \cdots (1 + x u_2) (1 + x u_1), \quad B(x) := 1/(1 - x u_1) 1/(1 - x u_2) \cdots,$$ $$[A(x), A(y)] = 0, \quad [B(x), B(y)] = 0, \quad [A(x), B(y)] = 0,$$ and the noncommutative analogs of the Cauchy identities hold: $$\begin{aligned} \label{cauchy} \cdots A(x_2)A(x_1) &= \sum_{\lambda} s_{\lambda'}(x_1, x_2, \ldots) s_{\lambda}(u), \\ \cdots B(x_2)B(x_1) &= \sum_{\lambda} s_{\lambda}(x_1, x_2, \ldots) s_{\lambda}(u).\end{aligned}$$ Schur operators --------------- \[schuro\] Define the linear operators $u_i, d_i : \mathbb{Z}_{\mu} \to \mathbb{Z}_{\mu}$, $i \in \mathbb{Z}_{>0}$ which act on bases as follows: $$u_i \cdot \lambda = \begin{cases} \lambda \cup \text{box in $i$th column}, & \text{ if possible,}\\ 0, & \text{ otherwise}; \end{cases}$$ $$d_i \cdot \lambda = \begin{cases} \lambda - \text{box in $i$th column}, & \text{ if possible,}\\ 0, & \text{ otherwise.} \end{cases}$$ It is known [@fomin; @fg] that both operators $u, d$ satisfy the relations ; the following [*local Knuth*]{} relations (which sum to ) $$\begin{aligned} \label{com2} u_{i+1} u_i u_i = u_i u_{i+1} u_{i}, \quad u_{i+1} u_i u_{i+1} = u_{i+1} u_{i+1} u_i;\end{aligned}$$ and the [duality]{} (or conjugate) relations [@fomin] $$\begin{aligned} \label{1} [d_j, u_i] = 0\ (i\ne j), \quad d_{i+1} u_{i+1} = u_i d_i\ (i \in \mathbb{Z}_{>0}), \quad d_1 u_1 = 1.\end{aligned}$$ These operators build Schur functions by its tableau interpretation (e.g., [@fg Example 2.4]). Note that for each $i \in \mathbb{Z}_{>0},$ the operator $u_i d_i$ simply gives $1$ (identity) if the box on $i$th column is removable, and $0$ otherwise. The elements $\{u_i d_i\}$ commute, it is easy to see that $$\label{c1} [u_i d_i, u_{i-1}d_{i-1}] = [d_{i+1}u_{i+1}, u_{i-1} d_{i-1}]= 0.$$ Moreover, we also have $$\label{c2} [u_id_i, d_iu_i] = [u_id_i, u_{i-1}d_{i-1}] = 0.$$ The relations , follow from . We use one more type of relations (that can easily be checked on bases) $$\label{c3} [u_i d_i, u_{i+1}u_i] = 0.$$ Operators for $G^{(\alpha, \beta)}_{\lambda}$ --------------------------------------------- Consider now the set $u^{(\alpha, \beta)}$ of linear operators defined (on the same spaces) using Schur operators: $$u^{(\alpha, \beta)}_i := u_i (1 + (\alpha + \beta) d_i) - \alpha = u_i + (\alpha + \beta)u_i d_i - \alpha, \quad i \in \mathbb{Z}_{>0}$$ To see effect of these operators, consider the case $\alpha = 0,$ i.e. $u^{(0, \beta)}_i = u_i (1 + \beta d_i) = u_i + \beta u_i d_i$ which means the following for a diagram. If possible, it adds a single box on $i$th column; [or]{} just multiplies by a scalar parameter $\beta$ if the box on $i$th column is removable (without removing it). This procedure allows to construct set-valued tableaux with a parameter $\beta$. As we will show, these deformations of Schur operators build the functions $G^{(\alpha, \beta)}_{\lambda/\mu}$. Using the relations – for the operators $u,d$, the following statement is a straightforward check. The operators $u^{(\alpha, \beta)}$ satisfy the relations , . Define the series (the set $u^{(\alpha, \beta)}$ is finite) $$\begin{aligned} C(x) &:= \cdots \left(\frac{1 + x u^{(\alpha, \beta)}_2}{1 - \alpha x} \right) \left(\frac{1 + x u^{(\alpha, \beta)}_1}{1 - \alpha x} \right), \quad D(x) := \left(\frac{1 + \alpha x}{1 - x u^{(\alpha, \beta)}_1} \right) \left(\frac{1 + \alpha x}{1 - x u^{(\alpha, \beta)}_2} \right) \cdots \end{aligned}$$ Then from the Lemma above we obtain $$[C(x), C(y)] = 0, \quad [D(x), D(y)] = 0.$$ Let $\langle\cdot, \cdot \rangle: \mathbb{Z}_\mu \times \mathbb{Z}_\mu \to \mathbb{Z}[\alpha, \beta]$ be a (non-degenerate) $\mathbb{Z}[\alpha, \beta]$-bilinear pairing defined on bases by $\langle \lambda, \nu \rangle = \delta_{\lambda, \nu}$. We have $$\langle \cdots C(x_2) C(x_1) \cdot \varnothing, \lambda \rangle = G^{(\alpha, \beta)}_{\lambda}, \quad \langle \cdots D(x_2) D(x_1) \cdot \varnothing, \lambda \rangle = G^{(\beta, \alpha)}_{\lambda'}.$$ We can rewrite $$\begin{aligned} C(x) &= \overleftarrow{\prod_{i \ge 1}} \left(1 + \frac{x}{1 - \alpha x} u_i(1 + (\alpha + \beta) d_i) \right) = \overleftarrow{\prod_{i \ge 1}} \left(1 + (x + \alpha x^2 + \cdots) u_i(1 + (\alpha + \beta) d_i) \right).\end{aligned}$$ For each particular $x_k$ taken from the product, the term $\alpha^\ell x_k^{\ell+1} (u_i+(\alpha + \beta)u_id_i)$ means the following procedure of building the hook-valued tableau: - we add a new box on $i$th column (if possible) and put $\ell + 1$ copies of $k$ in a row inside this box (each copy except the first has weight $\alpha$); - or if the last box in $i$th column is removable, then (applying $u_id_i$ means that the shape does not change) add $\ell$ copies of $k$ in a row of the hook inside this box and then add the remaining one copy of $k$ to either first row (with weight $\alpha$) or first column (with weight $\beta$). On can see from this procedure and order of operators, that we have inequalities exactly as in hook-valued tableaux: hooks inside the boxes weakly increase in rows and strictly increase in columns. Therefore, applying the operator series until we obtain $\lambda$ gives the symmetric function $G^{(\alpha, \beta)}_{\lambda}$, or $\langle \cdots C(x_2) C(x_1) \cdot \varnothing, \lambda \rangle = G^{(\alpha, \beta)}_{\lambda}$. For the second equality involving the $D$ series, we rewrite it as follows $$D(x) = \overrightarrow{\prod_{i \ge 1}} \left(\frac{1}{1 - \frac{x}{1 + \alpha x} u_i(1 + (\alpha + \beta) d_i)} \right).$$ Using a similar reasoning it is not hard to see that for the series given by $$\widetilde{D}(t) := \overrightarrow{\prod_{i \ge 1}} \left(\frac{1}{1 - t u_i(1 + (\alpha + \beta) d_i)} \right)$$ we have $$\langle \cdots \widetilde{D}(t_2) \widetilde{D}(t_1) \cdot \varnothing, \lambda \rangle = G_{\lambda'}^{(\alpha + \beta, 0)}(t_1, t_2, \ldots ).$$ Finally note that for $t_i = x_i/(1 + \alpha x_i)$ by Proposition we have $$G_{\lambda'}^{(\alpha + \beta, 0)}(t_1, t_2, \ldots ) = G^{(\beta, \alpha)}_{\lambda'}(x_1, x_2, \ldots).$$ We have $\omega(G^{(\alpha, \beta)}_{\lambda}) = G^{(\beta, \alpha)}_{\lambda'}.$ Let $N = \# u^{(\alpha, \beta)}$. From the previous Theorem applying the noncommutative Cauchy identities to the series $C, D$ we have $$\begin{aligned} \omega(G^{(\alpha, \beta)}_{\lambda}) &= \omega(\langle \cdots C(x_2) C(x_1) \cdot \varnothing, \lambda \rangle)\\ &= \omega \left\langle \prod_i {(1 - \alpha x_i)^{-N}} \sum_{\nu}s_{\nu'}(x) s_{\nu}(u^{(\alpha, \beta)}) \cdot\varnothing,\lambda\right\rangle\\ &= \sum_{\nu} \omega(\prod_i {(1 - \alpha x_i)^{-N}} s_{\nu'}(x)) \left\langle s_{\nu}(u^{(\alpha, \beta)}) \cdot\varnothing,\lambda\right\rangle\\ &= \sum_{\nu} \prod_i {(1 + \alpha x_i)^N} s_{\nu}(x) \left\langle s_{\nu}(u^{(\alpha, \beta)}) \cdot\varnothing,\lambda\right\rangle\\ &= \left\langle \prod_i {(1 + \alpha x_i)^N} \sum_{\nu} s_{\nu}(x) s_{\nu}(u^{(\alpha, \beta)}) \cdot\varnothing,\lambda\right\rangle\\ &= \langle \cdots D(x_2) D(x_1) \cdot \varnothing, \lambda \rangle\\ &= G^{(\beta, \alpha)}_{\lambda'}.\end{aligned}$$ Skew shapes ----------- For skew shapes, we can define the symmetric functions $G^{(\alpha, \beta)}_{\lambda/\mu}$ as $$\label{gskew} G^{(\alpha, \beta)}_{\lambda/\mu} := \langle \cdots C(x_2) C(x_1) \cdot \mu, \lambda \rangle,$$ for which we may similarly obtain that $$G^{(\beta, \alpha)}_{\lambda'/\mu'} = \langle \cdots D(x_2) D(x_1) \cdot \mu, \lambda \rangle$$ and hence the duality $\omega(G^{(\alpha, \beta)}_{\lambda/\mu}) = G^{(\beta, \alpha)}_{\lambda'/\mu'}.$ This definition of $G^{(\alpha, \beta)}_{\lambda/\mu}$ for skew shapes does not match exactly the combinatorial definition (i.e. if we extend directly hook-valued tableaux for skew shapes) as it has different boundary conditions. For example, for a single variable $t$ $$\begin{aligned} G^{(\alpha, \beta)}_{(2)/(1)}(t) &=\langle \overleftarrow{\prod_{i \ge 1}} \left(1 + \frac{t}{1 - \alpha t} u_i(1 + (\alpha + \beta) d_i) \right) \cdot (1), (2)\rangle \\ &=\left(1 + \frac{(\alpha + \beta) t}{1 - \alpha t} \right) \frac{t}{1 - \alpha t} \\ &= \frac{1 + \beta t}{1 - \alpha t} \frac{t}{1 - \alpha t},\end{aligned}$$ (the first factor comes from applying $u_1 d_1$ on the shape $(1)$) whereas it should be just $ \frac{t}{1 - \alpha t}$ if we compute hook-valued tableaux in the skew shape $(2)/(1)$. For $(\alpha, \beta) = (0, -1)$ it corresponds to the function denoted as $G_{\lambda /\!\!/ \mu}$ in [@buch]. Remarks ------- For $\alpha = \beta = 0$, the operators ${u^{(\alpha, \beta)}}$ reduce to the usual Schur operators, which build Schur functions. For $\alpha + \beta = 0$, we obtain the deformations of Schur functions $s_{\lambda}(x/(1- \alpha x))$ and $\omega s_{\lambda}(x/(1- \alpha x)) = s_{\lambda'}(x/(1 + \alpha x)).$ The case $(\alpha, \beta) = (0, \pm 1)$ or $(\pm 1,0)$ corresponds to the usual stable Grothendieck polynomials $G_{\lambda}$ and its image under $\omega$, the weak stable Grothendieck polynomial $J_{\lambda}$ [@lp]. The functions $G^{(\alpha, \beta)}_{\lambda}$ can also be built using another types of operators $v_i$ which add boxes by diagonals, an approach used in [@buch; @lp] for $G_{\lambda}$. Let us recall these operators. The box $(i,j)$ of the Young diagram of $\lambda$ lies on the $(j-i)$th diagonal. Say that $\lambda$ has an [*inner corner*]{} on $i$th diagonal if there is a partition $\mu$ so that $\lambda/\mu$ is a single box on $i$th diagonal. Similarly, $\lambda$ has an [*outer corner*]{} on $i$th diagonal if there is a partition $\mu$ so that $\mu/\lambda$ is a single box on the same diagonal. Define the linear operators $v_{i} : \mathbb{Z}_\mu \to \mathbb{Z}_\mu$ ($i\in \mathbb{Z}$) as follows $$v_i \cdot \lambda = \begin{cases} \nu, & \text{ if } \lambda \text{ has an outer corner } \nu/\lambda \text{ on } i\text{th diagonal;}\\ \lambda, & \text{ if } \lambda \text{ has an inner corner not contained in } \mu \text{ on $i$th diagonal;}\\ 0, & \text{ otherwise}. \end{cases}$$ These operators $v_i$ satisfy the following relations: $ v_i^2 = v_i,$ $v_{i} v_{i+1}v_{i} = v_{i+1} v_{i} v_{i+1},$ $v_{i} v_{j} = v_{j} v_{i}, |i-j| > 1, i,j \in \mathbb{Z}$ [@lp; @buch]. To obtain similar properties for the functions $G^{(\alpha, \beta)}_{\lambda/\mu}$ one could play with the series $$E^{}(x) := \cdots \left(\frac{1+\beta x v_1}{1 - \alpha x v_1}\right) \left(\frac{1+\beta x v_0}{1 - \alpha x v_0}\right) \left(\frac{1+\beta x v_{-1}}{1 - \alpha x v_{-1}}\right) \cdots$$ Let $v = v_i$ for any $i$. From the fact that $v^2 = v$ it is easy to see that $$\label{sw} \frac{1 + \beta x v}{1 - \alpha x v} = \frac{1 + ((\alpha + \beta)v - \alpha)x}{1 - \alpha x}.$$ Hence we apply the transformation $v_{i} \to v_i' = (\alpha + \beta) v_{i} - \alpha$ and use the theory of noncommutative Schur functions. Note that the operators $v'_i$ satisfy the properties of the (generalized) Hecke algebra: $v'^2_i = (\beta - \alpha)v'_i + \alpha \beta,$ $v'_i v'_{i+1} v'_i = v'_{i+1} v'_{i} v'_{i+1}$, and $v'_{i} v'_{j} = v'_{j} v'_{i}$ for $|i - j| > 1$. So Grothendieck functions, like Schur functions, can be build using both types of operators. In fact, similarly as we used Schur operators, the operators $v_i$ can be constructed via diagonal Schur operators as follows: $v_i = \bar v_i(1 + \bar d_i)$ where $\bar v_i$ adds a single box on $i$th diagonal if possible and returns $0$ otherwise; $\bar d_i$ removes a single box on $i$th diagonal if possible and returns $0$ otherwise. It is not hard to prove the following result: Let $\phi_{\alpha} : \hat\Lambda \to \hat\Lambda$ be an automorphism given by $\phi_{\alpha} f(x) = f(x/(1 - \alpha x))$. Then, $\phi_{\alpha} \omega \phi_{\alpha} = \omega$ or equivalently $\omega \phi_{\alpha} =\phi_{\alpha}^{-1} \omega$. In other words, let $f,g \in \hat\Lambda$ be symmetric power series satisfying $\omega f(x) = g(x).$ Then, $\omega f({x}/{(1 - \alpha x)}) = g({x}/{(1 + \alpha x)}).$ The dual canonical Grothendieck polynomials $g^{(\alpha, \beta)}_{\lambda}$ =========================================================================== Recall that $$G^{(\alpha, \beta)}_{\lambda} = s_{\lambda} + \text{ higher degree terms}.$$ We now want to introduce the dual family for $\{G^{(\alpha, \beta)}_{\lambda} \}$ via the Hall inner product $\langle \cdot, \cdot \rangle : \Lambda \times \hat\Lambda \to \mathbb{Z}[\alpha, \beta]$ defined on Schur basis by $\langle s_{\lambda}, s_{\mu} \rangle = \delta_{\lambda, \mu}$. The [*dual canonical stable Grothendieck polynomials*]{} $g^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots)$ is the dual basis for $G^{(-\alpha, -\beta)}_{\lambda}$ via the Hall inner product, $\langle g^{(\alpha, \beta)}_{\lambda}, G^{(-\alpha, -\beta)}_{\mu} \rangle = \delta_{\lambda, \mu}$. It is a priori clear that $g^{(\alpha, \beta)}_{\lambda} \in \Lambda$ is a symmetric function satisfying the duality $\omega(g^{(\alpha, \beta)}_{\lambda}) = g^{(\beta, \alpha)}_{\lambda'}$. We have $$g^{(\alpha, \beta)}_{\lambda} = s_{\lambda} + \text{ lower degree terms}$$ and for a finite number of variables, $g^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n)$ is a non-homegeneous symmetric polynomial. Later we will see that the function $g^{(\alpha, \beta)}_{\lambda}$ has a nice combinatorial description and that its expansion in the Schur basis is positive (i.e. exchange constants are polynomials in $\alpha, \beta$ with positive integer coefficients). The polynomials $g^{(\alpha, \beta)}_{\lambda}$ combine the dual stable Grothendieck polynomials $g^{(0, 1)}_{\lambda} = g_{\lambda}$, $g^{(1, 0)}_{\lambda} = j_{\lambda'} = \omega(g_{\lambda'})$ given in [@lp]; note that $g^{(0,0)}_{\lambda} = s_{\lambda}$. Another equivalent description is via the Cauchy identity $$\sum_{\lambda} g^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots) G^{(-\alpha, -\beta)}_{\lambda}(y_1, y_2, \ldots) = \prod_{i,j} \frac{1}{1 - x_i y_j}.$$ Let $\Gamma^{(\alpha, \beta)} = \bigoplus_{\lambda} \mathbb{Z}[\alpha, \beta] \cdot G^{(\alpha, \beta)}_{\lambda}$. The polynomials $g^{(\alpha, \beta)}_{\lambda}$ have multiplicative structure constants as by the comultiplication $\Delta : \Gamma^{(-\alpha, -\beta)} \to \Gamma^{(-\alpha, -\beta)} \otimes \Gamma^{(-\alpha, -\beta)}$ given by $$\begin{aligned} \Delta(G^{(-\alpha, -\beta)}_{\nu}) &= \sum_{\lambda, \mu} (-\alpha - \beta)^{|\nu| - |\lambda| - |\mu|} d^{\nu}_{\lambda \mu} G^{(-\alpha, -\beta)}_{\lambda} \otimes G^{(-\alpha, -\beta)}_{\mu}.\end{aligned}$$ In the case $(\alpha, \beta) \to (\beta/2, \beta/2)$ we obtain the polynomials $\widetilde{g}^\beta_{\lambda} := g^{(\beta/2, \beta/2)}_{\lambda}$ which form a basis of $\Lambda$ satisfying $\omega(\widetilde{g}^{\beta}_{\lambda}) = \widetilde{g}^{\beta}_{\lambda'}$. The map $\bar\phi : \Lambda \to \Lambda$ which sends $\widetilde{g}^{\beta}_{\lambda}$ to $g^{\beta}_{\lambda}$, is a ring automorphism. If we define the linear map $\bar\tau : \Lambda \to \Lambda$ given by $\bar\tau({g}^{\beta}_{\lambda}) = {g}^{\beta}_{\lambda'}$, then by the symmetry $d^{\nu}_{\lambda \mu} = d^{\nu'}_{\lambda' \mu'}$, this map is a ring automorphism. Then we have $\omega = \bar\phi \bar\tau \bar \phi^{-1}$ and $\Lambda \cong \mathbb{Z}[\beta][\widetilde{g}^{\beta}_{(1)}, \widetilde{g}^{\beta}_{(2)}, \ldots]$ as a polynomial ring. Once we specify these generators $\widetilde{g}^{\beta}_{(k)}$ we can say that the automorphism $\bar\phi$ (and hence $\phi$ for $\widetilde{G}^{\beta}_{\lambda}$) is unique. Note also that another automorphism which sends $g^{(\alpha, \beta)}_{\lambda}$ to $g^{(\alpha, \beta)}_{\lambda'}$ coincides with the canonical involution $\omega$ for $\alpha = \beta$. Combinatorial formulas for $g^{(\alpha, \beta)}_{\lambda}$ and rim border tableaux {#rrt} ================================================================================== In this section we give combinatorial formulas for the dual canonical Grothendieck polynomials $g^{(\alpha, \beta)}_{\lambda}$. A [*reverse plane partition*]{} (RPP) is a filling of a Young diagram so that each box contains a single positive integer and numbers weakly increase in rows (from left to right) and columns (from top to bottom). A [*rim hook*]{} (or [*ribbon*]{}) is a connected skew shape which contains no $2 \times 2$ square. \[rrpp\] Let $T$ be an RPP. For each integer $i$ written in $T$ let $T_i$ be the (skew shape) part of $T$ containing all elements $i$. Define the [*border*]{} $R_i$ ($R_i \subset T_i$) consisting of all boxes $b$ of $T_i$ for which no box on the same diagonal above and to the left of $b$ is in $T_i$. For example, in Figure \[on\] (a), the borders $R_1, R_2, R_3$ are shadowed. Let $I_i = T_i \setminus R_i$ be the [*inner part*]{} of $T_i$. (0.5,1.5) rectangle (12.5,0.5); (0.5,1.5) rectangle (1.5,-4.5); (4.5,0.5) rectangle (7.5,-0.5); (8.5,0.5) rectangle (9.5,-0.5); (3.5,-0.5) rectangle (5.5,-1.5); (6.5,-0.5) rectangle (9.5,-1.5); (2.5,-1.5) rectangle (4.5,-2.5); (6.5,-1.5) rectangle (7.5,-2.5); (0.5,-2.5) rectangle (5.5,-3.5); (0.5,-3.5) rectangle (3.5,-4.5); \(1) at ( 1, 1) [1]{}; (2) at ( 2, 1) [1]{}; (3) at ( 3, 1) [1]{}; (4) at ( 4, 1) [1]{}; (5) at ( 5, 1) [1]{}; (6) at ( 6, 1) [1]{}; (7) at ( 7, 1) [2]{}; (8) at ( 8, 1) [2]{}; (9) at ( 9, 1) [2]{}; (10) at ( 10, 1) [2]{}; (11) at ( 11, 1) [3]{}; (12) at ( 12, 1) [3]{}; \(a) at (-0.5, 1) [(a)]{}; \(21) at ( 1, 0) [1]{}; (22) at ( 2, 0) [1]{}; (23) at ( 3, 0) [1]{}; (24) at ( 4, 0) [1]{}; (25) at ( 5, 0) [2]{}; (26) at ( 6, 0) [2]{}; (27) at ( 7, 0) [2]{}; (28) at ( 8, 0) [2]{}; (29) at ( 9, 0) [3]{}; \(31) at ( 1, -1) [1]{}; (32) at ( 2, -1) [1]{}; (33) at ( 3, -1) [1]{}; (34) at ( 4, -1) [2]{}; (35) at ( 5, -1) [2]{}; (36) at ( 6, -1) [2]{}; (37) at ( 7, -1) [3]{}; (38) at ( 8, -1) [3]{}; (39) at ( 9, -1) [3]{}; \(41) at ( 1, -2) [1]{}; (42) at ( 2, -2) [1]{}; (43) at ( 3, -2) [2]{}; (44) at ( 4, -2) [2]{}; (45) at ( 5, -2) [2]{}; (46) at ( 6, -2) [2]{}; (47) at ( 7, -2) [3]{}; (48) at ( 8, -2) [3]{}; \(51) at ( 1, -3) [2]{}; (52) at ( 2, -3) [2]{}; (53) at ( 3, -3) [3]{}; (54) at ( 4, -3) [3]{}; (55) at ( 5, -3) [3]{}; \(61) at ( 1, -4) [3]{}; (62) at ( 2, -4) [3]{}; (63) at ( 3, -4) [3]{}; (0.5,1.5) to (6.5,1.5); (6.5,1.5) to (6.5,0.5); (6.5,0.5) to (1.5,0.5); (1.5,0.5) to (1.5,-2.5); (1.5,-2.5) to (0.5,-2.5); (0.5,-2.5) to (0.5,1.5); (6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5,-1.5); (5.5,-1.5) to (5.5-1,-1.5); (5.5-1,-1.5) to (5.5-1,-1.5-1); (5.5-1,-1.5-1) to (5.5-1-2,-1.5-1); (5.5-1-2,-1.5-1) to (5.5-1-2,-1.5-1+1); (5.5-1-2,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1); (5.5-1-2+1,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1+1); (0.5,-2.5) to (0.5,-2.5-1) to (0.5+2,-2.5-1) to (0.5+2,-2.5-1+1) to (0.5+2-1,-2.5-1+1); (0.5, -3.5) to (0.5, -4.5) to (0.5+3, -4.5) to (0.5+3, -4.5+1) to (0.5+3+2, -4.5+1) to (0.5+3+2, -4.5+1+1) to (0.5+3+2-1, -4.5+1+1); (6.5, -0.5) to (6.5, -2.5) to (6.5+1, -2.5) to (6.5+1, -2.5+1) to (6.5+1+1, -2.5+1) to (6.5+1+1+1, -2.5+1) to (6.5+1+1+1, -2.5+1+2) ; (6.5+1+1+1-1, -2.5+1+2) to (6.5+1+1+1-1, -2.5+1+2-1) to (6.5+1+1+1-1-1, -2.5+1+2-1); (6.5, -2.5) to (6.5-1, -2.5); (8.5, -2.5+1) to (8.5, -2.5) to (8.5-1, -2.5); (10.5, 0.5) to (10.5+2, 0.5) to (10.5+2, 0.5+1) to (10.5+2-2, 0.5+1); \(7) at ( 7, 1) [2]{}; (8) at ( 8, 1) [2]{}; (9) at ( 9, 1) [2]{}; (10) at ( 10, 1) [2]{}; \(25) at ( 5, 0) [2]{}; (26) at ( 6, 0) [2]{}; (27) at ( 7, 0) [2]{}; (28) at ( 8, 0) ; \(34) at ( 4, -1) [2]{}; (35) at ( 5, -1) [2]{}; (36) at ( 6, -1) ; \(43) at ( 3, -2) [2]{}; (44) at ( 4, -2) [2]{}; (45) at ( 5, -2) ; (46) at ( 6, -2) ; \(51) at ( 1, -3) [2]{}; (52) at ( 2, -3) [2]{}; \(b) at (-0.5, 1) [(b)]{}; (bb) at (0.5, -4.33) ; (6.5,1.5) to (6.5,0.5); (6.5,0.5) to (4.5,0.5); (1.5,-2.5) to (0.5,-2.5); (6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5,-1.5); (5.5,-1.5) to (5.5-1,-1.5); (5.5-1,-1.5) to (5.5-1,-1.5-1); (5.5-1,-1.5-1) to (5.5-1-2,-1.5-1); (5.5-1-2,-1.5-1) to (5.5-1-2,-1.5-1+1); (5.5-1-2,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1); (5.5-1-2+1,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1+1); (0.5,-2.5) to (0.5,-2.5-1) to (0.5+2,-2.5-1) to (0.5+2,-2.5-1+1) to (0.5+2-1,-2.5-1+1); (8.5,1.5) to (8.5,0.5); (5.5,0.5) to (5.5,-0.5); (0.5,1.5) rectangle (12.5,0.5); (0.5,1.5) rectangle (1.5,-4.5); (4.5,0.5) rectangle (7.5,-0.5); (8.5,0.5) rectangle (9.5,-0.5); (3.5,-0.5) rectangle (5.5,-1.5); (6.5,-0.5) rectangle (9.5,-1.5); (2.5,-1.5) rectangle (4.5,-2.5); (6.5,-1.5) rectangle (7.5,-2.5); (0.5,-2.5) rectangle (5.5,-3.5); (0.5,-3.5) rectangle (3.5,-4.5); \(1) at ( 1, 1) [1]{}; (2) at ( 2, 1) [1]{}; (3) at ( 3, 1) [1]{}; (4) at ( 4, 1) [1]{}; (5) at ( 5, 1) [1]{}; (6) at ( 6, 1) [1]{}; (7) at ( 7, 1) [2]{}; (8) at ( 8, 1) [2]{}; (9) at ( 9, 1) [2]{}; (10) at ( 10, 1) [2]{}; (11) at ( 11, 1) [3]{}; (12) at ( 12, 1) [3]{}; \(c) at (-0.5, 1) [(c)]{}; \(21) at ( 1, 0) [1]{}; (22) at ( 2, 0) [1]{}; (23) at ( 3, 0) [1]{}; (24) at ( 4, 0) [1]{}; (25) at ( 5, 0) [2]{}; (26) at ( 6, 0) [2]{}; (27) at ( 7, 0) [2]{}; (28) at ( 8, 0) [2]{}; (29) at ( 9, 0) [3]{}; \(31) at ( 1, -1) [1]{}; (32) at ( 2, -1) [1]{}; (33) at ( 3, -1) [1]{}; (34) at ( 4, -1) [2]{}; (35) at ( 5, -1) [2]{}; (36) at ( 6, -1) [2]{}; (37) at ( 7, -1) [3]{}; (38) at ( 8, -1) [3]{}; (39) at ( 9, -1) [3]{}; \(41) at ( 1, -2) [1]{}; (42) at ( 2, -2) [1]{}; (43) at ( 3, -2) [2]{}; (44) at ( 4, -2) [2]{}; (45) at ( 5, -2) [2]{}; (46) at ( 6, -2) [2]{}; (47) at ( 7, -2) [3]{}; (48) at ( 8, -2) [3]{}; \(51) at ( 1, -3) [2]{}; (52) at ( 2, -3) [2]{}; (53) at ( 3, -3) [3]{}; (54) at ( 4, -3) [3]{}; (55) at ( 5, -3) [3]{}; \(61) at ( 1, -4) [3]{}; (62) at ( 2, -4) [3]{}; (63) at ( 3, -4) [3]{}; (0.5,1.5) to (6.5,1.5); (6.5,1.5) to (6.5,0.5); (6.5,0.5) to (1.5,0.5); (1.5,0.5) to (1.5,-2.5); (1.5,-2.5) to (0.5,-2.5); (0.5,-2.5) to (0.5,1.5); (3.5,1.5) to (3.5,0.5); (8.5,1.5) to (8.5,0.5); (11.5,1.5) to (11.5,0.5); (5.5,0.5) to (5.5,-0.5); (1.5,-3.5) to (1.5,-4.5); (7.5,-0.5) to (7.5,-1.5); (6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5,-1.5); (5.5,-1.5) to (5.5-1,-1.5); (5.5-1,-1.5) to (5.5-1,-1.5-1); (5.5-1,-1.5-1) to (5.5-1-2,-1.5-1); (5.5-1-2,-1.5-1) to (5.5-1-2,-1.5-1+1); (5.5-1-2,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1); (5.5-1-2+1,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1+1); (0.5,-2.5) to (0.5,-2.5-1) to (0.5+2,-2.5-1) to (0.5+2,-2.5-1+1) to (0.5+2-1,-2.5-1+1); (0.5, -3.5) to (0.5, -4.5) to (0.5+3, -4.5) to (0.5+3, -4.5+1) to (0.5+3+2, -4.5+1) to (0.5+3+2, -4.5+1+1) to (0.5+3+2-1, -4.5+1+1); (6.5, -0.5) to (6.5, -2.5) to (6.5+1, -2.5) to (6.5+1, -2.5+1) to (6.5+1+1, -2.5+1) to (6.5+1+1+1, -2.5+1) to (6.5+1+1+1, -2.5+1+2) ; (6.5+1+1+1-1, -2.5+1+2) to (6.5+1+1+1-1, -2.5+1+2-1) to (6.5+1+1+1-1-1, -2.5+1+2-1); (6.5, -2.5) to (6.5-1, -2.5); (8.5, -2.5+1) to (8.5, -2.5) to (8.5-1, -2.5); (10.5, 0.5) to (10.5+2, 0.5) to (10.5+2, 0.5+1) to (10.5+2-2, 0.5+1); The inner parts $I_1, I_2, I_3$ correspond to white parts of $T$ in Figure \[on\] (a). Let us then arbitrarily partition each border $R_i$ into rim hooks by some [*vertical*]{} cuts (see Figure \[on\] (b)). We call this resulting tableau a [*rim border tableau*]{} (RBT). Example of the resulting RBT is given in Figure \[on\] (c). Let $RBT(\lambda)$ be the set of all RBT of shape $\lambda$. For each element $T \in RBT(\lambda)$ define the $(\alpha, \beta)$-weight $w_T(\alpha, \beta) = \alpha^{wt} \beta^{ht} (\alpha + \beta)^{in}$, where $wt$ is the sum of [*width*]{}$-1$ of all rim hooks in $T$, $ht$ is the sum of [*height*]{}$-1$ of all rim hooks in $T$, and $in$ is the total number of boxes in inner parts of $T$. The corresponding monomial $x^T = \prod_{i} x_i^{a_i}$ is defined so that $a_i$ is the number of rim hooks in $T$ containing $i$. See Figure \[on\] (c) for an RBT $T$ with $w(T) = \alpha^{14} \beta^{9} (\alpha + \beta)^{11}$ and $x^T = x_1^2 x_2^4 x_3^6$. We now state that generating series for these rim border tableaux define combinatorial presentation of the dual polynomials $g^{(\alpha, \beta)}_{\lambda}$. \[cg\] The dual polynomials $g^{(\alpha, \beta)}_{\lambda}$ satisfy the following formula $$g^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots) = \sum_{T \in RBT(\lambda)} w_T(\alpha, \beta) x^T.$$ We will prove this theorem in next section after giving Pieri and branching formulas for the functions $G^{(\alpha, \beta)}_{\lambda}, g^{(\alpha, \beta)}_{\lambda}$. We will also show that the polynomials $g^{(\alpha, \beta)}_{\lambda}$ are Schur-positive (for $\alpha, \beta > 0$, see Table 1 with some examples). Recall that by definition $g^{(\alpha, \beta)}_{\lambda}(x_1, x_2, \ldots)$ is a symmetric function satisfying the duality $\omega(g^{(\alpha, \beta)}_{\lambda}) = g^{(\beta, \alpha)}_{\lambda'}$. \(1) at ( 1, 1) [1]{}; (2) at ( 2, 1) [1]{}; (3) at ( 3, 1) [1]{}; (4) at ( 4, 1) [1]{}; (5) at ( 5, 1) [1]{}; (6) at ( 6, 1) [1]{}; (7) at ( 7, 1) [2]{}; (8) at ( 8, 1) [2]{}; (9) at ( 9, 1) [2]{}; (10) at ( 10, 1) [2]{}; (11) at ( 11, 1) [3]{}; (12) at ( 12, 1) [3]{}; \(c) at (-0.5, 1) [(a)]{}; \(21) at ( 1, 0) [1]{}; (22) at ( 2, 0) [1]{}; (23) at ( 3, 0) [1]{}; (24) at ( 4, 0) [1]{}; (25) at ( 5, 0) [2]{}; (26) at ( 6, 0) [2]{}; (27) at ( 7, 0) [2]{}; (28) at ( 8, 0) [2]{}; (29) at ( 9, 0) [3]{}; \(31) at ( 1, -1) [1]{}; (32) at ( 2, -1) [1]{}; (33) at ( 3, -1) [1]{}; (34) at ( 4, -1) [2]{}; (35) at ( 5, -1) [2]{}; (36) at ( 6, -1) [2]{}; (37) at ( 7, -1) [3]{}; (38) at ( 8, -1) [3]{}; (39) at ( 9, -1) [3]{}; \(41) at ( 1, -2) [1]{}; (42) at ( 2, -2) [1]{}; (43) at ( 3, -2) [2]{}; (44) at ( 4, -2) [2]{}; (45) at ( 5, -2) [2]{}; (46) at ( 6, -2) [2]{}; (47) at ( 7, -2) [3]{}; (48) at ( 8, -2) [3]{}; \(51) at ( 1, -3) [2]{}; (52) at ( 2, -3) [2]{}; (53) at ( 3, -3) [3]{}; (54) at ( 4, -3) [3]{}; (55) at ( 5, -3) [3]{}; \(61) at ( 1, -4) [3]{}; (62) at ( 2, -4) [3]{}; (63) at ( 3, -4) [3]{}; (0.5, 1.5) to (0.5, -4.5) to (3.5, -4.5) to (3.5, -3.5) to (5.5, -3.5) to (5.5, -2.5) to (8.5, -2.5) to (8.5, -1.5) to (9.5, -1.5) to (9.5, 0.5) to (12.5, 0.5) to (12.5, 1.5) to (0.5, 1.5); (1.5, 1.5) to (1.5, -4.5); (2.5, 1.5) to (2.5, -4.5); (3.5, 1.5) to (3.5, -3.5); (4.5, 1.5) to (4.5, -3.5); (5.5, 1.5) to (5.5, -2.5); (6.5, 1.5) to (6.5, -2.5); (7.5, 1.5) to (7.5, -2.5); (8.5, 1.5) to (8.5, -1.5); (9.5, 1.5) to (9.5, 0.5); (10.5, 1.5) to (10.5, 0.5); (11.5, 1.5) to (11.5, 0.5); (0.5, -2.5) to (5.5, -2.5); (0.5, -3.5) to (2.5, -3.5); (2.5, -1.5) to (3.5, -1.5); (3.5, -0.5) to (4.5, -0.5); (4.5, 0.5) to (6.5, 0.5); (6.5, -0.5) to (8.5, -0.5); (8.5, 0.5) to (9.5, 0.5); \(1) at ( 1, 1) [1]{}; (2) at ( 2, 1) [1]{}; (3) at ( 3, 1) [1]{}; (4) at ( 4, 1) [1]{}; (5) at ( 5, 1) [1]{}; (6) at ( 6, 1) [1]{}; (7) at ( 7, 1) [2]{}; (8) at ( 8, 1) [2]{}; (9) at ( 9, 1) [2]{}; (c) at (-0.5, 1) [(b)]{}; \(51) at ( 1, 0) [2]{}; (52) at ( 2, 0) [2]{}; (53) at ( 3, 0) [2]{}; (54) at ( 4, 0) [3]{}; (55) at ( 5, 0) [3]{}; (55) at ( 6, 0) [3]{}; (55) at ( 7, 0) [3]{}; \(61) at ( 1, -1) [3]{}; (62) at ( 2, -1) [3]{}; (63) at ( 3, -1) [3]{}; (0.5,1.5) to (6.5,1.5) to (6.5, 0.5) to (0.5, 0.5) to (0.5, 1.5); (2.5, 0.5) to (2.5, 1.5); (6.5, 0.5) to (9.5, 0.5) to (9.5, 1.5) to (6.5, 1.5); (8.5, 0.5) to (8.5, 1.5); (0.5, 0.5) to (0.5, -0.5) to (3.5, -0.5) to (3.5, 0.5); (3.5, -0.5) to (7.5, -0.5) to (7.5, 0.5); (5.5, -0.5) to (5.5, 0.5); (0.5, -0.5) to (0.5, -1.5) to (3.5, -1.5) to (3.5, -0.5); (1.5, -0.5) to (1.5, -1.5); \(66) at ( 3, -4.25) [ ]{}; Let us now look at some special cases. - If $\alpha = 0, \beta = 1$, RBT’s of nonzero weight correspond to RPP whose monomial weight is given by $x^T = \prod_i x_i^{a_i}$ where $a_i$ is the number of columns which contain $i$, see Figure \[ext\] (a). Therefore, $g^{(0,1)}_{\lambda} = g_{\lambda}$ recovers combinatorial presentation given in [@lp] for dual stable Grothendieck polynomials. - For $\alpha = 1, \beta = 0$, nonzero weight RBT will be SSYT, their borders are just horizontal strips; each horizontal strip consisting of the same element split into several parts which account the monomial weight; this setting corresponds to the valued-set tableaux given in [@lp] for the polynomials $j_{\lambda} = \omega(g_{\lambda})$, see Figure \[ext\] (b). - For $\alpha = \beta = 0$, RBT will count only SSYT with the usual monomial weight given as for Schur functions. - One more interesting case arises when $\alpha + \beta = 0$, i.e. we sum over tableaux which have no inner parts. Here we consider a kind of [*rim tableaux*]{} with a special signed weight. More details on this case will be discussed in the final section. The function $g^{(\alpha, \beta)}_{\lambda/\mu}$ extended for any skew shape $\lambda/\mu$ can be defined by this combinatorial formula (other ways are compatible with this definition). Lattice forests on plane partitions ----------------------------------- We give one more equivalent combinatorial definition for RBT. From any $T \in RBT(\lambda)$ we construct the following [*forest-type*]{} structure. Let us put in the center of each box a vertex and then connect (in a chain manner) the vertices inside each rim hook in $T$. Every vertex in inner parts of $T$ has two options: to be connected by a vertical edge to the vertex in the upper box or by a horizontal edge to the vertex in the left box. This will produce a certain [*lattice forest*]{} on RPP where each tree has the same label.[^4] The weight $\alpha^{a} \beta^b$ will correspond to the total number $a$ of horizontal edges and $b$ vertical edges. See Figure \[on2\]. (0.5,1.5) rectangle (12.5,0.5); (0.5,1.5) rectangle (1.5,-4.5); (4.5,0.5) rectangle (7.5,-0.5); (8.5,0.5) rectangle (9.5,-0.5); (3.5,-0.5) rectangle (5.5,-1.5); (6.5,-0.5) rectangle (9.5,-1.5); (2.5,-1.5) rectangle (4.5,-2.5); (6.5,-1.5) rectangle (7.5,-2.5); (0.5,-2.5) rectangle (5.5,-3.5); (0.5,-3.5) rectangle (3.5,-4.5); (1,1) to (2,1); (2,1) to (3,1); (4,1) to (5,1) to (6,1); (6,0) to (7,0) to (7,1) to (8,1) to (8,0); (9,1) to (10,1); (1,1) to (1,0) to (1,-1) to (1,-2); (2,0) to (1,0); (3,0) to (3,1); (4,0) to (3,0); (2,-1) to (2,0); (3,-1) to (3,0); (2,-2) to (1,-2); (5,0) to (5,-1) to (4,-1) to (4,-2) to (3,-2); (9,0) to (9,-1) to (8,-1) to (8,-2); (7,-2) to (7,-1); (6,-1) to (5,-1) to (5,-2) to (6,-2); (1,-3) to (2,-3); (2,-4) to (3,-4) to (3,-3) to (4,-3) to (5,-3); (0.5,1.5) to (6.5,1.5); (6.5,1.5) to (6.5,0.5); (6.5,0.5) to (1.5,0.5); (1.5,0.5) to (1.5,-2.5); (1.5,-2.5) to (0.5,-2.5); (0.5,-2.5) to (0.5,1.5); (3.5,1.5) to (3.5,0.5); (8.5,1.5) to (8.5,0.5); (11.5,1.5) to (11.5,0.5); (5.5,0.5) to (5.5,-0.5); (1.5,-3.5) to (1.5,-4.5); (7.5,-0.5) to (7.5,-1.5); (6.5,1.5) to (10.5,1.5); (10.5,1.5) to (10.5,0.5); (10.5,0.5) to (7.5,0.5); (7.5,0.5) to (7.5,-0.5); (7.5,-0.5) to (5.5,-0.5); (5.5,-0.5) to (5.5,-1.5); (5.5,-1.5) to (5.5-1,-1.5); (5.5-1,-1.5) to (5.5-1,-1.5-1); (5.5-1,-1.5-1) to (5.5-1-2,-1.5-1); (5.5-1-2,-1.5-1) to (5.5-1-2,-1.5-1+1); (5.5-1-2,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1); (5.5-1-2+1,-1.5-1+1) to (5.5-1-2+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1) to (5.5-1-2+1+1,-1.5-1+1+1+1); (0.5,-2.5) to (0.5,-2.5-1) to (0.5+2,-2.5-1) to (0.5+2,-2.5-1+1) to (0.5+2-1,-2.5-1+1); (0.5, -3.5) to (0.5, -4.5) to (0.5+3, -4.5) to (0.5+3, -4.5+1) to (0.5+3+2, -4.5+1) to (0.5+3+2, -4.5+1+1) to (0.5+3+2-1, -4.5+1+1); (6.5, -0.5) to (6.5, -2.5) to (6.5+1, -2.5) to (6.5+1, -2.5+1) to (6.5+1+1, -2.5+1) to (6.5+1+1+1, -2.5+1) to (6.5+1+1+1, -2.5+1+2) ; (6.5+1+1+1-1, -2.5+1+2) to (6.5+1+1+1-1, -2.5+1+2-1) to (6.5+1+1+1-1-1, -2.5+1+2-1); (6.5, -2.5) to (6.5-1, -2.5); (8.5, -2.5+1) to (8.5, -2.5) to (8.5-1, -2.5); (10.5, 0.5) to (10.5+2, 0.5) to (10.5+2, 0.5+1) to (10.5+2-2, 0.5+1); \(1) at ( 1, 1) [$\bullet$]{}; (2) at ( 2, 1) [$\bullet$]{}; (3) at ( 3, 1) [$\bullet$]{}; (4) at ( 4, 1) [$\bullet$]{}; (5) at ( 5, 1) [$\bullet$]{}; (6) at ( 6, 1) [$\bullet$]{}; (7) at ( 7, 1) [$\bullet$]{}; (8) at ( 8, 1) [$\bullet$]{}; (9) at ( 9, 1) [$\bullet$]{}; (10) at ( 10, 1) [$\bullet$]{}; (11) at ( 11, 1) [$\bullet$]{}; (12) at ( 12, 1) [$\bullet$]{}; \(21) at ( 1, 0) [$\bullet$]{}; (22) at ( 2, 0) [$\bullet$]{}; (23) at ( 3, 0) [$\bullet$]{}; (24) at ( 4, 0) [$\bullet$]{}; (25) at ( 5, 0) [$\bullet$]{}; (26) at ( 6, 0) [$\bullet$]{}; (27) at ( 7, 0) [$\bullet$]{}; (28) at ( 8, 0) [$\bullet$]{}; (29) at ( 9, 0) [$\bullet$]{}; \(31) at ( 1, -1) [$\bullet$]{}; (32) at ( 2, -1) [$\bullet$]{}; (33) at ( 3, -1) [$\bullet$]{}; (34) at ( 4, -1) [$\bullet$]{}; (35) at ( 5, -1) [$\bullet$]{}; (36) at ( 6, -1) [$\bullet$]{}; (37) at ( 7, -1) [$\bullet$]{}; (38) at ( 8, -1) [$\bullet$]{}; (39) at ( 9, -1) [$\bullet$]{}; \(41) at ( 1, -2) [$\bullet$]{}; (42) at ( 2, -2) [$\bullet$]{}; (43) at ( 3, -2) [$\bullet$]{}; (44) at ( 4, -2) [$\bullet$]{}; (45) at ( 5, -2) [$\bullet$]{}; (46) at ( 6, -2) [$\bullet$]{}; (47) at ( 7, -2) [$\bullet$]{}; (48) at ( 8, -2) [$\bullet$]{}; \(51) at ( 1, -3) [$\bullet$]{}; (52) at ( 2, -3) [$\bullet$]{}; (53) at ( 3, -3) [$\bullet$]{}; (54) at ( 4, -3) [$\bullet$]{}; (55) at ( 5, -3) [$\bullet$]{}; \(61) at ( 1, -4) [$\bullet$]{}; (62) at ( 2, -4) [$\bullet$]{}; (63) at ( 3, -4) [$\bullet$]{}; Pieri and branching formulas {#spieri} ============================ Let us fix some notation which we will repeatedly use in this section. For any skew shape $\mu/\lambda$ define: - $r(\mu/\lambda)$ as the number of rows; - $c(\mu/\lambda)$ as the number of columns; - $b(\mu/\lambda)$ as the number of connected components; - $i(\mu/\lambda) = |\mu/\lambda| - c(\mu/\lambda) - r(\mu/\lambda) + b(\mu/\lambda)$ as the number of boxes in inner part. For example, for a skew shape $\mu/\lambda = 665222/2222$ we have $r(\mu/\lambda) = 5,$ $c(\mu/\lambda) = 6,$ $b(\mu/\lambda) = 2,$ $i(\mu/\lambda) = 6$. See Figure \[fig11\]. (a) & &   &   &   &  \ & &   & \*(lightgray) & \*(lightgray) & \*(lightgray)\ & &   & \*(lightgray) & \*(lightgray)\ &\   &  \   & \*(lightgray) (b) & & \*(lightgray) & \*(lightgray) & \*(lightgray) & \*(lightgray)\ & &   &   &   &  \ & &   & &\ &\ \*(lightgray) & \*(lightgray)\   &   (c) & &   &   &   & \*(lightgray)\ & &   &   &   & \*(lightgray)\ & &   & & \*(lightgray)\ &\   & \*(lightgray)\   & \*(lightgray) Pieri type formulas for $G^{(\alpha, \beta)}_{\lambda}$ ------------------------------------------------------- \[pieri\]The following formulas hold. Type 1: $$\begin{aligned} G^{(\alpha, \beta)}_{(k)} G^{(\alpha, \beta)}_{\lambda} = \sum_{\mu/\lambda \text{ hor. strip}} (\alpha + \beta)^{|\mu/\lambda| - k} \binom{r(\mu/\lambda) - 1}{|\mu/\lambda| - k} G^{(\alpha, \beta)}_{\mu},\end{aligned}$$ $$\begin{aligned} G^{(\alpha, \beta)}_{(1^k)} G^{(\alpha, \beta)}_{\lambda} = \sum_{\mu/\lambda \text{ vert. strip}} (\alpha + \beta)^{|\mu/\lambda| - k} \binom{c(\mu/\lambda) - 1}{|\mu/\lambda| - k} G^{(\alpha, \beta)}_{\mu}.\end{aligned}$$ Type 2: $$\begin{aligned} h_k\left(\frac{x}{1 + \alpha x} \right) G^{(-\alpha, -\beta)}_{\lambda} &= \sum_{\nu:\ c(\nu/\lambda) = k} (\alpha + \beta)^{|\nu/\lambda| - k} G^{(-\alpha, -\beta)}_{\nu}\\ e_k\left(\frac{x}{1 + \alpha x} \right) G^{(-\alpha, -\beta)}_{\lambda} &= \sum_{\nu/\lambda \text{ vert. strip}} (\alpha + \beta)^{|\nu/\lambda| - k} \binom{|\nu/\lambda| - c(\nu/\lambda)}{k - c(\nu/\lambda)} G^{(-\alpha, -\beta)}_{\nu}\end{aligned}$$ Type 3: $$\begin{aligned} h_{k} G^{(-\alpha, -\beta)}_{\lambda} = \sum_{\mu} v^{k}_{\mu/\lambda}(\alpha, \beta) G^{(-\alpha, -\beta)}_{\mu},\end{aligned}$$ $$\begin{aligned} \label{ge} e_{k} G^{(-\alpha, -\beta)}_{\lambda} = \sum_{\mu} \bar v^{k}_{\mu/\lambda}(\alpha, \beta) G^{(-\alpha, -\beta)}_{\mu}, \end{aligned}$$ where $$\label{vi} v^{k}_{\mu/\lambda}(\alpha, \beta) = \begin{cases} \beta^{r(\mu/\lambda) - b(\mu/\lambda)}(\alpha + \beta)^{i(\mu/\lambda)} \alpha^{c(\mu/\lambda) - k} \binom{c(\mu/\lambda) - b(\mu/\lambda)}{c(\mu/\lambda) - k}, & \text{ if } \lambda \subseteq \mu,\\ 0, & \text{ otherwise}. \end{cases}$$ and $\bar v^{k}_{\mu/\lambda}(\alpha, \beta)$ defined similarly as $ v^{k}_{\mu/\lambda}(\alpha, \beta)$ but with $\alpha, \beta$ and $r, c$ being simultaneously switched. Type 1 formulas are finite sums and other types are infinite in general. Before proving Theorem \[pieri\] we prepare some lemmas. \[vc\] The coefficients of Type 3 formulas satisfy the identity [$$\begin{aligned} \label{vk} &v^{k}_{\mu/\lambda}(\alpha, \beta) = \sum_{\mu/\nu \text{ vert. strip}} \alpha^{c(\nu/\lambda) - k} \binom{c(\nu/\lambda)}{k} (\alpha + \beta)^{|\nu/\lambda| - c(\nu/\lambda)} (-\alpha)^{c(\mu/\nu)} \beta^{|\mu/\nu| - c(\mu/\nu)}.\end{aligned}$$ ]{} For a skew shape $\mu/\lambda$ define its [*upper boundary*]{} as the horizontal strip containing all boxes for which no box lies strictly above each of them. Similarly, define its [*right boundary*]{} as the vertical strip of all boxes for which no box lies strictly to the right. See Figure \[fig11\]. Let $\nu$ be a partition so that $\mu/\nu$ is a vertical strip. We interpret the r.h.s. of with weighted fillings of the shape $\mu/\lambda$ having the following properties: - each box has one of five weights $\{\alpha,-\alpha, \beta, \alpha + \beta, 1\}$; - the upper boundary of $\nu/\lambda$ (which has $c(\nu/\lambda)$ elements) has $k$ elements $1$ and $c(\nu/\lambda) - k$ elements $\alpha$; - other boxes in $\nu/\lambda$ have weight $\alpha + \beta$; - in the remaining part $\mu/\nu$, the bottom elements of each column have weight $-\alpha$ and each of the remaining boxes has weight $\beta$. The weight of any such tableau is the product of weights of its entries. It is easy to see that for different $\nu$ we obtain different tableau and the total sum of weights of all such tableau gives the r.h.s. of . We now explain how to cancel most of the elements in these tableaux so that the formula will match . Consider locally how varies (when $\nu$ runs) the total weight of a single column of the right boundary of $\mu/\lambda$. If no box in this column belongs to $\nu$ then the weight of the column is $-\alpha \beta^{h-1}$ where $h$ is the height of the column. When $\nu$ occupies $i$ ($1 \le i \le h - 1$) topmost boxes of the column, the weight is given by $-\alpha \beta^{h - 1 - i}(\alpha + \beta)^{i-1} - \alpha^2 \beta^{h - 1 - i}(\alpha + \beta)^{i-1}$ depending on whether the topmost box has weight $1$ or $\alpha$. If $\nu$ occupies the entire column, the weight is given by $(\alpha + \beta)^{h-1} + \alpha(\alpha + \beta)^{h-1}$ (again we sum depending on whether the topmost box has weight $1$ or $\alpha$). Therefore, the total weight (when $\nu$ varies on this column and is fixed everywhere else) is given by $$\begin{aligned} &-\alpha \beta^{h-1} - \sum_{i = 1}^{h-1}(\alpha \beta^{h - 1 - i}(\alpha + \beta)^{i-1} + \alpha^2 \beta^{h - 1 - i}(\alpha + \beta)^{i-1}) + (\alpha + \beta)^{h-1} + \alpha(\alpha + \beta)^{h-1}\\ &= -\alpha \beta^{h-1} + (\beta^{h-1} - (\alpha + \beta)^{h-1}) + \alpha (\beta^{h-1} - (\alpha + \beta)^{h-1}) + (\alpha + \beta)^{h-1} + \alpha(\alpha + \beta)^{h-1}\\ &= \beta^{h-1}.\end{aligned}$$ If the topmost box of this column does not belong to the upper boundary of $\mu/\lambda,$ then we have the sum with the same result $$\begin{aligned} &- \sum_{i = 1}^{h-1}\alpha \beta^{h - 1 - i}(\alpha + \beta)^{i-1} + (\alpha + \beta)^{h-1} = \beta^{h-1}.\end{aligned}$$ This means that we can transform this column into the column whose topmost box has weight $1$ and every other box has weight $\beta$, so that the total weight will be conserved. We can do this procedure on every column of the right boundary of $\mu/\lambda$ and therefore every column will have this property. Finally, note that these resulted tableaux correspond to the defining formula of $v^{k}_{\mu/\lambda}(\alpha, \beta)$. \[hh\] $$\begin{aligned} h_k\left(x \right) &=\sum_i (-\alpha)^i e_i\left(\frac{x}{1 + \alpha x} \right) \sum_j h_{j}\left(\frac{x}{1 + \alpha x} \right) \alpha^{j - k} \binom{j}{k}\end{aligned}$$ The proof is a standard manipulation with the generating series $\sum_{k \ge 0} h_k(x) t^k = \prod_{i} \frac{1}{1 - t x_i}$ applying the substitutions $x \to \frac{x}{1 - \alpha x}$ and then $x \to \frac{x}{1 + \alpha x}$. For $\alpha = 0, \beta = \pm 1$, Lenart [@lenart] proved Pieri formulas, which can easily be restated with a $\beta$ parameter: [$$\begin{aligned} G^{\beta}_{(k)} G^{\beta}_{\lambda} = \sum_{\mu/\lambda \text{ hor. strip}} \beta^{|\mu/\lambda| - k} \binom{r(\mu/\lambda) - 1}{|\mu/\lambda| - k} G^{\beta}_{\mu}, \quad G^{\beta}_{(1^k)} G^{\beta}_{\lambda} = \sum_{\mu/\lambda \text{ vert. strip}} \beta^{|\mu/\lambda| - k} \binom{c(\mu/\lambda) - 1}{|\mu/\lambda| - k} G^{\beta}_{\mu},\end{aligned}$$ $$\begin{aligned} h_k G^{-\beta}_{\lambda} = \sum_{\nu:\ c(\nu/\lambda) = k} \beta^{|\nu/\lambda| - k} G^{-\beta}_{\nu}, \quad e_k G^{-\beta}_{\lambda} = \sum_{\nu/\lambda \text{ vert. strip}} \beta^{|\nu/\lambda| - k} \binom{|\nu/\lambda| - c(\nu/\lambda)}{k - c(\nu/\lambda)} G^{-\beta}_{\nu}\end{aligned}$$ ]{} Type 1, 2 formulas for $G^{(\alpha, \beta)}_{\lambda}$ are followed then from the latter identities via substitutions $\beta \to \alpha + \beta$, $x \to \frac{x}{1 \pm \alpha x}$. We now prove Type 3 formulas. Applying Lemma \[hh\] and then Type 2 formulas, we have $$\begin{aligned} &h_{k} G^{(-\alpha, -\beta)}_{\lambda} \\ &= \sum_{i} (-\alpha)^i e_i\left(\frac{x}{1 + \alpha x} \right) \sum_{j \ge k} \alpha^{j - k} \binom{j}{k} h_j\left(\frac{x}{1 + \alpha x} \right) G^{(-\alpha, -\beta)}_{\lambda}\\ &= \sum_{i} (-\alpha)^i e_i\left(\frac{x}{1 + \alpha x} \right) \sum_{j \ge k} \alpha^{j - k} \binom{j}{k} \sum_{\nu: c(\nu/\lambda) = j} (\alpha + \beta)^{|\nu/\lambda| - j} G_{\nu}^{(-\alpha, -\beta)}\\ &= \sum_{\nu} (\alpha + \beta)^{|\nu/\lambda| - c(\nu/\lambda)} \alpha^{c(\nu/\lambda) - k}\binom{c(\nu/\lambda)}{k} \sum_{i} (-\alpha)^i e_i\left(\frac{x}{1 + \alpha x} \right) G_{\nu}^{(-\alpha, -\beta)}\\ &= \sum_{\nu} (\alpha + \beta)^{|\nu/\lambda| - c(\nu/\lambda)} \alpha^{c(\nu/\lambda) - k}\binom{c(\nu/\lambda)}{k}\\ &\qquad\times\sum_{i} (-\alpha)^i \sum_{\mu/\nu \text{ vert. strip}} (\alpha + \beta)^{|\mu/\nu| - i}\binom{|\mu/\nu| - c(\mu/\nu)}{i - c(\mu/\nu)} G_{\mu}^{(-\alpha, -\beta)}\\ &= \sum_{\nu} (\alpha + \beta)^{|\nu/\lambda| - c(\nu/\lambda)} \alpha^{c(\nu/\lambda) - k}\binom{c(\nu/\lambda)}{k}\sum_{\mu/\nu \text{ vert. strip}} G_{\mu}^{(-\alpha, -\beta)} (-\alpha)^{c(\mu/\nu)}\\ &\qquad\times\sum_{i \ge c(\mu/\nu)} (\alpha + \beta)^{|\mu/\nu| - i} (-\alpha)^{i - c(\mu/\nu)}\binom{|\mu/\nu| - c(\mu/\nu)}{i - c(\mu/\nu)}\\ &= \sum_{\nu} (\alpha + \beta)^{|\nu/\lambda| - c(\nu/\lambda)} \alpha^{c(\nu/\lambda) - k}\binom{c(\nu/\lambda)}{k}\\ &\qquad\times\sum_{\mu/\nu \text{ vert. strip}} G_{\mu}^{(-\alpha, -\beta)} (-\alpha)^{c(\mu/\nu)} (\alpha + \beta - \alpha)^{|\mu/\nu| - c(\mu/\nu)}\\ &= \sum_{\mu} G_{\mu}^{(-\alpha, -\beta)}\sum_{\mu/\nu \text{ vert. strip}}(\alpha + \beta)^{|\nu/\lambda| - c(\nu/\lambda)} \alpha^{c(\nu/\lambda) - k}\binom{c(\nu/\lambda)}{k} (-\alpha)^{c(\mu/\nu)} \beta^{|\mu/\nu| - c(\mu/\nu)}\\ &= \sum_{\mu} v^{k}_{\mu/\lambda}(\alpha, \beta) G_{\mu}^{(-\alpha, -\beta)}.\end{aligned}$$ The last step uses Lemma \[vc\]. The second Type 3 formula now implies by applying the involution $\omega$ to what we just proved. Branching formulas for $g^{(\alpha, \beta)}_{\lambda}$ and proof of Theorem \[cg\] ---------------------------------------------------------------------------------- Let us first compute the weight generating function for rim border tableaux (RBT, Definition \[rrpp\]) at single variable $z$, and as a result it will correspond to a formula for $g^{(\alpha, \beta)}_{\lambda/\mu}(z)$. \[gl\] For $\mu \subseteq \lambda$, we have $$\sum_{T \in RBT(\lambda/\mu)} w_T(\alpha, \beta) z^{\#\{\text{rim hooks in } T\}}=\beta^{r(\lambda/\mu) - b(\lambda/\mu)} (\alpha + \beta)^{i(\lambda/\mu)} z^{b(\lambda/\mu)} (z+\alpha)^{c(\lambda/\mu) - b(\lambda/\mu)}.$$ We use Definition \[rrpp\] of RBT and try to put an element ($z$) in a shape $\lambda/\mu$. Each connected component of $\lambda/\mu$ contains at least one rim hook (from the border), which gives the factor $z^{b(\lambda/\mu)}$. The factor $(\alpha + \beta)^{i(\lambda/\mu)}$ arises from the inner part, also counted in $w_T(\alpha, \beta)$. All rim hooks are produced by the arbitrary choice from the upper border (consisting of $c(\lambda/\mu) - b(\lambda/\mu)$ remaining elements) of factors $z$ or $\alpha$. The remaining factor $\beta^{r(\lambda/\mu) - b(\lambda/\mu)}$ corresponds to the heights of rim hooks. We have $$g^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n, x) = \sum_{\mu} g^{(\alpha, \beta)}_{\mu}(x_1, \ldots, x_n) g^{(\alpha, \beta)}_{\lambda/\mu}(x),$$ where the function $g^{(\alpha, \beta)}_{\lambda/\mu}(x)$ of single variable $x$ is defined as follows[^5]: $$g^{(\alpha, \beta)}_{\lambda/\mu}(x) = \begin{cases} \beta^{r(\lambda/\mu) - b(\lambda/\mu)} (\alpha + \beta)^{i(\lambda/\mu)} x^{b(\lambda/\mu)} (x+\alpha)^{c(\lambda/\mu) - b(\lambda/\mu)}, & \text{ if } \mu \subseteq \lambda,\\ 0, & \text{otherwise}. \end{cases}$$ Take the Cauchy identity and use Type 3 Pieri formula for $G^{(\alpha, \beta)}_{\lambda}$ (Theorem \[pieri\]), [$$\begin{aligned} \sum_{\lambda} G^{(-\alpha, -\beta)}_{\lambda}(y) &g^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n, x) \\ &= \prod_{1 \le i \le n, 1 \le j} \frac{1}{1-x_i y_j} \prod_{1 \le j} \frac{1}{1 - x y_j}\\ &= \prod_{1 \le i \le n, 1 \le j} \frac{1}{1-x_i y_j} \sum_{0 \le k} x^k h_k(y)\\ &= \sum_{\mu} g^{(\alpha, \beta)}_{\mu}(x_1, \ldots, x_n) G^{(-\alpha, -\beta)}_{\mu}(y) \sum_{k \ge 0} x^k h_k(y)\\ &= \sum_{\mu} g^{(\alpha, \beta)}_{\mu}(x_1, \ldots, x_n) \sum_{k \ge 0} x^k \sum_{\lambda} v^k_{\lambda/\mu}(\alpha, \beta) G^{(-\alpha, -\beta)}_{\lambda}(y)\\ &= \sum_{\lambda} G^{(-\alpha, -\beta)}_{\lambda}(y) \sum_{\mu} g^{(\alpha, \beta)}_{\mu}(x_1, \ldots, x_n) \sum_{k} x^k v^k_{\lambda/\mu}(\alpha, \beta)\end{aligned}$$ ]{} Using we obtain $$\begin{aligned} \sum_{k} x^k v^k_{\lambda/\mu}(\alpha, \beta) &= \sum_{k} x^k \beta^{r(\lambda/\mu) - b(\lambda/\mu)}(\alpha + \beta)^{i(\lambda/\mu)} \alpha^{c(\lambda/\mu) - k} \binom{c(\lambda/\mu) - b(\lambda/\mu)}{c(\lambda/\mu) - k}\\ &=\beta^{r(\lambda/\mu) - b(\lambda/\mu)} (\alpha + \beta)^{i(\lambda/\mu)} x^{b(\lambda/\mu)} (x+\alpha)^{c(\lambda/\mu) - b(\lambda/\mu)},\end{aligned}$$ which completes the proof. Combinatorial formula for $g_{\lambda}^{(\alpha, \beta)}$ now follows from the previous Theorem and Lemma \[gl\] by iteratively adding new variables. So RBT of shape $\lambda$ is constructed as $\emptyset = \lambda^{(0)} \subseteq \cdots \subseteq \lambda^{(k)} = \lambda$, where $\lambda^{(i + 1)}/\lambda^{(i)}$ is an RBT of single variable. By combinatorial formula we can extend the polynomials $g_{\lambda/\mu}^{(\alpha, \beta)}$ to any skew shape $\lambda/\mu$. It is then easy to obtain the following formulas. \[Branching formulas\] The following properties hold $$g^{(\alpha, \beta)}_{\lambda}(x, x') = \sum_{\mu} g^{(\alpha, \beta)}_{\mu}(x) g^{(\alpha, \beta)}_{\lambda/\mu}(x'),$$ where $x, x'$ are two sets of variables; $$g^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n, x) = \sum_{\lambda/\mu \text{ connected}} \alpha^{c(\lambda/\mu) - 1}\beta^{r(\lambda/\mu) - 1} (\alpha + \beta)^{i(\lambda/\mu)} x g_{\lambda/\mu}(x_1, \ldots, x_n).$$ Branching formulas for $G^{(\alpha, \beta)}_{\lambda}$ ------------------------------------------------------ We now state branching formulas for $G^{(\alpha, \beta)}_{\lambda}$. Similar formulas were given in [@buch] for $G_{\lambda}$. For a partition $\lambda = (\lambda_1, \lambda_2, \lambda_3, \ldots)$, denote $\bar\lambda = (\lambda_2, \lambda_3, \ldots)$. \[bg\] We have $$G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n, x) = \sum_{\lambda/\mu \text{ hor. strip}} G^{(\alpha, \beta)}_{\lambda/\mu}(x) G^{(\alpha, \beta)}_{\mu}(x_1, \ldots, x_n),$$ where for a horizontal strip $\lambda/\mu$ we have $$G^{(\alpha, \beta)}_{\lambda/\mu}(x) =\left(\frac{x}{1 - \alpha x}\right)^{|\lambda/\mu|} \left(\frac{1 + \beta x}{1 - \alpha x} \right)^{r(\mu/\bar\lambda)}.$$ The proof follows from the operator definition (, Section \[dfg\]) $$\begin{aligned} G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n, x) &= \langle C(x) C(x_n) \cdots C(x_1) \cdot\varnothing,\lambda\rangle\\ &=\langle C(x) \sum_{\mu} G^{(\alpha, \beta)}_{\mu}(x_1, \ldots, x_n) \cdot \mu,\lambda\rangle\\ &= \sum_{\mu} G^{(\alpha, \beta)}_{\lambda/\mu}(x) G^{(\alpha, \beta)}_{\mu}(x_1, \ldots, x_n).\end{aligned}$$ Note that for a single variable $x$, $G^{(\alpha, \beta)}_{\lambda/\mu}(x) = 0$ if $\lambda/\mu$ is not a horizontal strip. Otherwise, $r(\mu/\bar\lambda)$ corresponds to the number of removable boxes of $\mu$ for which there is no box of $\lambda$ below them. In other words, the operators $u_i d_i$ can be applied here and then it is not hard to compute from the operator expansion that $G^{(\alpha, \beta)}_{\lambda/\mu}(x) = \left(1 + \frac{(\alpha + \beta) x}{1 - \alpha x} \right)^{r(\mu/\bar\lambda)} \left(\frac{x}{1 - \alpha x}\right)^{|\lambda/\mu|}.$ The elements $g^{(\alpha, \beta)}_{(k)}, g^{(\alpha, \beta)}_{(1^k)}$ --------------------------------------------------------------------- We have the following formulas and the generating series $$\begin{aligned} \label{42} g^{(\alpha, \beta)}_{(k)} = \sum_{i = 1}^k \alpha^{k - i} \binom{k - 1}{i - 1} h_i, \qquad g^{(\alpha, \beta)}_{(1^k)} = \sum_{i = 1}^k \beta^{k - i} \binom{k - 1}{i - 1} e_i.\end{aligned}$$ $$\begin{aligned} \label{43} 1 + \sum_{k \ge 1} \left(\frac{t}{1 + \alpha t} \right)^k g^{(\alpha, \beta)}_{(k)} = \prod_{j \ge 1} \frac{1}{1 - x_j t}, \quad 1 + \sum_{k \ge 1} \left(\frac{t}{1 + \beta t} \right)^k g^{(\alpha, \beta)}_{(1^k)} = \prod_{j \ge 1} {(1 + x_j t)}.\end{aligned}$$ The formulas for $g^{(\alpha, \beta)}_{(k)}, g^{(\alpha, \beta)}_{(1^k)}$ are easy to derive from combinatorial interpretations. The generating series are then implied from these formulas. From , for any $n \in \mathbb{Z}_{\ge 0}$ we have $$\label{ggk} \sum_{i} g^{(\alpha, \beta)}_{(i)}(x) g^{(\beta, \alpha)}_{(1^{n - i})}(-x) = \delta_{n,0}.$$ From , the elements $g^{(\alpha, \beta)}_{(k)}$ (and $g^{(\alpha, \beta)}_{(1^k)}$) are free generators of the polynomial ring $\Lambda$, we have $$\Lambda \otimes \mathbb{Z}[\alpha] \cong \mathbb{Z}[\alpha][g^{(\alpha, \beta)}_{(1)}, g^{(\alpha, \beta)}_{(2)}, \ldots], \qquad \Lambda \otimes \mathbb{Z}[\beta] \cong \mathbb{Z}[\beta][g^{(\alpha, \beta)}_{(1)}, g^{(\alpha, \beta)}_{(1^2)}, \ldots].$$ Pieri formulas for $g^{(\alpha, \beta)}_{\lambda}$ -------------------------------------------------- The following formulas hold: Type 1: $$g^{(\alpha, \beta)}_{(k)} g^{(\alpha, \beta)}_{\mu} = \sum_{\lambda/\mu \text{ hor. strip}} (-(\alpha + \beta))^{k - |\lambda/\mu|} \binom{r(\mu/\bar\lambda)}{k - |\lambda/\mu|} g^{(\alpha, \beta)}_{\lambda},$$ $$g^{(\alpha, \beta)}_{(1^k)} g^{(\alpha, \beta)}_{\mu} = \sum_{\lambda/\mu \text{ vert. strip}} (-(\alpha + \beta))^{k - |\lambda/\mu|} \binom{c(\mu'/\bar\lambda')}{k - |\lambda/\mu|} g^{(\alpha, \beta)}_{\lambda}.$$ Type 2: $$h_{k} g^{(\alpha, \beta)}_{\mu} = \sum_{\lambda/\mu \text{ hor. strip}} q_{\lambda/\mu}{(\alpha, \beta)} g^{(\alpha, \beta)}_{\lambda}, \quad e_{k} g^{(\alpha, \beta)}_{\mu} = \sum_{\lambda/\mu \text{ vert. strip}} q_{\lambda'/\mu'}{(\beta, \alpha)} g^{(\alpha, \beta)}_{\lambda},$$ where for a horizontal strip $\lambda/\mu$ we define $$q_{\lambda/\mu}{(\alpha, \beta)} = \sum_{\ell} (-(\alpha + \beta))^{\ell - |\lambda/\mu|} \binom{r(\mu/\bar\lambda)}{\ell - |\lambda/\mu|}(-\alpha)^{k - \ell} \binom{k - 1}{\ell - 1}$$ Type 1,2 formulas can be obtained from the branching formulas for $G^{(\alpha, \beta)}_{\lambda}$ (Proposition \[bg\]). Note that Type 1 formulas are also followed from the known comultiplication formulas for $G^{(-\alpha, -\beta)}_{\lambda}$. Schur expansions {#schur} ================ In this section we describe expansions of the (dual) bases $\{G^{(-\alpha, -\beta)}_{\lambda}\}, \{g^{(\alpha, \beta)}_{\lambda}\}$ in the basis $\{ s_{\lambda}\}$ of Schur functions. We give determinantal formulas for connection constants and combinatorial interpretations using nonintersecting lattice paths. In particular, we show that $g^{(\alpha, \beta)}_{\lambda}$ are Schur-positive (for $\alpha, \beta > 0$). Cauchy-Binet formula for partitions ----------------------------------- We will repeatedly use the following adaptation of the Cauchy-Binet determinantal formula. \[cb\] Let $a^{(i)}_{p, q}, b^{(i)}_{p, q}$ (for all $p, q \in \mathbb{Z}$ and $i \in \mathbb{Z}_{> 0}$) be elements of a commutative ring. For any partitions $\nu \subseteq \mu \subseteq \lambda$ and $t \ge \ell(\lambda), \ell(\mu)$, let $$F_{\lambda/\mu} = \det \left[a^{(i)}_{\lambda_i - i, \mu_j - j}\right]_{1 \le i,j \le t}, \quad G_{\mu/\nu} = \det\left[b^{(j)}_{\mu_i - i, \nu_j - j}\right]_{1 \le i,j \le t}.$$ Let $ H_{\lambda/\nu} := \sum_{\mu} F_{\lambda/\mu} G_{\mu/\nu}, $ then $$H_{\lambda/\nu} = \det\left[c^{(i,j)}_{\lambda_i - i, \nu_j - j}\right]_{1 \le i,j \le t}, \quad \text{ where }\quad c^{(i,j)}_{p, q} = \sum_{k} a^{(i)}_{p, k} b^{(j)}_{k, q}.$$ Lattice paths and supplementary tableaux. ----------------------------------------- In this subsection we define the coefficients $f^{(\alpha, \beta)}_{\mu/\nu}$ which appear in Schur expansions of $G^{(\alpha, \beta)}_{\lambda}, g^{(\alpha, \beta)}_{\lambda}$. [*Type 1 grid.*]{} Consider the lattice grid $\mathbb{Z}^2$ with the following assignment of edge weights: - $w[(x,y) \to (x,y+1)] = 1$ for $x,y \in \mathbb{Z}$, all up steps; - $w[(x,y) \to (x+1,y)] = \alpha$ for $x < 0, y \ge 0$, right steps in 2nd quadrant; - $w[(x,y) \to (x+1,y+1)] = \beta$ for $x \ge 0, y \ge 0$, diagonal steps in 1st quadrant; - $w[(x,y) \to (x+1,y)] = \alpha + \beta$ for $x < 0, y < 0$, right steps in 3rd quadrant; - $w[(x,y) \to (x+1,y+1)] =\alpha+ \beta$ for $x \ge 0, y < 0$, diagonal steps in 4th quadrant; and all other weights are $0$. See Figure \[fig3\] (a). (0, -3.5) to (0,3.5); (-3.5,0) to (0,0); (3.5,0) to (0,0); in [1,...,3]{} (, -3.5) to (,3.5); in [1,...,3]{} (-, -3.5) to (-,3.5); in [1,...,3]{} (-3.5, ) to (0,); in [1,...,3]{} (-3.5, -) to (0,-); in [0,...,3]{} (0, ) to (3.5-,3.5); in [1,...,3]{} (0, -) to (3.5,3.5-); in [1,...,3]{} (-0.5, -3.5) to (3.5,-+0.5); (-2, 0) to (-1, 0); at (-1.5,0.3) [$\alpha$]{}; (-2, -2) to (-1, -2); at (-1.5,-1.6) [$\alpha + \beta$]{}; (1.0, 0) to (2.0, 1); at (1.4,0.9) [$\beta$]{}; (1.0, -2) to (2.0, -1); at (0.8,-1.3) [$\alpha+\beta$]{}; (0, -3.5) to (0,3.5); (-3.5,0) to (0,0); (3.5,0) to (0,0); in [1,...,3]{} (, -3.5) to (,3.5); in [1,...,3]{} (-, -3.5) to (-,3.5); in [1,...,3]{} (-3.5, ) to (0,); in [1,...,3]{} (-3.5, -) to (0,-); in [0,...,3]{} (0, ) to (3.5-,3.5); in [1,...,3]{} (0, -) to (3.5,3.5-); in [1,...,3]{} (-0.5, -3.5) to (3.5,-+0.5); \[fill\] at (1-3,1-1) ; \[fill\] at (2-3,1-2) ; \[fill\] at (3-3,1-3) ; \[fill\] at (1-2,2-1) ; \[fill\] at (2-2,2-2) ; \[fill\] at (3-1,3-1-1) ; (-2,0) to (1-3+1,1-1) to (1-3+1,1-1+1); (2-3,1-2) to (2-3 + 1,1-2) to (2-3+1,1-2+1); (3-3,1-3) to (3-3+1,1-3+1) to (3-3+1,1-3+1+1) to (3-3+1+1,1-3+1+1+1); at (-1.5,0.3) [$\alpha$]{}; at (-0.9,-0.6) [$\alpha+\beta$]{}; at (1.4,0.9) [$\beta$]{}; at (1.3,-1.6) [$\alpha+\beta$]{}; Let $\mu/\nu$ be a skew shape, $\ell = \ell(\mu)$ the length of $\mu$ and $d = d(\nu)$ the number of boxes on the main diagonal of $\nu$. Consider in this grid the system of nonintersecting lattice paths from the set of points $A = (A_1, \ldots, A_{\ell})$ to the set of points $B = (B_1, \ldots, B_{\ell})$, where $A_i = (i - \mu_i, 1 - i)$, $B_i = (i - \nu_i, \nu_i - i)$ for $i = 1, \ldots, d$ and $A_i = (i - \mu_i, 1 - \mu_i)$, $B_i = (i - \nu_i, i - \nu_i - 1)$ for $i = d+1, \ldots, \ell$. See examples in Figure \[fig3\] (b) and Figure \[fig33\] (a). For every lattice path system $\mathcal{P}$ from $A$ to $B$ the [*weight*]{} $w$ of $\mathcal{P}$ is defined as the product of weights of its edges. Define $$f^{}_{\mu/\nu}(\alpha, \beta) := \sum_{\mathcal{P}: A \to B} w(\mathcal{P}),$$ so that the sum runs over all nonintersecting path systems $\mathcal{P}$ from $A$ to $B$. In particular, $f^{}_{\mu/\nu}(\alpha, \beta)$ is a polynomial in $\alpha, \beta$ with positive integer coefficients. For a lattice path system in Figure \[fig3\] (b), it is easy to compute that $f^{}_{333/221}(\alpha, \beta) = \alpha(\alpha + \beta) (3\alpha^2 + 9 \alpha\beta + 4\beta^2)$. One can obtain the symmetry $f^{}_{\mu/\nu}(\alpha, \beta) = f^{}_{\mu'/\nu'}(\beta, \alpha)$. In next subsection we will show that the numbers $f^{}_{\mu/\nu}(\alpha, \beta)$ are connection constants in the Schur expansion of $g^{(\alpha, \beta)}_{\lambda}.$ We now collect some technical details which will be useful later. \[prop8\] We have $$f^{}_{\mu/\nu}(\alpha, \beta) = \det\left[ f^{(i)}_{\mu_i - i,\nu_j - j} \right]_{1 \le i,j\le \ell(\mu)},$$ where $$\label{f} f^{(i)}_{p,q} = \begin{cases} \sum_{k} (\alpha + \beta)^k \binom{p+i-1}{k} \alpha^{p - q - k} \binom{p - k}{q}, & \text{ if } q \ge 0, i \le d;\\ \sum_{k} (\alpha + \beta)^k \binom{p+i-1}{k} \beta^{-p - q - k} \binom{-q-1}{-p-q-k}, & \text{ if } q < 0, i > d;\\ \sum_{k} (\alpha + \beta)^p \binom{p+k}{k} \sum_{m} (\alpha + \beta)^m \binom{i - k - 1}{m} \beta^{-q-m} \binom{-q-1}{-q-m}, & \text{ if } q < 0, i \le d;\\ 0, & \text{ otherwise.} \end{cases}$$ From the given (Type 1) lattice grid it is easy to check that the total weight of paths going from $A_i$ to $B_j$ is exactly $f^{(i)}_{\mu_i - i,\nu_j - j}$ with $f^{(i)}_{p,q}$ given by the formula above (depending in which regions the points $A_i$ and $B_j$ lie). Then by the Lindström-Gessel-Viennot Lemma [@gv], the determinant $\det\left[ f^{(i)}_{\mu_i - i,\nu_j - j} \right]_{1 \le i,j\le \ell}$ computes the number of nonintersecting lattice path systems from the set of points $A$ to the set $B$. [*Type 2 grid.*]{} The given path systems can be implemented on another transformed grid, see Figure \[fig33\] (right). On this grid the left half-plane remains the same. The right half-plane allows only moves up and right; the weight of the right steps under $y = -x$ is $\alpha + \beta$ and the weight of the right steps up to that line is $\beta$. The points $A_i$ have coordinates $(i - \mu_i, 1 - i)$ and $B_i$ $(i - \nu_i, -1)$ for $i = d+1, \ldots, \ell.$ It is easy to see that this system generates the same weights and repeats nonintersecting path systems as on the previous grid (we just moved down some of the points $A_i$ and $B_i$ without affecting the total weight in each nonintersecting path system, see Figure \[fig33\]). (0, -7.5) to (0,7.5); (-7.5,0) to (0,0); (7.5,0) to (0,0); (-7.5,7.5) to (0,0); (1,0) to (7.5,6.5); in [1,...,7]{} (, -7.5) to (,7.5); in [1,...,7]{} (-, -7.5) to (-,7.5); in [1,...,7]{} (-7.5, ) to (0,); in [1,...,7]{} (-7.5, -) to (0,-); in [0,...,7]{} (0, ) to (7.5-,7.5); in [1,...,7]{} (0, -) to (7.5,7.5-); in [1,...,7]{} (-0.5, -7.5) to (7.5,-+0.5); \[fill\] at (-6,0) ; \[fill\] at (-4,-1) ; \[fill\] at (-1,-2) ; \[fill\] at (-0,-3) ; \[fill\] at (2,-2) ; \[fill\] at (4,-1) ; \[fill\] at (6,0) ; \[fill\] at (-3,3) ; \[fill\] at (-1,1) ; \[fill\] at (-0,0) ; \[fill\] at (2,1) ; \[fill\] at (4,3) ; \[fill\] at (6,5) ; \[fill\] at (7,6) ; (-6,0) to (-6+1,0) to (-6+1,0+1) to (-6+1+1,0+1) to (-6+1+1+1,0+1) to (-6+1+1+1,0+1+1) to (-6+1+1+1,0+1+1+1); (-4,-1) to (-4+1,-1) to (-4+1,-1+1) to (-4+1+1,-1+1) to (-4+1+1+1,-1+1) to (-4+1+1+1,-1+1+1); (-1,-2) to (-1,-2+1) to (-1+1,-2+1) to (-1+1,-2+1+1); (-0,-3) to (-0+1,-3+1) to (-0+1,-3+1+1) to (-0+1,-3+1+1+1) to (-0+1+1,-3+1+1+1+1); (2,-2) to (2,-2+1) to (2,-2+1+1) to (2+1,-2+1+1+1) to (2+1,-2+1+1+1+1) to (2+1+1,-2+1+1+1+1+1); (4,-1) to (4+1,-1+1) to (4+1,-1+1+2) to (4+1+1,-1+1+2+1) to (4+1+1,-1+1+2+1+2); (6,0) to (6,2) to (7,3) to (7,6); at (-3.5,-3.5) [$\alpha + \beta$]{}; at (3.5,-3.5) [$\alpha + \beta$]{}; at (-4.5,2.5) [$\alpha$]{}; at (4.2,1.8) [$\beta$]{}; (8.1,0) to (11,0); (0, -7.5) to (0,7.5); (-7.5,0) to (0,0); (7.5,0) to (0,0); (-7.5,7.5) to (7.5,-7.5); in [1,...,7]{} (, -7.5) to (,7.5); in [1,...,7]{} (-, -7.5) to (-,7.5); (2,-7.5) to (2,7.5); in [1,...,7]{} (-7.5, ) to (0,); in [1,...,7]{} (-7.5, -) to (7.5,-); \[fill\] at (-6,0) ; \[fill\] at (-4,-1) ; \[fill\] at (-1,-2) ; \[fill\] at (-0,-3) ; \[fill\] at (2,-2-2) ; \[fill\] at (4,-1-4) ; \[fill\] at (6,0-6) ; \[fill\] at (-3,3) ; \[fill\] at (-1,1) ; \[fill\] at (-0,0) ; \[fill\] at (2,-1) ; \[fill\] at (4,-1) ; \[fill\] at (6,-1) ; \[fill\] at (7,-1) ; (-6,0) to (-6+1,0) to (-6+1,0+1) to (-6+1+1,0+1) to (-6+1+1+1,0+1) to (-6+1+1+1,0+1+1) to (-6+1+1+1,0+1+1+1); (-4,-1) to (-4+1,-1) to (-4+1,-1+1) to (-4+1+1,-1+1) to (-4+1+1+1,-1+1) to (-4+1+1+1,-1+1+1); (-1,-2) to (-1,-2+1) to (-1+1,-2+1) to (-1+1,-2+1+1); (-0,-3) to (-0+1,-3) to (-0+1,-3+1) to (-0+1,-3+1+1) to (-0+1+1,-3+1+1); (2,-4) to (2,-4+2) to (2+1,-4+2) to (2+1,-4+2+1) to (2+1+1,-4+2+1); (4,-1-4) to (4+1,-1-4) to (4+1,-1-4+2) to (4+1+1,-1-4+2) to (4+1+1,-1-4+2+2); (6,-6) to (6,-6+2) to (6+1,-6+2) to (6+1,-6+2+3); at (-3.5,-3.5) [$\alpha + \beta$]{}; at (2.2,-5.5) [$\alpha + \beta$]{}; at (-4.5,2.5) [$\alpha$]{}; at (4.2,-2.2) [$\beta$]{}; From lattice path interpretation (Type 1 or 2 grid) it is standard to define tableaux interpretations with some (restricted) properties which we omit here. We will also need the following two supplementary types of tableaux. \[elegant\] Denote by $f_{\mu/\nu}$ the number of semistandard tableaux of skew shape $\mu/\nu$ so that the row $i$ (of $\mu$) contains integers from $[1,i-1]$. In particular, if $f_{\mu/\nu} > 0$, the first row of $\mu/\nu$ is empty. Elegant tableaux arise from nonintersecting lattice path interpretation [@lenart] and the following determinantal formulas hold $$\label{ff} f_{\mu/\nu} = \det\left[ \binom{\mu_i - \nu_j + j - 2}{\mu_i - i -\nu_j + j} \right]_{1 \le i,j \le \ell(\mu)} = \det\left[\binom{\mu'_i - 1}{\mu'_i - i - \nu'_j + j} \right]_{1 \le i,j \le \ell(\mu')}.$$ Note that $f^{}_{\mu/\nu}(0,1) = f_{\mu/\nu}$. \[dht\] Suppose that $\lambda$ and $\mu$ have the same number of boxes $b$ on the main diagonal. Denote by $\psi_{\lambda/\mu}$ the number of fillings of $\lambda/\mu$ so that the elements in the first $b$ rows strictly decrease in rows and weakly in columns, and all elements in row $i$ are from the set $\{0, -1, \ldots, i - \lambda_i + 1 \}$ for $i = 1, \ldots, b$; elements of the first $b$ columns strictly increase in columns and weakly increase in rows, and all elements in the column $j$ are from the set $\{1, 2, \ldots, \lambda'_j - 1 \}$ for $j = 1, \ldots, b$. Dual hook tableaux can also be described using nonintersecting lattice paths and they satisfy the following determinantal formula [@molev] $$\begin{aligned} \psi_{\lambda/\nu} &= \det\left[ \binom{\lambda_i - i}{\lambda_i - i -\nu_j + j} \right]_{1 \le i,j \le d} \det\left[ \binom{i - \nu_j + j - 1}{\lambda_i - i -\nu_j + j} \right]_{d+1 \le i,j},\end{aligned}$$ which can be composed into a single determinant $$\begin{aligned} \label{psii} \psi_{\lambda/\nu}&= \det\left[ \binom{\lambda_i - i}{\lambda_i - i -\nu_j + j} \right]_{1 \le i,j \le \ell(\lambda)}.\end{aligned}$$ The following (nonpositive) formulas for $f_{\nu/\mu}(\alpha, \beta)$ will be useful for us in proving the Schur expansions. \[ll\] The following formulas hold: $$\label{ll1} f_{\nu/\mu}(\alpha, \beta) = \sum_{\lambda} (\alpha + \beta)^{|\nu/\lambda|} \alpha^{|\lambda/\mu|} (-1)^{n(\lambda/\mu)} f_{\nu/\lambda} \psi_{\lambda/\mu},$$ where $\lambda$ and $\mu$ have the same number $b$ of boxes on the main diagonal (see Definition \[dht\] of $\psi$), $n(\lambda/\mu)$ is the number of boxes in the first $b$ columns of $\lambda/\mu$; $$\label{ll2} f_{\mu/\nu}(\alpha, \beta) = \det\left[\widetilde{f}^{(i)}_{\mu_i - i,\nu_j - j} \right]_{1 \le i,j \le \ell},$$ where $$\widetilde{f}^{(i)}_{p,q} = \sum_{k = -\infty}^{\infty} \binom{p+i- k - 1}{i-1}\binom{k}{k - q} (\alpha + \beta)^{p - k} \alpha^{k - q},$$ or equivalently $$\sum_{k \ge 0}\widetilde{f}^{(i)}_{q+k,q} t^k = \frac{1}{(1 - (\alpha + \beta)t)^i}\frac{1}{(1 - \alpha t)^{q + 1}}.$$ (0, -7.5) to (0,7.5); (-7.5,0) to (0,0); (7.5,0) to (0,0); (-7.5,7.5) to (0,0); (1,0) to (7.5,6.5); in [1,...,7]{} (, -7.5) to (,7.5); in [1,...,7]{} (-, -7.5) to (-,7.5); in [1,...,7]{} (-7.5, ) to (0,); in [1,...,7]{} (-7.5, -) to (7.5,-); in [0,...,7]{} (0, ) to (7.5-,7.5); in [1,...,7]{} (, 0) to (7.5,7.5-); \[fill\] at (-6,0) ; \[fill\] at (-4,-1) ; \[fill\] at (-1,-2) ; \[fill\] at (-0,-3) ; \[fill\] at (2,-2-2) ; \[fill\] at (4,-1-4) ; \[fill\] at (6,0-6) ; \[fill\] at (-3,3) ; \[fill\] at (-1,1) ; \[fill\] at (-0,0) ; \[fill\] at (2,1) ; \[fill\] at (4,3) ; \[fill\] at (6,5) ; \[fill\] at (7,6) ; (-6,0) to (-6+1,0) to (-6+1,0+1) to (-6+1+1,0+1) to (-6+1+1+1,0+1) to (-6+1+1+1,0+1+1) to (-6+1+1+1,0+1+1+1); (-4,-1) to (-4+1,-1) to (-4+1,-1+1) to (-4+1+1,-1+1) to (-4+1+1+1,-1+1) to (-4+1+1+1,-1+1+1); (-1,-2) to (-1,-2+1) to (-1+1,-2+1) to (-1+1,-2+1+1); (2,-4) to (2,-4+2); (2,-4+2) to (2+1,-4+2) to (2+1,-4+2+1); (2+1,-4+2+1) to (2+1,-4+2+1+3) to (2+1+1,-4+2+1+3+1); at (-3.5,-3.5) [$\alpha + \beta$]{}; at (4.3,-3.5) [$\alpha + \beta$]{}; at (-4.5,2.5) [$\alpha$]{}; at (2.0,2.5) [$-\alpha$]{}; Note that using determinantal formulas , , the formula is equivalent to the determinantal formula by applying the Cauchy-Binet formula indexed by partitions. We will perform one more transformation of the lattice grid (now with some negative weights) whose nonintersecting path systems compute the same function $f_{\mu/\nu}(\alpha, \beta)$. Consider the new [*Type 3*]{} lattice grid as in Figure \[fig333\]. Which is essentially the same as the Type 2 grid, but diagonal steps in its 1st quadrant have negative weights $-\alpha$ and all right steps in 4th quadrant have weight $\alpha + \beta$. Points $A_i$ have the same coordinates, but the points $B_i$ on the right half-plane move again as in Type 1 lattice grid, i.e. they have the coordinates $(i - \nu_i, i - \nu_i - 1)$. By the following two cases we obtain that $$w(A_i \to B_j) = \widetilde{f}^{(i)}_{\mu_i - i,\nu_j - j}.$$ If $\nu_j - j \ge 0$, then $$\widetilde{f}^{(i)}_{\mu_i - i,\nu_j - j} = \sum_k \binom{\mu_i - k - 1}{i-1}\binom{k}{k - \nu_j + j} (\alpha + \beta)^{\mu_i - i - k} \alpha^{k - \nu_j + j}.$$ If $ j - \nu_j > 0$, then $$\widetilde{f}^{(i)}_{\mu_i - i,\nu_j - j} = \sum_k \binom{k + \mu_i - 1}{i - 1} \binom{j -\nu_j - 1}{j - \nu_j - k} (\alpha + \beta)^{k + \mu_i - i} (-\alpha)^{j - \nu_j - k}.$$ It is then not hard to see that the total weighted sum from $A$ to $B$ on this new Type 3 grid gives the same function $f_{\mu/\nu}(\alpha, \beta)$. Schur expansions for $G^{(\alpha, \beta)}_{\lambda}$, $g^{(\alpha, \beta)}_{\lambda}$ ------------------------------------------------------------------------------------- [Table 1. Schur expansions of $g^{(\alpha, \beta)}_{\lambda}$ for some $\lambda$.]{} [$$\begin{aligned} g^{(\alpha, \beta)}_{(k)} =\ &\sum_{i = 1}^k \binom{k - 1}{i - 1} h_i \alpha^{k - i} \qquad g^{(\alpha, \beta)}_{(1^k)} = \sum_{i = 1}^k \binom{k - 1}{i - 1} e_i \beta^{k - i}\\ g^{(\alpha, \beta)}_{21} =\ &\alpha \beta s_{1} + \beta s_{2} + \alpha s_{11} + s_{21}\\ g^{(\alpha, \beta)}_{31} =\ &\alpha^2 \beta s_{1} + 2\alpha \beta s_{2} + \alpha^2 s_{11} + \beta s_{3} + 2\alpha s_{21} + s_{31}\\ g^{(\alpha, \beta)}_{22} =\ &\alpha \beta (\alpha + \beta) s_{1} + \beta (\alpha + \beta) s_{2} + \alpha(\alpha + \beta) s_{11} + (\alpha + \beta) s_{21} + s_{22}\\ g^{(\alpha, \beta)}_{32} =\ &\alpha^2\beta(\alpha + \beta) s_{1} + 2\alpha\beta(\alpha+\beta) s_{2} + \alpha^2 (\alpha + \beta) s_{11} + \beta(\alpha+\beta) s_{3} + 2\alpha(\alpha+\beta) s_{21} \\ &+ (\alpha + \beta) s_{31} + 2\alpha s_{22} + s_{32}\\ g^{(\alpha, \beta)}_{321} =\ &\alpha^2 \beta^2 (\alpha + \beta) s_{1} + 2\alpha \beta^2(\alpha + \beta) s_{2} + 2\alpha^2 \beta(\alpha+\beta) s_{11} + \beta^2(\alpha+\beta) s_{3} + 4\alpha\beta(\alpha + \beta) s_{21} \\ &+ \alpha^2(\alpha + \beta) s_{111}+ 2\beta(\alpha + \beta) s_{31} + 2\alpha (\alpha + \beta) s_{211} + 4\alpha \beta s_{22} + 2\beta s_{32} + 2\alpha s_{221} + (\alpha + \beta)s_{311}+ s_{321}\\ g^{(\alpha, \beta)}_{33} =\ &\alpha^2\beta(\alpha+\beta)^2 s_{1} + 2\alpha\beta(\alpha+\beta)^2 s_{2} + \alpha^2(\alpha+\beta)^2 s_{11} + \beta(\alpha +\beta)^2 s_{3} + 2\alpha (\alpha + \beta)^2 s_{21} \\ &+ (\alpha + \beta)^2 s_{31} + \alpha (3\alpha + 2\beta) s_{22} + (2\alpha + \beta) s_{32} + s_{33}\\ g^{(\alpha, \beta)}_{333} =\ & \alpha^2\beta^2(\alpha+\beta)^4 s_{1} + 2\alpha\beta^2(\alpha+\beta)^4 s_{2} + 2\alpha^2\beta(\alpha+\beta)^4 s_{11} + \beta^2(\alpha+\beta)^4 s_{3} + 4\alpha\beta(\alpha + \beta)^4 s_{21} \\ &+ \alpha^2(\alpha+\beta)^4 s_{111} + 2\beta(\alpha + \beta)^4 s_{31} + 2\alpha (\alpha+\beta)^4 s_{211} + 2\alpha\beta(\alpha+\beta)(3\alpha^2 + 7\alpha\beta + 3\beta^2) s_{22} \\ &+ \beta(\alpha+\beta)(4\alpha^2 + 9\alpha\beta + 3\beta^2) s_{32} + (\alpha+\beta)^4 s_{311} + \alpha(\alpha+\beta)(3\alpha^2 + 9\alpha\beta + 4\beta^2) s_{221}\\ &+ 2\beta(\alpha+\beta)(\alpha + 2\beta) s_{33} + 2(\alpha + \beta)(\alpha^2 + 3\alpha\beta + \beta^2) s_{321} + 2\alpha(\alpha+\beta)(2\alpha + \beta) s_{222} \\ &+(\alpha+\beta)(\alpha + 3\beta) s_{331}+(\alpha+\beta)(3\alpha + \beta) s_{322}+2(\alpha+\beta)s_{332} + s_{333}\end{aligned}$$ ]{} \[schur1\] The following dual formulas hold $$s_{\mu} = \sum_{\nu} f_{\nu/\mu}{(\alpha, \beta)} G^{(-\alpha, -\beta)}_{\nu}, \qquad g^{(\alpha, \beta)}_{\mu} = \sum_{\nu} f_{\mu/\nu}{(\alpha, \beta)} s_{\nu}.$$ First recall the following expansion given in [@lenart] (restated with parameter $\beta$ here) $$s_{\lambda} = \sum_{\mu} (-\beta)^{|\mu/\lambda|} f_{\mu/\lambda} G^{\beta}_{\mu},$$ where $f_{\mu/\lambda}$ is the number of elegant tableaux of $\mu/\lambda$ (see Definition \[elegant\]). Therefore, $$\label{x} s_{\lambda}\left(\frac{x}{1 + \alpha x} \right) = \sum_{\nu} (\alpha + \beta)^{|\nu/\lambda|} f_{\nu/\lambda} G^{(-\alpha, -\beta)}_{\nu}$$ We have (from [@molev] Theorem 3.20, specializing all $a_i = \alpha$) $$\label{x2} s_{\mu} = \sum_{\lambda} (-1)^{n(\lambda/\mu)} \alpha^{|\lambda/\mu|} \psi_{\lambda/\mu} s_{\lambda}\left(\frac{x}{1 + \alpha x} \right),$$ where $\psi_{\lambda/\mu}$ is the number of dual hook tableaux (see Definition \[dht\]); here $\lambda, \mu$ have the same number $b$ of boxes on the main diagonal, $n(\lambda/\mu)$ is the number of boxes in the lower component of $\lambda/\mu$. Combining , we thus have $$\begin{aligned} s_{\mu} &= \sum_{\lambda} (-1)^{n(\lambda/\mu)} \alpha^{|\lambda/\mu|} \psi_{\lambda/\mu} \sum_{\nu} (\alpha + \beta)^{|\nu/\lambda|} f_{\nu/\lambda} G^{(-\alpha, -\beta)}_{\nu} \\ &= \sum_{\nu} G^{(-\alpha, -\beta)}_{\nu} \sum_{\lambda} (\alpha + \beta)^{|\nu/\lambda|} (-1)^{n(\lambda/\mu)} \alpha^{|\lambda/\mu|} f_{\nu/\lambda} \psi_{\lambda/\mu} \\ &= \sum_{\nu} f_{\nu/\mu}{(\alpha, \beta)} G^{(-\alpha, -\beta)}_{\nu}.\end{aligned}$$ The last step followed by Lemma \[ll\]. The Schur expansion of the dual $g^{(\alpha, \beta)}_{\lambda}$ polynomials implies from the duality of families $\{g^{(\alpha, \beta)}_{\lambda}\}$ and $\{G^{(-\alpha, -\beta)}_{\lambda}\}$. The dual polynomials $g^{(\alpha, \beta)}_{\lambda}$ are Schur-positive ($f_{\mu/\nu}(\alpha, \beta) \in \mathbb{Z}_{\ge 0}[\alpha, \beta]$). Jacobi-Trudi type identities {#jt} ============================ In this section we prove Jacobi-Trudi type determinantal identities for the functions $G^{(\alpha, \beta)}_{\lambda}$, $g^{(\alpha, \beta)}_{\lambda}$. The following determinantal formulas hold: $$\label{56} G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n) = \det\left[ \widetilde{h}^{(i)}_{\lambda_i - i + j} \right]_{1 \le i,j \le n},$$ where $$\widetilde{h}^{(i)}_{p} = \sum_{k} (\alpha + \beta)^k \binom{i-1}{k} h_{p+k}\left(\frac{x_1}{1 - \alpha x_1}, \ldots, \frac{x_n}{1 - \alpha x_n} \right),$$ $$G^{(\alpha, \beta)}_{\lambda}(x_1, \ldots, x_n) = \det\left[ \widetilde{e}^{(i)}_{\lambda'_i - i + j} \right]_{1 \le i,j \le n},$$ $$\widetilde{e}^{(i)}_{p} = \sum_{k} (\alpha + \beta)^k \binom{i-1}{k} e_{p+k}\left(\frac{x_1}{1 + \beta x_1}, \ldots, \frac{x_n}{1 + \beta x_n} \right).$$ For $G^{\beta}_{\lambda}$, the formula was given in [@kir1] and hence implies here via $\beta \to \alpha + \beta$ and $x_i \to \frac{x_i}{1 - \alpha x_i}$. The second formula is a dual version. \[jtg\] The following determinantal identities hold: $$g^{(\alpha, \beta)}_{\lambda} = \det \left[\widetilde{g}^{(i)}_{\lambda_i - i, j} \right]_{1 \le i,j \le \ell(\lambda)}, \qquad \widetilde{g}^{(i)}_{p, j} = \sum_{k} \widetilde{f}^{(i)}_{p, k} h_{k+j},$$ $$g^{(\beta, \alpha)}_{\lambda} = \det \left[\bar{g}^{(i)}_{\lambda'_i - i, j} \right]_{1 \le i,j \le \ell(\lambda')}, \qquad \bar{g}^{(i)}_{p, j} = \sum_{k} \widetilde{f}^{(i)}_{p, k} e_{k+j},$$ where $$\widetilde{f}^{(i)}_{p, q} = \sum_{k = -\infty}^{\infty} \binom{p+i- k - 1}{i-1}\binom{k}{k - q} (\alpha + \beta)^{p - k} \alpha^{k - q}.$$ Note: Instead of $\widetilde{f}^{(i)}_{p, k}$ (Lemma \[ll\]) we can take positive expressions ${f}^{(i)}_{p, k}$ from Proposition \[prop8\], . The formulas imply from the Cauchy-Binet formula, Schur expansion in Theorem \[schur1\], and determinantal expressions for the coefficients $f_{\mu/\nu}(\alpha, \beta)$ (Lemma \[ll\], Proposition \[prop8\]). The second (dual) formula implies by applying the involution $\omega$. Let $$\begin{aligned} h^{(m)}_{n}(\beta) &:= h_{n}(\underbrace{\beta, \ldots, \beta}_{m \text{ times}}, x_1, x_2, \ldots) = \sum_{k \ge 0} \beta^k \binom{m + k -1}{k} h_{n - k}, \\ e^{(m)}_{n}(\beta) &:= e_{n}(\underbrace{\beta, \ldots, \beta}_{m \text{ times}}, x_1, x_2, \ldots) = \sum_{k \ge 0} \beta^k \binom{m}{k} e_{n - k}.\end{aligned}$$ For $(\alpha, \beta) = (0,\beta)$ the determinantal formulas above can be refined to the following $$\begin{aligned} g^{(0,\beta)}_{\lambda} = \det\left[ h^{(i -1)}_{\lambda_i - i + j}(\beta) \right]_{1 \le i,j \le \ell(\lambda)} = \det\left[ e^{(\lambda'_i -1)}_{\lambda'_i - i + j}(\beta) \right]_{1 \le i,j \le \ell(\lambda')}\end{aligned}$$ ($g^{(0,1)} = g_{\lambda}$ and for $g_{\lambda}$ the last formula was given in [@sz]); for $(\alpha, \beta) = (\alpha,0)$ we have $$\begin{aligned} g^{(\alpha,0)}_{\lambda} &= \omega(g^{(0, \alpha)}_{\lambda'})= \det\left[ \omega(h^{(i -1)}_{\lambda'_i - i + j}(\alpha)) \right]_{1 \le i,j \le \ell(\lambda')} = \det\left[ \omega(e^{(\lambda_i -1)}_{\lambda_i - i + j}(\alpha)) \right]_{1 \le i,j \le \ell(\lambda)}$$ For $\alpha + \beta = 0$ we have $$\begin{aligned} g^{(\alpha, -\alpha)}_{\lambda} &= \det\left[\sum_{k}\binom{\lambda_i - i}{k} \alpha^k h_{k + j} \right]_{1 \le i,j \le \ell(\lambda)} = \det\left[\sum_{k}\binom{\lambda'_i - i}{k} \alpha^k e_{k + j} \right]_{1 \le i,j \le \ell(\lambda')}\end{aligned}$$ Jacobi-Trudi identities usually imply Giambelli formulas (determinantal formulas indexed by hooks), which is not (directly) the case here for the $G, g$ functions. For example, $G^{(0,1)}_{22}(x_1, x_2) = x_1^2 x_2^2,$ but $$\det\begin{bmatrix} G^{(0, 1)}_{21}(x_1, x_2) & G^{(0, 1)}_{2}(x_1, x_2) \\ G^{(0, 1)}_{11}(x_1, x_2) & G^{(0, 1)}_{1}(x_1, x_2) \end{bmatrix} = x_1^2 x_2^2 (x_1 + 1)(x_2 + 1).$$ Similarly for dual $g$ polynomials computation shows that $g^{(\alpha,\beta)}_{22}=(\alpha + \beta) g^{(\alpha, \beta)}_{21} + s_{22},$ but $$\det\begin{bmatrix} g^{(\alpha, \beta)}_{21} & g^{(\alpha, \beta)}_{2} \\ g^{(\alpha, \beta)}_{11} & g^{(\alpha, \beta)}_{1} \end{bmatrix} = g^{(\alpha, \beta)}_{21} g^{(\alpha, \beta)}_{1} - g^{(\alpha, \beta)}_{2} g^{(\alpha, \beta)}_{11} = s_{22}.$$ (Schur expansions of $g^{(\alpha, \beta)}_{\lambda}$ are provided in Table 1.) Lascoux and Naruse in [@ln] gave the following analog of Giambelli’s formula for $g_{\lambda}$ $$g_{\lambda} = \det\left[\widetilde{g}^{(i,j)}_{(p_i | q_j)} \right]_{1 \le i,j \le d},$$ where $\lambda = (p_1, \ldots, p_d | q_1, \ldots, q_d)$ written in Frobenius notation, and $$\widetilde{g}^{(i,j)}_{(p | q)} = \sum_{k, s} \binom{k + i - 2}{k} \binom{s + j - 2}{s} g_{(p - k | q - s)} + \sum_{t} \binom{p + i - t}{p}\binom{q + j - t}{q}.$$ So it would be interesting to see what versions of Giambelli identities hold for $G^{(\alpha, \beta)}_{\lambda}, g^{(\alpha, \beta)}_{\lambda}$. Combinatorial proofs of determinantal formulas ---------------------------------------------- Roughly speaking, if connection constants in Schur expansions have determinantal expressions, then by the Cauchy-Binet formula, the whole expression is also some determinant (the Schur function satisfies the usual Jacobi-Trudi formula). So, if combinatorial lattice interpretation is known for connection coefficients, this approach also provides a combinatorial proof for the corresponding determinantal identity (the Cauchy-Binet formula can be proved using the Lindström-Gessel-Viennot Lemma [@gv] by ‘gluing’ two graphs together, sinks of the first to the sources of the second). For the dual stable Grothendieck polynomials $g_{\lambda},$ the determinantal formula $$g_{\lambda} =\det\left[ e^{(\lambda'_i -1)}_{\lambda'_i - i + j}(1) \right]_{1 \le i,j \le \ell(\lambda')} = \det\left[\sum_{k = 0}^{\lambda'_i - 1} \binom{\lambda'_i-1}{k} e_{\lambda'_i - i + j - k} \right]_{1 \le i,j \le \ell(\lambda')}$$ is implied from Theorem \[jtg\]. To illustrate the idea how to pass to combinatorial constructions, we now show the lattice path interpretation which proves this formula for $g_{\lambda}$. Something similar can be designed for a general case using the lattice path interpretation of the coefficients $f^{(\alpha, \beta)}_{\mu/\nu}$. Let us prove the identity for a finite number of variables $x_1, \ldots, x_n$. Consider the lattice grid with allowed up and diagonal unit steps. All up steps have weights $1$, the upper half-plane has diagonal weights $x_{n - j + 1}$ for passing from $y$-coordinate $j - 1$ to $j$, and the lower half-plane has diagonal weights $1$. Let $\ell = \ell(\lambda')$ and consider nonintersecting lattice path systems from the points $A = (A_1, \ldots, A_{\ell})$ to the points $B = (B_1, \ldots, B_{\ell})$, where $A_i$ has coordinates $(-\lambda'_i + i - 1, 1 - \lambda'_i)$ and $B_j$ has coordinates $(j - 1, n)$. See Figure \[fff\]. It is easy to see that $$w(A_i \to B_j) = \sum_{k = 0}^{\lambda'_i - 1} \binom{\lambda'_i-1}{k} e_{\lambda'_i - i + j - k}.$$ Hence, $\det\left[w(A_i \to B_j) \right]$ is a weighted sum over all nonintersecting lattice path systems from $A$ to $B$. (0,0) to (7.5,0); \[fill\] at (5+1,4) ; \[fill\] at (0+1,-4) ; (0+1,-4) to (0+2+1,-4+2) to (0+2+1,-4+2+1) to (0+2+1+1,-4+2+1+1); (0+2+1+1,-4+2+1+1) to (0+2+1+1,-4+2+1+1+1) to (0+2+1+1+1,-4+2+1+1+1+1) to (0+2+1+1+1,-4+2+1+1+1+1+1) to (0+2+1+1+1+1,-4+2+1+1+1+1+1+1); at (0.2,-4) [$A_i$]{}; at (6.7, 4) [$B_j$]{}; (1,0) to (4,0); at (2.2,0.5) [$\lambda'_i - 1 - k$]{}; at (6.7, 1.5) [$e_{k + 1 - i + j}$]{}; (1,0) to (1,-4); at (0,-2) [$\lambda_i' - 1$]{}; at (4.2,-2) [$\binom{\lambda'_i - 1}{k}$]{}; (5, -4.5) to (5,5.0); (0,0) to (7.5,0); in [0,...,7]{} (, -4.5) to (,4.5); in [0,...,4]{} (0, ) to (4.5-,4.5); in [1,...,3]{} (0, -) to (4.5+,4.5); (0, -4) to (7.5,3.5); (0.5, -4.5) to (7.5,2.5); (1.5, -4.5) to (7.5,1.5); (2.5, -4.5) to (7.5,0.5); (3.5, -4.5) to (7.5,-0.5); (4.5, -4.5) to (7.5,-1.5); (5.5, -4.5) to (7.5,-2.5); (6.5, -4.5) to (7.5,-3.5); \[fill\] at (5,4) ; \[fill\] at (6,4) ; \[fill\] at (7,4) ; \[fill\] at (0,-4) ; \[fill\] at (3,-2) ; \[fill\] at (5,-1) ; (0,-4) to (0+2,-4+2) to (0+2,-4+2+1) to (0+2+1,-4+2+1+1); (0+2+1,-4+2+1+1) to (0+2+1,-4+2+1+1+1) to (0+2+1+1,-4+2+1+1+1+1) to (0+2+1+1,-4+2+1+1+1+1+1) to (0+2+1+1+1,-4+2+1+1+1+1+1+1); (3,-2) to (3+1,-2+1) to (3+1,-2+1+1); (3+1,-2+1+1) to (3+1+1,-2+1+1+1) to (3+1+1,-2+1+1+1+1) to (3+1+2,-2+1+2+2) to (3+1+2,-2+1+2+2+1); (5,-1) to (5+1,-1+1); (5+1,-1+1) to (5+1,-1+1+1) to (5+1,-1+1+1+1) to (5+1+1,-1+1+1+1+1) to (5+1+1,-1+1+1+1+1+1); at (4.4,3.9) [1]{}; at (5.4,2.9) [2]{}; at (6.4,2.9) [2]{}; at (3.4,1.9) [3]{}; at (4.4,0.9) [4]{}; at (2.4, -0.1) ; at (5.4, -0.1) ; at (3.4, -1.1) ; at (1.4, -2.1) ; at (0.4, -3.1) ; (1,0.8) to (2.9,0.8); at (5.2,0) [1]{} & [2]{} & [2]{}\ [3]{} & [4]{} & [1]{}\ [1]{} & [2]{}\ [3]{}\ [4]{} ; The Schur expansion $$\label{gs} g_{\lambda} = \sum_{\mu} f_{\lambda/\mu} s_{\mu}$$ was proved in [@lp] bijectively by mapping each RPP with a given weight to a pair of SSYT of shape $\mu$ (with the same weight) and an elegant filling (see Definition \[elegant\]) of shape $\lambda/\mu$. Note that each nonintersecting lattice path system from $A$ to $B$ (geometrically one can see here that each $A_i$ should go to $B_i$) decomposes into two parts: below the $y = 0$ line (‘underwater’) and the upper part having weights $x_i$. (For instance, in Figure \[fff\] we have the weight $x_1 x_2^2 x_3 x_4$.) The upper part can be translated into an SSYT of some shape $\mu \subseteq \lambda$; and the ‘underwater’ part into an elegant tableau of the remaining shape $\lambda/\mu$ which repeats the decomposition . So, $g_{\lambda} = \det\left[ w(A_i \to B_j)\right]_{1 \le i,j \le \ell}$. $\square$ Instead of counting elegant tableaux with all weights $1$, we can similarly give weights to numbers in these fillings, or equivalently give some weights to the ‘underwater’ part. So, if we put the weight $t_i$ to the number $i$ in an elegant tableaux or to diagonal steps ‘underwater’, the resulting function will correspond to the [*refined*]{} dual stable Grothendieck polynomials $\widetilde{g}_{\lambda/\mu}(x,t)$ introduced by Galashin, Grinberg, and Liu in [@ggl] where they gave Bender-Knuth type involutions for proving that $g_{\lambda/\mu}$ is a symmetric function. From our setting, the Schur decomposition rewrites with $t = (t_1, t_2, \ldots)$ parameters as follows $$\widetilde{g}_{\lambda}(x,t) = \sum_{\mu} f_{\lambda/\mu}(t) s_{\mu},$$ where $f_{\lambda/\mu}(t) = \sum_{\text{elegant tableaux of } \lambda/\mu} \prod t_i^{a_i}$ ($a_i$ is the number of times $i$ occurs in an elegant tableau). By the same argument as the proof above (but with the $t$ parameters added), we obtain the following Jacobi-Trudi formula $$\widetilde{g}_{\lambda}(x,t) = \det \left[ \sum_{k = 0}^{\lambda'_i - 1} e_{k}(t_1, \ldots, t_{\lambda'_i - 1}) e_{\lambda'_i - i + j - k}(x) \right]_{1 \le i,j \le \ell(\lambda')}$$ which was recently conjectured by Darij Grinberg (personal communication) in a more general setting for skew shapes. More formulas for $G_{\lambda}$ ------------------------------- The functions $G_{\lambda}$ also satisfy the following Jacobi-Trudi type formula $$G_{\lambda} = \det\left[\sum_{k \ge 0} \binom{\lambda'_i - 1 + k}{k} e_{\lambda'_i - i + j + k} \right]_{1 \le i,j \le \ell(\lambda')},$$ which can be proved similarly as the formula for $g_{\lambda}$ above, taking into account the known Schur expansions of $G_{\lambda}$ [@lenart]. Hence we also obtain the following formulas for $G^{(\alpha, \beta)}_{\lambda}$ $$G^{(\alpha, \beta)}_{\lambda} = \det\left[\sum_{k \ge 0} \binom{\lambda'_i - 1 + k}{k} (\alpha + \beta)^k e_{\lambda'_i - i + j + k}\left(\frac{x}{1 - \alpha x}\right) \right]_{1 \le i,j \le \ell(\lambda')}.$$ Grothendieck polynomials indexed by permutations {#ggw} ================================================ The polynomials $G_{\lambda}$ arise as a special case of stable Grothendieck polynomials $G_{w}$ indexed by permutations $w \in S_n$. The symmetric group $S_n$ acts on $\mathbb{Z}[x_1,\ldots, x_n]$ by permuting the indices of variables. Let $s_i = (i, i+1)$ denote a simple transposition in $S_n$. Define the operators $\pi_i$ acting on polynomials in $\mathbb{Z}[x_1, \ldots, x_n]$ as $$\pi_i := \frac{1 - s_i}{x_i - x_{i+1}} (1 - x_{i+1})$$ and for any permutation $w \in S_n$ with a reduced decomposition $w = s_{i_1} \ldots s_{i_{\ell}}$ set $ \pi_{w} := \pi_{i_1} \cdots \pi_{i_{\ell}}, $ which is independent of the choice of the reduced word (since the operators $\pi_i$ satisfy the braid relations). The [*Grothendieck polynomials*]{} $\mathfrak{G}^{}_w$ ($w \in S_n$) introduced by Lascoux and Schützenberger [@ls] is defined as follows: $$\mathfrak{G}_{w_0} := x_1^{n-1} x_{2}^{n-2} \cdots x_{n-1}, \qquad \mathfrak{G}_{w} := \pi_{w^{-1}w_0} \mathfrak{G}_{w_0},$$ where $w_0 = n (n-1) \ldots 2 1 \in S_n$ is the longest permutation. The double Grothendieck polynomials $\mathfrak{G}_w = \mathfrak{G}_w(x; y)$ of two sets of variables $x, y$ are defined in the same way, but $$\mathfrak{G}_{w_0}(x;y) = \prod_{i + j \le n}(x_i + y_j - x_i y_j).$$ Note that double Grothendieck polynomials satisfy the duality $\mathfrak{G}_{w}(x;y) = \mathfrak{G}_{w^{-1}}(y;x)$. For any permutation $w \in S_n$ and $m \in \mathbb{Z}_{\ge 0}$, let $1^m \times w \in S_{n + m}$ be a permutation which fixes $1, \ldots, m$ and maps $i$ to $w(i - m) + m$ for $i > m$. Fomin and Kirillov [@fk; @fk1] showed that the polynomials $\mathfrak{G}_{1^m \times w}$ eventually stabilize to the symmetric power series $$G_{w} := \lim_{m \to \infty} \mathfrak{G}_{1^m \times w}$$ called [*stable Grothendieck polynomials*]{} (indexed by permutations). When $w$ is a [*Grassmannian permutation*]{}, i.e. has a single descent at some position $d$, the stable Grothendieck polynomial $G^{}_{\lambda}$ is indexed by an integer partition $\lambda = (\lambda_1 \ge \cdots \ge \lambda_d)$, where $\lambda_i = w(d+1 - i) - (d + 1 - i)$. Various dualities hold for the double versions: $G_{\lambda}(x;y) = G_{\lambda'}(y;x)$, $G_{w}(x;y) = G_{w^{-1}}(y;x)$ and $G_{w}(x;y) = G_{w_0 w w_0}(y;x)$, where $w_0 = n \cdots 21 \in S_n$ is the longest permutation [@buch]. The series $G_{w}(1 - e^x; 1 - e^{-y})$ are [*super-symmetric*]{}, i.e. if for any $i$ we put $x_i = y_i = t,$ the resulting function is independent of $t$ (hence, considering $G$ of one set of variables is enough). Similarly, we may define the symmetric functions $G^{(\alpha, \beta)}_w(x_1, x_2, \ldots)$. First, extend the Grothendieck polynomials to the polynomials $\mathfrak{G}^{(\alpha + \beta)}_{w}$ via the operators $$\tilde\pi_i = \frac{1 - s_i}{x_i - x_{i+1}}(1 + (\alpha + \beta) x_{i+1}).$$ Taking the stable limit we define the power series $G^{(\alpha + \beta)}_w$ and finally substituting $x_i \to \frac{x_i}{1 - \alpha x_i}$ we obtain the functions $G^{(\alpha, \beta)}_w(x_1, x_2, \ldots)$. Note that in the double versions $G_{w}(x;y)$ we can put $y_j \to \frac{y_j}{1 - \beta y_j}$ to obtain the double versions of $G^{(\alpha, \beta)}_{w}(x;y)$ that are [*super-symmetric*]{} functions. The functions $G^{(\alpha, \beta)}_{w}(x_1, x_2, \ldots)$ ($w \in S_n$) have the following properties: - $G^{(\alpha, \beta)}_{w}$ is a finite linear combination of the elements $\{G^{(\alpha, \beta)}_{\lambda} \}$; - $\omega(G^{(\alpha, \beta)}_{w}) = G^{(\beta, \alpha)}_{w^{-1}} = G^{(\beta, \alpha)}_{w_0 w w_0},$ where $w_0 \in S_n$ is the longest permutation. We use Buch’s result ([@buch], Theorem 6.13) that $$\label{gw} G_{w} = \sum_{\lambda} a_{w, \lambda} G_{\lambda} \in \Gamma$$ can be expressed as a finite linear combination of $G_{\lambda}.$ It is then easy to see (from the definition) that the same holds for $$G^{(\alpha, \beta)}_w = \sum_{\lambda} a^{(\alpha, \beta)}_{w, \lambda} G^{(\alpha, \beta)}_{\lambda} \in \Gamma^{(\alpha, \beta)},$$ where $a^{(\alpha, \beta)}_{w, \lambda} = (\alpha + \beta)^{|\lambda| - \ell(w)} a_{w,\lambda}$. Hence to prove (ii), we have to show that $\alpha_{w, \lambda} = \alpha_{w^{-1},\lambda'}$ and $\alpha_{w, \lambda} = \alpha_{w_0w w_0,\lambda'}$. It is known that $\mathfrak{G}_{w}(x;y) = \mathfrak{G}_{w^{-1}}(y;x)$ and therefore, $G_{w}(x;y) = G_{w^{-1}}(y;x)$ (since in the limit we have $(1^m \times w)^{-1} = 1^m \times w^{-1}$). It is also known that $G_{w}(x;y) = G_{w_0 w w_0}(y;x)$ (Fomin’s lemma [@buch]). Combining these properties with $G_{\lambda}(x;y) = G_{\lambda'}(y;x)$ and we conclude both symmetries $\alpha_{w, \lambda} = \alpha_{w^{-1},\lambda'}$ and $\alpha_{w, \lambda} = \alpha_{w_0w w_0,\lambda'}$. Theory of canonical bases {#skl} ========================= We now discuss the way how canonical stable Grothendieck polynomials can be put in context of the Kazhdan-Lusztig theory of canonical bases [@kl]. First we propose a general scheme how canonical bases can be derived for bases of symmetric functions. For any element $a \in \mathbb{Z}[\alpha, \beta]$,[^6] let $a \to \overline{a}$ be the involution switching $\alpha$ and $\beta$. Let $\{ u_{\lambda}\}$ be a $\mathbb{Z}[\alpha, \beta]$-basis of $\Lambda$. For $f \in \Lambda$ define the [*bar*]{} involution $f \to \overline{f}$ given by e.g. $\overline{f} = \sum \overline{a_{\lambda}} s_{\lambda}$ for $f=\sum a_\lambda s_\lambda$. Suppose that $$\label{wu} \omega(\overline{u_{\lambda'}}) \in u_{\lambda} + \sum_{\mu < \lambda} \mathbb{Z}[\alpha, \beta] u_{\mu}$$ for a (partial) order $<$ on partitions (say, compatible with the weight of partition, e.g. lexicographic, or by inclusion of diagrams). (Note that the standard involution $\omega$ commutes with the [bar]{} involution.) Define $$\mathbb{Z}[\alpha > \beta] := \{\sum_{j < i < \infty} a_{ij} \alpha^i \beta^j : a_{ij} \in \mathbb{Z} \},\quad \mathbb{Z}[\alpha \ge \beta] := \{\sum_{j \le i < \infty} a_{ij} \alpha^i \beta^j : a_{ij} \in \mathbb{Z} \} \subset \mathbb{Z}[\alpha, \beta]$$ (i.e. polynomials with terms having powers of $\alpha$ greater (or equal) than powers of $\beta$). \[kazl1\] There is at most one $\mathbb{Z}[\alpha, \beta]$-basis $\{C_{\lambda} \}$ of $\Lambda$ satisfying: - $\omega(C_{\lambda}) = \overline{C_{\lambda'}}$; - $C_{\lambda} \in u_{\lambda} + \sum_{\mu < \lambda} \mathbb{Z}[\alpha > \beta] u_{\mu}$. Such basis $\{C_{\lambda} \}$, if exists, is a candidate for being a [*canonical basis*]{} with the respect to the given [*pre-canonical structure*]{} consisting of the [*standard*]{} basis $\{ u_{\lambda} \}$ satisfying and the involution $\omega$ composed with the [*bar*]{} and index transposition. The proof of this property is similar to the proof of uniqueness of Kazhdan-Lusztig canonical bases (for Hecke algebras) [@kl]. We show this argument in a more specific situation in the proof of Theorem \[gab\] below. Consider now instead of the generic basis $\{u_{\lambda} \}$, the $\mathbb{Z}[\beta]$-basis of $\Lambda$ given by the dual stable Grothendieck polynomials $\{g^\beta_{\lambda} \}$. Since, $g^\beta_{\lambda} = s_{\lambda} + \text{lower degree terms,}$ we have $$\omega(\overline{g^{\beta}_{\lambda'}}) = g^{(\alpha, 0)}_{\lambda} = g^\beta_{\lambda} + \sum_{\mu \subset \lambda} q_{\lambda \mu}(\alpha, \beta) g^\beta_{\mu}, \qquad q_{\lambda \mu}(\alpha, \beta) \in \mathbb{Z}[\alpha, \beta].$$ For example, $$\begin{aligned} g^{(\alpha, 0)}_{21} &= g^\beta_{21} + \alpha g^\beta_{11} - \beta g^\beta_{2} - \alpha \beta g^\beta_{1}\\ g^{(\alpha, 0)}_{22} &= g^\beta_{22} + (\alpha - \beta) g^\beta_{21} + \alpha^2 g^\beta_{11} - \alpha\beta g^\beta_{2} - \alpha \beta (\alpha - \beta) g^\beta_{1}. \end{aligned}$$ The dual canonical polynomials $g^{(\alpha, \beta)}_{\lambda}$ satisfy the condition (i): we have $\omega(g^{(\alpha, \beta)}_{\lambda}) = \overline{g^{(\alpha, \beta)}_{\lambda'}}$. Let us consider their expansion in the (standard) $\{g^\beta_{\lambda}\}$ basis. Let $$g^{(\alpha, \beta)}_{\lambda} = g^\beta_{\lambda} + \sum_{\mu \subset \lambda} p_{\lambda \mu}(\alpha, \beta) g^\beta_{\mu}, \qquad p_{\lambda \mu}(\alpha, \beta) \in \mathbb{Z}[\alpha, \beta].$$ For example, some of the $g^\beta_{\lambda}$ expansions look as follows: $$\begin{aligned} g_{21}^{(\alpha, \beta)} &= g^\beta_{21} + \alpha g^\beta_{11} \\ g_{22}^{(\alpha, \beta)} &= g^\beta_{22} + \alpha g^\beta_{21} + \alpha(\alpha + \beta) g^\beta_{11}\\ g_{31}^{(\alpha, \beta)} &= g^\beta_{31} + 2\alpha g^\beta_{21} + \alpha^2 g^\beta_{11}\\ g_{32}^{(\alpha, \beta)} &= g^\beta_{32} + 2\alpha g^\beta_{22} + \alpha g^\beta_{31} + 2\alpha^2 g^\beta_{21} + \alpha^2(\alpha + \beta) g^\beta_{11}\\ g_{33}^{(\alpha, \beta)} &= g^\beta_{33} + 2\alpha g^\beta_{32} + \alpha(3\alpha + 2\beta) g^\beta_{22} + \alpha^2 g^\beta_{31} + \alpha^2(2\alpha + \beta) g^\beta_{21} + \alpha^2 (\alpha + \beta)^2 g^\beta_{11}.\end{aligned}$$ Here we see that $g^{(\alpha, \beta)}_{\lambda}$ do not fall into the characterization (ii) of Proposition \[kazl1\], e.g. $p_{(22),(11)}(\alpha, \beta) = \alpha^2 + \alpha\beta \not\in \mathbb{Z}[\alpha > \beta]$. However, we can change the condition (ii) and get a similar characterization, if the polynomials $p_{\lambda \mu}(\alpha, \beta)$ satisfy some additional properties which we ask below. Say that $p \in \mathbb{Z}[\alpha, \beta]$ has a [*free*]{} term, if it contains a term $(\alpha \beta)^i$ for some $i \ge 1$. Which of the following [*positivity*]{} properties hold for the polynomials $p_{\lambda \mu}(\alpha, \beta)$? - $p_{\lambda \mu}(\alpha, \beta) \in \mathbb{Z}_{\ge 0}[\alpha, \beta]$ (i.e. polynomials in $\alpha, \beta$ with positive integer coefficients); - $p_{\lambda \mu}(\alpha, \beta) \in \mathbb{Z}_{}[\alpha \ge \beta]$; - at most one of $p_{\lambda \mu}(\alpha, \beta)$, $p_{\lambda' \mu'}(\alpha, \beta)$ has a free term. If (a), (b), (c) hold, then $g^{(\alpha, \beta)}_{\lambda}$ is a unique polynomial satisfying these conditions and self-duality (i). Otherwise, it is perhaps needed to describe for which $\lambda, \mu$, the polynomial $p_{\lambda \mu}(\alpha, \beta)$ has a free term. Then if (b), (c) hold, $\{g^{(\alpha, \beta)}_{\lambda}\}$ is again a unique basis falling into the characterization. (The coefficients $p_{\lambda \mu}(\alpha, \beta)$ are some analogs of Kazhdan-Lusztig polynomials.) Note that (for stable Grothendieck polynomials) there is no other candidate for canonical basis with exchange coefficients in $\mathbb{Z}[\alpha > \beta]$; i.e. when we construct these canonical polynomials, we are forced to consider $\mathbb{Z}[\alpha \ge \beta]$. For example, computing the coefficients $p_{\lambda \mu}(\alpha, \beta)$ recursively, we obtain that $p_{(22), (11)}(\alpha, \beta) - p_{(22), (2)}(\beta, \alpha) = \alpha^2 + \alpha\beta$ and so if we know positivity properties, we may conclude that $p_{(22), (11)}(\alpha, \beta) = \alpha^2 + \alpha \beta$ and $p_{(22), (2)}(\beta, \alpha) = 0$, which gives a unique $g^{(\alpha, \beta)}_{22}$. However we can replay the same characterization on the ring [*generators*]{} $g^{(\alpha, \beta)}_{(k)}, g^{(\alpha, \beta)}_{(1^k)}$. \[gab\] There is a unique set $\{C_k, k \in \mathbb{Z}_{> 0} \}$ of generators of $\Lambda$ with coefficients in $\mathbb{Z}[\alpha, \beta]$ satisfying: $$C_k \in g^\beta_{(k)} + \sum_{i < k} \mathbb{Z}[\alpha > \beta] g^\beta_{(i)}, \qquad \omega(\overline{C_k}) \in g^\beta_{(1^k)} + \sum_{i < k} \mathbb{Z}[\alpha > \beta] g^\beta_{(1^i)}.$$ The elements $C_k = g^{(\alpha, \beta)}_{(k)}$ coincide with the dual canonical stable Grothendieck polynomials indexed by a single row. Let us first prove that there is at most one such set of elements $\{C_k \}$. Recall that $$\label{geh} g^{\beta}_{(k)} = h_{k}, \qquad g^{\beta}_{(1^k)} = \sum_{i = 1}^k \beta^{k - i} \binom{k - 1}{i - 1} e_i.$$ We will prove by induction on $k-i$ that the coefficients $p_{k,i}(\alpha, \beta), p'_{k,i}(\alpha, \beta) \in \mathbb{Z}[\alpha > \beta]$ given by $$\label{aaa} C_k = g^\beta_{(k)} + \sum_{i < k} p_{k,i}(\alpha, \beta) g^\beta_{(i)}, \qquad \omega(\overline{C_k}) = g^\beta_{(1^k)} + \sum_{i < k} p'_{k,i}(\alpha, \beta) g^\beta_{(1^i)}.$$ are unique. First for $i = k$, $p_{k,k}(\alpha, \beta) = p'_{k,k}(\alpha, \beta) = 1.$ Suppose that we know all coefficients $p_{k,j}, p'_{k,j}$ for $i < j \le k$. Combining both expansions in we obtain $$\label{a4} {C_k} =g^\beta_{(k)} + \sum_{i < k} p_{k,i}(\alpha, \beta) g^\beta_{(i)} = \omega(g^{\alpha}_{(1^k)}) + \sum_{i < k} p'_{k,i}(\beta, \alpha) \omega(g^{\alpha}_{(1^i)}),$$ where by , $$\omega(g^{\alpha}_{(1^k)}) = \sum_{i = 1}^k \alpha^{k - i} \binom{k - 1}{i - 1} h_i = \sum_{i = 1}^k \alpha^{k - i} \binom{k - 1}{i - 1} g^\beta_{(i)}.$$ Hence, from $$\label{a5} g^\beta_{(k)} + \sum_{i < k} p_{k,i}(\alpha, \beta) g^\beta_{(i)} = \sum_{i = 1}^k \alpha^{k - i} \binom{k - 1}{i - 1} g^\beta_{(i)} + \sum_{j < k} p'_{k,j}(\beta, \alpha) \sum_{i = 1}^j \alpha^{j - i} \binom{j - 1}{i - 1} g^\beta_{(i)}.$$ Since, $h_{i} = g^{\beta}_{(i)}$ is a homogeneous component, the coefficients in the term $[g^{\beta}_{(i)}]$ from both sides of coincide, from that we obtain the recurrence relation $$p_{k,i}(\alpha, \beta) - p'_{k,i}(\beta, \alpha) = \alpha^{k - i} \binom{k - 1}{i - 1} + \sum_{i < j < k} p'_{k,j}(\beta, \alpha) \alpha^{j - i} \binom{j - 1}{i - 1}.$$ The r.h.s of the latter equation is known by the induction hypothesis, and so both polynomials $p_{k,i}(\alpha, \beta)$, $p'_{k,i}(\beta, \alpha)$ are also uniquely recovered, since they belong to $\mathbb{Z}[\alpha > \beta]$, $\mathbb{Z}[\beta > \alpha]$ and split the terms accordingly (we do not have any cancellation of terms). Finally, one can easily see that the (algebraically independent) elements $C_k$ given explicitly by the formula $$C_k = \sum_{i = 1}^k \alpha^{k - i} \binom{k - 1}{i - 1} h_i = g^{(\alpha, \beta)}_{(k)}$$ are indeed the unique elements satisfying the given conditions. The set $\{g^{(\alpha, \beta)}_{(k)}, g^{(\alpha, \beta)}_{(1^k)} | k \in \mathbb{Z}_{> 0} \}$ is the unique set of elements of $\Lambda$ satisfying: - $\omega(g^{(\alpha, \beta)}_{(k)}) = g^{(\beta, \alpha)}_{(1^k)}$; - $g^{(\alpha, \beta)}_{(k)} \in g^\beta_{(k)} + \sum_{i < k} \mathbb{Z}[\alpha > \beta] g^\beta_{(i)}$ and $g^{(\alpha, \beta)}_{(1^k)} \in g^\beta_{(1^k)} + \sum_{i < k} \mathbb{Z}[\alpha > \beta] g^\beta_{(1^i)}$. Furthermore, the homomorphism $\varphi :\Lambda \to \Lambda$ defined on the generators by $$\varphi(h_k) = \sum_{i = 1}^k \alpha^{k - i} \binom{k - 1}{i - 1} h_i = g^{(\alpha, \beta)}_{(k)}$$ satisfies $\varphi(g^{(\alpha + \beta)}_{\lambda}) = g^{(\alpha, \beta)}_{\lambda}$. Therefore, under these properties we have the unique (dual) canonical polynomials $g^{(\alpha, \beta)}_{\lambda}$. In addition, the dual family of functions $G^{(\alpha, \beta)}_{\lambda}$ (via the Hall inner product) is also unique. Concluding remarks, special cases {#final} ================================= Dual Hopf algebras ------------------ There are Hopf algebra structures with the bases $\{G^{(\alpha, \beta)}_{\lambda}\},$ $\{g^{(\alpha, \beta)}_{\lambda}\}$. Let $\Gamma^{(\alpha, \beta)} = \bigoplus_{\lambda} \mathbb{Z}[\alpha, \beta] \cdot G^{(\alpha, \beta)}_{\lambda}$ with completion $\hat\Gamma^{(\alpha, \beta)}$ and $\bar\Gamma^{(\alpha, \beta)} = \bigoplus_{\lambda} \mathbb{Z}[\alpha,\beta] \cdot g^{(\alpha, \beta)}_{\lambda}$. Note that $\Gamma^{(0,0)} = \bar\Gamma^{(0,0)} = \Lambda$ as self-dual Hopf algebras with the Schur basis $\{s_{\lambda}\}.$ One can show that $\hat\Gamma^{(\alpha, \beta)}$ and $\bar\Gamma^{(\alpha, \beta)} = \bigoplus_{\lambda} \mathbb{Z}[\alpha,\beta] \cdot g^{(\alpha, \beta)}_{\lambda}$ are Hopf algebras with the antipodes given by $S(G^{(\alpha, \beta)}_{\lambda}(x)) = G^{(\beta, \alpha)}_{\lambda'}(-x),$ and $S(g^{(\alpha, \beta)}_{\lambda}(x)) = g^{(\beta, \alpha)}_{\lambda'}(-x),$ respectively. Then, $\hat\Gamma^{(\alpha, \beta)}$ and $\bar\Gamma^{(-\alpha, -\beta)}$ will be dual Hopf algebras via the Hall inner product. Furthermore, $\bar\Gamma^{(\alpha, \beta)} \cong \Lambda$ as an abstract Hopf algebra (with different bases). Dual noncommutative operators ----------------------------- Let us recall noncommutative operators which build stable Grothendieck polynomials (Section \[dfg\]). In case of the polynomials $G_{\lambda}, g_{\lambda}$, the operators $$\widetilde{u}_i = u_i(1 - d_i), \quad \text{ and }\quad \widetilde{d}_i = (1-d_i)^{-1}d_i = d_i + d_i^2 + \cdots$$ (written in terms of the Schur operators $u_i, d_i$, see section \[dfg\]) seem to be dual to each other. The operators $\widetilde{u}$ build $G_{\lambda/\mu}$ and $\widetilde{d}$ build the dual polynomials $g_{\lambda/\mu}$ (so $(1-d_i)^{-1}d_i$ is an implementation of column deleting (adding) operators used in [@lp].) Some specializations -------------------- By combinatorial formulas, the polynomials $g^{(\alpha, 0)}_{\lambda} = \omega(g^{(0, \alpha)}_{\lambda'})$ can be expressed as generating series similar to Schur polynomials. We have $$g^{(\alpha, 0)}_{\lambda} = \sum_{T \in SSYT(\lambda)} (x | \alpha)^T, \qquad (x | \alpha)^T = \prod_{i \in T} x_{i}^{r_i} (x_{i} + \alpha)^{a_i - r_i},$$ where $r_i$ is a number of rows which contain $i$ and $a_i$ is the total number of entries $i$ in $T$ (this obviously reduces to Schur polynomials when $\alpha = 0$). Combining this formula with the Jacobi-Trudi identity, one can see that $$g^{(1,0)}_{\lambda}((-1)^{n}) = (-1)^{|\lambda|}f^\lambda_n = \det\left[\binom{-n + \lambda_i - 1}{\lambda_i - i + j} \right]_{1 \le i,j \le \ell(\lambda)},$$ where $f^\lambda_n$ is the number of row and column strict Young tableaux of shape $\lambda$ filled with numbers from $\{ 1, \ldots, n\}$. The hook-content formula (e.g. [@ec2]) counts SSYT having maximal entry at most $n$ $$s_{\lambda}(1^n) = \det\left[\binom{n}{\lambda'_i - i + j} \right]_{1 \le i,j \le \ell(\lambda').} = \prod_{(i,j) \in \lambda} \frac{n + j - i}{h_{ij}},$$ where $h_{ij} = \lambda_i - i + \lambda'_j - j + 1$ is the hook-length of the box $(i,j)$. The formula derived from the Jacobi-Trudi identity for $g_{\lambda}$ $$g^{(0,1)}_{\lambda}(1^n) = g_{\lambda}(1^n) = \det\left[\binom{n + \lambda'_i - 1}{\lambda'_i - i + j} \right]_{1 \le i,j \le \ell(\lambda').}$$ gives the number of RPP having maximal entry at most $n$. It does not always factorise as nicely as the formula above, for example $$g_{(532)}(1^n) = \frac{n(n+1)^2(n+2)^2(n+3)(n+4)(15n^3 + 58n^2 + 71n + 24)}{120960}.$$ But in a special case when $\lambda = (k^m)$ has a rectangular shape $m \times k$, it corresponds to the hook-content formula with the shift $n \to n + m - 1$, i.e. the number of RPP of shape $m \times k$ with maximal entry at most $n$ is $$g_{(k^m)}(1^n) = \prod_{(i,j) \in (k^m)} \frac{n + m - 1 + j - i}{h_{ij}} = \prod_{i = 1}^{m} \prod_{j = 1}^k \frac{n + m - 1 + j - i}{m + k - i - j + 1}.$$ More generally, for a rectangular shape $$g_{(k^m)}(x_1, \ldots, x_n) = s_{(k^m)}(x_1, \ldots, x_n, 1^m)$$ is a specialization of Schur function (this can be seen e.g. from our lattice path construction; note that for any $\lambda$, $g_{\lambda}$ is a specialization of flagged Schur function [@ln]). The polynomials $G^{\beta}_{\lambda}$ have some nice specializations when $\lambda$ is a single row or column. For example, $$G^{\beta}_{(k)}(1, q, \ldots, q^{n-1}) + \beta G^{\beta}_{(k+1)}(1, q, \ldots, q^{n-1}) = \left[{n + k-1 \atop k} \right] \prod_{j =0}^{n-1} (1 + \beta q^{j}),$$ where $\left[{n\atop k} \right]$ is the $q$-binomial coefficient; when $n\to \infty$, $$G^{\beta}_{(k)}(1, q, q^2, \ldots) + \beta G^{\beta}_{(k+1)}(1, q, q^2, \ldots) = \frac{\prod_{j \ge 0} (1 + \beta q^{j})}{\prod_{j \ge 1}(1 - q^j)}.$$ The case $\alpha + \beta = 0$, deformed Schur functions ------------------------------------------------------- For $\beta = -\alpha$ we have $$G^{(\alpha, -\alpha)}_{\lambda}(x) = s_{\lambda}\left(\frac{x}{1 - \alpha x}\right), \qquad \omega(G^{(\alpha, -\alpha)}_{\lambda}(x)) = s_{\lambda'}\left(\frac{x}{1 + \alpha x}\right).$$ By Lemma \[ll\] we have $$f_{\nu/\mu}{(\alpha, -\alpha)} = \alpha^{|\lambda/\mu|} (-1)^{n(\lambda/\mu)} \psi_{\lambda/\mu}$$ which is nonzero when $\lambda$ and $\mu$ have the same number of boxes on the main diagonal. Hence, the dual polynomials $g$ have the following Schur expansion $$g^{(\alpha, -\alpha)}_{\lambda} = \sum_{\mu} f_{\lambda/\mu}{(\alpha, -\alpha)} s_{\mu} = \sum_{\mu} \alpha^{|\lambda/\mu|} (-1)^{n(\lambda/\mu)} \psi_{\lambda/\mu} s_{\mu},$$ where $\mu \subset \lambda$ and $\mu$, $\lambda$ have the same number of boxes on the main diagonal; the diagram $\lambda/\mu$ decomposes into two subdiagrams and $n(\lambda/\mu)$ is the number of boxes in a lower part; recall that $\psi_{\lambda/\mu}$ is the number of the so-called dual hook tableaux [@molev], see Definition \[dht\]. In particular, $g^{(\alpha, -\alpha)}_{\lambda} = s_{\lambda}$ (for generic $\alpha$) iff $\lambda$ has a square shape. From combinatorial presentation (RBT) of $g^{(\alpha, \beta)}_{\lambda}$, the property $\alpha + \beta = 0$ means that we do not count tableaux with some inner parts. These tableaux have property that no $2 \times 2$ square contains the same number. Let us call them [*rim tableaux*]{} (RT). We then tile these RT by rim hooks consisting of the same letter such that no two rim hooks are connected horizontally. The weight $w(\rho)$ of any such rim hook $\rho$ is given by $(-1)^{ht(\rho) - 1}\alpha^{|\rho| - 1} ,$ where $ht(\rho)$ is the height of $\rho$. So we have the expression $$g^{(\alpha, -\alpha)}_{\lambda} = \sum_{T} w(T) \prod_{i \ge 1} x_i^{\#\{\text{rim hooks in $T$ containing }i\}},$$ where the sum runs through all tilings of RT of shape $\lambda$ into rim hooks as described above and $w(T)$ is the product of weights of rim hooks in $T$. [100]{} J. Blasiak, S. Fomin, Noncommutative Schur functions, switchboards, and positivity, preprint arXiv:1510.00657, 2015. A. Buch, A Littlewood Richardson rule for the K-theory of Grassmannians, [Acta Math.]{} [**189**]{} (2002), 37–78. S. Fomin, Schur operators and Knuth correspondences, J. Combin. Theory Ser. A, [**72**]{} (1995), 277–292. S. Fomin, C. Greene, Noncommutative Schur functions and their applications, [Discrete Math.]{} **193** (1998), 179–200. S. Fomin, A. N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, [Proc. 6th Intern. Conf. on Formal Power Series and Algebraic Combinatorics]{}, DIMACS, (1994), 183–190. S. Fomin, and A. N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. [**153**]{} (1996), 123–143. P. Galashin, D. Grinberg, G. Liu, Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions, preprint arXiv:1509.03803, 2015. I. Gessel, X. Viennot, Determinants, paths, and plane partitions, preprint, 1989 T. Ikeda, and H. Naruse, K-theoretic analogues of factorial Schur P-and Q-functions, Adv. Math., [**243**]{} (2013), 22–66. D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. math., [**53**]{} (1979), 165–184. A. Kirillov, On some quadratic algebras I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and Reduced polynomials, preprint RIMS-1817, arXiv:1502.00426, 2015. T. Lam and P. Pylyavskyy, Combinatorial Hopf algebras and K-homology of Grassmannians, [Int. Math. Res. Not.]{} [**2007**]{} (2007), rnm 125. A. Lascoux, H. Naruse, Finite sum Cauchy identity for dual Grothendieck polynomials, Proc. Japan Acad., 90, Ser. A (2014) A. Lascoux and M.-P. Schutzenberger, Symmetry and flag manifolds, [Lecture Notes in Mathematics]{} Vol. 996, (1983), 118–144. C. Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb., [**4.1**]{} (2000), 67–82. I. G. Maconald, Symmetric functions and Hall polynomials, Oxford university press, 1998. A. Molev, Comultiplication rules for the double Schur functions and Cauchy identities, Electron. J. Combin, [**16**]{} (2009), R13. K. Motegi, K. Sakai, Vertex models, TASEP and Grothendieck polynomials, J. Phys. A, [**46**]{} (2013), no. 35, p. 26. R. Patrias, P. Pylyavskyy, K-theoretic Poirier-Reutenauer bialgebra, arXiv:1404.4340 preprint, 2014. R. Patrias, Antipode formulas for combinatorial Hopf algebras, preprint arXiv:1501.00710, 2015. M. Shimozono, M. Zabrocki, Stable Grothendieck symmetric functions and $\Omega$-calculus, preprint, 2003. R. Stanley, Enumerative Combinatorics, Vol 2, Cambridge, (1999). [^1]: We will usually write $F$ or $F(x)$ meaning that it is $F(x_1, x_2, \ldots)$. Similarly, $F(x/(1-\alpha x))$ refers to the function $F(x_1/(1 - \alpha x_1), x_2/(1 - \alpha x_2), \ldots)$. If the function $F(x)$ is of a single variable $x$ it will be stated or clear from the context. [^2]: There is another way of defining $s_{\lambda}(u)$, directly converting SSYT into monomials consisting of the $u$ variables. [^3]: The first relations can be changed to the non-local Knuth relations: $u_i u_k u_j = u_k u_i u_j, i \le j < k, |i - k| \ge 2$ and $u_j u_i u_k = u_j u_k u_i, i < j \le k, |i - k| \ge 2$ (which we do not use for our purposes). [^4]: These forests have special structure, so not all lattice forests will correspond to the objects that we define here. [^5]: The function $g^{(\alpha, \beta)}_{\lambda/\mu}(x)$ given by that explicit formula coincides exactly with combinatorial definition for $g^{(\alpha, \beta)}_{\lambda}$ extended for a skew shape. [^6]: One can take a more familiar specialization $(\alpha, \beta) = (q, q^{-1})$.

--- abstract: 'We show that the ideal fluid limit, defined as the existance of a flow frame $u_\mu$ with respect to which the fluid is homogeneus and isotropic, and the consequent independence of the equation of state on $u_\mu$, is incompatible with non-Abelian gauge theory. Instead, the equation of state becomes dependent on $u_\mu$ via modes which are roughly equivalent to ghost modes in the hydrodynamic limit. These modes can be physically imagined as a field of ”purcell swimmers” whose ”arms and legs” are outstretched in Gauge space. Also, vorticity should couple to the Wilson loop via the chromo-electro-magnetic field tensor, which in this limit is not a ”force” but instead represents the polarization tensor of the gluons. We show that because of this coupling vorticity also aquires swirling non-hydrodynamic modes. We then argue that these swirling and swimming non-hydodynamic modes are the manifestation of gauge redunancy within local equilibrium, and speculate on their role in quark-gluon plasma thermalization' author: - Giorgio Torrieri title: Swimming and swirling colorful ghosts --- Introduction ============ The problem of gauge invariance in the ideal hydrodynamic limit is, already at first thought, an involved one. Typically it is developed via an extension of a Vlasov-type equation to a Non-Abelian theory, or a charged ideal fluid coupled to a Yang-Mills field [@wong; @heinz; @jackiw; @surowka; @mrow], a solution to Yang-Mills equations. These approaches presuppose a mixture between a thermalized high-entropy fluid and a coherent field, which carries zero entropy. In a theory where the field is interacting, it is not clear weather this results in a converging effective theory, since there are quite a few correlated length-scales in the problem. It is therefore not surprising these approaches often lead to instabilities, which have also been argued to induce a rapid effective equilibration [@mrow; @asakawa; @strickland]. One can of course assume that a total thermalization of fluid and field degrees of freedom can be well described by a parton cascade, which can be made compatible with non-abelian gauge symmetry [@cascade]. However, it is not clear that in the strongly coupled limit the BBGKY hyerarchy can be effectively truncated. For example, in a non-Abelian gauge theory, even in the absence of Fermions, gauge bosons interact in a way that by construction must involve spin-orbit interactions. Hence, hydrodynamics self-consistently must include polarization, which is problematic to describe using a Boltzmann or a Vlasov equation [@gt1; @gt2; @gt3]. On the other hand, the whole point of gauge symmetry is the freedom to exchange spatial angular momentum for the longitudinal polarization of the Gauge boson. For abelian theory, this exchange is ”harmless” [@araujo; @goldman] since the mean free path of photons is infinite. However, for non-abelian theories, since spatial angular momentum is carried by vorticity, this ambiguity seems to contradict the very definition of local thermalization, where flow, and hence vorticity, are uniquely determined from initial conditions. Furthermore, the energy-momentum tensor $T_{\mu \nu}$ will typically be Gauge-dependent, since the pseudo-gauge transformations [@pseudo] will correspond to real gauge transformations for this theory. An effective theory based on $T_{\mu \nu}$ or in terms of a covariantized partition function in terms of $T_{\mu \nu}$,in analogy with the usual thermodynamics [@bec; @liu] will therefore meet difficoulties. In addition, it has long been known that the naive picture of deconfinement, positing that quarks and gluons above the deconfinement temperature become essentially free particles able to propagate everywhere is naive. Qualitative gedankenexperiments involving such QGP at large scales leads to seemingly contradictory conclusions such as “orphan quarks“ [@orphan] and outright paradoxes [@paradox]. Indeed, if examines the statistical mechanics of high temperature QCD, one finds infrared singularities [@linde] which lead to unexpected coupling constant dependences even in the region where the coupling constant is “small“ [@arnold]. The effect of these ambiguities on the hydrodynamic limit are still largely mysterious. It is the purpose of this work to try to poke at these questions by treating hydrodynamics as a ”bottom-up” effective theory, where the hydrodynamic limit is defined not in terms of an underlying theory but in terms of its symmetries. We can then explore how the symmetries present in the hydrodynamic limit square with gauge invariance. What is the ideal fluid dynamics limit? Usually, it means that there is a velocity frame field $u^\mu$ at rest with which the system is isotropic, homogeneus, and locally equilibrated. That fixes the energy-momentum tensor and any internal current $J_\mu$ to the form $$T_{\mu \nu} = u_\mu u_\nu (p+e) - p g_{\mu \nu} {\phantom{A},\phantom{A}}J_\mu = n u_\mu$$ where $p,e,n$ are scalars representing the pressure, energy density and conserved charge density in the comoving frame. Together with the local equilibrium condition $$\exists {\ln \mathcal{Z}}{\phantom{A},\phantom{A}}p = T {\ln \mathcal{Z}}{\phantom{A},\phantom{A}}e=\frac{d{\ln \mathcal{Z}}}{d(1/T)} {\phantom{A},\phantom{A}}n = T \frac{d{\ln \mathcal{Z}}}{d(\mu)}$$ these equations will be closed, i.e. solvable from initial conditions. A novel vision [@shift] is to see these conditions in terms of symmetries on the lagrangian field coordinates of a volume element and internal symmetries. In the next section we will explain how this works. Colorful swimming ghosts ======================== For a systematic look into this issue let us start from the first part of [@shift]. There, it is shown that for a general theory with continuus media (three fields $\phi_I$) and internal conserved currents (The $\phi_I$ aquire a complex phase $\alpha$) ideal hydrodynamics is equivalent of imposing a Lagrangian depending not on $\phi_I,\alpha$ but on $b,y$. $$\label{syms} F(\phi_I e^{i \alpha}) \rightarrow F(b,y) {\phantom{A},\phantom{A}}b=\left( \mathrm{Det}_{IJ}\left[ \partial_\mu \phi_I \partial^\mu \phi_J \right] \right)^{1/2} {\phantom{A},\phantom{A}}y=J^\mu \partial_\mu \alpha$$ where $$\label{shift} J^\mu \propto u^\mu {\phantom{A},\phantom{A}}u_\mu = \frac{1}{6b} \epsilon_{IJK} \epsilon_{\alpha \beta \gamma \mu} \partial^\alpha \phi^I \partial^\beta \phi^J \partial^\gamma \phi^J$$ Here, $b$ dependence in Eq. \[syms\] is equivalent to imposing invariance under all volume preserving diffeomorphisms and in addition Eq. \[shift\] imposes invariance under $\alpha \rightarrow \alpha +f(\phi_I)$, the chemical shift symmetry[@shift]. Physically, chemical shift imposes the fact that any gradient of either chemical potential and density is proportional to velocity. Mathematically, the phase in the internal symmetry becomes a function of $\phi_I$: Conservation laws ensure that any dynamics is a function of phase differences, and the gradient o the phase is exclusively in the $u_\mu$ direction. Since Gauge symmetry is a symmetry in internal space, it is this symmetry that we will have to expand. Let us therefore generalize $$\label{gauggen} y= J^\mu \partial_\mu \alpha \rightarrow \left[ J^\mu \right]_a \partial_\mu \left[ \alpha \right]_b = y_{ab}$$ Note the “matrix” of chemical potentials, which reflects the fact that any linear combination of conserved currents is conserved separately, leading to a free energy of the form $$\mathcal{Z} = Tr \exp\left[ -\hat{H}+\mu \hat{N} \right] \rightarrow Tr \exp\left[ -\hat{H}+\vec{\mu}_j \hat{U}_{ij} \hat{N}_i \right]$$ For a global symmetry, such as flavor, this implies an ambiguity of how chemical potentials are defined (is the chemical potential for the $s$ quark, the $d$ quark, or $\alpha s + \beta d$) which does not actually arise in physics, since $u,d,s$ have different charge and mass and hence live in different superselection sectors, so $U$ is diagonalized in the physical basis. Let us however consider color rather than flavor, and assume $y_{i}$ to be “color charge” chemical potentials, where potential redefinitions should occur. At first the idea of a “color chemical potential” appears crazy (although similar concepts have been explored [@cfl]) since it seems to violate gauge invariance. Chemical potentials, however, have to be gauge [*covariant*]{} rather than [*invariant*]{}, it is only the free energy that is gauge invariant. Within a locally equilibrated fluid, such a chemical potential corresponds, in analogy to electromagnetism, to the effect on the fluid of the chromo-electric potential (a magnetic potential would break isotropy). Let us therefore try to impose invariance under the gauge Symmetry. Throughout we shall assume a Lorentz-scalar gauge in order not to spoil isotropy explicitly $$\label{gaugedef} F(y,...) = F(U^{-1}(x) y U(x)) {\phantom{A},\phantom{A}}U_{ab}(x) \in SU(N) = \exp \left[ \sum_i \alpha_i (x) \hat{T}_i \right]$$ At first sight, any term dependent on $|y_{ab}|^2$ will do. One must remember,however, that ”color chemical potentials” $y_{ab}$ do not have to be gauge-invariant, but they have to be gauge-covariant, to allow for the lagrangian to be gauge invariant. Comparing Eq. \[gauggen\] and Eq. \[gaugedef\] one gets $$y_{ab} \rightarrow U^{-1}_{ac}(x) y_{cd} U_{bd}(x) = U^{-1}(x)_{ac} J^\mu_f U_{cf} U^{-1}_{fg} \partial_\mu \alpha_{g} U_{bg} =$$ $$\ = U^{-1}(x)_{ac} J^\mu_f U_{cf} \partial_\mu \left( U^{-1}_{fg} \alpha_d U_{bd}(x) \right) - J^\mu_a \left( U \partial_\mu U \right)_{fb} \alpha_f$$ the first term is automatically satisfied if $\alpha$ and $J$ transform in the fundamental representation under the gauge group. The second term is impossible to satisfy without introducing additional degrees of freedom, represented by Gauge fields $$F\left(b,J^\mu \partial_\mu \alpha \right) \rightarrow F\left( b, J^\mu \left( \partial_\mu - U(x) \partial_\mu U(x) \right) \alpha \right)$$ Continuing in this direction and building a gauge field out of the $U(x)$ will give us a pure gauge classical theory of the type examined in [@jackiw; @surowka]. However, we would like to define a local equilibrium state. Therefore, in addition to gauge symmetry we would like to impose the chemical shift symmetry, $$\label{surprise} J^\mu_a = \frac{\partial F}{\partial y_a} u^\mu {\phantom{A},\phantom{A}}L= F(b,y_{ab} \left(1-u_\mu \partial^\mu \alpha^{i} ) \right) \simeq F\left( b,Tr \left[ y_{ab} \left(1- (\hat{T}_{bc})_i u_\mu \partial^\mu \alpha^{i} \right) \right]^2,... \right)$$ The last term can be thought of as giving interactions between the different chemical potentials within the fluid. Any infinitesimal change in $U$ is always $\delta U \sim \sum_i \delta \alpha_i \hat{T}_i$ where $\hat{T}$ are the generators. Analogously to [@shift] local equilibrium ensures only $\delta \alpha_i$ in the direction of $u^\mu$ can change the dynamics. The number of independent components of $y_{ab}$ in gauge space is indeed equal to the number of generators. Note that this implies a Gauge-invariant theory is in a sense never locally equilibrated, since $u_\mu$ must enter the Lagrangian even in the ideal hydrodynamic limit. Physically, a manifestation of this idea has been known for a long time within non-relativistic fluid dynamics: A swimmer can [@swim1; @swim2] move themselves with no net force in a time-reversible fluid (for the non-relativistic limit this is a compressible highly viscous fluid) because, at each second, they move within the “gauge space of shapes” allowable to their body.This class of problems is popularized by the famous ”falling cat problem” [@cat]: a cat can always land on its feet despite not having anything to push against because, again, angular momentum conservation is not enough to ”fix the gauge”. The ”colorful swimming ghost” non-hydrodynamic modes derived here can be thought of as a field of such “swimmers”, each in a gauge adjacent configuration (the “arms and legs” are in gauge space) and each within a neighboring fluid cell. These modes will connect neighbouring cells with no advective flow, something impossible in the usual Euler equation. Colorful swirling ghosts ======================== Since $u^\mu \partial_\mu$ is in the Lagrangian, let us now investigate a situation where one of the currents experiences a non-zero vorticity $$\oint J_{i}^\mu dx_\mu \equiv \int_\Sigma d \Sigma_{\mu \nu} \omega^{\mu \nu} \ne 0 \rightarrow \omega_{i}^{\mu \nu}= \epsilon^{\mu \nu \alpha \beta} \partial_\alpha J_{\beta ab} \ne 0$$ The vorticity, like the current, is not invariant under a gauge transformation, but it transforms in the same way as the Wilson loop. In fact, the Wilson loop is nothing else but a vortex in gauge space rather than in flow space. $$\ \oint dx_\mu \partial^\mu U_{i} \equiv \int_\Sigma d \Sigma_{\mu \nu} (G^{\mu \nu}_{i})_i$$ here $G^{\mu \nu}_i$ is the field strength, the Yang-mills generalization of the elctromagnetic field, which is not gauge invariant. Thus, terms such as $Tr_{i} \left[ \omega_{\mu \nu} G^{\mu \nu} \right]$ can also enter the Lagrangian, and are at the same order as $u_\mu \alpha^\mu \alpha$. In fact, such terms are unavoidable for fluids with non-zero vorticity, since closed loops in the previous section always go to zero. In [@jackiw], these terms are interpreted as force terms in a Vlasov-type plasma. Here, this enters the free energy so there is no force, it is a degree of freedom w.r.t. entropy is maximized. It is therefore to be interpreted as the gluon polarization tensor, and such a term describes the ”chiral vortaic“ and ”chiral separation” effects [@kharzeev]. Hence, it is indeed true that a Gauge-invariant fluid is polarized. However, we rather unexpectdly found, via Eq. \[surprise\], that its free energy, via the “color chemical potentials”, must depend explicity on velocity. As a result, the polarization tensor, which in general has six independent components, is here determined by gauge structure, with $N^2-1$ redundant fields having 2 independent polarizations each. Unlike the general polarization tensor $y_{\mu \nu}$ explored in [@gt1], which has 6 degrees of freedom, here the equivalent is $N^2$ copies of $A_\mu^i$, which combine into $G^{\mu \nu}_i$ the usual way $$G^{\mu \nu}_i = \partial^\mu A^\nu_i - \partial^\nu A^\mu_i + f_{i j k} A^\mu_j A^\nu_k$$ The form of the equation of state in the small polarization limit should however be similar to that in [@gt1; @gt2], namely $$L= F(b,Tr_{i} \left[y_{ab} \left(1-u_\mu \hat{T}_{ab}^i \partial^\mu \alpha^{i} ) \right]^2,Tr_{i} \left[ w_{i}^{\mu \nu} G_{\mu \nu}^i\right] \right) \simeq$$ $$\ \simeq F(b\times \left[ 1-c\Omega^2+{ \mathcal{O} \left( \Omega^4 \right) } \right],Tr_{i} \left[y_{ab i} \left(1-u_\mu \hat{T}_{bc i} \partial^\mu \alpha^{i} \right) \right] {\phantom{A},\phantom{A}}\Omega^2=\sum_i \left( G^{\mu \nu}_i \omega_{\mu \nu i} \right)^2$$ This equation of state includes both swimming and swirling ghosts. In equilibrium, polarization and vorticity must point in the same direction according to the arguments made in [@gt1]. However, the local equilibrium limit is unstable as shown in [@gt3]. The resulting relaxation dynamics will be affected by this different number of effective polarization degrees of freedom. In analogy to [@gt3], we could then postulate that $G^{\mu \nu}_i$ would relax to $\omega^{\mu \nu}_i$ using an Israel-Stewart type equation. The naive equivalent is $$\tau u^\mu \partial_\mu G^i_{\mu \nu} + G^i_{\mu \nu} = \chi \omega^i_{\mu \nu} + { \mathcal{O} \left( f_{ijk} \omega_j \omega_k \right) }$$ where $$\ \chi \equiv \left| dF/d G^i_{\mu \nu} \right|$$ However, one has to be careful: $G^i_{\mu \nu}$ is not gauge invariant, but transforms in the same way as the vorticity current. Hence, unlike in [@gt3] the relaxation equation can only have a gauge invariant form, for instance $$\label{relax} \tau u^\mu \partial_\mu Tr_i \left[ G^i_{\mu \nu} \right]^2 + Tr_i \left[ G^i_{\mu \nu} \right]^2 = \chi Tr_i \left[ \omega^i_{\mu \nu} \right]^2 + { \mathcal{O} \left( f_{ijk} \omega_j \omega_k \right) }$$ This equation does not have a unique relaxation minimum, since “swirling” solutions which rotate in gauge space and in configuration space at the same frequency $$F^{\mu \nu}_i \propto U^{i j}(x^\mu) \omega^{\mu \nu}_j(x^\mu) {\phantom{A},\phantom{A}}U^{ij} = \exp \left[ \sum_i \alpha_i(x) \hat{T}_i \right] {\phantom{A},\phantom{A}}\nabla_X \wedge \alpha_i \ne 0$$ will never relax to a value parallel to $\omega$, since such a relaxation breaks gauge symmetry. One can think of such solutions as being the vortex equivalent of the swimming ghosts. Like real vortices they do not propagate. Discussion ========== What conclusions can we draw physically from all this? The swimming ghosts can be thought of as “sequences” of color space rotation along the direction of the flow. Even in perfect local equilibrium (the perfect fluid limit), gauge invariance ensures such “sequences” will have no energy gap, and an equivalent free energy to the hydrostatic limit. The presence of non-hydrodynamical fluctuations, which go away from local equilibrium and connect differntly flowing volume cells is therefore unavoidable even in the ”ideal hydrodynamic limit”. To draw a connection to statistical mechanics, we need to think about the meaning of gauge transformations. In the context of quantum mechanics, different gauge configurations are usually thought to be redundancies of the system. In a locally thermal enviroenment, however, they correspond to indistinguisheable configurations of the free energy. Since the fundamental postulate of statistical mechanics means that all microstates are equally probable, the [*number*]{} of such isentropic configurations must be discounted from the total number of microstates. As in quantum field theory, this is what ghosts do. However, unlike quantum field theory, gauge symmetries break the distinction between the “microscopic” and “macroscopic” system, central to hydrodynamics and transport. To see this, let us recall how sound waves are defined in the Lagrangian approach. We perturb the hydrostatic limit, where $\phi_I=X_I$ (the comoving and lab coordinates are the same), and isolate a transverse mode (vortex) and a longitudinal mode (sound wave. Note that the two are mixed when polarization is present [@gt1; @gt2; @gt3]) $$\label{soundfree} \phi_I = X_I + \vec{\pi}_I^{sound} + \vec{\pi}_I^{vortex} {\phantom{A},\phantom{A}}\nabla.\vec{\pi}_I^{vortex}=\nabla\times \vec{\pi}_I^{sound}=0$$ These perturbations will evolve as propagating sound-waves and non-propagating vortices. Since the derivative of the free energy w.r.t. $b$ is positive (entropy and energy density are correlated for thermodynamically normal systems), sound waves and vortices do “work”. Let us now assume the system has a “color chemical potential” in some direction “1” $\mu_1$ as well as an energy density. Let us change the color chemical potential in space according to $\mu_1+\Delta\mu(x)$ $$\label{colfree} \Delta \mu(x) = \sum_i \left( \mu_i^{swim}(x) + \mu_i^{swirl}(x) \right) \hat{T}_i {\phantom{A},\phantom{A}}\nabla_X . \mu^{swim}_i= \nabla_X \wedge \mu_i^{swirl} =0$$ Because of gauge redundancy, the derivatives of the free energy with respect to color (“color susceptibility”) will typically be negative (it is a “ghost” state), since more color chemical potential means more gauge-redundancy in the microstates. One can tune Eqs \[soundfree\] and \[colfree\] so that the free energy exactly cancels. Hence, a “hydrostatic state” with a color chemical potential is unstable against the formation of sound-waves as well as vortices. Now sound-waves and vortices are “heat” rather than “work”. In statistical mechanics, what normally distinguishes “work” from “heat” is coarse-graining, the separation between micro and macro states. Quantitatively, probability of thermal fluctuations is normalized by the heat capacity and temperature scale $1/(c_V T)$ and microscopic correlations due to viscosity are $\sim \eta/(Ts)$. Since for a usual fluid, there is a hierarchy between microscopic scale, Knudsen number and gradient $$\label{hyerarchy} \frac{1}{C_V T} \ll \frac{\eta}{(Ts)} \ll (\partial u_\mu)^{-1}$$ The first inequality defines the truncation of the BBGKY hyerarchy within transport and is equivalent to the planar limit in the gauge/gravity correspondence. Physically, the first inequality ensures that any thermal fluctuations (the left hand side of this inequality) dissipate too fast to create sound-waves that influence the fluid evolution (one could in principle connect such ”randomly generated sound-waves” to the unphysical ”wild solutions” studied in the context of non-relativistic hydrodynamics [@wild]. Were such solutions physical, all distinction between microscopic heat and macroscopic work within a hydrodynamics background would manifestly disappear). The second inequality is usually associated with the Knudsen number, it avoids microscopic correlations between different fluid cells and hence allows for an expansion in either gradients of conserved quantities or moments of the microscopic distribution function [@kodama]. Gauge invariance directly breaks this hyerarchy by introducing redundancies affecting both microscopic correlations (the “smallest” scale in Eq. \[hyerarchy\]) and sound-waves (the “largest” scale). In a sense, the first microscopic length scale becomes infinite, since ghost modes of arbitrarily low and high frequency and wavenumber appear. In elementary processes ghosts can be gotten rid of by choosing an axial gauge, but since in the hydrodynamic effective theory is explicitly built around isotropy, this option is not available here. Since in an expansion parameter the first term is “quantum-like” and the second is dissipative [@gt0; @burch], this hydrodynamics will be dominated by fluctuations, possibly necessitating a lattice study [@burch]. It is difficoult to see how such as system can be reduced to the type of ”interacting quasi-particle picture” for which a Boltzmann or a Boltzmann-Vlasov equation are appropriate. While one imagines perturbative gluons to be modeled via a Boltzmann equation [@cascade], and the effect of coherent fields to reduce to an ”anomalous viscosity” [@asakawa] when the direction of the fields is random enough, ghost fields cannot be pictured this way precisely because they have a ”negative” effect on the fluid at the level of the density matrix. The obvious question which must be asked is, why is there no trace of such non-hydrodynamic modes in holography, where everything converges to a Navier-Stokes expansion with a well-posed equation of state. The answer is that gauge/gravity duality, as done so far, requires a planar limit and a conformally invariant ultraviolet fixed point. Color flying ghosts are of order ${ \mathcal{O} \left( 1 \right) }$, just like thermal perturbations. Hence, they do not contribute to the planar limit. And, as shown in [@marcelo], a conformally invariant fixed point makes the gribov issue microscopically non-dynamical, leading to the suspicion it will not contribute to any locally equilibrated dynamics either. What is the role of the swimming and swirling ghosts in the dynamics of a close-to-ideal fluids in non-Abelian gauge theory? A linearization and causality analysis of this system is left for a forthcoming work. We note, however, that as in [@gt1; @gt2; @gt3] Ostrogradski’s theorem means that such non-hydrodynamic modes usually generate instabilities and causality violation. The only way to make such modes go away is to insure local color neutrality (zero chemical potential everywhere in the system), leading to the suspicion that these non-hydrodynamic modes quickly color-neutralize and locally thermalize the system. In conclusion, we find that the symmetries of ideal hydrodynamics do not comute with non-Abelian gauge theory. The ideal fluid limit of a theory whose microscopic dynamics has such a symmetry, therefore, is very different from the Euler equations, as it will be full of non-hydrodynamic “ghost“ modes carrying rotations of color space along the flow direction. The only fluid dynamic limit where something like an Euler equation, with an equation of state independent of flow emerges, is one where color neutrality is assured in each volume cell. The consequences of this for quark gluon plasma thermalization is likely to be profound. *Acknowledgements* GT acknowledges support from FAPESP proc. 2017/06508-7, partecipation in FAPESP tematico 2017/05685-2 and CNPQ bolsa de produtividade 301996/2014-8. DM was supported by CNPQ graduate fellowship n. 147435/2014-5. This work is a part of the project INCT-FNA Proc. No. 464898/2014-5. We wish to thank David Montenegro, Radoslaw Ryblewski, Saso Grozdanov and Henrique Sa Earp for fruitful discussions. [99]{} S. K. Wong, Nuovo Cim. A [**65**]{}, 689 (1970). doi:10.1007/BF02892134 U. W. Heinz, Phys. Rev. Lett.  [**51**]{}, 351 (1983). doi:10.1103/PhysRevLett.51.351 B. Bistrovic, R. Jackiw, H. Li, V. P. Nair and S. Y. Pi, Phys. Rev. D [**67**]{}, 025013 (2003) doi:10.1103/PhysRevD.67.025013 \[hep-th/0210143\]. J. J. Fernandez-Melgarejo, S. J. Rey and P. Surówka, JHEP [**1702**]{}, 122 (2017) doi:10.1007/JHEP02(2017)122 \[arXiv:1605.06080 \[hep-th\]\]. C. Manuel and S. Mrowczynski, Phys. Rev. D [**68**]{}, 094010 (2003) doi:10.1103/PhysRevD.68.094010 \[hep-ph/0306209\]. M. Asakawa, S. A. Bass and B. Muller, Phys. Rev. Lett.  [**96**]{}, 252301 (2006) doi:10.1103/PhysRevLett.96.252301 \[hep-ph/0603092\]. S. Mrowczynski, B. Schenke and M. Strickland, Phys. Rept.  [**682**]{}, 1 (2017) doi:10.1016/j.physrep.2017.03.003 \[arXiv:1603.08946 \[hep-ph\]\]. B. Muller, AIP Conf. Proc.  [**482**]{}, no. 1, 1 (1999) doi:10.1063/1.59571 \[nucl-th/9902065\]. D. Montenegro, L. Tinti and G. Torrieri, Phys. Rev. D [**96**]{}, no. 5, 056012 (2017) Addendum: \[Phys. Rev. D [**96**]{}, no. 7, 079901 (2017)\] doi:10.1103/PhysRevD.96.079901, 10.1103/PhysRevD.96.056012 \[arXiv:1701.08263 \[hep-th\]\]. D. Montenegro, L. Tinti and G. Torrieri, Phys. Rev. D [**96**]{}, no. 7, 076016 (2017) doi:10.1103/PhysRevD.96.076016 \[arXiv:1703.03079 \[hep-th\]\]. D. Montenegro and G. Torrieri, arXiv:1807.02796 \[hep-th\]. S.J. Van Enk ,G. Nienhuis (1994) Journal of Modern Optics, 41:5, 963-977, DOI: 10.1080/09500349414550911 X. S. Chen, X. F. Lu, W. M. Sun, F. Wang and T. Goldman, Phys. Rev. Lett.  [**100**]{}, 232002 (2008) doi:10.1103/PhysRevLett.100.232002 \[arXiv:0806.3166 \[hep-ph\]\]. W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski and E. Speranza, arXiv:1807.04946 \[nucl-th\]. J. Bhattacharya, S. Bhattacharyya and M. Rangamani, JHEP [**1302**]{}, 153 (2013) doi:10.1007/JHEP02(2013)153 \[arXiv:1211.1020 \[hep-th\]\]. P. Glorioso, M. Crossley and H. Liu, JHEP [**1709**]{}, 096 (2017) doi:10.1007/JHEP09(2017)096 \[arXiv:1701.07817 \[hep-th\]\]. S. Banerjee, A. Bhattacharyya, S. K. Ghosh, E. M. Ilgenfritz, S. Raha, B. Sinha, E. Takasugi and H. Toki, Nucl. Phys. A [**774**]{}, 769 (2006) doi:10.1016/j.nuclphysa.2006.06.133 \[hep-ph/0602202\]. D. Miskowiec, PoS CPOD [**07**]{}, 020 (2007) doi:10.22323/1.047.0020 \[arXiv:0707.0923 \[hep-ph\]\]. A. D. Linde, Phys. Lett.  [**96B**]{}, 289 (1980). doi:10.1016/0370-2693(80)90769-8 P. B. Arnold, Int. J. Mod. Phys. E [**16**]{}, 2555 (2007) doi:10.1142/S021830130700832X \[arXiv:0708.0812 \[hep-ph\]\]. S. Dubovsky, L. Hui, A. Nicolis and D. T. Son, Phys. Rev. D [**85**]{}, 085029 (2012) doi:10.1103/PhysRevD.85.085029 \[arXiv:1107.0731 \[hep-th\]\]. M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B [**422**]{}, 247 (1998) doi:10.1016/S0370-2693(98)00051-3 \[hep-ph/9711395\]. A.Shapere and F.Wilczek, “Self-Propulsion at Low Reynolds Number“, Phys. Rev. Lett. 58, 2051–2054 (1987) J. Avron, O.Raz, 2008 New J. Phys. 10 063016, 0712.2047 R.Montgomery, ”Gauge theory of the falling cat”, Dynamics and Control of Mechanical Systems, pp, 218 (1993) D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang, Prog. Part. Nucl. Phys.  [**88**]{}, 1 (2016) doi:10.1016/j.ppnp.2016.01.001 \[arXiv:1511.04050 \[hep-ph\]\]\ G. Torrieri, Phys. Rev. D [**85**]{}, 065006 (2012) doi:10.1103/PhysRevD.85.065006 \[arXiv:1112.4086 \[hep-th\]\]. Camillo De Lellis, Laszlo Szekelyhidi Jr, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 225-260, 0712.3288 R. Derradi de Souza, T. Koide and T. Kodama, Prog. Part. Nucl. Phys.  [**86**]{}, 35 (2016) doi:10.1016/j.ppnp.2015.09.002 \[arXiv:1506.03863 \[nucl-th\]\]. T. Burch and G. Torrieri, Phys. Rev. D [**92**]{}, no. 1, 016009 (2015) doi:10.1103/PhysRevD.92.016009 \[arXiv:1502.05421 \[hep-lat\]\]. M. A. L. Capri, M. S. Guimaraes, I. F. Justo, L. F. Palhares and S. P. Sorella, Phys. Lett. B [**735**]{}, 277 (2014) doi:10.1016/j.physletb.2014.06.035 \[arXiv:1404.7163 \[hep-th\]\].

--- abstract: | **Counting** is a fundamental operation. For example, counting the $\alpha$th frequency moment, $F_{(\alpha)} = \sum_{i=1}^D A_t[i]^\alpha$, of a streaming signal $A_t$ (where $t$ denotes time), has been an active area of research, in theoretical computer science, databases, and data mining. When $\alpha =1$, the task (i.e., counting the sum) can be accomplished using a counter. When $\alpha \neq 1$, however, it becomes non-trivial to design a small space (i.e., low memory) counting system. [*Compressed Counting (CC)*]{} is proposed for efficiently computing the $\alpha$th frequency moment of a data stream $A_t$, where $0<\alpha\leq2$. CC is applicable if the streaming data follow the [*Turnstile*]{} model, with the restriction that at the time $t$ for the evaluation, $A_t[i]\geq 0, \forall i\in[1,D]$, which includes the [*strict Turnstile*]{} model as a special case. For data streams in practice, this restriction is minor. The underlying technique is [*skewed stable random projections*]{}, which captures the intuition that, when $\alpha =1$ a simple counter suffices, and when $\alpha = 1\pm\Delta$ with small $\Delta$, the sample complexity should be low (continuously as a function of $\Delta$). We show the sample complexity (number of projections) [$k = G \frac{1}{\epsilon^2}\log\left(\frac{2}{\delta}\right)$]{}, where [$G = O\left(\epsilon\right)$]{} as [$\Delta\rightarrow 0$]{}. In other words, for small $\Delta$, $k = O\left({1}/{\epsilon}\right)$ instead of $O\left({1}/{\epsilon^2}\right)$. The case $\Delta\rightarrow 0$ is practically very important. It is now well-understood that one can obtain good approximations to the entropies of data streams using the $\alpha$th moments with $\alpha=1\pm\Delta$ and very small $\Delta$. For statistical inference using the [*method of moments*]{}, it is sometimes reasonable use the $\alpha$th moments with $\alpha$ very close to 1. As another example, $\Delta$ might be the “decay rate” or “interest rate,” which is usually small. Thus, [*Compressed Counting*]{} will be an ideal tool, for estimating the total value in the future, taking in account the effect of decaying or interest accruement. Finally, our another contribution is an algorithm for approximating the logarithmic norm, [$\sum_{i=1}^D\log A_t[i]$]{}, and the logarithmic distance, [$\sum_{i=1}^D\log\left|A_t[i] - B_t[i]\right|$]{}. The logarithmic norm arises in statistical estimations. The logarithmic distance is useful in machine learning practice with heavy-tailed data. author: - | Ping Li\ Department of Statistical Science\ Faculty of Computing and Information Science\ Cornell University, Ithaca, NY 14853\ pingli@cornell.edu title: Compressed Counting --- Introduction ============ This paper focuses on [*counting*]{}, which is among the most fundamental operations in almost every field of science and engineering. Computing the sum [$\sum_{i=1}^DA_t[i]$]{} is the simplest counting ($t$ denotes time). Counting the [$\alpha$th moment $\sum_{i=1}^DA_t[i]^\alpha$]{} is more general. When [$\alpha\rightarrow 0+$, $\sum_{i=1}^DA_i[i]^\alpha$]{} counts the total number of non-zeros in $A_t$. When [$\alpha = 2$, $\sum_{i=1}^DA_t[i]^\alpha$]{} counts the “energy” or “power” of the signal $A_t$. If $A_t$ actually outputs the power of an underlying signal $B_t$, counting the sum [$\sum_{i=1}^DB_t$]{} is equivalent to computing [$\sum_{i=1}^DA_t[i]^{1/2}$]{}. Here, $A_t$ denotes a time-varying signal, for example, [*data streams*]{}[@Book:Henzinger_99; @Proc:Feigenbaum_FOCS99; @Article:Indyk_JACM06; @Proc:Babcock_PODS02; @Article:Indyk_TKDE03; @Article:Muthukrishnan_05]. In the literature, the $\alpha$th frequency moment of a data stream $A_t$ is defined as Counting $F_{(\alpha)}$ for massive data streams is practically important, among many challenging issues in data stream computations. In fact, the general theme of “scaling up for high dimensional data and high speed data streams” is among the “ten challenging problems in data mining research.” Because the elements, $A_t[i]$, are time-varying, a naíve counting mechanism requires a system of $D$ counters to compute $F_{(\alpha)}$ exactly. This is not always realistic when $D$ is large and we only need an approximate answer. For example, $D$ may be $2^{64}$ if $A_t$ records the arrivals of IP addresses. Or, $D$ can be the total number of checking/savings accounts. ***Compressed Counting (CC)*** is a new scheme for approximating the $\alpha$th frequency moments of data streams (where $0<\alpha\leq 2$) using low memory. The underlying technique is based on what we call [*skewed stable random projections*]{}. The Data Models --------------- We consider the popular [*Turnstile*]{} data stream model [@Article:Muthukrishnan_05]. The input stream $a_t = (i_t, I_t)$, $i_t\in [1,\ D]$ arriving sequentially describes the underlying signal $A$, meaning $A_t[i_t] = A_{t-1}[i_t] + I_t$. The increment $I_t$ can be either positive (insertion) or negative (deletion). Restricting $I_t\geq 0$ results in the [*cash register*]{} model. Restricting $A_t[i]\geq 0$ at all $t$ (but $I_t$ can still be either positive or negative) results in the [*strict Turnstile*]{} model, which suffices for describing most (although not all) natural phenomena. For example[@Article:Muthukrishnan_05], in a database, a record can only be deleted if it was previously inserted. Another example is the checking/savings account, which allows deposits/withdrawals but generally does not allow overdraft. [*Compressed Counting (CC)*]{} is applicable when, at the time $t$ for the evaluation, $A_t[i]\geq 0$ for all $i$. This is more flexible than the [*strict Turnstile*]{} model, which requires $A_t[i]\geq 0$ at all $t$. In other words, CC is applicable when data streams are (a) insertion only (i.e., the [*cash register*]{} model), or (b) always non-negative (i.e., the [*strict Turnstile*]{} model), or (c) non-negative at check points. We believe our model suffices for describing most natural data streams in practice. With the realistic restriction that $A_t[i]\geq 0$ at $t$, the definition of the $\alpha$th frequency moment becomes [$$\begin{aligned} \label{eqn_def_F2}%\vspace{-0.25in} F_{(\alpha)} = \sum_{i=1}^DA_t[i]^\alpha; \end{aligned}$$]{} and the case $\alpha = 1$ becomes trivial, because [$$\begin{aligned} \vspace{-0.15in} F_{(1)} = \sum_{i=1}^DA_t[i]= \sum_{s=1}^t I_s\end{aligned}$$]{} In other words, for $F_{(1)}$, we need only a simple counter to accumulate all values of increment/decrement $I_t$. For $\alpha \neq 1$, however, counting (\[eqn\_def\_F2\]) is still a non-trivial problem. Intuitively, there should exist an intelligent counting system that performs almost like a simple counter when $\alpha = 1\pm\Delta$ with small $\Delta$. The parameter $\Delta$ may bear a clear physical meaning. For example, $\Delta$ may be the “decay rate” or “interest rate,” which is usually small. The proposed [*Compressed Counting (CC)*]{} provides such an intelligent counting systems. Because its underlying technique is based on [*skewed stable random projections*]{}, we provide a brief introduction to [*skewed stable distributions*]{}. Skewed Stable Distributions --------------------------- A random variable $Z$ follows a $\beta$-skewed $\alpha$-stable distribution if the Fourier transform of its density is[@Book:Zolotarev_86] [$$\begin{aligned} \notag\vspace{-0.1in} {\mathscr{F}}_Z(t) &= \text{E}\exp\left(\sqrt{-1}Zt\right) \hspace{0.3in} \alpha \neq 1,\\\notag &= \exp\left(-F|t|^\alpha\left(1-\sqrt{-1}\beta\text{sign}(t)\tan\left(\frac{\pi\alpha}{2}\right)\right)\right),\end{aligned}$$]{} where $-1\leq \beta\leq 1$ and $F>0$ is the scale parameter. We denote $Z \sim S(\alpha,\beta,F)$. Here $0<\alpha \leq 2$. When $\alpha<0$, the inverse Fourier transform is unbounded; and when $\alpha>2$, the inverse Fourier transform is not a probability density. This is why [*Compressed Counting*]{} is limited to $0<\alpha\leq2$. Consider two independent variables, $Z_1, Z_2 \sim S(\alpha, \beta,1)$. For any non-negative constants $C_1$ and $C_2$, the “$\alpha$-stability” follows from properties of Fourier transforms: [$$\begin{aligned} \notag\vspace{-0.1in} Z = C_1Z_1 + C_2Z_2 \sim S\left(\alpha,\beta, C_1^\alpha + C_2^\alpha\right).\end{aligned}$$]{} However, if $C_1$ and $C_2$ do not have the same signs, the above “stability” does not hold (unless $\beta = 0$ or $\alpha = 2$, $0+$). To see this, we consider $Z = C_1 Z_1 - C_2 Z_2$, with $C_1\geq 0$ and $C_2\geq 0$. Then, because $\mathscr{F}_{-Z_2}(t) = \mathscr{F}_{Z_2}(-t) $, [$$\begin{aligned} \notag\vspace{-0.1in} \mathscr{F}_Z =& \exp\left(-|C_1t|^\alpha\left(1-\sqrt{-1}\beta\text{sign}(t)\tan\left(\frac{\pi\alpha}{2}\right)\right)\right) \\\notag \times&\exp\left(-|C_2t|^\alpha\left(1+\sqrt{-1}\beta\text{sign}(t)\tan\left(\frac{\pi\alpha}{2}\right)\right)\right),\end{aligned}$$]{} which does not represent a stable law, unless $\beta = 0$ or $\alpha =2$, $0+$. This is the fundamental reason why [*Compressed Counting*]{} needs the restriction that at the time of evaluation, elements in the data streams should have the same signs. Skewed Stable Random Projections -------------------------------- Given $R\in\mathbb{R}^{D}$ with each element $r_i\sim S(\alpha, \beta, 1)$ i.i.d., then meaning $R^\text{T}A_t$ represents one sample of the stable distribution whose scale parameter $F_{(\alpha)}$ is what we are after. Of course, we need more than one sample to estimate $F_{(\alpha)}$. We can generate a matrix $\mathbf{R}\in\mathbb{R}^{D\times k}$ with each entry $r_{ij} \sim S(\alpha, \beta, 1)$. The resultant vector $X = \mathbf{R}^\text{T}A_t\in\mathbb{R}^k$ contains $k$ i.i.d. samples: $x_j \sim S\left(\alpha,\beta,F_{(\alpha)}\right)$, $j = 1$ to $k$. Note that this is a linear projection; and recall that the [*Turnstile*]{} model is also linear. Thus, [*skewed stable random projections*]{} can be applicable to dynamic data streams. For every incoming $a_t = (i_t, I_t)$, we update $x_j \leftarrow x_j + r_{i_tj} I_t$ for $j = 1$ to $k$. This way, at any time $t$, we maintain $k$ i.i.d. stable samples. The remaining task is to recover $F_{(\alpha)}$, which is a statistical estimation problem. Counting in Statistical/Learning Applications --------------------------------------------- The [*method of moments*]{} is often convenient and popular in statistical parameter estimation. Consider, for example, the three-parameter generalized gamma distribution $GG(\theta,\gamma,\eta)$, which is highly flexible for modeling positive data, e.g., [@Article:Li_SINR06]. If $X\sim GG(\theta,\gamma,\eta)$, then the first three moments are $\text{E}(X) = \theta\gamma$, $\text{Var}(X^2) = \theta\gamma^2$, $\text{E}\left(X - \text{E}(X)\right)^3 = (\eta+1)\theta\gamma^3$. Thus, one can estimate $\theta$, $\gamma$ and $\eta$ from $D$ i.i.d. samples $x_i\sim GG(\theta,\gamma,\eta)$ by counting the first three empirical moments from the data. However, some moments may be (much) easier to compute than others if the data $x_i$’s are collected from data streams. Instead of using integer moments, the parameters can also be estimated from any three [*fractional*]{} moments, i.e., $\sum_{i=1}^D x_i^\alpha$, for three different values of $\alpha$. Because $D$ is very large, any consistent estimator is likely to provide a good estimate. Thus, it might be reasonable to choose $\alpha$ mainly based on the computational cost. See Appendix \[app\_moments\] for comments on the situation in which one may also care about the relative accuracy caused by different choices of $\alpha$. The logarithmic norm $\sum_{i=1}^D\log x_i$ arises in statistical estimation, for example, the maximum likelihood estimators for the Pareto and gamma distributions. Since it is closely connected to the moment problem, Section \[sec\_log\] provides an algorithm for approximating the logarithmic norm, as well as for the logarithmic distance; the latter can be quite useful in machine learning practice with massive heavy-tailed data (either dynamic or static) in lieu of the usual $l_2$ distance. Entropy is also an important summary statistic. Recently [@Proc:Zhao_IMC07] proposed to approximate the entropy moment $\sum_{i=1}^D x_i\log x_i$ using the $\alpha$th moments with $\alpha = 1\pm\Delta$ and very small $\Delta$. Comparisons with Previous Studies --------------------------------- Pioneered by[@Proc:Alon_STOC96], there have been many studies on approximating the $\alpha$th frequency moment $F_{(\alpha)}$. [@Proc:Alon_STOC96] considered integer moments, $\alpha = 0$, 1, 2, as well as $\alpha>2$. Soon after, [@Proc:Feigenbaum_FOCS99; @Proc:Indyk_FOCS00] provided improved algorithms for $0<\alpha\leq 2$. [@Proc:Saks_STOC02; @Proc:Kumar_FOCS02] proved the sample complexity lower bounds for $\alpha >2$. [@Proc:Woodruff_SODA04] proved the optimal lower bounds for all frequency moments, except for $\alpha = 1$, because for non-negative data, $F_{(1)}$ can be computed essentially error-free with a counter[@Article:Morris_CACM78; @Article:Flajolet_BIT85; @Proc:Alon_STOC96]. [@Proc:Indyk_STOC05] provided algorithms for $\alpha >2$ to (essentially) achieve the lower bounds proved in [@Proc:Saks_STOC02; @Proc:Kumar_FOCS02]. Note that an algorithm, which “achieves the optimal bound,” is not necessarily practical because the constant may be very large. In a sense, the method based on [*symmetric stable random projections*]{}[@Article:Indyk_JACM06] is one of the few successful algorithms that are simple and free of large constants. [@Article:Indyk_JACM06] described the procedure for approximating $F_{(1)}$ in data streams and proved the bound for $\alpha =1$ (although not explicitly). For $\alpha \neq 1$, [@Article:Indyk_JACM06] provided a conceptual algorithm. [@Proc:Li_SODA08] proposed various estimators for [*symmetric stable random projections*]{} and provided the constants explicitly for all $0<\alpha\leq 2$. None of the previous studies, however, captures of the intuition that, when $\alpha = 1$, a simple counter suffices for computing $F_{(1)}$ (essentially) error-free, and when $\alpha = 1 \pm \Delta$ with small $\Delta$, the sample complexity (number of projections, $k$) should be low and vary continuously as a function of $\Delta$. ***Compressed Counting (CC)*** is proposed for $0<\alpha \leq 2$ and it works particularly well when $\alpha = 1\pm\Delta$ with small $\Delta$. This can be practically very useful. For example, $\Delta$ may be the “decay rate” or the “interest rate,” which is usually small; thus CC can count the total value in the future taking into account the effect of decaying or interest accruement. In parameter estimations using the [*method of moments*]{}, one may choose the $\alpha$th moments with $\alpha$ close 1. Also, one can approximate the entropy moment using the $\alpha$th moments with $\alpha = 1\pm\Delta$ and very small $\Delta$[@Proc:Zhao_IMC07]. Our study has connections to the Johnson-Lindenstrauss Lemma[@Article:JL84], which proved $k = O\left(1/\epsilon^2\right)$ at $\alpha = 2$. An analogous bound holds for $0<\alpha \leq 2$[@Article:Indyk_JACM06; @Proc:Li_SODA08]. The dependency on $1/\epsilon^2$ may raise concerns if, say, $\epsilon\leq 0.1$. We will show that CC achieves $k = O(1/\epsilon)$ in the neighborhood of $\alpha = 1$. Two Statistical Estimators -------------------------- Recall that [*Compressed Counting (CC)*]{} boils down to a statistical estimation problem. That is, given $k$ i.i.d. samples $x_j \sim S\left(\alpha, \beta=1, F_{(\alpha)}\right)$, estimate the scale parameter $F_{(\alpha)}$. Section \[sec\_gm\] will explain why we fix $\beta = 1$. Part of this paper is to provide estimators which are convenient for theoretical analysis, e.g., tail bounds. We provide the [*geometric mean*]{} and the [*harmonic mean*]{} estimators, whose asymptotic variances are illustrated in Figure \[fig\_comp\_var\_factor\]. - [The **geometric mean** estimator, $\hat{F}_{(\alpha),gm}$]{} $\hat{F}_{(\alpha),gm}$ is unbiased. We prove the sample complexity explicitly and show [$k = O\left(1/\epsilon\right)$]{} suffices for $\alpha$ around 1. - [The **harmonic mean** estimator, $\hat{F}_{(\alpha),hm,c}$, for $\alpha <1$]{} It is considerably more accurate than $\hat{F}_{(\alpha),gm}$ and its sample complexity bound is also provided in an explicit form. Here $\Gamma(.)$ is the usual gamma function. Paper Organization ------------------ Section \[sec\_gm\] begins with analyzing the moments of skewed stable distributions, from which the [*geometric mean*]{} and [*harmonic mean*]{} estimators are derived. Section \[sec\_gm\] is then devoted to the detailed analysis of the [*geometric mean*]{} estimator. Section \[sec\_hm\] analyzes the [*harmonic mean*]{} estimator. Section \[sec\_log\] addresses the application of CC in statistical parameter estimation and an algorithm for approximating the logarithmic norm and distance. The proofs are presented as appendices. The Geometric Mean Estimator {#sec_gm} ============================ We first prove a fundamental result about the moments of skewed stable distributions. \[lem\_moments\] If $Z \sim S(\alpha,\beta,F_{(\alpha)})$, then for any $-1<\lambda<\alpha$, [$$\begin{aligned} \notag%\label{eqn_moment_beta}%\notag &\textbf{E}\left(|Z|^\lambda\right) = F_{(\alpha)}^{\lambda/\alpha} \cos\left(\frac{\lambda}{\alpha}\tan^{-1}\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\right) \\\notag &\times\left(1+\beta^2\tan^2\left(\frac{\alpha\pi}{2}\right)\right)^{\frac{\lambda}{2\alpha}} \left(\frac{2}{\pi}\sin\left(\frac{\pi}{2}\lambda\right)\Gamma\left(1-\frac{\lambda}{\alpha}\right)\Gamma\left(\lambda\right)\right),\end{aligned}$$]{} which can be simplified when $\beta = 1$, to be [$$\begin{aligned} \notag%\label{eqn_moment}%\notag &\textbf{E}\left(|Z|^\lambda\right) = \\\notag &{F}_{(\alpha)}^{\lambda/\alpha} \frac{\cos\left(\frac{\kappa(\alpha)}{\alpha}\frac{\lambda\pi}{2}\right)} {\cos^{\lambda/\alpha}\left(\frac{\kappa(\alpha)\pi}{2}\right)}\left(\frac{2}{\pi}\sin\left(\frac{\pi}{2}\lambda\right)\Gamma\left(1-\frac{\lambda}{\alpha}\right)\Gamma\left(\lambda\right)\right), \\\notag&\kappa(\alpha) = \alpha \ \ \ \text{if} \ \ \ \alpha<1, \ \ \ \text{and} \ \ \kappa(\alpha)=2-\alpha\ \ \ \text{if} \ \ \alpha>1.\end{aligned}$$]{} For $\alpha <1$, and $-\infty<\lambda <\alpha$, [$$\begin{aligned} \notag%\label{eqn_moment}%\notag \textbf{E}\left(|Z|^\lambda\right) =\textbf{E}\left(Z^\lambda\right) = {F}_{(\alpha)}^{\lambda/\alpha} \frac{ \Gamma\left(1-\frac{\lambda}{\alpha}\right) } {\cos^{\lambda/\alpha}\left(\frac{\alpha\pi}{2}\right) \Gamma\left(1-\lambda\right)}.\end{aligned}$$]{} **Proof:** See Appendix \[proof\_lem\_moments\]. $\Box$\ Recall that [*Compressed Counting*]{} boils down to estimating $F_{(\alpha)}$ from these $k$ i.i.d. samples $x_j \sim S(\alpha,\beta,F_{(\alpha)})$. Setting $\lambda = \frac{\alpha}{k}$ in Lemma \[lem\_moments\] yields an unbiased estimator: [$$\begin{aligned} \notag &\hat{F}_{(\alpha),gm,\beta} = \frac{\prod_{j=1}^k |x_j|^{\alpha/k}} { D_{gm,\beta}},\\\notag &D_{gm,\beta} = \cos^k\left(\frac{1}{k}\tan^{-1}\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\right) \times\\\notag & \left(1+\beta^2\tan^2\left(\frac{\alpha\pi}{2}\right)\right)^{\frac{1}{2}}\left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}\right) \Gamma\left(1-\frac{1}{k}\right)\Gamma\left(\frac{\alpha}{k}\right)\right]^k.\end{aligned}$$]{} The following Lemma shows that the variance of $\hat{F}_{(\alpha),gm,\beta}$ decreases with increasing $\beta\in[0,1]$. The variance of $\hat{F}_{(\alpha),gm,\beta}$ [$$\begin{aligned} \notag &\text{Var}\left(\hat{F}_{(\alpha),gm,\beta}\right) = F_{(\alpha)}^2 V_{gm,\beta}\\\notag &V_{gm,\beta} = \frac{ \cos^k\left(\frac{2}{k}\tan^{-1}\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\right)} {\cos^{2k}\left(\frac{1}{k}\tan^{-1}\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\right)}\times\\\notag &\hspace{0.5in}\frac{ \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{k}\right) \Gamma\left(1-\frac{2}{k}\right)\Gamma\left(\frac{2\alpha}{k}\right)\right]^k} { \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}\right) \Gamma\left(1-\frac{1}{k}\right)\Gamma\left(\frac{\alpha}{k}\right)\right]^{2k}}-1,\vspace{-0.1in}\end{aligned}$$]{} is a decreasing function of $\beta \in [0,1]$. **Proof:**  The result follows from the fact that [$$\begin{aligned} \notag &\frac{\cos\left(\frac{2}{k}\tan^{-1}\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\right)} {\cos^2\left(\frac{1}{k}\tan^{-1}\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\right) } \\\notag =& 2 - \sec^2\left(\frac{1}{k}\tan^{-1}\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\right),\end{aligned}$$]{} is a deceasing function of $\beta\in[0,1]$. $\Box$ Therefore, for attaining the smallest variance, we take $\beta =1$. For brevity, we simply use $\hat{F}_{(\alpha),gm}$ instead of $\hat{F}_{(\alpha),gm,1}$. In fact, the rest of the paper will always consider $\beta =1$ only. We rewrite $\hat{F}_{(\alpha),gm}$ (i.e., $\hat{F}_{(\alpha),gm,\beta=1}$) as [$$\begin{aligned} \label{eqn_F_gm} &\hat{F}_{(\alpha),gm} = \frac{\prod_{j=1}^k |x_j|^{\alpha/k}} { D_{gm}}, \hspace{0.5in} (k\geq 2),\\\notag &D_{gm} = \left(\cos^k\left(\frac{\kappa(\alpha)\pi}{2k}\right)/\cos \left(\frac{\kappa(\alpha)\pi}{2}\right)\right)\\\notag &\hspace{0.3in}\times \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}\right) \Gamma\left(1-\frac{1}{k}\right)\Gamma\left(\frac{\alpha}{k}\right)\right]^k.\end{aligned}$$]{} Here, $\kappa(\alpha) = \alpha$, if $\alpha<1$, and $\kappa(\alpha) = 2-\alpha$ if $\alpha>1$. Lemma \[lem\_gm\_moments\] concerns the asymptotic moments of $\hat{F}_{(\alpha),gm}$. \[lem\_gm\_moments\] As $k\rightarrow\infty$ [$$\begin{aligned} \notag &\left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\frac{2}{\pi}\Gamma\left(\frac{\alpha}{k}\right)\Gamma\left(1-\frac{1}{k}\right)\sin\left(\frac{\pi}{2}\frac{\alpha}{k}\right)\right]^k \\\label{eqn_gm_asymp} \rightarrow& \exp\left(-\gamma_e\left(\alpha-1\right)\right),\end{aligned}$$]{} **monotonically** with increasing $k$ ($k\geq 2$), where $\gamma_e = 0.57724...$ is Euler’s constant. For any fixed $t$, as $k\rightarrow\infty$, [$$\begin{aligned} &\text{E}\left(\left( \hat{F}_{(\alpha),gm} \notag \right)^t\right)\\\notag =& F_{(\alpha)}^t\frac{ \cos^{k}\left(\frac{\kappa(\alpha)\pi}{2k}t\right) \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}t\right) \Gamma\left(1-\frac{t}{k}\right)\Gamma\left(\frac{\alpha}{k}t\right)\right]^{k}} { \cos^{kt}\left(\frac{\kappa(\alpha)\pi}{2k}\right) \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}\right) \Gamma\left(1-\frac{1}{k}\right)\Gamma\left(\frac{\alpha}{k}\right)\right]^{kt} }\\\notag =& F_{(\alpha)}^t\exp\left( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alpha^2+2-3\kappa^2(\alpha)\right)+O\left(\frac{1}{k^2}\right)\right).\end{aligned}$$]{} [$$\begin{aligned} \notag \text{Var}\left(\hat{F}_{(\alpha),gm}\right) = \frac{F_{(\alpha)}^2}{k}\frac{\pi^2}{12}\left(\alpha^2+2-3\kappa^2(\alpha)\right)+O\left(\frac{1}{k^2}\right).\end{aligned}$$]{} **Proof:** See Appendix \[proof\_lem\_gm\_moments\]. $\Box$ In (\[eqn\_F\_gm\]), the denominator $D_{gm}$ depends on $k$ for small $k$. For convenience in analyzing tail bounds, we consider an asymptotically equivalent [*geometric mean*]{} estimator: [$$\begin{aligned} \notag%\label{eqn_F_gm_b} \hat{F}_{(\alpha),gm,b} = \exp\left(\gamma_e(\alpha-1)\right)\cos\left(\frac{\kappa(\alpha)\pi}{2}\right)\prod_{j=1}^k |x_j|^{\alpha/k}.\end{aligned}$$]{} Lemma \[lem\_gm\_bounds\] provides the tail bounds for $\hat{F}_{(\alpha),gm,b}$ and Figure \[fig\_G\_gm\] plots the tail bound constants. One can infer the tail bounds for $\hat{F}_{(\alpha),gm}$ from the monotonicity result (\[eqn\_gm\_asymp\]). \[lem\_gm\_bounds\] The right tail bound: and the left tail bound: [$$\begin{aligned} \notag &\frac{\epsilon^2}{G_{R,gm}} = C_R \log(1+\epsilon) - C_R \gamma_e(\alpha-1) \\\notag & - \log\left(\cos\left(\frac{\kappa(\alpha)\pi C_R}{2}\right) \frac{2}{\pi}\Gamma\left(\alpha C_R\right)\Gamma\left(1-C_R\right)\sin\left(\frac{\pi\alpha C_R}{2}\right)\right),\\\notag &\frac{\epsilon^2}{G_{L,gm}} = -C_L \log(1-\epsilon) + C_L\gamma_e(\alpha-1)+\log\alpha\\\notag &\hspace{0.in} - \log\left(\cos\left(\frac{\kappa(\alpha)\pi}{2}C_L\right)\Gamma\left(C_L\right)\right)+\log\left( \Gamma\left(\alpha C_L\right)\cos\left(\frac{\pi\alpha C_L}{2}\right)\right).\end{aligned}$$]{} $C_R$ and $C_L$ are solutions to Here $\psi(z) = \frac{\Gamma^\prime(z)}{\Gamma(z)}$ is the “Psi” function. **Proof:** See Appendix \[proof\_lem\_gm\_bounds\]. $\Box$ It is important to understand the behavior of the tail bounds as $\alpha = 1\pm\Delta \rightarrow 1$. ($\alpha=1-\Delta$ if $\alpha<1$; and $\alpha=1+\Delta$ if $\alpha>1$.) See more comments in Appendix \[app\_moments\]. Lemma \[lem\_G\_gm\_rate\] describes the precise rates of convergence. \[lem\_G\_gm\_rate\] For fixed $\epsilon$, as $\alpha \rightarrow 1$ (i.e., $\Delta \rightarrow 0$), [$$\begin{aligned} \notag\vspace{-0.1in} &G_{R,gm}= \frac{\epsilon^2}{\log(1+\epsilon) - 2\sqrt{\Delta\log\left(1+\epsilon\right)}+o\left(\sqrt{\Delta}\right)},\\\notag &\text{If}\ \alpha>1, \ \text{then}\\\notag &G_{L,gm} = \frac{\epsilon^2}{-\log(1-\epsilon) - 2\sqrt{-2\Delta\log(1-\epsilon)} + o\left(\sqrt{\Delta}\right)},\\\notag &\text{If} \ \alpha<1, \ \text{then}\\\notag &G_{L,gm} = \frac{\epsilon^2}{ \Delta\left(\exp\left(\frac{-\log(1-\epsilon)}{\Delta} -1 - \gamma_e\right)\right)+o\left(\Delta\exp\left(\frac{1}{\Delta}\right)\right)}.\end{aligned}$$]{} **Proof:**    See Appendix \[proof\_lem\_G\_gm\_rate\]. $\Box$ Figure \[fig\_G\_gm\_approx\] plots the constants for small values of $\Delta$, along with the approximations suggested in Lemma \[lem\_G\_gm\_rate\]. Since we usually consider $\epsilon$ should not be too large, we can write, as $\alpha\rightarrow 1$, $G_{R,gm} = O\left(\epsilon\right)$ and $G_{L,gm} = O\left(\epsilon\right)$ if $\alpha>1$; both at the rate [$O\left(\sqrt{\Delta}\right)$]{}. However, if $\alpha<1$, [$G_{L,gm} = O\left(\epsilon \exp\left(-\frac{\epsilon}{\Delta}\right)\right)$]{}, which is extremely fast.\ The sample complexity bound is then straightforward. \[lem\_JL\] Using the geometric mean estimator, it suffices to let $k = G\frac{1}{\epsilon^2}\log\left(\frac{2}{\delta}\right)$ so that the error will be within a $1\pm\epsilon$ factor with probability $1-\delta$, where $G = \max(G_{R,gm}, G_{L,gm})$. In the neighborhood of $\alpha = 1$, $k = O\left(\frac{1}{\epsilon}\log \frac{2}{\delta}\right)$ only. The Harmonic Mean Estimator {#sec_hm} =========================== For $\alpha <1$, the [*harmonic mean*]{} estimator can considerably improve $\hat{F}_{(\alpha),gm}$. Unlike the [*harmonic mean*]{} estimator in [@Proc:Li_SODA08], which is useful only for small $\alpha$ and has no exponential tail bounds except for $\alpha=0+$, the [*harmonic mean*]{} estimator in this study has very nice tail properties for all $0<\alpha<1$. The [*harmonic mean*]{} estimator takes advantage of the fact that if $Z\sim S(\alpha<1, \beta =1, F_{(\alpha)})$, then $\text{E}\left(|Z|^\lambda\right)=\text{E}\left(Z^\lambda\right)$ exists for all $-\infty<\lambda <\alpha$. \[lem\_hm\] Assume $k$ i.i.d. samples $x_j \sim S(\alpha<1, \beta=1, F_{(\alpha)})$, define the harmonic mean estimator $\hat{F}_{(\alpha),hm}$, [$$\begin{aligned} \notag\vspace{-0.05in} \hat{F}_{(\alpha),hm} = \frac{k\frac{\cos\left(\frac{\alpha\pi}{2}\right)}{\Gamma(1+\alpha)}}{\sum_{j=1}^k|x_j|^{-\alpha}},\end{aligned}$$]{} and the bias-corrected harmonic mean estimator $\hat{F}_{(\alpha),hm,c}$, [$$\begin{aligned} \notag \hat{F}_{(\alpha),hm,c} = \frac{k\frac{\cos\left(\frac{\alpha\pi}{2}\right)}{\Gamma(1+\alpha)}}{\sum_{j=1}^k|x_j|^{-\alpha}} \left(1- \frac{1}{k}\left(\frac{2\Gamma^2(1+\alpha)}{\Gamma(1+2\alpha)}-1\right) \right).\end{aligned}$$]{} The bias and variance of $\hat{F}_{(\alpha),hm,c}$ are [$$\begin{aligned} \notag &\text{E}\left(\hat{F}_{(\alpha),hm,c}\right) = F_{(\alpha)}+O\left(\frac{1}{k^2}\right),\\\notag &\text{Var}\left(\hat{F}_{(\alpha),hm,c}\right) = \frac{F^{2}_{(\alpha)}}{k}\left(\frac{2\Gamma^2(1+\alpha)}{\Gamma(1+2\alpha)}-1\right) + O\left(\frac{1}{k^2}\right).\end{aligned}$$]{} The right tail bound of $\hat{F}_{(\alpha),hm}$ is, for $\epsilon>0$, [$$\begin{aligned} \notag &\mathbf{Pr}\left( \hat{F}_{(\alpha),hm} - F_{(\alpha)} \geq \epsilon F_{(\alpha)}\right) \leq\exp\left(-k\left(\frac{\epsilon^2}{G_{R,hm}} \right)\right), \\\notag &\frac{\epsilon^2}{G_{R,hm}} = -\log \left(\sum_{m=0}^\infty \frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}(-t_1^*)^m\right) -\frac{t_1^*}{1+\epsilon},\end{aligned}$$]{} where $t_1^*$ is the solution to [$$\begin{aligned} \notag \frac{\sum_{m=1}^\infty(-1)^m m (t_1^*)^{m-1}\frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)} }{\sum_{m=0}^\infty(-1)^m (t_1^*)^{m}\frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)} } + \frac{1}{1+\epsilon} = 0.\end{aligned}$$]{} The left tail bound of $\hat{F}_{(\alpha),hm}$ is, for $0<\epsilon<1$, [$$\begin{aligned} \notag &\mathbf{Pr}\left( \hat{F}_{(\alpha),hm} - F_{(\alpha)} \leq -\epsilon F_{(\alpha)}\right) \leq\exp\left(-k\left(\frac{\epsilon^2}{G_{L,hm}}\right)\right),\\\notag &\frac{\epsilon^2}{G_{L,hm}} = -\log \left(\sum_{m=0}^\infty \frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}(t_2^*)^m\right) +\frac{t_2^*}{1-\epsilon}\end{aligned}$$]{} where $t_2^*$ is the solution to [$$\begin{aligned} \notag -\frac{\sum_{m=1}^\infty m (t_2^*)^{m-1}\frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)} }{\sum_{m=0}^\infty (t_2^*)^{m}\frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)} } + \frac{1}{1-\epsilon} = 0\end{aligned}$$]{} **Proof:** See Appendix \[proof\_lem\_hm\]. $\Box$. The Logarithmic Norm and Distance {#sec_log} ================================= The logarithmic norm and distance can be important in practice. Consider estimating the parameters from $D$ i.i.d. samples $x_i \sim Gamma(\theta,\gamma)$. The density function is $f_X(x) = x^{\theta-1}\frac{\exp\left(-x/\gamma\right)}{\gamma^\theta\Gamma(\theta)}$, and the likelihood equation is [$$\begin{aligned} \notag\vspace{-0.1in} (\theta - 1)\sum_{i=1}^D\log x_i - \sum_{i=1}^D{x_i}/\gamma - D\theta\log(\gamma) - D\log\Gamma(\theta).\end{aligned}$$]{} If instead, $x_i \sim Pareto(\theta)$, $i = 1$ to $D$, then the density is $f_X(x) = \frac{\theta}{x^{\theta+1}}$, $x\geq 1$, and the likelihood equation is [$$\begin{aligned} \notag\vspace{-0.1in} D\log\theta - (\theta+1)\sum_{i=1}^D\log x_i.\end{aligned}$$]{} Therefore, the logarithmic norm occurs at least in the content of maximum likelihood estimations of common distributions. Now, consider the data $x_i$’s are actually the elements of data streams $A_t[i]$’s. Estimating $\sum_{i=1}^D\log A_t[i]$ becomes an interesting and practically meaningful problem. Our solution is based on the fact that, as $\alpha\rightarrow 0+$, [$$\begin{aligned} \notag \frac{D}{\alpha}\log\left(\frac{1}{D}\sum_{i=1}^DA_t[i]^\alpha\right)\rightarrow \sum_{i=1}^D\log A_t[i],\end{aligned}$$]{} which can be shown by L’Hópital’s rule. More precisely, [$$\begin{aligned} \notag &\left|\frac{D}{\alpha}\log\left(\frac{1}{D}\sum_{i=1}^DA_t[i]^\alpha\right) - \sum_{i=1}^D\log A_t[i]\right| \\\notag =&O\left(\frac{\alpha}{D}\left(\sum_{i=1}^D\log A_t[i]\right)^2\right) + O\left(\alpha\sum_{i=1}^D\log^2A_t[i]\right),\end{aligned}$$]{} which can be shown by Taylor expansions. Therefore, we obtain one solution to approximating the logarithmic norm using very small $\alpha$. Of course, we have assumed that $A_t[i]>0$ strictly. In fact, this also suggests an approach for approximating the logarithmic distance between two streams $\sum_{i=1}^D\log|A_t[i] - B_t[i]|$, provided we use [*symmetric stable random projections*]{}. The logarithmic distance can be useful in machine learning practice with massive heavy-tailed data (either static or dynamic) such as image and text data. For those data, the usual $l_2$ distance would not be useful without “term-weighting” the data; and taking logarithm is one simple weighting scheme. Thus, our method provides a direct way to compute pairwise distances, taking into account data weighting automatically. One may be also interested in the tail bounds, which, however, can not be expressed in terms of the logarithmic norm (or distance). Nevertheless, we can obtain, e.g., [$$\begin{aligned} \notag &\mathbf{Pr}\left(\left[\frac{D}{\alpha}\log\left(\frac{1}{D}\hat{F}_{(\alpha),hm}\right)\right]\geq (1+\epsilon) \left[\frac{D}{\alpha}\log\left(\frac{1}{D}F_{(\alpha)}\right)\right]\right)\\\notag \leq& \exp\left(-k\frac{\left(\left(F_{(\alpha)}/D\right)^{\epsilon}-1\right)^2}{G_{R,hm}}\right), \hspace{0.5in} \epsilon>0,\\\notag &\mathbf{Pr}\left(\left[\frac{D}{\alpha}\log\left(\frac{1}{D}\hat{F}_{(\alpha),hm}\right)\right]\leq (1-\epsilon) \left[\frac{D}{\alpha}\log\left(\frac{1}{D}F_{(\alpha)}\right)\right]\right)\\\notag \leq& \exp\left(-k\frac{\left(1-\left(D/F_{(\alpha)}\right)^{\epsilon}\right)^2}{G_{L,hm}}\right), \hspace{0.5in} 0<\epsilon<1\end{aligned}$$]{} If $\hat{F}_{(\alpha),gm}$ is used, we just replace the corresponding constants in the above expressions. If we are interested in the logarithmic distance, we simply apply [*symmetric stable random projections*]{} and use an appropriate estimator of the distance; the corresponding tail bounds will have same format. Conclusion ========== Counting is a fundamental operation. In data streams $A_t[i]$, $i\in[1,D]$, counting the $\alpha$th frequency moments $F_{(\alpha)} = \sum_{i=1}^DA_t[i]^\alpha$ has been extensively studied. Our proposed [*Compressed Counting (CC)*]{} takes advantage of the fact that most data streams encountered in practice are non-negative, although they are subject to deletion and insertion. In fact, CC only requires that at the time $t$ for the evaluation, $A_t[i]\geq 0$; at other times, the data streams can actually go below zero. [*Compressed Counting*]{} successfully captures the intuition that, when $\alpha = 1$, a simple counter suffices, and when $\alpha=1\pm\Delta$ with small $\Delta$, an intelligent counting system should require low space (continuously as a function of $\Delta$). The case with small $\Delta$ can be practically important. For example, $\Delta$ may be the “decay rate” or “interest rate,” which is usually small. CC can also be very useful for statistical parameter estimation based on the [*method of moments*]{}. Also, one can approximate the entropy moment using the $\alpha$th moments with $\alpha = 1\pm\Delta$ and very small $\Delta$. Compared with previous studies, e.g., [@Article:Indyk_JACM06; @Proc:Li_SODA08], [*Compressed Counting*]{} achieves, in a sense, an “infinite improvement” in terms of the asymptotic variances when $\Delta\rightarrow 0$. Two estimators based on the geometric mean and the harmonic mean are provided in this study, including their variances, tail bounds, and sample complexity bounds. We analyze our sample complexity bound $k = G\frac{1}{\epsilon^2}\log\frac{2}{\delta}$ at the neighborhood of $\alpha = 1$ and show $G = O\left(\epsilon\right)$ at small $\Delta$. This implies that our bound at small $\Delta$ is actually $k =O\left(1/\epsilon\right)$ instead of $O\left(1/\epsilon^2\right)$, which is required in the Johnson-Lindenstrauss Lemma and its various analogs. Finally, we propose a scheme for approximating the logarithmic norm and the logarithmic distance, useful in statistical parameter estimation and machine learning practice.\ We expect that new algorithms will soon be developed to take advantage of [*Compressed Counting*]{}. For example, via private communications, we have learned that a group is vigorously developing algorithms using projections with [$\alpha=1\pm\Delta$]{} very close to 1, where [$\Delta$]{} is their important parameter. An Example of Method of Moments {#app_moments} =============================== We provide a (somewhat contrived) example of the [*method of moments*]{}. Suppose the observed data $x_i$’s are from data streams and suppose the data follows a gamma distribution $x_i \sim Gamma(\theta,1)$, i.i.d. Here, we only consider one parameter $\theta$ so that we can analyze the variance easily. Suppose we estimate $\theta$ using the $\alpha$th moment. Because $\text{E}(x_i^\alpha) = \Gamma(\alpha+\theta)/\Gamma(\theta)$, we can solve for $\hat{\theta}$ from [$$\begin{aligned} \notag \frac{\Gamma(\alpha+\hat{\theta})}{\Gamma(\hat{\theta})} = \frac{1}{D}\sum_{i=1}^Dx_k^\alpha, \hspace{0.in} \Longrightarrow \text{Var}\left(\frac{\Gamma(\alpha+\hat{\theta})}{\Gamma(\hat{\theta})}\right) = \frac{1}{D}\left(\frac{\Gamma(2\alpha+\theta)}{\Gamma(\theta)} - \frac{\Gamma^2(\alpha+\theta)}{\Gamma^2(\theta)}\right)\end{aligned}$$]{} By the “delta method” (i.e., $\text{Var}(h(x))\approx \text{Var}(x)(h^\prime(\text{E}(x)))^2$) and using the implicit derivative of $\hat{\theta}$, we obtain [$$\begin{aligned} \notag \text{Var}\left(\hat{\theta}\right) \approx \frac{1}{D} \left(\frac{\Gamma(2\alpha+\theta)\Gamma(\theta)}{\Gamma^2(\alpha+\theta)} -1 \right)\frac{1}{\left(\psi(\alpha+\theta)-\psi(\theta)\right)^2}.\end{aligned}$$]{} One can verify [$\text{Var}(\hat{\theta})$]{} increases monotonically with increasing [$\alpha\in[0,\infty)$]{}. Because $x_i$’s are from data streams, we apply [*Compressed Counting*]{} for the $\alpha$th moment. Suppose we consider the difference in the estimation accuracy at different $\alpha$ is not important (because [$D$]{} is large). Then we simply let $\alpha =1$. In case we need to estimate two parameters, we might choose [$\alpha=1$]{} and another $\alpha$ close to 1. Now suppose we actually care about both the estimation accuracy (which favors smaller $\alpha$) and the computational efficiency (which favors $\alpha=1$), we then need to balance this trade-off by choosing $\alpha$. To do so, we need to know the precise behavior of [*Compressed Counting*]{} in the neighborhood of $\alpha =1$, as well as the precise behavior of $\hat{\theta}$, i.e., its tail bounds (not just variance). Thus, our analysis on the convergence rates in Lemma \[lem\_G\_gm\_rate\] will be very useful. Proof of Lemma \[lem\_moments\] {#proof_lem_moments} =============================== Assume $Z \sim S(\alpha,\beta,F_{(\alpha)})$. To prove $\textbf{E}\left(|Z|^\lambda\right)$ for $-1<\lambda <\alpha$, [@Book:Zolotarev_86 Theorem 2.6.3] provided only a partial answer: [$$\begin{aligned} \notag &\int_0^\infty z^\lambda f_Z(z;\alpha,\beta_B,F_{(\alpha)}) dz \\\notag =& F^{\lambda/\alpha}_{(\alpha)}\frac{\sin(\pi\rho\lambda)}{\sin(\pi\lambda)}\frac{\Gamma\left(1-\frac{\lambda}{\alpha}\right)}{\Gamma\left(1-\lambda\right)} \cos^{-\lambda/\alpha}\left(\pi\beta_B\kappa(\alpha)/2\right)\end{aligned}$$]{} where we denote [$$\begin{aligned} \notag \kappa(\alpha) = \alpha \ \ \ \text{if} \ \ \ \alpha<1, \ \ \ \text{and} \ \ \kappa(\alpha)=2-\alpha\ \ \ \text{if} \ \ \alpha>1,\end{aligned}$$]{} and according to the parametrization used in [@Book:Zolotarev_86 I.19, I.28]: [$$\begin{aligned} \notag \beta_B = \frac{2}{\pi\kappa(\alpha)}\tan^{-1}\left(\beta\tan\left(\frac{\pi\alpha}{2}\right)\right),\hspace{0.1in} \rho = \frac{1-\beta_B\kappa(a)/\alpha}{2}.\end{aligned}$$]{} Note that [$$\begin{aligned} \notag &\cos^{-\lambda/\alpha}\left(\pi\beta_B\kappa(\alpha)/2\right) = \left(1 + \tan^2\left(\pi\beta_B\kappa(\alpha)/2\right)\right)^{\frac{\lambda}{2\alpha}}\\\notag =& \left(1 + \tan^2\left(\tan^{-1}\left(\beta\tan\left(\frac{\pi\alpha}{2}\right)\right)\right)\right)^{\frac{\lambda}{2\alpha}}\\\notag =& \left(1 + \beta^2\tan^2\left(\frac{\pi\alpha}{2}\right)\right)^{\frac{\lambda}{2\alpha}}.\end{aligned}$$]{} Therefore, for $-1<\lambda<\alpha$, [$$\begin{aligned} \notag &\int_0^\infty z^\lambda f_Z(z;\alpha,\beta_B,F_{(\alpha)}) dz\\\notag =& F^{\lambda/\alpha}_{(\alpha)} \frac{\sin(\pi\rho\lambda)}{\sin(\pi\lambda)}\frac{\Gamma\left(1-\frac{\lambda}{\alpha}\right)}{\Gamma\left(1-\lambda\right)} \left(1 + \beta^2\tan^2\left(\frac{\pi\alpha}{2}\right)\right)^{\frac{\lambda}{2\alpha}}.\end{aligned}$$]{} To compute $\text{E}\left(|Z|^\lambda\right)$, we take advantage of a useful property of the stable density function[@Book:Zolotarev_86 page 65]: [$$\begin{aligned} \notag f_Z(- z;\alpha,\beta_B,F_{(\alpha)}) = f_Z(z;\alpha,-\beta_B,F_{(\alpha)}).\end{aligned}$$ $$\begin{aligned} \notag &\text{E}\left(|Z|^\lambda\right)\\\notag =& \int_{-\infty}^0 (-z)^\lambda f_Z(z;\alpha,\beta_B,F_{(\alpha)}) dz + \int_0^\infty z^\lambda f_Z(z;\alpha,\beta_B,F_{(\alpha)}) dz\\\notag =& \int_0^{\infty} z^\lambda f_Z(z;\alpha,-\beta_B,F_{(\alpha)}) dz + \int_0^\infty z^\lambda f_Z(z;\alpha,\beta_B,F_{(\alpha)}) dz\\\notag =&\frac{F^{\lambda/\alpha}_{(\alpha)}}{\sin(\pi\lambda)}\frac{\Gamma\left(1-\frac{\lambda}{\alpha}\right)}{\Gamma\left(1-\lambda\right)} \left(1 + \beta^2\tan^2\left(\frac{\pi\alpha}{2}\right)\right)^{\frac{\lambda}{2\alpha}}\\\notag &\times\left( \sin\left(\pi\lambda\frac{1-\beta_B\kappa(\alpha)/\alpha}{2}\right)+ \sin\left(\pi\lambda\frac{1+\beta_B\kappa(\alpha)/\alpha}{2}\right) \right)\\\notag =&\frac{F^{\lambda/\alpha}_{(\alpha)}}{\sin(\pi\lambda)}\frac{\Gamma\left(1-\frac{\lambda}{\alpha}\right)}{\Gamma\left(1-\lambda\right)} \left(1 + \beta^2\tan^2\left(\frac{\pi\alpha}{2}\right)\right)^{\frac{\lambda}{2\alpha}}\\\notag &\times\left( 2\sin\left(\frac{\pi\lambda}{2}\right)\cos\left(\frac{\pi\lambda}{2}\beta_B\kappa(\alpha)/\alpha\right)\right)\\\notag =&\frac{F^{\lambda/\alpha}_{(\alpha)}}{\cos(\pi\lambda/2)}\frac{\Gamma\left(1-\frac{\lambda}{\alpha}\right)}{\Gamma\left(1-\lambda\right)} \left(1 + \beta^2\tan^2\left(\frac{\pi\alpha}{2}\right)\right)^{\frac{\lambda}{2\alpha}}\\\notag &\times\cos\left(\frac{\lambda}{\alpha}\tan^{-1}\left(\beta\tan\left(\frac{\pi\alpha}{2}\right)\right) \right) \\\notag =&F^{\lambda/\alpha}_{(\alpha)}\left(1 + \beta^2\tan^2\left(\frac{\pi\alpha}{2}\right)\right)^{\frac{\lambda}{2\alpha}} \cos\left(\frac{\lambda}{\alpha}\tan^{-1}\left(\beta\tan\left(\frac{\pi\alpha}{2}\right)\right) \right)\\\notag &\hspace{0.5in}\times\left(\frac{2}{\pi}\sin\left(\frac{\pi}{2}\lambda\right)\Gamma\left(1-\frac{\lambda}{\alpha}\right)\Gamma\left(\lambda\right)\right),\end{aligned}$$]{} which can be simplified when $\beta = 1$, to be [$$\begin{aligned} \notag%\label{eqn_moment}\notag \textbf{E}\left(|Z|^\lambda\right) = {F}_{(\alpha)}^{\lambda/\alpha} \frac{\cos\left(\frac{\kappa(\alpha)}{\alpha}\frac{\lambda\pi}{2}\right)} {\cos^{\lambda/\alpha}\left(\frac{\kappa(\alpha)\pi}{2}\right)} \left(\frac{2}{\pi}\sin\left(\frac{\pi}{2}\lambda\right)\Gamma\left(1-\frac{\lambda}{\alpha}\right)\Gamma\left(\lambda\right)\right).\end{aligned}$$]{} The final task is to show that when $\alpha <1$ and $\beta=1$, $\text{E}\left(|Z|^\lambda\right)$ exists for all $-\infty<\lambda<\alpha$, not just $-1<\lambda<\alpha$. This is an extremely useful property. Note that when $\alpha<1$ and $\beta = 1$, $Z$ is always non-negative. As shown in the proof of [@Book:Zolotarev_86 Theorem 2.6.3], [$$\begin{aligned} \notag &\text{E}\left(|Z|^\lambda\right) = F_{(\alpha)}^{\lambda/\alpha} \cos^{-\lambda/\alpha}\left(\frac{\pi\alpha}{2}\right) \frac{1}{\pi} \text{Im} \int_0^\infty z^\lambda \int_0^\infty \\\notag &\exp\left( - zu \exp(\sqrt{-1}\pi/2) - u^\alpha \exp(-\sqrt{-1}\pi\alpha/2)+\frac{\sqrt{-1}\pi}{2}\right)dudz\\\notag =&F_{(\alpha)}^{\lambda/\alpha} \cos^{-\lambda/\alpha}\left(\frac{\pi\alpha}{2}\right) \frac{1}{\pi}\times\\\notag \text{Im} &\int_0^\infty \int_0^\infty z^\lambda \exp\left( - zu\sqrt{-1} - u^\alpha \exp(-\sqrt{-1}\pi\alpha/2)\right)\sqrt{-1}dudz.\end{aligned}$$]{} The only thing we need to check is that in the proof of [@Book:Zolotarev_86 Theorem 2.6.3], the condition for Fubini’s theorem (to exchange order of integration) still holds when $-\infty<\alpha <1$, $\beta = 1$, and $\lambda<-1$. We can show [$$\begin{aligned} \notag &\int_0^\infty \int_0^\infty \left| z^\lambda \exp\left( - zu\sqrt{-1} - u^\alpha \exp(-\sqrt{-1}\pi\alpha/2)\right)\sqrt{-1}\right|dudz\\\notag =&\int_0^\infty\int_0^\infty z^\lambda \left|\exp\left(-u^\alpha\cos(\pi\alpha/2)+ \sqrt{-1}u^\alpha\sin(\pi\alpha/2)\right)\right|dudz\\\notag =&\int_0^\infty\int_0^\infty z^\lambda \exp\left(-u^\alpha\cos(\pi\alpha/2)\right)dudz<\infty, \end{aligned}$$]{} provided $\lambda<-1$ ($\lambda \neq -1, -2, -3, ....$) and $\cos(\pi\alpha/2)>0$, i.e., $\alpha<1$. Note that $|\exp(\sqrt{-1}x)|=1$ always and Euler’s formula: $\exp(\sqrt{-1}x) = \cos(x) + \sqrt{-1}\sin(x)$ is frequently used to simplify the algebra. Once we show that Fubini’s condition is satisfied, we can exchange the order of integration and the rest follows from the proof of [@Book:Zolotarev_86 Theorem 2.6.3]. Because of continuity, the “singularity points” $\lambda = -1, -2, -3,...$ do not matter. Proof of Lemma \[lem\_gm\_moments\] {#proof_lem_gm_moments} =================================== We first show that, for any fixed $t$, as $k\rightarrow\infty$, [$$\begin{aligned} &\text{E}\left(\left( \hat{F}_{(\alpha),gm} \notag \right)^t\right) \\\notag =& F_{(\alpha)}^t\frac{ \cos^{k}\left(\frac{\kappa(\alpha)\pi}{2k}t\right) \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}t\right) \Gamma\left(1-\frac{t}{k}\right)\Gamma\left(\frac{\alpha}{k}t\right)\right]^{k}} { \cos^{kt}\left(\frac{\kappa(\alpha)\pi}{2k}\right) \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}\right) \Gamma\left(1-\frac{1}{k}\right)\Gamma\left(\frac{\alpha}{k}\right)\right]^{kt} }\\\notag =& F_{(\alpha)}^t\exp\left( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alpha^2+2-3\kappa^2(\alpha)\right)+O\left(\frac{1}{k^2}\right)\right).\end{aligned}$$]{} In [@Proc:Li_SODA08], it was proved that, as $k\rightarrow\infty$, [$$\begin{aligned} \notag &\frac{\left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}t\right) \Gamma\left(1-\frac{t}{k}\right)\Gamma\left(\frac{\alpha}{k}t\right)\right]^{k}} { \left[\frac{2}{\pi}\sin\left(\frac{\pi\alpha}{2k}\right) \Gamma\left(1-\frac{1}{k}\right)\Gamma\left(\frac{\alpha}{k}\right)\right]^{kt} }\\\notag =&1+\frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alpha^2+2\right)+O\left(\frac{1}{k^2}\right)\\\notag =&\exp\left( \frac{1}{k}\frac{\pi^2(t^2-t)}{24}\left(\alpha^2+2\right)+O\left(\frac{1}{k^2}\right)\right).\end{aligned}$$]{} Using the infinite product representation of cosine[@Book:Gradshteyn_00 1.43.3] [$$\begin{aligned} \notag \cos(z) = \prod_{s=0}^\infty\left(1 - \frac{4z^2}{(2s+1)^2\pi^2}\right),\end{aligned}$$]{} we can rewrite [$$\begin{aligned} \notag &\frac{\cos^{k}\left(\frac{\kappa(\alpha)\pi}{2k}t\right) } { \cos^{kt}\left(\frac{\kappa(\alpha)\pi}{2k}\right) } = \prod_{s=0}^\infty \left( 1 - \frac{\kappa^2(\alpha)t^2}{(2s+1)^2k^2}\right)^k\left( 1 - \frac{\kappa^2(\alpha)}{(2s+1)^2k^2}\right)^{-kt}\\\notag =&\prod_{s=0}^\infty \left( \left( 1 - \frac{\kappa^2(\alpha)t^2}{(2s+1)^2k^2}\right) \left( 1 + t\frac{\kappa^2(\alpha)}{(2s+1)^2k^2}+O\left(\frac{1}{k^3}\right)\right)\right)^k\\\notag =&\prod_{s=0}^\infty \left( 1 - \frac{\kappa^2(\alpha)(t^2-t)}{(2s+1)^2k^2} +O\left(\frac{1}{k^3}\right)\right)^k =\prod_{s=0}^\infty \left( 1 - \frac{\kappa^2(\alpha)(t^2-t)}{(2s+1)^2k} +O\left(\frac{1}{k^2}\right)\right)\\\notag =&\exp\left(\sum_{s=0}^\infty \log \left( 1 - \frac{\kappa^2(\alpha)(t^2-t)}{(2s+1)^2k} +O\left(\frac{1}{k^2}\right)\right) \right)\\\notag =&\exp\left(-\frac{\kappa^2(\alpha)}{k}(t^2-t) \sum_{s=0}^\infty \frac{1}{(2s+1)^2} +O\left(\frac{1}{k^2}\right)\right) \\\notag =&\exp\left(-\frac{\kappa^2(\alpha)}{k}(t^2-t) \frac{\pi^2}{8}+O\left(\frac{1}{k^2}\right)\right),\end{aligned}$$]{} which, combined with the result in [@Proc:Li_SODA08], yields the desired expression.\ The next task is to show [$$\begin{aligned} \notag \left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\frac{2}{\pi}\Gamma\left(\frac{\alpha}{k}\right)\Gamma\left(1-\frac{1}{k}\right)\sin\left(\frac{\pi}{2}\frac{\alpha}{k}\right)\right]^k \rightarrow \exp\left(-\gamma_e\left(\alpha-1\right)\right),\end{aligned}$$]{} monotonically as $k\rightarrow\infty$, where [$\gamma_e = 0.577215665...$]{}, is Euler’s constant. In [@Proc:Li_SODA08], it was proved that, as $k\rightarrow\infty$, [$$\begin{aligned} \notag \left[\frac{2}{\pi}\Gamma\left(\frac{\alpha}{k}\right)\Gamma\left(1-\frac{1}{k}\right)\sin\left(\frac{\pi}{2}\frac{\alpha}{k}\right)\right]^k \rightarrow \exp\left(-\gamma_e\left(\alpha-1\right)\right),\end{aligned}$$]{} monotonically. In this study, we need to consider instead [$$\begin{aligned} \notag\label{eqn_proof_gm_coefficent} &\left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\frac{2}{\pi}\Gamma\left(\frac{\alpha}{k}\right)\Gamma\left(1-\frac{1}{k}\right)\sin\left(\frac{\pi}{2}\frac{\alpha}{k}\right)\right]^k \\ =&\left[2\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\frac{\Gamma\left(\frac{\alpha}{k}\right)\sin\left(\frac{\pi\alpha}{2k}\right)}{\Gamma\left(\frac{1}{k}\right)\sin\left(\frac{\pi}{k}\right)} \right]^k\end{aligned}$$]{} (Note Euler’s reflection formula $\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$.) The additional term [$\left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\right]^k = 1 - O\left(\frac{1}{k}\right)$]{}. Therefore, [$$\begin{aligned} \notag &\left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\frac{2}{\pi}\Gamma\left(\frac{\alpha}{k}\right)\Gamma\left(1-\frac{1}{k}\right)\sin\left(\frac{\pi}{2}\frac{\alpha}{k}\right)\right]^k \rightarrow \exp\left(-\gamma_e\left(\alpha-1\right)\right).\end{aligned}$$]{} To show the monotonicity, however, we have to use some different techniques from [@Proc:Li_SODA08]. The reason is because the additional term $\left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\right]^k$ increases (instead of decreasing) monotonically with increasing $k$. First, we consider $\alpha>1$, i.e., $\kappa(\alpha) = 2-\alpha<1$. For simplicity, we take logarithm of (\[eqn\_proof\_gm\_coefficent\]) and replace $1/k$ by $t$, where $0\leq t\leq 1/2$ (recall $k\geq 2$). It suffices to show that $g(t)$ increases with increasing $t \in [0,1/2]$, where [$$\begin{aligned} \notag &g(t) =\frac{1}{t}W(t),\\\notag &W(t) = \log\left(\cos\left(\frac{\kappa(\alpha)\pi}{2}t\right)\right) + \log\left(\Gamma\left(\alpha t\right)\right) + \log\left(\sin\left(\frac{\pi\alpha}{2}t\right)\right)\\\notag &\hspace{0.5in}- \log\left(\Gamma\left(t\right)\right) - \log\left(\sin\left(\pi t\right)\right) + \log(2).\end{aligned}$$]{} Because $g^\prime(t) = \frac{1}{t}W^\prime(t) - \frac{1}{t^2}W(t)$, to show $g^\prime(t) \geq 0$ in $t \in [0,1/2]$, it suffices to show [$$\begin{aligned} \notag tW^\prime(t) - W(t) \geq 0.\end{aligned}$$]{} One can check that $tW^\prime(t)\rightarrow 0$ and $W(t)\rightarrow0$, as $t\rightarrow0+$. [$$\begin{aligned} \notag W^\prime(t) =& -\tan\left(\frac{\kappa(\alpha)\pi}{2}t\right)\left(\frac{\kappa\pi}{2}\right) + \psi\left(\alpha t\right)\alpha + \frac{1}{\tan\left(\frac{\pi\alpha}{2}t\right)}\left(\frac{\alpha\pi}{2}\right)\\\notag &\hspace{0.5in}- \psi(t) - \frac{1}{\tan(\pi t)}\pi.\end{aligned}$$]{} Here $\psi(x) = \frac{\partial \log(\Gamma(x))}{\partial x}$ is the “Psi” function. Therefore, to show $tW^\prime(t) - W(t) \geq 0$, it suffices to show that $tW^\prime(t) - W(t)$ is an increasing function of $t\in [0,1/2]$, i.e., [$$\begin{aligned} \notag &\left(tW^\prime(t) - W(t)\right)^\prime = W^{\prime\prime}(t) \geq 0, \ \ \ \text{i.e.,}\\\notag &W^{\prime\prime}(t) = -\sec^2\left(\frac{\kappa(\alpha)\pi}{2}t\right)\left(\frac{\kappa(\alpha)\pi}{2}\right)^2 + \psi^\prime(\alpha t)\alpha^2\\\notag &\hspace{0.5in}- \csc^2\left(\frac{\pi\alpha}{2}t\right)\left(\frac{\pi\alpha}{2}\right)^2 - \psi^\prime(t) +\csc^2(\pi t)\pi^2 \geq 0.\end{aligned}$$]{} Using series representation of $\psi(x)$ [@Book:Gradshteyn_00 8.363.8], we show [$$\begin{aligned} \notag \psi^\prime\left(\alpha t\right)\alpha^2 - \psi^{\prime}(t) =& \sum_{s=0}^\infty \frac{\alpha^2}{(\alpha t + s)^2} - \sum_{s=0}^\infty \frac{1}{(t + s)^2}\\\notag =& \sum_{s=0}^\infty \left(\frac{1}{(t + s/\alpha)^2} - \frac{1}{(t + s)^2}\right) \geq 0,\end{aligned}$$]{} because we consider $\alpha >1$. Thus, it suffices to show that [$$\begin{aligned} \notag & Q(t;\alpha) = -\sec^2\left(\frac{\kappa(\alpha)\pi}{2}t\right)\left(\frac{\kappa(\alpha)\pi}{2}\right)^2\\\notag &\hspace{0.5in}- \csc^2\left(\frac{\pi\alpha}{2}t\right)\left(\frac{\pi\alpha}{2}\right)^2 +\csc^2(\pi t)\pi^2 \geq 0.\end{aligned}$$]{} To show $Q(t;\alpha)\geq 0$, we can treat $Q(t;\alpha)$ as a function of $\alpha$ (for fixed $t$). Because both $\frac{1}{\sin(x)}$ and $\frac{1}{\cos(x)}$ are convex functions of $x\in[0,\pi/2]$, we know $Q(t;\alpha)$ is a concave function of $\alpha$ (for fixed $t$). It is easy to check that [$$\begin{aligned} \notag \underset{\alpha\rightarrow1+}{\lim}{Q(t;\alpha)} = 0, \hspace{0.5in} \underset{\alpha\rightarrow2-}{\lim}{Q(t;\alpha)} = 0.\end{aligned}$$]{} Because $Q(t;\alpha)$ is concave in $\alpha\in[1,2]$, we must have $Q(t;\alpha)\geq 0$; and consequently, $W^{\prime\prime}(t) \geq 0$ and $g^\prime(t)\geq 0$. Therefore, we have proved that (\[eqn\_proof\_gm\_coefficent\]) decreases monotonically with increasing $k$, when $1<\alpha\leq 2$. For $\alpha<1$ (i.e., $\kappa(\alpha)=\alpha<1$), we prove the monotonicity by a different technique. First, using infinite-product representations [@Book:Gradshteyn_00 8.322,1.431.1], [$$\begin{aligned} \notag &\Gamma(z) = \frac{\exp\left(-\gamma_ez\right)}{z}\prod_{s=1}^\infty \left(1+\frac{z}{s}\right)^{-1}\exp\left(\frac{z}{s}\right),\\\notag &\sin(z) =z\prod_{s=1}^\infty\left(1-\frac{z^2}{s^2\pi^2}\right),\end{aligned}$$]{} we can rewrite (\[eqn\_proof\_gm\_coefficent\]) as [$$\begin{aligned} \notag &\left[2\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\frac{\Gamma\left(\frac{\alpha}{k}\right)\sin\left(\frac{\pi\alpha}{2k}\right)}{\Gamma\left(\frac{1}{k}\right)\sin\left(\frac{\pi}{k}\right)} \right]^k%=\left[\frac{\Gamma\left(\frac{\alpha}{k}\right)\sin\left(\frac{\pi\alpha}{k}\right)}{\Gamma\left(\frac{1}{k}\right)\sin\left(\frac{\pi}{k}\right)} \right]^k = \exp\left(-\gamma_e(\alpha-1)\right)\times\\\notag &\left( \prod_{s=1}^\infty\exp\left(\frac{\alpha-1}{sk}\right)\left(1+\frac{\alpha}{ks}\right)^{-1}\left(1+\frac{1}{ks}\right) \left(1-\frac{\alpha^2}{k^2s^2}\right)\left(1-\frac{1}{s^2k^2}\right)^{-1}\right)^k.\end{aligned}$$]{} To show its monotonicity, it suffices to show for any $s\geq 1$ [$$\begin{aligned} \notag \left(\left(1+\frac{\alpha}{ks}\right)^{-1}\left(1+\frac{1}{ks}\right)\left(1-\frac{\alpha^2}{k^2s^2}\right)\left(1-\frac{1}{s^2k^2}\right)^{-1} \right)^k\end{aligned}$$]{} decreases monotonically, which is equivalent to show the monotonicity of $g(t)$ with increasing $t$, for $t\geq2$, where [$$\begin{aligned} \notag g(t) = t\log\left( \left(1+\frac{\alpha}{t}\right)^{-1}\left(1+\frac{1}{t}\right)\left(1-\frac{\alpha^2}{t^2}\right)\left(1-\frac{1}{t^2}\right)^{-1} \right) = t \log\left(\frac{t-\alpha}{t-1}\right).\end{aligned}$$]{} It is straightforward to show that $t \log\left(\frac{t-\alpha}{t-1}\right)$ is monotonically decreasing with increasing $t$ ($t\geq 2$), for $\alpha<1$. To this end, we have proved that for $0<\alpha\leq 2$ ($\alpha \neq 1$), [$$\begin{aligned} \notag &\left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}\right)\frac{2}{\pi}\Gamma\left(\frac{\alpha}{k}\right)\Gamma\left(1-\frac{1}{k}\right)\sin\left(\frac{\pi}{2}\frac{\alpha}{k}\right)\right]^k \rightarrow \exp\left(-\gamma_e\left(\alpha-1\right)\right),\end{aligned}$$]{} monotonically with increasing $k$ ($k\geq 2$). Proof of Lemma \[lem\_gm\_bounds\] {#proof_lem_gm_bounds} ================================== We first find the constant $G_{R,gm}$ in the right tail bound [$$\begin{aligned} \notag \mathbf{Pr}\left(\hat{F}_{(\alpha),gm,b} - F_{(\alpha)} \geq \epsilon F_{(\alpha)} \right) \leq \exp\left(-k\frac{\epsilon^2}{G_{R,gm}}\right), \ \ \epsilon>0.\end{aligned}$$]{} For [$0<t<k$]{}, the Markov moment bound yields [$$\begin{aligned} \notag%\label{eqn_moment_bound} &\mathbf{Pr}\left(\hat{F}_{(\alpha),gm,b} - F_{(\alpha)} \geq \epsilon F_{(\alpha)}\right) \leq \frac{\text{E}\left(\hat{F}_{(\alpha),gm}\right)^t}{(1+\epsilon)^tF_{(\alpha)}^t}\\\notag =& \frac{\left[\cos\left(\frac{\kappa(\alpha)\pi}{2k}t\right)\frac{2}{\pi}\Gamma\left(\frac{\alpha t}{k}\right)\Gamma\left(1-\frac{t}{k}\right)\sin\left(\frac{\pi\alpha t}{2k}\right)\right]^k}{(1+\epsilon)^{t}\exp\left(-t\gamma_e (\alpha -1)\right)}.\end{aligned}$$]{} We need to find the $t$ that minimizes the upper bound. For convenience, we consider its logarithm, i.e., [$$\begin{aligned} \notag &g(t) = t\gamma_e\left(\alpha-1\right) -t\log(1+\epsilon)+ \\\notag &\hspace{0.5in}k\log\left(\cos\left(\frac{\kappa(\alpha)\pi}{2k}t\right)\frac{2}{\pi}\Gamma\left(\frac{\alpha t}{k}\right)\Gamma\left(1-\frac{t}{k}\right)\sin\left(\frac{\pi\alpha t}{2k}\right)\right),\end{aligned}$$]{} whose first and second derivatives (with respect to $t$) are [$$\begin{aligned} \notag &g^\prime(t) = \gamma_e (\alpha -1)-\log(1+\epsilon)- \frac{\kappa(\alpha)\pi}{2}{\tan\left(\frac{\kappa(\alpha)\pi}{2k}t\right)} + \frac{\alpha\pi/2}{\tan\left(\frac{\alpha\pi t}{2k}\right)}\\\notag &\hspace{0.5in}+ \psi\left(\frac{\alpha t}{k}\right)\alpha - \psi\left(1-\frac{t}{k}\right),\\\notag &tg^{\prime\prime}(t) =-\left(\frac{\kappa(\alpha)\pi}{2}\right)^2{\sec^2\left(\frac{\kappa(\alpha)\pi}{2k}t\right)} - \left(\frac{\alpha\pi}{2}\right)^2\csc^2\left(\frac{\alpha\pi t}{2k}\right)\\\notag&\hspace{0.5in} +\alpha^2\psi^\prime\left(\frac{\alpha t}{k}\right) + \psi^\prime\left(1-\frac{t}{k}\right).\end{aligned}$$]{} We need to show that $g(t)$ is a convex function. By the following expansions: [@Book:Gradshteyn_00 1.422.2, 1.422.4, 8.363.8] [$$\begin{aligned} \notag &\sec^2\left(\frac{\pi x}{2}\right) = \frac{4}{\pi^2}\sum_{j=1}^\infty\left(\frac{1}{(2j-1-x)^2}+\frac{1}{(2j-1+x)^2}\right),\\\notag &\csc^2(\pi x) = \frac{1}{\pi^2x^2} +\frac{2}{\pi^2}\sum_{j=1}^\infty \frac{x^2+j^2}{(x^2-j^2)^2}, \hspace{0.2in} \psi^\prime(x) = \sum_{j=0}^\infty\frac{1}{(x+j)^2},\end{aligned}$$]{} we can rewrite [$$\begin{aligned} \notag &kg^{\prime\prime}(t) = -\kappa^2\sum_{j=1}^\infty\left(\frac{1}{(2j-1-\kappa t/k)^2}+\frac{1}{(2j-1+\kappa t/k)^2}\right) - \frac{k^2}{t^2}\\\notag & - \frac{\alpha^2}{2}\sum_{j=1}^\infty \frac{(\alpha t/2k)^2+j^2}{((\alpha t/2k)^2-j^2)^2} +\alpha^2\sum_{j=0}^\infty\frac{1}{(\alpha t/k + j)^2} + \sum_{j=0}^\infty \frac{1}{(1-t/k+j)^2}\\\notag =& -\kappa^2\sum_{j=1}^\infty\left(\frac{1}{(2j-1-\kappa t/k)^2}+\frac{1}{(2j-1+\kappa t/k)^2}\right)\\\notag &\hspace{0.2in}- \alpha^2\sum_{j=1}^\infty\left(\frac{1}{(\alpha t/k - 2j)^2} + \frac{1}{(\alpha t/k + 2j)^2}\right)\\\notag &\hspace{0.2in}+\alpha^2\sum_{j=1}^\infty\frac{1}{(\alpha t/k + j)^2} + \sum_{j=1}^\infty \frac{1}{(j-t/k)^2}.\end{aligned}$$]{} If $\alpha <1$, i.e., $\kappa(\alpha) = \alpha$, then [$$\begin{aligned} \notag &kg^{\prime\prime}(t)=- \alpha^2\sum_{j=1}^\infty\left(\frac{1}{(\alpha t/k - j)^2} + \frac{1}{(\alpha t/k + j)^2}\right) +\alpha^2\sum_{j=1}^\infty\frac{1}{(\alpha t/k + j)^2}\\\notag & + \sum_{j=1}^\infty \frac{1}{(j-t/k)^2} =- \alpha^2\sum_{j=1}^\infty\frac{1}{(j-\alpha t/k)^2} + \sum_{j=1}^\infty \frac{1}{(j-t/k)^2}\geq \ 0,\end{aligned}$$]{} because $\alpha <1$ and $0<t<k$. If $\alpha >1$, i.e., $\kappa(\alpha) = 2-\alpha<1$, then [$$\begin{aligned} \notag &kg^{\prime\prime}(t) = -\kappa^2\sum_{j=1}^\infty\left(\frac{1}{(2j-1-\kappa t/k)^2}+\frac{1}{(2j-1+\kappa t/k)^2}\right)\\\notag &- \alpha^2\sum_{j=1}^\infty\left(\frac{1}{(\alpha t/k - 2j)^2} + \frac{1}{(\alpha t/k + 2j)^2}\right) +\alpha^2\sum_{j=1}^\infty\frac{1}{(\alpha t/k + 2j)^2}\\\notag & + \alpha^2\sum_{j=1}^\infty\frac{1}{(\alpha t/k + 2j-1)^2} + \sum_{j=1}^\infty \frac{1}{(2j-t/k)^2} + \sum_{j=1}^\infty \frac{1}{(2j-1-t/k)^2} \\\notag \geq& -\kappa^2\sum_{j=1}^\infty\frac{1}{(2j-1+\kappa t/k)^2} - \alpha^2\sum_{j=1}^\infty\frac{1}{(2j-\alpha t/k )^2}\\\notag &+ \alpha^2\sum_{j=1}^\infty\frac{1}{(\alpha t/k + 2j-1)^2} + \sum_{j=1}^\infty \frac{1}{(2j-t/k)^2} \\\notag =&\left( -\sum_{j=1}^\infty\frac{1}{((2j-1)/\kappa+ t/k)^2}+ \sum_{j=1}^\infty\frac{1}{((2j-1)/\alpha + t/k )^2}\right)\\\notag &+\left(- \sum_{j=1}^\infty\frac{1}{(2j/\alpha-t/k )^2} + \sum_{j=1}^\infty \frac{1}{(2j-t/k)^2}\right) \geq 0, \hspace{0.2in} (\text{because} \ \alpha>\kappa). \end{aligned}$$]{} Since we have proved that $g^{\prime\prime}(t)$, i.e., $g(t)$ is a convex function, one can find the optimal $t$ by solving $g^\prime(t) = 0$: [$$\begin{aligned} \notag &\gamma_e (\alpha -1)-\log(1+\epsilon)- \frac{\kappa(\alpha)\pi}{2}{\tan\left(\frac{\kappa(\alpha)\pi}{2k}t\right)} + \frac{\alpha\pi/2}{\tan\left(\frac{\alpha\pi t}{2k}\right)}+ \\\notag &\hspace{1.0in} + \psi\left(\frac{\alpha t}{k}\right)\alpha - \psi\left(1-\frac{t}{k}\right)=0,\end{aligned}$$]{} We let the solution be $t = C_R k$, where $C_R$ is the solution to [$$\begin{aligned} \notag &\gamma_e (\alpha -1)-\log(1+\epsilon)- \frac{\kappa(\alpha)\pi}{2}{\tan\left(\frac{\kappa(\alpha)\pi}{2}C_R\right)} + \frac{\alpha\pi/2}{\tan\left(\frac{\alpha\pi }{2}C_R\right)}\\\notag&\hspace{1in}+ \psi\left(\alpha C_R\right)\alpha - \psi\left(1-C_R\right)=0.\end{aligned}$$]{} Alternatively, we can seek a “sub-optimal” (but asymptotically optimal) solution using the asymptotic expression for [$\text{E}\left(\hat{F}_{(\alpha),gm}\right)^t$]{} in Lemma \[lem\_gm\_moments\], i.e., the $t$ that minimizes [$$\begin{aligned} \notag (1+\epsilon)^{-t}\exp\left(\frac{1}{k}\frac{\pi^2}{24}\left(t^2-t\right)\left(2+\alpha^2-3\kappa^2(\alpha)\right)\right),\end{aligned}$$]{} whose minimum is attained at [$$\begin{aligned} \notag t = k \frac{\log(1+\epsilon)}{(2+\alpha^2-3\kappa^2(\alpha))\pi^2/12} + \frac{1}{2}.\end{aligned}$$]{} This approximation can be useful (e.g.,) for serving the initial guess for $C_R$ in a numerical procedure. Assume we know $C_R$ (e.g., by a numerical procedure), we can then express the right tail bound as [$$\begin{aligned} \notag &\mathbf{Pr}\left(\hat{F}_{(\alpha),gm,b} - F_{(\alpha)} \geq \epsilon F_{(\alpha)}\right) \leq \exp\left(-k \frac{\epsilon^2}{G_{R,gm}}\right),\\\notag &\frac{\epsilon^2}{G_{R,gm}} = C_R \log(1+\epsilon) - C_R \gamma_e(\alpha-1)\\\notag &\hspace{0.5in} - \log\left(\cos\left(\frac{\kappa(\alpha)\pi C_R}{2}\right) \frac{2}{\pi}\Gamma\left(\alpha C_R\right)\Gamma\left(1-C_R\right)\sin\left(\frac{\pi\alpha C_R}{2}\right)\right).\end{aligned}$$]{} Next, we find the constant $G_{L,gm}$ in the left tail bound [$$\begin{aligned} \notag \mathbf{Pr}\left(\hat{F}_{(\alpha),gm,b} - F_{(\alpha)} \leq -\epsilon F_{(\alpha)}\right) \leq \exp\left(-k\frac{\epsilon^2}{G_{L, gm}}\right), \ \ 0<\epsilon < 1.\end{aligned}$$]{} From Lemma \[lem\_gm\_moments\], we know that, for any $t$, where $0< t<k/\alpha$ if $\alpha>1$ and $t>0$ if $\alpha<1$, [$$\begin{aligned} \notag &\mathbf{Pr}\left(\hat{F}_{(\alpha),gm,b} \leq (1-\epsilon)F_{(\alpha)} \right) \\\notag =& \mathbf{Pr}\left(\hat{F}_{(\alpha),gm,b}^{-t} \geq (1-\epsilon)^{-t}F_{(\alpha)}^{-t} \right) \leq \frac{\text{E}\left(\hat{F}_{(\alpha),gm,b}^{-t}\right)}{(1-\epsilon)^{-t}F_{(\alpha)}^{-t}}\\\notag =& (1-\epsilon)^t\frac{\left[ -\cos\left(\frac{\kappa(\alpha)\pi}{2k}t\right) \frac{2}{\pi}\Gamma\left(-\frac{\alpha t}{k}\right)\Gamma\left(1+\frac{t}{k}\right)\sin\left(\frac{\pi\alpha t}{2k}\right)\right]^k} {\exp\left(t\gamma_e(\alpha-1)\right)}\\\notag =&(1-\epsilon)^t\exp\left(-t\gamma_e(\alpha-1)\right)\frac{\left[ \cos\left(\frac{\kappa(\alpha)\pi}{2k}t\right)\Gamma\left(1+\frac{t}{k}\right) \right]^k}{\left[ \Gamma\left(1+\frac{\alpha t}{k}\right)\cos\left(\frac{\pi\alpha t}{2k}\right) \right]^k}\\\notag =&(1-\epsilon)^t\exp\left(-t\gamma_e(\alpha-1)\right)\frac{\left[ \cos\left(\frac{\kappa(\alpha)\pi}{2k}t\right)\Gamma\left(\frac{t}{k}\right) \right]^k}{\left[\alpha \Gamma\left(\frac{\alpha t}{k}\right)\cos\left(\frac{\pi\alpha t}{2k}\right) \right]^k}\end{aligned}$$]{} whose minimum is attained at $t = C_L k$ (we skip the proof of convexity) such that [$$\begin{aligned} \notag &\log(1-\epsilon) - \gamma_e(\alpha-1) - \frac{\kappa(\alpha)\pi}{2} \tan\left(\frac{\kappa(\alpha)\pi}{2}C_L\right) +\frac{\alpha\pi}{2} { \tan\left(\frac{\alpha\pi}{2}C_L\right)}\\\notag &\hspace{0.5in} -\psi\left(\alpha C_L\right)\alpha+ \psi\left(C_L\right)=0.\end{aligned}$$]{} Thus, we show the left tail bound Proof of Lemma \[lem\_G\_gm\_rate\] {#proof_lem_G_gm_rate} =================================== First, we consider the right bound. From Lemma \[lem\_gm\_bounds\], [$$\begin{aligned} \notag &\frac{\epsilon^2}{G_{R,gm}} = C_R \log(1+\epsilon) - C_R \gamma_e(\alpha-1) \\\notag & - \log\left(\cos\left(\frac{\kappa(\alpha)\pi C_R}{2}\right) \frac{2}{\pi}\Gamma\left(\alpha C_R\right)\Gamma\left(1-C_R\right)\sin\left(\frac{\pi\alpha C_R}{2}\right)\right),\end{aligned}$$]{} and $C_R$ is the solution to $g_1(C_R,\alpha,\epsilon) = 0$, [$$\begin{aligned} \notag &g_1(C_R,\alpha,\epsilon) = -\gamma_e (\alpha -1)+\log(1+\epsilon)+ \frac{\kappa(\alpha)\pi}{2}{\tan\left(\frac{\kappa(\alpha)\pi}{2}C_R\right)}\\\notag &\hspace{0.5in}- \frac{\alpha\pi/2}{\tan\left(\frac{\alpha\pi }{2}C_R\right)}- \psi\left(\alpha C_R\right)\alpha + \psi\left(1-C_R\right)=0.\end{aligned}$$]{} Using series representations in [@Book:Gradshteyn_00 1.421.1,1.421.3,8.362.1] [$$\begin{aligned} \notag &\tan\left(\frac{\pi x}{2}\right) = \frac{4x}{\pi}\sum_{j=1}^\infty \frac{1}{(2j-1)^2-x^2},\hspace{0.05in} \frac{1}{\tan\left(\pi x\right)} = \frac{1}{\pi x} + \frac{2x}{\pi}\sum_{j=1}^\infty \frac{1}{x^2-j^2},\\\notag &\psi(x) %= -\gamma_e - \sum_{j=0}^\infty\left(\frac{1}{x+j} - \frac{1}{j+1}\right) = -\gamma_e - \frac{1}{x} + x\sum_{j=1}^\infty \frac{1}{j(x+j)},\end{aligned}$$]{} we rewrite $g_1$ as [$$\begin{aligned} \notag &g_1 = -\gamma_e (\alpha - 1) +\log(1+\epsilon)+ \frac{\kappa\pi}{2}\frac{4\kappa C_R}{\pi}\sum_{j=1}^\infty \frac{1}{(2j-1)^2-(\kappa C_R)^2}\\\notag &- \frac{\alpha\pi}{2} \left(\frac{2}{\pi \alpha C_R} + \frac{\alpha C_R}{\pi}\sum_{j=1}^\infty \frac{1}{(\alpha C_R/2)^2-j^2}\right)\\\notag &-\alpha\left( -\gamma_e - \frac{1}{\alpha C_R} + \alpha C_R\sum_{j=1}^\infty\frac{1}{j(\alpha C_R+j)}\right)\\\notag &+\left(-\gamma_e - \frac{1}{1-C_R} + (1-C_R) \sum_{j=1}^\infty\frac{1}{j(1-C_R+j)}\right) \\\notag =& \log(1+\epsilon)+ 2\kappa^2 C_R\sum_{j=1}^\infty \frac{1}{(2j-1)^2-(\kappa C_R)^2} + 2\alpha^2C_R \sum_{j=1}^\infty \frac{1}{(2j)^2 - (\alpha C_R)^2}\\\notag &-\alpha^2C_R\sum_{j=1}^\infty\frac{1}{j(\alpha C_R+j)} + (1-C_R)\sum_{j=1}^\infty\frac{1}{j(1-C_R+j)} -\frac{1}{1-C_R}\\\notag =& \log(1+\epsilon)+ \kappa\sum_{j=1}^\infty \left(\frac{1}{2j+1-\kappa C_R} - \frac{1}{2j-1+\kappa C_R}\right)\\\notag &+ \alpha\sum_{j=1}^\infty \left(\frac{1}{2j - \alpha C_R}-\frac{1}{2j+\alpha C_R}\right) -\alpha\sum_{j=1}^\infty\left(\frac{1}{j} - \frac{1}{\alpha C_R+j}\right)\\\notag &+ \sum_{j=1}^\infty\left(\frac{1}{j}-\frac{1}{1-C_R+j}\right) + \frac{\kappa}{1-\kappa C_R}-\frac{1}{1-C_R}\end{aligned}$$]{} We show that, as $\alpha\rightarrow 1$, i.e., $\kappa\rightarrow 1$, the term [$$\begin{aligned} \notag \lim_{\alpha\rightarrow 1}& \ \ \kappa\sum_{j=1}^\infty \left(\frac{1}{2j+1-\kappa C_R} - \frac{1}{2j-1+\kappa C_R}\right)\\\notag &+ \alpha\sum_{j=1}^\infty \left(\frac{1}{2j - \alpha C_R}-\frac{1}{2j+\alpha C_R}\right)\\\notag & -\alpha\sum_{j=1}^\infty\left(\frac{1}{j} - \frac{1}{\alpha C_R+j}\right) + \sum_{j=1}^\infty\left(\frac{1}{j}-\frac{1}{1-C_R+j}\right)\\\notag =\lim_{\alpha\rightarrow 1}& \ \ \sum_{j=1}^\infty \left(\frac{\kappa}{2j+1-\kappa C_R} + \frac{\alpha}{2j - \alpha C_R}\right)\\\notag & - \sum_{j=1}^\infty\left( \frac{\kappa}{2j-1+\kappa C_R} +\frac{\alpha}{2j+\alpha C_R}\right)\\\notag & -\alpha\sum_{j=1}^\infty\left(\frac{1}{j} - \frac{1}{\alpha C_R+j}\right) + \sum_{j=1}^\infty\left(\frac{1}{j}-\frac{1}{1-C_R+j}\right)\\\notag =\lim_{\alpha\rightarrow 1}& \ \ \sum_{j=1}^\infty\frac{\kappa}{1+j-\kappa C_R} - \sum_{j=1}^\infty \frac{\kappa}{j+\kappa C_R} -\alpha\sum_{j=1}^\infty\left(\frac{1}{j} - \frac{1}{\alpha C_R+j}\right)\\\notag &+ \sum_{j=1}^\infty\left(\frac{1}{j}-\frac{1}{1-C_R+j}\right)=0.\end{aligned}$$]{} From Lemma \[lem\_gm\_bounds\], we know $g_1 = 0$ has a unique well-defined solution for $C_R\in(0,1)$. We need to analyze this term [$$\begin{aligned} \notag \frac{\kappa}{1-\kappa C_R}-\frac{1}{1-C_R} = \frac{\kappa - 1}{(1-\kappa C_R)(1-C_R)} =\frac{-\Delta}{(1-\kappa C_R)(1-C_R)},\end{aligned}$$]{} which, when $\alpha \rightarrow 1$ (i.e., $\kappa \rightarrow 1$), must approach a finite limit. In other words, $C_R \rightarrow 1$, at the rate $O\left(\sqrt{\Delta}\right)$, i.e., [$$\begin{aligned} \notag C_R = 1 - \sqrt{\frac{\Delta}{\log (1+\epsilon)}} + o\left(\sqrt{\Delta}\right).\end{aligned}$$]{} By Euler’r reflection formula and series representations, [$$\begin{aligned} \notag \frac{\epsilon^2}{G_{R,gm}} =&C_R \log(1+\epsilon) - C_R \gamma_e(\alpha-1) + \log\left(\frac{\cos\left(\frac{\alpha\pi C_R}{2}\right) \Gamma(1-\alpha C_R)}{\cos\left(\frac{\kappa\pi C_R}{2}\right)\Gamma(1-C_R)} \right),\end{aligned}$$]{} [$$\begin{aligned} \notag &\frac{\cos\left(\frac{\alpha\pi C_R}{2}\right) \Gamma(1-\alpha C_R)}{\cos\left(\frac{\kappa\pi C_R}{2}\right)\Gamma(1-C_R)}\\\notag =&\exp(\gamma_e(\alpha-1) C_R)\frac{1- C_R}{1-\alpha C_R} \prod_{j=0}^\infty \left(1-\frac{\alpha^2 C_R^2}{(2j+1)^2}\right) \left(1-\frac{\kappa^2 C_R^2}{(2j+1)^2}\right)^{-1} \\\notag &\times\prod_{j=1}^\infty\exp\left(\frac{(1-\alpha) C_R}{j}\right) \left(1+\frac{1- C_R}{j}\right)\left(1+\frac{1-\alpha C_R}{j}\right)^{-1} \\\notag =&\exp(\gamma_e(\alpha -1) C_R)\frac{(1+\alpha C_R)(1- C_R)}{1-\kappa^2C_R^2} \prod_{j=1}^\infty \left(1-\frac{\alpha^2 C_R^2}{(2j+1)^2}\right)\\\notag &\times \left(1-\frac{\kappa^2 C_R^2}{(2j+1)^2}\right)^{-1} \exp\left(\frac{(1-\alpha) C_R}{j}\right) \left(1+\frac{1- C_R}{j}\right)\left(1+\frac{1-\alpha C_R}{j}\right)^{-1},\end{aligned}$$]{} taking logarithm of which yields [$$\begin{aligned} \notag &\log \frac{\cos\left(\frac{\alpha\pi C_R}{2}\right) \Gamma(1-\alpha C_R)}{\cos\left(\frac{\kappa\pi C_R}{2}\right)\Gamma(1-C_R)} =\gamma_e(\alpha-1) C_R + \log \frac{(1+\alpha C_R)(1- C_R)}{1-\kappa^2C_R^2}\\\notag &\hspace{0.5in}+\sum_{j=1}^\infty \log \frac{\left(1-\frac{\alpha^2 C_R^2}{(2j+1)^2}\right)}{\left(1-\frac{\kappa^2 C_R^2}{(2j+1)^2}\right)} + \left(\frac{(1-\alpha) C_R}{j}\right) + \log \frac{ \left(1+\frac{1- C_R}{j}\right)}{\left(1+\frac{1-\alpha C_R}{j}\right)}.\end{aligned}$$]{} If $\alpha <1$, i.e., $\kappa =\alpha = 1-\Delta$, then [$$\begin{aligned} \notag &\log \frac{\cos\left(\frac{\alpha\pi C_R}{2}\right) \Gamma(1-\alpha C_R)}{\cos\left(\frac{\kappa\pi C_R}{2}\right)\Gamma(1-C_R)}\\\notag =&-\gamma_e\Delta C_R + \log \frac{1- C_R}{1-\alpha C_R}+\sum_{j=1}^\infty \left(\frac{(1-\alpha) C_R}{j}\right) + \log \frac{ \left(1+\frac{1- C_R}{j}\right)}{\left(1+\frac{1-\alpha C_R}{j}\right)}\\\notag =&-\gamma_e\Delta C_R - \log\left(1+ \frac{\Delta C_R}{1-C_R}\right)+\sum_{j=1}^\infty \frac{1}{2}\left(\frac{1-\alpha C_R}{j}\right)^2 - \frac{1}{2}\left(\frac{1-C_R}{j}\right)^2 ...\\\notag =&-\gamma_e\Delta C_R - \log\left(1+ \frac{\Delta C_R}{1-C_R}\right)+ \frac{\pi^2}{12}C_R\Delta (2-\alpha C_R - C_R) +...\end{aligned}$$]{} Thus, for $\alpha <1$, as [$C_R = 1-\sqrt{\frac{\Delta}{\log (1+\epsilon)}} + o\left(\sqrt{\Delta}\right)$]{}, we obtain [$$\begin{aligned} \notag \frac{\epsilon^2}{G_{R,gm}} =& C_R \log(1+\epsilon) - \frac{\Delta C_R}{1-C_R} + \frac{\pi^2}{12}C_R\Delta (2-\alpha C_R - C_R) +... \\\notag =& \log(1+\epsilon) - 2\sqrt{\Delta\log\left(1+\epsilon\right)}+o\left(\sqrt{\Delta}\right)\end{aligned}$$]{} If $\alpha >1$, i.e., $\alpha = 1+\Delta$ and $\kappa = 1-\Delta$, then [$$\begin{aligned} \notag &\log \frac{\cos\left(\frac{\alpha\pi C_R}{2}\right) \Gamma(1-\alpha C_R)}{\cos\left(\frac{\kappa\pi C_R}{2}\right)\Gamma(1-C_R)}\\\notag =&\gamma_e\Delta C_R + \log \frac{(1+\alpha C_R)(1- C_R)}{1-\kappa^2C_R^2} +\sum_{j=1}^\infty \log \frac{\left(1-\frac{\alpha^2 C_R^2}{(2j+1)^2}\right)}{\left(1-\frac{\kappa^2 C_R^2}{(2j+1)^2}\right)} +...\end{aligned}$$]{} Also [$$\begin{aligned} \notag &\log \frac{(1+\alpha C_R)(1- C_R)}{1-\kappa^2C_R^2} = \log \frac{1+\alpha C_R}{1+\kappa C_R} - \log \frac{1-\kappa C_R}{1-C_R}\\\notag =&\log\left(1+ \frac{2\Delta C_R}{1+\kappa C_R}\right) - \log\left(1+\frac{\Delta C_R}{1-C_R}\right)\\\notag =& -\sqrt{\Delta\log(1+\epsilon)} + o\left(\sqrt{\Delta}\right),\end{aligned}$$]{} and [$$\begin{aligned} \notag &\sum_{j=1}^\infty \log \frac{\left(1-\frac{\alpha^2 C_R^2}{(2j+1)^2}\right)}{\left(1-\frac{\kappa^2 C_R^2}{(2j+1)^2}\right)} =\sum_{j=1}^\infty \log \frac {1+\frac{\alpha C_R}{2j+1}}{1+ \frac{\kappa C_R}{2j+1}} + \log \frac {1-\frac{\alpha C_R}{2j+1}}{1-\frac{\kappa C_R}{2j+1}} \\\notag =&\sum_{j=1}^\infty \log \left(1+\frac {\frac{2\Delta C_R}{2j+1}}{1+ \frac{\kappa C_R}{2j+1}}\right) + \log \left(1-\frac {\frac{2\Delta C_R}{2j+1}}{1-\frac{\kappa C_R}{2j+1}}\right) = O\left(\Delta\right). \end{aligned}$$]{} Therefore, for $\alpha >1$, we also have [$$\begin{aligned} \notag \frac{\epsilon^2}{G_{R,gm}} =& \log(1+\epsilon) - 2\sqrt{\Delta\log\left(1+\epsilon\right)}+o\left(\sqrt{\Delta}\right).\end{aligned}$$]{} In other words, as $\alpha \rightarrow 1$, the constant $G_{R,gm}$ converges to $\frac{\epsilon^2}{\log(1+\epsilon)}$ at the rate $O\left(\sqrt{\Delta}\right)$.\ Next, we consider the left bound. From Lemma \[lem\_gm\_bounds\], [$$\begin{aligned} \notag &\mathbf{Pr}\left(\hat{F}_{(\alpha),gm,b} - F_{(\alpha)} \leq -\epsilon F_{(\alpha)}\right)\leq \exp\left(-k \frac{\epsilon^2}{G_{L,gm}}\right),\end{aligned}$$]{} where [$$\begin{aligned} \notag &\frac{\epsilon^2}{G_{L,gm}} = -C_L \log(1-\epsilon) + C_L\gamma_e(\alpha-1)+\log\alpha\\\notag &\hspace{0.in} - \log\left(\cos\left(\frac{\kappa(\alpha)\pi}{2}C_L\right)\Gamma\left(C_L\right)\right)+\log\left( \Gamma\left(\alpha C_L\right)\cos\left(\frac{\pi\alpha C_L}{2}\right)\right).\end{aligned}$$]{} and $C_L$ is the solution to $g_2(C_L,\alpha,\epsilon) = 0$, [$$\begin{aligned} \notag &g_2(C_L,\alpha,\epsilon) =\log(1-\epsilon) - \gamma_e(\alpha-1) - \frac{\kappa(\alpha)\pi}{2} \tan\left(\frac{\kappa(\alpha)\pi}{2}C_L\right)\\\notag &\hspace{0.5in} +\frac{\alpha\pi}{2} { \tan\left(\frac{\alpha\pi}{2}C_L\right)} -\psi\left(\alpha C_L\right)\alpha+ \psi\left(C_L\right)=0.\end{aligned}$$]{} Using series representations, we rewrite $g_2$ as [$$\begin{aligned} \notag &g_2 = -\gamma_e (\alpha - 1) +\log(1-\epsilon)- \frac{\kappa\pi}{2}\frac{4\kappa C_L}{\pi}\sum_{j=1}^\infty \frac{1}{(2j-1)^2-(\kappa C_L)^2}\\\notag &+ \frac{\alpha\pi}{2}\frac{4\alpha C_L}{\pi}\sum_{j=1}^\infty \frac{1}{(2j-1)^2-(\alpha C_L)^2} \\\notag &-\alpha\left( -\gamma_e - \frac{1}{\alpha C_L}+ (\alpha C_L)\sum_{j=1}^\infty\frac{1}{j(\alpha C_L+j)}\right)\\\notag &+\left(-\gamma_e - \frac{1}{C_L} + C_L \sum_{j=1}^\infty\frac{1}{j(C_L+j)}\right) \\\notag =&- 2\kappa^2 C_L\sum_{j=1}^\infty \frac{1}{(2j-1)^2-(\kappa C_L)^2} + 2\alpha^2 C_L\sum_{j=1}^\infty \frac{1}{(2j-1)^2-(\alpha C_L)^2}\\\notag &-\alpha^2 C_L\sum_{j=1}^\infty\frac{1}{j(\alpha C_L+j)} + C_L\sum_{j=1}^\infty\frac{1}{j(C_L+j)}+ \log(1-\epsilon)\\\notag =& \log(1-\epsilon)- \kappa\sum_{j=1}^\infty \left(\frac{1}{2j-1-\kappa C_L} - \frac{1}{2j-1+\kappa C_L}\right)+\\\notag &\alpha\sum_{j=1}^\infty \frac{1}{2j-1 - \alpha C_L}-\frac{1}{2j-1+\alpha C_L} +(1-\alpha) C_L\sum_{j=1}^\infty\frac{\alpha C_L + j(1+\alpha)}{j(\alpha C_L + j)(C_L+j)}.\end{aligned}$$]{} We first consider $\alpha=1+\Delta>1$. In order for $g_2 = 0$ to have a meaningful solution, we must make sure that [$$\begin{aligned} \notag \frac{-\kappa}{1-\kappa C_L} + \frac{\alpha}{1-\alpha C_L} = \frac{2\Delta}{(1-\kappa C_L)(1-\alpha C_L)} = \frac{2\Delta}{1-2C_L +C_L^2-\Delta^2C_L^2}\end{aligned}$$]{} converges to a finite value as $\alpha\rightarrow 1$, i.e., $C_L\rightarrow 1$ also. This provides an approximate solution for $C_L$ when $\alpha>1$: [$$\begin{aligned} \notag C_L = 1 - \sqrt{\frac{2\Delta}{-\log(1-\epsilon)}} + o\left(\sqrt{\Delta}\right).\end{aligned}$$]{} Using series representations, we obtain [$$\begin{aligned} \notag &C_L\gamma_e(\alpha-1)+\log\alpha+\log\frac{ \Gamma\left(\alpha C_L\right)\cos\left(\frac{\pi\alpha C_L}{2}\right)}{ \cos\left(\frac{\kappa(\alpha)\pi}{2}C_L\right)\Gamma\left(C_L\right)}\\\notag =&\log\left( \prod_{s=1}^\infty\frac{1+\frac{C_L}{s}}{1+\frac{\alpha C_L}{s}}\exp\left(\frac{\Delta C_L}{s}\right)\prod_{s=0}^\infty\frac{1-\frac{\alpha^2C_L^2}{(2s+1)^2}}{1-\frac{\kappa^2C_L^2}{(2s+1)^2}}\right)\\\notag =&\sum_{s=1}^\infty\left(-\frac{\Delta C_L}{s+C_L}+\frac{\Delta C_L}{s}+o\left(\Delta\right)\right) + \log\left(\frac{1-\alpha^2C_L^2}{1-\kappa^2C_L^2}\right) +\sum_{s=1}^\infty\log \frac{1-\frac{\alpha^2C_L^2}{(2s+1)^2}}{1-\frac{\kappa^2C_L^2}{(2s+1)^2}}\\\notag =&-\sqrt{-2\Delta\log(1-\epsilon)} + O\left(\Delta\right).\end{aligned}$$]{} Therefore, for $\alpha >1$ [$$\begin{aligned} \notag G_{L,gm} = \frac{\epsilon^2}{ -\log(1-\epsilon) - 2\sqrt{-2\Delta\log(1-\epsilon)} + o\left(\sqrt{\Delta}\right)}.\end{aligned}$$]{} Finally, we need to consider $\alpha <1$. In this case, [$$\begin{aligned} \notag g_2 =& \log(1-\epsilon)+\Delta C_L\sum_{j=1}^\infty\frac{\alpha C_L + j(1+\alpha)}{j(\alpha C_L + j)(C_L+j)}\\\notag =&\log(1-\epsilon) + \Delta C_L\left(\sum_{j=1}^\infty\frac{1}{j(j+C_L)} + \sum_{j=1}^\infty\frac{1}{(1+C_L)^2}\right)+o\left(\Delta\right).\end{aligned}$$]{} Using properties of Riemann’s Zeta function and Bernoulli numbers[@Book:Gradshteyn_00 9.511,9.521.1,9.61] [$$\begin{aligned} \notag &\sum_{j=1}^\infty\frac{1}{(j+C_L)^2} = -\frac{1}{C_L^2} + \int_{0}^\infty\frac{t\exp(-C_L t)}{1-\exp(-t)}dt\\\notag =&-\frac{1}{C_L^2} + \int_0^\infty\left(1+\frac{t}{2} + \frac{t^2}{12}+...\right) \exp(-C_L t) dt = \frac{1}{C_L} + O\left(\frac{1}{C_L^2}\right).\end{aligned}$$]{} Using the integral relation[@Book:Gradshteyn_00 0.244.1] and treating $C_L$ as a positive integer (which does not affect the asymptotics) [$$\begin{aligned} \notag &\sum_{j=1}^\infty\frac{1}{j(j+C_L)} = \frac{1}{C_L}\int_0^1\frac{1-t^{C_L}}{1-t}dt\\\notag =&\frac{1}{C_L}\int_0^1 t^{C_L-1} + t^{C_L-2} + ...+ 1 dt \\\notag =&\frac{1}{C_L} \sum_{j=1}^{C_L} \frac{1}{j}= \frac{1}{C_L}\left(\gamma_e + \log C_L+O\left(C_L^{-1}\right)\right).\end{aligned}$$]{} Thus, the solution to $g_2 =0$ can be approximated by [$$\begin{aligned} \notag &\log(1-\epsilon) + \Delta \left(1+\gamma_e + \log C_L\right) +o(\Delta)= 0.\end{aligned}$$]{} Again, using series representations, we obtain [$$\begin{aligned} \notag &C_L\gamma_e(\alpha-1)+\log\alpha+\log\frac{ \Gamma\left(\alpha C_L\right)}{ \Gamma\left(C_L\right)}\\\notag =&\log\left( \prod_{j=1}^\infty\frac{1+\frac{C_L}{j}}{1+\frac{\alpha C_L}{j}}\exp\left(-\frac{\Delta C_L}{j}\right)\right)\\\notag =&\sum_{j=1}^\infty\left(\frac{\Delta C_L}{j+C_L}-\frac{\Delta C_L}{j}+...\right)\\\notag =&-\Delta C_L\left(\gamma_e + \log C_L\right) +...\end{aligned}$$]{} Combining the results, we obtain, when $\alpha<1$ and $\Delta\rightarrow 0$, [$$\begin{aligned} \notag G_{L,gm} = \frac{\epsilon^2}{ \Delta\left(\exp\left(\frac{-\log(1-\epsilon)}{\Delta} -1 - \gamma_e\right)\right)+o\left(\Delta\exp\left(\frac{1}{\Delta}\right)\right)}.\end{aligned}$$]{} Proof of Lemma \[lem\_hm\] {#proof_lem_hm} ========================== Assume $k$ i.i.d. samples $x_j \sim S(\alpha<1, \beta =1, F_{(\alpha)})$. Using the $(-\alpha)$th moment in Lemma \[lem\_moments\] suggests that [$$\begin{aligned} \notag \hat{R}_{(\alpha)} = \frac{\frac{1}{k}\sum_{j=1}^k|x_j|^{-\alpha}}{\frac{\cos\left(\frac{\alpha\pi}{2}\right)}{\Gamma(1+\alpha)} },\end{aligned}$$]{} is an unbiased estimator of $d^{-1}_{(\alpha)}$,whose variance is [$$\begin{aligned} \notag \text{Var}\left(\hat{R}_{(\alpha)}\right) = \frac{d^{-2}_{(\alpha)}}{k}\left(\frac{2\Gamma^2(1+\alpha)}{\Gamma(1+2\alpha)}-1\right).\end{aligned}$$]{} We can then estimate $F_{(\alpha)}$ by $\frac{1}{\hat{R}_{(\alpha)}}$, i.e., [$$\begin{aligned} \notag \hat{F}_{(\alpha),hm} = \frac{1}{\hat{R}_{(\alpha)}} = \frac{k\frac{\cos\left(\frac{\alpha\pi}{2}\right)}{\Gamma(1+\alpha)}}{\sum_{j=1}^k|x_j|^{-\alpha}}.\end{aligned}$$]{} which is biased at the order $O\left(\frac{1}{k}\right)$. To remove the $O\left(\frac{1}{k}\right)$ term of the bias, we recommend a bias-corrected version obtained by Taylor expansions [@Book:Lehmann_Casella Theorem 6.1.1]: [$$\begin{aligned} \notag \frac{1}{\hat{R}_{(\alpha)}} - \frac{\text{Var}\left(\hat{R}_{(\alpha)}\right)}{2} \left(\frac{2}{F_{(\alpha)}^{-3}}\right),\end{aligned}$$]{} from which we obtain the bias-corrected estimator [$$\begin{aligned} \notag \hat{F}_{(\alpha),hm,c} = \frac{k\frac{\cos\left(\frac{\alpha\pi}{2}\right)}{\Gamma(1+\alpha)}}{\sum_{j=1}^k|x_j|^{-\alpha}} \left(1- \frac{1}{k}\left(\frac{2\Gamma^2(1+\alpha)}{\Gamma(1+2\alpha)}-1\right) \right),\end{aligned}$$]{} whose bias and variance are [$$\begin{aligned} \notag &\text{E}\left(\hat{F}_{(\alpha),hm,c}\right) = F_{(\alpha)}+O\left(\frac{1}{k^2}\right),\\\notag &\text{Var}\left(\hat{F}_{(\alpha),hm,c}\right) = \frac{F^{2}_{(\alpha)}}{k}\left(\frac{2\Gamma^2(1+\alpha)}{\Gamma(1+2\alpha)}-1\right) + O\left(\frac{1}{k^2}\right).\end{aligned}$$]{} We now study the tail bounds. For convenience, we provide tail bounds for $\hat{F}_{(\alpha),hm}$ instead of $\hat{F}_{(\alpha),hm,c}$. We first analyze the following moment generating function: [$$\begin{aligned} \notag &\text{E}\left(\exp\left(\frac{F_{(\alpha)}|x_j|^{-\alpha}}{\cos\left(\alpha\pi/2\right)/\Gamma(1+\alpha)}t\right)\right)\\\notag =& 1+\sum_{m=1}^\infty \frac{t^m}{m!} \text{E}\left( F_{(\alpha)}\left(\frac{|x_j|^{-\alpha}}{\cos\left(\alpha\pi/2\right)/\Gamma(1+\alpha)}\right)^m \right)\\\notag =&1+\sum_{m=1}^\infty \frac{t^m}{m!}\frac{\Gamma(1+m)\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)} = \sum_{m=0}^\infty \frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}t^m.\end{aligned}$$]{} For the right tail bound, [$$\begin{aligned} \notag &\mathbf{Pr}\left( \hat{F}_{(\alpha),hm} - F_{(\alpha)} \geq \epsilon F_{(\alpha)}\right) \\\notag =& \mathbf{Pr}\left( \frac{k\frac{\cos\left(\frac{\alpha\pi}{2}\right)}{\Gamma(1+\alpha)}}{\sum_{j=1}^k|x_j|^{-\alpha}} \geq (1+\epsilon)F_{(\alpha)}\right)\\\notag =&\mathbf{Pr}\left(\exp\left(-t\left( \frac{\sum_{j=1}^kF_{(\alpha)}|x_j|^{-\alpha}}{ \cos\left(\alpha\pi/2\right)/\Gamma(1+\alpha)}\right)\right)\geq \exp\left(-t\frac{k}{(1+\epsilon)}\right)\right)\hspace{0.05in} (t>0)\\\notag \leq&\left(\sum_{m=0}^\infty \frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}(-t)^m\right)^k \exp\left(t\frac{k}{(1+\epsilon)}\right)\\\notag =&\exp\left(-k\left(-\log \left(\sum_{m=0}^\infty \frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}(-t_1^*)^m\right) -\frac{t_1^*}{1+\epsilon}\right)\right)\\\notag =&\exp\left(-k\frac{\epsilon^2}{G_{R,hm}}\right),\end{aligned}$$]{} where $t_1^*$ is the solution to [$$\begin{aligned} \notag \frac{\sum_{m=1}^\infty(-1)^m m (t_1^*)^{m-1}\frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)} }{\sum_{m=0}^\infty(-1)^m (t_1^*)^{m}\frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)} } + \frac{1}{1+\epsilon} = 0 .\end{aligned}$$]{} For the left tail bound, [$$\begin{aligned} \notag &\mathbf{Pr}\left( \hat{F}_{(\alpha),hm} - F_{(\alpha)} \leq -\epsilon F_{(\alpha)}\right) \\\notag =& \mathbf{Pr}\left( \frac{k\frac{\cos\left(\frac{\alpha\pi}{2}\right)}{\Gamma(1+\alpha)}}{\sum_{j=1}^k|x_j|^{-\alpha}} \leq (1-\epsilon)F_{(\alpha)}\right)\\\notag =&\mathbf{Pr}\left(\exp\left(t\left( \frac{\sum_{j=1}^kF_{(\alpha)}|x_j|^{-\alpha}}{ \cos\left(\alpha\pi/2\right)/\Gamma(1+\alpha)}\right)\right)\geq \exp\left(t\frac{k}{(1-\epsilon)}\right)\right)\hspace{0.2in} (t>0)\\\notag \leq&\left(\sum_{m=0}^\infty \frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}t^m\right)^k \exp\left(-t\frac{k}{(1-\epsilon)}\right)\\\notag =&\exp\left(-k\left(-\log \left(\sum_{m=0}^\infty \frac{\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}(t_2^*)^m\right) +\frac{t_2^*}{1-\epsilon}\right)\right),\end{aligned}$$]{} where $t_2^*$ is the solution to [$$\begin{aligned} \notag \sum_{m=1}^\infty \left\{\frac{(t_2^*)^{m-1}\Gamma^{m-1}(1+\alpha)}{\Gamma(1+(m-1)\alpha)} - m(1-\epsilon) \frac{ (t_2^*)^{m-1}\Gamma^m(1+\alpha)}{\Gamma(1+m\alpha)}\right\}=0.\end{aligned}$$]{} [10]{} Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In [*STOC*]{}, pages 20–29, Philadelphia, PA, 1996. Brian Babcock, Shivnath Babu, Mayur Datar, Rajeev Motwani, and Jennifer Widom. Models and issues in data stream systems. In [*PODS*]{}, pages 1–16, Madison, WI, 2002. Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. In [*FOCS*]{}, pages 209–218, Vancouver, BC, Canada, 2002. Graham Cormode, Mayur Datar, Piotr Indyk, and S. Muthukrishnan. Comparing data streams using hamming norms (how to zero in). , 15(3):529–540, 2003. Joan Feigenbaum, Sampath Kannan, Martin Strauss, and Mahesh Viswanathan. An approximate $l_1$-difference algorithm for massive data streams. In [*FOCS*]{}, pages 501–511, New York, 1999. Philippe Flajolet. Approximate counting: A detailed analysis. , 25(1):113–134, 1985. Izrail S. Gradshteyn and Iosif M. Ryzhik. . Academic Press, New York, sixth edition, 2000. Monika R. Henzinger, Prabhakar Raghavan, and Sridhar Rajagopalan. . American Mathematical Society, Boston, MA, USA, 1999. Piotr Indyk. Stable distributions, pseudorandom generators, embeddings and data stream computation. In [*FOCS*]{}, pages 189–197, Redondo Beach, CA, 2000. Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. , 53(3):307–323, 2006. Piotr Indyk and David P. Woodruff. Optimal approximations of the frequency moments of data streams. In [*STOC*]{}, pages 202–208, Baltimore, MD, 2005. William B. Johnson and Joram Lindenstrauss. Extensions of mapping into space. , 26:189–206, 1984. Erich L. Lehmann and George Casella. . Springer, New York, NY, second edition, 1998. Ping Li. Estimators and tail bounds for dimension reduction in $l_\alpha$ ($0<\alpha\leq 2$) using stable random projections. In [*SODA*]{}, 2008. Ping Li, Debashis Paul, Ravi Narasimhan, and John Cioffi. On the distribution of for the receiver and performance analysis. , 52(1):271–286, 2006. Robert Morris. Counting large numbers of events in small registers. , 21(10):840–842, 1978. S. Muthukrishnan. Data streams: Algorithms and applications. , 1:117–236, 2 2005. Michael E. Saks and Xiaodong Sun. Space lower bounds for distance approximation in the data stream model. In [*STOC*]{}, pages 360–369, Montreal, Quebec, Canada, 2002. David P. Woodruff. Optimal space lower bounds for all frequency moments. In [*SODA*]{}, pages 167–175, New Orleans, LA, 2004. Haiquan Zhao, Ashwin Lall, Mitsunori Ogihara, Oliver Spatscheck, Jia Wang, and Jun Xu. A data streaming algorithm for estimating entropies of od flows. In [*IMC*]{}, San Diego, CA, 2007. Vladimir M. Zolotarev. . American Mathematical Society, Providence, RI, 1986.

--- abstract: 'We generalize McShane’s identity for the length series of simple closed geodesics on a cusped hyperbolic surface [@mcshane1998im] to hyperbolic cone-surfaces (with all cone angles $\le \pi$), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotient of the one-holed torus by its elliptic involution, and the closed genus two surface by its hyper-elliptic involution, we obtain generalizations of the Weierstrass identities for the one-holed torus, and identities for the genus two surface, also obtained by McShane using different methods in [@mcshane2004blms], [@mcshane1998preprint] and [@mcshane2004preprint]. We also give an interpretation of the identity in terms of complex lengths, gaps, and the direct visual measure of the boundary.' address: | Department of Mathematics\ National University of Singapore\ 2 Science Drive 2\ Singapore 117543 author: - 'Ser Peow Tan, Yan Loi Wong, and Ying Zhang' title: 'Generalizations of McShane’s Identity to Hyperbolic Cone-Surfaces' --- \[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Addendum]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Remark]{} [**Introduction**]{} {#s:intro} ==================== Greg McShane discovered the following striking identity in his Ph.D. thesis: \[thm:mcshane torus\] [(McShane [@mcshane1991thesis])]{} In a once punctured hyperbolic torus $T$, $$\begin{aligned} \sum_{\gamma}\frac{1}{1 + \exp |\gamma|} = \frac{1}{2},\end{aligned}$$ where the sum extends over all simple closed geodesics on $T$ and where $|\gamma|$ denotes the length of $\gamma$ in the given hyperbolic structure. Throughout this paper we shall always use $|\gamma|$ to denote the hyperbolic length of $\gamma$ if $\gamma$ is a (generalized) simple closed geodesic or a simple geodesic arc on a hyperbolic (cone-)surface. All surfaces considered in this paper are assumed to be connected and orientable. Later McShane extended his identity to more general surfaces: \[thm:mcshane general\] [(McShane [@mcshane1998im])]{} In a finite area hyperbolic surface $M$ with cusps and without boundary, $$\begin{aligned} \sum \frac{1}{1 + \exp \frac{1}{2}(|\alpha|+|\beta|)} = \frac{1}{2},\end{aligned}$$ where the sum is over all unordered pairs of simple closed geodesics $\alpha, \beta$ (where $\alpha$ or $\beta$ might be a cusp treated as a simple closed geodesic of length $0$) on $M$ such that $\alpha, \beta$ bound with a distinguished cusp point an embedded pair of pants on $M$. Note that Theorem \[thm:mcshane torus\] can be regarded as a special case of Theorem \[thm:mcshane general\] where $\alpha, \beta$ are the same for each pair $\alpha, \beta$. In [@mcshane2004blms] McShane demonstrated three other closely related identities for the lengths of simple closed geodesics in each of the three Weierstrass classes on a hyperbolic torus. Recall that a hyperbolic torus $T$ has three Weierstrass points which are the fixed points of the unique elliptic involution which maps each simple closed geodesic on $T$ onto itself with orientation reversed, and for a Weierstrass point $x$ on $T$ the simple closed geodesics in the Weierstrass class which is dual to $x$ are precisely all the simple closed geodesics on $T$ which do not pass through $x$. \[thm:mcshane weierstrass\] [(McShane [@mcshane2004blms])]{} In a once punctured hyperbolic torus, $$\begin{aligned} \label{eqn:mcshane weierstrass} \sum_{\gamma \in \mathcal A} \sin^{-1} \left ( \frac{1}{\cosh \frac{1}{2}|\gamma|} \right )= \frac{\pi}{2},\end{aligned}$$ where the sum is over all simple closed geodesics in a Weierstrass class $\mathcal A$. On the other hand, B. H. Bowditch gave an alternative proof of Theorem \[thm:mcshane torus\] using Markoff triples [@bowditch1996blms] and extended the identity in Theorem \[thm:mcshane torus\] to the case of quasi-fuchsian representations of the torus group [@bowditch1998plms] as well as to the case of hyperbolic once punctured torus bundles [@bowditch1997t]. There are also some other generalizations along these directions, by Makoto Sakuma and his co-workers, see [@akiyoshi-miyachi-sakuma2002preprint], [@sakuma1999sk]. In this paper we further generalize McShane’s identity as in Theorem \[thm:mcshane general\] to the cases of hyperbolic cone-surfaces possibly with cusps and/or geodesic boundary. (See for example [@cooper-hodgson-kerckhoff2000book] for basic facts on cone-manifolds.) We assume that all cone points have cone angle $\le \pi$ (except for the one-cone torus where we allow the cone angle up to $ 2 \pi$). The ideas are related in spirit to those in [@basmajian1993ajm] while the method of proof follows closely that of McShane’s in [@mcshane1998im]. The key points are that the assumption that all cone angles are $\le \pi$ implies that all non-peripheral simple closed curves are essentially realizable as simple geodesics in their free (relative) homotopy classes; and that the Birman-Series result [@birman-series1985t] on the sparsity of simple geodesics carries over to this case, in particular to simple geodesic rays emanating (normally) from a fixed boundary component. It should be noted that our result shows that the assumption of discreteness of the holonomy group is unnecessary, and that it gives identities for all hyperbolic orbifold surfaces. We also show how the result can be formulated in terms of complex lengths (Theorem \[thm:complexified\]) even though the situation we consider here is real. This is particularly useful, and is explored further in [@tan-wong-zhang2004preprint], where we show how this approach allows us to generalize McShane’s identity to Schottky groups, and how the Markoff triples and analytic continuation methods adopted by Bowditch in [@bowditch1996blms] can be generalized as well. (See also [@goldmanGT2003] for related work on generalized Markoff triples.) This should also lead to generalizations of Bowditch’s interpretation [@bowditch1997t] of McShane’s identity for complete hyperbolic 3-manifolds which are once punctured torus bundles over the circle to identities for the hyperbolic 3-manifolds obtained by hyperbolic Dehn surgery on such manifolds. This will be explored in future work, and should tie up nicely with the work of Sakuma in [@sakuma1999sk], and Akiyoshi-Miyachi-Sakuma in [@akiyoshi-miyachi-sakuma2002preprint] and [@akiyoshi-miyachi-sakuma2004preprint]. To state the most general form of our generalized McShane’s identities, we need to introduce some new terminology. However, to let the reader get the flavor of the generalized identities, we first state the corresponding generalizations of Theorems \[thm:mcshane torus\] and \[thm:mcshane general\]. \[thm:mcshane cone hole torus\] Let $T$ be either a hyperbolic one-cone torus where the single cone point has cone angle $\theta \in (0, 2\pi)$ or a hyperbolic one-hole torus where the single boundary geodesic has length $l>0$. Then we have respectively $$\begin{aligned} \label{eqn:mcshane cone torus} \sum_{\gamma} 2 \tan^{-1} \left ( \frac {\sin \frac{\theta}{2}} {\cos \frac{\theta}{2}+ \exp |\gamma|} \right )= \frac{\theta}{2},\end{aligned}$$ $$\begin{aligned} \sum_{\gamma} 2 \tanh^{-1}\left ( \frac {\sinh \frac{l}{2}} {\cosh \frac{l}{2}+ \exp |\gamma|} \right )= \frac{l}{2},\end{aligned}$$ where the sum in either case extends over all simple closed geodesics on $T$. \[thm:mcshane cone hole surface\] Let $M$ be a compact hyperbolic cone-surface with a single cone point of cone angle $\theta \in (0, \pi]$ and without boundary or let $M$ be a compact hyperbolic surface with a single boundary geodesic having length $l>0$. Then we have respectively $$\begin{aligned} \sum 2 \tan^{-1} \left ( \frac {\sin \frac{\theta}{2}}{\cos\frac{\theta}{2}+\exp \frac{|\alpha|+|\beta|}{2}} \right )= \frac{\theta}{2},\end{aligned}$$ $$\begin{aligned} \sum 2 \tanh^{-1} \left ( \frac {\sinh \frac{l}{2}}{\cosh \frac{l}{2}+ \exp \frac{|\alpha|+|\beta|}{2}} \right )= \frac{l}{2},\end{aligned}$$ where the sum in either case extends over all unordered pairs of simple closed geodesics on $M$ which bound with the cone point (respectively, the boundary geodesic) an embedded pair of pants. For the purposes of this paper we make the following definition. A [**compact hyperbolic cone-surface**]{} $M$ is a compact (topological) surface $M$ with hyperbolic cone structure where each boundary component is a smooth simple closed geodesic and where there are a finite number of interior points which form all the cone points and cusps. Its [**geometric boundary**]{}, denoted $\Delta M$, is the union of all cusps, cone points and geodesic boundary components. (Note that $\Delta M$ is different from the usual topological boundary $\partial M$ when there are cusps or cone points.) Thus a [**geometric boundary component**]{} is either a cusp, a cone point, or a boundary geodesic. The [**geometric interior**]{} of $M$ is $M - \Delta M$. In this paper we consider a compact hyperbolic cone-surface $M = M(\Delta_{0}; k, \Theta, L)$ with $k$ cusps $C_{1}, C_{2}, \cdots, C_{k}$, with $m$ cone points $P_{1}, P_{2}, \cdots, P_{m}$, where the cone angle of $P_{i}$ is $\theta_{i} \in (0, \pi]$, $i=1, 2, \cdots, m$, and with $n$ geodesic boundary components $B_{1}, B_{2}, \cdots, B_{n}$, where the length of $B_{i}$ is $l_{i}>0$, $i=1, 2, \cdots, n$, together with an extra [*distinguished*]{} geometric boundary component $\Delta_{0}$. Thus $\Delta_{0}$ is either a cusp $C_{0}$ or a cone point $P_{0}$ of cone angle $\theta_{0} \in (0, \pi]$ or a geodesic boundary component $B_{0}$ of length $l_{0}>0$. Note that in the above notation $\Theta = (\theta_{1}, \theta_{2}, \cdots, \theta_{m})$ and $L = (l_{1}, l_{2}, \cdots, l_{n})$. We exclude the case where $M$ is a geometric pair of pants for we have only trivial identities in that case. We allow that some (even all) of the cone angles $\theta_{i}$ are equal to $\pi$, $i=0, 1, \cdots, m$; these are often cases of particular interest. However, for clarity of exposition, quite often in proofs/statements of lemmas/theorems we shall first consider the case where all the cone angles are less than $\pi$ and then point out the addenda that should be made when there are angle $\pi$ cone points. The advantage of this assumption of strict inequality is that every non-trivial, non-peripheral simple closed curve on such $M$ can be realized as a (smooth) simple closed geodesic in its free homotopy class in the geometric interior of $M$ under the given hyperbolic cone-structure (see §\[s:realizibility\] for the proof of this statement). We call a simple closed curve on $M$ [*peripheral*]{} if it is freely homotopic on $M$ to a geometric boundary component of $M$. \[defn:gscg\] By a [**generalized simple closed geodesic**]{} on $M$ we mean either - a simple closed geodesic in the geometric interior of $M$; or - a degenerate simple closed geodesic which is the double of a simple geodesic arc in the geometric interior of $M$ connecting two angle $\pi$ cone points; or - a geometric boundary component, that is, a cusp or a cone point or a boundary geodesic. In particular, generalized simple closed geodesics of the first two kinds are called [**interior generalized simple closed geodesics**]{}. For each pair of generalized simple closed geodesics $\alpha, \beta$ which bound with $\Delta_{0}$ an embedded geometric pair of pants we shall define in §\[s:dGf\] a [**gap function**]{} ${\rm Gap}(\Delta_{0};\alpha, \beta)$ when $\Delta_{0}$ is a cone point or a boundary geodesic as well as a [**normalized gap function**]{} ${\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)$ when $\Delta_{0}$ is a cusp. Now we are in a position to state the most general (real) form of our generalization of McShane’s identity. \[thm:mcshane most general\] Let $M$ be a compact hyperbolic cone-surface with all cone angles in $(0, \pi]$. Then one has either $$\begin{aligned} \label{eqn:001} \sum {\rm Gap}(\Delta_{0};\alpha, \beta) = \frac{\theta_{0}}{2},\end{aligned}$$ when $\Delta_{0}$ is a cone point of cone angle $\theta_{0}$; or $$\begin{aligned} \label{eqn:002} \sum {\rm Gap}(\Delta_{0};\alpha, \beta) = \frac{l_{0}}{2},\end{aligned}$$ when $\Delta_{0}$ is a boundary geodesic of length $l_{0}$; or $$\begin{aligned} \label{eqn:00'} \sum {\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta) = \frac{1}{2},\end{aligned}$$ when $\Delta_{0}$ is a cusp; where in each case the sum is over all pairs of generalized simple closed geodesics $\alpha, \beta$ on $M$ which bound with $\Delta_{0}$ an embedded pair of pants.  \ - In the case of the hyperbolic one-cone torus, the theorem holds for $\theta_0 \in (0, 2\pi)$. - In the special cases where the geometric boundary $\Delta M$ is a single cone point or a single boundary geodesic Theorem \[thm:mcshane most general\] gives all the previously stated generalized identities in Theorems \[thm:mcshane cone hole torus\] and \[thm:mcshane cone hole surface\]. - The cusp case (that is, $\Delta_{0}$ is a cusp) is the limit case of the other cases as the cone angle $\theta_{0}$ or the boundary geodesic length $l_{0}$ approaches $0$, and the identity in the cusp case can indeed be derived from the first order infinitesimal of the identities of the other cases. It is also interesting to note that McShane’s Weierstrass identities can be deduced as special cases of our general Theorem \[thm:mcshane most general\] by applying the theorem to the quotient of the once punctured torus by its elliptic involution and then lifting back to the torus. Thus we have the following generalized Weierstrass identities: \[cor:mcshane conical holed weierstrass\] Let $T$ be either a hyperbolic one-cone torus where the single cone point has cone angle $\theta \in (0, 2\pi)$ or a hyperbolic one-hole torus where the single boundary geodesic has length $l>0$. Then we have respectively $$\begin{aligned} \sum_{\gamma \in \mathcal A} \tan^{-1} \left ( \frac{\cos \frac{\theta}{4}}{\sinh \frac{|\gamma|}{2}} \right )= \frac{\pi}{2},\end{aligned}$$ $$\begin{aligned} \sum_{\gamma \in \mathcal A} \tan^{-1} \left ( \frac{\cosh \frac{l}{4}}{\sinh \frac{|\gamma|}{2}} \right )= \frac{\pi}{2},\end{aligned}$$ where the sum in either case is over all the simple closed geodesics $\gamma$ in a Weierstrass class $\mathcal A$. McShane’s original Weierstrass identity (\[eqn:mcshane weierstrass\]) then corresponds to the case $\theta = 0$ or $l = 0$ in the above two identities, noticing that $$\begin{aligned} \tan^{-1} \left ( \frac{1}{\sinh \frac{|\gamma|}{2}}\right )= \sin^{-1} \left ( \frac{1}{\cosh \frac{|\gamma|}{2}}\right ).\end{aligned}$$ As further corollaries, there are the following weaker but neater identities, each of which is obtained by summing the three McShane’s Weierstrass identities in the corresponding case. \[cor:mcshane combined weierstrass\] Let $T$ be a hyperbolic torus whose geometric boundary is either a single cusp, a single cone point of cone angle $\theta \in (0, 2\pi)$, or a single boundary geodesic of length $l>0$. Then we have respectively $$\begin{aligned} \sum_{\gamma} \tan^{-1} \left ( \frac{1}{\sinh \frac{|\gamma|}{2}} \right )= \frac{3\pi}{2},\end{aligned}$$ $$\begin{aligned} \sum_{\gamma} \tan^{-1} \left ( \frac{\cos \frac{\theta}{4}}{\sinh \frac{|\gamma|}{2}}\right )= \frac{3\pi}{2},\end{aligned}$$ $$\begin{aligned} \label{eqn:combinedWeierstrassonehole} \sum_{\gamma} \tan^{-1} \left ( \frac{\cosh \frac{l}{4}}{\sinh \frac{|\gamma|}{2}}\right )= \frac{3\pi}{2},\end{aligned}$$ where the sum in each case is over all the simple closed geodesics $\gamma$ on $T$. The identity (\[eqn:combinedWeierstrassonehole\]) was also obtained by McShane [@mcshane2004preprint] using Wolpert’s variation of length method. It seems likely his method can be extended to prove some of the other identities as well. Similarly, for a genus two closed hyperbolic surface $M$, one can consider the (six) identities on the quotient surface $M/\eta$ where $\eta$ is the unique hyper-elliptic involution on $M$ (note that $M/\eta$ is a closed hyperbolic orbifold of genus $0$ with six cone angle $\pi$ points, and we may choose any one of these cone points to be the distinguished geometric boundary component) and re-interpret them as Weierstrass identities on the original surface $M$ (see also McShane [@mcshane1998preprint] where the Weierstrass identities were obtained directly). Combining all the six Weierstrass identities for $M$, we then have the following very neat identity. \[thm:mcshane genus two global\] Let $M$ be a genus two closed hyperbolic surface. Then $$\begin{aligned} \sum \tan^{-1} \exp \left( -\frac{|\alpha|}{4}- \frac{|\beta|}{2} \right ) = \frac{3\pi}{2},\end{aligned}$$ where the sum is over all ordered pairs $(\alpha, \beta)$ of disjoint simple closed geodesics on $M$ such that $\alpha$ is separating and $\beta$ is non-separating. This is the only case that we know of where McShane’s identity extends in a nice way to a closed surface. We observe that the above identity for closed genus two surface $M$ also extends to quasi-Fuchsian representations of ${\pi}_{1}(M)$. More precisely, let $\rho: {\pi}_{1}(M) \rightarrow {\rm SL}(2, \mathbf C)$ be a quasi-Fuchsian representation, that is, $\pi \circ \rho: {\pi}_{1}(M) \rightarrow {\rm PSL}(2, \mathbf C)$ is a quasi-Fuchsian representation where $\pi: {\rm SL}(2, \mathbf C) \rightarrow {\rm PSL}(2, \mathbf C)$ is the projection map. For each essential simple closed curve $\gamma$, let $l_{\rho}(\gamma)/2 \in \mathbf C$ with positive real part and with imaginary part $\in (-\pi, \pi]$ be defined by $${\rm tr} \rho ([\gamma]) = 2 \cosh (l_{\rho}(\gamma)/2),$$ where $[\gamma] \in {\pi}_{1}(M)$ is the homotopy class of $\gamma$. Note that $l_{\rho}(\gamma)$ is also called the complex length of $\rho([\gamma])$, see for example [@fenchel1989book]. \[add:genus two\] For a quasi-Fuchsian representation $\rho: {\pi}_{1}(M) \rightarrow {\rm SL}(2, \mathbf C)$ for the closed genus two surface $M$, we have $$\begin{aligned} \label{eqn:genus two QF} \sum \tan^{-1} \exp \left( -\frac{l_{\rho}(\alpha)}{4}- \frac{l_{\rho}(\beta)}{2} \right ) = \frac{3\pi}{2},\end{aligned}$$ where the sum is over all the ordered pairs $[\alpha],[\beta]$ of homotopy classes of disjoint unoriented essential simple closed curves $\alpha, \beta$ on $M$ such that $\alpha$ is non-separating and $\beta$ is separating. In the statement of Theorem \[thm:mcshane most general\] we did not write down the explicit expression for the gap functions due to their “case by case” nature as can be seen in §\[s:dGf\]. The cone points and boundary geodesics as geometric boundary components seem to have different roles in the series in the generalized identities, hence making the identities not in a unified form. This difference can, however, be removed by assigning purely imaginary length to a cone point as a geometric boundary component. More precisely, for each generalized simple closed geodesic $\delta$, we define its [**complex length**]{} $|\delta|$ as: $|\delta|=0$ if $\delta$ is a cusp; $|\delta|=\theta i$ if $\delta$ is a cone point of angle $\theta \in (0, \pi]$; and $|\delta|=l$ if $\delta$ is a boundary geodesic or an interior generalized simple closed geodesic of length $l>0$. Then we can reformulate the generalized McShane’s identities in Theorem \[thm:mcshane most general\] as follows. \[thm:complexified\] Let $M$ be a compact hyperbolic cone-surface with all cone angles in $(0, \pi]$, and let all its geometric boundary components be $\Delta_0, \Delta_1, \cdots, \Delta_N$ with complex lengths $L_0, L_1, \cdots, L_N$ respectively. Then $$\begin{aligned} \label{eqn:reform of cp and gb cases} \lefteqn{\sum_{\alpha, \beta} 2 \tanh^{-1} \left ( \frac {\sinh \frac{L_0}{2}}{\cosh \frac{L_0}{2}+ \exp\frac{|\alpha|+|\beta|}{2}} \right )} \\ & & + \sum_{j=1}^{N}\sum_{\beta} \tanh^{-1} \left ( \frac{\sinh\frac{L_0}{2}\sinh\frac{L_j}{2} }{\cosh\frac{|\beta|}{2}+\cosh\frac{L_0}{2}\cosh\frac{L_j}{2} } \right ) = \frac{L_0}{2}\nonumber,\end{aligned}$$ if $\Delta_0$ is a cone point or a boundary geodesic; and $$\begin{aligned} \label{eqn:reform of cusp cases} \sum_{\alpha,\beta}\frac{1}{1+\exp\frac{|\alpha|+|\beta|}{2}}+ \sum_{j=1}^{N}\sum_{\beta}\frac{1}{2}\frac{\sinh\frac{L_j}{2}} {\cosh\frac{|\beta|}{2}+\cosh\frac{L_j}{2}}=\frac{1}{2},\end{aligned}$$ if $\Delta_0$ is a cusp; where in either case the first sum is over all (unordered) pairs of generalized simple closed geodesics $\alpha, \beta$ on $M$ which bound with $\Delta_0$ an embedded pair of pants on $M$ (note that one of $\alpha, \beta$ might be a geometric boundary component) and the sub-sum in the second sum is over all interior simple closed geodesics $\beta$ which bounds with $\Delta_j$ and $\Delta_0$ an embedded pair of pants on $M$. Furthermore, each series in (\[eqn:reform of cp and gb cases\]) and (\[eqn:reform of cusp cases\]) converges absolutely. [*Additional Remark.*]{} We were informed while writing this paper by Makoto Sakuma and Caroline Series of the recent striking results of Maryam Mirzakhani [@mirzakhani2004preprint] where she had generalized McShane’s identities to hyperbolic surfaces with boundary and used it to calculate the Weil-Petersson volumes of the corresponding moduli spaces. There is obviously an overlap of her results with ours, in particular, the identities she obtains are equivalent to ours in the case of hyperbolic surfaces with boundary (see §\[s:reformulation\] for further explanations). In fact, her expressions in terms of the $\log$ function seems particular well suited to her purpose of calculating the Weil-Petersson volumes. It also seems (as already observed by her in [@mirzakhani2004preprint]) that her methods should extend fairly easily to cover the case of volumes of the moduli spaces of compact hyperbolic cone-surfaces with all cone angles bounded above by $\pi$, as defined and used in our context, and that the formulas she exhibited for the volumes should hold in this case as well, using the convention that a cone point of angle $\theta$ corresponds to a geometric boundary component with purely imaginary length $\theta i$ . [*Acknowledgements.*]{} We would like to thank Caroline Series for helpful conversations; Makoto Sakuma for his encouragement to write up our results on the cone-manifold case (during conversations with the first named author at the Isaac Newton Institute in Aug, 2003) and also for bringing to our attention the recent works of McShane [@mcshane2004preprint] and Mirzakhani [@mirzakhani2004preprint]; and Greg McShane for helpful e-mail correspondence and also for bringing our attention to [@mcshane1998preprint]. [**The organization of the rest of this paper**]{} ================================================== The rest of this paper is organized as follows. In §\[s:dGf\] we define the gap functions used in Theorem \[thm:mcshane most general\] for the various cases. In §\[s:realizibility\] we deal with the problem of realization of simple closed curves by geodesics, and show that the assumption that all cone angles are less than or equal to $\pi$ is essential. In §\[s:gaps\] we analyze the so-called $\Delta_{0}$-geodesics, that is, the geodesics starting/emanating orthogonally from $\Delta_{0}$, and determine all the gaps between all simple-normal $\Delta_{0}$-geodesics. In §\[s:calculation\] we calculate the gap function which is the width of a combined gap measured suitably. In §\[s:gBS\] we generalize the Birman-Series theorem (which states that the point set of all complete geodesics with bounded self intersection numbers on a compact hyperbolic surface has Hausdorff dimension 1) to the case of compact hyperbolic cone-surfaces with all cone angles less than or equal to $\pi$. We prove the theorems in this paper in §\[s:proof\], except for Theorem \[thm:complexified\], which is deferred to the last section. Finally in §\[s:reformulation\] we restate the complexified generalized McShane’s identity (\[eqn:reform of cp and gb cases\]) (Theorem \[thm:complexified\]) using two functions of complex variables and hence unify the somewhat unattractive “case-by-case” definition of the gap functions. We interpret the geometric meanings of the complexified summands in the complexified generalized McShane’s identity and prove the absolute convergence of the complexified series in it by a simple use of the Birman–Series arguments in [@birman-series1985t]. [**Defining the Gap functions**]{} {#s:dGf} ================================== In this section, for a compact hyperbolic cone-surface $M = M(\Delta_{0}; k, \Theta, L)$ with all cone angles $\le \pi$ we define the gap function ${\rm Gap}(\Delta_{0};\alpha, \beta)$ (when $\Delta_{0}$ is a cone point or a boundary geodesic) and the normalized gap function ${\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)$ (when $\Delta_{0}$ is a cusp) where $\alpha, \beta$ are generalized simple closed geodesics on $M$ which bound with $\Delta_{0}$ a geometric pair of pants. Throughout this paper we use $|\alpha|$ to denote the length of $\alpha$ when $\alpha$ is an interior generalized simple closed geodesic or a boundary geodesic. In particular, when $\alpha$ is a degenerate simple closed geodesic (that is, the double cover of a simple geodesic arc which connects two angle $\pi$ cone points), its length $|\alpha|$ is defined as twice the length of the simple geodesic that it covers. Recall that an interior generalized simple closed geodesic is either a simple closed geodesic in the geometric interior of $M$ or a degenerate simple closed geodesic on $M$ which is the double cover of a simple geodesic arc which connects two angle $\pi$ cone points. [*Case*]{} 0. $\Delta_{0}$ is a cusp. [*Subcase*]{} 0.1. Both $\alpha$ and $\beta$ are interior generalized simple closed geodesics. In this case $$\begin{aligned} {\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)= \frac{1}{1+\exp\frac{1}{2}(|\alpha|+|\beta|)}.\end{aligned}$$ [*Subcase*]{} 0.2. One of $\alpha, \beta$, say $\alpha$, is a boundary geodesic and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)= \frac{1}{2}-\frac{1}{2} \frac{\sinh\frac{|\beta|}{2}}{\cosh\frac{|\alpha|}{2}+\cosh\frac{|\beta|}{2}}.\end{aligned}$$ [*Subcase*]{} 0.3. One of $\alpha, \beta$, say $\alpha$, is a cone point of cone angle $\varphi \in (0,\pi]$ and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)= \frac{1}{2}-\frac{1}{2} \frac{\sinh\frac{|\beta|}{2}}{\cos\frac{\varphi}{2}+\cosh\frac{|\beta|}{2}}.\end{aligned}$$ [*Subcase*]{} 0.4. One of $\alpha, \beta$, say $\alpha$, is also a cusp and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta) = \frac{1}{2}-\frac{1}{2} \frac{\sinh\frac{|\beta|}{2}}{1+\cosh\frac{|\beta|}{2}} = \frac{1}{1+\exp \frac{1}{2}|\beta|},\end{aligned}$$ which is the common value of ${\rm Gap}(\Delta_{0};\alpha, \beta)$ in Subcases 0.1 through 0.3 when $|\alpha|=0$. [*Case*]{} 1. $\Delta_{0}$ is a cone point of cone angle $\theta \in (0,\pi]$. [*Subcase*]{} 1.1. Both $\alpha$ and $\beta$ are interior generalized simple closed geodesics. In this case $$\begin{aligned} \label{eqn:dGf 1.1} {\rm Gap}(\Delta_{0};\alpha, \beta) = 2 \tan^{-1} \left (\frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}+ \exp\frac{|\alpha|+|\beta|}{2}} \right ).\end{aligned}$$ [*Subcase*]{} 1.2. One of $\alpha, \beta$, say $\alpha$, is a boundary geodesic and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{\theta}{2} - \tan^{-1} \left ( \frac{\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2} }{\cosh\frac{|\alpha|}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2} } \right ).\end{aligned}$$ [*Subcase*]{} 1.3. One of $\alpha, \beta$, say $\alpha$, is a cone point of cone angle $\varphi \in (0,\pi]$ and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} \label{subcase 1.3} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{\theta}{2} - \tan^{-1} \left ( \frac{\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2} }{\cos\frac{\varphi}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2} } \right ).\end{aligned}$$ Note that there is no gap when $\theta=\varphi=\pi$. [*Subcase*]{} 1.4. One of $\alpha, \beta$, say $\alpha$, is a cusp and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta) &=& 2 \tan^{-1} \left ( \frac {\sin \frac{\theta}{2}}{\cos\frac{\theta}{2}+ \exp\frac{|\beta|}{2}} \right )\\ &=& \frac{\theta}{2} - \tan^{-1} \left ( \frac{\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2} }{1+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2} } \right ),\end{aligned}$$ which is the common value of ${\rm Gap}(\Delta_{0};\alpha, \beta)$ in Subcases 1.1 through 1.3 when $|\alpha|=0$. [*Case*]{} 2. $\Delta_{0}$ is a boundary geodesic of length $l>0$. [*Subcase*]{} 2.1. Both $\alpha$ and $\beta$ are interior generalized simple closed geodesics. In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= 2 \tanh^{-1} \left ( \frac {\sinh \frac{l}{2}}{\cosh \frac{l}{2}+ \exp\frac{|\alpha|+|\beta|}{2}} \right ).\end{aligned}$$ [*Subcase*]{} 2.2. One of $\alpha, \beta$, say $\alpha$, is a boundary geodesic and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{l}{2} - \tanh^{-1}\left ( \frac{\sinh\frac{l}{2}\sinh\frac{|\beta|}{2} }{\cosh\frac{|\alpha|}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2} }\right ).\end{aligned}$$ [*Subcase*]{} 2.3. One of $\alpha, \beta$, say $\alpha$, is a cone point of cone angle $\varphi \in (0,\pi]$ and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{l}{2} - \tanh^{-1} \left ( \frac{\sinh\frac{l}{2}\sinh\frac{|\beta|}{2} }{\cos\frac{\varphi}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2} }\right ).\end{aligned}$$ [*Subcase*]{} 2.4. One of $\alpha, \beta$, say $\alpha$, is a cusp and the other, $\beta$, is an interior generalized simple closed geodesic. In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta) &=& 2 \tanh^{-1} \left ( \frac {\sinh \frac{l}{2}}{\cosh \frac{l}{2}+ \exp \frac{|\beta|}{2}} \right )\\ &=& \frac{l}{2} - \tanh^{-1}\left ( \frac{\sinh\frac{l}{2}\sinh\frac{|\beta|}{2} }{1+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2} }\right ),\end{aligned}$$ which is the common value of ${\rm Gap}(\Delta_{0};\alpha, \beta)$ in Subcases 2.1 through 2.3 when $|\alpha|=0$. [**Realizing simple curves by geodesics on hyperbolic cone-surfaces**]{} {#s:realizibility} ======================================================================== In this section we consider the problem of realizing essential simple curves in their free (relative) homotopy classes by geodesics on a compact hyperbolic cone-surface $M$ with all cone angles smaller than $\pi$. We show that each essential simple closed curve in the geometric interior of $M$ can be realized uniquely in its free homotopy class (where the homotopy takes place in the geometric interior of $M$) as either a geometric boundary component or a simple closed geodesic in the geometric interior of $M$. We also show that each essential simple arc which connects geometric boundary components of $M$ can be realized uniquely in its free relative homotopy class (where the homotopy takes place in the geometric interior of $M$ and the endpoints slide on the same geometric boundary components) as a simple geodesic arc which is normal to the geometric boundary components involved. We also make addenda for the cases when there are angle $\pi$ cone points. \[thm:realization\] Let $M$ be a compact hyperbolic cone-surface with all cone angles less than $\pi$. \(i) If $c$ is an essential non-peripheral simple closed curve in the geometric interior of $M$, then there is a unique simple closed geodesic in the free homotopy class of $c$ in the geometric interior of $M$. \(ii) If $c$ is an essential simple arc which connects geometric boundary components, then there is a unique simple normal geodesic arc in the free relative homotopy class of $c$ in the geometric interior of $M$ with endpoints varying on the respective geometric boundary components. \[add:realization\] If in addition $M$ has some cone angles equal to $\pi$, then - in Theorem \[thm:realization\](i), if the simple closed curve $c$ bounds with two angle $\pi$ cone points an embedded pair of pants, then the geodesic realization for $c$ is the double cover of the simple geodesic arc which connects these two angle $\pi$ cone points and is homotopic (relative to boundary) to a simple arc lying wholly in the pair of pants; - in Theorem \[thm:realization\](ii), if the simple arc $c$ connects a geometric boundary component $\Delta$ to itself and bounds together with $\Delta$ and an angle $\pi$ cone point $P$ an embedded cylinder then the geodesic realization for $c$ is the double cover of the normal simple geodesic arc which connects $\Delta$ to $P$ and is homotopic (relative to boundary) to a simple arc lying wholly in the cylinder. The simple geodesic in Theorem \[thm:realization\] and Addendum \[add:realization\] is called the geodesic realization of the given simple curve in the respective homotopy class. The proof is a well-known use of the Arzela-Ascoli Theorem as used in [@buser1992book] with slight modifications. \(i) Suppose $c$ is an essential non-peripheral simple closed curve in the geometric interior of $M$, parameterized on $[0, 1]$ with constant speed. Let the length of $c$ be $|c|>0$. Then for each cusp $C_{i}$, there is an embedded neighborhood $N(C_{i})$ of $C_{i}$ on $M$, bounded by a horocycle, such that each non-peripheral simple closed curve $c^{\prime}$ in the geometric interior of $M$ with length $\le |c|$ cannot enter $N(C_{i})$; for otherwise $c^{\prime}$ would be either peripheral or of infinite length. Now let $M_{0}$ be $M$ with all the chosen horocycle neighborhoods $N(C_{i})$ removed. Then $M_{0}$ is a compact metric subspace of $M$ with the induced hyperbolic metric. Now choose a sequence of simple closed curves $\{c_{k}\}_{1}^{\infty}$, where each $c_{k}$ is parameterized on $[0, 1]$ with constant speed, in the free homotopy class of $c$ (where the homotopy takes place in the geometric interior of $M$) such that their lengths $\le |c|$ and are decreasing with limit the infimum of the lengths of the simple closed curves in the free homotopy class of $c$. Then by the Arzela-Ascoli Theorem (c.f. [@buser1992book] Theorem A.19, page 429) there is a subsequence of $\{c_{k}\}_{1}^{\infty}$, assumed to be $\{c_{k}\}_{1}^{\infty}$ itself, such that it converges uniformly to a closed curve $\gamma$ in $M_{0}$. It is clear that $\gamma$ is a geodesic since it is locally minimizing. Note that $\gamma$ is away from cusps by the choice of $\{c_{k}\}_{1}^{\infty}$. We claim that $\gamma$ cannot pass through any cone point. For otherwise, suppose $\gamma$ passes through a cone point $P$. Then for sufficiently large $k$, $c_{k}$ can be modified in the free homotopy class of $c$ to have length smaller than $|\gamma|$ (since the cone point has cone angle smaller than $\pi$), which is a contradiction. Thus $\gamma$ must be a closed geodesic in the geometric interior of $M$. The uniqueness and simplicity of $\gamma$ can be proved by an easy argument since there are no bi-gons in the hyperbolic plane. \(ii) For an essential simple arc $c$ in the geometric interior of $M$ which connects geometric boundary components, the proof of case (i) applies without modifications when none of the involved geometric boundary components is a cusp. Now suppose at least one of the involved geometric boundary components is a cusp. For definiteness let us assume that $c$ connects cusps $C_{1}$ to $C_{2}$. Remove suitable horocycle neighborhoods $N(C_{1})$ and $N(C_{2})$ respectively for $C_{1}$ and $C_{2}$ where the two horocycles are $H_{1}$ and $H_{2}$ respectively. Choose a simple arc $c_{0}$ in $M - N(C_{1}) \cup N(C_{2})$ which goes along $c$ and connects $H_{1}$ to $H_{2}$. Let the length of $c_{0}$ be $|c_{0}|>0$. Now for all other cusps $C_{i}$, there is a horocycle neighborhood $N(C_{i})$ of $C_{i}$ on $M$ such that each non-peripheral simple closed curve $c^{\prime}$ in the geometric interior of $M$ with length $\le |c_{0}|$ cannot enter $N(C_{i})$. Again let $M_{0}$ be $M$ with all the chosen horocycle neighborhoods $N(C_{i})$ removed. By the same argument as in (i) we have a shortest simple geodesic realization $\gamma_{0}$ in the free relative homotopy class of $c_{0}$ in $M_{0}$ and $c_{0}$ does not pass through any cone point. Hence $\gamma_{0}$ must be perpendicular to both $H_{1}$ and $H_{2}$ at its endpoints. Thus $\gamma_{0}$ can be extended to a geodesic arc connecting $C_{1}$ to $C_{2}$. Again simplicity and uniqueness can be proved easily. The addendum can be verified easily since the realizations as degenerate simple geodesics in the respective cases are already known. We make a remark that the following fact, whose proof is easy and hence omitted, is implicitly used through out this paper: On a hyperbolic cone-surface for each cone point $P$ with angle less than $\pi$ there is a cone region $N(P)$, bounded by a suitable circle centered at $P$, such that if a geodesic $\gamma$ goes into $N(P)$ then either $\gamma$ will go directly to the cone point $P$ (hence perpendicular to all the circles centered at $P$) or $\gamma$ will develop a self-intersection in $N(P)$. The analogous fact for a cusp is used in [@birman-series1985t], [@haas1986actam] and [@mcshane1998im]. [**Gaps between simple-normal $\Delta_{0}$-geodesics**]{} {#s:gaps} ========================================================= \[defn:deltageodesic\] A [**$\Delta_{0}$-geodesic**]{} on $M$ is an [*oriented*]{} geodesic ray which starts from $\Delta_{0}$ (and is perpendicular to it if $\Delta_{0}$ is a boundary geodesic) and is fully developed, that is, it develops forever until it terminates at a geometric boundary component. We denote by ${\mathcal G}(\Delta_0)$ (or just ${\mathcal G}$) the set of $\Delta_0$-geodesics. A $\Delta_{0}$-geodesic is either non-simple or simple. It is regarded as [**non-simple**]{} if and only if it intersects itself transversely at an interior point (a cone point is not treated as an interior point) or at a point on a boundary geodesic. We shall see later that somewhat surprisingly, in some sense, the set of non-simple $\Delta_{0}$-geodesics is easier to analyze than the set of simple $\Delta_{0}$-geodesics. A simple $\Delta_{0}$-geodesic is either normal or not-normal in the following sense: A simple $\Delta_{0}$-geodesic is [**normal**]{} if when fully developed either it never intersects any boundary geodesic or it intersects (hence terminates at) a boundary geodesic perpendicularly. Note that a simple-normal $\Delta_{0}$-geodesic may terminate at a cusp or a cone point. Thus a simple $\Delta_{0}$-geodesic is [**not-normal**]{} if and only if it intersects a boundary geodesic (which might be $\Delta_{0}$ itself) obliquely. We shall analyze the structure of all non-simple and simple-not-normal $\Delta_{0}$-geodesics and show that they form gaps between simple-normal $\Delta_{0}$-geodesics. Furthermore, the naturally measured widths of the suitably combined gaps are given by the Gap functions defined before in §\[s:dGf\]. Note that McShane [@mcshane1998im] analyzes directly all simple $\Delta_{0}$-geodesics (there are no simple-not-normal $\Delta_{0}$-geodesics in his case since there are no geodesic boundary componenets). Our analysis of the structure of $\Delta_{0}$-geodesics is a bit different from and actually simpler than that of McShane’s. We shall analyze all non-simple and simple-not-normal $\Delta_{0}$-geodesics and show that they arise in the nice ways we expect. First we parameterize all the $\Delta_{0}$-geodesics and define the widths for gaps between simple-normal $\Delta_{0}$-geodesics. If $\Delta_{0}$ is a cusp let $\mathcal H$ be a suitably chosen small horocycle as in McShane [@mcshane1998im], see also [@haas1986actam]. If $\Delta_{0}$ is a cone point let $\mathcal H$ be a suitably chosen small circle centered at $\Delta_{0}$. Let $\mathcal H$ be $\Delta_{0}$ itself if $\Delta_{0}$ is a boundary geodesic. Then each $\Delta_{0}$-geodesic has a unique first intersection point with $\mathcal H$, which is the starting point when $\Delta_{0}$ is a boundary geodesic. Note that the $\Delta_{0}$-geodesics intersect $\mathcal H$ orthogonally at their first intersection points. Thus ${\mathcal G}$ can be naturally identified with ${\mathcal H}$, with the induced topology and measure. Let ${\mathcal H}_{\texttt{ns}}$, ${\mathcal H}_{\texttt{sn}}$, ${\mathcal H}_{\texttt{snn}}$ be the point sets of the first intersections of $\mathcal H$ with respectively all non-simple, all simple-normal, all simple-not-normal $\Delta_{0}$-geodesics. The set ${\mathcal H}_{\texttt{ns}} \cup {\mathcal H}_{\texttt{snn}}$ is an open subset of $\mathcal H$ and hence ${\mathcal H}_{\texttt{sn}}$ is a closed subset of $\mathcal H$. [*Proof*]{}. It is easy to see that the condition that either self-intersecting or ending obliquely at a boundary component is an open condition. For the open subset ${\mathcal H}_{\texttt{ns}} \cup {\mathcal H}_{\texttt{snn}}$ of $\mathcal H$, we determine its structure by determining its maximal open intervals (which are the gaps we are looking for). By a generalized Birman–Series Theorem (see §\[s:gBS\]), the subset ${\mathcal H}_{\texttt{sn}}$ of $\mathcal H$ has Hausdorff dimension $0$, and hence Lebesgue measure $0$. Therefore the open subset ${\mathcal H}_{\texttt{ns}} \cup {\mathcal H}_{\texttt{snn}}$ of $\mathcal H$ has full measure, and our generalized McShane’s identities (\[eqn:001\])-(\[eqn:00’\]) follow immediately. A $[\Delta_{0},\Delta_{0}]$-geodesic, $\gamma$, is an (oriented) $\Delta_{0}$-geodesic which terminates at $\Delta_{0}$ perpendicularly. (With the orientation one can refer to its starting point and ending point.) Hence the same geodesic with reversed orientation (hence with the starting and ending points interchanged) is also a $[\Delta_{0},\Delta_{0}]$-geodesic, denoted by $-\gamma$. We say that a $[\Delta_{0},\Delta_{0}]$-geodesic $\gamma$ is a [**degenerate simple**]{} $[\Delta_{0},\Delta_{0}]$-[**geodesic**]{} if $\Delta_0$ is not a $\pi$ cone point, and $\gamma$ is the double cover of a simple geodesic arc which connects $\Delta_{0}$ to an angle $\pi$ cone point, that is, $\gamma$ reaches the angle $\pi$ cone point along the simple geodesic arc and goes back to $\Delta_{0}$ along the same arc. Note that in this case $\gamma=-\gamma$. We show that each non-degenerate simple $[\Delta_{0},\Delta_{0}]$-geodesic $\gamma$ determines two maximal open intervals of ${\mathcal H}_{\texttt{ns}} \cup {\mathcal H}_{\texttt{snn}}$ as follows. (Their union is the [*main gap*]{}, defined later, determined by $\gamma$.) Consider the configuration $\gamma \cup \mathcal H$. Assume $\gamma$ is non-degenerate and let $\mathcal H_1$ and $\mathcal H_2$ be the two sub-arcs with endpoints inclusive that $\gamma$ divides $\mathcal H$ into. Note that $\gamma$ intersects $\mathcal H$ twice (if ${\mathcal H}$ is taken to be a suitably small circle about $\Delta_0$ when $\Delta_0$ is a cone point). Let $\gamma_0$ be the sub-arc of $\gamma$ between the two intersection points. Thus we have two simple closed curves $\mathcal H_1 \cup \gamma_0$ and $\mathcal H_2 \cup \gamma_0$ on $M$. Their geodesic realizations are disjoint generalized simple closed geodesics, denoted $\alpha, \beta$ respectively (except when $M$ is a hyperbolic torus with a single geometric boundary component, in which case $\alpha=\beta$). Note that $\alpha, \beta$ bound with $\Delta_{0}$ an embedded geometric pair of pants, denoted $\mathcal P(\gamma)$, on $M$. Let $\delta_{\alpha}$ be the simple $\Delta_{0}$-geodesic arc in $\mathcal P(\gamma)$ which terminates at $\alpha$ and is normal to $\alpha$. Similarly, let $\delta_{\beta}$ be the simple $\Delta_{0}$-geodesic arc in $\mathcal P(\gamma)$ which terminates at $\beta$ and is normal to $\beta$. Let $[\alpha, \beta]$ be the simple geodesic arc in $\mathcal P(\gamma)$ which connects $\alpha$ and $\beta$ and is normal to them. See Figure \[fig01\]. Cutting $\mathcal P(\gamma)$ along $\delta_{\alpha}, \delta_{\beta}$ and $[\alpha, \beta]$ one obtains two pieces; let the one which contains the initial part of $\gamma$ be denoted $\mathcal P^{+}(\gamma)$. There are two simple $\Delta_{0}$-geodesics, $\gamma_{\alpha}$ and $\gamma_{\beta}$, in $\mathcal P(\gamma)$ such that they are asymptotic to $\alpha$ and $\beta$ respectively, and such that their initial parts are contained in $\mathcal P^{+}(\gamma)$. See Figure \[fig01\]. Each $\Delta_{0}$-geodesic whose initial part lies in $\mathcal P^{+}(\gamma)$ between $\gamma_{\alpha}$ and $\gamma$ or between $\gamma$ and $\gamma_{\beta}$ is non-simple or simple-not-normal. The union of these two gaps between simple-normal $\Delta_{0}$-geodesics formed by non-simple and simple-not-normal $\Delta_{0}$-geodesics is called the [**main gap**]{} determined by $\gamma$. This lemma can be proved easily using a suitable model of the hyperbolic plane; see [@zhang2004thesis] for details. The idea is that a $\Delta_{0}$-geodesic ray whose initial part lies in $\mathcal P^{+}(\gamma)$ between $\gamma_{\alpha}$ and $\gamma$ will not intersect $\gamma_{\alpha}$ or $\gamma$ directly, so it must come back to intersect for first time either itself or $\Delta_0$, hence is either non-simple or simple but not-normal (that is, intersecting $\Delta_{0}$ obliquely). More precisely, if $\Delta_{0}$ is a cusp or a cone point all the $\Delta_{0}$-geodesics in the lemma are non-simple, while if $\Delta_{0}$ is a boundary geodesic then there is a (critical) $\Delta_{0}$-geodesic, $\rho_\gamma$, whose initial part lies in $\mathcal P^{+}(\gamma)$ between $\gamma_{\alpha}$ and $\gamma$ such that $\rho_\gamma$ is non-simple and its only self-intersection is at its starting point on $\Delta_{0}$ (and hence terminates there) and it has the property that each $\Delta_{0}$-geodesic whose initial part lies in $\mathcal P^{+}(\gamma)$ between $\gamma_{\alpha}$ and $\rho_\gamma$ is non-simple, while each $\Delta_{0}$-geodesic whose initial part lies in $\mathcal P^{+}(\gamma)$ between $\rho_\gamma$ and $\gamma$ is simple-not-normal terminating at $\Delta_{0}$. There is a similar dichotomy for the $\Delta_{0}$-geodesics whose initial parts lie in $\mathcal P^{+}(\gamma)$ between $\gamma$ and $\gamma_\beta$. Now suppose one of $\alpha, \beta$, say $\alpha$, is a boundary geodesic. Then there are two simple $\Delta_{0}$-geodesics in $\mathcal P(\gamma)$ which are asymptotes to $\alpha$. They are $\gamma_{\alpha}$ and $(-\gamma)_{\alpha}$. The following lemma tells us that there is an [**extra gap**]{} determined by $\gamma$ in $\mathcal P^{+}(\gamma)$ between simple-normal $\Delta_{0}$-geodesics formed by simple-not-normal $\Delta_{0}$-geodesics. Each $\Delta_{0}$-geodesic whose initial part lies in $\mathcal P^{+}(\gamma)$ between $\delta_{\alpha}$ and $\gamma_{\alpha}$ is simple-not-normal. This is almost self-evident from the geometry of the pair of pants $P(\gamma)$, and is similar to the proof of the previous lemma; see [@zhang2004thesis] for details. Note that there is a similar and symmetric picture for the $\Delta_{0}$-geodesics whose initial parts lie in $\mathcal P^{-}(\gamma)$. Hence (for non-degenerate $\gamma$) in the geometric pair of pants $\mathcal P(\gamma)$, which is the same as $\mathcal P(-\gamma)$, if none of $\alpha, \beta$ is a boundary geodesic then there are two main gaps determined by $\gamma$ and $-\gamma$ respectively; if (exactly) one of $\alpha, \beta$ is a boundary geodesic then there are two extra gaps determined by $\gamma$ and $-\gamma$. The case of a degenerate simple $[\Delta_0,\Delta_0]$-geodesic $\gamma$ is handled in a similar way. Recall that $\gamma$ is the double cover of a $\Delta_0$-geodesic arc $\delta$ from $\Delta_0$ to an angle $\pi$ cone point $\alpha$. Then there is a simple closed curve ${\beta}^{\prime}$, which is the boundary of a suitable regular neighborhood of $\Delta_0 \cup \delta$ on $M$, such that ${\beta}^{\prime}$ bounds with $\Delta_0$ and $\alpha$ an embedded (topological) pair of pants. If $\Delta_0$ is not itself an angle $\pi$ cone point, then ${\beta}^{\prime}$ can be realized as an interior generalized simple closed geodesic $\beta$ which bounds with $\Delta_0$ and $\alpha$ an embedded pair of pants $\mathcal H(\Delta_0, \alpha, \beta)$ on $M$ and we can carry out the analysis as above with suitable modifications. In this case $\gamma$ determines no gaps if $\Delta_0$ is itself an angle $\pi$ cone point. If $\Delta_0$ is not itself an angle $\pi$ cone point then there are two main gaps, between $\gamma$ and each of the two $\Delta_0$-geodesics which are asymptotic to $\beta$ in $\mathcal H(\Delta_0, \alpha, \beta)$. We say that one of the two main gaps is determined by $\gamma$ and the other by $-\gamma$ although $\gamma=-\gamma$ in this case. The [**width**]{} of an open subinterval $\mathcal H^{\prime}$ of $\mathcal H$ is defined respectively as: - $\Delta_{0}$ is a cusp: the normalized parabolic measure, that is, the ratio of the Euclidean length of $\mathcal H^{\prime}$ to the Euclidean length of $\mathcal H$; - $\Delta_{0}$ is a cone point: the elliptic measure, that is, the angle (measured in radians) that $\mathcal H^{\prime}$ subtends with respect to the cone point $\Delta_{0}$; - $\Delta_{0}$ is a boundary geodesic: the hyperbolic measure, that is, the hyperbolic length of $\mathcal H^{\prime}$ (recall that in this case $\mathcal H$ is the same as the distinguished boundary geodesic $\Delta_{0}$). The [**combined gap**]{} between simple-normal $\Delta_{0}$-geodesics determined by $\gamma$ is the union of the main gap and the extra gap (if there is any) determined by $\gamma$. The [**gap function**]{} ${\rm Gap}(\Delta_{0};\alpha, \beta)$ when $\Delta_{0}$ is a cone point or boundary geodesic or the [**normalized gap function**]{} ${\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)$ when $\Delta_{0}$ is a cusp is defined as the total width of the combined gap determined by $\gamma$, which is by symmetry the same as the total width of the combined gap determined by $-\gamma$. We shall calculate the the gap functions in §\[s:calculation\]. On the other hand, the following key lemma shows that the non-simple and simple-not-normal $\Delta_{0}$-geodesics obtained above are [*all*]{} the non-simple and simple-not-normal $\Delta_{0}$-geodesics. Each non-simple or simple-not-normal $\Delta_{0}$-geodesic lies in a main gap or an extra gap determined by some $[\Delta_{0},\Delta_{0}]$-geodesic $\gamma$. First let $\delta$ be a non-simple $\Delta_{0}$-geodesic, with its first self-intersection point $Q$, where $Q$ lies in the geometric interior of $M$ or in $\Delta_{0}$ when $\Delta_{0}$ is a boundary geodesic. Let $\delta_{1}$ be the part of $\delta$ from starting point to $Q$; note that $\delta_{1}$ has the shape of a lasso. Then in the boundary of a suitable regular neighborhood of $\delta_{1}$ there is a simple arc $\gamma^{\prime}$ which connects $\Delta_{0}$ to itself and is disjoint from $\delta_{1}$ (except at $\Delta_{0}$ when $\Delta_{0}$ is a cone point); there is also a simple closed curve $\alpha^{\prime}$ which is freely homotopic to the loop part of $\delta_{1}$. See Figure \[fig02\]. Let $\gamma$, $\alpha$ be the generalized simple closed geodesics on $M$ which realize $\gamma^{\prime}$, $\alpha^{\prime}$ in their respective free (relative) homotopy classes in the geometric interior of $M$. An easy geometric argument shows that $\alpha$ is disjoint from $\delta_{1}$ and that $\gamma$ is also disjoint from $\delta_{1}$ except at $\Delta_{0}$ when $\Delta_{0}$ is a cone point or a cusp. Furthermore, $\gamma$ and $\alpha$ cobound (together with $\Delta_{0}$ when $\Delta_{0}$ is a boundary geodesic) an embedded cylinder which contains $\delta_{1}$. Hence the point in $\mathcal H$ which corresponds to the $\Delta_{0}$-geodesic $\delta$ lies in the main gap determined by $\gamma$. See Figure \[fig03\] Next let $\delta$ be a simple-not-normal $\Delta_{0}$-geodesic which terminates at $\Delta_{0}$ itself; in this case $\Delta_{0}$ is a boundary geodesic and $\mathcal H$ is $\Delta_{0}$ itself. Then the boundary of a suitably chosen regular neighborhood of $\delta \cup \mathcal H$ consist of two disjoint simple closed curves in the geometric interior of $M$. Let their geodesic realizations be (disjoint) generalized simple closed geodesics $\alpha$ and $\beta$. Then $\alpha, \beta$ bound with $\Delta_{0}$ an embedded pair of pants which contains $\delta$ in a main gap determined by the $[\Delta_{0}, \Delta_{0}]$-geodesic $\gamma$ which is the geodesic realization of $\delta$ in its free relative homotopy class. Finally let $\delta$ be a simple-not-normal $\Delta_{0}$-geodesic which terminates at a boundary geodesic $\Delta_{1}$ which is different from $\Delta_{0}$. The boundary of suitably chosen regular neighborhood of $\delta \cup \Delta_{1}$ on $M$ is a simple arc connecting $\Delta_{0}$ to itself and is disjoint from $\delta$. Its geodesic realization is a $[\Delta_{0}, \Delta_{0}]$-geodesic, $\gamma$, which is disjoint from $\delta$. Now $\Delta_{1}, \gamma$ bound with $\Delta_{0}$ an embedded cylinder which contains $\delta$. Hence $\delta$ lies in the extra gap determined by $\gamma$ or $-\gamma$. [**Calculating the gap functions**]{} {#s:calculation} ===================================== In this section we calculate the gap function ${\rm Gap}(\Delta_{0};\alpha, \beta)$ when $\Delta_{0}$ is a cone point or a boundary geodesic, it is the width of the combined gap determined by a simple $[\Delta_{0},\Delta_{0}]$-geodesic $\gamma$ on $M$. Recall that $\alpha, \beta$ are the generalized simple closed geodesics determined by $\gamma$ and $\mathcal P(\gamma)$ is the geometric pair of pants that $\alpha, \beta$ bound with $\Delta_{0}$ on $M$. [*Case*]{} 1. $\Delta_{0}$ is a cone point of cone angle $\theta \in (0, \pi]$. In this case the width of the main gap determined by $\gamma$ is the angle between $\gamma_{\alpha}$ and $ \gamma_{\beta}$. Let $x$ be the angle between $\delta_{\alpha}$ and $\gamma_{\alpha}$ and let $y$ be the angle between $\delta_{\beta}$ and $\gamma_{\beta}$. [*Subcase*]{} 1.1. Both $\alpha$ and $\beta$ are interior generalized simple closed curves. In this case the width of the combined gap determined by $\gamma$ is the angle between $\gamma_{\alpha}$ and $ \gamma_{\beta}$ and is equal to $\frac{\theta}{2} - (x+y)$. By a formula in Fenchel [@fenchel1989book] VI.3.2 (line 10, page 87), $$\begin{aligned} \sinh |\delta_{\alpha}| = \frac{\cosh\frac{|\beta|}{2}+\cos\frac{\theta}{2}\cosh\frac{|\alpha|}{2}} {\sin\frac{\theta}{2}\sinh\frac{|\alpha|}{2}},\end{aligned}$$ $$\begin{aligned} \sinh |\delta_{\beta}| = \frac{\cosh\frac{|\alpha|}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2}} {\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2}}.\end{aligned}$$ Hence $$\begin{aligned} \label{eqn:1.1.x} \tan x = \frac{1}{\sinh |\delta_{\alpha}|} = \frac{\sin\frac{\theta}{2}\sinh\frac{|\alpha|}{2}} {\cosh\frac{|\beta|}{2}+\cos\frac{\theta}{2}\cosh\frac{|\alpha|}{2}},\end{aligned}$$ $$\begin{aligned} \label{eqn:1.1.y} \tan y = \frac{1}{\sinh |\delta_{\beta}|} = \frac{\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2}} {\cosh\frac{|\alpha|}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2}}.\end{aligned}$$ From these one can derive that $$\begin{aligned} \tan \, (x+y) = \frac{\sin\frac{\theta}{2}\sinh\frac{|\alpha|+|\beta|}{2}} {1+\cos\frac{\theta}{2}\cosh\frac{|\alpha|+|\beta|}{2}}\end{aligned}$$ and hence that $$\begin{aligned} \tan \frac {x+y}{2} = \tan \frac {\theta}{4} \tanh \frac {|\alpha|+|\beta|}{4}.\end{aligned}$$ Thus $$\begin{aligned} \tan \left(\frac {\theta}{4} - \frac {x+y}{2}\right) &=& \frac {\tan \frac{\theta}{4} \left(1-\tanh \frac{|\alpha|+|\beta|}{4}\right)} {1+ \tan^{2}\frac{\theta}{4} \, \tanh \frac{|\alpha|+|\beta|}{4}}\\ &=& \frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}+ \exp\frac{|\alpha|+|\beta|}{2}}.\end{aligned}$$ Hence in this case we have $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta) &=& \frac{\theta}{2} - (x+y)\\ &=& 2 \tan^{-1} \left(\frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}+ \exp\frac{|\alpha|+|\beta|}{2}} \right ).\end{aligned}$$ [*Subcase*]{} 1.2. $\alpha$ is a boundary geodesic and $\beta$ is an interior generalized simple closed geodesic. In this case the width of the combined gap determined by $\gamma$ is the angle between $\delta_{\alpha}$ and $ \gamma_{\beta}$ and is equal to $\frac{\theta}{2} - y$. Hence by (\[eqn:1.1.y\]) we have $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{\theta}{2} - \tan^{-1} \left ( \frac{\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2} }{\cosh\frac{|\alpha|}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2} } \right ).\end{aligned}$$ [*Subcase*]{} 1.3. $\alpha$ is a cone point of cone angle $\varphi \in (0, \pi]$ and $\beta$ is an interior generalized simple closed geodesic. Note that in this case $\gamma_{\alpha}$ coincides with $\delta_{\alpha}$ and hence $x=0$. Therefore the width of the combined gap determined by $\gamma$ is the angle between $\delta_{\alpha}$ and $ \gamma_{\beta}$ and is equal to $\frac{\theta}{2} - y$. Now by a formula in Fenchel [@fenchel1989book] VI.3.3 (line 13, page 88), $$\begin{aligned} \label{} \sinh |\delta_{\beta}| = \frac{\cos\frac{\varphi}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2}} {\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2}}.\end{aligned}$$ Hence $$\begin{aligned} \label{eqn:1.3.y} \tan y = \frac{1}{\sinh |\delta_{\beta}|} = \frac{\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2}} {\cos\frac{\varphi}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2}}.\end{aligned}$$ Thus in this case we have $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{\theta}{2} - \tan^{-1} \left ( \frac{\sin\frac{\theta}{2}\sinh\frac{|\beta|}{2} }{\cos\frac{\varphi}{2}+\cos\frac{\theta}{2}\cosh\frac{|\beta|}{2}} \right ).\end{aligned}$$ [*Case*]{} 2. $\Delta_{0}$ is a boundary geodesic of length $l>0$. In this case the width of the main gap determined by $\gamma$ is the distance between $\gamma_{\alpha}$ and $ \gamma_{\beta}$ along $\Delta_{0}$. Let $x$ be the distance between $\delta_{\alpha}$ and $\gamma_{\alpha}$ along $\Delta_{0}$ and let $y$ be the distance between $\delta_{\beta}$ and $\gamma_{\beta}$ along $\Delta_{0}$. We shall see that all calculations in this case are parallel to those in Case 1. [*Subcase*]{} 2.1. Both $\alpha$ and $\beta$ are interior generalized simple closed curves. In this case the width of the combined gap determined by $\gamma$ is the distance between $\gamma_{\alpha}$ and $ \gamma_{\beta}$ along $\Delta_{0}$ and is equal to $\frac{l}{2} - (x+y)$. By the cosine rule for right angled hexagons on the hyperbolic plane (c.f. Fenchel [@fenchel1989book] VI.3.1, page 86, or Beardon [@beardon1983book] Theorem 7.19.2, page 161), $$\begin{aligned} \cosh |\delta_{\alpha}| = \frac{\cosh\frac{|\beta|}{2}+\cosh\frac{l}{2}\cosh\frac{|\alpha|}{2}} {\cosh\frac{l}{2}\sinh\frac{|\alpha|}{2}},\end{aligned}$$ $$\begin{aligned} \cosh |\delta_{\beta}| = \frac{\cosh\frac{|\alpha|}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2}} {\sinh\frac{l}{2}\sinh\frac{|\beta|}{2}}.\end{aligned}$$ Hence $$\begin{aligned} \label{eqn:2.1.x} \tanh x = \frac{1}{\cosh |\delta_{\alpha}|} = \frac{\sinh\frac{l}{2}\sinh\frac{|\alpha|}{2}} {\cosh\frac{|\beta|}{2}+\cosh\frac{l}{2}\cosh\frac{|\alpha|}{2}},\end{aligned}$$ $$\begin{aligned} \label{eqn:2.1.y} \tanh y = \frac{1}{\cosh |\delta_{\beta}|} = \frac{\sinh\frac{l}{2}\sinh\frac{|\beta|}{2}} {\cosh\frac{|\alpha|}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2}}.\end{aligned}$$ From these one can derive that $$\begin{aligned} \tanh \, (x+y) = \frac{\sinh\frac{l}{2}\sinh\frac{|\alpha|+|\beta|}{2}} {1+\cosh\frac{l}{2}\cosh\frac{|\alpha|+|\beta|}{2}}\end{aligned}$$ and hence that $$\begin{aligned} \tanh \frac {x+y}{2} = \tanh \frac {l}{4} \tanh \frac {|\alpha|+|\beta|}{4}.\end{aligned}$$ Thus $$\begin{aligned} \tanh \left(\frac {l}{4} - \frac {x+y}{2}\right) &=& \frac {\tanh \frac{l}{4} \left(1-\tanh \frac{|\alpha|+|\beta|}{4}\right)} {1- \tanh^{2}\frac{l}{4} \, \tanh \frac{|\alpha|+|\beta|}{4}}\\ &=&\frac {\sinh \frac{l}{2}}{\cosh \frac{l}{2}+ \exp\frac{|\alpha|+|\beta|}{2}}.\end{aligned}$$ Hence in this case we have $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta) &=& \frac{l}{2} - (x+y)\\ &=& 2\tanh^{-1} \left ( \frac {\sinh \frac{l}{2}} {\cosh\frac{l}{2}+ \exp\frac{|\alpha|+|\beta|}{2}}\right ).\end{aligned}$$ [*Subcase*]{} 2.2. $\alpha$ is a boundary geodesic and $\beta$ is an interior generalized simple closed geodesic. In this case the width of the combined gap determined by $\gamma$ is the distance between $\delta_{\alpha}$ and $ \gamma_{\beta}$ along $\Delta_{0}$ and is equal to $\frac{l}{2} - y$. Hence by (\[eqn:2.1.y\]) we have $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{l}{2} - \tanh^{-1} \left ( \frac{\sinh\frac{l}{2}\sinh\frac{|\beta|}{2} }{\cosh\frac{|\alpha|}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2} } \right ).\end{aligned}$$ [*Subcase*]{} 2.3. $\alpha$ is a cone point of cone angle $\varphi \in (0, \pi]$ and $\beta$ is an interior generalized simple closed geodesic. Note that in this case $\gamma_{\alpha}$ coincides with $\delta_{\alpha}$ and hence $x=0$. Hence the width of the combined gap determined by $\gamma$ is the distance between $\delta_{\alpha}$ and $ \gamma_{\beta}$ along $\Delta_{0}$ and is equal to $\frac{l}{2} - y$. Now by a formula in Fenchel [@fenchel1989book] VI.3.2 (line 8, page 87), $$\begin{aligned} \cosh |\delta_{\beta}| = \frac{\cos\frac{\varphi}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2}} {\sinh\frac{l}{2}\sinh\frac{|\beta|}{2}}.\end{aligned}$$ Hence $$\begin{aligned} \label{eqn:2.3.y} \tanh y = \frac{1}{\cosh |\delta_{\beta}|} = \frac{\sinh\frac{l}{2}\sinh\frac{|\beta|}{2}} {\cos\frac{\varphi}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2}}.\end{aligned}$$ Thus in this case we have $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{l}{2} - \tanh^{-1} \left ( \frac{\sinh\frac{l}{2}\sinh\frac{|\beta|}{2} }{\cos\frac{\varphi}{2}+\cosh\frac{l}{2}\cosh\frac{|\beta|}{2}} \right ).\end{aligned}$$ We remark that the formulas in Case 0 for the normalized width ${\rm Gap}^{\prime}(\Delta_{0};\alpha, \beta)$ when $\Delta_{0}$ is a cusp can be derived by similar (and simpler) calculations or by considering the first order infinitesimal terms of those formulas with respect to $\theta$ in Case 1 or with respect to $l$ in Case 2. Hence all derivations in Case 0 are omitted. [**Generalization of the Birman–Series Theorem**]{} {#s:gBS} =================================================== The celebrated Birman–Series Theorem [@birman-series1985t] in its simplest form states that complete simple geodesics on a closed hyperbolic surface are sparsely distributed. More precisely, let $M$ be a hyperbolic surface possibly with boundary such that $M$ is either compact or obtained from a compact surface by removing a finite set of points which form the cusps and such that each boundary component of $M$ is a simple closed geodesic. A geodesic on $M$ is said to be [*complete*]{} if it is either closed and smooth, or open and of infinite length in both directions. Hence a complete geodesic never intersects $\partial M$. Let $G_{k}$ be the family of complete geodesics on $M$ which have at most $k$, counted with multiplicity, transversal self-intersections, $k \ge 0$. Then the main result in [@birman-series1985t] is: For each $k \ge 0$, the point set $S_{k}$ which is the union of all geodesics, as point sets, in $G_{k}$ is nowhere dense and has Hausdorff dimension one. In this section we show that this theorem extends to the case when $M$ is a compact hyperbolic cone-surface with geometric boundary where each cone point has cone angle in $(0, \pi]$, with complete geodesics replaced by complete-normal ones. This is the set of geodesics which are either complete, or intersect the boundary perpendicularly. Let $M$ be a compact hyperbolic cone-surface with geometric boundary where each cone point has cone angle in $(0, \pi]$, and let $G_{k}$ be the family of complete-normal geodesics on $M$ which have at most $k$ transversal self-intersections, $k \ge 0$. Then, for each $k \ge 0$, the point set $S_{k}$ which is the union of all geodesics, as point sets, in $G_{k}$ is nowhere dense and has Hausdorff dimension one. The proof of this generalization is is essentially the same as that of the original Birman–Series theorem given in [@birman-series1985t]. Hence for simplicity we shall only sketch the proof of the theorem for the case $k=0$, that is, for simple complete-normal geodesics; the reader is referred to [@birman-series1985t] for omitted details. We only need to consider the case where $M$ has no geodesic boundary components; for if $M$ has nonempty geodesic boundary we can replace $M$ by the double of $M$ along its geodesic boundary. We also assume for clarity that each cone point of $M$ has cone angle less than $\pi$. We decompose the set $G_0$ into finitely many subsets and prove the conclusion for each such subset. For the subset of simple complete geodesics on $M$, that is, the geodesics which never start from or terminate at cusps or cone points, the proof is the same as that in [@birman-series1985t] with little modification (which can be seen from the sketch below). For the subset of simple normal geodesics which connect a given cusp or cone point to another (possibly the same) given cusp or cone point, it is easy to see that in this subset each such geodesic is isolated in suitable neighborhoods of its endpoints and hence the conclusion follows. Thus it remains to prove the conclusion for the subset of simple complete-normal geodesics which starts from a given cusp or cone point $P$ and never terminates at any geometric boundary component. One can cut $M$ along normal geodesics connecting cusps or cone points to form a (convex) fundamental polygon $R$ for $M$ in the hyperbolic plane. Let $A=\{ a_1, a_2, \cdots, a_m \}$ denote the ordered set of vertices and oriented sides of $R$ with anti-clockwise ordering with some arbitrary but henceforth fixed initial element $a_1$. Let $J_0$ be the set of oriented simple-normal geodesic arcs $\gamma$ on $M$ such that the initial point and the ending point of $\gamma$ lie in $\partial R$. (Note that except at its initial point or ending point $\gamma$ cannot pass through a vertex of $R$.) For $\gamma \in J_0$, we call the components of $\gamma \cap R$ the [*segments*]{} of $\gamma$ and the points of $\gamma \cap \partial R$ the partition points of $\gamma$. We label the partition points $t_0, t_1, \cdots, t_n$ in the order in which they occur along $\gamma$ (note that we treat $t_i \in \partial R$ as the initial point of the segment of $\gamma$ from $t_i$ to $t_{i+1}$) and we set $\parallel \gamma \parallel = n$ as the combinatorial length of $\gamma$. For $\gamma \in J_0$, the segments of $\gamma$ give rise to a [*simple diagram*]{} on $R$ which is a collection of finitely many pairwise disjoint (geodesic) arcs joining pairs of distinct elements of $A$. Two simple diagrams are regarded as being identical if they agree up to isotopy supported on each side of $R$. For $a_i, a_j \in A, i \neq j$, let $n_{ij}$ denote the number of arcs joining $a_i$ to $a_j$ in the given simple diagram. The [*length*]{} of a simple diagram is $n = \sum n_{ij}, 1 \le i < j \le m$. The Birman–Series parameterization of elements of $J_0$ consists of two sets of data. The first is the ordered sequence $h_1(\gamma)=(n_{12}, n_{13}, \cdots, n_{m-1,m})$ which records for each pair of distinct elements $a_i, a_j$ of $R$ the number $n_{ij}$ of segments of $\gamma$ which join $a_i$ to $a_j$. The second set of data, $h_2(\gamma)$, records information about the position of the initial and final points $t_0, t_n$ of $\gamma$. Let $a(t_i)$ be the element of $A$ containing $t_i$ and let $j(t_i) \in \mathbf N$ be the position of $t_i$ among the partition points of $\gamma$ which lie along $a(t_i)$ counting in the anticlockwise direction round $\partial R$. Define $h_2(\gamma)=(a(t_0), j(t_0), a(t_n), j(t_n))$. The following lemmas and their proofs in [@birman-series1985t] still hold in our case. Suppose that $\gamma, \gamma^{\prime} \in J_0$ and that $h_1(\gamma)=h_1(\gamma^{\prime}), h_2(\gamma)=h_2(\gamma^{\prime})$. Let $t_0, t_1, \cdots, t_n$ and $t_0^{\prime}, t_1^{\prime}, \cdots, t_n^{\prime}$ be the partition points of $\gamma, \gamma^{\prime}$ respectively. Then $a(t_i)=a(t_i^{\prime})$ for each $i=0,1,\cdots,n$. Let $J_0(n)=\{\gamma \in J_0 : \, \parallel \gamma \parallel =n \}$. Then there is a polynomial $P_0(n)$ such that the number of simple diagrams of length $n$ $${\rm card} \{(h_1(\gamma), h_2(\gamma)): \gamma \in J_0(n)\} \le P_0(n).$$ The main idea of the proof of Birman–Series Theorem in [@birman-series1985t] is that geodeisc arcs in $J_0(n)$ (for sufficiently large $n$) with the same parameterization lie exponentially close in $M$. It relies on the following key lemma which is Lemma 3.1 in [@birman-series1985t]. There is a universal constant $\alpha >0$ (depending only on the choice of the fundamental polygon $R$) so that $$l(\gamma) \ge \alpha \parallel \gamma \parallel$$ for $\gamma \in J_0$ with $\parallel \gamma \parallel$ sufficiently large, where $l(\gamma)$ denotes the hyperbolic length of $\gamma$. There is a universal constant $\epsilon >0$ so that any segment of $\gamma$ which does not connect two consecutive sides of $R$ or does not intersect a suitably chosen disk neighborhood of each cusp or cone point has hyperbolic length at least $\epsilon$. Let $q$ be the maximum number of sides of $R$, projected to $M$, which meet at any cusp or cone point of $M$. Then at most $q-1$ consecutive segments of $\gamma$ can connect consecutive sides of $R$ around the same cusp or cone point and intersect the chosen disk neighborhood of that cusp or cone point; for otherwise there will be a self-intersection on $\gamma$. Hence in any $q$ consecutive segments of $\gamma$, at least one has hyperbolic length $\epsilon$, which gives the result. The following two lemmas then apply respectively to the set of all complete simple geodesics which never intersect any cusp or cone point and to the set of simple geodesics which start from a fixed cusp or cone point and never terminates at any cusp or cone point. (Recall that we assume that $M$ has no boundary geodesics.) \[lem:BS 3.2\] Let $\gamma, \gamma^{\prime} \in J_0(2n+1)$ and suppose that $h_1(\gamma)=h_1(\gamma^{\prime}), h_2(\gamma)=h_2(\gamma^{\prime})$. Let $\delta \subset \gamma, \delta^{\prime} \subset \gamma^{\prime}$ denote the segments of $\gamma, \gamma^{\prime}$ lying between the partition points $t_n, t_{n+1}$ and $t_n^{\prime}, t_{n+1}^{\prime}$ respectively. Then $\delta^{\prime} \subset B_{ce^{-\alpha n}}(\delta)$ where $c, \alpha$ are universal constants and where $B_\epsilon(\delta)$ denotes the tubular neighborhood of $\delta$ of hyperbolic radius $\epsilon>0$. \[lem:gBS 3.2\] Let $\gamma, \gamma^{\prime} \in J_0(n+k)$ be such that they start at the same vertex of $R$ and that $h_1(\gamma)=h_1(\gamma^{\prime}), h_2(\gamma)=h_2(\gamma^{\prime})$. Let $\delta \subset \gamma, \delta^{\prime} \subset \gamma^{\prime}$ denote the segments of $\gamma, \gamma^{\prime}$ lying between the partition points $t_i, t_{i+1}$ and $t_i^{\prime}, t_{i+1}^{\prime}$ respectively for some $1 \le i \le k$. Then $\delta^{\prime} \subset B_{ce^{-\alpha n}}(\delta)$ where $c, \alpha$ are universal constants and where $B_\epsilon(\delta)$ denotes the tubular neighborhood of $\delta$ of hyperbolic radius $\epsilon>0$. Note that Lemma \[lem:BS 3.2\] is Lemma 3.2 in [@birman-series1985t] and Lemma \[lem:gBS 3.2\] can be proved similarly. From these we have the following proposition which is Proposition 4.1 in [@birman-series1985t] from which the conclusion of the Birman–Series Theorem follows exactly as in the proofs in [@birman-series1985t] §5. There exist universal constants $L, c, \alpha >0$ and a polynomial $P_0(\cdot)$ such that for each $n$ there is a set $F_n$ of simple geodesic arcs, each of length at most $L$, so that ${\rm card}(F_n) \le P_0(n)$ and so that $$S_0 \subset \cup \{B_\epsilon(\gamma) \mid \gamma \in F_n\}, \epsilon = ce^{\alpha n}.$$ Finally we remark that the above Birman–Series’ arguments will give rough estimates on the distribution of simple closed geodesics on a compact hyperbolic cone-surface $M$ which is enough for proving the absolute convergence of the series appearing in various generalized McShane’s identities, as was observed and used in [@akiyoshi-miyachi-sakuma2004preprint] (for the case of complete hyperbolic surfaces) for similar purposes. \[lem:BS\] Let $M$ be a compact hyperbolic cone-surface with all cone angles in $(0,\pi]$. Then for any constant $c>0$ [(i)]{} the series $$\begin{aligned} \sum_{\beta} \frac{1}{\exp(c|\beta|)}\end{aligned}$$ converges absolutely, where the sum is over all generalized simple closed geodesics on $M$ and all simple normal geodesic arcs connecting geometric boundary components of $M$; the series $$\begin{aligned} \sum_{\alpha, \beta} \frac{1}{\exp[c(|\alpha|+|\beta|)]}\end{aligned}$$ converges absolutely, where the sum is over all pairs $\alpha, \beta$ of disjoint generalized simple closed geodesics on $M$ and/or simple normal geodesic arcs connecting geometric boundary components of $M$. [**Proof of Theorems**]{} {#s:proof} ========================= [*Proof of Theorem \[thm:mcshane most general\]*]{} Now the proof is obvious from the previous discussions. Suppose $\Delta_0$ is a cone point. Recall $\mathcal H$ is a suitably chosen small circle centered at $\Delta_0$, and ${\mathcal H}_{\texttt{ns}}$, ${\mathcal H}_{\texttt{sn}}$, ${\mathcal H}_{\texttt{snn}}$ are the point sets of the first intersections of $\mathcal H$ with respectively all non-simple, all simple-normal, all simple-not-normal $\Delta_{0}$-geodesics. The elliptic measure of each of these subsets of $\mathcal H$ is the radian measure that it subtends to the cone point $\Delta_0$. The generalized Birman–Series Theorem in §\[s:gBS\] implies that the closed subset ${\mathcal H}_{\texttt{sn}}$ has measure $0$. Hence the open subset ${\mathcal H}_{\texttt{ns}} \cup {\mathcal H}_{\texttt{snn}}$ has full measure, that is, $\theta_0$. Now the maximal open intervals of ${\mathcal H}_{\texttt{ns}} \cup {\mathcal H}_{\texttt{snn}}$, suitably combined, have measure $2 {\rm Gap}(\Delta_{0};\alpha, \beta)$ for each unordered pair of generalized simple closed geodesics $\alpha, \beta$ on $M$ which bound with $\Delta_0$ an embedded pair of pants on $M$. Hence their sum is equal to $\theta_0$ and the desired identity follows. The cases where $\Delta_0$ is a boundary geodesic or a cusp are similarly proved. [*Proof of Corollary \[cor:mcshane conical holed weierstrass\]*]{} Consider the case where $\Delta_0$ is a cone point. In this case $T$ admits a unique elliptic involution $\eta$ such that $\eta$ maps each oriented simple closed geodesics on $T$ onto itself with orientation reversed. Note that $\eta$ fixes the cone point $\Delta_0$ and three other interior points which are the so-called Weierstrass points of $T$. Each simple closed geodesics on $T$ passes exactly two Weierstrass points; hence there are three Weierstrass classes of simple closed geodesics on $T$. Now the quotient of $T$ under $\eta$ is a sphere with three angle $\pi$ cone points and a cone point with angle $\theta /2$. Then Theorem \[thm:mcshane most general\] applies to $M = T/\langle\eta\rangle$, with $\Delta_0$ the angle $\pi$ cone point whose inverse image under $\eta$ is the Weierstrass point that the Weierstrass class $\mathcal A$ misses. Note that each generalized simple closed geodesic on $M = T/\langle\eta\rangle$ is either a geometric boundary component or degenerate simple closed geodesic which is the double cover of a simple geodesic arc which connects two Weierstrass points. Hence the set of all pairs of generalized simple closed geodesics which bound with $\Delta_0$ an embedded pair of pants is exactly the set of pairs consisting of the angle $\theta /2$ cone point plus a degenerate simple closed geodesic $\gamma^{\prime}$ which is the double cover of the quotient simple geodesic arc of a simple closed geodesic $\gamma$ on $T$ in the given Weierstrass class $\mathcal A$ (note that by definition the length of $\gamma^{\prime}$ is the same as that of $\gamma$). Hence by (\[subcase 1.3\]) the summand in the summation is $$\begin{aligned} \frac{\pi}{2} - \tan^{-1} \left ( \frac{\sin\frac{\pi}{2}\sinh\frac{|\gamma|}{2} }{\cos\frac{\theta}{4}+\cos\frac{\pi}{2}\cosh\frac{|\gamma|}{2} } \right ) = \tan^{-1} \left ( \frac{\cos\frac{\theta}{4}}{\sinh\frac{|\gamma|}{2}} \right ).\end{aligned}$$ The proof for the case where $\Delta_0$ is a boundary geodesic is similar. Note that we can also choose $\Delta_0$ to be the angle $\theta /2$ cone point on $T/\langle\eta\rangle$, then we obtain (\[eqn:mcshane cone torus\]), the generalization of McShane’s original identity to the cone-torus $T$. This is one way of seeing why we can allow the cone angle of up to $2\pi$ in the cone torus case. [*Proof of Theorem \[thm:mcshane genus two global\]*]{} It is well known that $M$ admits a unique hyperelliptic involution $\eta$ (see for example [@haas-susskind1989pams]) such that $\eta$ maps each simple closed geodesic onto itself and preserves/reverses the orientation of separating/non-separating simple closed geodesics. Note that $\eta$ leaves six points on $M$ fixed; they are the six Weierstrass points on $M$. Consider the quotient $M^{\prime}=M/\langle\eta\rangle$ which is a sphere with six angle $\pi$ cone points. Each generalized simple closed geodesic on $M^{\prime}$ is either - an angle $\pi$ cone point; or - a degenerate simple closed geodesic ${\beta}^{\prime}$ which is the double cover of a simple geodesic arc $c$ connecting two angle $\pi$ cone points where the inverse image of $c$ under $\eta$ is a non-separating simple closed geodesic $\beta$ on $M$; or - a separating (non-degenerate) simple closed geodesic ${\alpha}^{\prime}$ whose inverse image under $\eta$ is a separating simple closed geodesic $\alpha$ on $M$. In this case ${\alpha}^{\prime}$ does not pass through any of the six angle $\pi$ cone points and there are three of them on each side of ${\alpha}^{\prime}$ on $M^{\prime}$. Hence $\alpha$ passes none of six Weierstrass points and there are three of them on each side of $\alpha$ on $M$. Now apply Theorem \[thm:mcshane most general\] to $M^{\prime}$ with $\Delta_0$ one of the six angle $\pi$ cone points. Then each pair of generalized simple closed geodesics on $M^{\prime}$ which bound with $\Delta_0$ an embedded pair of pants $\mathcal P$ consists of a separating simple closed geodesic $\alpha^{\prime}$ on $M^{\prime}$ and a degenerate simple closed geodesic $\beta^{\prime}$ on $M^{\prime}$ which lies on the same side of $\alpha^{\prime}$ as $\Delta_0$ and misses $\Delta_0$. Let the inverse image of $\alpha^{\prime}, \beta^{\prime}$ under $\eta$ be $\alpha, \beta$. Then $\alpha$ is a separating simple closed geodesic on $M$ and $\beta$ is a non-separating simple closed geodesic on $M$. Furthermore, $\beta$ and the Weierstrass point which is the inverse image of $\Delta_0$ lie on the same side of $\alpha$ on $M$. Note that the hyperbolic lengths of $\alpha^{\prime}, \beta^{\prime}$ are respectively $|\alpha|/2, |\beta|$. Hence by (\[eqn:dGf 1.1\]) in this case the summand in the resulting generalized McShane’s Weierstrass identity for $M^{\prime}$ with the chosen $\Delta_0$ is $$\begin{aligned} 2 \tan^{-1} \left (\frac{\sin\frac{\pi}{2}}{\cos\frac{\pi}{2}+ \exp\frac{|\alpha|/2+|\beta|}{2}} \right ) = 2 \tan^{-1} \exp\left(-\frac{|\alpha|}{4}-\frac{|\beta|}{2}\right).\end{aligned}$$ Note that each pair of disjoint simple closed geodesics $(\alpha,\beta)$ on $M$ such that $\alpha$ is separating and $\beta$ is non-separating arises as the inverse image of a unique pair of generalized simple closed geodesics on $M'$ as described above, where the chosen $\Delta_0$ is the angle $\pi$ cone point which is the image under $\eta$ of the Weierstrass point on $M$ that lies on the same side of $\alpha$ as $\beta$ and is missed by $\beta$. Summing all the six resulting Weierstrass identities we then have $$\begin{aligned} \sum 2 \tan^{-1} \exp \left( -\frac{|\alpha|}{4}- \frac{|\beta|}{2} \right ) = \frac{6\pi}{2},\end{aligned}$$ where the sum is over all ordered pairs $(\alpha, \beta)$ of disjoint simple closed geodesics on $M$ such that $\alpha$ is separating and $\beta$ is non-separating. [*Proof of Addendum \[add:genus two\]*]{} We first prove that the series in (\[eqn:genus two QF\]) converges absolutely and uniformly on compact set in the space $\mathcal Q \mathcal F$ of quasi-Fuchsian representations of ${\pi_1}(M)$ into ${\rm SL}(2, \mathbf C)$ by the same argument as used in [@akiyoshi-miyachi-sakuma2004preprint]. The identity (\[eqn:genus two QF\]) then follows by analytic continuation since each summand in it is an analytic function of the complex Fenchel–Nielsen coordinates for the quasi-Fuchsian space (see [@tan1994ijm]) and the identity holds when all the coordinates take real values (by Theorem \[thm:mcshane genus two global\]) and the space of quasi-Fuchsian representations of ${\pi_1}(M)$ into ${\rm PSL}(2, \mathbf C)$ is simply connected. As pointed out in [@akiyoshi-miyachi-sakuma2004preprint] Lemma 5.2, by [@jorgensen-marden1979qjm] Lemma 3, for any compact subset $\mathcal C$ of $\mathcal Q \mathcal F$, there is a constant $k = k(C) >0$ such that $$\begin{aligned} k l_{\rho_0}(\gamma) \le \Re l_{\rho}(\gamma) \le k^{-1} l_{\rho_0}(\gamma),\end{aligned}$$ for any essential simple closed curve $\gamma$, where $\rho_0$ is a fixed Fuchsian representation of ${\pi_1}(M)$ into ${\rm SL}(2, \mathbf C)$. Since $|\tan^{-1}(x)| \le 2|x|$ for $|x|$ sufficiently small, we have for all except a finitely many pairs of (free homotopy classes of) disjoint essential simple closed curves $\alpha, \beta$ on $M$ such that $\alpha$ is separating and $\beta$ is non-separating $$\begin{aligned} \left| \tan^{-1}\exp\Big(-\frac{l_{\rho}(\alpha)}{4}-\frac{l_{\rho}(\beta)}{2}\Big)\right| &\le& 2\left|\exp\Big(-\frac{l_{\rho}(\alpha)}{4}-\frac{l_{\rho}(\beta)}{2}\Big)\right|\\ &=& 2 \exp\Big(-\frac{\Re l_{\rho}(\alpha)}{4}-\frac{\Re l_{\rho}(\beta)}{2}\Big)\\ &\le& 2 \exp\Big(-k\Big(\frac{l_{\rho_0}(\alpha)}{4}+\frac{l_{\rho_0}(\beta)}{2}\Big)\Big).\end{aligned}$$ Thus the series in (\[eqn:genus two QF\]) converges absolutely and uniformly on the compact set $C$ of $\mathcal Q \mathcal F$ since the series $$\sum \exp\Big(-k\Big(\frac{l_{\rho_0}(\alpha)}{4}+\frac{l_{\rho_0}(\beta)}{2}\Big)\Big)$$ converges by Lemma \[lem:BS\]. [**Complexified reformulation of the generalized McShane’s identity** ]{} {#s:reformulation} ========================================================================= In this section we prove the unified version (\[eqn:reform of cp and gb cases\]) of our generalized McShane’s identity using complex arguments and interpret it geometrically. [**Two functions**]{} First we would like to define two functions $G, S: {\mathbf C}^3 \rightarrow \mathbf C$ as follows: $$\begin{aligned} G(x,y,z)=2 \tanh^{-1}\left(\frac{\sinh(x)}{\cosh(x)+\exp(y+z)}\right),\end{aligned}$$ $$\begin{aligned} S(x,y,z)=\tanh^{-1}\left(\frac{\sinh(x)\sinh(y)}{\cosh(z)+\cosh(x)\cosh(y)}\right).\end{aligned}$$ Note that here for a complex number $x$, $\tanh^{-1}(x)$ is defined to have imaginary part in $(-\pi/2, \pi/2]$. Using the identity $$x=\frac{1}{2}\log\frac{1+\tanh(x)}{1-\tanh(x)},$$ it is easy to check that the two functions have also the following expressions: $$\begin{aligned} G(x,y,z)=\log\frac{\exp(x)+\exp(y+z)}{\exp(-x)+\exp(y+z)},\end{aligned}$$ $$\begin{aligned} S(x,y,z)=\frac{1}{2}\log\frac{\cosh(z)+\cosh(x+y)}{\cosh(z)+\cosh(x-y)},\end{aligned}$$ as used by Mirzakhani in [@mirzakhani2004preprint]. (She uses different notations $\mathcal D, \mathcal R$ as explained below.) Here for a non-zero complex number $x$, $\log(x)$ assumes the main branch value with imaginary part in $(-\pi, \pi]$. We shall see that both expressions of the functions are useful. For $x,y,z >0$, the geometrical meanings of $G(x,y,z)$ and $S(x,y,z)$ are as follows. Let $\mathcal P(2x,2y,2z)$ be the unique hyperbolic pair of pants whose boundary components $X,Y,Z$ are simple closed geodesics of lengths $2x,2y,2z$ respectively. Then $S(x,y,z)$ is half the length of the orthogonal projection of the boundary geodesic $Y$ onto $X$ in $\mathcal P(2x,2y,2z)$ and $S(x,z,y)$ is half the length of the orthogonal projection of the boundary geodesic $Z$ onto $X$ in $\mathcal P(2x,2y,2z)$, and $G(x,y,z)$ is the length of each of the two gaps between these two projections on $X$. We have therefore the identity $$\begin{aligned} G(x,y,z)+S(x,y,z)+S(x,z,y)=x\end{aligned}$$ for all $x,y,z \ge 0$. Note that the same identity holds modulo $\pi i$ for all $x,y,z \in \mathbf C$. The relations of our functions $G,S$ with Mirzakhani’s functions $\mathcal D, \mathcal R$ are $$\begin{aligned} G(x,y,z)=\mathcal D(2x,2y,2z) /2,\end{aligned}$$ $$\begin{aligned} S(x,y,z)=(x-\mathcal R(2x,2z,2y)) /2.\end{aligned}$$ \[lem: Gap=G+S\] [(i)]{} For $x,z \ge 0$ and $y \in [0, \frac{\pi}{2}]$, $$\begin{aligned} \label{eqn:(x,yi,z)} \hskip 6pt G(x,yi,z)+S(x,yi,z)=x-\tanh^{-1}\left( \frac{\sinh(x)\sinh(z)}{\cos(y)+\cosh(x)\cosh(z)} \right).\end{aligned}$$ [(ii)]{} For $x,y \in [0, \frac{\pi}{2}]$ and $z \ge 0$, $$\begin{aligned} \label{eqn:(xi,yi,z)} \hskip 16pt G(xi,yi,z)+S(xi,yi,z)=\left[x-\tan^{-1}\left( \frac{\sin(x)\sinh(z)}{\cos(y)+\cos(x)\cosh(z)} \right)\right]i.\end{aligned}$$ [(i)]{} It follows from the following two identities since $\Re S(x,yi,z)=0$: $$\begin{aligned} \label{eqn:(i)1} \Re G(x,yi,z)=x-\tanh^{-1}\left( \frac{\sinh(x)\sinh(z)}{\cos(y)+\cosh(x)\cosh(z)} \right),\end{aligned}$$ $$\begin{aligned} \label{eqn:(i)2} \Im G(x,yi,z)+ \Im S(x,yi,z)=0.\end{aligned}$$ [*Proof of [(\[eqn:(i)1\])]{} and [(\[eqn:(i)2\])]{}*]{}: By definition, $$\begin{aligned} G(x,yi,z)&=&\log\frac{\exp(x)+\exp(yi+z)}{\exp(-x)+\exp(yi+z)}\\ &=&\log\frac{[\exp(x)+\cos(y)\exp(z)]+i[\sin(y)\exp(z)]}{[\exp(-x)+\cos(y)\exp(z)]+i[\sin(y)\exp(z)]}.\end{aligned}$$ Hence $$\begin{aligned} \Re G(x,yi,z)&=&\frac{1}{2}\log\frac{[\exp(x)+\cos(y)\exp(z)]^2+ [\sin(y)\exp(z)]^2}{[\exp(-x)+\cos(y)\exp(z)]^2+[\sin(y)\exp(z)]^2}\\ &=&\frac{1}{2}\log\frac{\exp(2x)+\exp(2z)+2\exp(x)\cos(y)\exp(z)}{\exp(-2x)+\exp(2z)+2\exp(-x)\cos(y)\exp(z)}\\ &=&\frac{1}{2}\log\left(\frac{\cosh(x-z)+\cos(y)}{\cosh(x+z)+\cos(y)}\frac{\exp(x+z)}{\exp(-x+z)}\right)\\ &=&x-\frac{1}{2}\log\frac{\cosh(x+z)+\cos(y)}{\cosh(x-z)+\cos(y)}\\ &=&x-\tanh^{-1}\left(\frac{\sinh(x)\sinh(z)}{\cos(y)+\cosh(x)\cosh(z)}\right).\end{aligned}$$ On the other hand, $$\begin{aligned} \Im G(x,yi,z) &=& \tan^{-1}\left(\frac{\sin(y)\exp(z)}{\exp(x)+\cos(y)\exp(z)}\right) - \tan^{-1}\left(\frac{\sin(y)\exp(z)}{\exp(-x)+\cos(y)\exp(z)}\right)\\ &=& \tan^{-1}\left(\frac{[\exp(-x)-\exp(x)]\sin(y)\exp(z)}{[\exp(x)+\cos(y)\exp(z)][\exp(-x)+\cos(y)\exp(z)]+[\sin(y)\exp(z)]^2}\right)\\ &=& \tan^{-1}\left(\frac{[\exp(-x)-\exp(x)]\sin(y)\exp(z)}{1+\exp(2z)+[\exp(x)+\exp(-x)]\cos(y)\exp(z)}\right)\\ &=& - \tan^{-1}\left(\frac{\sinh(x)\sin(y)}{\cosh(z)+\cosh(x)\cos(y)}\right)\\ &=& -\Im S(x,yi,z),\end{aligned}$$ since $$\begin{aligned} S(x,yi,z)&=&\tanh^{-1}\left(\frac{\sinh(x)\sinh(yi)}{\cosh(z)+\cosh(x)\cosh(yi)}\right)\\ &=& i \,\tan^{-1}\left(\frac{\sinh(x)\sin(y)}{\cosh(z)+\cosh(x)\cos(y)}\right).\end{aligned}$$ [(ii)]{} It will follow from the following two identities: $$\begin{aligned} \label{eqn:(ii)1} \Im G(xi,yi,z)=x-\tan^{-1}\left( \frac{\sin(x)\sinh(z)}{\cos(y)+\cos(x)\cosh(z)} \right),\end{aligned}$$ $$\begin{aligned} \label{eqn:(ii)2} \Re G(xi,yi,z)+ S(xi,yi,z)=0.\end{aligned}$$ [*Proof of [(\[eqn:(ii)1\])]{} and [(\[eqn:(ii)2\])]{}*]{}: By definition, $$\begin{aligned} G(xi,yi,z)&=&\log\frac{\exp(xi)+\exp(yi+z)}{\exp(-xi)+\exp(yi+z)}\\ &=&\log\frac{[\cos(x)+\cos(y)\exp(z)]+i[\sin(x)+\sin(y)\exp(z)]}{[\cos(x)+\cos(y)\exp(z)]+i[-\sin(x)+\sin(y)\exp(z)]}.\end{aligned}$$ Hence $$\begin{aligned} \Re G(xi,yi,z)&=&\frac{1}{2}\log\frac{[\cos(x)+\cos(y)\exp(z)]^2+ [\sin(x)+\sin(y)\exp(z)]^2}{[\cos(x)+\cos(y)\exp(z)]^2+[-\sin(x)+\sin(y)\exp(z)]^2}\\ &=&\frac{1}{2}\log\frac{1+\exp(2z)+\cos(x-y)\exp(z)}{1+\exp(2z)+\cos(x+y)\exp(z)}\\ &=&\frac{1}{2}\log\frac{\cosh(z)+\cos(x-y)}{\cosh(z)+\cos(x+y)}\\ &=&-\frac{1}{2}\log\frac{\cosh(z)+\cosh(xi+yi)}{\cosh(z)+\cosh(xi-yi)}\\ &=&-S(xi,yi,z).\end{aligned}$$ On the other hand, $$\begin{aligned} I&=&\Im G(xi,yi,z)\\ &=&\tan^{-1}\left(\frac{\sin(x)+\sin(y)\exp(z)}{\cos(x)+\cos(y)\exp(z)}\right) - \tan^{-1}\left(\frac{-\sin(x)+\sin(y)\exp(z)}{\cos(x)+\cos(y)\exp(z)}\right)\\ &=& \tan^{-1}\left(\frac{2\sin(x)[\cos(x)+\cos(y)\exp(z)]}{[\cos(x)+\cos(y)\exp(z)]^2-[\sin(x)]^2+[\sin(y)\exp(z)]^2}\right)\\ &=&\tan^{-1}\left(\frac{\sin(2x)+2\sin(x)\cos(y)\exp(z)}{\cos(2x)+\exp(2z)+2\cos(x)\cos(y)\exp(z)}\right).\end{aligned}$$ Hence $$\begin{aligned} iI=\tanh^{-1}\left(\frac{i\sin(2x)+2i\sin(x)\cos(y)\exp(z)}{\cos(2x)+\exp(2z)+2\cos(x)\cos(y)\exp(z)}\right),\end{aligned}$$ or $$\begin{aligned} \frac{\exp(2iI)-1}{\exp(2iI)+1}=\frac{i\sin(2x)+2i\sin(x)\cos(y)\exp(z)}{\cos(2x)+\exp(2z)+2\cos(x)\cos(y)\exp(z)}.\end{aligned}$$ Hence $$\begin{aligned} \exp(2iI)&=&\frac{\exp(2xi)+\exp(2z)+2\exp(xi)\cos(y)\exp(z)}{\exp(-2xi)+\exp(2z)+2\exp(-xi)\cos(y)\exp(z)}\\ &=&\frac{\cosh(xi-z)+\cos(y)}{\cosh(xi+z)+\cos(y)}\frac{\exp(xi+z)}{\exp(-xi+z)}\\ &=&\exp(2xi)\frac{\cosh(yi)+\cosh(xi-z)}{\cosh(yi)+\cosh(xi+z)}.\end{aligned}$$ Thus $$\begin{aligned} iI&=&xi-\frac{1}{2}\log\frac{\cosh(yi)+\cosh(xi+z)}{\cosh(yi)+\cosh(xi-z)}\\ &=&xi-\tanh^{-1}\left(\frac{\sinh(xi)\sinh(z)}{\cosh(yi)+\cosh(xi)\cosh(z)} \right)\\ &=&xi-i\tan^{-1}\left(\frac{\sin(x)\sinh(z)}{\cos(y)+\cos(x)\cosh(z)}\right).\end{aligned}$$ Therefore $$\begin{aligned} \Im G(xi,yi,z)=I= x-\tan^{-1}\left(\frac{\sin(x)\sinh(z)}{\cos(y)+\cos(x)\cosh(z)}\right).\end{aligned}$$ [**Restatement of the complexified identities**]{} Now we can restate the non-cusp cases of Theorem \[thm:complexified\] using the functions $G,S$ defined above. Recall that for each generalized simple closed geodesic $\delta$, we have defined in §\[s:intro\] its [**complex length**]{} $|\delta|$, that is, $|\delta|=0$ if $\delta$ is a cusp; $|\delta|=\theta i$ if $\delta$ is a cone point of angle $\theta \in (0, \pi]$; and $|\delta|=l$ if $\delta$ is a boundary geodesic or an interior generalized simple closed geodesic of hyperbolic length $l>0$. \[thm:complexified non-cusp cases\] For a compact hyperbolic cone-surface $M$ with all cone angles in $(0, \pi]$, let all its geometric boundary components be $\Delta_0, \Delta_1, \cdots, \Delta_n$ with complex lengths $L_0, L_1, \cdots, L_n$ respectively. If $\Delta_0$ is a cone point or a boundary geodesic then $$\begin{aligned} \label{eqn:reform of non-cusp cases with GS} \sum_{\alpha, \beta} G\left(\frac{L_0}{2},\frac{|\alpha|}{2},\frac{|\beta|}{2}\right) + \sum_{j=1}^{n}\sum_{\beta} S\left(\frac{L_0}{2},\frac{L_j}{2},\frac{|\beta|}{2}\right)= \frac{L_0}{2},\end{aligned}$$ where the first sum is over all (unordered) pairs of generalized simple closed geodesics $\alpha, \beta$ on $M$ such that $\alpha, \beta$ bound with $\Delta_0$ an embedded pair of pants on $M$ (note that one of $\alpha, \beta$ might be a geometric boundary component) and the sub-sum in the second sum is over all interior simple closed geodesics $\beta$ such that $\beta$ bounds with $\Delta_j$ and $\Delta_0$ an embedded pair of pants on $M$. Furthermore, all the series in (\[eqn:reform of non-cusp cases with GS\]) converge absolutely. We shall omit the proof of Theorem \[thm:complexified\] in the case where $\Delta_0$ is a cusp, for as remarked before, in the cusp case the identity (\[eqn:reform of cusp cases\]) can either be proved similarly or be derived by considering the first order infinitesimal terms of the corresponding identity (\[eqn:reform of cp and gb cases\]) in other cases. We first show that our generalized McShane’s identities (\[eqn:001\]) and (\[eqn:002\]) can be reformulated as (\[eqn:reform of non-cusp cases with GS\]) modulo convergence. First suppose that $\Delta_0$ is a boundary geodesic of hyperbolic length $l_0 >0$. For a pair of interior generalized simple closed geodesics $\alpha, \beta$ which bound with $\Delta_0$ an embedded pair of pants on $M$, we have directly by definition that $$\begin{aligned} {\rm Gap}(\Delta_0;\alpha,\beta)= G\left(\frac{l_0}{2},\frac{|\alpha|}{2},\frac{|\beta|}{2}\right).\end{aligned}$$ For a pair of generalized simple closed geodesics $\alpha, \beta$ such that $\alpha$ is a boundary geodesic and $\beta$ is an interior generalized simple closed geodesic and that they bound with $\Delta_0$ an embedded pair of pants on $M$, we have by definition and the geometric meanings of $G,S$ that $$\begin{aligned} {\rm Gap}(\Delta_0;\alpha,\beta)= G\left(\frac{l_0}{2},\frac{|\alpha|}{2},\frac{|\beta|}{2}\right)+ S\left(\frac{l_0}{2},\frac{|\alpha|}{2},\frac{|\beta|}{2}\right).\end{aligned}$$ For a pair of generalized simple closed geodesics $\alpha, \beta$ such that $\alpha$ is a cone point of angle $\varphi \in (0,\pi]$ and $\beta$ is an interior generalized simple closed geodesic and that they bound with $\Delta_0$ an embedded pair of pants on $M$, we have by (\[eqn:(x,yi,z)\]) with $x=l_0/2, y=\varphi/2, z=|\beta|/2$ that $$\begin{aligned} {\rm Gap}(\Delta_0;\alpha,\beta)= G\left(\frac{l_0}{2},\frac{\varphi i}{2},\frac{|\beta|}{2}\right)+ S\left(\frac{l_0}{2},\frac{\varphi i}{2},\frac{|\beta|}{2}\right).\end{aligned}$$ Next suppose that $\Delta_0$ is a cone point of angle $\theta_0 \in (0,\pi]$. For a pair of interior generalized simple closed geodesics $\alpha, \beta$ which bound with $\Delta_0$ an embedded pair of pants on $M$, we have by definition that $$\begin{aligned} {\rm Gap}(\Delta_0;\alpha,\beta)\,i= G\left(\frac{\theta_0i}{2},\frac{|\alpha|}{2},\frac{|\beta|}{2}\right).\end{aligned}$$ For a pair of generalized simple closed geodesics $\alpha, \beta$ such that $\alpha$ is a boundary geodesic and $\beta$ is an interior generalized simple closed geodesic and that they bound with $\Delta_0$ an embedded pair of pants on $M$, we have by the analysis in §\[s:calculation\] that $$\begin{aligned} & &{\rm Gap}(\Delta_0;\alpha,\beta)\,i\\&=&2i\tan^{-1} \frac{\sin\frac{\theta_0}{2}}{\cos\frac{\theta_0}{2}+\exp\frac{|\alpha|+|\beta|}{2}} +i\tan^{-1} \frac{\sin\frac{\theta_0}{2}\sinh\frac{|\alpha|}{2}} {\cosh\frac{|\beta|}{2}+\cos\frac{\theta_0}{2}\cosh\frac{|\alpha|}{2}}\\ &=&G\left(\frac{\theta_0i}{2},\frac{|\alpha|}{2},\frac{|\beta|}{2}\right)+ S\left(\frac{\theta_0i}{2},\frac{|\alpha|}{2},\frac{|\beta|}{2}\right).\end{aligned}$$ For a pair of generalized simple closed geodesics $\alpha, \beta$ such that $\alpha$ is a cone point of angle $\varphi \in (0,\pi]$ and $\beta$ is an interior generalized simple closed geodesic and that they bound with $\Delta_0$ an embedded pair of pants on $M$, we have by (\[eqn:(xi,yi,z)\]) with $x=\theta_0/2, y=\varphi/2, z=|\beta|/2$ that $$\begin{aligned} {\rm Gap}(\Delta_0;\alpha,\beta)\,i = G\left(\frac{\theta_0i}{2},\frac{\varphi i}{2},\frac{|\beta|}{2} \right)+ S\left(\frac{\theta_0i}{2},\frac{\varphi i}{2}, \frac{|\beta|}{2} \right).\end{aligned}$$ Finally we prove the absolute convergence of the series in (\[eqn:reform of non-cusp cases with GS\]). It is not hard to see that we only need to prove, for each $j=1, \cdots, n$, the absolute convergence of the series $$\begin{aligned} \sum_{\beta} S\left(\frac{L_0}{2},\frac{L_j}{2},\frac{|\beta|}{2}\right),\end{aligned}$$ where the sum is over all interior generalized simple closed geodesics $\beta$ which bounds with $\Delta_j$ and $\Delta_0$ an embedded pair of pants on $M$. The desired absolute convergence follows from Lemma \[lem:BS\] since $$\begin{aligned} S\left(\frac{L_0}{2},\frac{L_j}{2},\frac{|\beta|}{2}\right) \sim \frac{\sinh\frac{L_0}{2}\sinh\frac{L_j}{2}}{\cosh\frac{|\beta|}{2}} \sim {\rm const.}\exp\left(-\frac{|\beta|}{2}\right)\end{aligned}$$ as $|\beta| \rightarrow \infty$. [**Geometric interpretation**]{} We would like to explore the geometric meanings of the summands in the complexified formula (\[eqn:reform of non-cusp cases with GS\]). In the case that $M$ has no cone points, all its geometric boundary components (here cusps are treated as boundary geodesics of length $0$) $\Delta_0, \Delta_1, \cdots, \Delta_n$ are boundary geodesics with hyperbolic lengths $L_0, L_1, \cdots, L_n$ respectively. Assume $\Delta_0$ is not a cusp, that is, $L_0>0$. Then as explained in §\[s:calculation\], in the first sum the summand is the width of one of the main gaps in the pair of pants $\mathcal P(\Delta_0,\alpha,\beta)$ bounded by $\Delta_0$ and $\alpha, \beta$; while in the second sum the sub-summand is the width of one of the two extra gaps associated to $\Delta_j$ in the pair of pants $\mathcal P(\Delta_0,\Delta_j,\beta)$ bounded by $\Delta_0, \Delta_j$ and $\beta$. We would like to think of the union of the two extra gaps in $\mathcal P(\Delta_0,\Delta_j,\beta)$ as the orthogonal projection of $\Delta_i$ onto $\Delta_0$ along the common perpendicular $\delta$ of $\Delta_j$ and $\Delta_0$ in $\mathcal P(\Delta_0,\Delta_j,\beta)$ and think of its width as the [*direct visual measure*]{} of $\Delta_j$ at $\Delta_0$ along $\delta$. Hence the second part of the left hand side of (\[eqn:reform of non-cusp cases with GS\]) can be thought of as the total direct visual measure of all the non-distinguished geometric boundary components $\Delta_1, \cdots, \Delta_n$ at $\Delta_0$. In the case that $\Delta_0$ is a cone point of angle $\theta_0 \in (0, \pi]$ (hence $L_0=\theta_0 i$) and all other geometric boundary components of $M$ are boundary geodesics (here cusps treated as boundary geodesics of length $0$), for each pair of generalized simple closed geodesics $\alpha, \beta$ which bound with $\Delta_0$ an embedded pair of pants $\mathcal P(\Delta_0,\alpha,\beta)$ on $M$, each of $\alpha, \beta$ has a direct visual angle at the cone point $\Delta_0$; and the summand in the first sum is $i$ times the angle measure of one of the two gaps at $\Delta_0$ between the two $\Delta_0$-geodesic rays asymptotic to $\alpha^+, \beta^-$(respectively $\alpha^-$, $\beta^+$ ). The sub-summand in the second sum is $i$ times half the visual angle measure of $\Delta_j$ at $\Delta_0$ in the pair of pants $\mathcal P(\Delta_0,\Delta_j,\beta)$ on $M$. When $M$ has cone points other than $\Delta_0$, the similar formulations of the generalized McShane’s identities (\[eqn:001\])–(\[eqn:00’\]) in terms of ${\rm Gap}(\Delta_0; \alpha, \beta)$ will not be as neat as in the above two special cases. The problem lies in that a cone point (other than $\Delta_0$) [*seems*]{} to have direct visual measure zero at $\Delta_0$, causing the formulas to be non-uniform. However, this non-uniformity is caused by the (wrong) point of view that we treat a cone point as only a point. The correct point of view is (perhaps) that a cone point (as a geometric boundary component) should be a geodesic perpendicular to the surface at the very cone point when the surface is “imagined” as lying in the hyperbolic 3-space and hence one should use purely complex length instead of real one for a cone point. (The point of view of using complex translation length for an isometry of the hyperbolic 3-space is well discussed in details in [@fenchel1989book] and [@series2001pjm].) First assume that $\Delta_0$ is boundary geodesic of length $l_0 >0$ and consider a pair of generalized simple closed geodesics $\alpha, \beta$ on $M$ such that $\alpha$ is a cone point of angle $\varphi \in (0,\pi]$ and $\beta$ is an interior generalized simple closed geodesic and that they bound with $\Delta_0$ an embedded pair of pants $\mathcal P(\Delta_0,\alpha,\beta)$ on $M$. Let the (unoriented) geodesic arc in $\mathcal P(\Delta_0,\alpha,\beta)$ which is perpendicular to $\Delta_0$ and $\alpha$ (respectively, $\alpha$ and $\beta$, $\beta$ and $\Delta_0$) be denoted $[\Delta_0,\alpha]$ (respectively, $[\alpha, \beta]$, $[\beta,\Delta_0]$). We cut $\mathcal P(\Delta_0,\alpha,\beta)$ open along $[\Delta_0,\alpha]$, $[\alpha, \beta]$, $[\beta,\Delta_0]$ to obtain two congruent pentagons; lift one of them to a pentagon ${\mathbf P}(\Delta_0,\alpha,\beta)$ in the hyperbolic plane $H^2$. Then by Fenchel [@fenchel1989book] ${\mathbf P}(\Delta_0,\alpha,\beta)$ can be regarded as a right angled hexagon ${\mathbf H}(\Delta_0,\tilde{\alpha},\beta)$ spanned by straight lines $\Delta_0,\tilde{\alpha},\beta$ in a hyperbolic 3-space $H^3$ containing the hyperbolic plane $H^2$. See Figure \[fig08\] for an illustration. Here $\tilde{\alpha}$ is the straight line in $H^3$ which passes through the cone point $\alpha$ in $H^2$ and is perpendicular to $H^2$. Let the common perpendiculars in $H^3$ between pairs of $\Delta_0,\tilde{\alpha},\beta$ be $[\Delta_0,\tilde{\alpha}]$, $[\tilde{\alpha}, \beta]$, $[\beta,\Delta_0]$, where, as straight lines $[\Delta_0,\tilde{\alpha}]$, $[\tilde{\alpha}, \beta]$ are the same as $[\Delta_0,\alpha]$, $[\alpha, \beta]$ respectively. We orient the six straight lines in the cyclic order $\Delta_0,[\Delta_0,\tilde{\alpha}],\tilde{\alpha},[\tilde{\alpha}, \beta],\beta,[\beta,\Delta_0]$ as Fenchel did in [@fenchel1989book]; see Figure \[fig08\]. Then the three oriented sides $\Delta_0,\tilde{\alpha},\beta$ of the right angled hexagon ${\mathbf H}(\Delta_0,\tilde{\alpha},\beta)$ have complex lengths $\frac{l_0}{2} + \pi i, \frac{\varphi i}{2} + \pi i, \frac{|\beta|}{2} + \pi i$ respectively. Let the ideal points which are the starting and ending endpoints of an oriented straight line ${\bf l}$ in $H^3$ be denoted ${\bf l}^{-},{\bf l}^{+}$ respectively. Then we have in $H^3$ an oriented straight line $[\Delta_0,\tilde{\alpha}^{+}]$ which intersects $\Delta_0$ perpendicularly and has $\tilde{\alpha}^{+}$ as its ending ideal point, and similarly an oriented straight line $[\Delta_0, \beta^{-}]$ which intersects $\Delta_0$ perpendicularly and has $\beta^{-}$ as its ending ideal point. Then it can be verified that $$G\left(\frac{l_0}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i\right)=G\left(\frac{l_0}{2}, \frac{\varphi i}{2}, \frac{|\beta|}{2}\right)$$ is the complex length from $[\Delta_0, \beta^{-}]$ to $[\Delta_0,\tilde{\alpha}^{+}]$ measured along $\Delta_0$ and $$S\left(\frac{l_0}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i\right)=S\left(\frac{l_0}{2}, \frac{\varphi i}{2}, \frac{|\beta|}{2}\right)$$ is the the complex length from $[\Delta_0,\tilde{\alpha}^{+}]$ to $[\Delta_0,\tilde{\alpha}]$ measured along $\Delta_0$. Note that $S(\frac{l_0}{2}, \frac{\varphi i}{2}, \frac{|\beta|}{2})$ is purely imaginary, which is obvious from its geometric meaning. We remark that it is crucial that in $G(\frac{l_0}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i)$ and $S(\frac{l_0}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i)$ the value used for $\Delta_0$ is $\frac{l_0}{2}$ instead of $\frac{l_0}{2}+\pi i$. Next assume $\Delta_0$ is a cone point of angle $\theta_0 \in (0, \pi]$ and consider a pair of generalized simple closed geodesics $\alpha, \beta$ on $M$ such that $\alpha$ is a cone point of angle $\varphi \in (0,\pi]$ and $\beta$ is an interior generalized simple closed geodesic and that they bound with $\Delta_0$ an embedded pair of pants $\mathcal P(\Delta_0,\alpha,\beta)$ on $M$. In this case we cut $\mathcal P(\Delta_0,\alpha,\beta)$ open along $[\Delta_0,\alpha]$, $[\alpha, \beta]$, $[\beta,\Delta_0]$ to obtain two congruent quadrilaterals and lift one of them to a quadrilateral ${\mathbf Q}(\Delta_0,\alpha,\beta)$ in the hyperbolic plane $H^2$. As before, let $\tilde{\alpha}$ be the straight line in $H^3$ which passes the cone point $\alpha$ in $H^2$ and is perpendicular to $H^2$. Similarly for $\tilde{\Delta_0}$. Then we obtain a right angled hexagon ${\mathbf H}(\tilde{\Delta_0},\tilde{\alpha},\beta)$ in $H^3$. Let the six sides of ${\mathbf H}(\tilde{\Delta_0},\tilde{\alpha},\beta)$ be oriented as illustrated in Figure \[fig09\]. Then the three oriented sides $\tilde{\Delta_0},\tilde{\alpha},\beta$ of the right angled hexagon ${\mathbf H}(\tilde{\Delta_0},\tilde{\alpha},\beta)$ have complex lengths $\frac{\theta_0 i}{2} + \pi i, \frac{\varphi}{2}i + \pi i, \frac{|\beta|}{2} + \pi i$ respectively. Similarly, we have in $H^3$ an oriented straight line $[\tilde{\Delta_0},\tilde{\alpha}^{+}]$ which intersects $\tilde{\Delta_0}$ perpendicularly and has $\tilde{\alpha}^{+}$ as its ending ideal point, and another oriented straight line $[\tilde{\Delta_0}, \beta^{-}]$ which intersects $\tilde{\Delta_0}$ perpendicularly and has $\beta^{-}$ as its ending ideal point. Then $$G\left(\frac{\theta_0 i}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i\right)=G\left(\frac{\theta_0 i}{2}, \frac{\varphi i}{2}, \frac{|\beta|}{2}\right)$$ is the complex length from $[\tilde{\Delta_0}, \beta^{-}]$ to $[\tilde{\Delta_0},\tilde{\alpha}^{+}]$ measured along $\tilde{\Delta_0}$ and $$S\left(\frac{\theta_0 i}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i\right)=S\left(\frac{\theta_0 i}{2}, \frac{\varphi i}{2}, \frac{|\beta|}{2}\right)$$ is the the complex length from $[\tilde{\Delta_0},\tilde{\alpha}^{+}]$ to $[\tilde{\Delta_0},\tilde{\alpha}]$ measured along $\tilde{\Delta_0}$. Note that $S(\frac{\theta_0 i}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i)$ is real, which is obvious from its geometric meaning. Here it is crucial that in $G(\frac{\theta_0 i}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i)$ and $S(\frac{\theta_0 i}{2}, \frac{\varphi i}{2}+\pi i, \frac{|\beta|}{2} + \pi i)$ the value used for $\Delta_0$ is $\frac{\theta_0 i}{2}$ instead of $\frac{\theta_0 i}{2}+\pi i$. [99]{} Hirotaka Akiyoshi, Hideki Miyachi and Makoto Sakuma, *Variations of McShane’s identity for punctured surface groups*, preprint. Hirotaka Akiyoshi, Hideki Miyachi and Makoto Sakuma, *A refinement of McShane’s identity for quasifuchsian punctured torus groups*, preprint. Ara Basmajian, *The orthogonal spectrum of a hyperbolic manifold*, Amer. J. Math. [**115**]{} (1993), no. 5, 1139–1159. Alan F. Beardon, *The Geometry of Discrete Groups*, Graduate Texts in Mathematics [**91**]{}, Springer-Verlag, New York, 1983. Joan S. Birman and Caroline M. Series, *Geodesics with bounded intersection number on surfaces are sparsely distributed*, Topology [**24**]{} (1985), no. 2, 217–225. Brian H. Bowditch, *A proof of McShane’s identity via Markoff triples*, Bull. London Math. Soc. [**28**]{} (1996), no. 1, 73–78. Brian H. Bowditch, *A variation of McShane’s identity for once-punctured torus bundles*, Topology [**36**]{} (1997), no. 2, 325–334. Brian H. Bowditch, *Markoff triples and quasi-Fuchsian groups*, Proc. London Math. Soc. (3) [**77**]{} (1998), no. 3, 697–736. Peter Buser, *Geometry and Spectra of Compact Riemann Surfaces*, Birkhäuser, Boston, 1992. Daryl Cooper, Craig D. Hodgson and Steve P. Kerckhoff, *Three-dimensional orbifolds and cone-manifolds*, MSJ Memoirs, 5, Mathematical Society of Japan, Tokyo, 2000. Werner Fenchel, *Elementary Geometry in Hyperbolic Space*, Walter de Gruyter & Co., Berlin, 1989. William M. Goldman, *The modular group action on real ${\rm SL}(2)$-characters of a one-holed torus*, Geom. Topol. [**7**]{} (2003), 443–486. Andrew Haas, *Diophantine approximation on hyperbolic Riemann surfaces*, Acta Math. [**156**]{} (1986), no. 1-2, 33–82. Andrew Haas and Perry Susskind, *The geometry of the hyperelliptic involution in genus two*, Proc. Amer. Math. Soc. [**105**]{} (1989), no. 1, 159–165. Troels J[ø]{}rgensen and Albert Marden, *Two doubly degenerate groups*, Quart. J. Math. Oxford (2) [**30**]{} (1979), no. 118, 143–156. Greg McShane, *A remarkable identity for lengths of curves*, Ph.D. Thesis, University of Warwick, 1991. Greg McShane, *Simple geodesics and a series constant over Teichmuller space*, Invent. Math. [**132**]{} (1998), no. 3, 607–632. Greg McShane, *Weierstrass points and simple geogesics*, Bull. London Math. Soc. [**36**]{} (2004), no. 2, 181–187. Greg McShane, *Length series on Teichmuller space*, arXiv:math.GT/0403041 v1. Greg McShane, *Simple geodesics on surfaces of genus 2*, preprint. Maryam Mirzakhani, *Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces*, preprint. Makoto Sakuma, *Variations of McShane’s identity for the Riley slice and 2-bridge links*, Hyperbolic spaces and related topics (Japanese) (Kyoto, 1998), Surikaisekikenkyusho Kokyuroku No. 1104 (1999), 103–108. Paul Schmutz Schaller, *Geometry of Riemann surfaces based on closed geodesics*, Bull. Amer. Math. Soc. (N.S.) [**35**]{} (1998), no. 3, 193–214. Caroline Series, *An extension of Wolpert’s derivative formula*, Pacific J. Math. [**197**]{} (2001), no. 1, 223–239. Ser Peow Tan, *Complex Fenchel–Nielsen coordinates for quasi-Fuchsian structures*, Internat. J. Math. [**5**]{} (1994), no. 2, 239–251. Ser Peow Tan, Yan Loi Wong and Ying Zhang, *Schottky groups, generalized Markoff maps and McShane’s identity*, in preparation. Ying Zhang, *Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane’s identity*, Ph.D. Thesis, National University of Singapore, July 2004.

--- abstract: 'Supplemental material for paper: xxxx.' title: | [ Supplemental Material for\ Antideuteron production in [$\Upsilon(nS)$]{}decays and in [$\epem \to \qqbar$]{}]{} --- Monmentum range () 0.35 – 0.55 0.70 0.80 0.90 1.00 1.15 1.30 1.55 2.25 ---------------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ Events in Signal Region 81.0 105.0 97.0 93.0 103.0 95.0 67.8 71.4 57.4 Branching Ratio ($\times 10^{-6} / (\gevc)$) 15.66 17.43 23.93 25.07 31.25 19.89 17.16 12.35 3.13 Statistical error (%) 10.49 9.21 9.62 9.58 8.92 9.29 18.51 10.30 40.16 Fit Biases 0.62 0.65 1.11 1.11 0.50 0.51 2.27 0.55 3.99 Background Model 0.30 0.32 0.16 0.52 0.65 0.90 0.27 0.23 7.72 Reconstruction Efficiency 2.64 2.52 3.02 4.10 4.86 6.67 2.84 10.47 6.47 Kinematic Acceptance 0.53 2.16 2.51 3.91 4.68 6.76 2.74 10.33 5.92 Material Interaction 2.79 2.98 3.26 4.30 5.04 7.01 2.79 10.48 6.65 Fake antideuterons -9.77 -2.09 -2.14 -1.66 -2.31 -3.30 -3.43 -1.73 -0.45 DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 Event Selection 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 Normalization 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 Total systematic error (%) $^{ +7.49}_{ -10.85}$ $^{ +7.82}_{ -5.62}$ $^{ +8.24}_{ -6.21}$ $^{ +9.63}_{ -7.84}$ $^{ +10.59}_{ -9.15}$ $^{ +13.46}_{ -12.57}$ $^{ +8.32}_{ -6.86}$ $^{ +19.16}_{ -18.34}$ $^{ +15.40}_{ -14.27}$ Total error (%) $^{ +12.89}_{ -15.09}$ $^{ +12.08}_{ -10.79}$ $^{ +12.67}_{ -11.45}$ $^{ +13.58}_{ -12.38}$ $^{ +13.85}_{ -12.78}$ $^{ +16.35}_{ -15.63}$ $^{ +20.29}_{ -19.74}$ $^{ +21.76}_{ -21.03}$ $^{ +43.01}_{ -42.62}$ Bin range () 0.35 – 0.60 0.70 0.80 0.90 1.00 1.10 1.25 1.45 2.25 ---------------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ Events in Signal Region 89.2 93.0 92.0 61.5 91.0 70.2 75.5 71.2 118.5 Branching Ratio ($\times 10^{-6} / (\gevc)$) 12.63 19.18 20.42 13.80 24.14 19.39 13.68 10.34 5.78 Statistical error (%) 56.04 11.25 11.27 29.51 10.48 18.83 23.69 19.63 18.46 Fit Biases 2.78 0.10 0.70 9.57 0.64 3.41 6.55 5.31 4.89 Background Model 3.09 3.40 3.57 5.61 7.44 5.35 12.45 11.74 6.74 Reconstruction Efficiency 5.19 4.22 3.41 11.66 7.78 9.34 16.93 16.99 7.07 Kinematic Acceptance 4.42 3.62 3.59 10.82 7.34 9.38 16.34 16.39 4.85 Material Interaction 5.64 4.71 4.27 11.43 7.64 9.45 16.78 16.45 4.92 Fake antideuterons -2.90 -1.51 -1.07 -1.62 -1.99 -1.59 -2.94 -2.04 -0.97 DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 Event Selection 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 Normalization 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 Total systematic error (%) $^{ +11.66}_{ -10.52}$ $^{ +10.26}_{ -8.58}$ $^{ +9.83}_{ -7.99}$ $^{ +23.40}_{ -22.72}$ $^{ +16.41}_{ -15.47}$ $^{ +18.58}_{ -17.72}$ $^{ +32.77}_{ -32.38}$ $^{ +32.17}_{ -31.70}$ $^{ +14.41}_{ -13.22}$ Total error (%) $^{ +57.24}_{ -57.02}$ $^{ +15.23}_{ -14.15}$ $^{ +14.95}_{ -13.81}$ $^{ +37.66}_{ -37.24}$ $^{ +19.47}_{ -18.69}$ $^{ +26.46}_{ -25.86}$ $^{ +40.44}_{ -40.12}$ $^{ +37.68}_{ -37.29}$ $^{ +23.42}_{ -22.71}$ Bin range () 0.35 – 0.65 0.85 1.00 1.20 2.25 ---------------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ Events in Signal Region 11.6 22.6 18.9 18.3 19.5 Branching Ratio ($\times 10^{-6} / (\gevc)$) 12.09 27.73 34.22 27.98 6.00 Statistical error (%) 54.14 28.75 34.42 30.55 40.43 Fit Biases 1.16 0.07 0.22 0.31 2.03 Background Model 7.58 0.86 2.22 1.65 3.83 Reconstruction Efficiency 7.08 1.68 3.57 1.28 3.23 Kinematic Acceptance 2.36 0.59 2.49 0.75 2.90 Material Interaction 7.27 2.47 2.84 1.95 2.97 Fake antideuterons -32.00 -1.90 -5.52 -3.46 -4.85 DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 Event Selection 1.11 1.11 1.11 1.11 1.11 Normalization 0.24 0.24 0.24 0.24 0.24 Total systematic error (%) $^{ +14.22}_{ -34.53}$ $^{ +6.72}_{ -3.86}$ $^{ +8.20}_{ -7.99}$ $^{ +6.63}_{ -4.70}$ $^{ +9.04}_{ -8.44}$ Total error (%) $^{ +55.97}_{ -64.21}$ $^{ +29.52}_{ -29.01}$ $^{ +35.39}_{ -35.34}$ $^{ +31.26}_{ -30.91}$ $^{ +41.42}_{ -41.30}$ Bin range () 0.35 – 0.60 0.75 0.85 0.95 1.05 1.25 1.40 1.65 2.25 ---------------------------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- Events in Signal Region -25.5 1.20 -10.1 -16.1 -22.5 -27.9 -11.5 27.7 39.7 Branching Ratio ($\times 10^{-6} / (\gevc)$) 1.17 -1.06 -2.42 -3.62 -3.79 -1.69 -2.05 0.39 1.72 Statistical error (%) 161.00 222.17 142.97 100.48 105.69 149.04 150.48 445.82 556.71 Fit Biases 4.64 16.06 10.40 4.72 4.53 6.56 2.38 32.11 0.35 Background Model 4.06 14.91 7.49 3.85 0.70 6.81 8.98 2.12 1.56 Reconstruction Efficiency 71.78 122.04 86.46 61.57 57.38 62.42 62.28 187.82 3.35 Kinematic Acceptance 68.12 120.62 84.95 62.45 56.45 61.11 60.97 187.15 3.25 Material Interaction 74.08 168.33 89.82 64.72 59.46 63.95 64.36 198.58 3.51 Fake antideuterons -1.02 -3.91 -1.96 -1.90 -1.99 -6.13 -5.60 -16.17 -0.61 DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 Event Selection 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 Normalization 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 Total systematic error (%) $^{ +123.93}_{ -123.79}$ $^{ +241.45}_{ -241.41}$ $^{ +151.54}_{ -151.44}$ $^{ +109.34}_{ -109.21}$ $^{ +100.37}_{ -100.23}$ $^{ +108.85}_{ -108.87}$ $^{ +108.92}_{ -108.91}$ $^{ +332.88}_{ -333.22}$ $^{ +8.73}_{ -6.53}$ Total error (%) $^{ +203.17}_{ -203.09}$ $^{ +328.11}_{ -328.08}$ $^{ +208.34}_{ -208.27}$ $^{ +148.50}_{ -148.40}$ $^{ +145.76}_{ -145.66}$ $^{ +184.56}_{ -184.57}$ $^{ +185.77}_{ -185.76}$ $^{ +556.39}_{ -556.59}$ $^{ +556.78}_{ -556.75}$ Bin range () 0.35 – 0.60 0.75 0.85 0.95 1.05 1.25 1.40 1.65 2.25 --------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ Events in Signal Region 157 130 108 102 117 176 128 178 308 Cross Section ($0.1 \rm{fb}/(\gevc)$) 66.46 54.13 72.22 71.70 79.84 64.24 66.38 47.28 18.97 Statistical error (%) 9.05 9.97 11.16 11.28 10.63 8.34 10.05 9.44 51.94 Fit Biases 0.05 0.05 0.18 0.07 0.02 0.02 0.06 0.00 0.19 Background Model 0.22 8.86 5.05 3.61 6.91 6.43 5.23 6.40 0.04 Reconstruction Efficiency 3.77 6.28 6.95 7.34 6.46 4.22 5.35 4.88 3.02 Kinematic Acceptance 1.47 6.17 8.32 8.40 7.69 5.93 4.08 1.37 4.17 Material Interaction 4.57 6.71 7.29 7.36 6.80 4.78 5.47 4.80 2.88 Fake antideuterons -0.56 -2.37 -2.04 -2.98 -2.93 -5.01 -5.37 -4.15 -1.73 DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 Event Selection 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 Normalization 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 Total systematic error (%) $^{ +9.64}_{ -7.70}$ $^{ +16.02}_{ -15.11}$ $^{ +15.86}_{ -14.90}$ $^{ +15.72}_{ -14.91}$ $^{ +15.83}_{ -15.00}$ $^{ +13.15}_{ -12.81}$ $^{ +12.57}_{ -12.37}$ $^{ +12.05}_{ -11.34}$ $^{ +9.50}_{ -7.71}$ Total error (%) $^{ +13.22}_{ -11.88}$ $^{ +18.87}_{ -18.10}$ $^{ +19.40}_{ -18.62}$ $^{ +19.35}_{ -18.69}$ $^{ +19.06}_{ -18.39}$ $^{ +15.57}_{ -15.28}$ $^{ +16.09}_{ -15.94}$ $^{ +15.31}_{ -14.75}$ $^{ +52.80}_{ -52.51}$

--- author: - 'Vladimir B.Kopeliovich' - 'Andrei M.Shunderuk' title: Strange and Heavy Flavoured Hypernuclei in Chiral Soliton Models --- Main features of the chiral soliton approach. ============================================== The chiral soliton approach (CSA) is based on few principles and ingredients incorporated in the truncated [*effective chiral lagrangian*]{}: $$\begin{aligned} L^{eff} &=& -{F_\pi^2\over 16}Tr\,(l_\mu l_\mu) + {1\over 32e^2}Tr [l_\mu l_\nu]^2+ \\ \nonumber &+& {F_\pi^2m_\pi^2\over 16}Tr \big(U+U^\dagger -2\big)+...,\end{aligned}$$ $l_\mu = \partial_\mu U U^\dagger,$ $U\in SU(2)$ or $U\in SU(3)$- unitary matrix depending on chiral fields, $m_\pi$ is the pion mass, $F_\pi$-pion decay constant, $e$ - the only parameter of the model. The soliton (skyrmion) is coherent configuration of classical chiral fields, possessing topological charge (or winding number) identified with the baryon number $B$ (Skyrme, 1961). Important simplifying feature of this approach is that configurations with different baryon, or atomic numbers are considered on equal footing, when zero modes only are taken into account in the quantization procedure. Another feature is that baryons individuality is absent within the multiskyrmion, and can be recovered - as it is believed - due to careful consideration of the nonzero modes. The observed spectrum of baryon states is obtained by means of quantization procedure and depends on their quantum numbers (isospin, strangeness, etc) and static characteristics of classical configurations. For the $B=1$ case this was made first in the paper [@anw]. Masses, binding energies of classical configurations with baryon number $B\geq 2$, their moments of inertia $\Theta_I,\;\Theta_J$, $\Sigma$-term ($\Gamma$), and some other characteristics of the chiral solitons contain implicitly information about interaction between baryons. They are obtained usually numerically and depend on parameters of the model $F_\pi,\,e$ and masses of mesons which enter the mass term in the effective lagrangian [^1]. Ordinary ($S=0$) nuclei; symmetry energy as quantum correction ============================================================== In the $SU(2)$ case, which is relevant for description of nonstrange baryons and nuclei, the rigid rotator quantization model is most adequate when quantum corrections are not too large. The symmetry energy $E_{sym}=b_{sym}(N-Z)^2/(2A)$, $b_{sym}\simeq 50\,MeV$, within chiral soliton approach is described mainly by the isospin dependent quantum correction $$\delta E_I = {I(I+1)\over 2 \Theta_I},$$ $\Theta_I \sim A$ being isotopical moment of inertia, $I=(N-Z)/2$ for the ground states of nuclei. The $SU(2)$ quantization method - simplest and most reliable - is used here according to [@anw]. The moment of inertia $\Theta_I$ grows not only with increasing number of colours, but also with increasing baryon number ($\sim B$ approximately), therefore this correction decreases like $\sim 1/B$ and such estimates become more selfconsistent for larger $B$. In [**Fig.**]{}\[fig:1\] the differences of binding energies between states with integer isospins are shown, for the even-even or for the odd-odd nuclei (e.g. such differences for the $I=0$ and $I=2$ states or for the $I=1$ and $I=3$), calculated with the help of formula $(2)$ and for the value of the model parameter $e=3.0$ (for the $B=1$ case the value of $e$ is taken usually close to $e\simeq 4.1$ which allowed to describe the mass splittings of baryons octet and decuplet). Many uncertainties and some specific corrections introduced in the nuclear mass formula are cancelled in such differences (see [@ksm] for details and references). Similar differences for the odd-even or even-odd nuclei with half-integer isospins are shown in [**Fig.**]{}\[fig:2\]. The differences of binding energies of other nuclei, besides those shown in [**Fig.**]{}\[fig:1\] and [**Fig.**]{}\[fig:2\], are also described well, see [@ksm]. The change of the model parameter $e$ is a natural way to take into account effectively the nonzero modes - breathing, vibration - which lead to the increase of dimensions of multiskyrmions (the natural unity of length in the model is $\sim 1/(F_\pi e)$). Recently similar procedure has been used for description of the $^6Li$ nucleus [@mw] properties. The change of the pion decay constant $F_\pi$, also made in [@mw], is much more limited since it is directly measurable quantity, via pion decay. The mass and baryon number distributions of multiskyrmions have shell-like form [@basu], at variance from the real ones. However, skyrmions are easily deformable objects, as previous experience has shown [@zks] and recently has been observed, e.g. for the $B=7$ multiskyrmion [@mm]. Therefore, one can hope that transition to realistic shape of the mass distribution could proceed without large increase of the energy. Moreover, the important result obtained recently numerically by Battye, Manton and Sutcliffe [@bms] is that at large baryon numbers and large enough value of the chiral symmetry breaking mass in the lagrangian $(1)$ the transition to more realistic alpha-cluster shape takes place. The quantum correction due to collective rotation of the multiskyrmion in usual space equals to $\delta E_J= J(J+1)/(2\Theta_J)$, where $J$ is spin of the nucleus. It is technically complicated problem to define allowed values of $J$ for the classical configuration with definite symmetry properties. The experimentally observed value of spin of the nucleus’ ground state not always can be obtained when quantization of the lowest in energy classical configuration is made [@irwin; @krusch]. However, due to rich landscape of the classical energy local minima with the energy not much different from the lowest one, but different symmetry properties of the chiral field configurations[@basu], quantization of one of them could give the desired value of spin (the $B=7$ case has been considered recently in details in [@mm]), and we make here in fact a natural assumption that it is always possible. Since the orbital inertia $\Theta_J$ is considerably greater than isotopical one $\Theta_I$ ($\Theta_J \geq B\Theta_I$), this correction is not important for large enough baryon numbers and is not included in [@ksm] and here. The success of the CSA in description the differences of binding energies of known nuclei allows to make predictions for the binding energies of still unknown neutron-rich nuclides (some examples were considered in [@ksm]) and to go further to the consideration of different kinds of hypernuclei. Strange hypernuclei ($S=-1$); binding energies of ground states =============================================================== $_\Lambda A$ $\omega_s$ $\Delta \epsilon_s $ $\epsilon^{tot}_s$ $\epsilon^{tot}_{exp,s}$ $\omega_b^{r_b=1.5}$ $\Delta \epsilon_b$ $\epsilon^{tot}_b$ $\omega_b^{r_b=2}$ $\Delta \epsilon_b$ $\epsilon^{tot}_b$ ------------------- ------------ ---------------------- -------------------- -------------------------- ---------------------- --------------------- -------------------- -------------------- --------------------- -------------------- $1$ $306$ — — — $4501$ — — $4805$ — — $^3_\Lambda H$ $ 289$ $\;-3$ $\;5$ $2.35$ $4424$ $75$ $83$ $4751$ $53$ $61$ $^5_\Lambda He$ $ 287$ $\;-6$ $33 $ $31.4$ $4422$ $76$ $103$ $4749$ $54$ $81$ $^7_\Lambda Li$ $282 $ $\;-3$ $29$ $37.2$ $4429$ $81$ $119$ $4744$ $59$ $97$ $^9_\Lambda Be$ $291 $ $-13$ $40$ $62.5$ $4459$ $40$ $97$ $4773$ $31$ $88$ $^{11}_\Lambda B$ $294 $ $-16$ $59$ — $4478$ $21$ $96$ $4786$ $18$ $93$ $^{13}_\Lambda C$ $295$ $-18$ $78$ $104$ $4488$ $10$ $106$ $4793$ $11$ $107$ $_\Lambda A$ $\omega_s$ $\Delta\epsilon_s$ $\epsilon^{tot}_s$ $\epsilon^{tot}_{exp}$ $\omega_b^{r_b=2}$ $\Delta \epsilon_b$ $ \epsilon^{tot}_b$ ------------------------------------- ------------ -------------------- -------------------- ------------------------ -------------------- --------------------- --------------------- $^4_\Lambda H - ^4_\Lambda He$ $283$ $-23$ $5.3$ $10.5;\;\;10.1$ $4735$ $52$ $80$ $^6_\Lambda He - ^6_\Lambda Li$ $287$ $-22$ $10.3$ $31.7;\;\;30.8$ $4752$ $40$ $72$ $^8_\Lambda Li - ^8_\Lambda Be$ $288 $ $-20$ $36.5$ $46.0;\;\;44.4$ $4765$ $33$ $89$ $^{10}_\Lambda Be -^{10}_\Lambda B$ $292$ $-23$ $42$ $67.3;\;\;65.4$ $4778$ $20$ $85$ $^{12}_\Lambda B - ^{12}_\Lambda C$ $294$ $-24$ $67$ $87.6;\;\; 84.2$ $4788$ $11$ $103$ In the $SU(3)$ case invoking strangeness (or charm, beauty) the flavour symmetry breaking terms in the lagrangian $$\begin{aligned} & &L_{FSB}= \\ \nonumber &=&-{F_K^2m_K^2 - F_\pi^2m_\pi^2\over 24} Tr\left[\left(1-\sqrt 3 \,\lambda_8\right)\left(U+U^\dagger -2\right)\right]+...\end{aligned}$$ play the crucial role in calculating the spectrum of states with different flavours (strangeness first of all). Some terms proportional to the difference $F_K^2-F_\pi^2$ are omitted here (see [@kw; @vkh; @ksh] for details and references). Different quantization schemes have been used in literature: rigid rotator, soft rotator or bound state model. (5,6) (3,1.5)[(1,0)[2.5]{}]{} (3,1.5)[(0,1)[3.8]{}]{} (2.8,5.2)[$Y$]{} (5.5,1.2)[**$I_3$**]{} (1.5,5.7)[ a) $Odd \;B\,,\; J=1/2$]{} (2.3,4.2)[$^3 H$]{} (3.3,4.2)[$^3 He$]{} (2.8,3.3)[$^3_\Lambda H$]{} (2.5,4) (3.5,4) (2,3) (3,3) (3,3) (4,3) (1.5,2) (2.5,2) (3.5,2) (4.5,2) (5,6) (3,1.5)[(1,0)[2.3]{}]{} (3,1.5)[(0,1)[3.8]{}]{} (2.8,5.2)[$Y$]{} (5,1.2)[$I_3$]{} (1.5,5.7)[ b) $Even \;B\,,\; J=0$]{} (2.8,4.2)[$^4 He $]{} (2.3,3.3)[$^4_\Lambda H$]{} (3.3,3.3)[$^4_\Lambda He$]{} (3,4) (2.5,3) (2.5,3) (3.5,3) (3.5,3) (2,2) (3,2) (4,2) The version of the bound state soliton model proposed by Callan, Klebanov, Westerberg (1985 - 1996) and modified for the flavour symmetry breaking case $(F_K > F_\pi)$ allows to calculate the binding energy differences of ground states between flavoured and unflavoured nuclei. Combined with few phenomenological arguments it is very successful in some cases of light hypernuclei. Within bound state model (BSM) [@kw] (see also [@vkh; @ksh] for details and [**Fig.3**]{} where location of strange baryon states within minimal $SU(3)$ multiplets is shown) $$M = M_{cl} + \omega_F + \omega_{\bar F} + |F| \omega_F + \Delta M_{HFS}$$ where flavour and antiflavour excitation energies $$\omega_F= N_cB(\mu-1)/8\Theta_F,\;\;\omega_{\bar F}= N_cB(\mu+1)/8\Theta_F,$$ $\mu \simeq \sqrt{1+\bar m_F^2/M_0^2},\; \bar m_K^2=m_K^2F_K^2/F_\pi^2-m_\pi^2$, and similar for $D$ or $B$ mesons, $M_0^2\simeq N_c^2B^2/(16\Gamma\Theta_F)$, $\Theta_F$ (or $\Theta_K$), is the so called flavour moment of inertia for rotation of the skyrmion to “flavoured direction” - strange, or charmed, etc., the number of colours $N_c=3$ in all realistic calculations. The hyperfine splitting correction $\Delta M_{HFS}$ for the cases we consider here can be written in the form $$\begin{aligned} \Delta M_{HFS} &=& {J(J+1)\over 2\Theta_I} + (\bar c_F-c_F)\frac{ I_F(I_F+1)}{2\Theta_I}\\ \nonumber &+&(c_F-1)\frac{[I_r(I_r+1)-I(I+1)]}{2\Theta_I},\end{aligned}$$ with $I_F=|F|/2=1/2$, and the hyperfine splitting constants $c_F$ and $\bar c_F$ given by $$c_F =1-{\Theta_I\over 2\mu\Theta_F}(\mu -1); \qquad \bar c_F =1-{\Theta_I\over \mu^2\Theta_F}(\mu -1).$$ There is general qualitative agreement with data in the behaviour of the calculated binding energy of the ground states of $S=-1$ hypernucle1 with increasing atomic number, as can be seen from Tables 1,2 and [**Fig.**]{}4, but the binding energy is underestimated in most of cases. The tendency of decrease of binding energies with increasing $B$-number, beginning with $B\sim 10$, is connected with the fact that the rational map approximation, leading to the one-shell structure of the classical configuration, is not good for such values of $B$. Binding energies of charmed or beautiful hypernuclei ==================================================== Same method can be applied for the prediction of the binding energies of charmed and beautiful hypernuclei [@vkh]. Evident replacements should be made, $m_K\to m_D$ or $m_B$ and $F_K\to F_D$ or $F_B$. For beautiful hypernuclei the binding energies are presented in Table 1 (isoscalar states), where the first 3 columns correspond to $r_b=F_B/F_\pi =1.5$, and the last 3 - to $r_b = 2$, and Table 2 (isodoublets, $r_b=2$). $_\Lambda A$ $\omega_c^{r_c=1.5}$ $\Delta \epsilon_c$ $\epsilon^{tot}_c$ $\omega_c^{r_c=2}$ $\Delta \epsilon_c$ $\epsilon^{tot}_c$ ------------------- ---------------------- --------------------- -------------------- -------------------- --------------------- -------------------- $1$ $1535$ $-$ $-$ $1673$ $-$ $-$ $^3_\Lambda He$ $1504$ $27$ $35$ $1647$ $24$ $32$ $^5_\Lambda Li$ $1505$ $25$ $52$ $1646$ $25$ $52$ $^7_\Lambda Be$ $1497$ $32$ $70$ $1641$ $30$ $68$ $^9_\Lambda B$ $1518$ $11$ $68$ $1654$ $17$ $74$ $^{11}_\Lambda C$ $1525$ $\;4$ $79$ $1658$ $13$ $87$ $^{13}_\Lambda N$ $1529$ $\;0$ $96$ $1660$ $10$ $106$ : The binding energies of the charmed hypernuclei, isoscalars and isodoublets, with unit charm, $c=1$. $\Delta \epsilon_c$ and $\epsilon^{tot}$ (both in $Mev$) are the same as in [**Tables 1,2**]{}, for the charm quantum number. The results are shown for two values of charm decay constant $F_D$, corresponding to $r_c=1.5$ and $r_c=2$ (the last $3$ columns). The chemical symbol is ascribed to the nucleus according to its total electric charge.[]{data-label="table:3"} \ $_\Lambda A$ $\omega_c^{r_c=1.5}$ $\Delta \epsilon_c$ $\epsilon^{tot}_c$ $\omega_c^{r_c=2}$ $\Delta \epsilon_c$ $\epsilon^{tot}_c$ ------------------------------------- ---------------------- --------------------- -------------------- -------------------- --------------------- -------------------- $^4_\Lambda He - ^4_\Lambda Li$ $1493$ $12$ $40$ $1639$ $16$ $44$ $^6_\Lambda Li - ^6_\Lambda Be$ $1504$ $\;9$ $41$ $1646$ $14$ $46$ $^8_\Lambda Be - ^8_\Lambda B$ $1510$ $\;7$ $63$ $1648$ $15$ $71$ $^{10}_\Lambda B -^{10}_\Lambda C$ $1520$ $\;0$ $65$ $1655$ $10$ $75$ $^{12}_\Lambda C - ^{12}_\Lambda N$ $1526$ $-4$ $88$ $1659$ $\;7$ $99$ : The binding energies of the charmed hypernuclei, isoscalars and isodoublets, with unit charm, $c=1$. $\Delta \epsilon_c$ and $\epsilon^{tot}$ (both in $Mev$) are the same as in [**Tables 1,2**]{}, for the charm quantum number. The results are shown for two values of charm decay constant $F_D$, corresponding to $r_c=1.5$ and $r_c=2$ (the last $3$ columns). The chemical symbol is ascribed to the nucleus according to its total electric charge.[]{data-label="table:3"} For charmed hypernuclei the binding energies are presented in Table 3 [@vkh]. Without any new parameters, beautiful (or charmed) hypernuclei are predicted to be bound stronger than strange hypernuclei. Their binding energies only slightly depend on the poorly known values of the decay constants $F_D$ or $F_B$ [@vkh]. There is rough agreement of our results with some early estimates made within conventional (potential) approach first by C.Dover and S.Kahana [@dk] and later by several authors [@bb; @st]. The model we used overestimates the flavour excitation energies, especially for strangeness, but is more reliable for differences of energies which contribute to the differences of binding energies we calculate here, and for charm or beauty quantum numbers . The binding energies of states with flavour quantum numbers $|F|=2$ or greater have been estimated roughly in [@kz]. Theta-hypernuclei, strange, beautiful or charmed ================================================ Situation with observation of exotic baryons remains to be somewhat contradictive (see, in particular, the talk by K.Hicks at present conference). Apparently, experimental methods of observation of relatively narrow resonances, with a width about $1 MeV$ or smaller, need further development. Same approach as in previous sections can be applied for the estimates of the binding energies of Theta- hypernuclei. For anti-flavour (positive strangeness, beauty or negative charm) the same formula as above holds , but with certain changes for the hyperfine splitting constants, $c_F \to c_{\bar F}$ and $\bar c_F \to \bar c_{\bar F}$ in the last term $\Delta M_{HFS} $. $c_{\bar F}$ ($\bar c_{\bar F}$) is obtained from $c_F$ ($\bar c_F$) by means of substitution $\mu\to -\mu$: $$c_{\bar F} =1-{\Theta_I\over 2\mu \Theta_F}(\mu +1); \qquad \bar c_{\bar F} =1+{\Theta_I\over \mu^2\Theta_F}(\mu +1).$$ This change is crucially important for the link between rotator and bound state models of the $SU(3)$ quantization [@ksh2], but often it was not made in the literature. The mass of the $\Theta^+$ hyperon within this approach equals to about $1570\,MeV$ ($e=4.1, \; F_K/F_\pi =1.22$). $ A$ $\bar \omega_s$ $ \epsilon^{tot}_s $ $\bar \omega_c$ $\epsilon^{tot}_c$ $\bar \omega_b$ $\epsilon^{tot}_b$ ------ ----------------- ---------------------- ----------------- -------------------- ----------------- -------------------- $1$ $591$ — $1750$ — $4940$ — $3$ $564$ $76$ $1710$ $46$ $4890$ $57$ $5$ $558$ $108$ $1710$ $71$ $4880$ $82$ $7$ $559$ $120$ $1710$ $85$ $4880$ $100$ $9$ $550$ $152$ $1710$ $100$ $4900$ $100$ $11$ $547$ $173$ $1710$ $115$ $4900$ $110$ $13$ $546$ $196$ $1720$ $125$ $4910$ $120$ : The collective motion contributions to the binding energies of the Theta-hypernuclei with unit flavour, strangeness, charm or beauty, $S=+1$, $c=-1$ and $b=+1$. $\bar \omega_{s,c,b}$, in $Mev$, are the antiflavour excitation energies, $\epsilon^{tot}$ is the total binding energy of the ground state of hypernucleus with $|F|=1$. For charm $r_c=1.5$, for beauty $r_b = 2$.[]{data-label="table:4"} As can be seen from Table 4, presenting some of the results obtained in [@ksh], the binding energies for Theta-hypernuclei increase with increasing atomic number. If the $\Theta^+$ pentaquark is not narrow and has the width of several tens of $MeV$, as argued in [@ww], or even greater , the Theta-hypernucleus can have much smaller width, and even be bound relative to the strong interactions [^2]. For rescaled (nuclear) variant of the model with smaller value of the parameter $e$, which should be applied for larger atomic numbers, the binding energies of hypernuclei are greater. In view of these results being in qualitative agreement with more conventional approaches [@miller; @cabrera; @zhong], searches for such hypernuclear states are of interest. Conclusions =========== Chiral soliton models, based on few principles and ingredients incorporated in the effective lagrangian, allow to describe qualitatively, in some cases quantitatively, various chracteristics of nuclei spectra - from ordinary $(S=0)$ nuclei to known light hypernuclei. The symmetry energy of nuclei with isospin up to 4 or 9/2 is described for atomic numbers between 10 and 30 with only one fixed semifree parameter - Skyrme constant $e$. The binding energies of the ground states of strange hypernuclei have been described in qualitative and in some cases quantitative agreement with data for atomic numbers up to $\sim 15$. In view of this success, predictions of CSM are of interest, including the well bound heavy flavoured (charmed, beautiful) hypernuclei and so called Theta-hypernuclei, i.e. multibaryon states with positive strangeness or beauty, or negative charm. Essential advantage of this approach is that the case of $B>1$, within CSA, does not differ in principle from the case of baryons, until the nonzero modes are included into consideration. There are some obvious drawbacks of this approach, as continuation of this advantage. In particular, one-, two-, etc. baryons excitations are not included - this is related to a very complicated problem of detailed study of nonzero modes of multiskyrmions. Specific and partly technical problem is also a smooth transition from the $B=1$ to rescaled “nuclear variant” of the model with smaller value of the parameter $e$. Some scepticism concerning validity of the CSA - partly because of the unconfirmed narrow pentaquarks states - has no firm grounds. Still, the chiral soliton approach is not the complete theory (of course!), but may carry some important features of the true theory. The work is supported partly by the RFBR grant 07-02-00960-a.\ [25]{} G. Adkins, C. Nappi and E. Witten, Nucl. Phys. B228, 552 (1983) V. Kopeliovich, A. Shunderuk and G. Matushko, Phys. Atom. Nucl. 69, 120 (2006) N. Manton and S. Wood, Phys.Rev. D74, 125017 (2006) R.A. Battye and P.M. Sutcliffe, Rev. Math. Phys. 14, 29 (2002) S.V. Zenkin et al, Sov.J.Nucl.Phys. 45, 106 (1987) O. Manko and N. Manton, hep-th/0611179 R. Battye, N. Manton and P. Sutcliffe, Proc. Roy. Soc. Lond. A463, 261 (2007) P. Irwin, Phys.Rev. D61, 114024 (2000) S. Krusch, Annals Phys. 304, 103 (2003); Proc. Roy. Soc.Lond. A462, 2001 (2006); hep-th/0610176 I.R. Klebanov and K.M. Westerberg, Phys.Rev. D53, 2804 (1996); ibid. D50, 5834 (2004) V.B. Kopeliovich, JETP 96, 782 (2003); ibid. 93, 435 (2001); Nucl. Phys. A721, 1007c (2003) V.B. Kopeliovich and A.M. Shunderuk, JETP 100, 929 (2005) H. Bando, T. Motoba and J.Zofka, Int.J.Mod.Phys. 21, 4021 (1990) O. Hashimoto and H. Tamura, Prog.Part.Nucl.Phys. 57, 564 (2006) C. Dover and S. Kahana, Phys.Rev.Lett. 39, 1506 (1977) H. Bando and M. Bando, Phys.Lett. B109, 164 (1982); B. Gibson et al, Phys. Rev. C27, 2085 (1983), N. Starkov and V. Tsarev, Nucl.Phys. A450, 507 (1990) S. Bunyatov et al, Sov. J. Nucl. Phys. 23, 253 (1992) V.B. Kopeliovich and W.J. Zakrzewski, JETP Lett. 69, 721 (1999); Eur.Phys.J. C18, 369 (2000) H. Walliser and H. Weigel, Eur.Phys.J. A26, 361 (2005); H.Weigel, hep-ph/0610123 (2006) V.B. Kopeliovich and A.M. Shunderuk, Phys.Rev. D73, 094018 (2006) G.A. Miller, Phys.Rev. C70, 022202 (2004) D. Cabrera et al, Phys.Lett. B608, 231 (2005); E. Oset et al, Nucl.Phys. A755, 503 (2005) X.H. Zhong et al, Phys.Rev. C71, 015206 (2005); ibid. C72, 065212 (2005) [^1]: It is of interest that baryon interaction potentials depend on the weak decay constant $F_\pi$ and Skyrme parameter $e$. This connection of weak and strong interaction properties apparently needs deeper understanding. [^2]: It should be noted that there is a distinction between different quantization schemes in the next-to leading order of the $1/N_c$ expansion for the flavour symmetry breaking terms in the spectrum of baryon states [@ksh2] - a problem not resolved yet consistently. The same holds for the widths of baryon resonances, see also [@ww].

--- address: Unlisted author: - Milad Olia Hashemi bibliography: - 'diss.bib' title: | On-Chip Mechanisms\ to Reduce Effective Memory Access Latency --- Only a fraction of the work that has allowed me to write this dissertation is my own. I can’t imagine the strength that it must’ve taken my parents to immigrate to a new and unfamiliar country with no resources and then raise two kids. They always prioritized our education over any of their own needs. I was only able to write this dissertation because of their sacrifice. I thank my parents: Homa and Mohammad, and my sister Misha for their unwavering love. I would never have pursued a Ph.D. or arrived at UT without my wife, Kelley. Well before I had any idea, she knew that I wouldn’t be happy leaving graduate school without a doctorate. I thank her for her advice, clairvoyance, and patience throughout these last seven years. My time in graduate school has allowed me to meet and work with amazing people who have taught me far more than I could list here. This starts with my advisor, Professor Yale N. Patt. Despite his accomplishments, Professor Patt maintains a contagious passion for both teaching and research. He’s taught me how to learn, how to ask questions, how to attack problems, and tried to teach me how to share knowledge with others. I’m still not sure why he agreed to let me join HPS, but it’s one of the pivotal moments of my life. It’s an honor to be counted as a member of his research group, I thank him for his faith in me. I’ve had the incredible opportunity of learning from Doug Carmean for the past five years. Technically, Doug has taught me how to pay attention to details and more importantly, how to listen to everybody and not allow preconceptions to color your opinion of what they’re saying. Beyond work, Doug is one of the kindest people that I know and he has impacted my life in more ways than I can count. I thank him for putting up with my constant pestering and being so open with me when he had no reason to be. I’d like to thank Professor Derek Chiou, Professor Mattan Erez, and Professor Don Fussell for serving on my committee. Professor Erez and Professor Chiou are instrumental to my success at UT. The wealth of knowledge that they’ve shared with me has given me the foundation to work in computer architecture and motivated me to want to work in this field. The university is lucky to have such amiable and brilliant individuals. Many of the research directions that I’ve worked on have come as a result of discussions with Professor Onur Mutlu. Professor Mutlu is an incredibly motivational, hardworking, and intelligent person. I’d like to thank him for teaching me how to never be satisfied with the work that I’ve done, how to continuously strive for more, and for pushing me to not give up when things didn’t go my way. I’d like to thank him and Professor Moinuddin Qureshi for their advice, research discussions, and for always treating me like one of their own family. I wouldn’t have joined HPS without Eiman Ebrahimi. Eiman was the first person to teach me how to do research and how to strive towards writing high-quality papers. I’d like to thank him for his advice and support throughout my time at UT. I’d like to thank Carlos Villavieja for putting up with an obstinate young graduate student and showing him how to grow both as a person and a researcher. I’d also like to thank him for proof-reading this entire dissertation. I’d like to thank the entire HPS research group while I’ve been at UT, and in particular Khubaib for always being eager to talk about research, José Joao for his guidance and maintaining our computing systems, Rustam Miftakhutdinov for the insane amount of work that he put into our simulation infrastructure, and Faruk Guvenilir for maintaining our computing systems after José and for completing countless miscellaneous tasks without complaint. Finally, I’d like to thank my friends: Will Diel, Curtis Hickmott, Zack Smith, Trevor Kilgannon, and David Cate for keeping me sane for over a decade now. Milad Hashemi\ August 2016, Austin, TX This dissertation develops hardware that automatically reduces the effective latency of accessing memory in both single-core and multi-core systems. To accomplish this, the dissertation shows that all last level cache misses can be separated into two categories: dependent cache misses and independent cache misses. Independent cache misses have all of the source data that is required to generate the address of the memory access available on-chip, while dependent cache misses depend on data that is located off-chip. This dissertation proposes that dependent cache misses are accelerated by migrating the dependence chain that generates the address of the memory access to the memory controller for execution. Independent cache misses are accelerated using a new mode for runahead execution that only executes filtered dependence chains. With these mechanisms, this dissertation demonstrates a 62% increase in performance and a 19% decrease in effective memory access latency for a quad-core processor on a set of high memory intensity workloads.

--- abstract: 'We describe practical approaches to measuring flexion in observed galaxies. In particular, we look at the issues involved in using the Shapelets and HOLICs techniques as means of extracting 2nd order lensing information. We also develop an extension of HOLICs to estimate flexion in the presence of noise, and with a nearly isotropic PSF. We test both approaches in simple simulated lenses as well as a sample of possible background sources from ACS observations of A1689. We find that because noise is weighted differently in shapelets and HOLICs approaches, that the correlation between measurements of the same object is somewhat diminished, but produce similar scatter due to measurement noise.' author: - 'David M. Goldberg and Adrienne Leonard' title: Measuring Flexion --- Introduction ============ Motivation ---------- Flexion has recently been introduced as a means of measuring small scale variations in weak gravitational lens fields (Goldberg & Bacon, 2005; Bacon, Goldberg, Rowe & Taylor, 2006, hereafter BGRT). Rather than simply measuring the ellipticities of arclets, this technique aims to measure the “arciness” and “skewness” (collectively referred to as “flexion”) of a lensed image. Flexion is a complementary approach to shear analysis in that it uses the odd moments (3rd multipole moments, for example) to compute local gradients in a shear field. BGRT have discussed how flexion may be used to identify substructure in clusters, to normalize the matter power spectrum on sub-arcminute scales via “cosmic flexion” (as an analog to cosmic shear), and to estimate the ellipticity of galaxy-galaxy lenses. As a practical application, flexion has already been used to measure galaxy-galaxy lensing (Goldberg & Bacon, 2005), and is presently being used in cluster reconstruction (Leonard et al., in preparation). However, there have been several difficulties in the estimation of flexion on real objects. First, the flexion inversion is difficult to describe, contains an enormous number of terms, and thus, is rather daunting to code. Secondly, there has been little discussion of the explicit effects of PSF convolution or deconvolution. Finally, unlike shear, there has, until recently, been no simple form to even approximate what the “flexion” is. The remainder of this paper will thus be a practical guide to measuring flexion in real images. We begin, below, by reminding the reader of the basic terms involved in flexion analysis. In § \[sec:shapelets\], we review shapelet decomposition, and discuss some of the issues involved in using shapelets to measure flexion. In § \[sec:holics\], we discuss a new, conceptually simpler, form of flexion analysis developed by Okura et al. (2006), which uses moments, rather than basis functions to measure flexion. They call their technique Higher Order Lensing Image’s Characteristics, or HOLICs. We refine the HOLICs approach somewhat, and develop a KSB (Kaiser, Squires, & Broadhurst, 1995)-type approach using a Gaussian filter to perform an inversion, as well as describe a technique for PSF deconvolution. In § \[sec:simulate\], we discuss comparisons of the two techniques using simulated lenses and simulated PSFs. In § \[sec:measurement\], we compare shapelets and HOLICs inversions on HST images. Finally, in § \[sec:discuss\], we discuss the implications of this study. In Appendix A, we also present the explicit HOLICs inversion matrix, so the reader can write his/her own code. He/she need not do so, however, as all codes discussed herein are available from the flexion webpage.[^1] Flexion ------- What is flexion? Conceptually, flexion represents local variability in the shear field which expresses itself as second-order distortions in the coordinate transformation between unlensed and lensed images: $$\beta_i\simeq A_{ij}\theta_j +\frac{1}{2}D_{ijk} \theta_j \theta_k, \label{eq:transform}$$ with $$D_{ijk}=\partial_k A_{ij} \ ,$$ where $\partial_k$ is shorthand for $\partial/\partial x_k$. Here, ${\bf A}$ is the normal deprojection operator: $${\bf A}= \left( \begin{array}{cc} 1-\kappa-\gamma_1 & -\gamma_2 \\ -\gamma_2 & 1-\kappa+\gamma_1 \end{array} \right)\ , \label{eq:Adef}$$ and thus, the second term on right in equation (\[eq:transform\]) represents the flexion signal. ${\bf D}$ may be written as: $$\begin{aligned} D_{ij1}&=&\left( \begin{array}{cc} -2\gamma_{1,1}-\gamma_{2,2} & \ \ -\gamma_{2,1} \\ -\gamma_{2,1} & \ \ -\gamma_{2,2} \end{array} \right), \\ \nonumber D_{ij2}&=& \left( \begin{array}{cc} -\gamma_{2,1} & \ \ -\gamma_{2,2}\\ -\gamma_{2,2} & \ \ 2\gamma_{1,2}-\gamma_{2,1} \end{array} \right).\end{aligned}$$ These distortions create asymmetries in a lensed image – a skewness and a bending, depending on the values of individual coefficients. Irwin & Shmakova (2005;2006) describe a similar lensing analysis technique in which the elements of ${\bf D}$ are referred to as “Catenoids” and “Displacements.” BGRT describe an inversion whereby one can estimate the individual components, and thus measure two “flexions”: $${\cal F}\equiv \partial^\ast \gamma$$ $${\cal G}=\partial \gamma$$ where $\partial$ is the complex derivative operator: $$\partial = \partial_1+i\partial_2\ .$$ Figure \[fg:shapes\] is reproduced from BGRT and shows the effect of a first or second flexion on a circular source. An object with first flexion, ${\cal F}$, appears skewed, while an object with second flexion, ${\cal G}$, appears arced, especially if the image has an induced shear as well. The first flexion has an $m=1$ rotational symmetry, and thus behaves like a vector. In particular, it is a direct tracer of the gradient of the convergence: $${\cal F}_1 {\bf i}+{\cal F}_2 {\bf j}=\nabla \kappa$$ where ${\cal F}_1$ is the real component and ${\cal F}_2$ is the imaginary part (as with the second flexion and, as the standard convention, with shear). The second flexion has an $m=3$ rotational symmetry, though unlike the first flexion, it has no simple physical interpretation like that of the first flexion. It is, however, roughly proportional to the local derivative of the magnitude of the shear. A more complete discussion of flexion formalism can be found in BGRT. Shapelets Decomposition {#sec:shapelets} ======================= Review of Shapelets ------------------- Measurement of flexion ultimately requires very accurate knowledge of the distribution of light in an image. The shapelets (Refregier 2003; Refregier & Bacon 2003) method of image reconstruction decomposes an image into 2D Hermite polynomial bases: $$f({\bf \theta})=\sum_{n,m} {\cal B}_{nm}({\bf \theta}) f_{nm}\ .$$ This technique has a number of very natural advantages. In the absence of a PSF, all shapelet coefficients will have equal noise. Moreover, the basis set is quite localized (Hermite polynomials have a Gaussian smoothing filter), and thus is ideal for modeling galaxies. Furthermore, the generating “step-up” and “step-down” operators for the Hermite polynomials are simply combinations of the $x_i$, and $\partial_i$ operators. Refregier (2003) shows that if we decompose a source image, $f$, into shapelet coefficients, the transformation to a lensed image may be expressed quite simply as: $$f'=(1+\kappa \hat{K}+\gamma_j \hat{S}_j)f$$ where the various lensing operators are: $$\begin{aligned} \hat{K}&=&1+\frac{1}{2}\left(\hat{a}_1^{\dagger 2}+\hat{a}_2^{\dagger 2}-\hat{a}_1^2-\hat{a}_2^2\right)\nonumber \\ \hat{S}_1&=&\frac{1}{2}\left(\hat{a}_1^{\dagger 2}-\hat{a}_2^{\dagger 2}-\hat{a}_1^2+\hat{a}_2^2\right)\nonumber \\ \hat{S}_2&=&\frac{1}{2}\left(\hat{a}_1^{\dagger}\hat{a}_2^{\dagger}-\hat{a}_1\hat{a}_2\right)\ ,\end{aligned}$$ $\hat{a}^{\dagger}$ and $\hat{a}$ are the normal step-up and step-down operators, and the subscript refers to the directional component of the coefficient (i.e. 1 for the first or x-component, and 2 for the second, or y-component). Note that in the weak field limit, these operators indicate that power will be transferred between coefficients with indices $|\Delta n|+|\Delta m|=2$, which preserves symmetry as well as keeping the image representation in shapelet space compact. In Goldberg & Bacon (2005), similar (albeit more complicated) transforms were found relating the derivatives of shear. We will not reproduce the full second order operators here, as they are written in full in the earlier work, but we will point out some key features. First, some of the elements in the operators have an explicit dependence on the (unlensed) quadrupole moments of the light distribution. This is due to a relatively subtle effect not present in shear analysis. Since the flexion signal is asymmetric, the center of brightness in the image plane will no longer necessarily correspond to the center of brightness in the source plane, and since the shapelet decomposition is performed around the center of light, we need to correct for this. Most important, though, is the fact that second order lensing terms yield transfer of power between indices with $|\Delta n|+|\Delta m|=$ 1 or 3. To second order, then, a lensed image can be expressed as: $$f'=(1+\kappa \hat{K}+\gamma_j \hat{S}_j +\hat{S}_{ij}^{(2)}\gamma_{i,j})f \ .$$ Flexion analysis assumes (as does shear analysis) that the intrinsic flexion is random, and thus all “odd” (defined as n+m) moments are expected to be zero. Thus, from a set of shapelet coefficients, a best estimate for the flexion signal may be found via $\chi^2$ minimization, where: $$\chi^2\equiv\left[\mu_{n_1m_1}-f_{n_1m_1}+(\gamma_i \hat{S}_i^{(1)}+\gamma_{i,j}S_{ij}^{(2)}) \overline{f}_{n'_1m'_1}\right]V^{-1}_{n_1m_1 n_2 m_2} \left[(\mu_{n_2m_2}-f_{n_2m_2}+(\gamma_i \hat{S}_i^{(1)}+\gamma_{i,j}S_{ij}^{(2)}) \overline{f}_{n'_2m'_2}\right], \label{eq:chi2}$$ $V_{n_1m_1 n_2m_2}$ is the covariance matrix of the shapelet coefficients, and $\mu_{nm}$ is the “unlensed” estimate of a shapelet coefficient. For odd modes, this is zero. For even modes, the relative effect of shear is typically much smaller than the intrinsic ellipticity of an image, thus it makes sense to set $\mu_{nm}=f_{nm}$. Effective Estimation of the Flexion ----------------------------------- Though the form looks quite complicated, conceptually, computing the flexion is very straightforward. A simplified pipeline may be written as follows: 1. Generate a catalog of objects and, for each, excise an isolated postage stamp. 2. Compute the shapelet coefficients of the postage stamp. 3. Deconvolve the postage stamp with a known PSF kernel. 4. Compute the transformation matrices associated with each of the four flexion operators, solve the $\chi^2$ minimization (equation \[eq:chi2\]) for $\gamma_{i,j}$, and estimate the flexion. We discuss each of these steps in turn below. The data used for this analysis was taken using HST and the Advanced Camera for Surveys, and in the particular context of cluster lensing. In this context, the galaxies in which we are interested are potentially blended with much larger and brighter foreground objects. We discuss the specific properties of our data catalog in § \[sec:measurement\], but many of the issues involved are quite generic. ### Catalog Generation and Postage Stamp Cutout The first step in the process, the generation of a catalog and postage stamps seems quite straightforward. For some datasets, such as the SDSS (York et al. 2000), the data release includes an atlas of pre-cut postage stamps. For other applications, such as in relatively shallow galaxy-galaxy or cosmic shear/flexion studies, fields will be relatively uncrowded and thus simple application of widely used packages such as SExtractor (Bertin & Arnouts, 1996) can be used. When fields are crowded, however, and contain a wide range of brightnesses and sizes, the catalog generation becomes more complicated. It has been noted by Rix et al. (2004) that in general, a single set of SExtractor parameters is insufficient for detection of all the objects of interest within an image; setting the source detection threshold too low will result in excessive blending near bright objects, whereas a high threshold results in a failure to detect fainter sources. Rix et al. describe a two-pass strategy for object detection and deblending involving an initial (“cold”) pass to identify large, bright objects, followed by a lower-threshold (“hot”) pass to pick up dimmer objects. Their final catalog consists of all the objects detected in the cold run, plus any objects detected in the hot run that do not lie within the isophotal area of any object detected in the first pass. This technique works well to prevent spurious deblending by SExtractor in images in which there is significant substructure. However, when dealing with crowded fields (such as clusters of galaxies) the largest problem in catalog generation is excessive blending of sources, particularly in the central region. To remedy this, we use a modified version of this hot/cold technique. Our method consists of a primary SExtractor run to detect only the brightest objects. In a lensing field, especially in a lensing cluster, these bright objects will tend to be the lenses. Making use of the RMS maps generated during this SExtractor run, we mask out the bright objects by setting the pixel values to background noise, and thus simulate an emptier field. We then run SExtractor on the masked image, using a much lower detection threshold, to create a catalog of background objects. Since shape estimation including both flexion and shear have a minimum of 10 degrees of freedom, we require at least 10 connected pixels above the detection threshold, though in reality, we are unlikely to be able to get a reliable measurement from an image with fewer than 15 included pixels. We then discard all objects for which reliable shape estimates cannot be found. For each remaining object, a postage stamp is generated. Ideally, this should identify any neighboring objects and mask them out (by setting their pixel values equal to background noise). Our postage stamp code also identifies objects which are blended by using a friends-of-friends algorithm to find sets of connected pixels that are a certain threshold (typically 2-3$\sigma$ for the stacked images described below) above the background. If there is any overlap between the object of interest and another object within the field of the postage stamp, we consider the source to be excessively blended and exclude it from further analysis. ### Shapelet Decomposition Shapelets can be an extremely compact representation of an individual image. However, in reality, they are a [*family*]{} of basis functions. There is a characteristic scaling parameter, $\beta$, which represents the width of the Gaussian kernel in the basis function Hermite polynomials: $${\cal B}_{nm}({\bf \theta})\propto \exp\left(-\frac{\theta_1^2+\theta_2^2}{2\beta^2}\right) \ .$$ In principle, while all values of $\beta$ will yield an orthonormal basis set, some values produce a dramatically faster reconstruction in terms of the number of coefficients required to reach convergence. Moreover, in reality we don’t [*want*]{} to reconstruct all details in an image. Structure on the individual pixel scale may simply represent noise. From a practical perspective, our goal is to optimize selection of $\beta$, and the maximum coefficient index, $n_{max}$. Refregier (2003) suggests the following parameters: $$\begin{aligned} \beta\simeq\sqrt{\theta_{min}\theta_{max}}\nonumber\\ n_{max}\simeq\frac{\theta_{max}}{\theta_{min}} -1\ ,\end{aligned}$$ where $\theta_{min}$ and $\theta_{max}$ represent the minimum and maximum scales of image structure, respectively. R. Massey (private communication) has found that rather than performing overlap integrals to solve for the shapelet coefficients (as was done, for example, in the analysis of Goldberg & Bacon 2005), the ideal approach is to do a $\chi^2$ minimization of the reconstructed image with the original postage stamp. This may seem complicated, and it is. Fortunately, a shapelets package is available in IDL at the shapelets webpage.[^2] For our sample of co-added, background-subtracted, HST ACS images of Abell 1689, we find that $\theta_{min} = 0.4$ pixels and $\theta_{max} = 1.8\sqrt{a^2+b^2}$ give good shapelet reconstructions, where $a$ and $b$ are the semi-major and semi-minor axes of the galaxy as measured by SExtractor. However, it is important to note that these parameters are somewhat dependent on the noise level in the images. For a sky-limited sample, we have found that the optimal choice of $\theta_{min}$ scales approximately linearly as the ratio of the flux to the RMS sky noise. We have looked at this scaling in a sample of galaxies detected with ACS (and which we describe in greater detail below), each of which was imaged in 4 frames. Prior to stacking of these frames, we found that $\theta_{min} = 0.75$ produced low $\chi^2$ and convergence with small values of $n_{max}$. After stacking, $\theta_{min}=0.4$ was required. This makes sense, since the noisier our image, the more prone we might otherwise be to fitting complex polynomials to what is, essentially, noise. Roughly, the processing time for a decomposition scales as $\theta_{max}^4$, as $\theta_{max}$ determines both the postage stamp size and the maximum order of the shapelet decomposition. Due to the high resolution of our images, we encountered a number of objects for which $n_{max}$ was so large that the decomposition time became prohibitive. We opted to re-grid images with $n_{max} > 50$ into larger pixels by taking the mean of the pixel values in square bins, the size of which is determined by $binsize = n_{max}/50$. This number was rounded up for objects with $50 < n_{max} \le 75$. A flexion measurement is then carried out using the $\chi^2$ minimization technique described previously. However, we have found that truncating the shapelet series prior to the flexion measurement yields a more accurate and robust measure of the flexion than using the full series. Excluding the higher order shapelet modes avoids contamination of the flexion signal by small scale substructure and by noise (particularly in dimmer objects). We exclude all shapelet modes with $n > n_{max}/5$ in our flexion measurement. This effectively increases $\theta_{min}$ to 2 pixels, without affecting the accuracy of the reconstruction. ### PSF Deconvolution One of the complications in measuring properties of lensed images is that, in practice, they are convolved with a PSF: $$f({\bf \theta})=\int d^2\theta' P({\bf \theta-\theta'}) f^{(0)}({\bf \theta})\ . \label{eq:PSFdef}$$ In principle, the PSF can be estimated through measurement of stars, but in deep, small-field, high galactic-latitude observations, stars may be scarce, and thus PSF estimation may rely partly on numerical analysis of the instrument (e.g. the Tiny Tim algorithm, Krist, J. 1993). In reality, though, this should rarely be an issue. Estimations of the PSF flexion from Tiny Tim yield values of $\sigma_{aF,psf}\simeq 7\times 10^{-4}$. This represents the maximum induced flexion which can arise from convolution with the PSF, and is still several orders of magnitude lower than the scatter in intrinsic flexion of galaxies. We are not surprised by this since, for example, PSF distortions arising from variable sizes in chips is likely to scale as the variation in PSF ellipticity. In ACS, chip distortions produce ellipticities of order $1\%$, and vary on scales of 100’s of pixels, producing an induced flexion of $\sim 10^{-4}\ pix.^{-1}$. From the ground, the atmospheric distortions are expected, on average, to be even more isotropic. There is another reason to suppose that PSF flexion contributions will be unimportant. In shear measurements, the PSF ellipticity typically varies smoothly and somewhat symmetrically around the center of a field, mimicking (or partially reversing) the overall behavior of the expected shear field. Since flexion probes smaller scale effects, the induced flexion by the PSF will, on average, cancel out. This is not to say that we cannot deal with PSF flexion inversion. Refregier (2003) describes an explicit deconvolution algorithm (see also Refregier and Bacon 2003, and references therein). In shapelet space, equation (\[eq:PSFdef\]) can be re-written as: $$f_{nm} = \sum_{n'm'n''m''} C_{nmn'm'n''m''} P_{n'm'}f^{(0)}_{n''m''}$$ Where $C_{nmn'm'n''m''}$ is the 2-dimensional convolution tensor: $$C_{nmn'm'n''m''}(\gamma,\alpha,\beta)=2\pi(-1)^{n+m}i^{n+m+n'+m'+n''+m''}\int d^2{\bf x}{\cal B}_{n''m''}({\bf x}/\gamma){\cal B}_{n'm'}({\bf x}/\alpha){\cal B}_{nm}({\bf x}/\beta)\ ,$$ and $\alpha$, $\beta$ and $\gamma$ are the characteristic scales of $f^{(0)}$, $P$ and $f$, respectively. We may then define a PSF convolution matrix as: $$P_{nmn'm'}\equiv\sum_{n''m''} C_{nmn'm'n''m''}P_{n''m''}\ .$$ If only low order terms in the convolution matrix are included, it may be inverted to perform a deconvolution via: $$f^{(0)}_{nm}=\sum_{n'm'} (P^{-1})_{nmn'm'}f_{n'm'}\ .$$ This provides a good estimate of the low order coefficients, but high order information is lost. An alternative inversion scheme involves fitting the observed galaxy coefficients using a $\chi^2$ minimization scheme. Refregier and Bacon (2003) note that the $\chi^2$ scheme may be more robust numerically, and can take full account of variations in the noise characteristics across an image (although it is strictly only valid in the case of Gaussian noise). It is this scheme that is implemented in the shapelets IDL software. ### Flexion Inversion If the shapelet coefficients are statistically independent (as they will be in the absence of an explicit PSF deconvolution), formal inversion of the flexion operator is quite straightforward. Under these circumstances, we also have the benefit that the measurement error for each moment is identical (see Refregier 2003 for discussion). Noting that, in most galaxies, the coefficients corresponding to the $n+m=$even moments will be much larger than the odd moments (and, indeed, upon random rotations, the latter will necessarily average to zero) we can dramatically simplify equation (\[eq:chi2\]). First, we define the susceptibility of each odd moment as: $$\Delta f_{n' m',ij}=\hat{S}^{(2)}_{ij}f_{nm}$$ where $f_{nm}$ represents all of the “even” coefficients, and $n'm'$ represents all of the odd coefficients. Thus, we wish to solve for the relation: $$\sum_{n'm'}(f_{n'm'}-\gamma_{i,j}\Delta f_{n' m',ij})^2=min.$$ where the first term is taken directly from measurement. Taking the derivatives and rearranging, we find: $$\sum_{n'm'} f_{n'm'}\Delta f_{n'm',ij}=\gamma_{i,j}\sum_{n'm'}(\Delta f_{n'm',ij}\Delta f_{n'm',i'j'})$$ which can readily be inverted to solve for $\gamma_{i,j}$. In practice, however, there are a number of issues which must be considered. First, if the PSF or pixel scale are relatively large compared to the minimum resolution scale of an image then many of the high-order moments returned by shapelets decomposition will, in fact, not have any information. Thus, the above inversion will yield a systematic underestimate of the true image flexion. Above, we describe a truncation which minimizes this effect. While the flexion inversion is, at its core, linear algebra, it involves an enormous number of terms. We have thus provided an inversion code for shapelets estimates of flexion along with examples on the flexion webpage. HOLICs Analysis {#sec:holics} =============== Higher Order Moments -------------------- Okura et al. (2006) recently related flexion directly to the 3rd moments of observed images. This is a significant extension of flexion, and very much along the lines of Goldberg & Natarajan’s (2002) original work which talked about “arciness” in terms of the measured octopole moments. Throughout our discussion, we will use the notation: $$Q_{ij}=\frac{1}{F}\int (\theta_i-\overline{\theta_i}) (\theta_j-\overline{\theta_j}) f({\bf \theta}) d^2{\bf \theta}$$ to refer, in this case, to the unweighted quadrupole moments, with all higher moments being defined by exact analogy. In this context, $F$ refers to the unweighted integrated flux. They define the complex terms: $$\zeta\equiv\frac{(Q_{111}+Q_{122})+i(Q_{112}+Q_{222})}{\xi}$$ and $$\delta\equiv\frac{(Q_{111}-3Q_{122})+i(3Q_{112}-Q_{222})}{\xi}$$ where $$\xi\equiv Q_{1111}+2Q_{1122}+Q_{2222}\ .$$ These terms are collectively referred to as HOLICs. If a galaxy is otherwise perfectly circular (i.e. no ellipticity), and in the absence of noise, then the HOLICs may be directly related to estimators of the flexion (subject to an unknown bias of $1-\kappa$). Namely: $${\cal F}\simeq \frac{4 \zeta\xi}{9\xi-6(Q_{11}^2+Q_{22}^2)} \label{eq:skewness}$$ $${\cal G}\simeq \frac{4 \delta}{3} \label{eq:arciness}$$ where the latter term in the denominator of ${\cal F}$ does not appear in the Okura et al. analysis. Bacon and Goldberg (2005) show that a flexion induces a shift in the centroid proportional to the quadrupole moments. In order to correctly invert the HOLICs, this term needs to be incorporated explicitly. The simplicity of the extra term results from an approximation of near circularity. The beauty of this approach is that it gives us a very intuitive feel for what flexion means in an observational way. We thus introduce the term “skewness” to the intrinsic properties of a galaxy as measured from equation (\[eq:skewness\]) whether or not the galaxy is otherwise circular, and whether or not it is lensed. The skewness may be thought of as the intrinsic property, much as the “ellipticity” is the intrinsic property related to the “shear.” Likewise, the intrinsic property associated with equation (\[eq:arciness\]) will be referred to as the “arciness.” In reality, however, equations (\[eq:skewness\]) and (\[eq:arciness\]) are not sufficient to perform a flexion estimate even if a galaxy has an ellipticity of only a few percent. Okura et al provide a general relationship between estimators for flexion and HOLICs, though the relation is best expressed in matrix form: $${\cal M} \left( \begin{array}{c} {\cal F}_1 \\ {\cal F}_2 \\ {\cal G}_1 \\ {\cal G}_2 \end{array} \right) = \left( \begin{array}{c} \zeta_1 \\ \zeta_2 \\ \delta_1 \\ \delta_2 \end{array} \right) \label{eq:Fsolve}$$ where ${\cal M}$ is a $4\times 4$ matrix consisting of elements proportional to sums of $Q_{ijkl}$ and $Q_{ij}Q_{kl}$, the former of which can be found by explicitly expanding the expressions in Okura et al., and the latter of which is again derived from the shift in the centroid. For the convenience of the reader, we write out the explicit form of ${\cal M}$ in Appendix A. It may be seen by examining the elements of ${\cal M}$ why this inversion must be done explicitly for even mildly elliptical sources. For fully circular sources, it may be seen by inspection that ${\cal M}$ is diagonal. However, when a source has an ellipticity even as small as $10\%$, it can be shown that $|M_{11}|\simeq |M_{12}|$, and thus equations (\[eq:skewness\]) and (\[eq:arciness\]) are no longer even approximately correct. Gaussian Weighting with HOLICs ------------------------------ The application of the HOLICs technique would be trivial if there were no measurement noise. In the presence of noise, and especially, when the sky dominates, measurement of unweighted moments is inherently quite noisy. In a case where we are measuring the 3rd and 4th moments, it is even more so. Kaiser, Squires & Broadhurst (1995; see also a nice review by Bartelmann & Schneider, 2001) developed perhaps the most comprehensive approach to dealing with the second moments (the ellipticity) with noisy observing, and with a (potentially anisotropic) PSF. Our approach is similar. We have only worked with a Gaussian window thus far, but the approach is generalizable for any circularly symmetric weighting. We thus define a window function: $$W({\bf \theta})=\frac{1}{2\pi\sigma_{W}}\exp\left(-\frac{\theta_1^2+\theta_2^2}{2\sigma_W^2}\right)$$ where the origin is taken to be the center of light, and the integral is normalized to unity. Further, we define the weighted moments as, for example: $$\hat{Q}_{11}=\frac{1}{\hat{F}}\int (\theta_1-\overline{\theta_1})^2 f({\bf \theta}) W({\bf \theta}) d^2{\bf \theta}\ .$$ We can thus redefine all HOLICs and moments similarly. We have found through experimentation (see below) that for a sky noise limited source, a reasonable value of $\sigma_W$ is 1.5 times the half-light radius. If we were to simply replace all elements in ${\cal M}$, $\zeta$, and $\delta$ from equation (\[eq:Fsolve\]) with their weighted counterparts we would not get an unbiased estimate of the flexion. There are two corrections. One has to do with the fact that centroid shift will differ from the unweighted case to the weighted case. Consider an extreme scenario in which the window width is arbitrarily small and in which the unlensed image was circularly symmetric with a peak at the center. In that case, the centroid will essentially remain at the center (peak brightness) even if the unweighted moments shift. Thus, compared to the unweighted moments, the centroid will shift: $$\Delta \overline\theta_l=\frac{\sum_{ijk}D_{ijk}\hat{Q}_{ijkl}}{\sigma_W^2}$$ where we have used the explicit fact that for a Gaussian: $$\frac{dW({\bf \theta})}{d\theta_i}=-\frac{\theta_i}{\sigma_W^2}W({\bf \theta})\ ,$$ The other correction has to do with the fact that though lensing preserves surface brightness, it does not preserve total flux. This is normally related by the Jacobian of the coordinate transformation. However, when considering a window function, we need consider that transformation explicitly: $$W(\beta)d^2\beta=\left|\frac{\partial \beta}{\partial \theta}\right|W(\beta)d^2\theta$$ as used by Okura et al. (2006), and where we have simply multiplied both sides by the factor $W(\beta)$. In this context, ${\bf \beta}$ refers to the image coordinate in the source plane. Ignoring the terms proportional to shear (which cannot be directly addressed by this method at any rate), we have the approximate relation: $$W(\beta)\simeq W(\theta)+\frac{1}{2}D_{ijk}\theta_i\theta_j \frac{dW}{d\theta_k}$$ or, as we have already asserted: $$W(\beta)\simeq W(\theta)-\frac{1}{2}D_{ijk}\frac{\theta_i\theta_j\theta_k}{\sigma_W^2} W(\theta)\ .$$ Note that the latter term contains an odd number of position elements, and thus, coupling to the generating equations for $\zeta$ and $\delta$ produces contributions of $6^{th}$ moments in ${\cal M}$: $$\Delta \hat{Q}_{ijk}=-\frac{1}{2}D_{lmn}\frac{\hat{Q}_{ijklmn}}{\sigma_W^2}\ ,$$ which, in turn, must be corrected for. We may thus say that: $$\hat{\cal M}={\cal M}(\hat{Q}_{ij},...)+\Delta{\cal M}\ ,$$ where the latter expression can also be found in Appendix A. PSF Correction in HOLICs {#sec:psfcorrection} ------------------------ As with our discussion of shapelets, above, we must also consider PSF deconvolution in our HOLICs pipeline. We define the PSF function in equation (\[eq:PSFdef\]), and all unweighted moments of the PSF are denoted by $P_{ij}$, etc. In principle, because of the higher signal-noise of the PSF, the unweighted moments are easier to estimate than the moments of the detected image. While we argued, above, that the induced flexion from a PSF is likely to be small, it is still the case, as with shear, that the PSF will reduce the measured flexion. Let’s first consider the case in which we were able to measure the unweighted moments of both the PSF and the observed image. It is straightforward to show that: $$Q_{ij}=Q_{ij}^{(0)}+P_{ij}\ . \label{eq:quadrel}$$ Thus may be computed via the relation: $$Q_{ij}=\frac{1}{F}\int \theta_i \theta_j f^{(0)}(\theta')P(\theta-\theta') d^2\theta d^2\theta'$$ Making the substitution, $$\theta''=\theta-\theta'$$ yields $$Q_{ij}=\frac{1}{F}\int (\theta_i'\theta_j'+\theta_i''\theta_j''+ \theta_i'\theta_j''+ \theta_i''\theta_j')f^{(0)}(\theta')P(\theta'') d^2\theta'd^2\theta''$$ It is straightforward to show that this yields equation (\[eq:quadrel\]). Similarly, it may be shown that: $$Q_{ijk}=Q_{ijk}^{(0)}+P_{ijk}\ .$$ However, $$Q_{1111}=Q_{1111}^{(0)}+P_{1111}+6Q_{11}^{(0)}P_{11}\ ,$$ with a similar expression for $Q_{2222}$, and $$Q_{1122}=Q_{1122}^{(0)}+P_{1122}+Q_{11}^{(0)}P_{22}+Q_{22}^{(0)}P_{11}$$ provided we assume the PSF is nearly circular. If we further look only at nearly circular sources, then we may estimate the flexion using the forms in equations (\[eq:skewness\]) and (\[eq:arciness\]). Again, assuming unweighted moments, and zero PSF and intrinsic flexion we find: $$\tilde{{\cal F}}_i={\cal F}_i \frac{9\xi^{(0)}-6(Q^{(0)2}_{11}+Q^{(0)2}_{22})}{9\xi-6(Q_{11}^2+Q_{22}^2)} \label{eq:fcorrect}$$ Where ${\cal F}_i$ is an unbiased estimate of the flexion, and $\tilde{{\cal F}}_i$ is the estimated flexion if one does not include the correction for the PSF. However, the normalization constant may be estimated directly from combinations of the PSF 2nd and 4th moments, and the unweighted moments of the image. Since this term represents something like the overall radial profile of the source, the unweighted moments can be estimated even under noisy conditions. Similarly, the second flexion may be estimated as: $$\tilde{{\cal G}}_i={\cal G}_i \frac{\xi^{(0)}}{\xi}$$ Though we have derived these relations for a nearly circular source, we have found they provide a good correction even when the PSF and intrinsic image size are comparable, and when ellipticities for the source image are $\varepsilon\simeq 0.2$. Simulated Lensing {#sec:simulate} ================= Which approach is better, shapelets or HOLICs? From a signal perspective, the shapelets technique is better. It is designed to provide optimal weighting and return optimal signal-noise. Moreover, as described above, inversion of the PSF is a straightforward and well-designed process. In the absence of noise, the two techniques produce very similar results. On the other hand, the HOLICs technique has several practical advantages, especially for large surveys. For one, the HOLICs code is typically much faster than shapelets. For an N pixel image, the HOLICs technique is an ${\cal O}(N)$ calculation, whereas the shapelets is ${\cal O}(N^2)$. Additionally, some values of $\beta$ produce very bad reconstructions, and hence, minimization of $\chi^2$ can be time-consuming and may not converge to a minimum. As a simple test, we created images with brightness profiles of: $$I(r)\propto \exp[-(r/r_0)^n]$$ and though we found similar results for a reasonable range of exponents, the results presented below are for $n=1.5$. We have used a constant source ellipticity, typical of those observed in the field, $\varepsilon=0.2$, and had measurement errors which were dominated by sky brightness. In each case, we had no intrinsic arciness or skewness (that is, the flexion of the unlensed objects were zero), since our aim was to measure the response of each of the estimators to lensing. We then artificially lensed each of our simulated images, added sky noise, and measured the flexion using both the HOLICs and shapelets techniques. The noise is fixed throughout this discussion, as is the strength of the flexion signal. It is clear, however, that all relevant signal-noise values will scale linearly with the strength of the lensing signal and inversely with sky noise. Optimzing the HOLICs Scale Factor --------------------------------- Our first questions is, what is the optimal value of $C_W$, such that: $$\sigma_W=C_W\times r_{half-light} \ ?$$ Ideally, we would like an unbiased estimator of the flexion which also has very little scatter. It is clear that the larger the value of $C_W$, the larger the scatter will be (in general), since we will be measuring more and more of the noisy sky. However, the smaller the $C_W$, the less accurate will be our measure of the real shape of the galaxy. Figure \[fg:findC\] bears this out. There is an optimal value of $C_W$ around 1.5, which reflects a balance between minimizing measurement errors as well as any measurement bias inherent in the technique. With shapelets, we find a systematic underestimate of 11% in the first flexion, and an overestimate of 12% in the second flexion. We find a scatter of about 12% in both. This is very similar in magnitude to the results found by an “optimal” HOLICs analysis. Correlation of HOLICs and Shapelets Measurement Error ----------------------------------------------------- Since both HOLICs and shapelets give similar measurement errors at fixed sky noise, it is worth considering whether we expect measurement errors between the two techniques to be correlated. Even in these idealized circumstances, uncorrelated errors would mean that there is significant information in the images which is not being used. In Fig. \[fg:corrsim\], we show the correlation in uncertainty between our $C_W=1.5$ HOLICs estimates, and our shapelets estimates. For the first flexion, in particular, the correlation is quite high, with a Pearson correlation coefficient of $0.86$. The correlation in measurements of the second flexion is much lower, with $\rho_G=0.23$. Why don’t they have perfect correlation? The two techniques weight various components of the signal (and thus, the noise) differently, and therefore have a slightly different response to the noise. This general trend is borne out with observed objects as well, in which we will see much higher correlation between measurements of the first flexion than the second flexion between the two techniques. PSF Deconvolution ----------------- Finally, we can simulate PSF deconvolution. Using a Gaussian PSF with a characteristic size somewhat larger than the intrinsic image (the correction factor described in equation \[eq:fcorrect\] is 2.7), we distorted and then recovered the flexion estimates from images of increasing intrinsic ellipticity. This analysis is done in the absense of sky noise, and thus any errors in shape recovery represent a systematic effect. We show the fractional errors in measurement of the first and second flexion in Figure \[fg:psf\]. Since it is possible to estimate the systematic error for a combination of measured shear and PSF shape, it is advisable to those wishing to make high-precision flexion measurements to take this empirical correction into account. We find that, despite the fact that the PSF correction is based on an assumption of circularity, it continues to produce a good result even if the image has an intrinsic ellipticity as high as 0.3. Measurement of Flexion on HST Images {#sec:measurement} ==================================== Sample Selection and Pipeline ----------------------------- We also compare the two approaches to flexion inversion on real objects. Our data consists of 4 HST ACS cosmic-ray rejected (CRJ) images of Abell 1689 using the F625W WFC filter (hereafter “R-band”). Each image was taken by H. Ford during HST Cycle 11, and has an exposure time between 2300-2400 seconds. The observations are described in detail in Broadhurst et al. (2005). Using the SWarp software package [^3], these four images were co-added to create a single “full” R-band image. We also generated 2 independent “split” images for comparison purposes by combining only two of the original images. The images are background-subtracted, aligned and re-sampled, then projected into subsections of the output frame using a gnomonic (or tangential) projection, and combined using median pixel values. Each image undergoes a primary SExtractor run designed to detect only the foreground objects (cluster members and known stars). This detection is carried out using the cross-correlation utility in SExtractor, which allows us to specify the locations of the foreground objects. Our foreground object catalog was generated using a combination of spectroscopically confirmed cluster members (Duc et al. 2002), and identification by eye of foreground objects that were later confirmed as such by use of the NASA/IPAC Extragalactic Database (NED), as well as clearly identifiable stars in the field. These objects are then masked out as described previously, and a second SExtractor run carried out. A catalog of objects is then generated, using only those objects that were detected in both of the split R-band images. We measure the flexion in our catalog objects using both the truncated shapelets method (described above) and the HOLICs approach, and then compare the measurements by computing Pearson correlation coefficients between the different estimates in the full image. We also compute correlation coefficients between measurements taken using the same technique in the two split images. This gives us an estimate of the robustness of the measurement technique. When computing the correlation coefficients, we include only objects with $a > 3$ pixels, and consider only the brightest half of our catalog objects. In order to exclude extreme or erroneous measurements, we require $(a|\cal{F}|) <$ 0.2 and $(a|\cal{G}|)<$ 0.5. Results ------- Figure \[fullcorr\] shows a comparison of the HOLICs and shapelets estimates of flexion in the full image. Both $\cal{F}$ and $\cal{G}$ have a positive correlation, with a Pearson correlation coefficient of 0.17 for $\cal{F}$ and 0.12 for $\cal{G}$. Additionally, both methods yield similar standard deviations for both first and second flexion. This is what we expected from our simulated results above. Clearly, if flexion represents any real signal, the two techniques should be correlated, and, as we showed in our simulated results, the correlation in first flexion is higher than in second flexion. But the correlation in our measured results is lower than in the simulated ones. Why? In part, this is due to a relatively noisy field. We’ve found that selections on brighter magnitudes and larger objects improves the correlation somewhat. In part, however, this is due to what we mean by “flexion.” Recall that the shapelets and HOLICs analysis of flexion involve weighting different modes in different ways. Real, unlensed, galaxies will have odd modes which are not necessarily correlated in a simple or obvious way. Lensing, of course, produces a significant correlation, and thus, a population of significantly lensed objects (for which the majority of the flexion is due to lensing) would be expected to have a more correlated flexion. This is similar to the case with weak shear analysis in that the S/N from a typical object is usually less than 1. We can test this hypothesis directly by comparing the measurements in the split images and estimating the flexion in both using the same technique. Any discrepancies between the two ought to be the result of photon noise rather than intrinsic complexity in the structure of the 3rd moments. Figure  \[split\_moments\] shows a comparison of the HOLICs measurements made on each of the split images. These measurements are well correlated: the Pearson correlation coefficient here is 0.37 for $\cal{F}$ and 0.23 for $\cal{G}$. In Figure  \[split\_shapelets\], we see a comparison of the shapelets measurements in these images, which appear to be more strongly correlated (particularly for $\cal{F}$). The Pearson correlation coefficients here are 0.58 for $\cal{F}$ and 0.18 for $\cal{G}$. As motivated above, most of the “noise” in our measurements comes from the intrinsic distribution of flexion within our sample. Indeed, using the HOLICs approach, we find: $$\sigma_{a|F|+Noise}=0.05$$ $$\sigma_{a|G|+Noise}=0.08$$ The distribution function may be seen in Fig. \[fg:fhist\]. Note that this result includes noise. However, we may estimate the relative effect of photon noise on this scatter by using correlation between frames. That is: $$\sigma_{a|F|}=\sqrt{\rho}\sigma_{a|F|+Noise}$$ And thus, we find that our best estimate of the intrinsic scatter in first flexion is: $$\sigma_{a|F|}=0.03$$ (as found in Goldberg & Bacon 2005), and $$\sigma_{a|G|}=0.04\ .$$ The combination $a|{\cal F}|$ represents a dimensionless term, and thus is independent of distance. It should also be noted that since these measurements are taken within a cluster, the signal is included as well, and one might question whether it is reasonable to estimate the intrinsic variability of flexion from lensed images. The intrinsic scatter in flexion was originally measured in Goldberg & Bacon (2005), and we merely confirm the result here. However, this is a reasonable thing to do, as flexion drops off much more quickly than shear, and thus, even within a rich cluster, the flexion signal is dominated by individual galaxies. Even at a separation of $1''$, the flexion from even a very massive $300 km/s$ galaxy on an $a=0.4''$ source is about 0.05, approximately the level of the intrinsic flexion. Such separations are relatively rare, however. Discussion {#sec:discuss} ========== We have endeavored to present a detailed guide to measuring flexion in real observations, with a focus on space-based imaging. In the process, we have taken a look at two different approaches to measuring flexion: shapelets and HOLICs, with an eye toward which approach is “better.” From an idealized perspective of maximal signal-noise, the answer is simple. Shapelets produces a mode-by-mode comparison which optimally averages to produce a unique estimate of flexion. However, this result is complicated somewhat in two limits: blending, which affects larger objects, and PSF convolution, which affects smaller ones. When images are blended, it is clear that we benefit by giving extra weighting to those pixels near the center of the the object. In that sense, HOLICs can be said to produce more robust results. Likewise, despite an explicit PSF deconvolution algorithm, applying the flexion inversion using shapelets results in inclusion of small scale power which has been blended away through the atmosphere or instrument. We have discussed, above, how this might be alleviated by only using relatively low order modes from the reconstruction in the estimate of flexion. However, doing so comes at the expense of some (but by no means all), of the signal-noise advantage from shapelets. Indeed, even using a relatively truncated form of the shapelets analysis still produced greater correlation between independent images of the same objects, and thus, cleaner estimates of the flexion. However, one complication in the shapelets analysis is producing a good shapelets decomposition in the first place. While R. Massey’s shapelet code comes with an optimization routine to find the best fit scaling parameter, $\beta$, the shapelet decomposition runs several orders of magnitude slower than HOLICs. For very large lensing fields, this may prove a significant limitation, and thus, HOLICs provides a fast, physically motivated, reasonably reliable alternative. This work was supported by NASA ATP NNG05GF61G and HST Archival grant 10658.01-A. The authors would like to gratefully acknowledge useful conversations with Jason Haaga, David Bacon & Sanghamitra Deb, and thank Richard Massey for thoughtful comments and the use of his shapelets code. We would also like to thank the anonymous referee whose comments greatly improved the final draft. [99]{} Bacon, D. J., Goldberg, D.M., Rowe, B.T.P. & Taylor, A.N., 2006, MNRAS 365, 414 Bartelmann, M. & Schneider, P., 2001, Physics Reports 340, 291 Bertin, E. & Arnouts, S., 1996, A&AS 117, 393 Broadhurst, T., et al. 2005, ApJ 621, 53 Duc, P. et al., 2002, A$\&$A 382, 60 Goldberg, D.M. & Bacon, D.J., 2005 ApJ 619, 741 Goldberg, D.M. & Natarajan , P. 2002, ApJ 564, 65 Irwin, J. & Shmakova, M., 2005, NewAR 49, 83 Irwin, J. & Shmakova, M., 2006, ApJ, 645, 17 Kaiser N., Squires G., Broadhurst T., 1995, ApJ, 449, 460. Krist, J. 1993, ASPC 52, 536 Okura, Y., Umetsu, K., & Futamase, T., 2006, submitted to ApJ, http://xxx.lanl.gov/abs/astro-ph/0607288 Refregier, A., 2003, MNRAS 338, 35 Refregier, A. & Bacon, D.J., 2003, MNRAS 338 48 Rix, H. et al., 2004, ApJS 152, 163 York, D.G. et al. 2000, AJ, 120, 1579 Expanded Coefficients for HOLICs Analysis of Flexion ==================================================== In equation (\[eq:Fsolve\]), we state that the flexion may be solved via inversion of the relation: $$y={\cal M}x$$ where $x$ is a vector of the desired flexion estimators, and $y$ is the measure of the 3rd order HOLICs. Here, we show the explicit form of [M]{}. $$\begin{aligned} M_{11}&=&\frac{1}{4}(9+8\eta_1)-\frac{33Q_{11}^2+14Q_{11}Q_{22}+Q_{22}^2+20Q_{12}^2}{4\xi} \nonumber \\ M_{12}&=&2\eta_2-\frac{32Q_{12}Q_{22}+32Q_{11}Q_{12}}{4\xi} \nonumber \\ M_{13}&=&\frac{1}{4}(2\eta_1+\lambda_1)-\frac{3Q_{11}^2-2Q_{11}Q_{22}-Q_{22}^2-4Q_{12}^2}{4\xi} \nonumber \\ M_{14}&=&\frac{1}{4}(2\eta_2+\lambda_2)-\frac{2Q_{11}Q_{12}}{\xi} \nonumber \\ M_{21}&=&2\eta_2-\frac{32Q_{12}Q_{22}+32Q_{11}Q_{12}}{4\xi} \nonumber \\ M_{22}&=&\frac{1}{4}(-8\eta_1+9)-\frac{Q_{11}^2+14Q_{11}Q_{22}+20Q_{12}^2+33Q_{22}^2}{4\xi} \nonumber \\ M_{23}&=&\frac{1}{4}(-2\eta_2+\lambda_2)-\frac{-2Q_{12}Q_{22}}{/\xi} \nonumber \\ M_{24}&=&\frac{1}{4}(2\eta_1-\lambda_1)-\frac{(Q_{11}^2+4Q_{12}^2+Q_{11}Q_{22}-3Q_{22}^2)}{4\xi} \nonumber \\ M_{31}&=&\frac{1}{4}(10\eta_1+7\lambda_1)-\frac{3(11Q_{11}^2-10Q_{11}Q_{22}-Q_{22}^2-20Q_{12}^2)}{4\xi} \nonumber \\ M_{32}&=&\frac{1}{4}(-10\eta_2+7\lambda_2)-\frac{3(8Q_{11}Q_{12}-32Q_{12}Q_{22})}{4\xi} \nonumber \\ M_{33}&=&\frac{3}{4}-\frac{3(-2Q_{11}Q_{22}+Q_{11}^2+Q_{22}^2+4Q_{12}^2)}{4\xi} \nonumber \\ M_{34}&=&0 \nonumber \\ M_{41}&=&\frac{1}{4}(10\eta_2+7\lambda_2)-\frac{3(32Q_{11}Q_{12}-8Q_{12}Q_{22})}{4\xi} \nonumber \\ M_{42}&=&\frac{1}{4}(10\eta_1-7\lambda_1)-\frac{3(Q_{11}^2+20Q_{12}^2+10Q_{11}Q_{22}-11Q_{22}^2)}{4\xi} \nonumber \\ M_{43}&=&0 \nonumber \\ M_{44}&=&\frac{3}{4}-\frac{3(-2Q_{11}Q_{22}+Q_{11}^2+Q_{22}^2+4Q_{12}^2)}{4\xi}\end{aligned}$$ where, as defined in Okura et al. (2006), we use: $$\eta\equiv \frac{(Q_{1111}-Q_{2222})+2i(Q_{1112}+Q_{1222})}{\xi}$$ $$\lambda\equiv\frac{(Q_{1111}-6Q_{1122}+Q_{2222})+4i(Q_{1112}-Q_{1222})}{\xi}$$ Note that $\eta=0$ and $\lambda=0$ for all circularly symmetric distributions and even those with no ellipticity but with flexion. If we apply a Gaussian weighting with width, $\sigma_W$, to our moment measurements, then ${\cal M}$ should be computed using the weighted moments. In addition, the following terms must be added: $$\begin{aligned} \Delta M_{11}&=&\frac{-3Q_{111111}-6Q_{111122}-3Q_{112222}+3Q_{22}Q_{1122}+9Q_{11}Q_{1111}+ 6Q12Q_{1112}+9Q_{11}Q_{1122}+6Q_{12} Q_{1222}-3Q_{22}Q_{1111}}{4\xi\sigma_W^2}\nonumber \\ \Delta M_{12}&=&\frac{-3Q_{111112}-6Q_{111222}-3Q_{122222}+3Q_{22}Q_{1112}+ 9Q_{11}Q_{1222}+3Q_{22}Q_{1222}+ 6Q_{12}Q_{1122}+ 9Q_{11}Q_{1112}+6Q_{12}Q_{2222}} {4\xi\sigma_W^2}\nonumber \\ \Delta M_{13}&=&\frac{-Q_{111111}+2Q_{111122}+3Q_{112222}-3Q_{22}Q_{1122}+3Q_{11}Q_{1111}+ 2Q_{12}Q_{1112}-9Q_{11}Q_{1122}-6Q_{12}Q_{1222}+Q_{22}Q_{1111}} {4\xi\sigma_W^2}\nonumber \\ \Delta M_{14}&=&\frac{-3Q_{111112}-2Q_{111222}+Q_{122222}+6Q_{12}Q_{1122}+9Q_{11}Q_{1112}-3Q_{11}Q_{1222}+3Q_{22}Q_{1112}- Q_{22}Q_{1222}-2Q_{12}Q_{2222}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{21}&=&\frac{-3Q_{111112}-6Q_{111222}-3Q_{122222}+6Q_{12}Q_{1122}+3Q_{11}Q_{1112}+9Q_{22}Q_{1112}+3Q_{11}Q_{1222}+ 9Q_{22}Q_{1222}+6Q_{12}Q_{1111}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{22}&=&\frac{-3Q_{111122}-6Q_{112222}-3Q_{222222}+6Q_{12}Q_{1112}+3Q_{11}Q_{2222}+6Q_{12}Q_{1222}+9Q_{22}Q_{1122}+ 3Q_{11}Q_{1122}+9Q_{22}Q_{2222}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{23}&=&\frac{-Q_{111112}+2Q_{111222}+6Q_{122222}-6Q_{12}Q_{1122}+Q_{11}Q_{1112}+3Q_{22}Q_{1112}-3Q_{11}Q_{1222}- 9Q_{22}Q_{1222}+2Q_{12}Q_{1111}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{24}&=&\frac{-3Q_{111122}-2Q_{112222}+Q_{222222}+9Q_{22}Q_{1122}+3Q_{11}Q_{1122}-Q_{11}Q_{2222}+6Q_{12}Q_{1112}- 2Q_{12}Q_{1222}-3Q_{22}Q_{2222}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{31}&=&\frac{-3Q_{111111}+6Q_{111122}+9Q_{112222}-9Q_{22}Q_{1122}+9Q_{11}Q_{1111}-18Q_{12}Q_{1112}+9Q_{11}Q_{1122}- 18Q_{12}Q_{1222}-9Q_{22}Q_{1111}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{32}&=&\frac{-3Q_{111112}+6Q_{111222}+9Q_{122222}-9Q_{22}Q_{1112}+9Q_{11}Q_{1222}-9Q_{22}Q_{1222}-18Q_{12}Q_{1122}+ 9Q_{11}Q_{1112}-18Q_{12}Q_{2222}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{33}&=&\frac{-Q_{111111}+6Q_{111122}-9Q_{112222}+9Q_{22}Q_{1122}+3Q_{11}Q_{1111}-6Q_{12}Q_{1112}-9Q_{11}Q_{1122}+ 18Q_{12}Q_{1222}-3Q_{22}Q_{1111}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{34}&=&\frac{-3Q_{111112}+10Q_{111222}-3Q_{122222}-18Q_{12}Q_{1122}+9Q_{11}Q_{1112}-3Q_{11}Q_{1222}-9Q_{22}Q_{1112}+ 3Q_{22}Q_{1222}+6Q_{12}Q_{2222}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{41}&=&\frac{-9Q_{111112}-6Q_{111222}+3Q_{122222}+18Q_{12}Q_{1122}+9Q_{11}Q_{1112}-9Q_{22}Q_{1112}+9Q_{11}Q_{1222}- 9Q_{22}Q_{1222}+18Q_{12}Q_{1111}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{42}&=&\frac{-9Q_{111122}-6Q_{112222}+3Q_{222222}+18Q_{12}Q_{1112}+9Q_{11}Q_{2222}+18Q_{12}Q_{1222}-9Q_{22}Q_{1122}+ 9Q_{11}Q_{1122}-9Q_{22}Q_{2222}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{43}&=&\frac{-3Q_{111112}+10Q_{111222}-3Q_{122222}+18Q_{12}Q_{1122}+3Q_{11}Q_{1112}-3Q_{22}Q_{1112}-9Q_{11}Q_{1222}+ 9Q_{22}Q_{1222}+6Q_{12}Q_{1111}} {4\xi\sigma_W^2} \nonumber \\ \Delta M_{44}&=&\frac{-9Q_{111122}+6Q_{112222}-Q_{222222}+9Q_{22}Q_{1122}+9Q_{11}Q_{1122}-3Q_{11}Q_{2222}+18Q_{12}Q_{1112}- 6Q_{12}Q_{1222}+3Q_{22}Q_{2222}}{4\xi\sigma_W^2} \end{aligned}$$ [^1]: http://www.physics.drexel.edu/\~goldberg/flexion [^2]: http://www.astro.caltech.edu/\~rjm/shapelets/ [^3]: http://terapix.iap.fr

--- abstract: 'The isothermal gravitational collapse and fragmentation of a molecular cloud region and the subsequent formation of a protostellar cluster is investigated numerically. The clump mass spectrum which forms during the fragmentation phase can be well approximated by a power law distribution $dN/dM \propto M^{-1.5}$. In contrast, the mass spectrum of protostellar cores that form in the centers of Jeans unstable clumps and evolve through accretion and $N$-body interaction is best described by a log-normal distribution. Assuming a star formation efficiency of $\sim\!10\;\!$%, it is in excellent agreement with the IMF of multiple stellar systems.' author: - RALFKLESSEN - ANDREASBURKERT title: 'Fragmentation in Molecular Clouds: The Formation of a Stellar Cluster' --- Introduction {#sec:intro} ============ Understanding the processes leading to the formation of stars is one of the fundamental challenges in astronomy and astrophysics. However, theoretical models considerably lag behind the recent observational progress. The analytical description of the star formation process is restricted to the collapse of isolated, idealized objects (Whitworth & Summers 1985). Much the same applies to numerical studies (e.g. Boss 1997, Burkert et al. 1997 and reference therein). Previous numerical models that treated cloud fragmentation on scales larger than single, isolated clumps were strongly constrained by numerical resolution. Larson (1978), for example, used just 150 particles in an SPH-like simulation. Whitworth et al. (1995) were the first who addressed star formation in an entire cloud region using high-resolution numerical models. However, they studied a different problem: fragmentation and star formation in the shocked interface of colliding molecular clumps. While clump-clump interactions are expected to be abundant in molecular clouds, the rapid formation of a whole star cluster requires gravitational collapse on a size scale which contains many clumps and dense filaments. Here, we present a high-resolution numerical model describing the dynamical evolution of an entire [*region*]{} embedded in the interior of a molecular cloud. We follow the fragmentation into dense protostellar cores which form a hierarchically structured cluster. Numerical Technique and Initial Condition {#sec:num-technique-and-initial-cond} ========================================= To follow the time evolution of the system, we use smoothed particle hydrodynamics (SPH: for a review see Monaghan 1992) which is intrinsically Lagrangian and can resolve very high density contrasts. We adopt a standard description of artificial viscosity (Monaghan & Gingold 1983) with the parameters $\alpha_v = 1$ and $\beta_v = 2$. The smoothing lengths are variable in space and time such that the number of neighbors for each particle remains at approximately fifty. The system is integrated in time using a second order Runge-Kutta-Fehlberg scheme, allowing individual timesteps for each particle. Once a highly condensed object has formed in the center of a collapsing cloud fragment and has passed beyond a certain density, we substitute it by a ‘sink’ particle which then continues to accrete material from its infalling gaseous envelope (Bate et al. 1995). By doing so we prevent the code time stepping from becoming prohibitively small. This procedure implies that we cannot describe the evolution of gas inside such a sink particle. For a detailed description of the physical processes inside a protostellar core, i.e. its further collapse and fragmentation, a new simulation just concentrating on this single object with the appropriate initial conditions taken from the larger scale simulation would be necessary (Burkert et al. 1998). To achieve high computational speed, we have combined SPH with the special purpose hardware device GRAPE (Ebisuzaki et al. 1993), following the implementation described in detail by Steinmetz (1996). Since we wish to describe a region in the interior of a molecular cloud, we have to prevent global collapse. Therefore, we use periodic boundaries, applying the Ewald method in an PM-like scheme (Klessen 1997). The structure of molecular clouds is very complex, consisting of a hierarchy of clumps and filaments on all scales (e.g. Blitz 1993). Many attempts have been made to identify the clump structure and derive its properties (Stutzki & G[ü]{}sten 1990, Williams et al. 1994). We choose as starting conditions Gaussian random density fluctuations with a power spectrum $P(k) \propto 1/k^N$ and $0 \le N \le 3$. The fields are generated by applying the Zel’dovich (1970) approximation to an originally homogeneous gas distribution: we compute a hypothetical field of density fluctuations in Fourier space and solve Poisson’s equation to obtain the corresponding self-consistent velocity field. These velocities are then used to advance the particles in one single big timestep $\delta t$. We present simulations with $50\;000$ and $500\;000$ SPH particles, respectively. 1.2cm (12,10) ( 0.0, 4.9)[=6.5cm ]{} ( 5.5, 4.9)[=6.5cm ]{} ( 0.0,-0.0)[=6.5cm ]{} ( 5.5,-0.0)[=6.5cm ]{} ( 1.0, 5.2)[$t=0.0$]{} ( 6.5, 5.2)[$t=0.7$]{} ( 1.0, 0.3)[$t=1.3$]{} ( 6.5, 0.3)[$t=2.0$]{} A Case Study {#sec:case-study} ============ As a case study, we present the time evolution of a region in the interior of a molecular cloud with $P(k) \propto 1/k^2$ and containing a total mass of 222 Jeans masses determined from the temperature and mean density of the gas. Figure \[fig:3D-plots\] depicts snapshots of the system initially, and when 10, 30 and 60 per cent of the gas mass has been accreted onto the protostellar cores. Note that the cube has to be seen periodically replicated in all directions. At the beginning, pressure smears out small scale features, whereas large scale fluctuations start to collapse into filaments and knots. After $t \!\approx\! 0.3$, the first highly-condensed cores form in the centers of the most massive and densest Jeans unstable gas clumps and are replaced by sink particles. Soon clumps of lower mass and density follow, altogether creating a hierarchically-structured cluster of accreting protostellar cores. For a realistic timing extimate, the Zel’dovich shift interval $\delta t = 2.0$ has to be taken into account. In dimension-less time units, the free-fall time of the isolated cube is $\tau_{\rm ff} = 1.4$. Scaling Properties ------------------ The gas is isothermal. Hence, the calculations are scale free, depending only on one parameter: the dimensionless temperature $T\equiv E_{\rm int}/|E_{\rm pot}|$, which is defined as the ratio between the internal and gravitational energy of the gas. The model can thus be applied to star-forming regions with different physical properties. In the case of a dark cloud with mean density $n(H_2)\simeq100\,$cm$^{-3}$ and a temperature $T\simeq10\,$K like Taurus-Auriga, the computation corresponds to a cube of length 10$\,$pc and a total mass of $6\,300\,$M$_{\odot}$. The dimensionless time unit corresponds to $2.2 \times 10^6\,$yrs. For a high-mass star-forming region like Orion with $n(H_2) \simeq 10^5\,$cm$^{-3}$ and $T\simeq 30\,$K these values scale to $0.5\,$pc and $1\,000\,$M$_{\odot}$, respectively. The time scale is $6.9\times10^4\,$yrs. The Importance of Dynamical Interaction and Competitive Accretion ----------------------------------------------------------------- The location and the time at which protostellar cores form, is determined by the dynamical evolution of their parental gas clouds. Besides collapsing individually, clumps stream towards a common center of attraction where they merge with each other or undergo further fragmentation. The formation of dense cores in the centers of clumps depends strongly on the relation between the timescales for individual collapse, merging and sub-fragmentation. Individual clumps may become Jeans unstable and start to collapse to form a condensed core in their centers. When clumps merge, the larger new clump continues to collapse, but contains now a [*multiple*]{} system of cores in its center. Now sharing a common environment, these cores compete for the limited reservoir of gas in their surrounding (see e.g. Price & Podsiadlowski 1995, Bonnell et al. 1997). Furthermore, the protostellar cores interact gravitationally with each other. As in dense stellar clusters, close encounters lead to the formation of unstable triple or higher order systems and alter the orbital parameters of the cluster members. As a result, a considerable fraction of “protostellar” cores get expelled from their parental clump. Suddenly bereft of the massive gas inflow from their collapsing surrounding, they effectively stop accreting and their final mass is determined. Ejected objects can travel quite far and resemble the weak line T Tauri stars found via X-ray observation in the vicinities of star-forming molecular clouds (e.g. Neuh[ä]{}user et al. 1995). Mass Spectrum – Implications for the IMF ---------------------------------------- Figures \[fig:mass-distr\]a – d describe the mass distribution of identified gas clumps (thin lines) and of protostellar cores (thick lines) that formed within unstable clumps in a simulation analogous to Fig. \[fig:3D-plots\], but with 10 times higher resolution. To identify individual clumps we have developed an algorithm similar to the method described by Williams et al. (1994), but based on the framework of SPH. As reference, we also plot the observed canonical form for the clump mass spectrum, $dN/dM \propto M^{-1.5}$ (Blitz 1993), which has a slope of $-0.5$ when plotting $N$ versus $M$. Note that our initial condition does not exhibit a clear power law clump spectrum. The Zel’dovich approximation generates an overabundance of small scale fluctuations. However, in the subsequent evolution, these small clumps are immediately damped by pressure forces and non-linear gravitational collapse begins to create a power-law like mass spectrum. 0.85cm (13.6,5.8) ( 1.00, 2.70)[=2.7625cm ]{} ( 4.25, 2.70)[=2.7625cm ]{} ( 1.00, 0.10)[=2.7625cm ]{} ( 4.25, 0.10)[=2.7625cm ]{} ( 9.00, 0.10)[=5.525cm ]{} A common feature in all our simulations is the broad mass spectrum of protostellar cores which peaks slightly above the overall [*Jeans mass*]{} of the system. This is somewhat surprising, given the fact that the evolution of individual cores is highly influenced by complex dynamical processes. In a statistical sense, the system retains ‘knowledge’ of its (initial) average properties. The present simulations cannot resolve subfragmentation in condensed cores. Since detailed simulations show that perturbed cores tend to break up into multiple systems (e.g. Burkert et al. 1997), we can only determine the mass function of multiple systems. Our simulations predict an initial mass function with a [*log-normal*]{} functional form. Figure \[fig:mass-distr\]e compares the results of our calculations with the observed IMF for multiple systems from Kroupa et al. (1990). Assuming a typical Jeans mass $M_{\rm J} \approx 1\,$M$_{\odot}$ and a star formation efficiency of individual cores of  10$\:\!$%, the agreement between the numerically-calculated mass function and the observed IMF for multiple systems (thick dashed line; from Kroupa et al. 1990) is excellent. For comparison, also the IMF corrected for binary stars (Kroupa et a. 1993) is indicated as thin solid line, together with the mass function from Salpeter (1955) as thin dashed line. Discussion {#sec:summary} ========== Large-scale collapse and fragmentation in molecular clouds leads to a hierarchical cluster of condensed objects whose further dynamical evolution is extremely complex. The agreement between the numerically-calculated mass function and the observations strongly suggests that gravitational fragmentation and accretion processes dominate the origin of stellar masses. The final mass distribution of protostellar cores in isothermal models is a consequence of the chaotic kinematical evolution during the accretion phase. Our simulations give evidence, that the star formation process can best be understood in the frame work of a probabilistic theory. A sequence of statistical events may naturally lead to a log-normal IMF (see e.g. Zinnecker 1984, Adams & Fatuzzo 1996; also Price & Podsiadlowski 1995, Murray & Lin 1996, Elmegreen 1997). Adams, R.C., Fatuzzo, M., 1995, [*ApJ*]{}, [**464**]{}, 256 Bate, M.R., Burkert, A., 1997, [*MNRAS*]{}, [**288**]{}, 1060 Bate, M.R., Bonnell, I.A, Price, N.M., 1995, [*MNRAS*]{}, [**277**]{}, 362 Blitz, L., 1993, in [*Protostars and Planets III*]{}, eds. E.H. Levy & J.J. Lunine, Univ. of Arizona Bonnell, I.A., Bate, M.R., Clarke, C.J., Pringle, J.E., 1997, [*MNRAS*]{}, [**285**]{}, 201 Boss, A., 1997, [*ApJ*]{}, [**483**]{}, 309 Burkert, A., Bate, M.R., Bodenheimer, P., 1997, [*MNRAS*]{}, [**289**]{}, 497 Burkert, A., Klessen, R.S., Bodenheimer, P., 1998, in [*The Orion Complex Revisited*]{}, eds. M. McCaughrean & A. Burkert, ASP Conference Series, in press Ebisuzaki, T., Makino, J., Fukushige, T., Taiji, M., Sugimoto, D., Ito, T., Okumura, S., 1993, [*PASJ*]{}, [**45**]{}, 269 Elmegreen, B.G., 1997, [*ApJ*]{}, [**486**]{}, 944 Klessen, R.S., 1997, [*MNRAS*]{}, [**292**]{}, 11 Kroupa, P., Tout, C.A. & Gilmore, G. 1990, [*MNRAS*]{}, [**244**]{}, 76 Kroupa, P., Tout, C.A. & Gilmore, G. 1993, [*MNRAS*]{}, [**262**]{}, 545 Larson, R.B., 1978, [*MNRAS*]{}, [**184**]{}, 69 Monaghan, J.J., 1992, [*ARAA*]{}, [**30**]{}, 543 Monaghan, J.J., Gingold, R.A., 1983, [*J. Comp. Phys.*]{},[**52**]{}, 135 Murray, S.D., Lin, D.N.C., 1996, [*ApJ*]{}, [**467**]{}, 728 Neuh[ä]{}user, R., Sterzik, M.F., Schmitt, J.H.M.M., Wichmann, R., Krautter, J., 1995, [*A&A*]{}, [**297**]{}, 391 Price, N.M., Podsiadlowski, Ph., 1995, [*MNRAS*]{}, [**273**]{}, 1041 Salpeter, E.E. 1955, [*ApJ*]{}, [**121**]{}, 161 Steinmetz, M., 1996, [*MNRAS*]{}, [**278**]{}, 1005 Stutzki, J., G[ü]{}sten, R., 1990, [*ApJ*]{}, [**356**]{}, 513 Williams, J.P, De Geus, E.J., Blitz, L., 1994, [*ApJ*]{}, [**428**]{}, 693 Whitworth, A.P., Summers, D., 1985, [*MNRAS*]{}, [**214**]{}, 1 Whitworth, A.P., Chapman, S.J., Bhattal, A.S., Disney, M.J., Pongracic, H., Turner, J.A., 1995, [*MNRAS*]{}, [**277**]{}, 727 Zel’dovich, Y.B., 1970, [*A&A*]{}, [**5**]{}, 84 Zinnecker, H., 1984, [*MNRAS*]{}, [**210**]{}, 43

--- abstract: | Recently we calculated relativistic recoil corrections to the energy levels of the low lying states in muonic hydrogen induced by electron vacuum polarization effects. The results were obtained by Breit-type and Grotch-type calculations. The former were described in our previous papers in detail, and here we present the latter. The Grotch equation was originally developed for pure Coulomb systems and allowed one to express the relativistic recoil correction to order $(Z\alpha)^4m^2/M$ in terms of the relativistic non-recoil contribution $(Z\alpha)^4m$. Certain attempts to adjust the method to electronic vacuum polarization took place in the past, however, the consideration was incomplete and the results were incorrect. Here we present a Groth-type approach to the problem and in a series of papers consider relativistic recoil effects in order $\alpha(Z\alpha)^4m^2/M$ and $\alpha^2(Z\alpha)^4m^2/M$. That is the first paper of the series and it presents a general approach, while two other papers present results of calculations of the $\alpha(Z\alpha)^4m^2/M$ and $\alpha^2(Z\alpha)^4m^2/M$ contributions in detail. In contrast to our previous calculation, we address now a variety of states in muonic atoms with a certain range of the nuclear charge $Z$. author: - 'Savely G. Karshenboim' - 'Vladimir G. Ivanov' - 'Evgeny Yu. Korzinin' title: 'Relativistic recoil effects in a muonic atom within a Grotch-type approach: General approach' --- Introduction ============ Spectroscopy of light muonic atoms was used for a while and provided us with certain important data on the nuclear structure. It was based on a study of the emission lines and had limited accuracy. Recently, the first successful laser-spectroscopy measurement on muonic hydrogen has opened a new generation of experiments. The experiment performed at PSI delivered the value of the Lamb shift in muonic hydrogen and allowed to determine the proton charge radius with unprecedented accuracy. Unexpectedly, that measurement has led to one of the currently largest controversies in QED related experiments. A strong discrepancy between the value of the proton charge radius obtained from muonic hydrogen [@nature] and that in ordinary hydrogen [@codata2010] is of about 5 standard deviations. Meantime, the latter value is in perfect agreement with a recent electron-proton scattering result [@mainz]. That circumstance has renewed interest in spectroscopy of muonic atoms. The low $l$ states and, mostly, the $1s$ and $2s$ states are sensitive to the finite-nuclear-size effects and have been used for a while to determine the charge radius for a broad range of nuclei from hydrogen [@nature] to uranium [@zumbro]. Higher-$l$ states are also of interest for more “metrological” measurements. In particular, the $3d_{5/2}{-}2p_{3/2}$ transition in muonic $^{24}$Mg and $^{28}$Si was used in [@mumass] to determine $m_\mu/m_e$. A similar measurement was also performed in pionic atoms to determine the pion mass. In such experiments one has to deal with X-ray transitions and then there is a problem in calibration of the X-ray standards. In [@pimass] the $5f{-}4g$ transition in pionic nitrogen and the $6h{-}5f$ one in pionic neon were compared with $5f{-}4g$ transitions in muonic oxygen. Higher $l$ states can also be of interest due to antiprotonic helium spectroscopy. At present, highly accurate data are available only for a three-body system, which includes a nucleus, antiproton and electron [@mep2:1; @mep2:2]. While the antiproton in a circular or a near circular state is rather immune against annihilation, the electron “protects” the antiprotonic state from collision quenching. Still, a possibility for a two-body antiprotonic helium ion has not been given up and such a system may be of experimental interest in the future. In this situation a theoretical study of low-lying states of circular states, such as $2p, 3d, 4g, 5f, 6h$ is of practical interest. Since the muon mass is substantially higher than the electron mass, one has to pay attention to recoil effects. To find recoil contributions to energy levels of a hydrogenic atom one can apply various approaches and, in particular, a Grotch-type one. A calculation of recoil corrections to order $m/M$ is possible in hydrogenic atoms exactly [@shabaev] (see also [@yelkhovsky]) without any expansion in $Z\alpha$. The result consists of two contributions, one is a result of one-photon exchange in an effective Dirac equation, while the other takes into account multi-photon exchanges. It is the one-photon exchange that was first derived in [@Gro67] without any expansion in $Z\alpha$. The Grotch equation is an efficient way to derive from the one-photon-exchange term the result which allows one to combine a few important features of theory of the energy levels and to obtain a result which incorporates - the leading nonrelativistic term (i.e. a result of the Schrödinger-Coulomb problem) exactly in $m/M$; - the complete relativistic series for infinitely heavy nucleus (i.e. a result of the Dirac-Coulomb problem) exactly in $(Z\alpha)$; - the leading relativistic recoil correction to energy in order $(Z\alpha)^4m^2/M$. On the other hand, the electronic vacuum polarization (eVP) effects and, in particular, the Uehling potential, play a crucial role in the theory of energy levels in muonic atoms. It is important to be able to calculate relativistic and recoil corrections to them for a variety of levels. Recently, such a relativistic recoil contribution of order $\alpha(Z\alpha)^4m^2/M$ was considered in various approaches for low-lying states in light muonic atoms [@jentschura; @borie11; @pra] (see also [@pach96; @pach04; @borie05] for earlier evaluations). Results on $\alpha^2(Z\alpha)^4m^2/M$ can be found in [@a2Za4]. Here, we rederive the Grotch equation for a pure Coulomb problem and generalize it for a broad class of potentials. The generalized approach allows one to find relativistic recoil eVP corrections in the first and second order in $\alpha$, which are studied in subsequent papers [@II; @III]. One-photon exchange in two-body bound systems\[s:ope\] ====================================================== The Coulomb bound two-body systems have a binding energy of order $(Z\alpha)^2m$, where $\alpha$ is the fine structure constant, $Z$ is the nuclear charge, $m$ is the mass of the orbiting particle, i.e. the lighter one in the bound system. Throughout the paper we apply relativistic units in which $\hbar=c=1$. These energy levels have various corrections due to the relativistic, recoil and QED effects and due to the nuclear structure. The $(Z\alpha)^2m$ term can be found by many different methods, while the methods to derive the corrections often depend on the nature of those corrections. A certain class of the corrections can be expressed in terms of the potentials and one can expect that for their evaluation it is possible to adjust approaches used for pure Coulomb calculations. The potential corrections and, in particular, those presented by the Uehling potential, are dominant QED effects for light and medium-$Z$ muonic atoms. Here we develop an effective approach to study relativistic recoil corrections in the first order in the electronic vacuum polarization. Electronic vacuum polarization (eVP) effects are responsible for the Uehling potential, but even for the relativistic recoil contribution one has to go somewhat beyond just the Uehling potential, just as for the calculation of the $(Z\alpha)^4m^2/M$ term one has to go beyond a pure Coulomb field. Here $M$ is the nuclear mass and appearance of the $m/M$ ratio indicates that recoil effects are involved. Throughout the paper we consider a point-like nucleus; however, in many situations the finite-nuclear-size effects can be treated as a small perturbation and, specifically, for low-$Z$ calculations and for a high $l$ medium-$Z$ case. In any case, the finite nuclear size affects the interaction between the muon and the nucleus; however, the effect can be still described as a kind of potential and the results obtained below can be in part adjusted for the extended nuclei. Relativistic recoil effects contribute to one-photon exchange as well as to many-photon exchanges. The Coulomb and Uehling potentials correspond to a dominant contribution in one-photon exchange. The one-photon contribution for the Coulomb case and Uehling term are depicted in Figs. \[f:c1gamma\] and \[f:u1gamma\], respectively. They are responsible for the entire nonrelativistic contribution to orders $(Z\alpha)^2m$ and $\alpha(Z\alpha)^2m$, respectively. Those contributions can be described by a potential. They partly include recoil effects in a sense, that one has to use the reduced mass $m_R = m M/(m+M)$ in calculations. The result for the Uehling correction can be achieved analytically in terms of elementary functions [@uehl_cl; @uehl_an]. The potential approach can be also applied for a relativistic evaluation with the Dirac wave functions. For the Uehling potential the energy with the Dirac wave functions is known in closed analytic terms [@rel1; @rel2]. Indeed, as far as the wave functions for Schrödinger-Coulomb and Dirac-Coulomb problems and the dispersion presentation of the Uehling potential, such as $$\label{e:uehling} V_U(r) = -\frac{\alpha{(Z\alpha)}}{\pi} \int_0^1 dv \, \rho_e(v) \frac{e^{-\lambda r}}{r} \,,$$ where $$\begin{aligned} \lambda&=&\frac{2m_e}{\sqrt{1-v^2}} \,,\nonumber\\ \rho_e(v)&=&\frac{v^2(1-v^2/3)}{1-v^2}\,,\end{aligned}$$ are well known, a numerical calculation has never been a problem (see, e.g., [@pach96; @jentschura; @borie11]). Nevertheless, analytic evaluations allow one to find various useful asymptotics [@rel1; @uehl_an; @rel2]. We note that the Uehling potential is smaller than the Coulomb potential roughly by a factor of $\alpha/\pi$ in any kinematic area. Similarly, we see that the eVP potential related to the second-order correction possesses the same property — it is smaller than the Coulomb exchange in any kinematic area by a factor of $(\alpha/\pi)^2$. Since general behavior of the eVP-induced potentials is somewhat similar to the $(\alpha/\pi)V_C(r)$ and $(\alpha/\pi)^2V_C(r)$, we can hope that whatever we use for a pure Coulomb problem, it may be adjusted for eVP effects, including the relativistic recoil. Meanwhile, neither a complete calculation of the one-photon exchange (Figs. \[f:c1gamma\] and \[f:u1gamma\]) can be identically presented in terms of a potential, nor can the two-photon one exchange (Figs. \[f:c2gamma\] and \[f:u2gamma\]) be in general ignored. The approach developed by Grotch and Yennie [@Gro67] allowed one to resolve this problem for exchange by free photons (Figs. \[f:c1gamma\] and \[f:c2gamma\]) and here [@pra] we generalize it, following our previous paper, for the case of the eVP contributions. At first, we have to address a question of a possibility to use a certain relativistic equation with a kind of an effective potential for a calculation of recoil effects. The one-photon-exchange contribution can be evaluated with the help of the photon propagator, which in the Coulomb gauge takes the form $$\begin{aligned} \label{g:coul} D_{00}^C&=&-\frac{1}{{\mathbf k^2}}\;, \nonumber\\ D_{i0}^C&=&0\;,\nonumber\\ D_{ij}^C&=&-\left(\delta_{ij}-\frac{k_ik_j}{{\mathbf k^2}}\right)\;\frac{1}{k^2}\;,\end{aligned}$$ where $k^2=k_0^2-{\mathbf k}^2$. Note, that only the static part of $D_{00}$ produces a contribution in the non-recoil limit and thus is responsible for an electrostatic potential. The other components of the photon propagators in general depend on the choice of the gauge and they are not directly related to $D_{00}$. For this reason the complete one-photon contribution cannot in general be expressed in terms of an electrostatic potential. The one-photon contribution in the Coulomb gauge can be reduced for the $m/M$ correction to its static approximation (i.e. neglecting the $k_0$ dependence) and thus to several potential-like terms because - there is no $k_0$ dependence in $D_{00}$ and thus no retardation effects are involved (if they were involved, that still would be of reduced importance because they are proportional to $k_0^2/{\mathbf k}^2$ and for atomic energy levels that would lead to relativistic corrections proportional to $(m/M)^2$, while here we are interested in the $m/M$ correction only); - $D_{i0}=0$; - $D_{ij}$ involves lower components of the spinor for the nucleus and thus the contribution is proportional to at least $m/M$, which means that the retardation effects in the $D_{ij}$ term are of order $(m/M)^2$ or higher and negligible. In the next sections we apply the static approximation to the one-photon exchange and develop an effective potential equation, first for a pure Coulomb problem and next for a perturbed Coulomb problem. The remaining question is about two-photon-exchange contributions for the $(Z\alpha)^4m^2/M$ correction. (In any effective Dirac equation approach, and we follow such an approach since we are to find a Grotch-type effective Dirac equation, it is assumed that certain two-photon-exchange subtractions take place (see, e.g., [@ede] for detail).) This question was reviewed, e.g. in [@pra]. The two-photon contributions are of at least order $(Z\alpha)^5m^2/M$ in the Coulomb gauge because - there is no $k_0$ dependence in $D_{00}$ and thus there is no photon pole in two-photon exchange with two $D_{00}$ components; - $D_{i0}=0$, and thus there is no contribution which involves one $D_{i0}$ photon and one $D_{00}$ photon. That is sufficient to avoid any potential $(Z\alpha)^4m^2/M$ contribution. Grotch equation and its solution for the Coulomb bound systems\[s:gc\] ====================================================================== Once we are limiting our consideration to one-photon-contribution in a static approximation (i.e. at $k_0=0$), we can derive the Grotch equation for the free one-photon exchange (Fig. \[f:c1gamma\]) in order, after that, to generalize it step by step for a more general case, including the eVP contributions. Our consideration closely follows the original one by Grotch and Yennie [@Gro67]. Here we give a brief reminder of the derivation of the Grotch equation and its solution in order to describe every step which we will need to adjust to eVP contributions. The Grotch equation [@Gro67] is one of several effective Dirac equations for a two-particle system. It is important to reproduce the two most important features of any system of two fermions with an orbiting particle much lighter than the nucleus. The electron in ordinary hydrogen and the muon in muonic hydrogen are such particles. It is useful to consider the orbiting particle within a full relativistic consideration, while treating the nucleus in the leading nonrelativistic approximation. As a result, we may derive an equation, which correctly reproduces its limits both the Schrödinger-Coulomb equation with the reduced mass and the Dirac-Coulomb equation with the original mass of the muon (or electron). Indeed, the equation is also supposed to take into account certain relativistic recoil corrections. The uncertainty in the calculation of the static one-photon contribution is of order $(Z\alpha)^4(m/M)^2m$. The two-photon contribution is of order $(Z\alpha)^5m^2/M$. The desired equation is of the form of Dirac equation for a muon $$\left[\widehat{P}_n-\widehat{p}_N-m - \widetilde{V}_{1\gamma}\right]\Psi_n=0\,.$$ where $\widehat{A}=\gamma^\nu_{[\mu]} A_\nu$ and $P_n = (E_n, {\bf 0 )}$ (here $A^\nu$ is an arbitrary vector, $\nu$ is a relativistic 4-index, while $\mu$ stands for a muon.) This is an equation in the center-of-mass system. While the equation is for the muon energy and wave function, the quantized energy $E_n$ is for the two-body system and we should subtract from the whole 4-momentum $P_n$ the nuclear 4-momentum $p_N=\left(\sqrt{M^2+{\bf p}^2}, -{\bf p}\right)$, where ${\bf p}$ is the muon momentum. To obtain a one-particle equation from a two-body one it was suggested that one can present the two-body wave function $\Psi_{\mu N}$ in terms of the free nuclear spinor and the muon wave function $\psi$ $$\label{e:psi} \Psi_{\mu N}={1 \choose -\frac{{\bf p}\cdot{\mbox{\boldmath $\sigma$}}_N}{2M}} \psi\;.$$ This suggestion is not just an approximation in a sense that one can construct a perturbation theory and systematically take into account all the corrections required for a certain level of accuracy. The nuclear on-shell corrections are of relativistic nature for the nucleus and thus they are of higher order in $m/M$ and $Z\alpha$ than the leading recoil effects we study. The off-shell corrections can be found through many-photon exchange diagrams and a proper choice of gauge can eliminate them in the leading recoil order. The effective potential $ \widetilde{V}_{1\gamma}$ results from the static part of the one-photon exchange averaged over the nuclear part of the wave function in (\[e:psi\]). In the momentum space we find $$\begin{aligned} \label{wv} \widetilde{V}_{1\gamma}({\bf q},{\bf p})&=& -i\gamma_{\mu}^0 \gamma_{N}^0(Z\alpha) \left(1, -\frac{{\mbox{\boldmath $\sigma$}_N}\cdot{\bf q}}{2M}\right) \nonumber\\ &&\times \left[i\gamma_{N}^0\gamma_{\mu}^0D_{00}({\bf k})+i\gamma_{N}^i\gamma_{\mu}^jD_{ij}({\bf k})\right] {1 \choose -\frac{{\mbox{\boldmath $\sigma$}_N}\cdot{\bf p}}{2M}} \nonumber\\\nonumber\\ &=&-\frac{Z\alpha}{{\bf k}^2} \Bigg\{1+\frac{1}{2M}\bigg[{\mbox{\boldmath $\alpha$}}_\mu\cdot({\bf p}+{\bf q}) \nonumber\\ &&\phantom{9} -\frac{({\mbox{\boldmath $\alpha$}}_\mu\cdot{\bf k})\ ({\bf k}\cdot({\bf p}+{\bf q}))}{{\bf k}^2}\bigg] \nonumber \\ &&-\frac{1}{2M}[{\bf k}\times i{\mbox{\boldmath $\sigma$}}_N]\cdot{\mbox{\boldmath $\alpha$}}_\mu\nonumber\\ &&+{{\cal O}\left( (Z\alpha)^4\left(\frac{m}{M}\right)^2m \right)}\Bigg\}\,,\end{aligned}$$ where ${\bf k} = {\bf p} - {\bf q}$. Here, the neglected term is not ${{\cal O}\left( (Z\alpha)^4({m}/{M})^2m \right)}$ by itself, but it represents an operator, the matrix element of which over the atomic wave function is ${{\cal O}\left( (Z\alpha)^4({m}/{M})^2m \right)}$. This effective potential includes a nuclear-spin-dependent term which is responsible for the hyperfine splitting. It is of order $(Z\alpha)^4m^2/M$. However, experimentally and theoretically the hyperfine structure effects are well separated from the Lamb shift effects. We consider this term as a perturbation and neglect the hyperfine-interaction term (i.e. we average over the nuclear spin). Once we average the results over the nuclear spin, i.e. over the hyperfine structure, we note that all the remaining nuclear-spin effects appear only in order $(m/M)^2$ (see, e.g., [@spin0; @spin1; @spin_nov; @spin_r]) and thus this derivation, started for the nuclear spin 1/2, is now valid for a nucleus with an arbitrary spin. That is the last crucial step to obtain the Grotch equation [@Gro67] and we arrive at that in coordinate space $$\begin{aligned} \label{Dirac_C} &&\biggl( {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p} + \beta m + \frac{\mathbf{p}^2}{2M} + V_C + \frac{1}{2M} \left\{ {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p}, V_C \right\}\nonumber\\ &~& +\frac{1}{4M} \left[ {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p}, [\mathbf{p}^2, W_C] \right] \biggr) \psi(r) = E \psi(r) \,.\end{aligned}$$ where the operator $W_C$ appears due to taking into account the $D_{ij}^C$ components of the photon propagator. It is essential that it can be expressed in a certain way through $V_C$, which is defined through $D_{00}^C$. In particular, for free one-photon-exchange (Fig. \[f:c1gamma\]) and the relation between the Coulomb gauge is of the form$$\label{c_wc} W_C({\mathbf k}) = -\frac{2V_C({\mathbf k})}{{\mathbf k}^2} \,.$$ For the case of the Coulomb gauge one finds in coordinate and momentum space $$\begin{aligned} V_C(r) &=& -\frac{{Z\alpha}}{r}\,,\nonumber\\ V_C({\mathbf k}) &=& -\frac{4\pi{Z\alpha}}{{\mathbf k}^2} \,,\end{aligned}$$ and $$\begin{aligned} \label{wc} W_C(r) &=& -{Z\alpha}r \,,\nonumber\\ W_C({\mathbf k}) &=& \frac{8\pi{Z\alpha}}{{\mathbf k}^4} \,.\end{aligned}$$ While the leading part of $D_{00}^C$ in any gauge should produce the Coulomb term $V_C$, the shape of the Hamiltonian in Eq. (\[Dirac\_C\]) and a particular shape of $W_C$ depends on the gauge chosen. The effective equation above can be solved in a closed analytic form after applying a series of transformations [@Gro67]. We start with rearranging the Hamiltonian $$\begin{aligned} \label{Dirac_CH} H&=&\biggl( {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p} + \beta m + \frac{\mathbf{p}^2}{2M} + V_C + \frac{1}{2M} \left\{ {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p}, V_C \right\}\nonumber\\ &~&~~~ +\frac{1}{4M} \left[ {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p}, [\mathbf{p}^2, W_C] \right] \biggr)\end{aligned}$$ as following $$\label{Dirac2_C} H = H_0+\delta H + {\cal O} \left( {(Z\alpha)}^4 \frac{m^3}{M^2}\right) \,,$$ where $$\label{c:hoh1} H_0=H_1 + \frac{H_1^2-m^2}{2M} + \frac{1}{4M} [H_1,[\mathbf{p}^2,W_C]] \,,$$ $$\label{c:h1} H_1 = {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p} + \beta m + V_C \frac{1-\beta m/M}{1-(m/M)^2} \,,$$ and $$\label{c:dh} \delta H = -\left( \frac{V_C^2}{2M} + \frac{1}{4M} [V_C,[\mathbf{p}^2,W_C]] \right) \,.$$ The correction, neglected in (\[Dirac2\_C\]), is indeed an operator; its matrix elements over bound states are of order ${\cal O} \left( {(Z\alpha)}^4\frac{m^3}{M^2} \right)$, which is explicitly shown in (\[Dirac2\_C\]). In this sense Eq. (\[Dirac2\_C\]) is not correct as an operator identity, but it is sufficiently valid for all matrix elements for the bound states. We note that due to the relation between $V_C$ and $W_C$ (\[c\_wc\]) the last term vanishes for the Coulomb potential in the Coulomb gauge $$\label{c:dh0} \delta H=0\;.$$ To solve Eq. (\[Dirac\_C\]) within the required accuracy is the same as to solve equation $$H_0\psi_0=E_0 \psi_0 \,,$$ where $E=E_0$ and $\psi=\psi_0$ for the pure Coulomb case. To deduce $E_0$ and $\psi_0$ we should first find a solution of equation $$\label{c:eh1} H_1 \psi_1 = E_1 \psi_1\;.$$ Looking for it in the form $$\psi_1 = ( 1 + \beta \xi ) \widetilde\psi\;,$$ one finds that $\widetilde\psi$ is a solution of an effective one-particle Dirac-Coulomb equation $$\label{c:h:tilde} \left[ {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p} + \beta \widetilde m - \frac{\widetilde{Z\alpha}}{Z\alpha} V_C(r) \right] \widetilde \psi = \widetilde E \widetilde \psi$$ with an effective mass $$\label{c:effm} \widetilde m = \frac{m\left( 1-\frac{E_1}{M} \right)}{\sqrt{1-\left(\frac{m}{M}\right)^2}}$$ and an effective Coulomb coupling constant $$\begin{aligned} \label{c:effa} \widetilde{{Z\alpha}} &=& \frac{Z\alpha}{\sqrt{1-\left(\frac{m}{M}\right)^2}}\nonumber\\ &=& Z\alpha \left[ 1 + {{\cal O}\left( \left( \frac{m}{M} \right)^2 \right)} \right]\label{c:effa1} \,,\end{aligned}$$ where $$\begin{aligned} \label{c:xi} \xi &=& \frac{M}{m} \left( 1-\sqrt{1-\left(\frac{m}{M}\right)^2} \right)\nonumber\\ &=& \frac{m}{2M} \left[ 1 + {\cal O} \left( \left( \frac{m}{M} \right)^2 \right) \right]\label{c:xi1} \,.\end{aligned}$$ The solutions of Eq. (\[c:h:tilde\]) are similar to the well-known solutions of the conventional Dirac-Coulomb equation (see, e.g. [@IV]), with the only difference being that the parameters $m$ and $Z\alpha$ must be replaced by effective values $\widetilde m$ and $\widetilde{{Z\alpha}}$, defined in (\[c:effm\]) and (\[c:effa\]). The energies $E_1$ are related to the known eigenvalues of the effective equation (\[c:h:tilde\]), $\widetilde E$, by the equation $$\label{c:ee1} \widetilde E = \frac{E_1-\frac{m^2}{M}}{\sqrt{1-(\frac{m}{M})^2}} \,.$$ The eigenvalues and eigenfunctions of Hamiltonian $H_0$ in Eq. [(\[Dirac\_C\])]{}, according to [(\[Dirac2\_C\])]{}), are related to $E_1$ and $\psi_1$, as $$\begin{aligned} \label{c:e0} E_0&=&E_1 + \frac{E_1^2-m^2}{2M}\\ &=& \widetilde E + \frac{{\widetilde E}^2+m^2}{2M} + {{\cal O}\left( \frac{m^3}{M^2} \right)} \;,\\ \label{c:psi0} \psi_0 &=& N\, \left[ 1-\frac{1}{4M} [ \mathbf{p}^2, W_C ] + {{\cal O}\left( \left(\frac{m}{M}\right)^2 {(Z\alpha)}^4 \right)} \right]\nonumber\\ &~&\times ( 1 + \beta \xi ) \widetilde\psi \,,\end{aligned}$$ where $N$ is a normalization constant, for which one can find (see, e.g., [@decay]) $$\begin{aligned} N^2 &=& \frac{1}{1+2\xi \widetilde E/\widetilde m+\xi^2}\\ &=& 1-\frac{m}{M} + \frac{{(Z\alpha)}^2}{2n^2} \frac{m}{M}\nonumber\\ &&+ {\cal O} \left( \left( \frac{m}{M} \right)^2 \right) + {\cal O} \left( \frac{m}{M} {(Z\alpha)}^4 \right) \,.\end{aligned}$$ This evaluation is not yet completed. We note that the energy $E_0$ is expressed in terms of $E_1$ (\[c:e0\]), and the latter in terms of $\widetilde{E}$ (\[c:ee1\]). Meanwhile, $\widetilde{E}$ is a function of $\widetilde{m}$ (\[c:effm\]) and $\widetilde{Z\alpha}$ (\[c:effa\]). The effective mass $\widetilde{m}$ in its turn depends on $E_1$ as follows from Eq. (\[c:effm\]). To proceed further, we note that for the Dirac-Coulomb problem $$\label{tm:dc} E_{DC} = f_{C}(Z\alpha)\,m \,,$$ and thus the value of $$\label{tildeF:def}\widetilde{F} = \frac{\widetilde{E}}{\widetilde{m}}\,, $$ being equal to $f_{C}(\widetilde{Z\alpha})$, does not depend on the effective mass of the orbiting particle $\widetilde{m}$, while the effective charge $\widetilde{Z\alpha}$, as follows from Eq. (\[c:effa\]), does not depend on energy. This allows simplifications. Applying Eqs. (\[tildeF:def\]) and (\[c:effm\]) to (\[c:ee1\]), we obtain $$\label{tm:e1f} E_1= m \frac{\widetilde{F}+\frac{m}{M}}{1+\frac{m}{M}\widetilde{F}} \,,$$ and, using (\[c:e0\]), $$\begin{aligned} \label{tm:exa1} E_0 &=& m + m\left(1-\frac{m}{M}\right)(\widetilde{F}-1) \nonumber\\ && - \frac{m^2}{2M} (\widetilde{F}-1)^2 \frac{\left(1-\frac{m}{M}\right)\left(1+\frac{m}{M}+2\frac{m}{M} \widetilde{F}\right)} {\left(1+\frac{m}{M} \widetilde{F}\right)^2}\nonumber\;.\end{aligned}$$ Since $$\widetilde{F}-1={\cal O}\left((Z\alpha)^2\right)\;,$$ we can efficiently expand $$\begin{aligned} \label{tm:exp1} E_0 &=& m +m\left(1-\frac{m}{M}\right)(\widetilde{F}-1) \nonumber\\ && - \frac{m^2}{2M}(\widetilde{F}-1)^2 \frac{\left(1-\frac{m}{M}\right)\left(1+3\frac{m}{M}\right)}{\left(1+\frac{m}{M}\right)^2} \nonumber\\ && + {{\cal O}\left( m\left(\frac{m}{M}\right)^3{(Z\alpha)}^6 \right)}\;,\\ \widetilde m &=& m\sqrt{\frac{1-\frac{m}{M}}{1+\frac{m}{M}}} \left[ 1 -\frac{\frac{m}{M}}{1+\frac{m}{M}} (\widetilde{F}-1)\right.\nonumber\\ && +\frac{\left(\frac{m}{M}\right)^2}{\left(1+\frac{m}{M}\right)^2} (\widetilde{F}-1)^2 \nonumber\\ && + \left.{{\cal O}\left( \left(\frac{m}{M}\right)^3{(Z\alpha)}^6 \right)} \right]\label{tm:exp2} \,.\end{aligned}$$ For the pure Coulomb problem it is sufficient to transform Eq. (\[tm:exp1\]), neglecting terms of order $(Z\alpha)^4(m/M)^2m$. We note, comparing $\widetilde{F}$ and $$F=f_{C}({Z\alpha})\;,$$ that we have to distinguish between $\widetilde{Z\alpha}$ and ${Z\alpha}$ only in the leading term of $(\widetilde{F}-1)$ $$F= 1 + \left( \frac{{Z\alpha}}{\widetilde{Z\alpha}} \right)^2 ({\widetilde F}-1) + {\cal O} \left( {(Z\alpha)}^4\left(\frac{m}{M}\right)^2m\right)\;.$$ As a result, we eventually find for the Coulomb problem $$\begin{aligned} \label{tm:masc} E &=& m + m_R (F-1)-\frac{m_R^2}{2M}\left(F-1\right)^2 \,,\end{aligned}$$ which has corrections only of order $(Z\alpha)^4(m/M)^2m$. Here we have taken into account that for a pure Coulomb problem $\delta H=0$ and thus the eigenvalues of the Hamiltonians $H$ in Eq. (\[Dirac\_CH\]) and $H_0$ in Eq. (\[c:hoh1\]) are the same, i. e. $E=E_0$. This evaluation, following [@Gro67], eventually presents eigenvalues and eigenfunctions of the Grotch equation (\[Dirac\_C\]) in terms of the well-known solution of the Dirac-Coulomb problem (see, e.g., [@IV]), but with effective parameters $\widetilde m$ and $\widetilde{{Z\alpha}}$. We briefly overview those solutions in Appendix \[s:dc\] (see, e.g., [@IV] for details). We note that the Grotch equation (\[Dirac\_C\]) and its solution (\[tm:masc\]) is a complete account of the static one-photon-exchange, once we average over the nuclear spin. The relativistic energies (see, e.g., [@IV]) are listed in Appendix \[s:dc\]. We have not evaluated the wave functions, but it is more appropriate to perform such an evaluation once we clarify what accuracy is required. The energy levels (\[tm:masc\]) by themselves are obtained without any need for explicit expressions for the wave functions. However, once we step out from a pure Coulomb case the wave functions will be required; however, they are to appear in calculations of a small perturbation and do not need a high accuracy. We remind that the energy levels (\[c:e0\]) and (\[tm:masc\]) and wave functions (\[c:psi0\]) obtained above reproduce correctly - the leading nonrelativistic term (i.e. a result of the Schrödinger-Coulomb problem with the reduced mass) exactly in $m/M$; - the relativistic corrections (exactly in $Z\alpha)$ for a infinitely heavy nucleus (i.e. a result of the Dirac-Coulomb problem); - the leading relativistic recoil correction to energy in order $(Z\alpha)^4m^2/M$. The result for the energy has to contain also various higher-order contributions $(Z\alpha)^km^2/M$ ($k\ge6$), which, without being a complete result, still have a certain sense, since it is sometimes clear how to upgrade them to a complete result [@shabaev; @yelkhovsky]. Consideration of an arbitrary nonrelativistic-type potential ============================================================ Let us consider now a potential, which is a sum of the Coulomb potential and a “nonrelativistic-type potential” $$V=V_C + V_{N}\;.$$ The “nonrelativistic-type potential” $V_{N}(r)$ is such a potential that the leading nonrelativistic correction to energy is of order ${\varepsilon}(Z\alpha)^2m$ and the leading relativistic correction is of order of ${\varepsilon}(Z\alpha)^4m$, while the leading correction to the wave function is of relative order ${\varepsilon}$ both for nonrelativistic and relativistic behavior. It is understood that ${\varepsilon}$ is a small but finite parameter, such as $\alpha/\pi$, and that the potential $V_{N}(r)$ is smaller than the Coulomb potential in any area by a factor of ${\varepsilon}$. We consider such a potential as a nonrelativistic-type potential, because its relativistic correction, similarly to the case of pure Coulomb potential, can be found through a relativistic expansion, which treats relativistic corrections as additional effective terms of a Hamiltonian of a nonrelativistic Schrödinger equation. Such a consideration is valid, e.g., for the eVP effects in muonic atoms, but not valid for eVP effects in ordinary atoms. What is different for consideration of $V_C + V_{N}$ in comparison with a pure Coulomb problem [@Gro67], reviewed in the previous section: - It is not necessary that ${\varepsilon}(Z\alpha)^4m^2/M$ contributions can be calculated in the one-photon exchange approximation. We suggest that it is valid for all ${\varepsilon}(Z\alpha)^4m^2/M$ terms, and that sets a constraint on effects which may be taken into account by the method developed here. This question is common for Grotch-type and Breit-type calculations and was discussed for one-loop eVP corrections in [@pra]. As explained there, there is a gauge, where the eVP contribution can be calculated within such an approximation. - Rigorously speaking, there is no such a thing as just “potential”. One has to deal with a generalized one-photon exchange. The correction can be due to the photon propagator correction (as it is in the case of eVP effects), nuclear structure etc. While its $D_{00}$ component in a static regime is related to a “potential” for the external field approximation, the result for the other terms depends on the nature of the correction. There is no single rule on how to express the complete effect in terms of $V_{N}$. Here, we suggest that the expression (\[Dirac\_C\]) holds for the one-photon contribution in order up to ${\varepsilon}(Z\alpha)^4m^2/M$. The Hamiltonian is of the form $$\begin{aligned} \label{Dirac_NH} H&=&\biggl( {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p} + \beta m + \frac{\mathbf{p}^2}{2M} + V + \frac{1}{2M} \left\{ {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p}, V \right\}\nonumber\\ &~&~~~ +\frac{1}{4M} \left[ {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p}, [\mathbf{p}^2, W] \right] \biggr)\;,\end{aligned}$$ where $$W = W_C + W_N\;,$$ and an appropriate $W_N$ term is to be found. Furthermore, we suggest that in general the behavior of $W_N$ is somewhat similar to that of ${\varepsilon}W_C$, and the order of magnitude of related matrix elements can be found from that similarity. It is essential that in some way the last term in the Hamiltonian resulted from the lower (smaller) component of the nuclear spinor, so the related matrix elements are of order $(Z\alpha)^4m^2/M$ and may additionally contain ${\varepsilon}$. All that apparently sets another constraint on interactions which can be described by means of a Grotch-type equation. What is important for our purposes is that for the eVP contributions we deal with a certain correction to the photon propagator, the equation (\[Dirac\_C\]) is valid and the appropriate function $W_N$ can be explicitly found (see [@pra; @II; @III]). - Next we note that the addition to the Hamiltonian, defined in Eq. (\[c:dh\]), which vanishes in the pure Coulomb case, does not in the general situation: $$\label{n:dh} \delta H = -\left( \frac{V^2}{2M} + \frac{1}{4M} [V,[\mathbf{p}^2,W]] \right)\neq 0 \,.$$ In the former, pure Coulomb, case this addition was equal to zero. It consisted of two operators, matrix elements of which are of order $(Z\alpha)^4m^2/M$: $$\bigg\langle \frac{V^2}{2M} \bigg\rangle -\bigg\langle \frac{1}{4M} [V,[\mathbf{p}^2,W]] \bigg\rangle = {\cal O} \left( {(Z\alpha)}^4\frac{m^2}{M}\right)\;.\nonumber$$ These are operators which have a non-vanishing matrix element between upper-upper (large-large) components of the muon spinor. To obtain the leading term of order $(Z\alpha)^4m^2/M$ it is sufficient to work with the nonrelativistic wave functions, $\psi_{\rm NR}$. So, the equations for the Hamiltonian and the energy are now $$\begin{aligned} \label{Dirac_N1} H &=& H_0+\delta H \,,\\ \label{Dirac_N2} H_0&=&H_1 + \frac{H_1^2-m^2}{2M} + \frac{1}{4M} [H_1,[\mathbf{p}^2,W]] \,,\\ \label{Dirac_N3} H_1 &=& {\mbox{\boldmath$\alpha$}}\cdot {\mathbf p} + \beta m + V \frac{1-\beta m/M}{1-(m/M)^2} \,,\end{aligned}$$ where we neglect the terms of order $(Z\alpha)^4(m/M)^2m$, and $$\begin{aligned} \label{Dirac_N4} E &=& E_0+\delta E\nonumber\\ \delta E&=&\langle \psi_{\rm NR} \vert \delta H \vert \psi_{\rm NR} \rangle \,.\end{aligned}$$ Since $\delta E$ is already of order ${\varepsilon}(Z\alpha)^4m^2/M$, only the linear corrections are necessary and the nonrelativistic wave function is that of the problem with $H_0$. In the first order in ${\varepsilon}$ we need only pure Coulomb wave functions (see Appendix \[s:dc\]), since we explicitly took into account that $\delta H$, which vanishes in the pure Coulomb case, has to be proportional to ${\varepsilon}$. To second order in ${\varepsilon}$ we have to construct the nonrelativistic wave function perturbatively. Such a problem can be successfully resolved for many problems numerically. - The solution suggests that the effective energy $\widetilde{E}$ depends on the effective mass $\widetilde{m}$, and the actual energy $E_0$ is expressed in terms of $\widetilde{E}$. Meantime, the effective mass $\widetilde{m}$ depends on the energy $E_0$. In the case of the pure Coulomb problem, the ratio $$\frac{\widetilde{E}}{\widetilde{m}}=\widetilde{F}$$ does not depend on the effective mass and as a result we can disentangle $\widetilde{E}$ and $\widetilde{m}$. In general case, the ratio $\widetilde{E}/\widetilde{m}$ depends on $\widetilde{m}$ and, through it, it depends on energy $E_0$. This can be resolved only through expansion over the relativistic effects. We have to apply expressions (\[tm:exa1\]) and (\[tm:exp2\]) studied above, where now the solution of the Dirac equation with potential $V$ is of the form $$\label{tm:n:dc} E = f_{D}(Z\alpha, Z\alpha m/\mu)\,m \,,$$ and $$\widetilde{F}=f_{D}(\widetilde{Z\alpha},\widetilde{Z\alpha}\widetilde{m}/\mu)\;$$ where $f_{D}$ is a dimensionless energy of the Dirac equation with $V$ and $$\widetilde{F}-1={\cal O}\left((Z\alpha)^2\right)\;.$$ In contrast to the pure Coulomb case the dimensionless energy $f_{D}$ depends on the effective mass through a dimensionless parameter $\widetilde{Z\alpha}\widetilde{m}/\mu$. This is possible if the potential $V_N$ depends on the dimensional parameter $\mu$. While calculating various integrals over the wave function the scale parameter of the potential, say, “radius” ($\sim 1/\mu$), is naturally compared with the atomic Bohr radius ($\sim 1/Z\alpha m$). For instance, in the case of eVP corrections in muonic atoms $\mu=m_e$ and the related parameter is $\sim 1.5 Z$. Next we note (see Eq. (\[tm:exp2\])) that $$\widetilde{m} = m_0\left(1 + {\cal O}\left(\frac{m}{M}(\widetilde{F}-1)\right) + \dots\right)$$ where $m_0$ is the result in the limit $Z\alpha\to0$. The relativistic part is already proportional to $(Z\alpha)^4m$ and it is sufficient to apply $m_0$ there. The nonrelativistic part is of order $(Z\alpha)^2m$ and a correction of relative order $(Z\alpha)^2m/M$ is important in the leading approximation, while higher powers of $m/M$ are to be neglected here. The result of the expansion with all terms required is $$\begin{aligned} \widetilde m &=& m_R\sqrt{1-\left(\frac{m}{M}\right)^2} \left[ 1 -\frac{m}{M}(\widetilde{F}-1)\right]\nonumber\\ &=& m_R\sqrt{1-\left(\frac{m}{M}\right)^2} -\frac{m}{M} E_{\rm NR} \label{tm:n:exp2} \,,\end{aligned}$$ where $E_{\rm NR}$ is the nonrelativistic part of the energy for the Schrödinger problem with $V$. As we mentioned, any further $m/M$ corrections in the second term are unimportant and in particular, we can choose to calculate $E_{\rm NR}$ with a muon mass $m$ or with the reduced mass $m_R$. The effective mass is not included in $\widetilde F$ and $F$ directly, but only in a combination $$\begin{aligned} \widetilde{Z\alpha} \widetilde m &=& Z\alpha m_R\left[1-\frac{E_{\rm NR}}{M} \right] \label{tm:n:exp3} \,.\end{aligned}$$ Thus we find $$\begin{aligned} \label{tm:n:F} \widetilde{F}-1&=&\frac{(Z\alpha)^2}{(\widetilde{Z\alpha})^2} (f_{D}(\widetilde{Z\alpha},\widetilde{Z\alpha}\widetilde{m}/\mu)-1)\nonumber\\ &=& \frac{(Z\alpha)^2}{(\widetilde{Z\alpha})^2} \Biggl\{ f_{D}\left(\widetilde{Z\alpha},Z\alpha m_R/\mu\right)-1\nonumber\\ &&-\frac{E_{\rm NR}}{M}\,\kappa\frac{\partial}{\partial \kappa} f_{D}\left(\widetilde{Z\alpha},\kappa\right) \Biggr\} \;,\end{aligned}$$ where for the following it is useful to introduce $$\kappa=\frac{Z\alpha m_R}{\mu} \;.$$ One can treat the first two terms in (\[tm:n:F\]) separately, introducing $$F_0-1= \frac{(Z\alpha)^2}{(\widetilde{Z\alpha})^2} \left\{ f_{D}\left(\widetilde{Z\alpha},Z\alpha m_R/\mu\right)-1 \right\} \;,$$ which now does not depend on $\widetilde m$. The energy can also be split into two terms $$E_0=E^{(1)}+E^{(2)}\;,$$ with the first term similar to the one for the pure Coulomb case (cf. Eq. \[tm:masc\]) $$\begin{aligned} \label{tm:n:masc1} E^{(1)} &=& m + m_R (F_0-1)-\frac{m_R^2}{2M}\left(F_0-1\right)^2\;.\end{aligned}$$ For the second term we note that $(f_D-1)$ is the leading nonrelativistic contribution to the energy and with a sufficient accuracy we can approximate $$f_D\left(\widetilde{Z\alpha},Z\alpha m_R/\mu\right)-1 = \frac{E_{\rm NR}}{m_R}$$ and thus $$\begin{aligned} \label{tm:n:masc2} E^{(2)} &=& -\frac{m_R^2}{2M}\frac{\partial}{\partial \ln\kappa}\left(\frac{E_{\rm NR}}{m_R}\right)^2\,.\end{aligned}$$ Eventually we arrive at the identity for the complete energy $$\begin{aligned} \label{tm:n:masc} E &=& m + m_R (F_0-1)-\frac{m_R^2}{2M}\left(F_0-1\right)^2\nonumber\\ &-&\frac{m_R^2}{2M}\frac{\partial}{\partial \ln\kappa}\left(\frac{E_{\rm NR}}{m_R}\right)^2\nonumber\\ &-&\langle \psi_{\rm NR} \vert \left( \frac{V^2}{2M} + \frac{1}{4M} [V,[\mathbf{p}^2,W]] \right) \vert \psi_{\rm NR} \rangle\,,\end{aligned}$$ which is valid for our purposes and have corrections to order $(Z\alpha)^4m^3/M^2$ and ${\varepsilon}(Z\alpha)^4m^3/M^2$. For the relativistic recoil term $(Z\alpha)^4m^2/M$ we choose between applying $m_R$ and $m$ in such a way that it would simplify a comparison with Breit-type calculations of the same corrections (see [@pra]) for details. A difference between $m_R$ and $m$ in relativistic recoil corrections produces only terms of order $(Z\alpha)^4(m/M)^2m$. - In contrast to the pure Coulomb problem in the external field approximation, for which we know the energy and wave functions in closed analytic form, we indeed cannot know them for an arbitrary potential. For the main term in (\[tm:n:masc\]) we need to be able to find the energy of the Dirac equation with potential $V$ and the reduced mass $m_R$ with a required accuracy. For two other terms we need to know only the nonrelativistic results for the related problem of a Schrödinger equation with potential $V$ and the reduced mass $m_R$. Both relativistic and nonrelativistic problems can be considered at this stage perturbatively since${\varepsilon}\ll1$ and $V_N$ is a small correction to $V_C$. Conclusions =========== The main result of this paper is that there is a certain kind of potential $V$ for which a calculation of the relativistic effects can be split into two parts. One is a calculation of the energy in the external field approximation for a muon with the mass equal to its reduced mass in the atom. That is a “standard” problem of a Dirac equation for a particle with the mass equal to the reduced mass. This calculation can be, in principle, performed by various means, including numerical solutions. The second part, which is a non-trivial part of the relativistic recoil correction, can be obtained once we know the nonrelativistic results for the atom with a muon with the reduced mass. That includes certain derivatives. Such a reduction of the relativistic correction to nonrelativistic calculations essentially simplifies the problem. Roughly speaking, the essential two-body effects are less complicated than the one-particle relativistic problem. Apparently, a number of problems to be solved for a relativistic muon is limited and we do not expect that a Dirac equation with potential $V$ can be solved exactly. As far as the non-Coulomb term is a perturbation, i.e. for ${\varepsilon}\ll1$, we can find all required elements perturbatively. In particular, in the subsequent paper [@II] we apply the developed approach to the eVP corrections in the first order in $\alpha$, i.e. to the relativistic Uehling correction. In this case, one can expand (\[tm:n:masc\]) in ${\varepsilon}=\alpha/\pi$ and find that all required terms are known in a closed form. In the other subsequent paper [@III] the same master equation is applied to the relativistic recoil Källen-Sabry correction, however, none of the eVP related terms are known analytically. So, they are calculated by means of numerical integration. Here, it is still sufficient to work in the first order in ${\varepsilon}=(\alpha/\pi)^2$. However, the relativistic recoil results of the same order, namely $\alpha^2(Z\alpha)^4m^2/M$ arise also from double iteration of the Uehling potential, for this case ${\varepsilon}=\alpha/\pi$, and the second order in ${\varepsilon}$ terms are required in (\[tm:n:masc\]). The recoil effects are obtained for these corrections also by means of numerical integration [@III]. To conclude, we mention that the condition ${\varepsilon}\ll1$ was set only because we are interested in developing a framework for perturbative calculations of the eVP relativistic recoil effects, which are performed in subsequent papers [@II; @III]. In principle, one can consider any “nonrelativistic-type potential”, but the related Dirac equation should be solved numerically. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported in part by DFG under grant GZ: HA 1457/7-2 and RFBR under grant No. 12-02-31741. A part of the work was done during a stay of VGI and EYK at the Max-Planck-Institut für Quantenoptik, and they are grateful to it for its warm hospitality. Solution of the Dirac equation with Coulomb potential\[s:dc\] ============================================================= The exact relativistic energy for a pure Dirac-Coulomb problem $E_C(nl_j)$ for the $nl_j$ state is of the form (see, e.g., [@IV]) $$\begin{aligned} \label{ECnlj} E_C(nl_j) &=& f_C(Z\alpha)\,m\\ f_C(Z\alpha)&=& \frac{1}{\sqrt{1+\frac{{(Z\alpha)}^2}{(n_r+\zeta)^2}}}\end{aligned}$$ and[^1] $$\begin{aligned} \nu &=& (-1)^{j+l+1/2} (j+1/2) \,,\nonumber\\ \zeta &=& \sqrt{\nu^2 -(Z\alpha)^2} \,,\nonumber\\ n_r &=& n - |\nu| \,.\nonumber\end{aligned}$$ The wave functions of the Dirac-Coulomb problem are (see, e.g., [@IV]) $$\begin{aligned} \psi_{njlm}^{(C)}({\bf r}) &=& \left( \begin{array}{c} \Omega_{j,l,m}({\bf r}/r) \; f(r) \\ (-1)^{\frac{1+2l-2j}{2}} \Omega_{j,2j-l,m}({\bf r}/r) \; g(r) \end{array} \right) \,,\\ &&\nonumber\end{aligned}$$ where the radial components are $$\begin{aligned} \left.\begin{array}{c}f\\g\end{array}\right\} &=& \pm \frac{(2m\eta)^{3/2}}{\Gamma(2\zeta+1)} \sqrt{ \frac{(m\pm E_C)\Gamma(2\zeta+n_r+1)} {\frac{4{Z\alpha}m}{\eta} \left( \frac{{Z\alpha}}{\eta} - \nu \right) n_r!} } \nonumber\\ &\times& e^{-m\eta r} (2m\eta r)^{\zeta-1} \nonumber\\ &\times& \Biggl\{ \left( \frac{{Z\alpha}}{\eta} - \nu \right) {}_1F_1(-n_r,2\zeta+1\,;2m\eta r) \nonumber\\ &&\mp\; n_r \times {}_1F_1(1-n_r,2\zeta+1\,;2m\eta r) \Biggr\}\;.\end{aligned}$$ Here the upper signs correspond to the large component $f$ and lower ones are for the small components $g$; ${}_1F_1(a,b;z)$ are confluent hypergeometric functions, $\Omega_{jlm}$ is a spherical spinor and $$\begin{aligned} \eta &=& \sqrt{1 - (E_{nl_j}/m)^2} \nonumber\\ &=&\frac{Z\alpha}{\sqrt{(n_r+\zeta)^2+{(Z\alpha)}^2}} \,.\nonumber\end{aligned}$$ The leading nonrelativistic contribution to the Dirac-Coulomb wave functions can be expressed in terms of the eigen functions of the Schrödinger-Coulomb problem $$\Phi_{nlm}^{(C)}({\bf r}) = Y_{lm}({\bf r}/r) R_{nl}(r) \;,$$ where $$\begin{aligned} R_{nl}(r) &=& \frac{2({Z\alpha}m)^{3/2}}{n^{l+2}(2l+1)!} \, \sqrt{\frac{(n+l)!}{(n-l-1)!}} \nonumber\\ &\times& (2{Z\alpha}m r)^l \, e^{-\frac{{Z\alpha}m r}{n}} \nonumber\\ &\times& {}_1F_1\left( -n+l+1,2l+2; \frac{2{Z\alpha}m r}{n} \right)\end{aligned}$$ and $Y_{lm}$ are spherical functions. [19.]{} R. Pohl, A. Antognini, F. Nez, Fernando D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L.M.P. Fernandes, A. Giesen, T. Graf, T.W. Hänsch, P. Indelicato, L. Julien, Cheng-Yang Kao, P. Knowles, E.-O. Le Bigot, Yi-Wei Liu, J.A.M. Lopes, L. Ludhova, C.M.B. Monteiro, F. Mulhauser, T. Nebel, P. Rabinowitz, J.M.F. dos Santos, L.A. Schaller, K. Schuhmann, C. Schwob, D. Taqqu, J.F.C.A. Veloso and F. Kottmann, Nature (London) [**466**]{}, 213 (2010);\ A. Antognini, F. Nez, K. Schuhmann, F.D. Amaro, F. Biraben, J.M.R. Cardoso, D.S. Covita, A. Dax, S. Dhawan, M. Diepold, L.M.P. Fernandes, A. Giesen, A.L. Gouvea, T. Graf, T.W. Hänsch, P. Indelicato, L. Julien, C.-Y. Kao, P. Knowles, F. Kottmann, E.-O. Le Bigot, Y.-W. Liu, J.A.M. Lopes, L. Ludhova, C.M.B. Monteiro, F. Mulhauser, T. Nebel, P. Rabinowitz, J.M.F. dos Santos, L.A. Schaller, C.Schwob, D.Taqqu, J.F.C.A. Veloso, J. Vogelsang, and R. Pohl, Science, [**339**]{}, 417 (2013). P.J. Mohr, B.N. Taylor, and D.B. Newell, Rev. Mod. Phys. [**84**]{}, 1527 (2012). J.C. Bernauer, P. Achenbach, C. Ayerbe Gayoso, R. Böhm, D. Bosnar, L. Debenjak, M.O. Distler, L. Doria, A. Esser, H. Fonvieille, J.M. Friedrich, J. Friedrich, M. Gómez Rodríguez de la Paz, M. Makek, H. Merkel, D.G. Middleton, U. Müller, L. Nungesser, J. Pochodzalla, M. Potokar, S. Sńchez Majos, B.S. Schlimme, S. Širca, Th. Walcher, and M. Weinriefer, Phys. Rev. Lett. [**105**]{}, 242001 (2010);\ Phys. Rev. Lett. [**107**]{}, 119102 (2011). J.D. Zumbro, E.B. Shera, Y. Tanaka, C.E. Bemis, Jr., R.A. Naumann, M.V. Hoehn, W. Reuter, and R.M. Steffen, Phys. Rev. Lett. [**53**]{}, 1888 (1984). I. Beltrami, B. Aas, W. Beer, G. De Chambrier, P.F.A. Goudsmit, T. v. Ledebur, H.J. Leisi, W. Ruckstuhl, W.W. Sapp, G. Strassner, and A. Vacchi, Nucl. Phys. A[**451**]{}, 679 (1986). N. Nelms, D.F. Anagnostopoulos, M. Augsburger, G. Borchert, D. Chatellard, M. Daum, J.-P. Egger, D. Gotta, P. Hauser, P. Indelicato, E. Jeannet, K. Kirch, O.W.B. Schult, T. Siems, L.M. Simons, and A. Wells, NIM A[**477**]{}, 461 (2002);\ D. Gotta, Prog. Part. Nucl. Phys. [**52**]{}, 133 (2004). M. Hori, A. Dax, J. Eades, K. Gomikawa, R.S. Hayano, N. Ono, W. Pirkl, E. Widmann, H.A. Torii, B. Juhász, D. Barna, and D. Horváth, Phys. Rev. Lett. [**96**]{}, 243401 (2006). M. Hori, A. Sótér, D. Barna, A. Dax, R. Hayano, S. Friedreich, B. Juhász, T. Pask, E. Widmann, D. Horváth, L. Venturelli, and N. Zurlo, Nature [**475**]{}, 484 (2011). V.M. Shabaev, Theor. Math. Phys., [**63**]{} 588 (1985). A.S. Yelkhovsky, arXiv:hep-th/9403095v2. H. Grotch and D.R. Yennie, Z. Phys., [**202**]{}, 425 (1967). U. D. Jentschura, Phys. Rev. A [**84**]{}, 012505 (2011). E. Borie, Annals of Physics [**327**]{}, 733 (2012). S.G. Karshenboim, V.G. Ivanov, and E.Yu. Korzinin, Phys. Rev. A [**85**]{}, 032509 (2012). K. Pachucki, Phys. Rev.A, **53**, 2092, (1996). A. Veitia and K. Pachucki, Phys. Rev. A. **69**, 042501 (2004). E. Borie, Phys. Rev. A **71**, 032508 (2005) E.Yu. Korzinin, V.G. Ivanov, and S.G. Karshenboim, Phys. Rev. D, [**88**]{}, 125019 (2013); arXiv:1311.5784. V.G. Ivanov, E.Yu. Korzinin, and S.G. Karshenboim Phys. Rev. A **89**, 022103 (2014); arXiv:1311.5790. S.G. Karshenboim, E.Yu. Korzinin, and V.G. Ivanov, Phys. Rev. A, [**89**]{}, 032129 (2014); arXiv:1311.5822. A.B. Mickelwait and H.C. Corben, Phys. Rev. [**96**]{}, 1145 (1954);\ G.E. Pustovalov, Zh. Eksp. Teor. Fiz. [**32**]{}, 1519 (1957) \[in Russian\]; Sov. Phys. JETP [**5**]{}, 1234 (1957);\ D.D. Ivanenko and G.E. Pustovalov. Usp. Fiz. Nauk [**61**]{}, 27 (1957) \[in Russian\]; Adv. Phys. Sci. [**61**]{}, 1943 (1957). S.G. Karshenboim, V.G. Ivanov, and E.Yu. Korzinin, Eur. Phys. J. D [**39**]{}, 351 (2006). S.G. Karshenboim, Can. J. Phys. [**76**]{}, 169 (1998);\ ZhETF [**116**]{}, 1575 (1999) \[in Russian\]; JETP [**89**]{}, 850 (1999). E.Yu. Korzinin, V.G. Ivanov, and S.G. Karshenboim, Eur. Phys. J. D [**41**]{}, 1 (2007). D. A. Owen, Phys. Rev. D [**42**]{}, 3534 (1990); D [**46**]{}, 4782 (E) (1992);\ D. A. Owen, Found. Phys. [**24**]{}, 273 (1994);\ M. Halpert and D. A. Owen, J. Phys. G [**20**]{}, 51 (1994). K. Pachucki and S. G. Karshenboim, J. Phys. B [**28**]{}, L221 (1995). I. B. Khriplovich, A. I. Milstein, and R. A. Sen’kov, Phys. Lett. A [**221**]{}, 370 (1996);\ JETP [**84**]{}, 1054 (1997). J. L. Friar, J. Martorel, D. W. L. Sprung, Phys. Rev. A[**56**]{}, 6, 4579 (1997). G.P. Lepage. Phys. Rev. A [**16**]{}, 863 (1977);\ G.T. Bodwin, D.R. Yennie and M.A. Gregorio. Rev. Mod. Phys. [**57**]{}, 723 (1985);\ J. Sapirstein and D.R. Yennie, in T. Kinoshita (Ed.), [*Quantum Electrodynamics*]{} (World Sci., Singapore, 1990), p. 560;\ M.I. Eides, S.G. Karshenboim and V.A. Shelyuto, Ann. Physics [**205**]{}, 231 (1991). L.D. Landau and E.M. Lifshitz. *Course of Theoretical Physics*. Vol. 4: V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii. *Quantum Electrodynamics*. (Pergamon Press, Oxford, 1982). V. G. Ivanov and S. G. Karshenboim, Phys. Rev. A **79**, 032518 (2009). [^1]: It is customary to use $\kappa$ for $(-1)^{j+l+1/2} (j+1/2)$ (cf. [@IV]), however, $\kappa$ is used in our papers on muonic atoms for something else.

--- abstract: 'Recently, semi-supervised learning methods based on generative adversarial networks (GANs) have received much attention. Among them, two distinct approaches have achieved competitive results on a variety of benchmark datasets. Bad GAN learns a classifier with unrealistic samples distributed on the complement of the support of the input data. Conversely, Triple GAN consists of a three-player game that tries to leverage good generated samples to boost classification results. In this paper, we perform a comprehensive comparison of these two approaches on different benchmark datasets. We demonstrate their different properties on image generation, and sensitivity to the amount of labeled data provided. By comprehensively comparing these two methods, we hope to shed light on the future of GAN-based semi-supervised learning. [^1]' author: - | Wenyuan Li\ Electrical and Computer Engineering\ University of California, Los Angeles\ [liwenyuan.zju@gmail.com]{}[^2] - | Zichen Wang\ Bioengineering\ University of California, Los Angeles\ [zcwang0702@g.ucla.edu]{} - | Jiayun Li\ Bioengineering\ University of California, Los Angeles\ [jiayunli@ucla.edu]{} - | Jennifer Polson\ Bioengineering\ University of California, Los Angeles\ [jpolson@g.ucla.edu]{} - | William Speier\ Radiology\ University of California, Los Angeles\ [speier@ucla.edu]{} - | Corey Arnold\ Radiology, Pathology, Bioengineering\ University of California, Los Angeles\ [cwarnold@ucla.edu]{} bibliography: - 'egbib.bib' title: 'Semi-supervised learning based on generative adversarial network: a comparison between good GAN and bad GAN approach' --- Introduction ============ Semi-supervised learning (SSL) aims to make use of large amounts of unlabeled data to boost model performance, typically when obtaining labeled data is expensive and time-consuming. Various semi-supervised learning methods have been proposed using deep learning and proven to be successful on several standard benchmarks. Weston [@weston2012deep] employed a manifold embedding technique using the pre-constructed graph of unlabeled data; Rasmus [@rasmus2015semi] used a specially designed auto-encoder to extract essential features for classification; Kingma and Welling [@kingma2013auto] developed a variational auto encoder in the context of semi-supervised learning by maximizing the variational lower bound of both labeled and unlabeled data; Miyato [@miyato2018virtual] proposed virtual adversarial training (VAT) that tied to find a deep classifier, which had a good prediction accuracy on training data and meanwhile was less sensitive to data perturbation towards the adversarial direction. Recently, generative adversarial networks (GANs) [@goodfellow2014generative], have demonstrated their capability in SSL frameworks [@salimans2016improved; @dai2017good; @gan2017triangle; @chongxuan2017triple; @kumar2017semi; @lecouat2018semi; @li2018semi]. GANs are a powerful class of deep generative models that are able to model data distributions over natural images [@radford2015unsupervised; @mirza2014conditional]. Salimans first proposed to use GANs to solve a $(K + 1)$-class classification problem, where the dataset contained $K$ class originally and the additional $(K + 1)$th class consisted of the synthetic images generated by the GAN’s generator. Later on, Li [@chongxuan2017triple] realized that the generator and discriminator in [@salimans2016improved] may not be optimal at the same time (*i.e.*, the discriminator was able to achieve good performance in SSL, while the generator may generate visually unrealistic images). They proposed a three-player game (Triple-GAN) to simultaneously achieve good classification results and obtained a good image generator. Dai [@dai2017good] realized the same problem, but instead gave theoretical justifications of why using bad samples from the generator was able to boost SSL performance. Their model is called Bad GAN, which achieves state-of-the-art performance on multiple benchmark datasets. Another line of work focused on manifold regularization [@belkin2006manifold]. Kumar [@kumar2017semi] estimated the manifold gradients at input data points and added an additional regularization term to a GAN, which promoted invariance of the discriminator to all directions in the data space. Lecouat [@lecouat2018semi] performed manifold regularization by approximating the Laplacian norm that was easily computed within a GAN and achieved competitive results. {width="16cm" height="4cm"} In this paper, we focus on two GAN-based SSL models, Triple GAN and Bad GAN, and perform a comprehensive comparison between them. As both of models attempt to solve a similar issue in the original setting [@salimans2016improved] but are motivated by dissimilar perspectives, we believe that our comparison will provide insight for future SSL research. For simplicity, we refer to Triple GAN as Good GAN in contrast to Bad GAN. In Section \[Section2\], we briefly review the two models and their different approaches for solving loss function incompatibility; in Section \[Section3\], we show the network architecture we employed, benchmark datasets we used, and hyperparameters we selected in order to perform a fair comparison between these two models; in Section \[Section4\], we demonstrate our comparison results and discuss several important aspects we found for these two models; we conclude our paper in Section \[Section5\]. Related Work {#Section2} ============ Bad GAN ------- Suppose we have a classification problem that requires classifying a data point $\boldsymbol{x}$ into one of $K$ possible classes. A standard classifier takes in $\boldsymbol{x}$ as input and outputs a $K$-dimensional vector of logits $\{l_1, ..., l_K\}$. Salimens [@salimans2016improved] extend the standard classifier *C* by simply adding samples from the GAN generator *G* to the dataset, labeling them as a new “generated” class $y = K + 1$, and correspondingly increasing the dimension of *C* output from $K$ to $K + 1$. The loss function $L_{C/D}$ for training *C* (*i.e.*, the extended discriminator *D* from the GAN’s perspective) then becomes $$\label{badgan_discriminator} \begin{aligned} L_{\textit{C/D}} &= L_{\text{supervised}} + L_{\text{unsupervised}}\\ L_{\text{supervised}} &= \mathop{\mathbb{E}}_{\boldsymbol{x}, y\sim p_{l}(\boldsymbol{x}, y)}[-\log (p_{C/D}(y|\boldsymbol{x}, y < K + 1))]\\ L_{\text{unsupervised}} &=\mathop{\mathbb{E}}_{\boldsymbol{x}\sim p_{u}(\boldsymbol{x})}[-\log (1 - p_{C/D}(y = K + 1|\boldsymbol{x}))]\\ &+ \mathop{\mathbb{E}}_{\boldsymbol{x}\sim p_g(\boldsymbol{x})}[-\log (p_{C/D}(y = K + 1|\boldsymbol{x}))] \end{aligned}$$ The supervised loss term $L_\text{supervised}$ is a traditional cross-entropy loss that is applied to labeled data $(\boldsymbol{x}, y)\sim p_l(\boldsymbol{x}, y)$. The unsupervised loss requires *C/D* to put the synthetic data from generator $\boldsymbol{x} \sim p_{g}(\boldsymbol{x})$ into the $(K+1)$th class, while putting the unlabeled data $\boldsymbol{x} \sim p_u(\boldsymbol{x})$ into the real $K$ classes. For the generator, [@salimans2016improved] found feature matching loss in Eq. \[feature-matching\] is the best in practice, though they generated visually unrealistic images. The feature matching loss is, $$\label{feature-matching} \begin{aligned} L_{\textit{G}} &= \left\| \mathop{\mathbb{E}}_{\boldsymbol{x}\sim p_{u}}(\boldsymbol{f}(\boldsymbol{x})) - \mathop{\mathbb{E}}_{\boldsymbol{z}_g\sim p_{z}(z)}(\boldsymbol{f}(G(\boldsymbol{z}_g))) \right\|_2^2 \end{aligned}$$ where $\boldsymbol{z}_g \sim p_z(\boldsymbol{z})$ is drawn from a simple distribution such as uniform. On the basis of this formulation, Dai [@dai2017good] give a theoretical justification on why the visually unrealistic images (*i.e.*, “bad” samples) from the generator could help with SSL. Loosely speaking, the carefully generated “bad” samples along with the loss function design in Eq. \[badgan\_discriminator\] could force *C*’s decision boundary to lie between the data manifolds of different classes, which in turn improves generalization of the classifier. Based on this analysis, they propose a Bad GAN model that learns a bad generator by explicitly adding a penalty term to generate “bad” samples. Their objective function of the generator becomes: $$\label{badgan_generator} \begin{aligned} L_{\textit{G}} &= -\mathcal{H}[p_{g}(\boldsymbol{x})] + \mathop{\mathbb{E}}_{\boldsymbol{x}\sim p_{g}(\boldsymbol{x})}(\log p^{pt}(x) \mathop{\mathbb{I}}[p^{pt}(x) > \epsilon]\\ &+ \left\| \mathop{\mathbb{E}}_{\boldsymbol{x}\sim p_{u}(\boldsymbol{x})}(\boldsymbol{f}(\boldsymbol{x})) - \mathop{\mathbb{E}}_{\boldsymbol{z}_g\sim p_{z}(z)}(\boldsymbol{f}(G(\boldsymbol{z}_g))) \right\|_2^2 \\ \end{aligned}$$ where the first term measures the negative entropy of the generated samples and tries to avoid collapsing while increasing the coverage of the generator. The second term explicitly penalizes generated samples that are in high density areas by using a pre-trained model, and the third term is the same feature matching term as in Eq. \[feature-matching\]. Good GAN -------- Li [@chongxuan2017triple] also noticed the same problem in [@salimans2016improved] as the generator and the discriminator have incompatible loss functions, but took a different approach to tackling this issue. Intuitively, assume the generator can generate good samples in the original settings of [@salimans2016improved], the discriminator should identify these samples as fake samples as well as predict the correct class for the generated samples. To address the problem, [@chongxuan2017triple] present a three-player game called Triple-GAN that consists of a generator *G*, a discriminator *D*, and a separate classifier *C*. *C* and *D* are two conditional networks that generate pseudo labels given real data and pseudo data given real labels respectively. To jointly evaluate the quality of the samples from the two conditional networks, a single discriminator *D* is used to distinguish whether a data–label pair is from the real labeled dataset or not. We refer this model as Good GAN because one of the aims for this formulation is to obtain a good generator. The authors prove that instead of competing equilibrium states as in [@salimans2016improved], Good GAN has the unique global optimum for both *C* and *G*, *i.e.*, $p(\boldsymbol{x}, y) = p_g(\boldsymbol{x}, y) = p_c(\boldsymbol{x}, y)$, the three joint distributions match one another. In other words, a good classifier will result in a good generator and vice versa. Furthermore, Good GAN is trained using the REINFORCE algorithm, in which it generates pseudo labels through *C* for some unlabeled data and uses these pairs as positive samples to feed into *D*. This is a key to the success of the model, as one of the crucial problems of SSL is the limited size of the labeled data. Figure \[Figure1\] shows the network architecture of Good GAN and Bad GAN. Comparison Method {#Section3} ================= Network Architecture -------------------- In Bad GAN, the discriminator has two roles: to classify the real data into the right class and to distinguish the real samples from the fake samples. For clarity, we refer to Bad GAN’s discriminator as the classifier, since its input and output are exactly the same as the classifier in Good GAN due to the over-parameterization of the softmax layer [@salimans2016improved]. To perform a fair comparison between Good GAN and Bad GAN, we use the same network architecture for the generator *G* and the classifier *C* in both models. We follow the architecture closely in [@chongxuan2017triple] to set up the additional discriminator *D* in Good GAN. Both of them use Leaky-Relu activation and weight normalization to ease the difficulty of GAN’s training. Implementing them using same architecture ideally avoids the possibility of using an architecture that is custom-tailored to work well with one or the other. Detailed model architectures can be found in the Appendix \[appA\]. Datasets -------- Using the above-defined network architectures, we compare the two models on the widely adopted MNIST [@lecun1998gradient], SVHN [@netzer2011reading], and CIFAR10 [@krizhevsky2009learning] datasets. MNIST consists of 50,000 training samples, 10,000 validation samples, and 10,000 testing samples of handwritten digits of size $28 \times 28$. SVHN consists of 73,257 training samples and 26,032 testing samples. Each sample is a colored image of size $32 \times 32$, containing a sequence of digits with various backgrounds. CIFAR10 consists of colored images distributed across 10 general classes – *airplane*, *automobile*, *bird*, *cat*, *deer*, *dog*, *frog*, *horse*, *ship* and *truck*. It contains 50,000 training samples and 10,000 testing samples of size $32 \times 32$. Following [@chongxuan2017triple], we reserve 5,000 training samples from SVHN and CIFAR10 for validation if needed. For our CIFAR10 experiment, we perform zero-based component analysis (ZCA) [@laine2016temporal] as suggested in [@chongxuan2017triple] for the input of *C*, but still generate and estimate the raw images using *G* and *D*. We perform an extensive investigation by varying the amount of labeled data. Following common practice, this is done by throwing away different amounts of the underlying labeled dataset [@salimans2016improved; @pu2016variational; @sajjadi2016mutual; @tarvainen2017mean]. The labeled data used for training are randomly selected stratified samples unless otherwise specified. We perform our experiments on setups with 20, 50, 100, and 200 labeled examples in MNIST, 500, 1000, and 2000 labeled examples in SVHN, and 1000, 2000, 400, 8000 examples in CIFAR10. Hyperparameter Selection ------------------------ For the hyperparameter selection such as learning rate and beta for Adam optimization, and the coefficient for each cost function term, we closely follow [@chongxuan2017triple; @dai2017good]. In addition, we perform extensive study of the effects of batch size on performance for Bad GAN. As reported by [@lecouat2018semi], Bad GAN training is sensitive to training batch size, and thus we vary batch size in the training phase and compare their final performances on MNIST and SVHN. Experimental Results and Discussion {#Section4} =================================== ----------------------------------- ------------------------------- --------------------------- --------------------------- --------------------------- Model 20 50 100 200 Bad GAN [@dai2017good] - - $\mathbf{99.21\pm0.01\%}$ - Triple GAN [@chongxuan2017triple] $95.19\pm4.95\%$ $98.44\pm0.72\%$ $99.09\pm0.58\%$ $\mathbf{99.33\pm0.16\%}$ Bad GAN (ours) $68.12\pm0.60\%$ $96.24\pm0.16\%$ $99.17\pm0.03\%$ $99.20\pm0.03\%$ Good GAN (ours) $\mathbf{95.93\pm4.45\%^{*}}$ $\mathbf{98.68\pm1.12\%}$ $99.07\pm0.46\%$ $99.17\pm0.08\%$ ----------------------------------- ------------------------------- --------------------------- --------------------------- --------------------------- ---------------------------------- ----------------------------- ----------------------------- ----------------------------- Model 500 1000 2000 Bad GAN[@dai2017good] - $\mathbf{95.75 \pm 0.03\%}$ - Triple GAN[@chongxuan2017triple] - $94.23 \pm 0.17\%$ - Bad GAN (ours) $94.21 \pm 0.45\%$ $95.32 \pm 0.07 \%$ $\mathbf{95.47 \pm 0.39\%}$ Good GAN (ours) $\mathbf{94.67 \pm 0.12\%}$ $95.30 \pm 0.38\%$ $95.37 \pm 0.09\%$ ---------------------------------- ----------------------------- ----------------------------- ----------------------------- ----------------------------------- ----------------------------- ----------------------------- ---------------------------- ----------------------------- Model 1000 2000 4000 8000 Bad GAN [@dai2017good] - - $\mathbf{85.59\pm 0.03\%}$ - Triple GAN [@chongxuan2017triple] - - $83.01 \pm 0.36\%$ - Bad GAN (ours) $77.58 \pm 0.17\%$ $81.36 \pm 0.08\%$ $82.89 \pm 0.13\%$ $\mathbf{85.47 \pm 0.10\%}$ Good GAN (ours) $\mathbf{81.08 \pm 0.57\%}$ $\mathbf{81.79 \pm 0.37\%}$ $82.82 \pm 0.41\%$ $85.37 \pm 0.18\%$ ----------------------------------- ----------------------------- ----------------------------- ---------------------------- ----------------------------- We implement Good GAN based on Tensorflow 1.10 [@girija2016tensorflow] and Bad GAN based on Pytorch 1.0 [@paszke2017automatic]. The generated images from $gG$ is not applied until the number of epochs reach a threshold that $gG$ could generate reliable image-lable pairs. We choose 200 in all three cases. All of the other hyperparameters including initial learning rate, maximum epoch number, relative weights and parameters in Adam [@kingma2014adam] are fixed according to [@salimans2016improved; @chongxuan2017triple; @dai2017good] across all of the experiments. Classification -------------- We report our classification accuracy on the test set in Table \[Table1\], Table \[Table2\] and Table \[Table3\] for MNIST, SVNH and CIFAR10, respectively, along with the results reported in the original papers. The similarity of our results to those reported in the original papers suggests that our reproduced models are accurate instantiations of Good GAN and Bad GAN. Furthermore, we perform extensive study by varying the amount of labeled data and observe that Good GAN and Bad GAN behave quite differently under various circumstances. First, with a medium amount of labeled data (*e.g.*, MNIST with 100 or 200 labeled data, SVHN with more than 2000 labeled data, or CIFAR10 with more than 2000 labeled data), Bad GAN performs better than Good GAN. In fact, to the best of our knowledge, Bad GAN achieves the current state-of-the-art performance on those benchmark datasets. However, with low amounts of labeled data, Good GAN performs better, which demonstrates that Good GAN is less sensitive to the amount of labeled data than Bad GAN. One possible explanation is due to the use of the REINFORCE algorithm in Good GAN, because it generates pseudo labels through *C* for some unlabeled data and use these pairs as positive samples of *D*. Since *C* converges quickly, this trick provides a clever way to enable the generator to explore a much larger data manifold that includes both the labeled and unlabeled data information. In other words, the classifier is able to provide pseudo labels for the unlabeled data, while the discriminator will judge if the pseudo labels are reliable or not throughout the training. This in return will affect the evolution of the generator, which will take advantage of the unlabeled data to generate good images. Generated good image-label pairs that implicitly contain unlabeled data information will eventually benefit the classifier. This works extremely well for relatively simple datasets like MNIST, as Good GAN is able to model the class-awarded data distribution through weak supervision. On the other hand, Bad GAN yields decreased performance when the amount of labeled data is low, as it does not have any mechanism to augment the information that could be used to train the classifier in this case. Generated Images ---------------- {width="14cm"} {width="13cm"} In Figure \[Figure2\], we compare the quality of images generated by Good GAN and Bad GAN. As can be seen, Good GAN is able to generate clear images and meaningful samples conditioned on class labels, while Bad GAN generates “bad” images that look like a fusion of samples from different classes. In addition, Good GAN is able to disentangle classes and styles. In Figure \[Figure2\] bottom, we vary the class label $y$ in the vertical axis and the latent vectors $z$ in the horizontal axis to generate the images. As shown in the figure, the latent vector $z$ encodes meaningful physical appearances, such as scale, intensity, orientation, color and so on, while the label $y$ controls the semantics of the generated images. Furthermore, Good-GAN can transition smoothly from one style to another with different visual factors without losing the label information as shown in Figure \[Figure3\]. This proves that Good GAN can learn meaningful latent space representations instead of simply memorizing the training data. Importance of Selection of Labeled Data {#label_data_importance} --------------------------------------- Another interesting observation is that the selection of labeled data plays a crucial role for training Good GAN model in the low labeled data scenario. As mentioned above, the labeled data used for the training are randomly selected stratified samples, except for the MNIST-20 case. In this case, we found selecting representative labeled data to train is the key to achieving good performance. The reported accuracy in Table \[Table1\] is averaged over 10 runs where we manually selected different representative labeled data in a stratified way. Figure \[Figure4\] (a) shows a single run that uses randomly selected labeled data and does not achieve good results, while Figure \[Figure4\] (b) shows another run that is able to achieve higher accuracy. The failure of the first run is due to the initial selections for digit 4 being similar to 9, causing the generator to generate many 9s when conditioned on label 4. The generator also generates low-quality images. We also report that with a random selection of 20 labeled data, the Good GAN was able to achieve $76.78 \pm 6.47 \%$ accuracy over 3 runs. ![Two-runs of Good GAN model on MNIST dataset. (a) A single run where we randomly select 20 labeled data. The generator generates a lot of wrong images conditioned on the label and the classifier has lower performance. (b) Another run where we manually select 20 representative labeled examples. This time the generator is able to generate correct images, and the classifier achieves good classification performance.[]{data-label="Figure4"}](Figure/Figure4.png){width="8cm"} Importance of Batch Size ------------------------ We found that batch size largely affect the final training results, in both Good GAN and Bad GAN. To investigate the effect of batch size on Bad GAN performance, we performed experiments with different batch size on MNIST (with 100 labeled samples) and SVHN (with 1000 labeled samples) using Bad GAN. As shown in Table \[Table4\], we empirically show that the performance of Bad GAN is sensitive to training batch size, and the optimal performance for each dataset is achieved with a batch size of 100. To further understand the effect of the batch size on Bad GAN training, we present the generator loss with different batch sizes for MNIST and SVHN in Figure \[Figure5\]. The results indicate that smaller batch sizes lead to larger generator loss in the final stage of training. As that generator loss mainly depends on the first-order feature matching loss in Bad GAN, an intuitive explanation could be that larger batch sizes reduce the variance of the sample mean, allowing the generator to quickly approximate the entire training set. This leads to smaller generator loss, especially when model training becomes more stable in the final stage. As noted by [@dai2017good], feature matching is performing distribution matching in a weak manner, which could be largely affected by batch size. On one extreme, when the batch size is too small, the power of the generator in distribution matching is weak due to the excessive generator loss. Generated samples are therefore more likely to diverge from the manifold. Especially when data complexity increases, it is more difficult to minimize the KL divergence between the generator distribution and a desired complement distribution in Bad GAN, which could be one possible reason why model degradation is more significant on SVHN when using 20 batch size. On the other extreme, larger batch size leads to smaller generator loss, which comes with reduced diversity of generated samples. When the batch size is too large, the small generator loss will lead to a collapsed generator which fails to generate diverse samples that cover complement manifolds. As a result, the decision boundary between such missing manifolds becomes under-determined, which will also degrades model performance. We plot Bad GAN performance under different batch sizes for MNIST and SVHN in Appendix \[appB\]. Based on our experience, Good GAN is best when we use a large batch size. Intuitively, a small batch size is not good for the REINFORCE algorithm adopted in Good GAN because a single wrong prediction of the unlabeled data will have a big impact on the weight update in each iteration. We perform Good GAN experiments on SVHN using different batch size. The results are shown in Table \[Table5\]. Empirically, we find that with small batch size, Good GAN is not able to generate good image-label pairs, hence the generated image-label pairs even hurt the classifier’s performance when we use them to train. (See more details in Appendix \[appB\]). ![Batch size effect on generator loss in Bad GAN. The experiments are performed on (a) MNIST using 100 labeled samples and (b) SVHN using 1000 labeled samples.[]{data-label="Figure5"}](Figure/Figure5.png){width="8cm"} Batch size 20 50 100 200 400 ------------ -------------------- -------------------- ----------------------------- -------------------- -------------------- MNIST-100 $98.90 \pm 0.04\%$ $99.10 \pm 0.03\%$ $\mathbf{99.17 \pm 0.03\%}$ $99.16 \pm 0.03\%$ $98.89 \pm 0.02\%$ SVHN-1000 $93.35 \pm 0.05\%$ $95.29 \pm 0.03\%$ $\mathbf{95.56 \pm 0.02\%}$ $95.19 \pm 0.02\%$ $94.20 \pm 0.04\%$ Batch size 20 50 100 ------------ ----------- ----------- -------------------- SVHN-1000 $92.47\%$ $92.59\%$ $\mathbf{95.30\%}$ : Good GAN performance versus batch size on SVHN. The results are achieved using 1000 labeled samples in SVHN.[]{data-label="Table5"} Conclusion {#Section5} ========== In this paper, we systematically and extensively compared two GAN-based SSL methods, Good GAN and Bad GAN, by applying these two models with commonly-used benchmark datasets. We illustrate the distinct characteristics of the images they generated, as well as each model’s sensitivity to varying the amount of labeled data used for training. In the case of low amounts of labeled data, model performance is contingent on the selection of labeled samples; that is, selecting non-representative samples results in generating incorrect image-label pairs and deteriorating classification performance. Furthermore, selecting the optimal batch size is crucial to achieve good results in both models. Notably, Good GAN and Bad GAN models can be used for complementary purposes; Good GAN generates good image-label pairs to train the classifier, while Bad GAN generates samples that force the decision boundary between data manifold of different classes. We envision that combining these two methods should yield further performance improvement in SSL. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to acknowledge support from the UCLA Radiology Department Exploratory Research Grant Program (16-0003) and NIH/NCI R21CA220352. This research was also enabled in part by GPUs donated by NVIDIA Corporation. Network Architecture {#appA} ==================== We list the detailed architecture we used to compare Good GAN and Bad GAN on MNIST, SVHN, and CIFAR10 datasets in Table \[Table1app\], Table \[Table2app\] and Table \[Table3app\] respectively. [c|c|c]{} **Generator G** & **Classifier C** &\ Input Label $y$, Noise $z$ & Input $28 \times 28$ Gray Image& Input $28 \times 28$ Gray Image, Label $y$\ [@c@]{}MLP 500 units, softplus, batch norm\ \ \ MLP 500 units, softplus, batch norm\ \ \ MLP 500 units, softplus, batch norm & ----------------------------- MLP 1000 units, lRelu, Gaussian noise, weight norm MLP 500 units, lRelu, Gaussian noise, weight norm MLP 250 units, lRelu, Gaussian noise, weight norm MLP 250 units, lRelu, Gaussian noise, weight norm MLP 250 units, lRelu, Gaussian noise, weight norm MLP 10 units, softmax, Gaussian noise, weight norm ----------------------------- & ----------------------------- MLP 1000 units, lRelu, Gaussian noise, weight norm MLP 500 units, lRelu, Gaussian noise, weight norm MLP 250 units, lRelu, Gaussian noise, weight norm MLP 250 units, lRelu, Gaussian noise, weight norm MLP 250 units, lRelu, Gaussian noise, weight norm MLP 12 units, sigmoid, Gaussian noise, weight norm ----------------------------- \ [c|c|c]{} **Generator G**& **Classifier C**& **Discriminator D (Good GAN only)**\ Input Label $y$, Noise $z$ & Input $32 \times 32$ Colored Image & Input $32 \times 32$ Colored Image, Label $y$\ [@c@]{}MLP 8192 units,\ Relu, batch norm\ Reshape $512 \times 4 \times 4$\ \ \ $5 \times 5$ deconv. 256. stride 2,\ Relu, batch norm & ------------------------------------------- Gaussian noise, 0.2 dropout $3 \times 3$ conv. 64. lRelu, weight norm $3 \times 3$ conv. 64. lRelu, weight norm $3 \times 3$ conv. 64. lRelu, stride 2, weight norm 0.5 dropout ------------------------------------------- & ------------------------------------------- 0.2 dropout $3 \times 3$ conv. 32. lRelu, weight norm $3 \times 3$ conv. 32. lRelu, stride 2, weight norm 0.2 dropout ------------------------------------------- \ ------------------------------------- $5 \times 5$ deconv. 128. stride 2, Relu, batch norm ------------------------------------- & -------------------------------------------- $3 \times 3$ conv. 128. lRelu, weight norm $3 \times 3$ conv. 128. lRelu, weight norm $3 \times 3$ conv. 128. lRelu, stride 2, weight norm 0.5 dropout -------------------------------------------- & ------------------------------------------- $3 \times 3$ conv. 64. lRelu, weight norm $3 \times 3$ conv. 64. lRelu, stride 2, weight norm 0.2 dropout ------------------------------------------- \ ----------------------------------- $5 \times 5$ deconv. 3. stride 2, sigmoid, weight norm ----------------------------------- & [@c@]{}$3 \times 3$ conv. 128. lRelu, weight norm\ $3 \times 3$ conv. 128. lRelu, weight norm\ $3 \times 3$ conv. 128. lRelu, weight norm\ \ Global pool\ MLP 10 units, softmax, weight norm & [@c@]{}$3 \times 3$ conv. 128. lRelu, weight norm\ $3 \times 3$ conv. 128. lRelu, weight norm\ \ Global pool\ MLP 1 unit, sigmoid, weight norm \ [c|c|c]{} **Generator G**& **Classifier C**& **Discriminator D (Good GAN only)**\ Input Label $y$, Noise $z$ & Input $32 \times 32$ Colored Image & Input $32 \times 32$ Colored Image, Label $y$\ [@c@]{}MLP 8192 units,\ Relu, batch norm\ Reshape $512 \times 4 \times 4$\ \ \ $5 \times 5$ deconv. 256. stride 2,\ Relu, batch norm & ------------------------------------------- Gaussian noise, 0.2 dropout $3 \times 3$ conv. 96. lRelu, weight norm $3 \times 3$ conv. 96. lRelu, weight norm $3 \times 3$ conv. 96. lRelu, stride 2, weight norm 0.5 dropout ------------------------------------------- & ------------------------------------------- 0.2 dropout $3 \times 3$ conv. 32. lRelu, weight norm $3 \times 3$ conv. 32. lRelu, stride 2, weight norm 0.2 dropout ------------------------------------------- \ ------------------------------------- $5 \times 5$ deconv. 128. stride 2, Relu, batch norm ------------------------------------- & -------------------------------------------- $3 \times 3$ conv. 192. lRelu, weight norm $3 \times 3$ conv. 192. lRelu, weight norm $3 \times 3$ conv. 128. lRelu, stride 2, weight norm 0.5 dropout -------------------------------------------- & ------------------------------------------- $3 \times 3$ conv. 64. lRelu, weight norm $3 \times 3$ conv. 64. lRelu, stride 2, weight norm 0.2 dropout ------------------------------------------- \ ----------------------------------- $5 \times 5$ deconv. 3. stride 2, sigmoid, weight norm ----------------------------------- & [@c@]{}$3 \times 3$ conv. 192. lRelu, weight norm\ $3 \times 3$ conv. 192. lRelu, weight ntheirorm\ $3 \times 3$ conv. 192. lRelu, weight norm\ \ Global pool\ MLP 10 units, softmax, weight norm & [@c@]{}$3 \times 3$ conv. 128. lRelu, weight norm\ $3 \times 3$ conv. 128. lRelu, weight norm\ \ Global pool\ MLP 1 unit, sigmoid, weight norm \ Batch Size Effect in Bad GAN {#appB} ============================ Figure \[Figure6\] shows the classification accuracy under different batch size of Bad GAN during the first 400 epochs of training. As can be seen, the model performance is very sensitive to batch size. Figure \[Figure7\] shows the generated images of Good GAN under different batch size. With small batch size, Good GAN is not able to generate good image-label pairs. ![Batch size effect in Bad GAN. The classification accuracy over the initial 400 training epochs under different batch size. (a) The experiments are performed on MNIST dataset, using 100 labeled data. (b) The experiments are performed on SVHN dataset, using 1000 labeled data.[]{data-label="Figure7"}](Figure/Figure6.png){width="\textwidth"} ![Batch size effect in Good GAN. With small batch size, Good GAN is not able to generate good image-label pairs. Experiments are performed on SVHN with $n = 1000$. All the images are generated at epoch $= 200$ when we start to use the generated image to train.[]{data-label="Figure6"}](Figure/Figure7.png){width="13cm"} [^1]: This paper appears at CVPR 2019 Weakly Supervised Learning for Real-World Computer Vision Applications (LID) Workshop. [^2]: W. Li and C. Arnold are the corresponding authors.

--- abstract: 'Measurements of the surface brightness distribution and of the velocity dispersion profile have been so far used to infer the inner dynamics of globular clusters. We show that those observations do not trace back the dark matter potentially concealed in these systems in the form of low-mass compact objects. We have built Michie models of globular clusters which contain both massive and low-mass stars. An analytic expression for the stellar mass densities has been explicitely derived in terms of the usual error function and Dawson’s integral. While the heavy population is kept fixed, the abundance of the light species of our models is varied. When stellar velocities are anisotropic, both the surface brightness and the velocity dispersion profiles of the cluster become insensitive to the abundance of low-mass stars. This suggests that the actual stellar mass function of many globular clusters is still to be discovered.' author: - 'Fran[ç]{}ois Roueff' - Pierre Salati - Richard Taillet date: 'Received / Accepted ' title: 'The velocity dispersion profile of globular clusters : a closer look' --- Introduction ============ Among the different kinds of astrophysical star conglomerates, globular clusters (GC) have a very peculiar status. First, as we go down the mass-scale of observed structures, from super clusters through galaxies down to dwarf spheroidals, they are the first objects whose dynamics can be understood without resorting to the presence of dark matter. Second, and this may be related to the first point, they are the only star clusters dense enough for a statistical equilibrium to be reached in a Hubble timescale. The dark matter content of globular clusters may be an important clue to the understanding of the formation of our galaxy. However, the observational situation is not very clear yet as regards stellar counts and the related direct detection of a hidden population of light and faint stars. Since the claim by Richer and Fahlman that the mass function steeply rises towards low masses in some GCs (Fahlman et al. 1989, Richer et al. 1991), the Hubble Space Telescope (HST) has been used to measure the mass function in several globular clusters, and the opposite conclusion was reached (Demarchi and Paresce 1995, Elson et al. 1995, Paresce, De Marchi and Romaniello 1995). Part of the discrepancy is due to the conversion scheme used to translate the measured luminosity functions into mass functions. There is actually no consensus on the mass-to-luminosity relation of low-mass objects. In a recent investigation, Santiago et al. (1996) found a steadily increasing mass function for , using HST observations. This suggests that the dark matter content may depend on the cluster. It seems therefore important to fully understand what we can also learn from other types of measurement, such as in particular the brightness and the velocity dispersion profiles. There is a widespread belief that globular clusters cannot contain large amounts of dark matter, because of the Virial theorem. The latter relates the velocity dispersion $\sigma$ at the center of the cluster to the total mass $M_\mathrm{t}$ and to the half-mass radius $r_\mathrm{h}$ of the system. For a one-component globular cluster, that relation may be expressed as (Spitzer 1987) : $$\begin{aligned} \langle \sigma^2 \rangle \approx 0.4 \frac{G M_\mathrm{t}}{r_\mathrm{h}} \label{viriel}\end{aligned}$$ However, if a dark component is also present, in the form of low-mass objects for example, the virial theorem must be modified to $$\begin{aligned} \sum_{i} M_\mathrm{t}(i) \langle \sigma^2 \rangle_{i} \; = \; E_p \;\; ,\end{aligned}$$ where the index $i$ refers to the different stellar species, and where $E_{p}$ is the potential energy of the cluster. In previous papers (Taillet et al. 1995 and 1996) focusing on isotropic globular clusters, we showed that in the presence of dark matter, relation (\[viriel\]) still holds for the visible component, and that most of the visible properties of the cluster are approximately the same as for a one-component system. In particular, the brightness profile is nearly unchanged even if dark matter is abundant. This is due to the phenomenon of mass segregation. Because of gravitational interactions, stars tend to share their kinetic energy. Thus, the light objects gain velocity from the heavier stars, which sink to the center of the cluster, while the light objects have a much more diffuse distribution. The heavier stars then form a self-gravitating system in the low-density background of the light objects, and their properties are not significantly altered. It was concluded that even a dark-matter dominated cluster looks very much the same as a dark-matter free system, and that the point of the dark matter content could not be made by the mere observation of the brightness profile. It was also noted that the presence of dark matter would be betrayed by the velocity dispersion profile of bright stars which flattens when dark matter is present. However, those profiles need to be accurately measured, which is not yet the case. The results of our analysis were obtained for King models where the stellar distribution is isotropic in velocity space. Their validity must be reassessed when the velocity distribution is no longer isotropic. Anisotropy is likely to be present at some level because stars mainly interact in the cluster core. At least part of those orbiting at the outskirts of the system gained the required energy by interacting at the center. Then, orbits should be predominantly radial at large radii. In that respect, note that for which the HST observations provide a steadily rising mass function may also have an anisotropic velocity distribution (Meylan 1987), even though an isotropic distribution is not excluded (Merrit et al. 1996). Such a cluster may be modeled by a Michie distribution in phase space, where an angular-momentum exponential cut-off is applied to a King distribution (see section 2). It is well known that when anistropy is present, the velocity dispersion profile is less flat. Thus we suspect that the flattening of the velocity dispersion profile that occurs in the presence of dark matter will be partially cancelled by the effect of anisotropy. A given velocity dispersion profile could therefore be interpreted either with a one-component isotropic GC or with a two-component anisotropic GC. However, anisotropy may also affect a priori the brightness profile. The main goal of this paper is therefore to make these statements more quantitative. We will investigate how anisotropy affects the detectability of dark matter in globular clusters. In section 2, we present a new analytic form for the equation that relates the gravitational potential to the stellar mass densities. The Michie distribution is expressed with the usual error function and Dawson’s integral. Section 3 is devoted to our set of two-component Michie models whose heavy component is kept fixed while the abundance in low-mass stars is varied. Various velocity anisotropies are explored. Finally, the surface brightness and the velocity dispersion profiles of these models are discussed in section 4 and conclusions are drawn. The Michie distribution functions ================================= A Michie model of a globular cluster is spherically symmetric as regards its distribution of stars in space. However, its velocity ellipsoid tends to be elongated towards the center of the cluster for radii larger than some critical value $r_a$. The phase space distribution of a stellar species with mass $m$ and whose one-dimensional velocity dispersion is $\sigma$, may be expressed as $$\begin{aligned} f(r,\vec{v}) \; = \; %\frac{n_{1}}{\left( 2 \pi \sigma^2 \right)^{3/2}} \; k \; e^{\displaystyle - L^2 / 2 r_a^2 \sigma^2} \; \left\{e^{\displaystyle {\cal E} / \sigma^2} \, - \, 1 \right\} \;\; , \label{MICHIE_1}\end{aligned}$$ where $\vec{L} = \vec{OM} \wedge \vec{v}$ is the orbital momentum of the stars located at point M and whose velocities are $\vec{v}$. That orbital momentum is defined with respect to the cluster center O. The quantity $\cal{E}$ is related to the stellar energy per unit mass $E = \Phi(r) + v^{2}/2$ through $$\begin{aligned} {\cal E} \; = \; \Phi_\mathrm{t} - E \;\; ,\end{aligned}$$ where $\Phi_\mathrm{t} = \Phi(r_\mathrm{t})$ is the gravitational potential at the tidal boundary $r_\mathrm{t}$ of the cluster. Whenever $\cal{E}$ is negative, the distribution function $f$ vanishes. Inside the radius of anisotropy $r_{a}$, the distribution of stellar velocities is spherical. Beyond $r_{a}$, it straightens in the direction of the cluster center. At remote distances, trajectories are almost radial. This is a major difference with King models for which velocities are spherically distributed. The mass density is now given by a double integral. The orbital momentum $L = r \, v \, \sin \theta$ depends actually on both the magnitude $v$ of the velocity and on the angle $\theta$ between the latter and the radial direction $\vec{OM}$. Defining the variables $x = v / \sqrt{2} \sigma$ and $y = \cos \theta$ yields a mass density $$\begin{aligned} \rho(r) \; = \; 4 \pi k \; m\, \left( 2 \sigma^2 \right)^{3/2} \, {\cal H} (u,\alpha)\end{aligned}$$ proportional to the double integral $$\begin{aligned} {\cal H} (u,\alpha) \; = \; {\displaystyle \int_{0}^{1}} dy \, {\displaystyle \int_{0}^{\displaystyle \sqrt{u}}} dx \, x^{2} \, h(x,y) \;\; , \label{MICHIE_2}\end{aligned}$$ where the function $h(x,y)$ is defined as $$\begin{aligned} h(x,y) \; = \; e^{\displaystyle - \alpha^{2} x^{2} \left( 1 - y^{2} \right)} \, \left\{ e^{\displaystyle \left( u - x^{2} \right)} \, - \, 1 \right\} \;\; .\end{aligned}$$ The parameter $u = \left( \chi - \psi \right)$ is the difference between the reduced potential $\chi = \Phi_\mathrm{t} / \sigma^2$ at the tidal boundary $r_\mathrm{t}$ of the system and its counterpart $\psi = \Phi(r) / \sigma^2$ at distance $r$ from the cluster center. The ratio $r / r_{a}$ is denoted by $\alpha$. The scaling factor between the mass density $\rho(r)$ and the integral (\[MICHIE\_2\]) may also be inferred from the relation $$\begin{aligned} {\displaystyle \frac{\rho(r)}{\rho(0)}} \; = \; {\displaystyle \frac{{\cal H} (u,\alpha)}{{\cal H} (\chi,0)}} \;\; ,\end{aligned}$$ where $\rho(0)$ denotes the stellar mass density at the cluster center. So far, expression (\[MICHIE\_2\]) was estimated numerically through a direct integration. We have derived here an analytic development of ${\cal H} (u,\alpha)$ in terms of the functions $$\begin{aligned} d(x) = e^{\displaystyle - x^{2}} \, {\displaystyle \int_{0}^{x}} \, e^{\displaystyle t^{2}} \, dt\end{aligned}$$ and $$\begin{aligned} e(x) = e^{\displaystyle x^{2}} \, {\displaystyle \int_{0}^{x}} \, e^{\displaystyle - t^{2}} \, dt \;\; .\end{aligned}$$ The first integral is called the Dawson’s function and may be easily computed thanks to a convenient approximation due to Rybicki (Rybicki 1989 and Numerical Recipes). The function $e(x)$ is related to the error function $$\begin{aligned} e(x) \; = \; \frac{\sqrt{\pi}}{2} \; e^{\displaystyle x^{2}} \; {\rm erf}(x) \;\; .\end{aligned}$$ The distribution function $f(r,\vec{v})$ can be averaged over the angle $\theta$ to yield a King distribution up to a correction factor $$\begin{aligned} f(r,v) \; = \; k \; \left\{e^{\displaystyle {\cal E} / \sigma^2} \, - \, 1 \right\} \; \left\{ \frac{d(\alpha x)}{\alpha x} \right\} \;\; . \label{MICHIE_KING}\end{aligned}$$ The parameter $\alpha x$ stands for the ratio $r v / \sqrt{2} r_{a} \sigma$. Inside the radius of anisotropy, $\alpha x$ is small compared to unity and the correction factor $d(\alpha x) / \alpha x$ of relation (\[MICHIE\_KING\]) reduces to 1. A King distribution is therefore recovered when the radius $r$ is smaller than $r_{a}$. After an integration on the magnitude $v$ of the velocity, relation (\[MICHIE\_2\]) may be expressed as $$\begin{aligned} {\cal H} (u,\alpha) = \frac{1}{2 (1 + \alpha^{2})} \left[ e(\sqrt{u}) - \sqrt{u} + \frac{d(\alpha \sqrt{u}) - \alpha \sqrt{u}}{\alpha^{3}} \right] \label{H_u_a}\end{aligned}$$ Several approximations to that expression are given in the Appendix depending on whether $\sqrt{u}$ and $\alpha \sqrt{u}$ are large or not with respect to unity. The dispersion velocity profile of globular clusters may be defined in various ways. What astronomers actually measure is the velocity of some stellar species along the line of sight. Suppose that a specific pixel contains $N$ stars with same spectral type. The radial velocity of the i$^{\rm th}$ object of the sample is denoted by $v_{i}$. A variety of statistical averages can be performed, each yielding a different value for the dispersion. In this article, the dispersion velocity $\bar{v}_\mathrm{rad}$ along the line of sight will be defined as $$\begin{aligned} \bar{v}_\mathrm{rad} \; = \; \left\{ {\displaystyle \sum_{i = 1}^{N}} \, \frac{v_{i}^{2}}{N} \right\}^{1/2} \;\; . \label{VR_OBS}\end{aligned}$$ In our two component models, we will be interested in the velocity dispersion profile of the heavy and bright stellar population for which spectral measurements can be performed. Let us consider now a line of sight with direction set by the unit vector $\vec{e}_{\rm s}$ as shown in Fig. \[figure1\]. At the pericluster H, the distance to the center O is minimal. For any given point M along that line of sight, a convenient frame may be defined with its unit vector $\vec{e}_{\rm z}$ pointing outwards in the same direction as the vector $\vec{OM}$. The unit vector $\vec{e}_{\rm x}$ is perpendicular to $\vec{e}_{\rm z}$ and is in the plane defined by the three points O, H and M. Finally, the last basis vector $\vec{e}_{\rm y}$ is defined as the product $\vec{e}_{\rm z} \wedge \vec{e}_{\rm x}$. The position of point M may be traced by the angle $\eta$ between the vectors $\vec{OH}$ and $\vec{OM}$. The projection $v_{s}$ of the velocity along the line of sight is then given by $$\begin{aligned} \vec{e}_{\rm s} \cdot \vec{v} \; = \; v_{x} \cos \eta \, + \, v_{z} \sin \eta \;\; .\end{aligned}$$ Its square $v_s^2$ may be averaged locally to yield $$\begin{aligned} \left\langle v_s^2 \right\rangle \; = \; \left\langle v_x^2 \right\rangle \cos^{2} \eta \, + \, \left\langle v_z^2 \right\rangle \sin^{2} \eta \;\; .\end{aligned}$$ Because the velocity distribution is axisymmetric around the direction connecting point M to the cluster center O, the contribution of the product $v_{x} v_{z}$ to the local average of the line of sight velocity $v_{s}^{2}$ vanishes. The latter may be expressed in terms of the functions ${\cal I} (u,\alpha)$ and ${\cal J} (u,\alpha)$ $$\begin{aligned} \rho (M) \, \left\langle \left( \frac{v_s}{\sigma} \right)^2 \right\rangle = 4 \pi k \; m\, \left( 2 \sigma^2 \right)^{3/2} \; \left\{ {\cal I} + {\cal J} \, \sin^2 \eta \right\} \end{aligned}$$ where $\rho (M)$ denotes the mass density of the stellar species with mass $m$ under consideration at the point $M$. The functions ${\cal I}$ and ${\cal J}$ are respectively defined by the integrals $$\begin{aligned} {\cal I} (u,\alpha) \; = \; {\displaystyle \int_{0}^{1}} dy \, {\displaystyle \int_{0}^{\displaystyle \sqrt{u}}} dx \, x^{4} \, \left( 1 - y^{2} \right) \, h(x,y) \;\; , \label{ISO_I}\end{aligned}$$ and $$\begin{aligned} {\cal J} (u,\alpha) \; = \; {\displaystyle \int_{0}^{1}} dy \, {\displaystyle \int_{0}^{\displaystyle \sqrt{u}}} dx \, x^{4} \, \left( 3 y^{2} - 1 \right) \, h(x,y) \;\; . \label{ANI_J}\end{aligned}$$ The functions ${\cal H}$ and ${\cal I}$ are related by $$\begin{aligned} {\cal I}(u,\alpha) \; = \; - \, \frac{\partial {\cal H}}{\partial \alpha^{2}} \;\; ,\end{aligned}$$ so that ${\cal I}$ may be expressed as $$\begin{aligned} \left( 1 + \alpha^{2} \right) \, {\cal I}(u,\alpha) &=& {\cal H} \, + \, \frac{3}{4 \alpha^{5}} \left\{ d \left( \alpha \sqrt{u} \right) - \alpha \sqrt{u} \right\} \nonumber \\ &+& \frac{u}{2 \alpha^{3}} d \left( \alpha \sqrt{u} \right) \;\; .\end{aligned}$$ The calculation of the function ${\cal J}$ is slightly more involved and yields $$\begin{aligned} \left( 1 + \alpha^{2} \right)^{2} \, {\cal J}(u,\alpha) & = & \frac{\alpha{2}}{2} \left\{ e \left( \sqrt{u} \right) - \sqrt{u} \right\} \nonumber \\ &+& \left( \alpha^{2} + u + u \alpha^{2} \right) \frac{u^{3/2}}{3} \nonumber \\ & - & {\cal K}(u,\alpha) \left\{ d \left( \alpha \sqrt{u} \right) - \alpha \sqrt{u} \right\} \;\; ,\end{aligned}$$ where $$\begin{aligned} {\cal K}(u,\alpha) \; = \; \frac{2 \alpha^{2} \, + \, \left( 1 + \alpha^{2} \right) \left( 5 + 2 u \alpha^{2} \right)} {4 \alpha^{5}} \;\; .\end{aligned}$$ Notice that $v_{s}^{2}$ needs eventually to be averaged along the entire line of sight. The end result is therefore the radial average $$\begin{aligned} \bar{v}_\mathrm{rad}^2 \; = \; {\displaystyle \frac{\displaystyle \int \, \rho (M) \, \left\langle v_{s}^{2} \right\rangle \, ds} {\displaystyle \int \, \rho (M) \, ds} } \;\; , \label{velocity_dispersion}\end{aligned}$$ to be compared with the observational definition (\[VR\_OBS\]). The models ========== To test the sensitivity of observable quantities to the amount of dark matter, we built two-component Michie models of globular clusters, with both a heavy and a light stellar population. One of the components encapsulates the visible, solar mass stars while the other component is dark matter in the form of low-mass objects with mass $m_{2} = 0.1\, \mbox{M}_\odot$. The indexes 1 and 2 respectively refer to the bright and dark populations. We built 25 globular clusters, with anisotropy radii $z_{a}$ = 1000, 100, 50, 20 and 10, and dark matter content $M_{2}/M_{1}$ = 0, 1, 3, 6 and 10. The parameter $z_{a}$ stands for the ratio of the anisotropy radius $r_{a}$ to the typical scale length $a$ yet to be defined. Models with $z_{a}$ = 1000 may be considered as King models. In that case, the anisotropy radius is so large that it contains most of the cluster, hence a spherical velocity distribution almost everywhere. A two-component Michie model is completely specified by six parameters, namely the anisotropy radius $r_{a}$, the velocity dispersions $\sigma_1$ and $\sigma_2$ of the two species, the normalization factors $k_1$ and $k_2$ (or alternatively the central densities $\rho_{c1}$ and $\rho_{c2}$), and finally the depth of the gravitational potential well $\chi = \Phi_\mathrm{t} / \sigma_1^2$. These parameters are entirely determined by the following conditions : - [(i)]{} The velocity dispersion for the luminous stars is measured. We will adopt the typical value $\sigma_1 = 7 \, \mbox{km/s}$. - [(ii)]{} The two species are in thermal equilibrium, imposing the relation $m_1 \sigma_1^2 \, = \, m_2 \sigma_2^2 $ which sets $\sigma_2$ equal to $\sqrt{10} \times \sigma_1$. - [(iii)]{} The visible central density is set to $\rho_{c1} = 8000 \, \mbox{M}_\odot / \mbox{pc}^3$. This leads to models having almost the same central brightness which, as a matter of fact, is a well measured quantity. - [(iv)]{} The anisotropy radius has been varied as explained above. The smaller $r_{a}$, the stronger the anisotropy. - [(v)]{} The total visible luminosity is also a well determined quantity. Here, it has been set equal to a typical value $L_{1} = 3\times 10^{5} \,\mbox{L}_\odot$. Throughout our set of models, $\rho_{c2}$ and $\chi$ have been chosen to yield that total luminosity, as well as the above mentioned ratios $M_{2}/M_{1}$. Remember that $M_{1}$ and $M_{2}$ respectively stand for the total mass in heavy and in low mass stars. Visible stars are assumed to have a standard solar $L/M$ ratio. Low mass objects are too faint to shine and their luminosity is negligible. The structure of the cluster is determined by the Poisson equation where the local mass density is given by $$\begin{aligned} \rho (r) \; = \; \rho_{c1} \, \frac{{\cal H} (u , \alpha)}{{\cal H} (\chi , 0)} \; + \; \rho_{c2} \, \frac{{\cal H} (m_{2} u / m_{1} , \alpha)}{{\cal H} (m_{2} \chi / m_{1} , 0)} \;\; .\end{aligned}$$ Introducing the dimensionless radius $z = r / a$ where the typical scale length $a$ is defined as $$\begin{aligned} a \; = \; \frac{\sigma_1}{\sqrt{4\pi G \rho_{c1}}} \;\; ,\end{aligned}$$ leads to the differential equation $$\begin{aligned} \frac{1}{z^{2}} \frac{d}{dz} \left( z^{2} \frac{du}{dz} \right) \; = \; - \, \frac{{\cal H} (u , \alpha)}{{\cal H} (\chi , 0)} \, - \, \left( \frac{\rho_{c2}}{\rho_{c1}} \right) \frac{{\cal H} (m_2 u / m_1 , \alpha)}{{\cal H} (m_2 \chi / m_1 , 0)} \nonumber\end{aligned}$$ Once that equation is solved and the structure of the cluster is determined, the velocity dispersion and the surface brightness profiles are computed. The former is evaluated with the help of relation (\[velocity\_dispersion\]). The latter is just the surface mass density of heavy stars $$\begin{aligned} \Sigma_1 \; = \; {\displaystyle \int} \, \rho_1(m) ds \;\; ,\end{aligned}$$ up to the factor $(L_{\odot} / M_{\odot}) / 4 \pi$. Discussion and conclusions ========================== Profiles of the surface mass density $\Sigma_1$ and of the velocity dispersion $\bar{v}_\mathrm{rad}$ for the heavy component of our models are presented in Fig. \[figure2\] and \[figure3\] respectively. Low mass stars do not shine in the V band and therefore do not contribute to the surface luminosity of the cluster. Figure 2 shows that whatever the anisotropy radius $z_{a}$, the brightness profiles are almost not affected by the dark matter content of the cluster. At constant $z_{a}$, the various curves are superimposed while the light to heavy component mass ratio $M_{2} / M_{1}$ increases from 0 to 10. This result was already obtained in the case of King models (Taillet et al. 1995 and 1996) and its validity is now extended to Michie clusters. For $z_{a}$ = 10 and 20, the surface luminosity drops like $r^{- 5/2}$ as expected for a strongly anisotropic system. In Fig. \[figure3\], the velocity dispersion profiles are presented. For $z_{a}$ = 1000, velocities are almost spherically distributed and the curves are different when the dark matter abundance $M_{2} / M_{1}$ varies. However, the velocity dispersion along the line of sight becomes insensitive to the dark matter content of the cluster when the anisotropy of the velocity distribution increases. For $z_{a}$ smaller than 20, the various profiles start to be fairly similar and become hard to disentangle observationally. Beyond the radius of anisotropy $r_{a}$, the stellar orbital velocity $v_{\theta} \, = \, v \, \sin \theta$ is dominated on average by the Michie exponential cut-off on the orbital momentum. The orbital velocity is in that case of order $v_{\theta} \sim \sigma \, r_a / r$. It is no longer sensitive to the gravitational potential. The velocity dispersion $\bar{v}_\mathrm{rad}$ measured along the line of sight, to which the orbital velocity contributes most, is therefore blind to the mass content of the cluster. As $z_{a}$ decreases, the effect is more and more pronounced. When it becomes dominant everywhere outside the core radius of the cluster, the gravitational potential and therefore the dark matter of the system have little effect on $\bar{v}_\mathrm{rad}$, hence the degeneracy of the profiles in Fig. \[figure3\] for $z_{a}$ = 10. Note that as velocities become radially distributed, light stars need more space to contribute the same dark mass as a result of the conservation of the phase space volume. At fixed $M_{2} / M_{1}$ ratio, low mass stars extend further away as the velocity anisotropy increases. However, there is still a substantial amount of them inside the visible part of the cluster. When $z_{a}$ = 10 for instance, the dark mass inside the inner 60 pc is respectively $4.7 \times 10^{4}, 1.3 \times 10^{5}, 2.2 \times 10^{5}$ and $2.9 \times 10^{5} \; {\rm M}_{\odot}$ for a light to heavy stellar mass ratio $M_{2} / M_{1}$ of $1, 3, 6$ and $10$. The dashed curve that appears in three of the plots of Fig. \[figure3\] is the velocity dispersion profile for a King model with no low mass stars. That case is degenerate with the entire set of Michie clusters with $z_{a} = 20$, whatever their dark matter content. The same velocity dispersion profile may therefore be interpreted either with a one-component isotropic cluster or, alternatively, with an anisotropic system containing dark matter. If so, the degeneracy may be lifted by the distribution of surface luminosity. Note however that the brightness profiles become different far from the center, in a region where the surface mass density $\Sigma_1$ has already dropped by four orders of magnitude with respect to its central value. Figures 2 and 3 illustrate actually the difficulty to estimate the amount of low mass stars potentially concealed in globular clusters from their velocity dispersion profiles and from the distributions of their surface luminosities. Both measurements have been used so far to determine the mass content of GCs. We therefore conclude that the same set of observations may be interpreted by quite different models. This suggests that the actual stellar mass function of many globular clusters needs to be reinvestigated. Dark matter in globular clusters should be traced by the three dimensional velocity distribution of bright stars. We have just shown that the projection along the line of sight alone does not lead to a unique answer. Another possibility is the search for tidal tails. If present, dark matter dominates the gravity of the outskirts of clusters and prevents heavy stars from evaporating (Moore 1996). Tidal tails exist whenever dark matter is not present to play that inhibition effect. As mentioned by Taillet et al. (1995 and 1996), the presence of low mass stars should also lead to an infrared halo surrounding the bright central part. Finally, if the cluster lies against a rich stellar background, low mass stars should also induce a few gravitational microlensing events. This work has been carried out under the auspices of the Human Capital and Mobility Programme of the European Economic Community, under contract number CHRX-CT93-0120 (DG 12 COMA). De Marchi, G., Paresce, F., 1995, A& A 304, 202 Elson, R.A.W., Gilmore, G., Santiago, G.X., Casertano, S., AJ 110, 682 Fahlman, G.G., Richer, H.B., Searle, L., Thompson, I.B., 1989, ApJ 343, L49 Merrit, D., Meylan, G., Mayor, M., 1996, preprint astro-ph/9612184 Meylan, G., 1987, A&A 184, 144 Moore, B., 1996, ApJ 461, L13 Press, W.H, 1995, Numerical Recipes in C : the Art of Scientific Computing, 2nd edition, Cambridge University Press Paresce, F., De Marchi, G., Romaniello, M., 1995, ApJ 440, 216 Richer, H.B., Fahlman, G.G., Buonnano, R., Fusi Pecci, F., Searle, L., Thompson, I.B., 1991, ApJ 381, 147 Rybicki ,G.B., 1989, Computers in Physics 3, 85 Santiago, B.X., Elson, R.A.W., Gilmore, G.F., MNRAS 281, 1363 Spitzer, L.Jr, 1987, Dynamical evolution of globular clusters, Princeton University Press Taillet, R., Longaretti, P.-Y., Salati, P., 1995, Astroparticle Physics 4, 87 Taillet, R., Salati, P., Longaretti, P.-Y., 1996, ApJ 461, 104 Useful expansions ================= The functions $d(x)$ and $e(x)$ may be expanded as power series of the variable $x$ when the latter is small. The Dawson’s function is given by $$\begin{aligned} d(x) &=& x \, - \, A_{1} x^{3} \, + \, A_{1} A_{2} \, x^{5} \, + \, ... \nonumber \\ &+& (-1)^{n} A_{1} A_{2} ... A_{n} \, x^{2n+1} \, + \, ... \;\; , \label{DEL_d}\end{aligned}$$ whereas $$\begin{aligned} e(x) &=& x \, + \, A_{1} x^{3} \, + \, A_{1} A_{2} \, x^{5} \, + \, ... \nonumber \\ &+& A_{1} A_{2} ... A_{n} \, x^{2n+1} \, + \, ... \;\; . \label{DEL_e}\end{aligned}$$ The constants $A_{n}$ stand for the ratios $2 / (2n+1)$. Whenever $\alpha \sqrt{u}$ or $\sqrt{u}$ are small compared to 1, expansions (\[DEL\_d\]) or (\[DEL\_e\]) should be used in expression (\[H\_u\_a\]). If both variables are small at the same time, the expansion becomes $$\begin{aligned} {\cal H} (u,\alpha) & = & \frac{u^{5/2}}{2} A_{1} A_{2} \left\{ 1 \, + \, A_{3} u \left( 1 - \alpha^{2} \right) \right. \\ &+& A_{3} A_{4} u^{2} \left( 1 - \alpha^{2} + \alpha^{4} \right) \nonumber \\ & + & A_{3} A_{4} A_{5} u^{3} \left( 1 - \alpha^{2} + \alpha^{4} - \alpha^{6} \right) \nonumber \\ &+& A_{3} A_{4} A_{5} A_{6} u^{4} \left( 1 - \alpha^{2} + \alpha^{4} - \alpha^{6} + \alpha^{8} \right) \nonumber \\ &+& \left. ... \; \right\} \nonumber\end{aligned}$$ In the same limit, the function ${\cal I}$ may be expanded as $$\begin{aligned} {\cal I} (u,\alpha) & = & \frac{u^{7/2}}{2} A_{1} A_{2} A_{3} \left\{ 1 \, + \, A_{4} u \left( 1 - 2 \alpha^{2} \right) \right. \nonumber \\ &+& A_{4} A_{5} u^{2} \left( 1 - 2 \alpha^{2} + 3 \alpha^{4} \right) \nonumber \\ & + & A_{4} A_{5} A_{6} u^{3} \left( 1 - 2 \alpha^{2} + 3 \alpha^{4} - 4 \alpha^{6} \right) \\ &+& A_{4} A_{5} A_{6} A_{7} u^{4} \left( 1 - 2 \alpha^{2} + 3 \alpha^{4} - 4 \alpha^{6} + 5 \alpha^{8} \right) \nonumber \\ &+& \left. ... \; \right\} \nonumber\end{aligned}$$ whereas ${\cal J}$ is a sum of three series expansions $$\begin{aligned} \left( 1 + \alpha^{2} \right)^{2} \, {\cal J} (u,\alpha) & = & \frac{u^{9/2}}{15} \alpha^{2} A_{3} A_{4} \; \left\{ 2 {\cal J}_{a}(u) \right. \nonumber \\ &-& \left( 5 + 7 \alpha^{2} \right) \alpha^{2} {\cal J}_{b}(u \alpha^{2}) \nonumber \\ & + & \left. 9 \left( 1 + \alpha^{2} \right) \alpha^{2} {\cal J}_{c}(u \alpha^{2}) \right\} \;\; .\end{aligned}$$ The functions ${\cal J}_{a}$, ${\cal J}_{b}$ and ${\cal J}_{c}$ are respectively defined by $$\begin{aligned} {\cal J}_{a}(x) \; = \; 1 \, + \, A_{5} \, x \, + \, A_{5} A_{6} \, x^{2} \, +\, A_{5} A_{6} A_{7} \, x^{3} \, + \, ...\end{aligned}$$ $$\begin{aligned} {\cal J}_{b}(x) \; = \; 1 \, - \, A_{5} \, x \, + \, A_{5} A_{6} \, x^{2} \, - \, A_{5} A_{6} A_{7} \, x^{3} \, + \, ...\end{aligned}$$ while $$\begin{aligned} {\cal J}_{c}(x) \; = \; 1 \, - \, A_{4} x {\cal J}_{b}(x) \;\; .\end{aligned}$$

--- abstract: 'We study the dynamics of a nonlinear one-dimensional disordered system from a spectral point of view. The spectral entropy and the Lyapunov exponent are extracted from the short time dynamics, and shown to give a pertinent characterization of the different dynamical regimes. The chaotic and self-trapped regimes are governed by log-normal laws whose origin is traced to the exponential shape of the eigenstates of the linear problem. These quantities satisfy scaling laws depending on the initial state and explain the system behaviour at longer times.' address: 'Université Lille 1 Sciences et Technologies, CNRS; F-59655 Villeneuve d’Ascq Cedex, France. ' author: - B Vermersch and J C Garreau bibliography: - 'ArtDataBase.bib' - 'library.bib' title: Spectral description of the dynamics of ultracold interacting bosons in disordered lattices --- Introduction ============ The motion of non-interacting particles in disordered lattices has been intensely studied over the last decades. In one and two dimensions, and for a sufficient amount of disorder in three dimensions, it has been shown that the spreading of quantum wavepackets is suppressed, a phenomenon known as Anderson localization [@Anderson:LocAnderson:PR58; @Abrahams:Scaling:PRL79]. However, the celebrated Anderson model that leads to this prediction is a highly simplified model which in particular neglects particle-particle interactions, so that a crucial question is: Do interactions destroy (or, on the contrary, enhance) Anderson localization? A burst of interest on the subject was recently driven by the demonstration that these questions could be studied experimentally and theoretically with an unprecedented degree of cleanness and precision using ultracold atoms. This has produced an impressive number of results, concerning both Anderson localization (AL) [@Moore:AtomOpticsRealizationQKR:PRL95; @Billy:AndersonBEC1D:N08; @Roati:AubryAndreBEC1D:N08; @Kondov:ThreeDimensionalAnderson:S11; @Jendrzejewski:AndersonLoc3D:NP12; @Lignier:Reversibility:PRL05] and the Anderson transition [@Chabe:Anderson:PRL08; @Lemarie:AndersonLong:PRA09; @Lemarie:CriticalStateAndersonTransition:PRL10; @Lopez:ExperimentalTestOfUniversality:PRL12], observed in 3D systems. Moreover, systems of ultracold bosons turned out to be very well modeled by mean-field approaches [@Stringari:BECRevTh:RMP99; @Bloch:ManyBodyUltracold:RMP08], in contrast to fermionic systems where such a simplification of the corresponding many-body problem is not possible. From the theoretical and numerical point of view, ultracold bosons in a 1D optical disordered lattice can be described rather realistically by a simple generalization of the original Anderson model including a nonlinear term taking into account interactions in a mean-field description. Numerical studies of the model suggest that the long term motion of the particles is subdiffusive [@Shepelyansky:DisorderNonlin:PRL08; @Flach:DisorderNonlin:PRL09; @Flach:DisorderNonlineChaos:EPL10; @Pikovsky:ScalingPropertiesWeakChaos:PRE11; @Basko2011; @Ivanchenko2011; @Vermersch:AndersonInt:PRE12]. These studies pointed out in particular the central role of chaotic dynamics in the destruction of AL. This approach has however two important drawbacks. The first is that the timescale of subdiffusion is larger by several orders of magnitude than the timescale of the single-particle dynamics, implying very long computer calculations which make difficult a full study of the interplay between disorder and interactions. The second is that the nonlinearity leads to a strong dependence on the initial conditions which also makes it difficult to give a “global” characterization of the different dynamical regimes. In previous works, we have shown the existence of scaling laws *with respect to the width of the initial state*, allowing, to some extent, such a global characterization [@Vermersch:AndersonInt:PRE12], and demonstrated that these scaling laws are robust with respect to decoherence effects  [@Vermersch2012a] (which can also destroy AL). In the present work we tackle the problem of interacting ultracold bosons in a 1D disordered lattice using a spectral analysis that does not require such long computational times, as the information on the chaotic behaviour is “inscribed” even in early times in the spectrum of the dynamics. This allows us to perform a more complete and precise study of the problem over a large range of parameters. A central quantity in our study, the spectral entropy, proves very useful to characterize the dynamic behaviours, confirmed by comparing the information extracted from the spectral entropy to the Lyapunov exponent, another well-known measure of chaotic behaviour. Moreover, we show that such behaviours are described by “log-normal” laws which can be scaled with respect to the initial conditions, and we propose a simple physical interpretation of our findings. The model ========= We use here the discrete nonlinear Schrödinger equation with diagonal disorder. It is essentially the same model used in previous works [@Vermersch:AndersonInt:PRE12; @Vermersch2012a], so we only give an outline of it here. The mean-field theory applied to ultracold bosons in an optical (ordered) lattice leads to the so-called Gross-Pitaevskii equation: $$i\hbar\dot{\phi}(x)=\left[\frac{p^{2}}{2m}+V(x)+g_{1D}N|\phi(x)|^{2}\right]\phi(x)$$ where $\phi$ is the macroscopic wavefunction (or order parameter) describing a Bose-Einstein condensate well below the critical temperature, $V(x)=-V_{0}\cos(2k_{L}x)$ is the optical potential and $g_{\mathrm{1D}}$ the 1D coupling constant [@Petrov:LowDimensionalTrappedGases:JP404]. Tight-binding equations are obtained by decomposing the wavefunction $\phi(x)=\sum_{n}c_{n}(t)w_{n}(x)$ onto the set of localized Wannier functions of the first band $w_{n}(x)$ associated to the $n^{\mathrm{th}}$ lattice site: $$i\dot{c}_{n}=v_{n}c_{n}-c_{n-1}-c_{n+1}+g\left|c_{n}\right|^{2}c_{n},\label{eq:DANSE}$$ where we kept only (symmetric) nearest-neighbours couplings, which is justified if the Wannier functions are strongly localized, that is, for large enough $V_{0}$. According to usual conventions, we measure distances in steps of the lattice and write energies in units of the coupling constant of neighbour sites $T=-\int dxw_{n}(x)[p^{2}/2m+V(x)]w_{n+1}(x)$ [^1]. Finally, times are written in units of $\hbar/T$. The coefficient $v_{n}=\int dxw_{n}(x)[p^{2}/2m+V(x)]w_{n}(x)/T$ is the diagonal on-site energy and the effect of interactions is taken into account, in a mean-field approach, by adding the nonlinear term $g\left|c_{n}\right|^{2}$ where the dimensionless interacting strength is $$g=\frac{g_{\mathrm{1D}}N\int d_{x}|w_{0}|^{4}}{T}.$$ In the absence of disorder, the on-site energy $v_{n}$ does not depend on the site index $n$ and can be thus set to $0$. According to the Anderson’s postulate [@Anderson:LocAnderson:PR58], we introduce diagonal disorder by picking random on-site energies $v_{n}$ uniformly in an interval $[-W/2,W/2]$. For $g=0$, describes the standard Anderson model, and we shall call the corresponding eigenstates (eigenvalues) “Anderson” eigenstates (eigenvalues). A few facts about it will be useful in what follows. The eigenstates are exponentially localized in average $\overline{c_{n}^{\nu}}\sim\exp\left(-|n|/l_{\nu}\right)$ (the overbar indicates the averaging over realizations of the disorder) with a localization length $l_{\nu}(W)\sim96(1-\epsilon_{\nu}^{2}/4)/W^{2}$ [@MuellerDelande:DisorderAndInterference:arXiv10]. For $g\neq0$ the equation becomes nonlinear, and it is useful, if not strictly rigorous, to interpret the nonlinear term as a “dynamical correction” to the on-site energy $v_{n}^{\mathrm{NL}}=g|c_{n}|^{2}$. Previous works [@Shepelyansky:DisorderNonlin:PRL08; @Flach:DisorderNonlin:PRL09; @Flach:DisorderNonlineChaos:EPL10; @Ivanchenko2011; @Pikovsky:ScalingPropertiesWeakChaos:PRE11; @Vermersch:AndersonInt:PRE12] put into evidence the existence of three main dynamical regimes: For $g\ll W$, Anderson localization is expected to survive for very long times, a regime that we shall call “quasi-localized”. For $g\sim W$, the nonlinear correction $v_{n}^{\mathrm{NL}}$ induces chaotic dynamics leading to subdiffusion and the destruction of AL. For $g\gg W$, the very large nonlinear term $v_{n}^{\mathrm{NL}}$ decouples all sites whose populations are not nearly equal (even in the absence of the disorder), suppressing the diffusive behaviour and leading to another type of localization, called self-trapping. This dynamics does not rely on quantum interference and is therefore very robust against external perturbations, including decoherence [@Vermersch2012a]. Our aim is to characterize the global dynamics on a relatively short timescale. We thus study the evolution according to of bosons in a 1D box containing $L$ sites (typically, $L=101$) and put an exponential absorber at each end of the box in order to prevent wavepacket reflection. The norm of the wavepacket is thus not anymore conserved as soon as it “touches” the borders, and we characterize the diffusive behaviour by calculating the survival probability $p(t)=\sum_{n}|c_{n}|^{2}$; a value $p(t)<1$ indicates that the packet has diffused outside the box. As in ref. [@Vermersch:AndersonInt:PRE12], we restrict the analysis to initial wavepackets of the form$$c_{n}(t=0)=\cases{L_0^{-1/2}\exp\left(i\theta_{n}\right)&$|n|\le\left(L_{0}-1\right)/2$\\0&overwise} %\cases{L_0^{-1/2}\exp\left(i\theta_{n}\right)&$}$$with $L_{0}\ll L$ [^2]. ![\[fig:pvsg\]Survival probability $p$ at time $t=10^{5}$ *vs* interaction strength $g$ for $W=4$ and different widths of the initial state : $L_{0}=$3 (blue squares), 7 (green triangles), 13 (red diamonds), 21 (cyan stars), 31 (magenta circles), 41 (yellow inverted triangles).](figure1){width="0.4\columnwidth"} The typical behaviour of the survival probability is illustrated in , which represents $p(g,t=10^{5})$ as a function of the interaction strength $g$ for $W=4$ and for various values of $L_{0}$. To obtain such smooth curves, the survival probability was averaged over typically $500$ realizations of the disorder and of the initial phases $\theta_{n}$. Such long times are necessary in order to clearly put into evidence the three different regimes mentioned above, which are then easily identifiable: At low $g$, the survival probability is close to one, corresponding to the quasi-localized regime in which AL survives. For intermediate values of $g$, the decrease of $p$ indicates that the wave packet has spread along the box and part of it has been absorbed at the borders, corresponding to a chaotic regime induced by the nonlinearity that destroys AL. For large values of $g$, the survival probability increases again and gets close to $1$, indicating that the packet is self-trapped [@Shepelyansky:DisorderNonlin:PRL08; @Flach:DisorderNonlin:PRL09; @Vermersch:AndersonInt:PRE12] and has never touched the borders if initially it was thin enough (e.g. in the $L_{0}=3$ case). In the next section, we show that a complementary way to describe the dynamics allows to a very precise characterization within much shorter computation times. Characterizing chaotic dynamics with the spectral entropy\[sec:SpecralEntropy\] =============================================================================== Spectral analysis is a very useful way of analyzing a chaotic dynamics, be it in classical, “quantum” [^3] or quantum nonlinear chaotic systems [@Lepers:QuasiClassTrack:PRL08]. The spectral entropy is a measure of the “richness” of a spectrum. Chaotic behaviours are associated to continuous spectra, and thus to a high spectral entropy. Given a quantity $M(t)$, its power spectrum is defined as $$S_{f}[M(t)]=\frac{\left|\tilde{M}(f)\right|^{2}}{\int_{0}^{f_{\max}}df\left|\tilde{M}(f)\right|^{2}}\quad,f\in[0,f_{\max}]$$ where $\tilde{M}(f)$ is the Fourier transform of $M(t)$ for $t\in[0,t_{\max}]$. The choice of the value of $t_{\max}$ determines the resolution of the spectrum. In our case, we chose $t_{\max}=200$, which is a very small value compared to the typical time of emergence of the interacting regime ($t\sim10^{5}$) but which will be shown to be sufficiently high to characterize the dynamics from a spectral point a view. As we do not expect excitations whose timescale is inferior to the tunneling time ($1$ in our rescaled units), we set $f_{\mathrm{max}}=1$ [^4]. From the power spectrum, one defines the spectral entropy as $$H=-\frac{\int df\ S(f)\log S(f)}{\log(f_{\mathrm{max}})}.\label{eq:H}$$ For a perfectly monochromatic signal $S_{f}=\delta(f-f_{0})$, the spectral entropy is zero, whereas for a white noise ($S_{f}=1/f_{\mathrm{max}}$) $H=1$. The spectral entropy “counts” the number of frequencies which are present in the signal, and is a good indicator of the chaoticity of the system [@Rezek:StochasticComplexityMeasures:ITBE98]. The spectral entropy obviously strongly depends on the choice of the observable, and its usefulness as a dynamics indicator is reliant on this choice. A good observable in the present problem is the so-called “participation number” $P$ with respect to the Anderson eigenstates, which is defined, in the present case, as follows. For a given realization of disorder $\{v_{n}\}$, we calculate the Anderson eigenstates in the Wannier basis, $\phi_{\nu}(x)=\sum_{n}d_{n}^{(\nu)}w_{n}(x)$ corresponding to an energy $\epsilon_{\nu}$, which are solutions of with $g=0$ (the $d_{n}^{(\nu)}$ replacing the $c_{n}$). Back to the $g\neq0$ case, allows us to calculate the evolution of the wavepacket $\psi(t)=\sum c_{n}(t)w_{n}(x)$ under the action of both disorder and nonlinearity[^5]. At any time, we can express the wavepacket in the Anderson eigenstates basis $$\psi(t)=\sum_{\nu}q_{\nu}(t)\phi_{\nu}(x),\label{eq:WavepacketAndersonEigeinbasis}$$ from which, trivially, $q_{\nu}(t)=\sum_{n}d_{n}^{(\nu)}c_{n}(t)$. The participation number is then defined as: $$P=\frac{\sum_{\nu}|q_{\nu}|^{2}}{\sum_{\nu}|q_{\nu}|^{4}}.\label{eq:ParticipationNb}$$ If $g=0$, the $\phi_{\nu}$ are the exact eigenstates of the problem, so that the populations $|q_{\nu}|^{2}$ are constant. In the non-interacting case $g\ne\text{0}$, the $|q_{\nu}|^{2}$ evolve under the action of the nonlinearity. The participation number roughly “counts” the number of Anderson eigenstates participating significantly in the dynamics[^6] and its time evolution thus reflects the apparition of Anderson eigenstates that were not initially populated. shows the spectral power $S(f)$ of the participation number $P$ in the three interacting regimes: (a) $g=1$ in the quasi-localized regime, (b) $g=100$ in the chaotic regime, and (c) $g=1000$ in the self-trapping regime. For $g=1$, the dynamics is very similar to the linear case: The populations of Anderson eigenstates practically do not evolve in time, as well as the participation number $P$ and the spectrum is dominated by low frequencies. As a consequence, the spectral entropy, calculated according to , is relatively small: $H=10^{-2}$. For $g=100$, Anderson eigenstates are strongly coupled by the nonlinear term, each pair of coupled states generating a Bohr frequency (shifted by the nonlinear correction), that is $\sim\epsilon_{\nu}+g\left|q_{\nu}\right|^{2}-\epsilon_{\nu'}-g\left|q_{\nu^{\prime}}\right|^{2}$. In this regime, most Anderson eigenstates are coupled to each other, so that the power spectrum is almost flat (with important local fluctuations) at high frequencies, and the spectral entropy increases by almost an order of magnitude $H=10^{-1}$ with respect to the preceding case. For $g=1000$ the wavepacket is self-trapped and only a relatively small number of Anderson eigenstates whose populations happen to be close enough can interact. The spectrum is therefore dominated by a finite number of frequencies and the spectral entropy is reduced, $H=2\times10^{-2}$. However, although populations are stable in this case, quantum phases may evolve chaotically under the action of the nonlinearity [@Thommen:ChaosBEC:PRL03]. Our definition of the spectral entropy from the participation number – which does not directly depends on the phases – excludes this “phase dynamics” from the corresponding spectrum. {width="0.3\columnwidth"}$\quad${width="0.3\columnwidth"}$\quad${width="0.3\columnwidth"} We display in a the averaged spectral entropy as a function of the nonlinearity parameter $g$ for different widths $L_{0}$ of the initial state. One clearly sees the crossover from the quasi-localized to the chaotic regime, signalled by a marked increase of $H$. The smaller the value of $L_{0}$, the smaller the value of the crossover. This is easily understandable, as a more concentrated wavepacket leads to a stronger nonlinear term $v^{\mathrm{NL}}$. On the right side of the plot, one also sees, especially for low values of $L_{0}$ the beginning of a decrease of $H$ due to self-trapping. It is interesting to compare the information obtained from the spectral entropy to another relevant quantity characterizing chaos, the Lyapunov exponent, which indicates how exponentially fast neighbour trajectories diverge. This quantity is usually defined for classical systems, but can be extended, with a little care, for quantum nonlinear systems [@Lepers:QuasiClassTrack:PRL08]. We present in \[app:Lyapunov\] a method for calculating the Lyapunov exponent of a quantum trajectory defined by amplitudes $c_{n}$ [\[]{}. b displays the Lyapunov exponent $\lambda$ obtained with the same parameters as in a. It displays a monotonous increase with the nonlinear parameter, even in the region where the spectral entropy decreases due to self-trapping, evidencing the presence of a regime of “phase chaos” mentioned above. This clearly shows that $H$ and $\lambda$ provide different information on the dynamics of the system, and that $H$ is clearly more adapted to distinguish the three dynamical regimes. In we will discuss the shape of these curves, their scaling properties, and suggest a physical mechanism explaining such properties.  Spectral entropy $H$ and (b) Lyapunov exponent $\lambda$ *vs* interaction strength $g$ for $W=4$ and different widths of the initial state : $L_{0}=$3 (blue squares), 7 (green triangles), 13 (red diamonds), 21 (cyan stars), 31 (magenta circles), 41 (yellow inverted triangles).](figure3a "fig:"){width="0.4\columnwidth"} Spectral entropy $H$ and (b) Lyapunov exponent $\lambda$ *vs* interaction strength $g$ for $W=4$ and different widths of the initial state : $L_{0}=$3 (blue squares), 7 (green triangles), 13 (red diamonds), 21 (cyan stars), 31 (magenta circles), 41 (yellow inverted triangles).](figure3b "fig:"){width="0.4\columnwidth"} Log-normal law and scaling\[sec:LogNormalLawScaling\] ===================================================== As we shall see below, log-normal functions are ubiquitous in the dynamics described in the present work, and so in various contexts which are not obviously related to each other. A first example can be found in , where each curve representing $p(\log g)$ can be fitted with a rather good accuracy by a inverted Gaussian function. More generally, a log-normal function is defined as $$\begin{aligned} f(x) & = & \frac{1}{x}\exp\left[-\frac{\left(\log x-\mu\right)^{2}}{2\sigma^{2}}\right]\\ & = & \exp(-\mu+\frac{\sigma^{2}}{2})\exp\left[-\frac{(\log x-\mu+\sigma^{2})^{2}}{2\sigma^{2}}\right]\end{aligned}$$ In physics, such a function appears more often as a “log-normal distribution”, related to the statistics of quantities which are a *product* of randomly distributed terms [@Limpert:LogNormalDistibutions:BSC01]. Log-normal statistics is very different from normal (Gaussian) statistics; for example, the most probable value of a log-normally distributed quantity is different from its average value. In the following, we shall not only consider statistical distributions over the realizations of the disorder: does not display the distribution of $p$ over the realizations of disorder but its average as a function of the interacting strength, so does a for the spectral entropy. However and more interestingly, we now show that the statistical distribution of the spectral entropy and of the Lyapunov exponent are indeed log-normal. Let us thus study the *distribution* of these two quantities over the realizations of the disorder $v_{n}$ and of the initial phases $\theta_{n}$. a displays the distributions of values of $H$ over $10^{4}$ realizations of the disorder for two values of $g$, and b the corresponding distribution for the Lyapunov exponent $\lambda$. As shown by the black fitting lines, both curves are perfectly fitted by a log-normal function. ![\[fig:histograms\]Histograms of the spectral entropy $H$ (a) and the Lyapunov exponent $\lambda$ (b) for $W=3$, $L_{0}=41$ and two interacting strengths : $g=10$ (left hand-side blue histogram), $g=100$ (right hand-side red histogram). Black solid lines correspond to log-normal fits.](figure4a "fig:"){width="0.4\columnwidth"}![\[fig:histograms\]Histograms of the spectral entropy $H$ (a) and the Lyapunov exponent $\lambda$ (b) for $W=3$, $L_{0}=41$ and two interacting strengths : $g=10$ (left hand-side blue histogram), $g=100$ (right hand-side red histogram). Black solid lines correspond to log-normal fits.](figure4b "fig:"){width="0.4\columnwidth"} In order to have an idea of the origin of these shapes, one can consider a simple, heuristic model. The destruction of AL is due to the nonlinear coupling between Anderson eigenstates. Initially unpopulated eigenstates, which, in the absence of nonlinearity, would never be populated, can be thus excited thanks to a nonlinear transfer of population. The goal of the simple model developed below is to characterize the statistical distribution of such excitations. Projecting the wavepacket in the Anderson eigenbasis [\[]{}cf. , then reads: $$i\dot{q}_{\nu}=\epsilon_{\nu}q_{\nu}+g\sum_{\nu_{1},\nu_{2},\nu_{3}}q_{\nu_{1}}^{*}q_{\nu_{2}}q_{\nu_{3}}I(\nu,\nu_{1},\nu_{2},\nu_{3})\label{eq:DANSE-nu}$$ with $I(\nu,\nu_{1},\nu_{2},\nu_{3})=\sum_{n}d_{n}^{(\nu)}d_{n}^{(\nu_{1})}d_{n}^{(\nu_{2})}d_{n}^{(\nu_{3})}$. Populations exchanges are controlled by the overlap $I$ of four coefficients. The population transfer from eigenstate $\mu$ to eigenstate $\nu$ depends on $J_{1}=I(\mu,\mu,\nu,\nu)$, $J_{2}=I(\mu,\mu,\mu,\nu)$ and $J_{3}=I(\mu,\nu,\nu,\nu)$ , the term due to $J_{1}$ being for instance : $$\frac{d\left|q_{\nu}\right|^{2}}{dt}=2gJ_{1}\,\mathrm{Im}\left(q_{\nu}^{*2}q_{\mu}^{2}\right)$$ In order to evaluate the probability distribution of $J_{1}$, we make the assumption that the coupled Anderson eigenstates are exponentially localized with the same localization length $\xi$. The exponential localization is valid on the average and the localization length is the same if the eigenstates have close enough eigenenergies. We thus write $$\begin{aligned} d_{n}^{(\nu)}{}^{2} & = & \tanh\left(\frac{1}{\xi}\right)\exp\left(-\frac{2|n|}{\xi}\right)\\ d_{n}^{(\mu)}{}^{2} & = & \tanh\left(\frac{1}{\xi}\right)\exp\left(-\frac{2|n-l(\mu,\nu)|}{\xi}\right).\end{aligned}$$ The overlap sum $J_{1}$ can then be written as $$J_{1}=\tanh^{2}\left(\frac{1}{\xi}\right)e^{-2l(\mu,\nu)\text{/\ensuremath{\xi}}}\left[\frac{2}{1-e^{-4/\xi}}+l(\mu,\nu)-1\right]\label{eq:J}$$ where, for simplicity, we have supposed that the spatial distance between the eigenstates, $l(\mu,\nu)$, is an integer. The most important term in is $e^{-2l(\mu,\nu)/\xi}$. In the limit $\xi\to0$: $$J_{1}=e^{-2l(\mu,\nu)/\xi}\left[l(\mu,\nu)+1\right]$$ The inverse localization length $\Lambda=1/\xi$ follows a *normal* distribution [@Starykh:DynLocCavities:PRE00; @MuellerDelande:DisorderAndInterference:arXiv10] $$P(\Lambda)\propto\exp[-(\Lambda-\Lambda_{0})^{2}/2\sigma^{2}]$$ we obtain for the distribution of values of the overlap $J_{1}$ $$\begin{aligned} P(J_{1}) & = & P(\Lambda)\left|\frac{d\Lambda}{dJ_{1}}\right|\\ & \propto & \frac{1}{J_{1}}\exp\left[-\frac{(\Lambda-\Lambda_{0})^{2}}{2\sigma^{2}}\right]\\ & \propto & \frac{1}{J_{1}}\exp\left[-\frac{\left(\log J_{1}-G\right)^{2}}{2\tilde{\sigma}^{2}}\right]\end{aligned}$$ where $G=\log\left[l(\mu,\nu)+1\right]-2l(\mu,\nu)\Lambda_{0}$ and $\tilde{\sigma}^{2}=4l(\mu,\nu)^{2}\sigma^{2}$. The overlap sum $J_{1}$ thus obeys a log-normal distribution. We conjecture that overlap sums like $J_{1}$, controlling the coupling between Anderson eigenstates, in fact control the destruction of the Anderson localization, and thus explain the log-normal shapes we observed for $H$ and $\lambda$. The above heuristic argument undoubtedly presents various assumptions that are not rigorously justified, but it has the merit of putting into evidence the intimate relation between the log-normal distribution and the exponential localization of the Anderson eigenstates. The link between the exponential shape and the emergence of log-normal statistics has also been studied in the case of the conductance of disordered systems [@vanLangen:DisorderedWaveguide:PRE96; @Evers:AndersonTransitions:RMP08; @MuellerDelande:DisorderAndInterference:arXiv10]. Let us now consider the averaged spectral entropy. In previous works [@Vermersch:AndersonInt:PRE12; @Vermersch2012a], we showed that suitable scaling with respect to the initial state width $L_{0}$ allowed a classification of dynamic regimes independently of the shape of the initial state. In particular the interacting strength $g$ was scaled as $\tilde{g}=gL_{0}^{-s}$ with $s\approx3/4$. This scaling is meaningless for low values of $L_{0}$ because in this case, the initial participation number is not of the order of $L_{0}$ but is of the order of the maximum Anderson localization length $\ell_{0}(W)\sim96/W^{2}$. a shows that scaling with $\tilde{g}$ and $\tilde{H}=HL_{0}^{s}$ make the curves $H(g)$ corresponding to $L_{0}\gtrsim20$ collapse to a single curve, except in the strong self-trapped region. b shows the Lyapunov exponent as a function of $\tilde{g}$; the curves also collapse for $L_{0}\gtrsim12$. The Lyapunov exponent itself is independent of $L_{0}$ (with no scaling of the variable $\lambda$), which is not surprising as it does not measure the absolute distance between two quantum trajectories but the timescale of their exponential divergence. ![\[fig:Atvsgt\] (a) rescaled spectral entropy $\tilde{H}=HL_{0}^{s}$ with $s\approx3/4$ and (b) Lyapunov exponent $\lambda$ *vs* the rescaled interaction strength $\tilde{g}$ for $W=4$ and for $L_{0}=$3 (blue squares), 7 (green triangles), 13 (red diamonds), 21 (cyan stars), 31 (magenta circles), 41 (yellow inverted triangles).](figure5a "fig:"){width="0.4\columnwidth"}![\[fig:Atvsgt\] (a) rescaled spectral entropy $\tilde{H}=HL_{0}^{s}$ with $s\approx3/4$ and (b) Lyapunov exponent $\lambda$ *vs* the rescaled interaction strength $\tilde{g}$ for $W=4$ and for $L_{0}=$3 (blue squares), 7 (green triangles), 13 (red diamonds), 21 (cyan stars), 31 (magenta circles), 41 (yellow inverted triangles).](figure5b "fig:"){width="0.4\columnwidth"} As shown by the black solid line in a, the scaled spectral entropy $\tilde{H}$ is very well fitted (outside the strong self-trapped region $\tilde{g}>100$) by the log-normal law: $$\tilde{H}=\frac{h_{0}}{\tilde{g}\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(\log\tilde{g}-G)^{2}}{2\sigma^{2}}\right]\label{eq:HtildeLogNorm}$$ with three free parameters, the amplitude $h_{0}$, the center $G$ of the distribution and the “standard deviation” $\sigma$. The study of these fitting parameters as a function of the disorder $W$ provides a full characterization of the dynamic regime. Instead of representing the fit parameters $h_{0}$ and $G$, we prefer to use more physical quantities, namely the maximum value of $\tilde{H}$, $\tilde{H}_{\max}=\left[h_{0}\left(2\pi\sigma^{2}\right)^{-1/2}\right]\exp\left(\sigma^{2}/2-G\right)$ (a) and the rescaled interaction strength $\tilde{g}_{c}=\exp\left(G-\sigma^{2}\right)$ (b) corresponding to this maximum. The dependence of the standard deviation $\sigma$ on $W$ is displayed in c. The quantity $\tilde{H}_{\max}$ is a decreasing function of $W$: as the localization length decreases with $W$, so does the overlap between two neighbour Anderson eigenstates. On the contrary, $\tilde{g}_{\mathrm{c}}$ is an increasing function of $W$: The number of resonances is maximum when $v_{n}^{\mathrm{NL}}$ is comparable to the typical energy between two neighbour states, itself of the order of the bandwidth $\sim4+W$; $\tilde{g}_{\mathrm{c}}$ is thus independent of $W$ at low disorders and increases with $W$ for larger disorders. Finally, in c, one can notice that the log-normal curve becomes sharper when the disorder increases, which can be attributed to the fact that Anderson Localization is more robust against interactions at high disorders. {height="0.15\textheight"}$\quad${height="0.15\textheight"}$\quad${height="0.15\textheight"} The fact that the spectral entropy can be determined from a relatively short time interval also allows one to follow the *evolution* of the dynamics. shows the evolution of the spectral entropy $H(t)$, defined as the spectral entropy calculated in an interval $[t,t+200]$ , that we shall call, for short, the *dynamic spectral entropy*. For the low value of $g=10$ (a), after the destruction of AL, the diffusion of the wavepacket produces a dilution and a consequent diminution of the nonlinearity that reduces the chaoticity, and thus the spectral entropy of the system. For the high value of $g=1000$ (b) one sees a more complex interplay of different regimes: The initial state is initially frozen in a self-trapping regime, but this regime is unstable (for this particular set of parameters): If a fraction of the packet escapes the self-trapped region, the nonlinear contribution $v_{n}^{\mathrm{NL}}$ decreases which leads to a weakening of the trapping. The destruction of the self-trapping regime takes a much longer time that the destruction of the AL. One first observes a small decrease of the spectral entropy, as some eigenstates leave the box and do not interact anymore. Then, when $v_{n}^{\mathrm{NL}}$ has decreased sufficiently in the center of the packet, the system enters the chaotic regime. ![\[fig:Hvst\]Evolution of the spectral entropy $H(t)$ for $W=4,$ $L_{0}=41$. For (a) $g=10$ one essentially sees the effect of dilution, which progressively weakens the chaoticity of the system. For (b) $g=10^{3}$, there is a first phase, in which the self-trapping is progressively destroyed which is followed by a slow transition towards the chaotic regime.](figure7a "fig:"){width="0.4\columnwidth"}![\[fig:Hvst\]Evolution of the spectral entropy $H(t)$ for $W=4,$ $L_{0}=41$. For (a) $g=10$ one essentially sees the effect of dilution, which progressively weakens the chaoticity of the system. For (b) $g=10^{3}$, there is a first phase, in which the self-trapping is progressively destroyed which is followed by a slow transition towards the chaotic regime.](figure7b "fig:"){width="0.4\columnwidth"} In plots a and b of we represented the dynamic spectral entropy $H(t)$ for three values of $L_{0}$, corresponding to a *same* value of $\tilde{g}$ (0.62 in a, 6.17 in b and c). The curves converge for long times, putting into evidence the existence of a *universal* asymptotic regime, independent of the initial state, once the proper scaling on $\tilde{g}$ is applied. Additional scaling can even be used to describe the transition from the self-trapping regime to the chaotic regime, as shown in c. Although we have presently no physical explanation for the exponents appearing in these scaling laws, these results strongly support the idea that the features of the nonlinear dynamics should be scaled again with respect to the initial state [@Vermersch:AndersonInt:PRE12]. {height="0.15\textheight"}$\quad${height="0.15\textheight"}$\quad${height="0.15\textheight"} Conclusion ========== We have shown that the spectral entropy is a very good indicator of the chaoticity of the dynamics, and allows a very good characterization of the dynamics regimes. Moreover, it can be calculated dynamically and also gives information on the *evolution* of the dynamics. The Lyapunov exponent gives, in the present context, a less complete characterization of the dynamics. It describes very well the progressive destruction of the Anderson Localization by the onset of chaotic behaviour, exactly as the spectral entropy, but it does not give information in the self trapping regime, where “phase chaos” is still present. The argument presented in suggests that the log-normal shape is linked to the exponential localization of the Anderson eigenstates. It is a bit surprising to find this link between the behaviour of a strongly nonlinear system and the eigenstates of the corresponding linear system. From a mathematical viewpoint, spectral analysis (in the sense of the determination of eigenvalues and eigenvectors) cannot be directly applied to nonlinear systems, and, presently, there is no alternative analytical method for analyzing nonlinear systems. The results discussed in the present work and in refs. [@Vermersch:AndersonInt:PRE12; @Vermersch2012a] indicate that *scaling over the initial state* is a promising tool for the study of the emerging field of nonlinear quantum mechanics. In this context, it is worth noting that we have presently no convincing explanation for the precise values of the exponents appearing in these scaling laws, and that we have considered only a particularly simple family of initial states. Moreover, there is no experimental verification of these scaling laws. It thus appears that a large field of investigations is opened for both theoreticians and experimentalists in the near future. Lyapunov exponent of a quantum trajectory\[app:Lyapunov\] ========================================================= We are interested in calculating the Lyapunov exponent of the quantum trajectory $c\equiv(c_{n})$. Considering two initial conditions $c^{a}$, $c^{b}$, which are separated by an infinitesimal distance $d_{0}$, the Lyapunov exponent measures the rate of their exponential divergence. $$\lambda=\lim_{t\to\infty}\frac{1}{t}\log\frac{d(t)}{d_{0}}\label{eq:lambda}$$ where $d(t)=|c^{b}(t)-c^{a}(t)|$. In principle, one could use directly but it turns out that the trajectories do not evolve in an ensemble of infinite volume. As a consequence the distance $d(t)$ rapidly saturates and $\lambda$ tends to $0$. The common method to get rid of this drawback is to let trajectories evolve during a short time period $d_{t}$ and then evaluate the corresponding Lyapunov exponent $$\lambda_{0}=\frac{1}{d_{t}}\log\frac{d_{1}}{d_{0}}\label{eq:lambda2}$$ where $d_{1}=|c^{b}(d_{t})-c^{a}(d_{t})|$. Before letting the system evolve for another time interval $d_{t}$, we rescale trajectories so that the distance between $a$ and $b$ is set to $d_{0}$ keeping their relative orientation unchanged. Usually, people rescale the second trajectory $\tilde{c}^{b}=c^{b}+\frac{d_{0}}{d_{1}}(c^{b}-c^{a})$[^7] satisfying immediately both requirements. We then obtain $\lambda_{1},\lambda_{2},..$ by iterating the three-steps operation : (i) evolution during a short time $d_{t}$ (ii) calculation of the corresponding Lyapunov exponent from (iii) renormalization of the trajectories. We finally deduce the final Lyapunov exponent : $$\lambda=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\lambda_{i}$$ where $N$ represents the number of iterations. Unfortunately, the crucial rescaling operation changes the norm of the second trajectory $c_{b}$ : in the case of quantum systems, the procedure is therefore based on non-physical states. Here we propose to modify the step (iii) rescaling both trajectories using the following scheme : $$\begin{aligned} \tilde{c}^{a} & = & \alpha c^{a}+\beta c^{b}\\ \tilde{c}^{b} & = & \gamma c^{a}+\delta c^{b}\end{aligned}$$ where $(\alpha,\beta,\gamma,\delta)$ satisfy the following conditions[^8] : $$\begin{aligned} \tilde{c}^{b}-\tilde{c}^{a} & = & \frac{d_{0}}{d_{1}}\left(c^{b}-c^{a}\right)\label{eq:rescale1}\\ |\tilde{c}^{a}|^{2} & = & 1\label{eq:rescale2}\\ |\tilde{c}^{b}|^{2} & = & 1\label{eq:rescale3}\end{aligned}$$ Given that $d_{1}=|c^{b}-c^{a}|$, the first equation is nothing but the usual scaling condition which is used for classical systems. Projecting on $c^{a}$ and on $c^{b}$, one immediately obtains $$\begin{aligned} \gamma & = & \alpha-\frac{d_{0}}{d_{1}}\\ \delta & = & \beta+\frac{d_{0}}{d_{1}}\end{aligned}$$ Subtracting from , and using $2\Re\langle c^{a}|c^{b}\rangle=2-d_{1}^{2}$, we obtain $$\beta=\alpha-\frac{d_{0}}{d_{1}}$$ and finally from , we can choose $$\alpha=\frac{d_{0}}{2d_{1}}+\frac{1}{2}\sqrt{\frac{4-d_{0}^{2}}{4-d_{1}^{2}}}$$ to finally deduce $\beta,\gamma,\delta$. This method allows an accurate calculation of the Lyapunov exponent in the discrete system considered in the present work, but it can in principle be also applied to the continuous Gross-Pitaevskii equation. References {#references .unnumbered} ========== [^1]: Wannier functions have the translation property $w_{n}(x)=w_{0}(x-n)$. [^2]: As discussed in more detail in ref. [@Vermersch:AndersonInt:PRE12], this form of wavepacket has the advantage of, on the average, projecting onto all Anderson eigenstates, thus rendering the dynamics roughly independent of the wavepacket energy. [^3]: Traditionally the term “quantum chaos” designates the behaviour of a quantum (linear) system whose classical (nonlinear) counterpart is chaotic. [^4]: None of our results is modified if we set $f_{\max}=2$. [^5]: We use a standard Crank-Nicholson scheme with time-step $0.01<dt<0.1$. [^6]: To see this intuitively, consider two limit cases: If only one $\nu=\nu_{0}$ Anderson eigenstate is populated, $|q_{\nu}|^{2}=\delta_{\nu,\nu_{0}}$ and thus $P=1$; if $L_{0}$ eigenstates are equally populated, $|q_{\nu}|^{2}=L_{0}^{-1}$ and $P=L_{0}$. [^7]: see for examplehttp://sprott.physics.wisc.edu/chaos/lyapexp.htm. [^8]: We calculate the Lyapunov exponent without using an absorber potential so that the norm $|c|^{2}$ is a conserved quantity.

--- abstract: 'A wide range of mechanisms have been proposed to supply the energy for gamma-ray bursts (GRB) at cosmological distances. It is a common misconception that some of these, notably NS-NS mergers, cannot meet the energy requirements suggested by recent observations. We show here that GRB energies, even at the most distant redshifts detected, are compatible with current binary merger or collapse scenarios involving compact objects. This is especially so if, as expected, there is a moderate amount of beaming, since current observations constrain the energy per solid angle much more strongly and directly than the total energy. All plausible progenitors, ranging from NS-NS mergers to various hypernova-like scenarios, eventually lead to the formation of a black hole with a debris torus around it, so that the extractable energy is of the same order, $10^{54}$ ergs, in all cases. MHD conversion of gravitational into kinetic and radiation energy can significantly increase the probability of observing large photon fluxes, although significant collimation may achieve the same effect with neutrino annihilation in short bursts. The lifetime of the debris torus is dictated by a variety of physical processes, such as viscous accretion and various instabilities; these mechanisms dominate at different stages in the evolution of the torus and provide for a range of gamma-ray burst lifetimes.' author: - ', P.$^1$, Rees, M.J.$^2$ & Wijers, R.A.M.J.$^{2,3}$' title: ' Energetics and Beaming of Gamma Ray Burst Triggers [^1] ' --- 52[E\_[52]{}]{} 13[r\_[13]{}]{} 2[\_2]{} 2[\^2]{} 5[t\_5]{} 3[\_3]{} $^1$Dpt. of Astronomy & Astrophysics, Pennsylvania State University, University Park, PA 16803\ $^2$Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.K.\ $^3$Dpt. of Physics & Astronomy, SUNY, Stony Brook, NY 11794-3800\ Date :  [ 9/03/98]{} Introduction ============ The discovery of afterglows in the last year has moved the investigation of gamma-ray bursts (GRB) to a new plane. It not only has opened the field to new wavelengths and extended observations to longer time scales, making the identification of counterparts possible, but also provided confirmation for much of the earlier work on the fireball shock model of GRB, in which the $\gamma$-ray emission arises at radii of $10^{13}-10^{15}$ cm (Rees & 1992, 1994, & Rees 1993, & Xu 1994, Katz 1994, Sari & Piran 1995). In particular, this model led to the prediction of the quantitative nature of the signatures of afterglows, in substantial agreement with subsequent observations ( & Rees 1997a, Costa et al. 1997, Vietri 1997a, Tavani 1997, Waxman 1997; Reichart 1997, Wijers et al. 1997). More recently, significant interest was aroused by the report of an afterglow for the burst GRB971214 at a redshift $z=3.4$, whose fluence corresponds to a $\gamma$-ray energy of $10^{53.5} (\Omega_\gamma /4\pi)$ erg (Kulkarni et al. 1998). There is also possible evidence that some fraction of the detected afterglows may arise in relatively dense gaseous environments. This is suggested, e.g. by evidence for dust in GRB970508 (Reichart 1998), the absence of an optical afterglow and presence of strong soft X-ray absorption in GRB 970828 (Groot et al. 1997, Murakami et al. 1997), the lack an an optical afterglow in the (radio-detected) afterglow of GRB980329 (Taylor et al. 1998), etc. This has led to the suggestion that “hypernova" models ( 1998, Fryer & Woosley 1998) may be responsible, since hypernovae are thought to involve the collapse of a massive star or its merger with a compact companion, both of which would occur on time scale short enough to imply a burst within the star forming region. By contrast, neutron star - neutron star (NS-NS) or neutron star - black hole (NS-BH) mergers would lead to a similar BH plus debris torus system and roughly the same total energies (a point not generally appreciated), but the mean distance traveled from birth is of order several kpc (Bloom, Sigurdsson & Pols 1998), leading to a burst presumably in a less dense environment. The fits of Wijers & Galama (1998) to the observational data on GRB 970508 and GRB 971214 in fact suggest external densities in the range of 0.04–0.4 cm$^{-1}$, which would be more typical of a tenuous interstellar medium. In any case, while it is at present unclear which, if any, of these progenitors is responsible for the bulk of GRB, or whether perhaps different progenitors represent different subclasses of GRB, there is general agreement that they all would be expected to lead to the generic fireball shock scenario mentioned above. Trigger Mechanisms and Black Hole/Debris Torus Systems ======================================================= The first detailed investigations of the disruption of a NS in a merger with another NS or a BH were carried out by Lattimer & Schramm (1976), and the significance of this work for GRB has only recently started to be appreciated. It has become increasingly apparent in the last few years that [*all*]{} plausible GRB progenitors suggested so far (e.g. NS-NS or NS-BH mergers, Helium core - black hole \[He/BH\] or white dwarf - black hole \[WD-BH\] mergers, and a wide category labeled as hypernova or collapsars including failed supernova Ib \[SNe Ib\], single or binary Wolf-Rayet \[WR\] collapse, etc.) are expected to lead to a BH plus debris torus system. An important point is that the overall energetics from these various progenitors do not differ by more than about one order of magnitude. Two large reservoirs of energy are available in principle: the binding energy of the orbiting debris, and the spin energy of the black hole (& Rees, 1997b). The first can provide up to 42% of the rest mass energy of the disk, for a maximally rotating black hole, while the second can provide up to 29% of the rest mass of the black hole itself. The $\nu\bar\nu \to e^+ e^-$ process (Eichler et al. 1989) can tap the thermal energy of the torus produced by viscous dissipation. For this mechanism to be efficient, the neutrinos must escape before being advected into the hole; on the other hand, the efficiency of conversion into pairs (which scales with the square of the neutrino density) is low if the neutrino production is too gradual. Typical estimates suggest a fireball of $\siml 10^{51}$ erg (Ruffert et al 1997, Popham, Woosley & Fryer 1998), except perhaps in the “collapsar" or failed SN Ib case where Popham et al. (1998) estimate $10^{52.3}$ ergs for optimum parameters. If the fireball is collimated into a solid angle $\Omj$ then of course the apparent “isotropized" energy would be larger by a factor $(4\pi/\Omj)$ , but unless $\Omj$ is $\siml 10^{-2} -10^{-3}$ this may fail to satisfy the apparent isotropized energy of $10^{53.5}$ ergs implied by a redshift $z=3.4$ for GRB 971214. An alternative way to tap the torus energy is through dissipation of magnetic fields generated by the differential rotation in the torus ( 1991, Narayan, & Piran 1992, & Rees 1997b, Katz 1997). Even before the BH forms, a NS-NS merging system might lead to winding up of the fields and dissipation in the last stages before the merger (& Rees 1992, Vietri 1997a). The above mechanisms tap the energy available in the debris torus or disk. However, a hole formed from a coalescing compact binary is guaranteed to be rapidly spinning, and, being more massive, could contain more energy than the torus; the energy extractable in principle through MHD coupling to the rotation of the hole by the Blandford & Znajek (1977) effect could then be even larger than that contained in the orbiting debris (& Rees 1997b,  1998). Collectively, any such MHD outflows have been referred to as Poynting jets. The various progenitors differ only slightly in the mass of the BH and that of the debris torus they produce, and they may differ more markedly in the amount of rotational energy contained in the BH. Strong magnetic fields, of order $10^{15}$ G, are needed needed to carry away the rotational or gravitational energy in a time scale of tens of seconds (Usov 1994, Thompson 1994). If the magnetic fields do not thread the BH, then a Poynting outflow can at most carry the gravitational binding energy of the torus. For a maximally rotating and for a non-rotating BH this is 0.42 and 0.06 of the torus rest mass, respectively. The torus or disk mass in a NS-NS merger is $M_d\sim 0.1\msun$ (Ruffert & Janka 1998), and for a NS-BH, a He-BH, WD-BH merger or a binary WR collapse it may be estimated at $M_d \sim 1\msun$ ( 1998, Fryer & Woosley 1998). In the HeWD-BH merger and WR collapse the mass of the disk is uncertain due to lack of calculations on continued accretion from the envelope, so $1\msun$ is just a rough estimate. The largest energy reservoir is therefore, ‘prima facie’, associated with NS-BH, HeWD-BH or binary WR collapse, which have larger disks and fast rotation, the maximum energy being $\sim 8 \times 10^{53} \eps (M_d/\msun)$ ergs; for the failed SNe Ib (which is a slow rotator) it is $\sim 1.2\times 10^{53}\eps (M_d/\msun)$ ergs, and for the (fast rotating) NS-NS merger it is $\sim 0.8\times 10^{53} \eps (M_d/0.1 \msun) $ ergs, where $\eps$ is the efficiency in converting gravitational into MHD jet energy. Conditions for the efficient escape of a high-$\Gamma$ jet may, however, be less propitious if the “engine" is surrounded by an extensive envelope. If the magnetic fields in the torus thread the BH, the rotational energy of the BH can be extracted via the B-Z (Blandford & Znajek 1977) mechanism (& Rees 1997b). The extractable energy is $\eps f(a)\Mbh c^2$, where $\eps$ is the MHD efficiency factor and $a = Jc/G M^2$ is the rotation parameter, which equals 1 for a maximally rotating black hole. $f(a)=1-\sqrt{\frac{1}{2}[1+\sqrt{1-a^2}]}$ is small unless $a$ is close to 1, where it sharply rises to its maximum value $f(1)=0.29$, so the main requirement is a rapidly rotating black hole, $a \simg 0.5$. For a maximally rotating BH, the extractable energy is therefore $0.29 \eps \Mbh c^2 \sim 5\times 10^{53} \eps (\Mbh/\msun)$ ergs. Rapid rotation is essentially guaranteed in a NS-NS merger, since the radius (especially for a soft equation of state) is close to that of a black hole and the final orbital spin period is close to the required maximal spin rotation period. Since the central BH will have a mass of about $2.5 \msun$ (Ruffert & Janka 1998), the NS-NS system can thus power a jet of up to $\sim 1.3 \times 10^{54} \eps (\Mbh/2.5\msun)$ ergs. The scenarios less likely to produce a fast rotating BH are the NS-BH merger (where the rotation parameter could be limited to $a \leq M_{ns}/\Mbh$, unless the BH is already fast-rotating) and the failed SNe Ib (where the last material to fall in would have maximum angular momentum, but the material that was initially close to the hole has less angular momentum). A maximal rotation rate may also be possible in a He-BH merger, depending on what fraction of the He core gets accreted along the rotation axis as opposed to along the equator (Fryer & Woosley 1998), and the same should apply to the binary fast-rotating WR scenario, which probably does not differ much in its final details from the He-BH merger. For a fast rotating BH of $3\msun$ threaded by the magnetic field, the maximal energy carried out by the jet is then $\sim 1.6\times 10^{54} \eps (\Mbh/3\msun)$ ergs. Thus in the accretion powered jet case the total energetics between the various models differs at most by a factor 20, whereas in the rotationally (B-Z) powered cases they differ by at most a factor of a few, depending on the rotation parameter. For instance, even allowing for low total efficiency (say 30%), a NS-NS merger whose jet is powered by the torus binding energy would only require a modest beaming of the $\gamma$-rays by a factor $(4\pi/\Omj)\sim 20$, or no beaming if the jet is powered by the B-Z mechanism, to produce the equivalent of an isotropic energy of $10^{53.5}$ ergs. The beaming requirements of BH-NS and some of the other progenitor scenarios are even less constraining. Intrinsic Time scales ======================= A question which has remained largely unanswered so far is what determines the characteristic duration of bursts, which can extend to tens, or even hundreds, of seconds. This is of course very long in comparison with the dynamical or orbital time scale for the “triggers" described in section 2. While bursts lasting hundreds of seconds can easily be derived from a very short, impulsive energy input, this is generally unable to account for a large fraction of bursts which show complicated light curves. This hints at the desirability for a “central engine" lasting much longer than a typical dynamical time scale. Observationally (Kouveliotou et al. 1993) the short ($\siml 2$ s) and long ($\simg 2$ s) bursts appear to represent two distinct subclasses, and one early proposal to explain this was that accretion induced collapse (AIC) of a white dwarf (WD) into a NS plus debris might be a candidate for the long bursts, while NS-NS mergers could provide the short ones (Katz & Canel 1996). As indicated by Ruffert et al. (1997), $\nu\bar\nu$ annihilation will generally tend to produce short bursts $\siml 1$ s in NS-NS systems, requiring collimation by $10^{-1}-10^{-2}$, while Popham, Woosley & Fryer (1998) argued that in collapsars and WD/He-BH systems longer $\nu\bar\nu$ bursts may be possible. Longer bursts however imply lower $e^\pm$ conversion efficiency, so the observed fluxes could then be explained only if the jets were extremely collimated, by at least $10^{-3}-10^{-4}$. We outline here several possible mechanisms, within the context of the basic compact merger or collapse scenario leading to a BH plus debris torus, which can lead to an adequate energy release on such time scales. If the trigger of a long-duration burst involves a black hole, then an acceptable model requires that the surrounding torus should not completely drain into the hole, or be otherwise dispersed, on too short a time scale. There have been some discussions in the literature of possible ’runaway instabilities’ in relativistic tori (Nishida et al.1996, Abramowicz, Karas & Lanza 1997, Daigne & Mochkovitch 1997): these are analogous to the runaway Roche lobe overflow predicted, under some conditions, in binary systems. These instabilities can be virulent in a torus where the specific angular momentum is uniform throughout, but are inhibited by a spread in angular momentum. In a torus that was massive and/or thin enough to be self-gravitating, bar-mode gravitational instabilities could lead to further redistribution of angular momentum and/or to energy loss by gravitational radiation within only a few orbits. Whether a torus of given mass is dynamically unstable depends on its thickness and stratification, which in turn depends on internal viscous dissipation and neutrino cooling. The disruption of a neutron star (or any analogous process) is almost certain to lead to a situation where violent instabilities redistribute mass and angular momentum within a few dynamical time scales (i.e. in much less than a second). A key issue for gamma ray burst models is the nature of the surviving debris after these violent processes are over: what is the maximum mass of a remnant disc/torus which is immune to very violent instabilities, and which can therefore in principle survive for long enough to power the bursts? Magnetic torques and viscosity ------------------------------ Differential rotation may amplify magnetic fields until magnetic viscosity dominates neutrino viscosity. Moreover, the torques associated with a large scale magnetic field may also extract energy and angular momentum by driving a relativistic outflow. If the trigger is to generate the burst energy, over a period 10–100 sec, via Poynting flux — either through a relativistic wind ’spun off’ the torus or via the Blandford-Znajek mechanism — the required field is a few times $10^{15}$G. A weaker field would extract inadequate power; on the other hand, if the large-scale field were even stronger, then the energy would be dumped too fast to account for the longer complex bursts. How plausible are fields of this strength? and Ruderman (1998) point out that, starting with $10^{12}$ G, it only takes of order a second for simple winding to amplify the field to $10^{15}$ G; they argue further that magnetic stresses would then be strong enough for flares to break out. But amplification in a newly-formed torus could well occur more rapidly, for instance via convective instabilities, as in a newly formed neutron star (cf. Duncan & Thompson 1992, Thompson 1994). Such fields can build up on very short time scales, or order $\sim$ few ms; however, convective overturning motions should stop after the disk has cooled by neutrino emission below a few MeV. The latter is generally estimated to be of order a few seconds (Ruffert et al, 1997). But azimuthal magnetic fields can also be generated via the Balbus-Hawley mechanism. The nonlinear evolution and/or reconnection of such fields as they become buoyant can then lead to poloidal components at least of order $\simg 10^{15}$ G. Indeed, it is not obvious why the fields cannot become even higher. Note that the virial limit is $B_v \sim 10^{17}$ G. After magnetic fields have built up to some fraction of the equipartition value with the shear motion, a magnetic viscosity develops. Assuming that $B_rB_\phi\sim B^2$, it can be characterized in the usual way by the parameter $\alpha\sim B^2/(4\pi \rho v_s^2 ) \sim 10^{-1} B_{15}^2 \rho_{13}^{-1} T_9^{-1}$. This viscosity continues operating also after cooling has led to the disappearance of neutrino viscosity. Assuming a value of $\alpha=0.1$, a BH mass 3 $\msun$ and outer disk radius equal to the Roche lobe size, Popham et al. (1998) estimate “viscous" life times of 0.1 s for NS/BH-NS, 10–20 s for a collapsar (failed SN Ib or rotating WR), and 15–150 s for WD-BH and He-BH systems (although fields of $10^{15}$ G may be more difficult to support in He-BH systems). A magnetic field configuration capable of powering the bursts is likely to have a large scale structure. Flares and instabilities occurring on the characteristic (millisecond) dynamical time scale would cause substantial irregularity or intermittency in the overall outflow that would manifest itself in internal shocks (Rees & , 1994) There is thus no problem in principle in accounting for sporadic large-amplitude variability, on all time scales down to a millisecond, even in the most long-lived bursts. Note also that it only takes a residual cold disk of $10^{-3}\msun$ to confine a field of $10^{15}$ G, which can extract energy from the black hole via the Blandford-Znajek mechanism. Even if the evolution time scale for the bulk of the debris were no more than a second, enough may remain to catalyse the extraction of enough energy from the hole to power a long-lived burst. Double peaked bursts -------------------- There are at least two mechanisms which might lead to a delayed “second" burst (or a double humped burst). One possibility is that a merger leads to a central NS, temporarily stabilized by its fast rotation, with a disrupted debris torus around it, which produces a burst powered by the accretion energy and the magnetic fields generated by the shear motions. After the NS has radiated enough of its angular momentum, and accreted enough matter to overcome its centrifugal support, it collapses to a BH, leading to a second burst, and second cycle of energy extraction (either from the disk or from the BH via B-Z). In both cases, the time scale between bursts should be between a few to few tens of seconds. The other possibility for a delayed second burst may arise in merging NS of very unequal masses. As the smaller one fills its Roche lobe and losses mass, the larger NS (which may also collapse to a BH) is surrounded by the gas acquired from its companion, producing a burst as above. Eventually the less massive donor comes under the critical mass for deleptonization, and this leads to an explosion (e.g. Eichler et al. 1989). Starting from a configuration with about $0.1\msun$ which losses mass to its companion, Sumiyoshi et al.. (1998) (see also & Lee 1998, Portegies Zwart 1998) find that the explosion occurs in a time scale of about 20 s. The importance of this process depends on the poorly known distribution of NS-NS binary mass ratios, and on whether the mass transfer between neutron stars of nearly equal mass can be stable. Isotropic or Beamed Outflows? =============================== [*Conversion into relativistic outflow.* ]{} Even if the outflow is not narrowly beamed, the energy of a fireball would be channeled preferentially along the rotation axis. Moreover, we would expect baryon contamination to be lowest near the axis, because angular momentum flings material away from the axis, and any gravitationally-bound material with low angular momentum falls into the hole. In hypernova and SNe Ib cases without a binary companion, however, the envelope is rotating only slowly and thus would not initially have a marked centrifugal funnel; a funnel might however develop after low angular momentum matter falls into the hole along the axis on a free-fall time scale measured from the outer radius of the envelope, $t\sim 10^4-10^5$ s.\ The dynamics are complex. Computer simulations of compact object mergers and black hole formation can address the fate of the bulk of the matter, but there are some key questions that they cannot yet tackle. In particular, high resolution of the outer layers is needed because even a tiny mass fraction of baryons loading down the outflow severely limits the attainable Lorentz factor — for instance a Poynting flux of $10^{52}$ ergs could not accelerate an outflow to $\Gamma > 100$ if it had to drag more than $\sim 10^{-4}$ solar masses of baryons with it. Further 2D numerical simulations of the merger and collapse scenarios are under way (Fryer & Woosley 1998, Eberl, Ruffert & Janka 1998, McFayden & Woosley 1998), largely using Newtonian dynamics, and the numerical difficulties are daunting. There may well be a broad spread of Lorentz factors in the outflow — close to the rotation axis $\Gamma$ may be very high; at larger angles away from the axis, there may be an increasing degree of entrainment, with a corresponding decrease in $\Gamma$. This picture suggests, indeed, that the variety of burst phenomenology could be largely attributable to a standard type of event being viewed from different orientations. As discussed in the last section, a variety of progenitors can lead to a very similar end result, whose energetics are within one order of magnitude from each other. [*Basic spherical afterglow model.*]{} Just as we can interpret supernova remnants even without fully understanding the initiating explosion, so we may hope to understand the afterglows of gamma ray bursts, despite the uncertainties recounted in the previous section. The simplest hypothesis is that the afterglow is due to a relativistic expanding blast wave. The complex time structure of some bursts suggests that the central trigger may continue for up to 100 seconds. However, at much later times all memory of the initial time structure would be lost: essentially all that matters is how much energy and momentum has been injected; the injection can be regarded as instantaneous in the context of the much longer afterglow. The simplest spherical afterglow model has been remarkably successful at explaining the gross features of the GRB 970228, GRB970508 and other afterglows (e.g. Wijers et al. 1997). This has led to the temptation to take the assumed sphericity for granted. For instance, the lack of a break in the light curve of GRB 970508 prompted Kulkarni et al. (1998a) to infer that all afterglows are essentially isotropic, leading to their very large (isotropic) energy estimate of $10^{53.5}$ ergs in GRB 971214. The multi-wavelength data analysis has in fact advanced to the point where one can use observed light curves at different times and derive, via parametric fitting, physical parameters of the burst and environment, such as the total energy $E$, the magnetic and electron-proton coupling parameters ${\eps}_B$ and ${\eps}_e$ and the external density $n$ (Waxman 1997, Wijers & Galama 1998). However, as emphasized by Wijers & Galama, 1998, what these fits constrain is only the energy per unit solid angle $\calE= (E/\Omj)$. [*Properties of a Jet Outflow.*]{} An argument for sphericity that has been invoked by observers is that, if the blast wave energy were channeled into a solid angle $\Omj$ then, as correctly argued by Rhoads (1997, 1998), one expects a faster decay of $\Gamma$ after it drops below $\Omj^{-1/2}$. A simple calculation using the usual scaling laws leads then to a steepening of the flux power law in time. The lack of such an observed afterglow downturn in the optical has been interpreted as further supporting the sphericity of the entire fireball. There are several important caveats, however. The first one is that the above argument assumes a simple, impulsive energy input (lasting $\siml$ than the observed $\gamma$-ray pulse duration), characterized by a single energy and bulk Lorentz factor value. Estimates for the time needed to reach the non-relativistic regime, or $\Gamma < \Omega_j^{-1/2} \siml$ few, could then be under a month (Vietri 1997, Huang, Dai & Lu 1998), especially if an initial radiative regime with $\Gamma\propto r^{-3}$ prevails. It is unclear whether, even when electron radiative time scales are shorter than the expansion time, such a regime applies, as it would require strong electron-proton coupling (, Rees & Wijers 1998). Waxman, et al. (1998) have also argued on observational grounds that the longer lasting $\Gamma \propto r^{-3/2}$ (adiabatic regime) is more appropriate. Furthermore, even the simplest reasonable departures from a top-hat approximation (e.g. having more energy emitted with lower Lorentz factors at later times, which still do not exceed the gamma-ray pulse duration) would drastically extend the afterglow lifetime in the relativistic regime, by providing a late “energy refreshment" to the blast wave on time scales comparable to the afterglow time scale (Rees & 1998). The transition to the $\Gamma < \Omega_j^{-1/2}$ regime occurring at $\Gamma\sim$ few could then occur as late as six months to more than a year after the outburst, depending on details of the brief energy input. Even in a simple top-hat model, more detailed calculations show that the transition to the non-relativistic regime is very gradual ($\delta t/t \simg 2$) in the light curve. Also, even though the flux from the head-on part of the remnant decreases faster, this is more than compensated by the increased emission measure from sweeping up external matter over a larger angle, and by the fact that the extra radiation, which arises at larger angles, arrives later and re-fills the steeper light curve. The sideways expansion thus actually can slow down the flux decay (Panaitescu & 1998), rather than making for a faster decay. As already noted by Katz & Piran (1997), the ratio $L_\gamma/L_{opt}$ (or $L_\gamma / L_x$) can be quite different from burst to burst. The fit of Wijers & Galama for GRB 970508 indicates an afterglow (X-ray energies or softer) energy per solid angle ${\cal E}_{52} =3.7$, while at $z=0.835$ with $h_{70}=1$ the corresponding $\gamma$-ray ${\cal E}_{52\gamma} =0.63$. On the other hand for GRB 971214, at $z=3.4$, the numbers are ${\cal E}_{52} = 0.68$ and ${\cal E}_{52\gamma}=20$. The bursts themselves require ejecta with $\Gamma > 100$. The gamma-rays we receive come only from material whose motion is directed within one degree of our line of sight. They therefore provide no information about the ejecta in other directions: the outflow could be isotropic, or concentrated in a cone of angle (say) 20 degrees (provided that the line of sight lay inside the cone). At observer times of more than a week, the blast wave would be decelerated to a moderate Lorentz factor, irrespective of the initial value. The beaming and aberration effects are less extreme so we observe afterglow emission not just from material moving almost directly towards us, but from a wider range of angles. The afterglow is thus a probe for the geometry of the ejecta — at late stages, if the outflow is beamed, we expect a spherically-symmetric assumption to be inadequate; the deviations from the predictions of such a model would then tell us about the ejection in directions away from our line of sight. It is quite possible, for instance, that there is relativistic outflow with lower $\Gamma$ (heavier loading of baryons) in other directions (e.g. Wijers, Rees & 1997); this slower matter could even carry most of the energy (, 1997). This hypothesis is, if anything, further reinforced by the fits of Wijers & Galama (1998) mentioned above. [*Observational constraints on beaming.*]{} As discussed above, anisotropy in the burst outflow and emission affects the light curve at the time when the inverse of the bulk Lorentz factor equals the opening angle of the outflow. If the critical Lorentz factor is less than 3 or so (i.e. the opening angle exceeds 20$^\circ$) such a transition might be masked by the transition from ultrarelativistic to mildly relativistic flow, so quite generically it would difficult to limit the late-time afterglow opening angle in this way if it exceeds 20$^\circ$. Since some afterglows are unbroken power laws for over 100 days (e.g. GRB970228), if the energy input were indeed just a a simple impulsive top-hat the opening angle of the late-time afterglow at long wavelengths is probably greater than 1/3, i.e. $\Omega_{\rm opt}\simg 0.4$. However, even this still means that the energy estimates from the afterglow assuming isotropy could be 30 times too high. The gamma-ray beaming is much harder to constrain directly. The ratio of $\Omega_\gamma /\Omega_x$ has been considered by Grindlay (1998) using data from Ariel V and HEAO-A1/A2 surveys, who did not find evidence for a significant difference between the deduced gamma-ray and X-ray rates, and concluded that higher sensitivity surveys would be needed to provide significant constraints. More promising for the immediate future, the ratio $\Omega_\gamma/\Omega_{\rm opt}$ can also be investigated observationally (see also Rhoads 1997). The rate of GRB with peak fluxes above 1 phcm$^{-2}$s$^{-1}$ as determined by BATSE is about 300/yr, i.e. 0.01/sq. deg/yr. According to Wijers et al. (1998) this flux corresponds to a redshift of 3. If the gamma rays were much more narrowly beamed than the optical afterglow there should be many ‘homeless’ afterglows, i.e. ones without a GRB preceding them. The transient sky at faint magnitudes is poorly known, but there are two major efforts under way to find supernovae down to about $R=23$ (Garnavich et al. 1998, Perlmutter et al. 1998). These searches have by now covered a few tens of square degree years of exposure and would be sensitive to afterglows of the brightness levels thus far observed. It therefore appears that the afterglow rate is not more than a few times 0.1/sq. deg/yr. Since the magnitude limit of these searches allows detection of optical counterparts of GRB brighter than 1 ph cm$^{-2}$ s$^{-1}$ it is fair to conclude that the ratio of homeless afterglows to GRB is at most a few tens, say 20. It then follows that $\Omega_\gamma>0.05\Omega_{\rm opt}$, which combined with our limit to $\Omega_{\rm opt}$ yields $\Omega_\gamma>0.02$. The true rate of events that give rise to GRB is therefore at most 600 times the observed GRB rate, and the opening angle of the ultrarelativistic, gamma-ray emitting material is no less than $5^\circ$. Combined with the most energetic bursts, this begins to pose a problem for the neutrino annihilation type of GRB energy source. Obviously, the above calculation is only sketchy and should be taken as an order of magnitude estimate at present. However, with the current knowledge of afterglows a detailed calculation of the sensitivity of the high-redshift supernova searches to GRB afterglows is feasible, and a precise limit can be set by such a study. Conclusions and Prospects ========================== Simple blast wave models seem able to accommodate the present data on afterglows. However we can at present only infer the energy per solid angle; as yet the constraints on the angle-integrated $\gamma$-ray energy are not strong. We must also remain aware of other possibilities. For instance, we may be wrong in supposing that the central object becomes dormant after the gamma-ray burst itself. It could be that the accretion-induced collapse of a white dwarf, or (for some equations of state) the merger of two neutron stars, could give rise to a rapidly-spinning, temporarily rotationally stabilized pulsar. The afterglow could then, at least in part, be due to a pulsar’s continuing power output. It could also be that mergers of unequal mass neutron stars, or neutron stars with other compact companions, lead to the delayed formation of a black hole. Such events might also lead to repeating episodes of accretion and orbit separation, or to the eventual explosion of a neutron star which has dropped below the critical mass, all of which would provide a longer time scale, episodic energy output. We need to be open minded, yet also not too sanguine, about the possibility of there being more subclasses of classical GRB than just short ones and long ones. For instance, GRB with no high energy pulses (NHE) appear to have a different (but still isotropic) spatial distribution than those with high energy (HE) pulses (Pendleton et al. 1996). Some caution is needed in interpreting this, since selection effects could lead to a bias against detecting HE emission in dim bursts (Norris, 1998). Then, there is the apparent coincidence of GRB 980425 with the SN Ib/Ic 1998bw (Galama et al. 1998). A simple but radical interpretation (Wang & Wheeler 1998) is that all GRB may be associated with SNe Ib/Ic and differences arise only from different viewing angles relative to a very narrow jet. The difficulties with this are that it would require extreme collimations by factors $10^{-3}-10^{-4}$, and that the statistical association of any subgroup of GRB with SNe Ib/Ic (or any other class of objects, for that matter) is so far not significant (Kippen et al. 1998). If however the GRB 980425/1998bw association is real, as argued by Woosley, Eastman & Schmidt (1998), Iwamoto et al. (1998) and Bloom et al. (1998), then we may be in the presence of a new subclass of GRB with lower energy $E_\gamma \sim 10^{48} (\Omj /4\pi )$ erg, which is only rarely observable even though its comoving volume density could be substantial. In this, more likely interpretation, the great majority of the observed GRB would have the energies $E_\gamma \sim 10^{54}(\Omj/4\pi)$ ergs as inferred from high redshift observations. Much progress has been made in understanding how gamma-rays can arise in fireballs produced by brief events depositing a large amount of energy in a small volume, and in deriving the generic properties of the long wavelength afterglows that follow from this (Rees 1998). There still remain a number of mysteries, especially concerning the identity of their progenitors, the nature of the triggering mechanism, the transport of the energy and the time scales involved. Nevertheless, even if we do not yet understand the intrinsic gamma-ray burst central engine, they may be the most powerful beacons for probing the high redshift ($z > 5$) universe. Even if their total energy is reduced by beaming to a “modest" $\sim 10^{52}-10^{52.5}$ ergs in photons, they are the most extreme phenomena that we know about in high energy astrophysics. The modeling of the burst itself — the trigger, the formation of the ultrarelativistic outflow, and the radiation processes — is a formidable challenge to theorists and to computational techniques. It is, also, a formidable challenge for observers, in their quest for detecting minute details in extremely faint and distant sources. And if the class of models that we have advocated here turns out to be irrelevant, the explanation of gamma-ray bursts will surely turn out to be even more remarkable and fascinating. Abramowicz, M.A., Karas, V., and Lanza, A., 1998 A&A (in press) (astro-ph/9712245) Blandford, R.D. & Znajek, R.L., 1977, MNRAS, 179, 433 Bloom, J., Sigurdsson, S. & Pols, O., 1998, MNRAS in press (astro-ph/9805222) Bloom, J., et al., 1998, ApJ(Letters), subm (astro-ph/9807050) Costa, E., et al., 1997, Nature, 387, 783 Daigne, F., and Mochkovitch, R., 1997, MNRAS, 285, L15 Fryer, C., & Woosley, S., 1998, subm to Ap.J.(Lett.), (astro-ph/9804167) Galama, T. et al., 1998, Nature, in press (astro-ph/9806175) Garnavich, P., et al., 1998, ApJ, 493, L53 Grindlay, J., 1998, preprint Groot, P. et al., 1997; in [*Gamma-Ray Bursts*]{}, Meegan, C., Preece, R. & Koshut, T., eds., 1997 (AIP: New York), p. 557 Huang, Y.F., Dai, Z.G. & Lu, T., 1998, A&A in press (astro-ph/9807061) Iwamoto, K. et al., 1998, Nature, in press (astro-ph/9806382) Katz, J., & Canel., 1996, ApJ, 471, 915 Katz, J., 1994, ApJ, 422, 248 Katz, J. & Piran, T., 1997, Ap.J., 490, 772 Kippen, R.M. et al., 1998, ApJ subm (astro-ph/9806364) , W. & Ruderman, M., 1998, preprint , W. & Lee, W, ApJ(Letters), 494, L53 Kouveliotou, C. et al., 1993, ApJ, 413, L101 Kulkarni, S., et al., 1998, Nature, 393, 35 Lattimer, J.M. & Schramm, D.N., 1976, ApJ, 210, 549 , P. and Rees, 1992, M.J., ApJ, 397, 570 , P. and Rees, M.J., 1993, ApJ, 405, 278 , P. & Rees, M.J., 1997a, ApJ, 476, 232 , P. & Rees, M.J., 1997b, ApJ, 482, L29 in press (astro-ph/9804119) , P., Rees, M.J. & Wijers, R.A.M.J., 1998, ApJ, 499, 301 (astro-ph/9709273) Metzger, M. et al., 1997, Nature, 387, 878 Murakami, T. et al., 1997, in [Gamma-Ray Bursts]{}, Meegan, C., Preece, R. & Koshut, T., eds., 1997 (AIP: New York), p. 435 Narayan, R., Paczyński, B. & Piran, T. 1992, ApJ, 395, L83 Nishida, S., Lanza, A., Eriguchi, Y., and Abramawicz, M.A., 1996, MNRAS 278, L41 Norris, J, 1998, private comm. Paczyński, B., 1991, Acta Astron. 41, 257 Paczyński, B. & Xu, G., 1994, ApJ, 427, 708 Paczyński, B., 1997, in [Gamma-Ray Bursts]{}, Meegan, C., Preece, R. & Koshut, T., eds., 1997 (AIP: New York), p. 783 Paczyński, B., 1998, ApJ, 494, L45 Panaitescu, A. & , P., 1998, ApJ, subm. (astro-ph/9806016) Perlmutter, S, et al., 1998, Nature, 391, 51 Pendleton, G, et al., 1996, Ap\[J, 464, 606 Popham, R., Woosley, S. & Fryer, C., 1998, ApJ, subm. (astro-ph/9807028) Portegies-Zwart, S., 1998, ApJ, Letters, in press (astro-ph/9804296) Rees, M.J. 1998, Nucl. Phys. B Sup., in press Rees, M.J. & , P., 1992, MNRAS, 258, 41P Rees, M.J. & , P., 1994, ApJ, 430, L93 Rees, M.J. & , P., 1998, ApJ, Letters, in press (astro-ph/9712252) Reichart, D., 1997, ApJ, 485, L57 Reichart, D., 1998, ApJ, 495, L99 Rhoads, J.E., 1997, ApJ, 487, L1 Ruffert, M., Janka, H.-T., Takahashi, K., Schaefer, G., 1997, A& A, 319, 122 Ruffert, M.& Janka, H.-T., 1998, preprint Sari, R. & Piran, T., 1995, ApJ, 455, L143 Sumiyoshi, K., Yamada, S., Suzuki, H. & Hillebrandt, W., 1998, A& A, 334, 159 Tavani, M., 1997, ApJ, 483, L87 Taylor, G.B., et al., 1997, Nature, 389, 263 Thompson, C., 1994, MNRAS, 270, 480 Thompson, C., Duncan, R.C., 1993, ApJ, 408, 194 Usov, V.V., 1994, MNRAS, 267, 1035 Vietri, M., 1997a, ApJ, 471, L95 Vietri, M., 1997b, ApJ, 488, L105 Wang, L. & Wheeler, J.C., 1998, ApJ subm (astro-ph/9806212) Waxman, E., 1997, ApJ, 485, L5 Waxman, E., Kulkarni, S. & Frail, D., 1998, ApJ, 497, 288 Wijers, R.A.M.J., Rees, M.J. & , P., 1997, MNRAS, 288, L51 Wijers, R.A.M.J. & Galama, T., 1998, ApJ, subm. (astro-ph/9805341) Wijers, R.A.M.J., Bloom, J.S., Bagla, J.S., and Natarajan, P., 1998, MNRAS, 294, L13 Woosley, S., Eastman, R. & Schmidt, B., 1998, ApJ, subm (astro-ph/9806299) [^1]: Manuscript prepared for New Astronomy, David N. Schramm memorial volume

--- author: - 'S. Pilling' - 'A. C. F. Santos' - 'H. M. Boechat-Roberty' date: 'Received / Accepted' title: | Photodissociation of organic molecules\ in star-forming regions II: Acetic acid --- Introduction ============ Acetic acid (CH$_3$COOH) has been detected toward various astrophysical regions, including hot molecular cores (HMCs) associated with low- and high-mass star-forming regions (Remijan et al. 2002, 2003, 2004; Cazaux et al. 2003; Mehringer et al. 1997; Wotten et al. 1992 and Wlodarczak & Demaiso 1988) and in comets (Crovisier et al. 1999, 2004). In these environments the radiation field can promote several photophysical and photochemical processes, including the photodissociation. The products of organic molecule dissociation (such as reactive ions and radicals) can form interstellar complex molecules like long carbon chain molecules and amino acids. The simplest amino acid, glycine (NH$_2$CH$_2$COOH), was recently detected in the molecular clouds SgrB2, Orion KL and W51 (Kuan et al. 2003, 2004), although this result was questioned by Snyder et al. (2005). In these objects, precursor molecules like ammonia (NH$_3$), methylamine (CH$_3$NH$_2$), formic acid (HCOOH) and acetic acid have been observed (Nummelin et al. 2000; Turner 1991 and Sutton et al. 1985). Liu et al. (2002) pointed out the importance of performing studies on carboxyl acids since they share common structural elements with biologically important species such as amino acids. Sgr B2, Orion KL and W51 are massive star-forming regions where the presence of widespread UV and X-ray fields could trigger the formation of photodissociation regions (PDRs). X-ray photons are capable of traversing large column densities of gas before being absorbed. X-ray-dominated regions (XDRs) in the interface between the ionized gas and the self-shielded neutral layers could influence the selective heating of the molecular gas. The complexity of these regions possibly allows a combination of different scenarios and excitation mechanisms to coexist(Goicoechea et al. 2004). The formation of interstellar acetic acid occurs both in the gas phase and on grain surfaces. Huntress & Mitchell (1979) proposed a radiative association mechanism followed by dissociative recombination with an electron in the gas-phase: $$CH_3CO^+ + H_2O \longrightarrow CH_3COOH_2^+ \stackrel{e^-}{\longrightarrow} CH_3COOH + H.$$ Ehrenfreund & Charnley (2000) have proposed a warm gas-phase route in which reactions with protonated methanol (CH$_3$OH$_2^+$) and formic acid, evaporated from grain surfaces, can result in CH$_3$COOH formation. Sorrel (2001) has proposed that accretion of gas-phase H, O, OH, H$_2$O, CH$_4$, NH$_3$ and CO onto dust grains sets up a carbon-oxygen-nitrogen chemistry in the grain. As a consequence, a high concentration of free OH, CH$_3$ and NH$_2$ radicals is created in the grain mantle mainly by photolysis. Once these radicals are created, they remain frozen in position until the grains heat up. As this occurs, the radicals become mobile and undergo chemical reactions amongst themselves and with other adsorbed molecules to produce complex organic molecules including CH$_3$COOH. Remijan et al. (2002) pointed out the importance of grain-surface chemistry to form acetic acid, since the location of acetic acid emission is nearly coincident with the emission from other large molecular species formed by grain-surface chemistry in HMCs. Two isomers of acetic acid have also been detected toward HMC, associated with star-forming regions, methyl formate (HCOOCH$_3$) and glycolaldehyde (CH$_2$OHCHO). Hollis et al. (2001) have measured the column density of both isomers in the Sgr B2 (LMH) source near the galactic center and the relative abundances were quite different, about 1:26:0.5 (CH$_3$COOH:HCOOCH$_3$:CH$_2$OHCHO). This suggests a different pathway of formation and/or different stability against the radiation field, which will be discussed in a future publication. Remijan et al. (2004) have proposed that CH$_3$COOH formation seems to favor HMCs with well-mixed N and O, despite the fact that CH$_3$COOH does not contain an N atom. If this is proved to be true, this is an important constraint on CH$_3$COOH formation and, possibly, other structurally similar biomolecules. Despite the differences in molecular structure and chemical formation mechanisms, methyl cyanide, ethyl cyanide (CH$_3$CH$_2$CN) and acetic acid are found to have similar abundances toward the W51 e1 and e2 sources. The photodissociation of acetic acid has been studied experimentally and theoretically in the ultraviolet region (Bernstein et al. 2004; Maçôas et al. 2004, Fang et al. 2002; Naik et al. 2001; Hunnicutt et al. 1989; Blake & Jackson 1969). However, despite some photoabsorption studies in the X-ray range (Robin et al. 1998 and references therein) there are no studies focusing on the photodissociation pathways due to soft X-rays. The present work aims to examine the photoionization and photodissociation of acetic acid by soft X-rays, from 100 eV up to 310 eV, including the energies around the carbon K edge ($\sim$ 290 eV). Experimental setup ================== The experiment was performed at the Brazilian Synchrotron Light Laboratory (LNLS) in Campinas, São Paulo, Brazil. Soft X-rays photons ($\sim10^{12}$ photons/s) from a toroidal grating monochromator (TGM) beamline (100-310 eV), perpendicularly intersect the gas sample inside a high vacuum chamber. The gas needle was kept at ground potential. The emergent photon beam flux was recorded by a light sensitive diode. The sample was bought commercially from Sigma-Aldrich with purity greate than 99.5%. No further purification was performed other than degassing the liquid sample by multiple freeze-pump-thaw cycles before admitting the vapor into the chamber. Conventional time-of-flight mass spectra (TOF-MS) were obtained using the correlation between one Photoelectron and a Photoion Coincidence (PEPICO). The ionized recoil fragments produced by the interaction with the photon beam were accelerated by a two-stage electric field and detected by two micro-channel plate detectors in a chevron configuration, after mass-to-charge (m/q) analysis by a time-of-flight mass spectrometer (297 mm long). They produced up to three stop signals to a time-to-digital converter (TDC) started by the signal from one of the electrons accelerated in the opposite direction and recorded without energy analysis by two micro-channel plate detectors. A schematic diagram of the time of flight spectrometer inside the experimental vacuum chamber is shown in Figure \[fig-diagram\], where A1 and A2 are the pre-amplifiers and D1 and D2 are the discriminators. The connection to the time-to-digital converter is also shown. Besides PEPICO spectra, other two kinds of coincidence mass spectra were obtained simultaneously, PE2PICO spectra (PhotoElectron Photoion Photoion Coincidence) and PE3PICO spectra (PhotoElectron Photoion Photoion Photoion Coincidence), which will be presented in a future publication. These spectra have ions coming from double and triple ionization processes, respectively, that arrive coincidentally with photoelectrons. Of all signals received by the detectors only about 10% come from PE2PICO and 1% from PE3PICO spectra, reflecting that the majority contribution is indeed due to single event coincidence. Nonetheless, PEPICO, PE2PICO and PE3PICO signals were taken into account for normalization purposes. Recoil ion and ejected electron detection efficiencies of 0.23 and 0.04, respectively, were assumed. In addition, we adopted the efficiencies of 0.54 and 0.78 to detect at least one of the photoelectrons from double ionization and triple ionization events, respectively (Cardoso 2001). The first stage of the electric field (708 V/ cm) consists of a plate-grid system crossed at the center by the photon beam . The TOF-MS was designed to have a maximized efficiency for ions with energies up to 30 eV. The secondary electrons produced in the ionization region are focused by an electrostatic lens polarizing the electron grid with 800 V, designed to focus electrons at the center of the micro-channel plate detector. Negative ions may also be produced and detected, but the corresponding cross-sections are negligible. The base pressure in the vacuum chamber was in the $10^{-8}$ Torr range. During the experiment the chamber pressure was maintained below $10^{-5}$ Torr. The pressure at the interaction region (volume defined by the gas beam and the photon beam intersection) was estimated to be $\sim$ 1 Torr (10$^{16}$ mols cm$-3$). The measurements were made at room temperature. Results and discussion ====================== Figure \[fig-ms\] shows the mass spectrum of acetic acid obtained at photon energy of 288.3 eV. The C1s$\rightarrow\pi^{\ast}$ resonance energy from C=O bond is 288.6 eV (Robin et al 1988). We can see the methyl fragment group (mass from 12 to 15 a.m.u), the C$_2$ group (24 to 27 a.m.u), the HCO group (28 to 31 a.m.u) and the CCO group (40 to 44 a.m.u). The carbon dioxide ion (CO$_2^+$) is also produced. At mass 45 a.m.u. we see the carboxyl ion (COOH$^+$). $^{13}$CH$_3$COOH$^+$ (or CH$_3^{13}$COOH$^+$) is also seen. The ions with the largest yields are the ionized acetyl radical (CH$_3$CO$^+$), ionized methyl (CH$_3^+$) and the ionized carboxyl radical (COOH$^+$). The first two may be interpreted as a result of neutral OH and COOH liberation, due to the bond rupture near the carbonyl in the dissociation process. The CH$_3^+$ and COOH$^+$ fragments present similar relative partial yield which may indicate that the molecular dissociation due to photoionization of the C1s inner shell-electron at 288.3 eV does not have a preferential carbon atom target. In this case, both molecular fragments could retain the core hole (charge). However, as we can see in Figure \[fig-piy\] and in Table \[tab-piy\], there was a small enhancement ($\leq$ 20%) in the production of COOH$^+$ with respect to CH$_3^+$ in the photon energy range of 288 to 300 eV. This behavior may be associated with some instabilities of CH$_3^+$ ion leading to subsequence fragmentation and/or charge resonance in the COOH$^+$ ion, increasing its stability. Figure \[fig-piy\] shows the partial ion yields (PIY) for the most significant outcomes (CH$_3$CO$^+$, COOH$^+$, H$^+$ and CH$_3^+$) in the acetic acid dissociation in 100-310 eV photon energy range. The yields of some minor products like HCO$^+$, CH$_3$COOH$^+$, CO$_2^+$ and C$_2$O$^+$ are also shown. A clear bump can be seen in the fractions of CH$_3$CO$^+$, COOH$^+$, CH$_3^+$ and CH$_3$COOH$^+$ near the C1s resonance and decreasing to higher photon energies. The HCO$^+$ and H$^+$ fragments show a distinct behavior and also exhibit a gradual increase toward higher energies which indicates that these ions are preferentially formed after the normal Auger decay. The C$_2$O$^+$ (0.9%) and CO$_2^+$ (1.3%) fragments do not show any clear energy dependence. The C1s resonance and the ionization potential of each carbon are also indicated. The statistical uncertainties are below 10%. In Figure \[fig-piycomp\] we compared the partial ion yield in soft X-rays (292 eV) obtained at LNLS and UV (by 70 eV electrons)[^1] from NIST[^2]. The molecular ion CH$_3$COOH$^+$ is more destroyed by soft X-rays than by UV photons, as expected. The partial ion yields of several fragments are also different in X-ray and in UV fields, for example, the enhancement of COOH$^+$ and CH$_3$CO$^+$ produced by UV radiation. The opposite occurs with HCO$^+$ and all lower mass ions, which seem to be more efficiently produced by X-ray photons. Practically no production of HCOOH$^+$ (formic acid ion) has been identified in the fragmentation by soft X-rays and, while UV photons produce only a small amount of this ion, as we can see in Figure \[fig-piy\] and \[fig-piycomp\]. As was noted before, the CO$_2^+$ and the C$_2$O$^+$ yields remain unchanged in the UV-X-ray photon energy range. The same applies to H$_2$CO$^+$ (formaldehyde) despite the extremely low yield (Figure \[fig-piycomp\]). The largest production of ions due to the photodissociation by soft X-ray photons near the C1s resonance energy is for CH$_3$CO$^+$, followed by COOH$^+$, CH$_3^+$ and H$^+$. The former, as mentioned before, was suggested by Huntress & Mitchell (1979) to be the main precursor of acetic acid together with the water molecule. Therefore, the continuous formation-destruction cycle of acetic acid could act as a catalytic agent converting water molecules in OH + H or (O + 2H) and possibly promoting a reduction in the local H$_2$O abundance. For higher photon energies, as we can see in Figure \[fig-piy\], the proton production is the major outcome. The carboxyl radical, as has been discussed by several authors (Woon 2002, Largo et al. 2004 and Mendonza et al. 2004), plays an important role in the formation of large biomolecules including amino acids. Its ionic counterpart was one of the most ionic products released from the photodissociation of acetic acid by a moderate/strong ionizing photon field. This may suggest that never though acetic acid undergoes considerable photodissociation in star-forming region, it could still produce glycine and other carboxylated biomolecules via barrier-free ion-molecule reactions involving some of its reactant fragments like COOH$^+$. Kinetic energy release (heating) of the ionic fragments ------------------------------------------------------- Several authors have recently focused on the pathway of formation of biomolecules present in the star-forming region and other gaseous-dusty astronomical media (Largo 2004, Woon 2002, and references therein). Despite the success of ab initio theoretical calculations, the endothermic ion-molecule reactions have been neglected and only exothermic reactions have been accepted as a viable mechanism. However, with the knowledge of the kinetic energy (or at least with its value range) of some radical and ionic fragments, some endothermic ion-molecule reactions could be likely and, in extreme situations, or even become more efficient than exothermic reactions. We have determined the kinetic energy of all cationic fragments from the photodissociation of acetic acid. The present time-of flight spectrometer was designed to fulfil the Wiley-McLaren conditions for space focusing (Wiley & McLaren 1955). Within the space focusing conditions, the observed broadening of peaks in spectra is mainly due to kinetic energy release of fragments. Considering that the electric field in the interaction region is uniform, we can determine the released energy in the fragmentation process ($U_0$) from each peak width used by Simon et al. (1991), Hansen et al. (1998) and Santos, Lucas & de Souza (2001) $$\label{eq-U0} U_0 = \Big(\frac{qE \Delta t}{2} \Big)^2 \frac{1}{2m}$$ where $q$ is the ion fragment charge, $E$ the electric field in the interaction region, $m$ is the mass of the fragment, and $\Delta t$ is the time peak width (FWHM) taken from PEPICO spectra. In order to test the above equation we have measured the argon mass spectrum under the same conditions. The calculated values for kinetic energy release ($U_0$) for acetic acid fragmentation are shown in Table \[tab-piy\]. We observe that the highest kinetic energy release was associated with the lightest fragment H$^+$ ($m/q=1$) followed by $H_2^+$ ($m/q=2$), as expected. Differently from formic acid photodissociation results (Boechat-Roberty et al. 2005), extremely fast ionic fragments ($U_0> 10$ eV), usually associated with dissociation of doubly or multiply-charged ions, were not observed at high photon energies. The study of the decay of core-excited molecules provides information about the bonding or antibonding nature of the molecular orbitals. Generally, the final electronic states of a core excited molecule are unknown due to the fact that the densities of the states are very high, and the bond distances and angles differ from their ground state configuration. The surface potentials of the ionic states are extremely repulsive. For core excited molecules that dissociate into one charged and one or more neutral fragments, the dissociation is primarily controlled by chemical (non-Coulomb) forces originating from the residual valence electrons of the system (Nenner & Morin, 1996). From Table 1, one can see that the mean kinetic energy release $U_0$ of some acetic acid fragments increases as the photon energy approaches the C 1s edge (288 eV). This enhancement is due to the repulsive character of the $\sigma$\* ($\pi$\*) resonance. ------- ----------------------------- -- ------------- ------------- ------------- ------------- ------------- ------------- ------------- $m/q$ Attribution 100 eV 200 eV 287.5 eV 288.3 eV 292 eV 300 eV 310 eV 1 $H^+$ 9.16 / 2.9 14.0 / 2.9 10.5 / 2.9 10.6 / 2.9 13.5 / 3.8 16.6 / 3.8 17.6 / 4.9 2 $H_2^+$ - 0.59 / 3.0 0.53 / 2.4 0.41 / 1.5 0.58 / 3.6 0.77 / 1.9 0.63 / 3.0 12 $C^+$ 1.98 / 0.24 2.69 / 0.31 2.10 / 0.31 2.08 / 0.24 2.71 / 0.40 3.71 / 0.32 3.66 / 0.50 13 $CH^+$ 3.48 / 0.22 4.47 / 0.22 3.81 / 0.22 3.54 / 0.29 4.20 / 0.29 5.24 / 0.37 4.87 / 0.29 14 $CH_2^+$; $CO^{++}$ 6.71 / 0.21 7.38 / 0.21 6.60 / 0.21 6.54 / 0.20 6.84 / 0.27 7.71 / 0.20 7.61 / 0.34 15 $CH_3^+$ 13.9 / 0.14 12.6 / 0.19 12.6 / 0.10 12.6 / 0.14 10.9 / 0.09 9.35 / 0.14 9.54 / 0.14 16 $O^+$ 2.19 / 0.13 2.88 / 0.37 2.31 / 0.13 2.29 / 0.23 2.81 / 0.13 3.42 / 0.73 3.28 / 0.63 17 $OH^+$ 2.55 / 0.68 2.81 / 1.14 2.34 / 0.79 2.21 / 1.1 2.91 / 0.35 3.26 / 0.69 2.89 / 1.86 18 $H_2O^+$ 0.61 / 0.27 0.76 / 0.12 0.69 / 0.08 0.56 / 0.05 1.43 / 0.05 1.47 / 0.53 0.60 / 0.11 24 $C_2^+$ - 0.88 / 0.56 0.63 / 0.35 0.58 / 0.48 1.03 / 0.42 1.50 / 0.25 1.17 / 0.56 25 $CHC^{+}$ ? - 1.14 / 0.40 0.89 / 0.40 0.80 / 0.40 1.07 / 0.40 1.50 / 0.95 1.17 / 0.54 26 $CH_2C^{+}$ ? - 0.87 / 0.27 0.68 / 0.23 0.65 / 0.45 0.84 / 0.33 - 0.77 / 0.58 28 $CO^+$ 1.98 / 0.14 2.72 / 0.30 2.26 / 0.26 2.13 / 0.26 2.69 / 0.26 3.96 / 0.10 3.36 / 0.42 29 $COH^+$; $HCO^+$ 6.13 / 0.16 6.95 / 0.25 6.32 / 0.21 6.42 / 0.21 6.66 / 0.29 7.15 / 0.17 7.06 / 0.40 30 $CH_2O^+$; $CH3COOH^{++}$ ? 1.33 / 0.23 - 0.27 / 0.24 0.55 / 0.05 0.22 / 0.45 - 0.21 / 0.16 31 $CH_3O^{+}$ ? - 1.27 / 0.23 1.42 / 0.33 1.41 / 0.27 1.29 / 0.23 1.43 / 0.07 0.92 / 0.27 40 $CCO^+$ 0.37 / 0.01 0.56 / 0.18 0.54 / 0.21 0.52 / 0.22 0.64 / 0.15 1.15 / 0.18 0.71 / 0.18 41 $CHCO^+$ 1.17 / 0.12 1.67 / 0.12 1.41 / 0.14 1.39 / 0.18 1.49 / 0.12 1.78 / 0.05 1.59 / 0.24 42 $CH_2CO^+$ 3.47 / 0.09 3.69 / 0.14 3.70 / 0.14 3.63 / 0.11 3.56 / 0.14 3.21 / 0.07 3.31 / 0.17 43 $CH_3CO^+$; 18.1 / 0.06 12.4 / 0.07 16.4 / 0.07 16.7 / 0.07 13.6 / 0.07 10.1 / 0.05 10.6 / 0.06 44 $CO_2^+$;$CH_3COH^+$ 1.31 / 0.03 1.34 / 0.07 1.28 / 0.11 1.34 / 0.11 1.19 / 0.09 1.67 / 0.14 1.28 / 0.11 45 $COOH^+$ 15.3 / 0.06 12.4 / 0.08 14.1 / 0.08 14.1 / 0.08 13.1 / 0.05 11.0 / 0.11 9.75 / 0.08 46 $HCOOH^+$ - - 0.15 / 0.06 0.16 / 0.03 - - 0.12 / 0.06 60 $CH_3COOH^+$ 8.94 / 0.01 4.73 / 0.01 6.45 / 0.01 6.53 / 0.01 6.03 / 0.01 3.94 / 0.02 3.58 / 0.02 61 $^{13}CH_3COOH^+$ 1.13 / 0.02 1.16 / 0.06 1.71 / 0.03 2.26 / 0.03 0.21 / 0.02 - 2.45 / 0.05 ------- ----------------------------- -- ------------- ------------- ------------- ------------- ------------- ------------- ------------- Photodissociation and formation pathways ---------------------------------------- Acetic acid is one of the simplest carboxylic acids. Its underlying decomposition dynamics (dissociation pathways) have been extensively investigated in the UV range, from both experimental and theoretical points of view. On the basis of the stable products observed in UV photolysis gas-phase of acetic acid, several possible dissociation processes (Fang et al. 2002; Mackie & Doolan 1984; Satio et al. 1990; Blake & Jackson 1969; Ausloss & Steacie 1955 ) were suggested:\ $$\begin{aligned} % \nonumber to remove numbering (before each equation) CH_3COOH + h\nu &\longrightarrow& CH_3CO + OH \\ &\longrightarrow& CH_3 + COOH \\ &\longrightarrow& CH_3CO_2 + H \\ &\longrightarrow& CH_2COOH + H \\ &\longrightarrow& CH_4 + CO_2 \\ &\longrightarrow& CH_2CO + H_2O\end{aligned}$$ This was also confirmed by Hunnicutt et al. (1989) using both room-temperature and jet-cooled conditions. The major pathway for acetic acid UV photodissociation is reaction 3. Moreover, the observed isotropic distribution of OH fragments from acetic acid dissociation and the parent molecule fluorescence is indicative of a moderately slow dissociation. Reactions 4 and 5 could also lead to CH$_3$ + CO + OH products (Hunnicutt et al. 1989). The behavior of acetic acid in the ice phase under the influence of UV photons was recently studied by Maçôas et al. (2004) in an experimental/theoretical approach. The UV photolysis of the Ar matrix-isolated acetic acid reveals very different products from the gas phase. As a result, 37% of the acetic acid molecules yield methanol (CH$_3$OH) plus carbon monoxide complexes, 17% yield carbon monoxide complexed with formaldehyde and molecular hydrogen, 20% yield quaternary complexes of two carbon monoxides molecules and two hydrogen molecules and 21% dissociate into carbon dioxide and methane (CH$_4$). Ketene (CH$_2$CO), which is the main product of thermal decomposition, was detected only in small amounts ($\leq$ 5%). The CO/CO$_2$ product ratio is $\sim$ 5. Bernstein et al. (2004) also studied acetic acid in the ice-phase, using both Ar and water matrices and determined the half-life in a UV radiation field and revealed a relatively low survival of acetic acid (large yield of photoproducts) when compared to other organic molecules, like aminoacetonitrile (H$_2$NCH$_2$CN). These experiments demonstrate that organic nitriles (cyanide compounds) survive 5 to 10 times longer exposure to UV photolysis than do the corresponding acids. The present work shows significative differences between the photoproducts of acetic acid due to low ionizing radiation (UV) and soft X-ray photons. The inner shell photoionization process may produce instabilities in the molecular structure (nuclear rearrangements) leading to dissociation. From Table \[tab-piy\] we determine the main photodissociation pathways from single ionizations in 100-300 eV photon energy range. These photodissociation pathways are shown in Table \[tab-path1\]. Only events with greater than 1% yields were considered here. The main dissociation leads to production of CH$_3$CO$^+$ due to the bond rupture of OH near the carbonyl. Herbst & Leung (1986) have presented several pathway syntheses of complex molecules in dense interstellar clouds via gas-phase chemistry models. The authors presented a significant amount of normal ion-molecule reactions including the ions C$^+$, C$_2^+$, CH$^+$, OH$^+$, CO$^+$, CH$_2^+$, H$_2$O$^+$, HCO$^+$, C$_2$H$^+$, CO$_2^+$, CH$_3^+$. In another set of reactions they have shown several radiative associations including the ions C$^+$, CH$_3^+$, HCO$^+$ leading to the production of high molecular complexity species. For example, the possible route to interstellar acetaldehyde, involving the HCO$^+$ ion and methane by $$HCO^+ + CH_4 \longrightarrow C_2H_5O^+ + h\nu \stackrel{e^-}{\longrightarrow} CH_3CHO + H.$$ Combes et al. (1987) have suggested that the gas phase acetone ((CH$_3$)$_2$CO) present in the Sgr B2 region could be formed by the radiative association reaction between the methyl ion and interstellar acetaldehyde, followed by dissociative recombination with electrons by $$CH_3CHO + CH_3^+ \rightarrow CH_3COCH_4^+ + h\nu \stackrel{e^-}{\rightarrow} (CH_3)_2CO + H.$$ Other radiative association reactions with the methyl ion could produce the simplest interstellar ether. In this situation, as proposed by Herbst & Leung (1986), the CH$_3^+$ reacts with methanol, and after dissociative recombination leads to dimethyl ether by $$CH_3^+ + CH_3OH \rightarrow CH_3OCH_4^+ + h\nu \stackrel{e^-}{\rightarrow} CH_3OCH_3 + H$$ These works point out the importance of the ionic species in the increase of interstellar molecular complexity. In star-forming regions many ions could be produced by the photodissociation of large organic molecules. Therefore, the knowledge of the photodissociation processes and its ion yields plays an essential role in interstellar chemistry. The absence of more doubly ionized fragments in the PEPICO spectra indicates that the doubly ionized acetic acid dissociates preferentially via charge separation. The dynamics of the doubly and triple ionized acetic acid molecule will be studied in a future publication. From Table \[tab-path1\] we can see that two of the main photodissociation pathways of acetic acid occurs with the rupture of the C-C bond, releasing CH$_3$$^+$ and COOH$^+$ ions, however the yield of COOH$^+$ (12.8%) undergoes little enhancement with respect to CH$_3^+$ (11.6%). This small excess could be associated with the high stabilization of COOH$^+$ due to charge resonance (charge migration) (Silverstein & Webster 1998) or a possible dissociation of CH$_3^+$ into minor fragments. Other possibility is that the methyl carbon IP occurs at 291.6, 5 eV less than for the carboxyl carbon (see Robin et al. 1988). As a consequence, the number of resonances to excited orbitals involving the carbon atom in the COOH site is higher than for the other carbon site at photon energies above the IP of the methyl carbon. This scenario could lead to a small preference for the carboxyl carbon during photoelectron excitation/ionization at these energies (and energies somewhat higher) which, after dissociation, retain the charge. $CH_3COOH + h\nu$ $\longrightarrow$ $CH_3COOH^+ + e^-$ ------------------- -------------------------------------- -------------------------------------------- $CH_3COOH^+$ $\stackrel{14.1\%}{\longrightarrow}$ $CH_3CO^+ + OH$ or ($O + H$) $\stackrel{13.1\%}{\longrightarrow}$ $H^+ + $ neutrals $\stackrel{12.8\%}{\longrightarrow}$ $COOH^+ + CH_3$ or ($CH_2 + H$) $\stackrel{11.6\%}{\longrightarrow}$ $CH_3^+ + COOH$ or ($CO_2 + H$; $CO + OH$) $\stackrel{7.1\%}{\longrightarrow}$ $CH_2^+ + HCOOH$ or ($H + COOH$) $\stackrel{6.7\%}{\longrightarrow}$ $COH^+ + CH_3 + O $ $\stackrel{4.2\%}{\longrightarrow}$ $CH^+ + $ neutrals $\stackrel{3.5\%}{\longrightarrow}$ $CH_2CO^+ + H_2O$ or ($OH + H$) $\stackrel{2.7\%}{\longrightarrow}$ $CO^+ + CH_3 + OH$ $\stackrel{2.7\%}{\longrightarrow}$ $O^+ +$ neutrals $\stackrel{2.7\%}{\longrightarrow}$ $OH^+ + CH_3CO$ or ($CH_3 + CO$) $\stackrel{1.5\%}{\longrightarrow}$ $CHCO^+ + $ neutrals $\stackrel{1.4\%}{\longrightarrow}$ $CO_2^+ + H + CH_3 $ or ($CH_2 + H$) : **Main photodissociation pathways from single ionization**[]{data-label="tab-path1"} Absolute photoionization and photodissociation cross sections ------------------------------------------------------------- The absolute cross section values for both photoionization and photodissociation processes of organic molecules are extremely important as the input for molecular abundances models (Sorrell 2001). In these theoretical model biomolecules are formed inside the bulk of icy grain mantles photoprocessed by starlight (ultraviolet and soft X-rays photons). Its main chemistry route was based on radical-radical reactions followed by chemical explosion of the processed mantle that ejects organic dust into the ambient gaseous medium. For example, the number density of a given biomolecule in a steady state regime of creation and destruction inside a gaseous-dusty cloud is given by $$N_{Mol} = \frac{\dot{N}_{Mol} n_d}{<\sigma_{ph-d}> I_{0}}$$ were $\dot{N}_{Mol}$ is the molecule ejection rate which depends mainly on the molecule mass, the properties of the grains and the cloud (see eq. 21 of Sorrell 2001). $n_d$ is the dust space density, $I_{0}$ is the flux of ionizing photons (photons cm$^{-2}$ s$^{-1}$) inside the cloud and $<\sigma_{ph-d}>$ is the average photodissociation cross section in the wavelength range of the photon flux density. As mentioned by Sorrell (2001), the main uncertainty in this equilibrium abundance model comes from $\sigma_{ph-d}$. Therefore the precise determination of $\sigma_{ph-d}$ of biomolecules is very important to estimate the molecular abundance of those molecules in the interstellar medium. Moreover, knowing the photon dose $\phi$ and $\sigma_{ph-d}$ values its is possible to determine the half-life of a given molecule, as discussed by Bernstein et al. (2004). The photodissociation rates, $R$, of a molecule dissociated by the interstellar radiation field $I_\varepsilon$ in the energy range $\varepsilon_2 - \varepsilon_1$ is given by $$\label{eq-R} R = \int_{\varepsilon_1 }^{\varepsilon_2} \sigma_{ph-d}(\varepsilon) I_0(\varepsilon) d\varepsilon$$ where $\sigma_{ph-d}(\varepsilon)$ is the photodissociation cross section as a function of photon energy (cm$^2$), $I_0(\varepsilon)$ is the photon flux as a function of energy (photons cm$^{-2} eV^{-1} s^{-1}$) (see discussion in Cottin et al. 2003 and Lee 1984). The half-life may be obtained from Eq. \[eq-R\] by writing $t_{1/2}=ln 2 / R$, which does not depend on the molecular number density. In order to put our data on an absolute scale, after subtraction of the linear background and false coincidences coming from aborted double and triple ionization (see Simon et al 1991), we have summed the contributions of all cationic fragments detected and normalized them to the photoabsorption cross sections measured by Robin et al (1988). Assuming a negligible fluorescence yield (due to the low carbon atomic number (Chen et al 1981)) and anionic fragments production in the present photon energy range, we adopted that all absorbed photons lead to cationic ionizing process. Therefore the non-dissociative single ionization (photoionization) cross section $\sigma_{ph-i}$ and the dissociative single ionization (photodissociation) cross section $\sigma_{ph-d}$ of acetic acid can be determined by $$\sigma_{ph-i} = \sigma^{+} \frac{PIY_{CH_3COOH^+}}{100}$$ and $$\sigma_{ph-d} = \sigma^{+} \Big( 1 - \frac{PIY_{CH_3COOH^+}}{100} \Big)$$ where $\sigma^{+}$ is the cross section for single ionized fragments (see description in Boechat-Roberty et al. 2005). Both cross sections can be seen in Figure \[fig-sigma\] as a function of photon energy. The absolute absorption cross section of acetic acid (Robin et al. 1988) is also shown for comparison. Those values are also shown in Table \[tab-sigma\]. ------------- -- ------------------------ ------------------------ ------------------------ Photon energy (eV) $\sigma_{ph-d}$ $\sigma_{ph-i}$ $\sigma_{ph-abs}$ 285 3.61 $\times 10^{-19}$ 2.49 $\times 10^{-20}$ 4.12 $\times 10^{-19}$ 288.3 2.23 $\times 10^{-18}$ 1.56 $\times 10^{-19}$ 2.58 $\times 10^{-18}$ 292 7.40 $\times 10^{-19}$ 4.75 $\times 10^{-20}$ 8.52 $\times 10^{-19}$ 300 1.19 $\times 10^{-18}$ 4.89 $\times 10^{-20}$ 1.37 $\times 10^{-18}$ 310 7.32 $\times 10^{-18}$ 2.72 $\times 10^{-20}$ 9.31 $\times 10^{-19}$ ------------- -- ------------------------ ------------------------ ------------------------ : Values of non-dissociative single ionization (photoionization) cross section ($\sigma_{ph-i}$) and dissociative ionization (photodissociation) cross section ($\sigma_{ph-d}$) of acetic acid as a function of photon energy. The estimated total error is 30%. The photoabsorption cross section ($\sigma_{ph-abs}$) from Robin et al. (1988) is also shown.[]{data-label="tab-sigma"} Summary and conclusions ======================= The goal of this work was to experimentally study ionization and photodissociation processes of a glycine precursor molecule, CH$_3$COOH (acetic acid). The measurements were taken at the Brazilian Synchrotron Light Laboratory (LNLS), employing soft X-ray photons from a toroidal grating monochromator (TGM) beamline (100 - 310 eV). The experimental set-up consists of a high vacuum chamber with a time-of-flight mass spectrometer (TOF-MS). Mass spectra were obtained using coincidence techniques. We have shown that X-ray photon interactions with acetic acid release a considerable number of energetic fragments, some of them with high kinetic energy (ex. H$^+$, H$_2^+$ and OH$^+$). Unlike the previous work with formic acid performed in the same spectral range (Boechat-Roberty et al. 2005), no very high kinetic energy fragments have been observed due the single photoionization of acetic acid. Several ionic fragments released from acetic acid photodissociation have considerable kinetic energy. An extension of this scenario to interstellar medium conditions suggests the possibility endothermic ion-molecule (or radical-molecule) reactions and this becomes important in elucidating the pathways of formation of complex molecules (Largo et al. 2004). Dissociative and non-dissociative photoionization cross sections were also determined. We found that about 4-6% of CH$_3$COOH survive the soft X-ray ionization field. CH$_3$CO$^+$ and COOH$^+$ were the main fragments produced by high energy photons. The former may indicate that the production-destruction cycle of acetic acid in hot molecular cores could decrease the H$_2$O abundance, since the net result of this process is the conversion of water into OH + H. The latter ion plays an important role in ion-molecule reactions to form large biomolecules like glycine. The authors would like to thank the staff of the Brazilian Synchrotron Facility (LNLS) for their valuable help during the experiments. We are particularly grateful to Dr. R. L. Cavasso and Professor A. N. de Brito for the use of the Time-of-Flight Mass Spectrometer. This work was supported by LNLS, CNPq and FAPERJ. Ausloos P. & Steacie E. W. R. 1955, Can. J. Chem., 33, 1530. Bernstein M. P., Ashbourn S. F. M., Sandford S. A. & Allamandola L. J. 2004, ApJ, 601, 365. Blake P. G., Jackson G. E., 1969, J. Chem Soc. B, 94. Boechat-Roberty H. M., Pilling S. & Santos A. C. F. 2005, A&A 438, 915. Cazaux, S., Tielens, A. G. G. M., Ceccarelli, C., Castets, A., Wakelam, V., Caux, E., Parise, B. & Teyssier, D. 2003, ApJ, 593, L51. Cardoso E. S. 2001, MSc. thesis, IF/UNICAMP, São Paulo, Brazil. Chen M. H., Crasemann B. & Mark, H., 1981, Phys. Rev. A, 24, 177. Combes F., Gerin M., Wooten A. S., et al., 1987, A&A, 180, L13. Cottin H., Moore M., H. & Bénilan, Y. 2003, ApJ, 590, 874. Crovisier J., Bockelée-Morvan D. 1999 Space Scien. Rev., 90, 19. Crovisier J., Bockelée-Morvan D., Colom P., Biver N., Despois D. & Lis D. C. 2004, A&A, 418, 1141. Ehrenfreund P. & Charnley S. B. 2000, ARA&A, 38, 427. Fang W.-H., Liu R.-Z., Zheng X. & Phillips D. L. 2002, J. Org. Chem., 67, 8470. Goicoechea J. R., Rodriguez-Fernandez N. J. & Cernicharo J. 2004 ApJ, 600, 214. Hansen D. L. et al, 1998, Phys. Rev. A, 58, 5. Herbst E. & Leung C. M., 1986, MNRAS, 222, 689. Hollis J. M., Vogel S. N., Snyder L. E., Jewell P. R. & Lovas F. J. 2001, ApJ, 554, L81. Huntress W.& Mitchell G. 1979, ApJ, 231, 456. Hunnicutt S. S., Waits L. D. & Guest J. A., 1989, J. Phys. Chem., 93, 5188. Kuan Y.-J., Charnley S. B., Huang H.-C., Tseng W.-L., Kisiel Z. 2003, ApJ, 593, Issue 2, 848. Kuan, Y.J., Charnley, S. B., Huang, H.C., Kisiel, Z., Ehrenfreund, P., Tseng, W.L. & Yan, C.H., 2004, Adv. Space Res., 33, 31. Lago A. F., Santos A. C. F. & de Souza G. G. B. 2004, J. of Chem. Phys., 120, 9547. Largo A., Redondo P. & Barrientos C. 2004, Int. J. Quant. Chem., 98, 355. Lee L. C. 1984, ApJ, 282, 172. Liu S. Y., Girard J. M., Remijan A. & Snyder L. E. 2002, ApJ, 576, 255. Mackie J. C. & Doolan K. R. 1984, Int. J. Chem. Kinet., 16, 525. Maçôas A. M. S., Khriachtchev L., Fausto R. & Räsänen M. 2004, J. Phys. Chem., 108, 3380. Mendoza, C., Ruette, F., Martorell, G., & Rodrígez, L. S. 2004, ApJ, 601, L59. Mehringer D. M., Snyder L. E. Miao Y. & Lovas F. J. 1997, ApJ, 480, L71. Naik P. D., Upadhya P. H., Kumar A. Sapre A. V. & Mittal J. P. 2001, Chem. Phys. Letters, 340, 116. Nenner, I., & Morin, P., 1996, VUV and Soft X-Ray Photoinization, Ed. Uwe Becker and D. A. Shirley, Plenum Press, New York. Nummelin A., Bergman P., Hjalmarson Å., Friberg P., Irvine W. M., Millar T. J., Ohishi M. & Saito S. 2000, ApJS, 128, 213. Remijan A. J., Snyder L. E., Liu S.-Y. Mehringer D. & Kuan Y. -J. 2002, ApJ, 576, 264. Remijan A. J., Snyder L. E., Friedel D. N., Liu S.-Y. & Shah R. Y. 2003, ApJ, 590, 314. Remijan A. J., Shiao Y. -S., Friedel D. N., Meier D. S. & Snyder L. E. 2004, ApJ, 617, 384. Robin, M. B., Ishii I., McLaren R. & Hitchcock A. P. 1988, J. Electron Spectrosc. Relat. Phenom., 47, 53. Santos A. C. F., Lucas C. A. & de Souza G. G. B. 2001, J. Electron Spectrosc. Relat. Phenom., 114, 115. Satio K., Sasaki T., Yoshinobu I. & Imamura A. 1990, Chem. Phys. Lett., 170, 385. Silverstein R. M. & Webster F. X., 1998, “Spectrometric Identification of Organic Compounds”, 6th edition., John Wiley & sons, Inc. Simon M., LeBrun T., Morin P., Lavolée M. & Maréchal J. L. 1991, Nucl. Instrum. Methods B, 62, 167. Snyder L. E., Lovas F. J., Hollis J. M. et al. 2005, ApJ, 619, 914. Sorrell, W. H. 2001, ApJ, 555, L129. Sutton E. C., Blake G. A., Masson C. R. & Phillips T. G. 1985, ApJS, 58, 341 Turner B. E. 1991, ApJS, 76, 617. Wiley W. E., & McLaren I. W. 1955 Rev. Sci. Instrum. 26, 1150. Wlodarczak G. & Demaiso J. 1988, A&A, 192, 313. Woon D. E. 2002, ApJ, 571, L177. Wotten A., Wlodarczak G., Mangnum J. G., Combes F., Encrenaz P. J. & Gerin M. 1992, A&A, 257, 740. [^1]: The effect of 70 eV electrons is very similar to 21.21 eV (He I Lamp) photons in both cases the main ionization occurs in valence shells (see discussion in Lago et al. 2004) [^2]: National Institute of Standards and Technology http://webbook.nist.gov/chemistry/

--- abstract: 'Multi-fidelity Gaussian process is a common approach to address the extensive computationally demanding algorithms such as optimization, calibration and uncertainty quantification. Adaptive sampling for multi-fidelity Gaussian process is a changing task due to the fact that not only we seek to estimate the next sampling location of the design variable, but also the level of the simulator fidelity. This issue is often addressed by including the cost of the simulator as an another factor in the searching criterion in conjunction with the uncertainty reduction metric. In this work, we extent the traditional design of experiment framework for the multi-fidelity Gaussian process by partitioning the prediction uncertainty based on the fidelity level and the associated cost of execution. In addition, we utilize the concept of Believer which quantifies the effect of adding an exploratory design point on the Gaussian process uncertainty prediction. We demonstrated our framework using academic examples as well as a industrial application of steady-state thermodynamic operation point of a fluidized bed process' author: - | Sayan Ghosh[^1], Jesper Kristensen, Yiming Zhang, Waad Subber, Liping Wang\ Probabilistics and Optimization Group\ General Electric Research\ Niskayuna, New York 12309\ \ bibliography: - 'asme2e.bib' title: 'A STRATEGY FOR ADAPTIVE SAMPLING OF MULTI-FIDELITY GAUSSIAN PROCESS TO REDUCE PREDICTIVE UNCERTAINTY' --- INTRODUCTION {#introduction .unnumbered} ============ Effective design and optimization of mechanical systems usually require extensive simulation runs and costly physical experiments. As a cost effective alternative, surrogate models have been introduced to approximate the response of the mechanical systems [@shan2010survey; @queipo2005surrogate; @viana2014special]. Numerical simulations and/or physical experiments are first performed at a given design variables and collected as samples. Surrogate models are then developed based on the scattered samples and serve as the basis for further optimization and uncertainty quantification. With development in the past decades, surrogate models have proved to be a major scheme for effective design optimization and uncertainty quantification of various mechanical systems including but not limited to composite laminates [@zhang2016approaches], thermodynamic modeling [@honarmandi2019bayesian], chemo-thermal modeling of composites [@subber2019], computational fluid dynamics [@bui2008model], high-performance computing [@zhang2017multi], structural prognosis [@an2015practical], crashworthiness-based lightweight design [@zhang2013crashworthiness], time-dependent reliability design optimization [@hu2016single], flapping wing design [@chaudhuri2015experimental]. The accuracy of the surrogate model relies on the sampling scheme which systematically determines the location and number of samples. The sampling scheme could be performed all-at-once or adaptively [@haftka2016parallel]. A challenge for sampling scheme is the mixture of dataset with varying fidelity. Fidelity refers to the degree to which the simulations reproduces the response of physical tests. Models with different fidelity could be finite element simulations with changing mesh density, computer simulations with simplified mathematical governing equations [@zhang2017multi] or simulation versus experiments [@zhang2016approaches]. The cost for sampling (i.e. data acquisition) increases with model fidelity. Allocating samples between multi-fidelity models under a fixed budget is expected to enhance the prediction accuracy by making most use of existing information. Multi-fidelity surrogate models are based on the idea that the high-fidelity experiments can be approximated as a tuned or corrected functions of low-fidelity models [@Toropov2001; @keane_book; @haftka1991; @Forrester2007; @ghosh2018bayesian; @ghosh2015multi]. A commonly and well known approach to fuse multi-fidelity dataset is by adding a correction term (discrepancy function) on the low-fidelity (LF) dataset towards the high-fidelity (HF) dataset [@KennedyOHagan; @zhang2018multifidelity]. The form of discrepancy function assumes the difference between LF and HF models is easier to approximate than the HF model. More details for a comprehensive discussion on MFS could be found in [@park2017remarks; @fernandez2016review]. The emerging schemes for MFS sampling could be also produced all-at-once or adaptively as the scheme for single-fidelity. All-at-once sampling usually generates first the low-fidelity samples and determines the HF samples as a subset of LF samples [@Gratiet2013]. Reference [@haaland2010approach] proposed a nested design scheme for categorical and mixed factors. Reference [@zheng2015difference] performs a comparison study between nested and non-nested design scheme for the effect on MFS accuracy. In the context of adaptive sampling, a few initial samples are first produced based on one-time sampling. Then MFS is developed with uncertainty estimation. New samples are recommended from the MFS based on certain infilling criterion to balance between accuracy and cost. Therefore, sampling proceeds alternately between multi-fidelity model. Reference [@huang2006sequential] proposes adaptive optimization using multi-fidelity kriging. The expected improvement was modified with multiplicative terms to account for fidelity, cost ratio and noise effect. Reference [@peherstorfer2016multifidelity] adopts LF model to fit the bias distribution for importance sampling. HF model are used to infer the unbiased estimation. Reference [@chaudhuri2017multifidelity] deals with coupled multidisciplinary system using LF model to approximate the coupling variables and HF model to refine surrogate. We found that the adaptive sampling schemes for the multi-fidelity GP surrogate are two step process, where in the first step location or design point on input parameters are determined by finding the location of maximum predictive uncertainty. Then in the second step, decision is made whether to run high-fidelity or low-fidelity analysis based on the cost ratio of these analyses [@pellegrini2016multi]. In this work a new criteria is proposed where the selection of next design point as well as fidelity is done in a single step. In multi-fidelity GP, the uncertainty on response is coming from both low-fidelity model as well as discrepancy model. Adding a low-fidelity data reduces uncertainty on low fidelity model and adding a high fidelity model improves the uncertainty on discrepancy model. Therefore hypothesis is that, rather than selecting the next analysis point at maximum overall predictive uncertainty (low-fidelity uncertainty + discrepancy uncertainty), selecting a design point and fidelity where reduction in uncertainty per unit cost is maximum will yield a efficient solution. The proposed criteria is also expanded with GP “believer” [@ginsbourger2010kriging]. The GP believer strategy is based on quantifying the effect of adding an exploratory design point on the multi-fidelity GP uncertainty prediction. For the numerical demonstrations, we consider two illustrative example with different complexity and input dimensions. To illustrate the usefulness of our proposed adaptive schemes, we consider a real industrial application of a steady-state thermodynamic modeling of a fluidized bed process. Gaussian Process Surrogate models {#GPSM .unnumbered} ================================= For many industrial applications, optimization for some operational conditions, model calibration and uncertainty quantification may require many calls to computationally expansive simulation codes. The computational burden can be overcome by utilizing surrogate models. The surrogate models require a limited carefully designed simulations through the Design of Experiments (DoE) techniques. Gaussian Process (GP) surrogate model is a common approach for metamodeling of a wide range of industrial problems [@KennedyOHagan; @rasmussen2006; @Arendt2012]. The estimation of the prediction uncertainty in GP can be considered as its major desirable property [@KennedyOHagan; @rasmussen2006; @Arendt2012]. This section outlines the framework surrounding single and multi-fidelity GP. The uncertainty associated with building the GP is discussed in detail. Including properties of this uncertainty is useful in performing adaptive uncertainty sampling. In adaptive uncertainty sampling technique a new sampling point is added to the training data set where the surrogate model uncertainty is largest. Single-Fidelity Gaussian Process {#single-fidelity-gaussian-process .unnumbered} -------------------------------- Consider a GP surrogate model of the form: $$y(x) \sim GP(m(x),k(x,x' )),$$ where $m(x)$, assumed to be zero here, is the mean function of the input vector $x$, and $k(x,x’)$ is the covariance function. In this work, the covariance function is assumed to be the squared exponential kernel [@rasmussen2006]: $$k(x,x')=\sigma^2 \exp\left(-\beta (x-x' )^2 \right) + I \lambda^2, \label{eq:gp_kernel}$$ where $\beta$ are the (inverse) length scale parameters collected in a vector, one per input dimension, $\sigma^2$ captures the data variance as the amount of data variance captured by the model and $\lambda^2$ quantifies the amount of variance captured by the residuals. This GP models an output $y(x)$ given an input vector $x$. Observe now a set of inputs and outputs and collect these in a training data set of $N$ elements $D=\left\{x_i,y_i \right\}_{i=1}^N$. One property of the GP is that, on any finite set of samples, such as the training data set collected, it reduces to a multivariate Gaussian distribution. Thus, specifically, the GP fitting process translates to fitting the hyperparameters associated with the matrix: $$K_{(i,j)} = k(x_i,x_j), \label{eq:gp_cov}$$ where $x_i$ is the $i^{th}$ training datum. The hyperparameters of the GP are defined as the vector $\theta =(\sigma,\beta,\lambda)$ and need to be fitted to the training data set $D$. In this work, priors are placed on the hyperparameters to incorporate the initial belief into the data modeling before seeing the data itself, such as smoothness. By combining the priors with the likelihood function, the problem of fitting $\theta$ boils down to sampling from the extrema of the posterior distribution $p(\theta|D)$ since, of course, larger values of $p(\theta|D)$ implies more likely models $\theta$. The Markov Chain Monte Carlo (MCMC) method both seeks out the extrema and provides a way to sample from it, even in cases where the normalization constant of the posterior probability distribution is unknown. Incidentally, note that while the GP does have hyperparameters, it is still considered a non-parametric surrogate model since it does not assume any functional form of the data being modeled, such as, e.g., assuming a polynomial. The MCMC method produces an array, or a chain, of samples of the hyperparameters. In short, typically, the first $20-50\%$ of the samples are discarded in order to “lose the memory” of the starting point. After the burn-in, each sample is considered a valid hyperparameter sample from the posterior distribution $p(\theta|D)$. To simplify the approach ahead, the chain is now condensed into a “lean form” where the median of the chain over each hyperparameter is used to represent the best hyperparameter. Limitations arise in this method if the distribution over the hyperparameter is multi-modal or cannot well be captured by the median. In any case, we can now think of having a single sample from $p(\theta|D)$ and thus a single GP which in an absolute-deviation sense best fits the data. Multi-Fidelity Gaussian Process {#multi-fidelity-gaussian-process .unnumbered} ------------------------------- Constructing the GP requires limited runs of the computational model at a designed input set. Nevertheless, computational budget allocation might be limited to only a handful of the afforded expansive runs of a high-fidelity simulation code. On the other hand, access to simplified models (low-fidelity) may provide a useful information that at least can capture the general trend of the high-fidelity model. The multi-fidelity GP surrogate can be trained to bridge the information from various levels of the models complexity [@AlexanderBook]. Specifically, consider the case where we are aiming at modeling two distinct data sets each of a different level of fidelity. We may have a low-fidelity computer simulation that models a given phenomenon, say, the performance of an engine, and the ability to run the real-world experiment. It might be also the case instead of the field experiment, a high-fidelity model is utilized to simulate the engine performance. We follow closely Kennedy O’Hagan’s (KOH) methodology [@KennedyOHagan] where the observed data ( i.e., the high-fidelity data), $y(x)$, is represented as a linear combination of a low-fidelity and model a discrepancy term [@KennedyOHagan]: $$y(x)=\eta(x,\theta)+\delta(x)+ \epsilon, \label{eq:koh}$$ where $\theta$ are calibration parameters, i.e., parameters of the low-fidelity model that may or may not have a physical meaning, that can be tuned in order to better match the observed data $y(x)$. Note that Eq. \[eq:koh\] contains of two separate GPs . Namely $\eta(x,\theta)$ is a GP for the simulator data and $\delta(x)$ is a GP for capturing the discrepancy between the simulator and the observed data which is collected at the [color[red]{}independent variable $x$]{} locations. In this work, the task of calibrating $\theta$ is not considered so these parameters are not included in the model, but future work could include this as well. The noise term $\epsilon$ in Eq. \[eq:koh\] is assumed as independent and identically distributed zero-mean with constant finite variance Gaussian random variable. Importantly, each GP in Eq. \[eq:koh\] is fitted to its own data set. For example, the low-fidelity model $\eta(x,\theta)$ is fitted to a data set $D_{\eta}=\left\{z_i,w_i \right\}_{i=1}^{N_\eta}$ where $z$ is the independent variable and $w$ is the dependent variable (output from the computer simulator). The discrepancy $\delta(x)$ GP is fitted by using information from both the low and the high-fidelity data $D_y=\left\{x_i,y_i \right\}_{i=1}^{N_y}$. The KOH method is a two-part solution: build a base model of the simulation data and a discrepancy model that maps the simulation model to experimental data. Generally, $z$ and $x$ are not located at the same points, and most typically will not have the same size, i.e., the simulator (low-fidelity model) is run at different input points than the observed (high-fidelity data), but nothing prevents them from being the same. The covariance matrix on a finite sample of points from both the simulator and the real-world experiment is now given by the following overall structure (the subscript $mf$ refers to multi-fidelity): $$K_{mf} = \left[ \begin{matrix} K_y & 0 & 0 \\ 0 & K_u & K_{uw} \\ 0 & K_{uw}^T & K_w \end{matrix} \right],$$ where each covariance matrix has been labeled with a subscript identifying which dependent variable data is being modelled. $K_y$ is the covariance matrix of the high-fidelity data, $K_w$ is the covariance of the low-fidelity data and the newly introduced variable $u$ refers to the low-fidelity data predicted on the high-fidelity points and thus $K_{uw}$ is the covariance matrix between the low-fidelity data and the low-fidelity model predicting high-fidelity data. This covariance matrix contains multiple hyperparameters via the covariance matrices which in turn are defined from the covariance function in Eq. \[eq:gp\_kernel\]. The parameters are fitted with MCMC in the same way as previously discussed. Gaussian Process Uncertainty {#gaussian-process-uncertainty .unnumbered} ----------------------------- Next, we discuss the uncertainty estimation associated with the GP predictions. The prediction uncertainty is a crucial aspect of GP and sets it apart from many other surrogate modeling techniques. The following observations regarding the GP prediction uncertainty (variance) will be important for the developments to follow: 1. The prediction uncertainty increases to the variance of the training data far away from the training data. Thus, there is a lower bound of 0 and an upper bound given by the variance of the data. 2. The prediction uncertainty is small near training data points. On a training datum the uncertainty is as small as it gets anywhere else. 3. The prediction uncertainty can be non-zero at a training datum in our formulation, but GPs can be used as interpolators if modifying the kernel in Eq. \[eq:gp\_kernel\]. Incidentally, consider another surrogate modeling techniques such as the Neural Network (NN) [@bishop2006pattern]. One way to obtain the uncertainty in NNs is by randomly selecting multiple subsets of the data fitting an NN on each and combining all the NNs into a “mean NN” and its associated uncertainty. In this case, there is no guarantee that the properties above hold true, but as we shall see, property 2 is especially helpful. To predict the mean and uncertainty of the GP at any point, generally referred to as an unseen point (but is allowed to be in the training set), $x^{*}$, the first step is to compute the covariance vector between the new point and the existing training data $k(x^{*},x)=k^{*}$. Then, the predicted mean $m^{*}$ value and uncertainty $V^{*}$ associated with the GP are then given by: $$\begin{aligned} m^*&={k^*}^T K^{-1} y, \\ V^*(x^*)&=k(x^*,x^* )-{k^*}^T K^{-1} k^*, \label{eq:gp_var}\end{aligned}$$ where $K$ is given in Eq. \[eq:gp\_cov\] and $k(x^*,x^* )$ is the covariance of the unseen point. Sampling Strategies {#sampling-strategies .unnumbered} =================== Constructing a multi-fidelity GP surrogate requires a careful experimental design of the sampling strategy. The design of the experiment (DoE) should provide an optimal selection of the design variables $x \in D$ such that it emulates the behavior of the expansive response of the high-fidelity model while minimizing a computational cost or performance metrics. Note that the selection criterion is defined by an objective function that can be maximized or minimized depending on the goal of the experiment. For multi-fidelity GP, the design space $D=\{D_\eta \cup D_y\}$ consists of low and high fidelity input variables, and thus an adaptive sampling strategy is more appropriate to achieve the aim of the experiment. In adaptive sampling, the design space is augmented sequentially by adding a new set of the variables that optimally satisfy the objective criterion. In addition for the multi-fidelity framework, the next design set not only consists of the best location to perform the computer experiment, but also the level of fidelity (i.e, execution of low or high fidelity model). The decomposition property of the prediction variance into a contribution from the low fidelity model and the discrepancy term provides a strategy to determine the next level of model fidelity to simulate at the next optimal design point [@Gratiet2013]. In the next sections, we discuss the adaptive sampling strategy for the single-fidelity GP, and we propose a new adaptive sampling techniques for the multi-fidelity GP. Adaptive sampling for the single-fidelity GP {#adaptive-sampling-for-the-single-fidelity-gp .unnumbered} -------------------------------------------- In adaptive sampling for the single-fidelity GP, typically, we start with a space-filling DoE with a number of points depending on the size of the problem [@AlexanderBook]. Then, iteratively, one or multiple points are added to the initial design and the GP is re-fitted in each iteration. For example, say we start with 10 points in the design and add points until our training data has 50 points. The question is which points should be chosen in each iteration? If the task is to learn the posterior distribution of the hyperparameters as well as possible, or in other words, learn the response surface to some threshold degree of accuracy throughout the design space, then a method called uncertainty sampling is an approach to take [@liu2009self]. In each iteration of uncertainty sampling, the next point to be added to the design space is the point with the largest predictive variance of the GP. The predictive variance is given in Eq. \[eq:gp\_var\], and thus uncertainty sampling picks the next point $x^*$ believed to be the most informative point as: $$x^* = \operatorname*{arg\,max}_x V^*(x),$$ where the right hand side seeks to find the input point $x$ that maximizes a utility function, here simply $V$, but utility functions can be more complex. Other approaches, for example, include variance reduction in which the total volume of the confidence band is sought to be reduced the most, worst-case variance reduction in which the change in maximum GP uncertainty is the utility function among others [@kristensen2016expected]. Assume that the resources required to obtain a new point is constant across the input space. In this case, the concept of the cost associated with selecting a specific point is not relevant. This will change in the sections to follow as we consider multi-fidelity GPs. Adaptive Sampling in Multi-fidelity GP {#adaptive-sampling-in-multi-fidelity-gp .unnumbered} -------------------------------------- A typical multi-fidelity adaptive sampling process is shown in Fig. \[fig:mfs\_process\]. The process starts by generating initial samples of input variables for both low and high-fidelity analysis. Next, is to carry of low and high-fidelity analysis to evaluate the response and generating the initial database. The number of initial samples is another factor which can affect the speed of convergence and total cost of the process, however that is not considered to be the part of the current study. Next, a multi-fidelity model is built by building a model for low-fidelity model ($\eta(x)$) and the discrepancy model ($\delta(x)$) for the discrepancy between low and high-fidelity data. Then the model convergence is checked with provided validation metric to evaluated if the model is accurate enough as per requirement. If not, then multi-fidelity adaptive sampling strategy is applied. ![A TYPICAL MULTI-FIDELITY ADAPTIVE SAMPLING PROCESS[]{data-label="fig:mfs_process"}](figs/flowchart.pdf){width="4in"} Multi-fidelity sampling strategy tries to answer two new issues: 1. In a given iteration, should we choose to run the low-fidelity model or the high-fidelity model? How do we decide? 2. How do we factor in the difference in costs of running the models of varying fidelities? Once the next sampling point, $x^*$, and the fidelity of analysis is determiner, new analysis is carried out at $x^*$ to evaluate $y_{LF}(x^*)$ or $y_{HF}(x^*)$. The new data is stored in the database and process is repeated by building a new multi-fidelity GP with the updated database. The focus of the current work is the multi-fidelity adaptive sampling strategy as shown by the bold box in Fig. \[fig:mfs\_process\]. Typically, a two step strategy is used in multi-fidelity adaptive sampling strategy [@pellegrini2016multi], where in the first step $x^*$ is determined without considering the cost impact. Once $x^*$ is determined, then fidelity of analysis is decided based on the cost ratio between high and low fidelity analyses. In this paper we refer this method as sampling at Maximum Multi-Fidelity Uncertainty to Cost Ratio (Max MF-UCR) and as considered as baseline method to compare the new proposed approach. The proposed approach, sampling at Maximum Individual-Fidelity Uncertainty to Cost Ratio (Max IF-UCR), carries out one step process where it determines the next sampling point as well the fidelity of analysis in a single step by considering the cost of analyses during the selection process of $x^*$. The proposed approach is also extended to sampling at Maximum Individual-Fidelity Uncertainty to Cost Ratio using Believer (Max IF-UCR Bel) to study if any benefit can be achieved by using GP believer. Details of each strategy is given below: ### Sampling at Maximum Multi-Fidelity Uncertainty to Cost Ratio (Max MF-UCR) {#sampling-at-maximum-multi-fidelity-uncertainty-to-cost-ratio-max-mf-ucr .unnumbered} In this strategy, the sampling at next iteration is carried out at design $x^*$ where the predictive uncertainty of multi-fidelity response ($\sigma_{y_{mf}})$ is maximum, i.e. $$x^* = \operatorname*{arg\,max}_x \sigma_{y_{mf}},$$ where $\sigma_{y_{mf}}$ is the standard deviation of response $y_{mf}$ of multi-fidelity GP. Let’s say $C_H$ is the cost of each high-fidelity analysis and $C_L$ is the cost of low fidelity analysis then, if $\frac{\sigma_{\eta}(x^*) }{C_L} \ge \frac{\sigma_{\delta}(x^*) }{C_H}$, then low fidelity analysis is carried out at $x^*$ during the next iteration, otherwise high-fidelity is carried out. ### Sampling at Maximum Individual-Fidelity Uncertainty to Cost Ratio (Max IF-UCR) {#sampling-at-maximum-individual-fidelity-uncertainty-to-cost-ratio-max-if-ucr .unnumbered} In this strategy, the sampling at next iteration is carried out at design $x^*$ where the uncertainty reduction per unit cost is maximum. At a given $x$, uncertainty reduction per unit cost for low-fidelity is function of $\eta(x)$, i.e. $\sigma_{\eta}(x)/C_L$. Similarly, for high-fidelity uncertainty reduction per unit cost is function of $\delta(x)$, i.e. $\sigma_{\delta}(x)/C_H$. The next sampling point is chosen as: $$x^* = \operatorname*{arg\,max}_x \left[ \max \left( \frac{\sigma_{\eta}(x) }{C_L}, \frac{\sigma_{\delta}(x) }{C_H} \right) \right],$$ where $\sigma_{y_{mf}}$ is the standard deviation of response $y_{mf}$ of multi-fidelity GP. If $\frac{\sigma_{\eta}(x^*) }{C_L} \ge \frac{\sigma_{\delta}(x^*) }{C_L}$, then low fidelity analysis is carried out at $x^*$ during the next iteration, otherwise high-fidelity is carried out. ### Sampling at Maximum Individual-Fidelity Uncertainty to Cost Ratio using Believer (Max IF-UCR Bel) {#sampling-at-maximum-individual-fidelity-uncertainty-to-cost-ratio-using-believer-max-if-ucr-bel .unnumbered} GP believer is the concept of quantifying the impact on the GP model from adding a hypothetical new data point. In other words, consider a multi-fidelity GP which has been built on the joint training data set $D=(D_{\eta},D_{y})$. Consider an unseen point $x$: how much does the overall uncertainty reduce by if running the low-fidelity code at $x$? How much does this compare to running the high-fidelity code? To gauge the effect of adding a hypothetical point to $D$, we observe that the variance of the multi-fidelity model does not need the observed value $y^*$ of the unseen datum $(x^*, y^*)$. In fact: $$V_{mf}^* (x^* )=k_{mf} (x^*,x^* )- k{*T}_{mf}K_{mf}^{(-1)} k_{mf}^*.$$ One can add $x^*$ to the covariance matrices and evaluate the new variance. However, this ignores the fact that adding a datum technically requires re-fitting of the GP since the hyperparameters depend on the training data. The fitting process is typically not very expensive compared to running the low- or high-fidelity models so can be done. The issue is that the number of unseen points we have to gauge can be on the order of $10-100,000$. If the fitting takes 10 seconds the process of selecting a single new point can take on the order of $1-10$ days. We are looking to spend much less time, on the order of seconds, on this task. A solution to this can be to assume that the hyperparameters fitted with MCMC are practically speaking unchanged temporarily while searching for the next point. Thus, re-fitting is not required. More advanced methods could be envisioned here leveraging the MCMC chain in other ways. When gauging the effect of adding a low-fidelity point, only the matrices involving low-fidelity data are of course changed and similarly for a hypothetical high-fidelity point. Similar to Max IF-UCR, Max IF-UCR-Bel uses GP believer to determine how much is the uncertainty reduction if a low-fidelity or high-fidelity analysis is carried out at given point $x$. The next sample and the fidelity is then chosen for which the uncertainty reduction per unit cost is maximum as: $$\begin{aligned} x^* &= \operatorname*{arg\,max}_x \left[ \max \left( \frac{\sigma_{y}(x) - \sigma_y^{Bel}(x | x^{LF}_{bel} =x)}{C_L}, \right.\right. \\ & \left. \left. \frac{\sigma_{y}(x) - \sigma_y^{Bel}(x | x^{HF}_{bel} =x)}{C_H} \right) \right], \end{aligned}$$ where $\sigma_y^{Bel}(x | x^{LF}_{bel} =x)$ is the standard deviation of multi-fidelity predictor $y$ at $x$ when a low-fidelity “believer” is added at $x^{LF}_{bel} =x$. Similarly, $\sigma_y^{Bel}(x | x^{HF}_{bel} =x)$ is the standard deviation of multi-fidelity predictor $y$ at $x$ when a high-fidelity “believer” is added at $x^{HF}_{bel} =x$. In the next iteration, sampling is done using low fidelity analysis if $ \frac{\sigma_{y}(x^*) - \sigma_y^{Bel}(x^* | x^{LF}_{bel} =x^*)}{C_L} \ge \frac{\sigma_{y}(x^*) - \sigma_y^{Bel}(x^* | x^{HF}_{bel} =x^*)}{C_H}$. else high-fidelity sampling is carried out at $x^*$. Test Problem {#test-problem .unnumbered} ------------ ### 1-D Forrester Function {#d-forrester-function .unnumbered} The first test problem consists of analytical functions [@forrester2008engineering] to define both high and low fidelity analyses. The high fidelity equation is given as $$f_H(x) = (6x - 2)^2 \sin(12x-4).$$ The function is one-dimensional and is multi-modal in nature and has been used in literature for validating and testing surrogate models for single and multi-fidelity. The low fidelity equation is given as $$f_L(x) = Af_H(x) + B(x-0.5) - C,$$ where $A=0.6$, $B = 10$ and $C = 7$ has been used in the current study. Both the high and low fidelity functions are evaluated for $x\in [0,1]$. To start the multi-fidelity adaptive sampling, $4$ samples for low-fidelity and $2$ samples for high-fidelity are randomly generated in domain on $x\in [0,1]$ and evaluated using the respective functions. Additionally, two “prospective” databases are generated for each of the high-fidelity and low-fidelity analyses. These databases contains prospective $100$ random samples of input $x$, which will be used to evaluate adaptive sampling criteria to pick the next sampling point and fidelity at each iteration. It should be noted that the design point in each of these databases are not collocated. Also, additional $100$ samples are generated and evaluated using high-fidelity analysis to estimate error statistics. Multi-fidelity adaptive sampling process is started by building multi-fidelity GP using the initial samples. At the end of each iteration. new sample $x^*$ either from low-fidelity or high-fidelity is picked from the “prospective” databases based on the aforementioned strategy. Based on the determined fidelity, analyses is carried out at $x^*$ and is added to GP database. The multi-fidelity GP is then retrained with updated database. The process is carried out for $10$ sampling iterations. The overall process is repeated $10$ times, where in each case different initial sample is chosen to carry out the multi-fidelity adaptive sampling. This is done to verify that overall selection criteria is robust to different initial samples. Also, three scenarios are studied with different high-fidelity to low-fidelity cost ratio ($C_H:C_L$) of $2:1, 5:1$, and $10:1$. The results of these scenarios are shown in Fig. \[fig:1D\_1\_2\], \[fig:1D\_1\_5\] and \[fig:1D\_1\_10\]. In each of these figures, the first plot shows the convergence of root means squared error (RMSE) with respect to adaptive sampling iterations, with error bars showing the uncertainty across $10$ different runs. The second subplot shows the total cost of analysis with respect to sampling iteration. As observed in all the cases the RMSE converges (Fig. \[fig:1D\_1\_2a\], \[fig:1D\_1\_5a\] and \[fig:1D\_1\_10a\]) to similar value after $10$ iteration for each of the multi-fidelity selection criteria. For the scenario of $C_H:C_L = 2:1$ (Fig. \[fig:1D\_1\_2b\]), although we found the total cost at the end of $10$ iteration of was not significant different. However, for the scenario of $C_H:C_L = 5:1$ and $10:1$ (Fig. \[fig:1D\_1\_5b\] and \[fig:1D\_1\_10b\]), both Max IF-UCR and Max IF-UCR Bel criteria did better than Max MF-UCR. It was also observed that Max IF-UCR to be better than Max IF-UCR Bel criteria for both these scenarios. We also found that the as the $C_H:C_L$ increases, the cost reduction for both Max IF-UCR and Max IF-UCR increases when compared to Max MF-UCR. [0.45]{} ![COMPARISON OF METHODS FOR 1-D FORRESTER FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 2:1[]{data-label="fig:1D_1_2"}](figs/forrester_cost_Ratio_1_2_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 1-D FORRESTER FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 2:1[]{data-label="fig:1D_1_2"}](figs/forrester_cost_Ratio_1_2_cost.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 1-D FORRESTER FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 5:1[]{data-label="fig:1D_1_5"}](figs/forrester_cost_Ratio_1_5_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 1-D FORRESTER FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 5:1[]{data-label="fig:1D_1_5"}](figs/forrester_cost_Ratio_1_5_cost.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 1-D FORRESTER FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 10:1[]{data-label="fig:1D_1_10"}](figs/forrester_cost_Ratio_1_10_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 1-D FORRESTER FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 10:1[]{data-label="fig:1D_1_10"}](figs/forrester_cost_Ratio_1_10_cost.pdf "fig:"){width="\textwidth"} ### 4-D Park Function {#d-park-function .unnumbered} The second numerical experiment is carried out with $4$-dimensional analytical test functions. The high fidelity function is given in Eq. \[eq:park\_hf\] and was used by Park and Cox [@park1991tuning; @cox2001statistical] for testing method for tuning computer code. $$f_H(x) = \frac{x_1}{2} \left[\sqrt{1 + (x_2 +x_3^2)\frac{x_4}{x_1^2}} - 1 \right] + (x_1 + 3x_4)\exp\left[1 + \sin(x_3)\right] \label{eq:park_hf}$$ The low-fidelity analysis is represented by Eq. \[eq:park\_lf\] and was used by Xiong et al. [@xiong2013sequential]. The input domain for both low and high-fidelity analysis is $x \in (0,1)$. $$f_L(x) = \left[1 + \frac{\sin(x_1)}{10}\right] f_H(x) - 2x_1 + x_2^2 + x_3^2 + 0.5 \label{eq:park_lf}$$ At first, two “prosective” databases are generated for each of the high-fidelity and low-fidelity analyses. These database contains $100$ random samples of input $x$, which will be used to evaluate adaptive sampling criteria to pick the design point and fidelity at each iteration. Also, additional $100$ samples are generated and evaluated using high-fidelity analysis to estimate error statistics. The multi-fidelity experiments begins with using $2$ samples from high-fidelity analyses and $4$ samples from low-fidelity analysis, randomly selected in $x \in (0,1)$, and building an initial mult-fidelity GP on these. Similar to $1-D$ test problem, three scenarios are studied with different high-fidelity to low-fidelity cost ratio ($C_H:C_L$) of $2:1, 5:1$, and $10:1$, and for each scenario $10$ cases were carried out with different initial sampling. The results for each scenario are shown in Fig. \[fig:4D\_1\_2\], \[fig:4D\_1\_5\], and \[fig:4D\_1\_10\]. In all the scenarios, RMSE converged to similar value after $15$ iterations for all the adaptive sampling approaches. In terms of cost, total cost was significantly different for any of the methods for $C_H: C_L = 2:1$. At $C_H: C_L = 5:1$, Max IF-UCR performed the best, followed by Max IF-UCR Bel. At $C_H: C_L = 10:1$, both Max IF-UCR and Max IF-UCR Bel performed similarly and much better than Max MF-UCR. h\] [0.45]{} ![COMPARISON OF METHODS FOR 4-D PARK FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 2:1[]{data-label="fig:4D_1_2"}](figs/park_function_1_2_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 4-D PARK FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 2:1[]{data-label="fig:4D_1_2"}](figs/park_function_1_2_cost.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 4-D PARK FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 5:1[]{data-label="fig:4D_1_5"}](figs/park_function_1_5_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 4-D PARK FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 5:1[]{data-label="fig:4D_1_5"}](figs/park_function_1_5_cost.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 4-D PARK FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 10:1[]{data-label="fig:4D_1_10"}](figs/park_function_1_10_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR 4-D PARK FUNCTION WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 10:1[]{data-label="fig:4D_1_10"}](figs/park_function_1_10_cost.pdf "fig:"){width="\textwidth"} ### 6-D Fluidized-Bed Problem {#d-fluidized-bed-problem .unnumbered} For this test, dataset was taken from the study carried out by Dewettinck et al. [@dewettinck1999modeling] on thermodynamic modeling of top-spray fluidized bed, which was used to understand the impact of process variables and ambient changes known as the so-called weather effect. The quantity of interest is the temperature of steady-state operation of fluidized-bed, which is function of six variables: humidity ($H_R$), room temperature ($T_R$), temperature of the air from the pump ($T_a$), flow rate of the coating solution ($R_f$), pressure of atomized air ($P_a$), and fluid velocity of the fluidization air ($V_f$). In that study, three different fidelity of the model has been used and the results were validated with experimental results for $28$ different operating conditions. In the present work, $8$ out $28$ data points were kept aside to carry out validation and estimating the error statistics. From the remaining data, $2$ data were randomly chosen from experimental data as high-fidelity analysis and $4$ data from mid-fidelity data were used as low-fidelity data as starting point (first iteration) of adaptive sampling process. During the adaptive sampling process, the new designed point were selected from remaining data in the database. Ten cases were runs to analyze the robustness of the method with respect to adaptive sampling strategy, where in each case initial samples were randomly chosen. Although, in the original work, experimental data and simulation data were collocated at the same operating conditions, for adaptive sampling method this is not a required condition. Three scenarios are studied with different high-fidelity to low-fidelity cost ratio ($C_H:C_L$) of $2:1$, $5:1$, and $10:1$ and $15$ iterations of adaptive sampling were carried out. The results are shown in Fig. \[fig:FB\_1\_2\], \[fig:FB\_1\_5\], and \[fig:FB\_1\_10\]. For all the cases, the RMSE converges to similar range for all the adaptive sampling strategy. As in the previous tests, difference in total cost has been found to be not very significant for $C_H:C_L = 2:1$. With the increase in cost ratio, the difference in cost has been found to be increasing. For the scenario, Max IF-UCR has been found to give the best results in terms of cost, however for higher cost ratio both Max IF-UCR and Max IF-UCR bel has been found to give close result. [0.45]{} ![COMPARISON OF METHODS FOR FLUIDIZED BED PROCESS WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 2:1[]{data-label="fig:FB_1_2"}](figs/fluidized_bed_1_2_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR FLUIDIZED BED PROCESS WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 2:1[]{data-label="fig:FB_1_2"}](figs/fluidized_bed_1_2_cost.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR FLUIDIZED BED PROCESS WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 5:1[]{data-label="fig:FB_1_5"}](figs/fluidized_bed_1_5_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR FLUIDIZED BED PROCESS WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 5:1[]{data-label="fig:FB_1_5"}](figs/fluidized_bed_1_5_cost.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR FLUIDIZED BED PROCESS WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 10:1[]{data-label="fig:FB_1_10"}](figs/fluidized_bed_1_10_rmse.pdf "fig:"){width="\textwidth"} [0.45]{} ![COMPARISON OF METHODS FOR FLUIDIZED BED PROCESS WITH COST RATIO OF HIGH-FIDELITY AND LOW-FIDELITY OF 10:1[]{data-label="fig:FB_1_10"}](figs/fluidized_bed_1_10_cost.pdf "fig:"){width="\textwidth"} CONCLUSION {#conclusion .unnumbered} ========== Multi-fidelity Gaussian Process has been commonly used to incorporate cheap low-fidelity data with expensive high-fidelity data to improve the prediction capability of Gaussian process. If the low-fidelity analysis is very cheap when compared to high-fidelity analysis, then a straightforward strategy is run large number of low-fidelity analyses to build a very accurate model of low-fidelity analysis ($\eta(x)$) and then adaptively sample only high-fidelity analysis until required accuracy of high fidelity ($y_{mf}(x) = \eta(x) + \delta(x)$) is achieved or cost is within some budget. On the other hand, if the cost of low-fidelity and high-fidelity analyses are not different, then it is apparent to run only high-fidelity analysis and build only a single-fidelity GP using adaptive sampling strategy. In both these cases, adaptive sampling strategy for a single-fidelity GP is sufficient to work. However, for scenarios when the high-fidelity to low-fidelity cost ratio is not very high or not close to one, then a multi-fidelity adaptive sampling strategy is required to efficiently selecting input space as well as fidelity of the analyses to maximize the information gain with minimum cost. In this work a adaptive sampling criteria for selecting the new design point and fidelity of analysis using Maximum Individual Fidelity Uncertainty to Cost Ratio (Max IF-UCR) is demonstrated. The criteria is also extended by using Multi-fidelity GP “Beleiver” (Max IF-UCR bel) to carry out adaptively sampling and is compared with baseline case of Max Multi-fidelity Uncertainty to Cost Ratio (Max MF-UCR). The method is tested with two analytical test problem and one engineering problem. It has been found that when high-fidelity to low-fidelity cost ratio is low ($2:1$), then the proposed approach does not give much cost benefit. However, at higher cost ratio ($5:1$ and $10:1$), both Max IF-UCR and Max IF-UCR bel are significantly better in terms of total cost when compared to baseline case of Max MF-UCR. Although Max IF-UCR method has been found to be best in all the cases, but Max IF-UCR bel converges to similar trend for higher cost ratios. [1]{} Songqing Shan and G Gary Wang. Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. , 41(2):219–241, 2010. Nestor V Queipo, Raphael T Haftka, Wei Shyy, Tushar Goel, Rajkumar Vaidyanathan, and P Kevin Tucker. Surrogate-based analysis and optimization. , 41(1):1–28, 2005. Felipe AC Viana, Timothy W Simpson, Vladimir Balabanov, and Vasilli Toropov. Special section on multidisciplinary design optimization: metamodeling in multidisciplinary design optimization: how far have we really come? , 52(4):670–690, 2014. Yiming Zhang, John Meeker, Jaco Schutte, Nam Kim, and Raphael Haftka. On approaches to combine experimental strength and simulation with application to open-hole-tension configuration. In [*Proceedings of the American Society for Composites: Thirty-First Technical Conference*]{}, 2016. Pejman Honarmandi, Thien Chi Duong, S Fatheme Ghoreishi, Dou Allaire, and Raymundo Arroyave. Bayesian uncertainty quantification and information fusion in calphad-based thermodynamic modeling. , 164:636–647, 2019. M. Shabouei, W. Subber, C. Williams, K. Matouš, and J. Powers. Chemo-thermal model and gaussian process emulator for combustion synthesis of ni/al composites. , 2019. Under review, Ref. No.: CNF-D-19-00026. Tan Bui-Thanh, Karen Willcox, and Omar Ghattas. Model reduction for large-scale systems with high-dimensional parametric input space. , 30(6):3270–3288, 2008. Yiming Zhang, Aravind Neelakantan, Nalini Kumar, Chanyoung Park, Raphael T Haftka, Nam H Kim, and Herman Lam. Multi-fidelity surrogate modeling for application/architecture co-design. In [*International Workshop on Performance Modeling, Benchmarking and Simulation of High Performance Computer Systems*]{}, pages 179–196. Springer, 2017. Dawn An, Nam H Kim, and Joo-Ho Choi. Practical options for selecting data-driven or physics-based prognostics algorithms with reviews. , 133:223–236, 2015. Siliang Zhang, Ping Zhu, and Wei Chen. Crashworthiness-based lightweight design problem via new robust design method considering two sources of uncertainties. , 227(7):1381–1391, 2013. Zhen Hu and Sankaran Mahadevan. A single-loop kriging surrogate modeling for time-dependent reliability analysis. , 138(6):061406, 2016. Anirban Chaudhuri, Raphael T Haftka, Peter Ifju, Kelvin Chang, Christopher Tyler, and Tony Schmitz. Experimental flapping wing optimization and uncertainty quantification using limited samples. , 51(4):957–970, 2015. Raphael T Haftka, Diane Villanueva, and Anirban Chaudhuri. Parallel surrogate-assisted global optimization with expensive functions–a survey. , 54(1):3–13, 2016. V. V. Toropov. . In J. Blachut and H. Eschenauer, editors, [*Emerging Methods for Multidisciplinary Optimization*]{}, chapter 5, pages 205–256. Vol. 425, CISM Courses and Lectures, Springer-Verlag, New York, 2001. Andy Keane and Prasanth Nair. . John Wiley & Sons, 2005. Raphael T. Haftka. . , 29(9):1523–1525, September 1991. A. I. J. Forrester, András Sóbester, and Andy J. Keane. . , 463(2088):3251–3269, 2007. Sayan Ghosh, Isaac Asher, Jesper Kristensen, You Ling, Kevin Ryan, and Liping Wang. Bayesian multi-source modeling with legacy data. In [*2018 AIAA Non-Deterministic Approaches Conference*]{}, page 1663, 2018. Sayan Ghosh, Ryan Jacobs, and Dimitri N Mavris. Multi-source surrogate modeling with bayesian hierarchical regression. In [*17th AIAA Non-Deterministic Approaches Conference*]{}, page 1817, 2015. Marc C. Kennedy and Anthony O’Hagan. Bayesian calibration of computer models. , 63(3):425–464, 2001. Yiming Zhang, Nam H Kim, Chanyoung Park, and Raphael T Haftka. Multifidelity surrogate based on single linear regression. , 56(12):4944–4952, 2018. Chanyoung Park, Raphael T Haftka, and Nam H Kim. Remarks on multi-fidelity surrogates. , 55(3):1029–1050, 2017. M Giselle Fern[á]{}ndez-Godino, Chanyoung Park, Nam-Ho Kim, and Raphael T Haftka. Review of multi-fidelity models. , 2016. Loic Le Gratiet. . PhD thesis, Université Paris-Diderot - Paris VII, 2013. Ben Haaland and Peter ZG Qian. An approach to constructing nested space-filling designs for multi-fidelity computer experiments. , 20(3):1063, 2010. Jun Zheng, Xinyu Shao, Liang Gao, Ping Jiang, and Haobo Qiu. Difference mapping method using least square support vector regression for variable-fidelity metamodelling. , 47(6):719–736, 2015. Deng Huang, Theodore T Allen, William I Notz, and R Allen Miller. Sequential kriging optimization using multiple-fidelity evaluations. , 32(5):369–382, 2006. Benjamin Peherstorfer, Tiangang Cui, Youssef Marzouk, and Karen Willcox. Multifidelity importance sampling. , 300:490–509, 2016. Anirban Chaudhuri, Remi Lam, and Karen Willcox. Multifidelity uncertainty propagation via adaptive surrogates in coupled multidisciplinary systems. , pages 235–249, 2017. Riccardo Pellegrini, Cecilia Leotardi, Umberto Iemma, Emilio F Campana, and Matteo Diez. A multi-fidelity adaptive sampling method for metamodel-based uncertainty quantification of computer simulations. In [*Proceedings of the VII European congress on computational methods in applied sciences and engineering, ECCOMAS*]{}, 2016. David Ginsbourger, Rodolphe Le Riche, and Laurent Carraro. Kriging is well-suited to parallelize optimization. In [*Computational intelligence in expensive optimization problems*]{}, pages 131–162. Springer, 2010. C. E. Rasmussen and C. K. I. Williams. . MIT Press, 2006. P. D. Arendt, D. W. Apley, and W. Chen. Quantification of model uncertainty: Calibration, model discrepancy, and identifiability. , 134(10):100908–100908, 2012. Alexander I. J. Forrester, Andras Sobester, and Andy J. Keane. Wiley, 2008. Christopher M Bishop. . springer, 2006. Alexander Liu, Goo Jun, and Joydeep Ghosh. A self-training approach to cost sensitive uncertainty sampling. , 76(2-3):257–270, 2009. Jesper Kristensen, You Ling, Isaac Asher, and Liping Wang. Expected-improvement-based methods for adaptive sampling in multi-objective optimization problems. In [*ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference*]{}, pages V02BT03A024–V02BT03A024. American Society of Mechanical Engineers, 2016. Alexander Forrester, Andras Sobester, and Andy Keane. . John Wiley & Sons, 2008. Jeong Soo Park. . PhD thesis, University of Illlinois, Champaign-Urbana, 1991. Dennis D Cox, Jeong-Soo Park, and Clifford E Singer. A statistical method for tuning a computer code to a data base. , 37(1):77–92, 2001. Shifeng Xiong, Peter ZG Qian, and CF Jeff Wu. Sequential design and analysis of high-accuracy and low-accuracy computer codes. , 55(1):37–46, 2013. Koen Dewettinck, Alex De Visscher, Lidewij Deroo, and Andr[é]{} Huyghebaert. Modeling the steady-state thermodynamic operation point of top-spray fluidized bed processing. , 39(2):131–143, 1999. [^1]: Address all correspondence to this author, email: `sayan.ghosh1@ge.com`

--- abstract: 'Non-orientable nanostructures are becoming feasable today.  This lead us to the study of spin in these geometries.  Hence a physically sound definition of spin is suggested.  Using our definition, we study the question of the number of different ways to define spin.  We argue that the possibility of having more than one spin structure should be taken into account energetically.  The effect of topology on spin is studied in detail using cohomological arguments.  We generalize the definition of equivalence among (s)pin structures to include non-orientable spaces.' author: - 'A. Rebei' title: 'A generalized definition of spin in non-orientable geometries' --- INTRODUCTION ============ It has recently been possible to realize new small-size materials with nontrivial topologies.  Tanda et al. were able to have Mobius bands formed by crystals of Niobium and Selenium, $NbSe_3$ [@japan].  Given the recent interest in spintronics, it seems therefore worth the effort to study the possible effect of geometry on spin, especially the combined effect of non-orientability and non-simple connectedness of the space.  Geometric effects in physics are often hidden in terms of constraints.  One good example of this is the $\theta$ vacua of QCD.  Here, the different topological sectors are due to the Gauss constraint [@ba].  The Chern number also appears due to constraints either in physical space or momentum space.  While studying charge density waves in a torus geometry, Thouless found that in this case, the conductivity is expressed in terms of the Chern number of the manifold and differs from the periodic lattice case in Euclidean space [@thouless].  Spin currents in semiconductors is still another problem where the Chern number can be used to explain the universality of the spin conductivity in the Rashba model [@rebei].  In this latter case, the constraint is in momentum space which is homotopic to the plane without the origin, a non-simply connected space.  A final example, we give, is the quantization of the spin of the $SU(2)$ Skyrmion.  The solution to this problem was possible only after extending $SU(2)$ to group $SU(3)$ [@witten].  However in this extension, a new term is needed in the Lagrangian, the Wess-Zumino term, which has a topological significance and it is related to the disconnectedness of $\mathbf{SO}(3)$.  This latter example shows the inter-connectedness of topology of fields and spin.  In this work, we address similar issues between spin and topology of the physical space of electrons on non-orientable manifolds such as a Mobius band.  In this case there is no global well defined spin structure for the manifold [@alvarez].  It is well known that quantum mechanical wave functions in non-simply connected spaces can be multivalued and are therefore well defined only on their simply connected covering spaces [@ba].  For non-orientable manifolds, we show that a definition of spin is possible by going to the orientable double cover of the initial space.  This work was motivated by Tanda’s group and a simple calculation that is presented in the application section.  For thin-film rings, we observed that there is a critical radius at which the trivial (or commonly used) spin representation becomes higher in energy than the non-trivial (twisted) spin representation.  This twisted representation should correspond to the trivial representation on a Mobius band.  The critical radius is estimated to be around $10 \; nm$ for a clean conductor.  The typical ring sizes today is in the $100-50 \; nm$ range, but it is expected that much smaller sizes will be available in the future [@saitoh].   Therefore, we claim that at these small scales the spin in the Mobius band and in the ring should ’behave’ the same way, e.g., as it interacts with an external magnetic field or in a ferromagnetic material.  However, before a physical analysis of our claim is possible, a consistent definition of spin structures in non-orientable manifolds is needed.   Spin is an inherently relativistic effect of the electron and follows from requiring Lorentz-invariance of the Schrodinger equation [@dirac].  The relativistic treatment of the spin is not necessary but it considerably simplifies the formal discussion.  In this paper, we study in some detail, the different spin structures that are possible in non-trivial geometries.  For non-orientable manifolds, the spin group is extended to a pin group where parity is violated similar to the extension of the special rotation group $\mathbf{SO}(3)$ to the full rotation group $\mathbf{O}(3)$.  In this latter case, the $\mathbf{Pin}(3)$ group double covers $\mathbf{O}(3)$ with the group $\mathbf{Spin}(3)$ being a connected component of $\mathbf{Pin}(3)$ which double covers the connected component of $\mathbf{O}(3)$, i.e., $\mathbf{SO}(3)$.  So far only Ref. [@petry] makes use of pin structures to give a viable alternative explanation of a physical theory such as superconductivity.  In this latter work, cooper pairs can be substituted for a more geometrical interpretation which is the existence of a two-inequivalent spin structures in a ring.  Hence in any non-trivial geometry, knowing the number of inequivalent pin structures is important to know before writing a Lagrangian for the dynamics [@shulman2].  For the sake of generality, we will allow even time non-orientable manifolds to be part of the discussion.  The work in the literature that covers questions related to the existence and the counting of the different Pin structures is mostly recent.  A comprehensive introduction to Pin group structures can be found in Refs. [@dabrowski] and [@cecile].  Pin groups first appeared in Ref.[@atiyah] and were derived from Clifford algebras in relation to the K-theory of vector bundles.  After that Karoubi [@karoubi] studied the obstructions to these Pin structures within fiber bundles theory and hence was confined to Pin structures that are only derivable from a Clifford algebra.   As is well known, there are eight non-isomorphic **Z**$_{2}$ extensions of the full Lorentz group [@ebner]. This is a direct consequence of the disconnectedness of the full Lorentz group and hence there is no unique universal two-cover as is the case for the special orthocronous Lorentz group.  The paper is organized as follows. In section II, we review the results on the orientable case and set the notation for the rest of the paper. In section III we introduce the Pin group through the Clifford algebra. In section IV we introduce a new definition for equivalence among Pin structures that works in non-orientable spaces and study its meaning on the level of representations of the Pinor field. Section V addresses the question of the counting of the inequivalent Pin structures defined on a non-orientable manifold in light of the new definition.  In section VI, we apply the results to two cases; a nano-circuit that has the geometry of a torus and a non-orientable de Sitter space. For the nanocircuit we argue that it is the nontrivial spin representation that must be used instead of the trivial one.  In section VII, we summarize our results. SPIN STRUCTURES ===============  In what follows, it is assumed that we are dealing with manifolds with metrics that have the signature $\left(+,-,-,-\right) $ and we will not address any questions that are dependent on the metric propre.  A good review on the mathematics involved here can be found in ref. [@eguchi].  In this section, we review the spin structures on orientable manifolds and set the notation for the rest of the paper. For an orientable manifold $\mathcal{\mathbf{B}}$, such as the de Sitter space $\mathbf{R}\times \mathbf{S}% ^{3}$, a Spin structure exists whenever the second Stiefel-Whitney class vanishes [@milnor]. This is the same as saying that the transition functions of the Lorentzian frame bundle lift up to new transition functions with values in the $\mathbf{Spin}(1,3)$ group. The number of inequivalent Spin structures is given by the number of classes in ${H}^{1}({\mathcal{\mathbf{B}}}, \mathbf{Z}_{2})$.  It is well known that $\mathbf{Spin}(1,3)$ is a double cover for the Lorentz group $\mathbf{SO}_{0}(1,3)$, which is the connected component of the identity of the orthogonal group $\mathbf{O}(1,3)$. This double covering induces some restrictions on the transition functions of a Spin structure. For a manifold to be orientable, the $\mathbf{O}(1,3)$ group structure of the frame bundle should be reducible to $\mathbf{SO}(1,3)$ by choosing an orientation. Since the sequence $$1\longrightarrow \mathbf{SO}(1,3)\longrightarrow \mathbf{O}(1,3)\longrightarrow \mathbf{Z}_{2}\longrightarrow 1$$ is a short exact sequence, we get the following long exact sequence [@hirzb], $$\begin{aligned} 0 &\longrightarrow & H^{0}({\mathcal{\mathbf{B}}},\mathbf{SO}(1,3))\longrightarrow H^{0}({\mathcal{\mathbf{B}}} ,\mathbf{O}(1,3))\longrightarrow H^{0}({\mathcal{\mathbf{B}}},\mathbf{Z}_{2}) \nonumber \\ &\longrightarrow &H^{1}({\mathcal{\mathbf{B}}},\mathbf{SO}(1,3))\longrightarrow H^{1}({\mathcal{\mathbf{B}}} ,\mathbf{O}(1,3))\longrightarrow H^{1}({\mathcal{\mathbf{B}}},\mathbf{Z}_{2}).\end{aligned}$$ Recalling that $H^{1}({\mathcal{\mathbf{B}}},\mathbf{SO}(1,3))$ is the set of equivalence classes of Principal $\mathbf{SO}(1,3)$-bundles, then ${\mathcal{\mathbf{B}}}$ is orientable if and only if $\left( iff \right)$ the last map is null. This map is by definition the first Stiefel-Whitney class $\omega_{1}$. Similarly, the short exact sequence $$1\longrightarrow \mathbf{Z}_{2}\longrightarrow \mathbf{Spin}(1,3)\longrightarrow \mathbf{SO}(1,3)\longrightarrow 1$$ induces a long exact sequence, $$...\longrightarrow H^{1}({\mathcal{\mathbf{B}}},\mathbf{Z}_{2})\longrightarrow H^{1}({\mathcal{\mathbf{B}} },\mathbf{Spin}(1,3))\longrightarrow H^{1}({\mathcal{\mathbf{B}}},\mathbf{SO}(1,3))\longrightarrow H^{2}({ \mathcal{\mathbf{B}}},\mathbf{Z}_{2}).$$ Hence ${\mathcal{\mathbf{B}}}$ has a spin structure iff the last map is null. This latter map is the second Stiefel-Whitney class $\omega _{2}( {\mathcal{\mathbf{B}}})$ and hence it is the obstruction to a spin structure on the space $\mathbf{B}$.  For non-orientable spaces, $w_1 \neq 0$, we need to establish similar sequences to the respective groups and this will lead us naturally to the pin group. PINOR STRUCTURES ================ Pinor groups are better understood from Clifford algebras [@atiyah]. Given a vector space $\mathcal{V}$ of dimension $n=4$ that is tangent to a point $p\in {\mathcal{\mathbf{B}}}$ and $g(x,y)$ a bilinear non-degenerate quadratic form on $\mathcal{V}$ associated with the metric $(+,-,-,-)$. Let $T(\mathcal{V})=\sum_{i=0}^{\infty }T^{i}(\mathcal{V})$, where $$T^{i}(\mathcal{V})=\underset{i-times}{\underbrace{\mathcal{V}\times ...\times \mathcal{V}}},$$ is the tensor algebra of $\mathcal{V}$. The set $\mathcal{I}$ generated by the set$\{x\otimes x-g(x,x),x\in \mathcal{V}\}$ is an ideal of $T(\mathcal{V}).$  The quotient space $$Cl(\mathcal{V},Q)=\frac{T(\mathcal{V})}{\mathcal{I}}$$ is the Clifford algebra of the vector space $\mathcal{V}$ equipped with the quadratic form $Q(x)=g(x,x).$  Clearly, if $Q(x)=0, Cl(\mathcal{V})$ is simply the Grassmann algebra of forms on $\mathcal{V}$. The dimension of $Cl(\mathcal{V},Q)$ is 2$^{4}.$  The multiplication in this algebra is induced by the tensor product in $T(\mathcal{V}).$ Let $\left\{ e_{0},e_{1},e_{2},e_{3}\right\} $ be an orthonormal basis for $% \mathcal{V} $.  Then, the following vectors of $Cl(\mathcal{V},Q)$, $$\begin{aligned} &&e_{0},e_{1},e_{2},e_{3} \nonumber \\ &&e_{0}e_{1},e_{0}e_{2},e_{0}e_{3},e_{1}e_{2},e_{1}e_{3},e_{2}e_{3} \nonumber \\ &&e_{0}e_{1}e_{2},e_{0}e_{1}e_{3},e_{0}e_{2}e_{3},e_{1}e_{2}e_{3} \nonumber \\ &&e_{0}e_{1}e_{3},1\end{aligned}$$ form a basis for the algebra.  Moreover, since $$\begin{aligned} Q(e_{\mu }+e_{\nu }) &=&(e_{\mu }+e_{\nu })\otimes (e_{\mu }+e_{\nu }) \nonumber \\ &=&e_{\mu }\otimes e_{\mu }+e_{\mu }\otimes e_{\nu }+e_{\nu }\otimes e_{\mu }+e_{\nu }\otimes e_{\nu },\end{aligned}$$ we have, on the level of algebra, $$\begin{aligned} e_{\mu }e_{\nu }+e_{\nu }e_{\mu } &=&0\qquad \text{\ \ \ \ \ \ \ if }\mu \neq \nu \nonumber \\ \ e_{\mu }^{2} &=&2e^{\mu }e_{\mu }\qquad \mu =0,1,2,3 .\end{aligned}$$ Clearly Dirac $\gamma _{\mu }$ matrices form a representation of this algebra ( Majorana representation).  The involution $x\rightarrow -x$ in $\mathcal{V}$ extends naturally to an involution $\alpha $ of the algebra which in turn induces a $\mathbf{Z}_{2}$ grading of $Cl(\mathcal{V},Q)$, i.e., $$Cl(\mathcal{V},Q)=Cl^{+}(\mathcal{V},Q)+Cl^{-}(\mathcal{V},Q),$$ where $Cl^{+}(\mathcal{V},Q) \left( Cl^{-}(\mathcal{V},Q)\right)$ contains the even (odd) elements of the algebra. A norm function can be defined on $Cl(\mathcal{V},Q)$ by first defining conjugation on the generators: $$(e_{i_{1}}....e_{i_{p}})^{\ast }=e_{i_{p}}....e_{i_{1}} \: .$$ Then the norm of $x$ is defined by $$\left\| x\right\| ^{2}=x\alpha (x^{\ast }) .$$ Let $Cl^{\ast }(\mathcal{V},Q)$ be the subset of all invertible elements of $Cl(\mathcal{V},Q).$  Actually in our case where the dimension is even we might as well define the norm without the inversion $\alpha .$  The Clifford group $\Gamma (\mathcal{V},Q)$ is the subgroup of $Cl^{\ast }(\mathcal{V},Q)$  defined by $$\Gamma (\mathcal{V},Q)=\left\{ x\in Cl^{\ast }(\mathcal{V},Q): v \in \mathcal{V},\alpha (x)vx^{-1}\in \mathcal{V}\right\}.$$ Given $x\in \Gamma (\mathcal{V},Q)$, then it can be represented by an orthogonal transformation $$\begin{aligned} \rho (x) &:&\mathcal{V}\rightarrow \mathcal{V} \nonumber \\ e &\rightarrow &\alpha (x)ex^{-1}\end{aligned}$$ If we were to drop $\alpha $ from the definition, the map $\rho $ fails to be a representation in the odd dimensional case.  Finally, the [*Pin group*]{} is the subgroup of the Clifford group with elements of modulus one, $$\mathbf{Pin}(\mathcal{V},Q)=\left\{ x\in \Gamma (\mathcal{V},Q):\qquad ||x||=1\right\} .$$ This group doubly covers the orthogonal group $\mathbf{O}(1,3)$. The sequence $$1\longrightarrow \mathbf{Z}_{2}\longrightarrow \mathbf{Pin}(1,3)\longrightarrow \mathbf{O}(1,3)\longrightarrow 1$$ is then a short exact sequence. We say that $\mathbf{Pin}(1,3)$ is a $\mathbf{Z}_{2}-$extension of $\mathbf{O}(1,3)$.  Finally, it is important to realize that not all pin groups are derived from a Clifford algebra.  Chamblin [@chamblin] studied the obstructions to non-Cliffordian Pin structures.  Starting from Dabrowski’s semidirect product formula for the Pin group  [@dabrowski],  Chamblin found an obstruction to Pin structures through the use of Sheaf cohomology [@hirzb]. The starting short exact sequence that was fundamental to his construction is however not correct.  This is easily seen by applying the second homomorphism theorem. Dabrowski’s formula is $$\mathbf{Pin}^{a,b,c}(p,q)=\frac{\mathbf{Spin}_{0}(p,q)\odot \mathbf{C}^{a,b,c}}{\mathbf{Z}_{2}},$$ where $\mathbf{C}^{a,b,c}$ stand for the discrete group of order 8 that is a double covering of the group $\mathbf{G}=\left\{ 1,T,P,T P\right\} $. The rotation group $ \mathbf{O}(p,q)=\mathbf{SO}(p,q)\odot G$ is double covered by the Pin group and there are eight non-isomorphic such groups.  This latter eight is unrelated to the order of the group $\mathbf{C}^{a,b,c}$ . The group $\mathbf{G}$ is isomorphic to $\mathbf{Z}_{2}\otimes \mathbf{Z}_{2}.$ The group $\mathbf{C}^{a,b,c}$ is isomorphic to either of the following groups. The quaternion group $\mathbf{Q}$, the dihedral group $\mathbf{D}_{4}$, the group $\mathbf{Z}_{2}\times \mathbf{Z}_{2}\times \mathbf{Z}_{2}$ and the group $\mathbf{Z}_{2}\times \mathbf{Z}_{4}$. The a, b and c indexes stand for the signs of the squares of the elements of the cover $T, P$ and $P T.$ For example in the quaternion case we can map $i$ to $T$, $j$ to $P$ and $k$ to $ T P$. In this case $a=b=c=-$, and so on.  In the rest of the paper, we will be only interested in determining the degrees of freedom on the spin structures for a given topology. We hope to address the question of obstructions in the future. WEAKLY-EQUIVALENT PIN STRUCTURES ================================ In this section, we introduce a new definition for equivalence among (s)pin structures that takes into account the possible non-orientability of the physical space of the system.  The definition is suggested such that the non-orientability is linked to the orientable cover of the space.  This is in analogy with relating non-simply connected spaces to their simply connected covers. Pinor-Frames ------------ Let $\mathcal{\mathbf{B}}$ a manifold with a metric of signature $(+,-,-,-)$ and covered by a simple cover $\left\{ U_{i}\right\} ,\Gamma $ is the $\mathbf{Pin}(1,3)$ group and $\mathbf{G}$ is the orthogonal group $\mathbf{O}(1,3)$. $(\mathcal{\mathcal{\mathbf{P}}},\pi ,\mathcal{\mathbf{B}},\mathbf{G})$ is a principal bundle. $( \widetilde{\mathcal{\mathbf{P}}},\widetilde{\pi },\mathcal{\mathbf{B}},\Gamma )$ is the principal bundle induced by the double covering $$\rho :\Gamma \rightarrow \mathbf{G} \: .$$ There is a bundle map $\Phi $ between $\widetilde{\mathcal{\mathbf{P}}}$ and $\mathcal{\mathbf{P}}$ such that $$\begin{aligned} \Phi \pi &=&\widetilde{\pi } \nonumber \\ \Phi (\upsilon \cdot \gamma ) &=&\Phi (\upsilon )\Phi (\gamma )\end{aligned}$$ for $\upsilon \in $ $\widetilde{\mathcal{\mathbf{P}}}$ and $\gamma \in \Gamma $ . Consider now two $\Gamma -$structures $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{\mathcal{\mathbf{P}}^{\prime}}$ over $\mathcal{\mathbf{P}}$ that differ only through an automorphism $\Psi $ of $\Gamma $, in other words, we have a bundle isomorphism $\Theta $ such that $$\begin{array}{llll} \Theta : & \ \widetilde{\mathcal{\mathbf{P}}} & \rightarrow & \ \widetilde{\mathcal{\mathbf{P}}}^{\prime } \\ & \Phi \searrow & & \swarrow \Phi ^{\prime } \\ & & \mathcal{\mathbf{P}} & \end{array}$$ commutes and $$\Theta (\upsilon \cdot \gamma )=\Theta (\upsilon )\cdot \Psi (\gamma )\text{ \ \ \ \ \ \ }\upsilon \in \widetilde{\mathcal{\mathbf{P}}},\gamma \in \Gamma \: .$$ $\widetilde{\mathcal{\mathbf{P}}}$ and $\mathcal{\mathbf{P}}$ are said to be *weakly-equivalent*. Because of the double covering, the map $\Psi $ is involutive. Next, we state the following definition of a pinor field which is a generalization of the usual spinor field. [*[A Pinor field $\psi $ of type ($\xi ,\mathbf{Y})$ on $\mathcal{\mathbf{B}}$ is a section of $ \widetilde{\mathcal{\mathbf{P}}}\times _{\Gamma } \mathbf{Y}$  with $$\xi :\Gamma \rightarrow Hom(\mathbf{Y},\mathbf{Y})$$ and is a representation map of $\Gamma $ .]{}*]{} ### **Theorem I** {#theorem-i .unnumbered} [*[The sets of Pinor fields representations in $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{\mathcal{\mathbf{P}} }^{\prime }$ are $\Psi -$ related.]{}*]{} To prove this we first take note of the fact that a section $\sigma $ of $ \widetilde{\mathcal{\mathbf{P}}}\times _{\Gamma }Y$  can be represented as a map $$\begin{aligned} S &:&\widetilde{\mathcal{\mathbf{P}}}\rightarrow \mathbf{Y} \nonumber \\ S(\widetilde{u}) &=&\widetilde{u}^{-1}(\sigma (x))\end{aligned}$$ such that $$S(\widetilde{u}\cdot \gamma )=\gamma ^{-1}\cdot S(\widetilde{u})$$ Here we have used the fact that an element $\widetilde{u}$ of $\widetilde{\mathcal{\mathbf{P}}}$ represents a map from $\mathbf{Y}$ to $Y_{x}$ . Moreover, the bundle map $\Theta $ can be taken to be the identity map on fibers, i.e., $$\Theta \equiv 1:Y_{x}\rightarrow Y_{x}$$ Let $\left\{ \gamma _{ij}\right\} $ and  $\left\{ \gamma _{ij}^{\prime }\right\} $ be the transition functions of $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{ \mathcal{\mathbf{P}}}^{\prime }$ , respectively. Then it is obvious that the following diagram commutes: $$\begin{tabular}{llllll} $\gamma _{ji}:$ & $U_{i}\smallfrown U_{j}$ & & $\rightarrow $ & & $\Gamma $ \\ & & & & $\Psi $ & $\downarrow $ \\ & & $\gamma _{ji}^{\prime }$ & $\searrow $ & & \\ & & & & & $\Gamma $% \end{tabular}$$ i.e., $$\Psi (\gamma _{ji})=\gamma _{ji}^{\prime } \: .$$ Therefore given a section $\sigma _{i} : U_{i}\rightarrow \widetilde{\mathcal{\mathbf{P}}}\times _{\Gamma }\mathbf{Y}$, there is a corresponding one $\sigma _{i}^{\prime }~: U_{i}\rightarrow \widetilde{\mathcal{\mathbf{P}}}^{\prime }\times _{\Gamma }\mathbf{Y}$ such that $$\begin{aligned} \sigma _{i}(x) &=&[x,y]=[x\cdot \gamma ,\gamma ^{-1}\cdot y] \nonumber \\ \sigma _{i}^{\prime }(x) &=&[x,y]=[x\cdot \Psi (\gamma ),\Psi (\gamma ^{-1})\cdot y]\end{aligned}$$ An element $u$ of $\widetilde{\mathcal{\mathbf{P}}}$ can be represented by an equivalence class $[i,x,\gamma ] = [j,x,\gamma _{ji}\gamma ]$ with a similar expression for $u^{\prime }$ of $\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ with $\gamma _{ji}$ replaced by $\Psi (\gamma _{ji})$ . This is equivalent to saying that $u$ is a map $$\mathbf{Y}\rightarrow Y_{x}$$ with $$u(y)=\varphi _{i}(x,\gamma \cdot y) \: .$$ The $\varphi _{i}$’s are the local trivializations of $\widetilde{\mathcal{\mathbf{P}}}$. For the cross sections we have the following diagram $$\begin{array}{ccccc} S: & \widetilde{\mathcal{\mathbf{P}}} & \rightarrow & & \mathbf{Y} \\ \Theta & \downarrow & & \nearrow & S^{\prime } \\ & \widetilde{\mathcal{\mathbf{P}}}^{\prime } & & & \end{array}$$ so $S(u)=S^{\prime }(\Theta (u))=S^{\prime }(u^{\prime })$. This in turn implies that $$\begin{aligned} S^{\prime }(u^{\prime }\cdot \gamma ^{\prime }) &=&\gamma ^{^{\prime }-1}\cdot S^{\prime }(u^{\prime }) \nonumber \\ &=&\Psi (\gamma ^{-1})\cdot S(u) \: .\end{aligned}$$ If $\xi $ is a representation of $\Gamma $ , the above relation trivially extends to $$S^{\prime }(u^{\prime }\cdot \xi (\gamma ^{\prime }))=\Psi (\xi (\gamma ^{-1}))\cdot S(u) \: .$$ We conclude that a pinor field defined on $\mathcal{\mathbf{B}}$ is a quantity independent of the action group within an isomorphism. On frame-Bundles ---------------- Let as above $\mathcal{\mathbf{B}}$ be a non-orientable manifold and $\mathcal{\mathbf{B}}^c$ its [*orientable double cover*]{}. Here, we propose to treat the question of what happens if the frame bundle $\overline{\mathcal{\mathbf{P}}}$, with group of action taken to be $\mathbf{O}_{+}^{\uparrow }(1,3)$ , is pushed forward with the covering map $p : \mathcal{\mathbf {B}}^c\rightarrow \mathcal{\mathbf{B}}$. We take $\{U_{i}\}$ as the cover of $\mathcal{\mathbf{B}}$. First it should be realized that the transition functions $\overline{g_{ji}} (x^{\prime })$ of $\overline{\mathcal{\mathbf{P}}}$  are the same as those of the tangent bundle $T\mathcal{\mathbf{B}}^c$ . By definition, these transition functions are the Jacobian of the transition functions of the coordinate functions: $$\varphi _{i}^{\prime }:E_{i}\rightarrow V_{i}=p^{-1}(U_{i}) \: ,$$ where $E_{i} \subset \mathbf{R^{4}}$ . The transition functions $t_{ji}(x^{\prime })$ of the chart $(V_{i},\varphi _{i})$ are given by $$t_{ji}^{\prime } = \varphi _{i}^{\prime -1}\varphi _{i}^{\prime}: E_{i}\smallfrown E_{j}\rightarrow E_{i}\smallfrown E_{j} \: .$$ Therefore the transition functions $a_{ji}(x^{\prime })$ of $T\mathcal{\mathbf{B}}^c$ are given by : $$\begin{aligned} a_{ji}^{\prime } &:&V_{i}\smallfrown V_{j}\rightarrow \mathbf{O}_{+}^{\uparrow }(1,3) \nonumber \\ x^{\prime } &\longmapsto &a_{ji}^{\prime }(x^{\prime })=J(t_{ji}^{\prime })|_{\varphi _{i}^{-1}(x^{\prime })} \: .\end{aligned}$$ Hence we have $$\overline{g_{ji}}(x^{\prime })\ =a_{ji}^{\prime }(x^{\prime }) \: .$$ This can be easily shown through the coordinate functions. Now, we describe in more detail the map $p:\mathcal{\mathbf{B}}^c$ $\rightarrow \mathcal{\mathbf{B}}$ . The orientable double cover is defined as follows. The Jacobian of the transition functions of $\mathcal{\mathbf{B}}$, $a_{ji}(x)$, that corresponds to $ a_{ji}^{\prime }$ in $\mathcal{\mathbf{B}}^c$ are defined similarly. From these transition functions, we can form 1-cochains $\theta $ $$\begin{aligned} \theta _{ij} &:&U_{i}\smallfrown U_{j}\rightarrow \mathbf{Z}_{2} \nonumber \\ x &\longmapsto &\theta _{ji}(x)=\det [J(a_{ji})|_{\varphi _{i}^{-1}(x)}].\end{aligned}$$ Therefore to each point $x\in \mathcal{\mathbf{B}}$ , we can associate to it two points $(x,1)$ and $(x,-1)$ where $\pm 1$ is the values of $\theta _{ij}(x)$ . The manifold $\mathcal{\mathbf{B}}^c$ is the set of all these points. First we note that $\mathcal{\mathbf{B}}^c$ is connected. A curve in $\mathcal{\mathbf{B}}^c$ that connects $(x,1)$ to $(x,-1)$ can be given through the unique lifting of a non-orientable closed loop at $x$ [@spanier]. $\mathcal{\mathbf{B}}^c$ is also orientable since lifting the $\theta _{ij}$’s will give the determinant of the Jacobian of the $a_{ij}$’s. The manifold $\mathcal{\mathbf{B}}^c$ can in fact be interpreted as a bundle structure over $\mathcal{\mathbf{B}}$ with fiber and group $\mathbf{Z}_{2}$ . Diagrammatically we have $$\begin{tabular}{lll} $\overline{\mathcal{\mathbf{P}}}$ & & \\ $\downarrow \overline{\pi }$ & & \\ $\mathcal{\mathbf{B}}^c$ & $\rightarrow $ & $\mathcal{\mathbf{B}}$ \\ & $p$ & \end{tabular}$$ Therefore for the principal bundle $(\mathcal{\mathbf{B}}^c,p,\mathcal{\mathbf{B}},\mathbf{Z}_{2})$ the local trivializations are given by $$\phi _{i}:U_{i}\times \mathbf{Z}_{2}\rightarrow p^{-1}(U_{i})=V_{i}$$ and the transition functions $g_{ji}(x):\mathbf{Z}_{2}\rightarrow \mathbf{Z}_{2}$ are elements of $\mathbf{Z}_{2}$ , they act as a permutation group of the points that cover $x$ . Now define a new set $P=\smallsmile _{x\in \mathcal{\mathbf{B}}}P_{x}$ where $P_{x}=\overline{\mathcal{\mathbf{P}} }_{x_{1}^{\prime }}\smallsmile $ $\overline{\mathcal{\mathbf{P}}}_{x_{2}^{\prime }}$ if $ p(x_{1}^{\prime })=$ $p(x_{2}^{\prime })=x$ and distinct. We claim that the map $$\begin{aligned} p_{\ast } &:&\mathcal{\mathbf{P}}\rightarrow \mathcal{\mathbf{B}} \nonumber \\ p_{\ast }(u) &=&x\text{ where }u\in \overline{\mathcal{\mathbf{P}}}\text{ and }p(\overline{\pi } (u))=x\end{aligned}$$ induces a bundle structure on $\mathcal{\mathbf{B}}$. Therefore we expect the following diagram to commute $$\begin{tabular}{llll} $\Lambda :$ & $\overline{\mathcal{\mathbf{P}}}$ & $\rightarrow $ & $\mathcal{\mathbf{P}}$ \\ & $\overline{\pi }\downarrow $ & & $\downarrow p_{\ast }$ \\ & $\mathcal{\mathbf{B}}^c$ & $\rightarrow$ & $\mathcal{\mathbf{B}}$ \\ & & $p$ & \end{tabular}$$ besides taking fiber to fiber, the map $\Lambda $ should respect the action of the respective groups in both manifolds: $$\begin{tabular}{lll} $\overline{\mathcal{\mathbf{P}}}\times \mathbf{O}_{+}^{\uparrow }(1,3)$ & $\rightarrow $ & $\overline{\mathcal{\mathbf{P}}}$ \\ $\Lambda \downarrow \tau $ & & $\downarrow $ \\ $\mathcal{\mathbf{P}}\times \mathbf{G}$ & $\rightarrow $ & $\mathcal{\mathbf{P}}$% \end{tabular}$$ $$\Lambda (u\cdot \lambda )=\Lambda (u)\cdot \tau (\lambda )$$ We would like to find out the group structure $\mathbf{G}$ of this bundle. Assuming that $\mathcal{\mathbf{B}}$ has a metric with signature $(+,-,-,-)$ then the functions $ a_{ji}(x)$ defined above are in $\mathbf{O}(1,3)$ . Since $\mathcal{\mathbf{B}}$ is not orientable then $\det a_{ji}(x)=\pm 1$. The elements that cover $x$ differ by an element in $\mathbf{Z_{2}}$ which can be represented by $T$ or $P$ or any other element with determinant -1 and involutive. The first matrix is related to time non-orientability, the second to space non-orientability. These elements describe global actions on the manifold. By construction, the fiber $Y_{x}\simeq \mathbf{O}_{+}^{\uparrow }\smallsmile \mathbf{O}_{+}^{\uparrow }$. So from the above discussion we should expect that the two copies differ by an element of determinant -1. In fact let $a$ be such an element with $a^{2}=1$. Using the map $\mathbf{Z}_{2}\times $ $\mathcal{\mathbf{B}}^c\rightarrow \mathcal{\mathbf{B}}$ , a well defined multiplication, then we can write $$x_{2}^{\prime }=a\cdot x_{1}^{\prime } \: ,$$ where the local coordinates are used , that is $$\begin{aligned} x_{1}^{\prime } &:&V_{i}\rightarrow E_{i} \nonumber \\ x_{2}^{\prime } &:&V_{j}\rightarrow E_{j} \: .\end{aligned}$$ On the manifold $\mathcal{\mathbf{B}}$ these charts get projected to a single chart around $x$: $$\begin{aligned} x_{1}^{\prime } &:&\text{ \ \ \ }y_{1}:U_{i}\rightarrow E_{i} \nonumber \\ x_{2}^{\prime } &:&\text{ \ \ \ }y_{2}:U_{j}\rightarrow E_{j} \: .\end{aligned}$$ Therefore, the frames  $u_{1}(x_{1}^{\prime })$ and $u_{2}(x_{2}^{\prime })$ become two frames at the same point $x$. They are related through the transformation $\frac{\partial y_{1}^{\mu }}{\partial y_{1}^{\upsilon }} |_{x} $. But $y_{1}^{\mu }(x)=x_{1}^{\prime \mu }(x)=a_{\epsilon }^{-1}z_{1}^{^{\prime }\varepsilon }(x_{2}^{\prime })$ and $y_{2}^{\upsilon }(x)=y_{2}^{^{\prime }\upsilon }(x_{2}^{\prime })$, this implies that $$\frac{\partial y_{1}^{\mu}}{\partial y_{2}^{\nu}} \; = \; \left ( a^{-1} \right )_{\epsilon}^{\mu} \; \frac{\partial {z_{2}^{\prime }}^{\epsilon}}{\partial {y_{2}^{\prime }}^{\nu} \; \mid _{x_{2}^\prime} }\: .$$ Since $\frac{\partial z_{2}^{\prime }}{\partial y_{2}^{\prime }}\in \mathbf{O}_{-}^{\uparrow }(1,3)$, we conclude that the transition functions of $\mathcal{\mathbf{P}}$ are in $\mathbf{O}_{+}^{\uparrow }(1,3)\smallsmile a\cdot \mathbf{O}_{+}^{\uparrow }(1,3)$ . There are two possible choices either $a\in \mathbf{O}_{-}^{\uparrow }(1,3)$ or $a\in \mathbf{O}_{-}^{\downarrow }(1,3)$ . Depending on which element $a$ we choose, we end up with different actions on $\mathcal{\mathbf{P}}$ which are equivalent. So the group $\mathbf{G}$ can be either $\mathbf{O}_{+}^{\uparrow }(1,3)\smallsmile $ $\mathbf{O}_{-}^{\uparrow }(1,3)$ or $ \mathbf{O}_{+}^{\uparrow }(1,3)\smallsmile $ $\mathbf{O}_{-}^{\downarrow }(1,3)$. Hence, if we have started with $\mathbf{O}_{+}^{\uparrow }(1,3)\smallsmile $ $\mathbf{O}_{+}^{\downarrow }(1,3)$ as the group of symmetry of $\overline{\mathcal{\mathbf{P}}}$ , we would have obtained $\mathbf{O}(1,3)$ as the group of action of $\mathcal{\mathbf{P}}$. On the level of fibers, the map $\Lambda$ is easily seen to be a 2-1 map similar to $p$ by construction. WEAKLY-INEQUIVALENT PIN STRUCTURES =================================== In the following , the topological group $\Gamma $ can be taken to be $ \mathbf{Pin}(p,q)$ and the group $\mathbf{G}$ the orthogonal group $\mathbf{O}(p,q)$. Let $(\mathcal{\mathbf{P}}$,$\pi ,\mathcal{\mathbf{B}},\mathbf{G})$ be a Principal bundle. Let $\Gamma $ be a double covering for $\mathbf{G}$. Then we have the following exact sequence: $$1\rightarrow K\rightarrow \Gamma \rightarrow \mathbf{G}\rightarrow 1 \: ,$$ where $K=\mathbf{Z}_{2}$. And let $$\rho :\Gamma \rightarrow \mathbf{G}$$ be the 2-1 map. We denote by $g_{ji} : U_{i}\cap U_{j} \rightarrow \mathbf{G}$ the transition functions of the bundle $\mathcal{\mathbf{P}}$.  The principal bundle $(\widetilde{\mathcal{\mathbf{P}}}$,$\widetilde{\pi },\mathcal{\mathbf{B}},\mathbf{G})$ is called a [*$\Gamma$ -structure on $\mathcal{\mathbf{P}}$*]{} iff there is a bundle map $$\begin{aligned} \Phi &:&\widetilde{\mathcal{\mathbf{P}}}\rightarrow \mathcal{\mathbf{P}} \nonumber \\ \Phi (\widetilde{u}\cdot \gamma ) &=&\Phi (\widetilde{u})\cdot \rho (\gamma ) \: \: \: \: \text{ for }\widetilde{u}\in \widetilde{\mathcal{\mathbf{P}}}\text{\ and }\gamma \in \Gamma\end{aligned}$$ and the functions $g_{ji}$ can be lifted to the transition functions of $ \widetilde{\mathcal{\mathbf{P}}}$ , i.e., we have $$\begin{tabular}{lll} $\gamma _{ji}$ & & $\Gamma $ \\ & $\nearrow $ & $\downarrow $ \\ $U_{ji}$ & $\overset{g_{ji}}{\rightarrow }$ & $\mathbf{G}$ \end{tabular}$$ $$\begin{aligned} \rho (\gamma _{ji}) &=& g_{ji} \nonumber \\ \gamma _{ij}\gamma _{kj}^{-1}\gamma _{ki} &=&1 \: .\end{aligned}$$ The last relation is the cocycle condition. This enables us to define a Ĉech-Cohomology on $\mathcal{\mathbf{B}}$ with coefficients in $\Gamma $: $$\begin{aligned} d &:&C^{1}\longrightarrow C^{2} \nonumber \\ (d\gamma )(ijk) &=&\gamma _{jk}\gamma _{ik}^{-1}\gamma _{ij} \: .\end{aligned}$$ Hence $\left[ \gamma _{ij}\right] \in H^{1}(\mathcal{\mathbf{B}},\Gamma )$ and $\left[ g_{ij} \right] \in H^{1}(\mathcal{\mathbf{B}},\mathbf{G}).$ The equivalence class $\left[ \gamma _{ij}\right] $ is defined by $$\left[ \gamma _{ij}\right] =\{\gamma _{jk}^{\prime }:\lambda _{i}\gamma _{ij}\lambda _{j}^{-1}\}$$ with $$\lambda _{i}:U_{i}\rightarrow \Gamma \: .$$ It is immediately clear from the above, that $\rho $ induces a map $\rho ^{\ast }$ between cohomologies: $$\begin{aligned} \rho ^{\ast } &:&H^{1}(\mathcal{\mathbf{B}},\Gamma )\rightarrow H^{1}(\mathcal{\mathbf{B}},\Gamma ) \nonumber \\ \rho ^{\ast }\left[ \gamma _{ij}\right] &=&[\rho \left[ \gamma _{ij}\right] ]=\left[ g_{ij}\right] \: .\end{aligned}$$ This map is 1-1 and onto if a $\Gamma -$ structure exists. Now if we assume that the manifold $\mathcal{\mathbf{B}}$ is non-orientable. Then $\mathcal{\mathbf{B}}$ has an orientable double cover $\mathcal{\mathbf{B}}^c$. Let {$U_{1}\}$ be a cover for $\mathcal{\mathbf{B}}$ , and $$\varphi _{i}:E_{i}\rightarrow U_{i}$$ be the coordinate maps of $\mathcal{\mathbf{B}}$ where $E_{i}\subset R^{n}$. The transition maps $a_{ij}$ are given by $$a_{ij}:\varphi _{j}\varphi _{i}^{-1}:E_{i}\rightarrow E_{j}\rightarrow E_{i}\rightarrow E_{j} \: .$$ Now let $x\in U_{i}\smallfrown U_{j}$ and define $\theta _{ij}$ to be the normalized determinant of the Jacobian of the transition functions $a_{ij}:$ $$\begin{aligned} \theta _{ij} &:&U_{i}\smallfrown U_{j}\rightarrow \mathbf{Z}_{2} \nonumber \\ \theta _{ij}(x) &=&\det J[a_{ij}(x)] \: .\end{aligned}$$ Using the properties of the determinant, we can see that $\theta _{ij}$ is a representative of an element of $H^{1}(\mathcal{\mathbf{B}},\mathbf{Z}_{2})$ . We construct $ \mathcal{\mathbf{B}}^c$ through these cocycles: $$p:\mathcal{\mathbf{B}}^c\rightarrow \mathcal{\mathbf{B}} \: ,$$ where $$\mathcal{\mathbf{B}}^c=\left\{ \left( x,\theta _{ij}(x)\right) ,x\in U_{ij}\right\} \: .$$ Therefore, $p$ induces a bundle structure. In fact $(\mathcal{\mathbf{B}}^c ,p,\mathcal{\mathbf{B}},\mathbf{Z}_{2})$ is a principal bundle. Note that this fiber bundle is connected if the base space $\mathcal{\mathbf{B}}$ is connected. If $\mathcal{\mathbf{B}}$ were orientable then $ \mathcal{\mathbf{B}}^c$ would simply be a trivial double covering.  Next, we let $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ be two $\Gamma $-structures on $\mathcal{\mathbf{P}}$ and $\Psi $ is a group isomorphism of $\Gamma $ such that $$\begin{tabular}{lllll} $\Psi :$ & $\Gamma $ & $\rightarrow $ & & $\Gamma $ \\ & $\rho \searrow $ & & $\swarrow \rho $ & \\ & & $\mathbf{G}$ & & \end{tabular}$$ commutes. $\widetilde{\mathcal{\mathbf{P}}}$  and $\widetilde{\mathcal{\mathbf{P}}}^{\prime }$  are said to be [*weakly equivalent*]{}, $\widetilde{\mathcal{\mathbf{P}}}$ $\simeq _{W}\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ , iff we have the following commutative diagram $$\begin{tabular}{lllll} $\widetilde{\mathcal{\mathbf{P}}}$ & & $\overset{\Theta \simeq }{\rightarrow }$ & & $ \widetilde{\mathcal{\mathbf{P}}}^{\prime }$ \\ & $\Phi \searrow $ & & $\swarrow \Phi ^{\prime }$ & \\ & & $\mathcal{\mathbf{P}}$ & & \end{tabular}$$ and $$\Theta (\widetilde{u}\cdot \gamma )=\Theta (\widetilde{u})\cdot \Psi (\gamma )\text{ \ \ \ \ \ \ }\widetilde{u}\in \widetilde{\mathcal{\mathbf{P}}},\gamma \in \Gamma \: .$$ If $\Psi =1,$ we say the [*equivalence is strong*]{} [@greub].  We believe that weak equivalence is the concept most appropriate from a physical point of view. We have seen above that such fields are $\Psi $-related, moreover this will enable us to concern ourselves only with fields defined on orientable manifolds.  Therefore if we are interested in knowing how many possible physical fields we can have on a given manifold, it is irrelevant how we represent the group of actions on the spinor field as long as they differ by an isomorphism.  We show next that the map $\Psi $ is an involution.  To show this, we observe that the map $\rho $ is a 2-1 map. Let $\gamma ^{1}$ and $\gamma ^{2}$ the two elements that cover $g$. Since $\rho \Psi (\gamma ^{1})=g$ and $ \rho (\gamma ^{2})=g$, we must either have $\Psi = 1$ or $$\begin{aligned} \Psi (\gamma ^{1}) &=&\gamma ^{2} \nonumber \\ \text{and }\Psi (\gamma ^{2}) &=&\gamma ^{1} \: .\end{aligned}$$ The map $\Psi $ acts as a permutation among the couple that cover a given element in $\mathbf{G}$.   Hence the involution follows immediately. A result that follows from the previous statement is that $\Psi |_{K}=1$ .  This is immediate from the fact that $K=\mathbf{Z}_{2}$. ### **Theorem II** {#theorem-ii .unnumbered} ** Let $\widetilde{\mathcal{\mathbf{P}}},\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ be $\Gamma$ -structures on $\mathcal{\mathbf{P}}$, 1. Let $\widetilde{\mathcal{\mathbf{P}}}\simeq _{W}\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ . If $ \left\{ \gamma _{ij}\right\} $ are transition functions for $\widetilde{\mathcal{\mathbf{P}}}$ associated with $\left\{ g_{ij}\right\} $ for $\mathcal{\mathbf{P}}$, then $\gamma _{ji}^{\prime }=$ $\Psi \circ \gamma _{ji}$ are transition functions for $\widetilde{\mathcal{\mathbf{P}}}^{\prime }$. 2. If $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ have transition functions $\left\{ \gamma _{ij}\right\} $ and $\left\{ \gamma _{ij}^{\prime }\right\} $ with $\gamma _{ij}^{\prime }=$ $\Psi \circ \gamma _{ji}$ then $\widetilde{\mathcal{\mathbf{P}}}\simeq _{W}\widetilde{\mathcal{\mathbf{P}}}^{\prime }$. To show this we observe that: 1. Associated with $\{ \gamma_{ij} \}$ is a system $\widetilde{\varphi }_{i}$ of charts $$\begin{aligned} \widetilde{\varphi }_{i} &:&U_{i}\times \Gamma \rightarrow \widetilde{\mathcal{\mathbf{P}}} \text{, with } \nonumber \\ \gamma _{ij} &:&U_{i}\smallfrown U_{j}\rightarrow \mathbf{G}\ \nonumber \\ \text{given by } \widetilde{\varphi }_{i}\widetilde{\varphi }_{j}^{-1}(\widetilde{e}) &=& \widetilde{e}\cdot \widetilde{\gamma }_{ij} \: .\end{aligned}$$ Then charts $\widetilde{\varphi }_{i}^{\prime }:$ $U_{i}\times \Gamma \rightarrow \widetilde{\mathcal{\mathbf{P}}}^{\prime }$ defined by $\widetilde{\varphi } _{i}^{\prime } = \Theta \circ \widetilde{\varphi }_{i}$ have transition functions $\gamma_{ij}^{\prime} : U_{i} \cap U_{j} \rightarrow \mathbf{G}$ given by $$\begin{aligned} \gamma _{ij}^{\prime } &=&\widetilde{\varphi }_{i}\widetilde{\varphi }% _{j}^{-1}(\widetilde{e}^{\prime }) \nonumber \\ &=&\Theta \circ \widetilde{\varphi }_{i}\circ \widetilde{\varphi }% _{j}^{-1}\circ \Theta ^{-1}(\widetilde{e}^{\prime }) \nonumber \\ &=&\Theta \lbrack \widetilde{e}\cdot \gamma _{ij}] \nonumber \\ &=&\widetilde{e}^{\prime }\cdot \Psi (\gamma _{ij})\end{aligned}$$ $$\Longrightarrow \gamma _{ij}^{\prime }=\Psi (\gamma _{ij}) \: .$$ That $\left\{ \gamma _{ij}^{\prime }\right\} $* satisfy consistency relations and* $\rho \circ \gamma _{ij}^{\prime }=g_{ij}$  is immediate. 2. Proof relies here on showing that $$\Theta |_{\widetilde{\pi }^{-1}U_{i}}\equiv \widetilde{\varphi }_{i}% \widetilde{\varphi }_{i}^{-1} \: ,$$ is well defined. We set, $$\begin{aligned} \widetilde{\varphi }_{1}^{\prime }(u,1) &=&e_{1}^{\prime }, \: \: \: \widetilde{\varphi }_{1}(u,1)=e_{1} \\ \widetilde{\varphi }_{2}^{\prime }(u,1) &=&e_{2}^{\prime }=e_{1}^{\prime }\cdot \gamma ^{\prime },\gamma ^{\prime }=\pm 1;\text{ \ }\widetilde{% \varphi }_{2}(u,1)=e_{2}=e_{1}\cdot \gamma ,\gamma =\pm 1\end{aligned}$$ $$\widetilde{\varphi }_{1}^{\prime }\widetilde{\varphi }_{2}^{\prime -1}( \widetilde{e}_{2}^{\prime })=\widetilde{e}_{2}^{\prime }\cdot \gamma _{12}^{\prime }=\widetilde{e}_{1}^{\prime }\Longrightarrow \gamma _{12}^{\prime }=\gamma .$$ Similarly $\gamma _{_{12}}=\gamma .$ Then on $ \widetilde{\pi }^{-1}\left( U_{i}\smallfrown U_{j}\right) $* ,* $ \Theta $* is well defined if* $\widetilde{\varphi }_{1}\widetilde{% \varphi }_{1}^{-1}=\widetilde{\varphi }_{2}\widetilde{\varphi }_{2}^{-1}.$$$\begin{aligned} \widetilde{\varphi }_{1}\widetilde{\varphi }_{1}^{-1}(\widetilde{e}_{1}) &=&% \widetilde{e}_{1} \nonumber \\ \widetilde{\varphi }_{2}\widetilde{\varphi }_{2}^{-1}(e_{1}) &=&\widetilde{ \varphi }_{2}^{\prime }\widetilde{\varphi }_{2}^{-1}\left( \widetilde{e} _{2}\cdot \gamma \right) =\widetilde{\varphi }_{2}^{\prime }(u,\gamma )= \widetilde{e}_{2}^{\prime }\cdot \gamma =\widetilde{e}_{1}^{\prime }\cdot \gamma ^{\prime }\gamma\end{aligned}$$ Finally, $\gamma _{12}=\gamma _{12}^{\prime }$ , since $ \Psi $ is an isomorphism : $$\begin{aligned} &\Longrightarrow &\Psi (1)=1,\Psi (-1)=-1 \\ &\Longrightarrow &\gamma ^{\prime }\gamma =\gamma _{12}^{2}=1, \end{aligned}$$ which implies that $\Theta$ is well defined. Hence we proved our statement. Now we try to show that if only weak equivalence is imposed on $\Gamma $-structures on $\mathcal{\mathbf{P}}$, then the number of inequivalent $\Gamma $-structures will no longer be given by $H^{1}(\mathcal{\mathbf{B}},\mathbf{Z}_{2})$ but instead by $H^{1}({ \mathcal{\mathbf{B}}}^{c},\mathbf{Z}_{2})$ . Note that the covering induced by $p$ is a regular covering. Hence $p_{\ast }\left( \pi _{1}(\mathcal{\mathbf{B}}^c,\ast )\right) \lhd \pi _{1}(\mathcal{\mathbf{B}},\ast )$ and this covering is a $\mathbf{Z}_{2}$-covering. We can see right away that the map $\Psi $ has similar properties on the group action level. As an example, take $\mathcal{\mathbf{B}}=RP_{2}$ . Then $\mathcal{\mathbf{B}}^c=\mathbf{S}^{2}.$ $$\begin{aligned} \pi _{1}(\mathcal{\mathbf{B}}) &=&\mathbf{Z}_{2}, \\ \pi _{1}(\mathcal{\mathbf{B}}^c) &=&1 .\end{aligned}$$ The space $\mathcal{\mathbf{B}}$ has in this case two strongly-inequivalent $\mathbf{Pin}^{-}(2)$ structures. If we let $w$ denotes the volume element of the Clifford algebra associated to $ \mathbf{Pin}^{-}(3)$, then the two $\mathbf{Pin}^{-}(2)$ structures are obtained through the following coverings $$\mathbf{Pin}^{-}(3)/\left\{ 1,\pm w\right\} \rightarrow \mathbf{O}(3)/\left\{ 1,-1\right\}$$ If we require only weak equivalence, then both structures become equivalent with $$\begin{aligned} \Psi (w) &=&-w \\ \Psi (1) &=&1 \: ,\end{aligned}$$ and the above two coverings reduce to a single one, namely $$\mathbf{Pin}^{-}(3)\rightarrow \mathbf{O}(3)\rightarrow \mathbf{S}^{2} \: .$$ So imposing weak-equivalence is equivalent to factoring out the effect of the non-orientability of the manifold $\mathcal{\mathbf{B}}$. Hence we should expect that $H^{1}( \mathcal{\mathbf{B}}^c,\mathbf{Z}_{2})$ gives distinct physical $\Gamma $-structures on $\mathcal{\mathbf{B}}$. ### **Theorem III** {#theorem-iii .unnumbered} [*[$\mathcal{\mathbf{P}}^{c}$ is a $\Gamma $-structure on $\mathcal{\mathbf{P}}$. The induced Principal bundle $p^{-1}\mathcal{\mathbf{P}}\equiv \mathcal{\mathbf{P}}^{c}$  is a trivial double covering for $\mathcal{\mathbf{P}}$. Similarly,  $p^{-1}\widetilde{\mathcal{\mathbf{P}}}\equiv \widetilde{\mathcal{\mathbf{P}}^{c}}$ is a trivial double covering for $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{\mathcal{\mathbf{P}}^{c}}$ is a $\Gamma $-structure on $\mathcal{\mathbf{P}}^{c}$ . ]{}*]{} We start first by showing that $\mathcal{\mathbf{P}}^{c}$ is a trivial covering of $\mathcal{\mathbf{P}}$. $\mathcal{\mathbf{P}}^{c}$ is by definition the Principal bundle induced by $p$, $$\begin{tabular}{llll} $F:$ & $\mathcal{\mathbf{P}}^{c}$ & $\longrightarrow $ & $\mathcal{\mathbf{P}}$ \\ $\overline{\pi }$ & $\downarrow $ & & $\downarrow \pi $ \\ & $\mathcal{\mathbf{B}}^c$ & $\overset{p}{\longrightarrow }$ & $\mathcal{\mathbf{B}}$ \end{tabular}$$ Therefore, $\mathcal{\mathbf{P}}^{c}$ and $\mathcal{\mathbf{P}}$ have the same group structure [@steenrod]. Moreover, we have $$\overline{g}_{ji}(x^{\prime })=g_{ji}\left( p(x^{\prime })\right) \text{ \ \ for }x^{\prime }\in \mathcal{\mathbf{B}}^c$$ The map F is a 2-1 map. Now consider the bundle $\mathcal{\mathbf{P}}\times \mathbf{Z}_{2}$ defined such that $$\begin{tabular}{lll} $\mathcal{\mathbf{P}}\times \mathbf{Z}_{2}$ & $\overset{\Phi }{\longrightarrow }$ & $\mathcal{\mathbf{P}}^{c}$ \\ $\downarrow p^{\prime }$ & & $\downarrow p$ \\ & & \\ $\mathcal{\mathbf{B}}^c$ & $\overset{id}{\rightarrow }$ & $\mathcal{\mathbf{B}}^c$ \end{tabular}$$ such that $$\begin{aligned} p^{\prime }(u,1) &=&x_{1}^{\prime } \\ p^{\prime }(u,-1) &=&x_{2}^{\prime } \\ p(x_{1}^{\prime }) &=&p(x_{2}^{\prime })=x\end{aligned}$$ We claim that $\mathcal{\mathbf{P}}\times \mathbf{Z}_{2}\simeq \mathcal{\mathbf{P}}^{c}$ , i.e., $\Phi $ is a bundle isomorphism. We construct $\Phi $ explicitly. Locally, we have $$\begin{aligned} \Phi &:&\mathcal{\mathbf{P}}\times \mathbf{Z}_{2}\longrightarrow \mathcal{\mathbf{P}}^{c} \nonumber \\ & & \left( x,u(x),1\right) \longrightarrow \left( x_{1}^{^{\prime }},u\right) \nonumber \\ & & \left( x,u,-1\right) \longrightarrow \left( x_{2}^{^{\prime }},u\right)\end{aligned}$$ Therefore, $\overline{\pi }\Phi (x,u,1)=\overline{\pi }($ $x_{1}^{\prime },u)=x_{1}^{\prime }=p^{\prime }(x,u,1)$  and a similar relation holds for $ (x,u,1)$. Hence, the above diagram commutes and $\Phi $ carries fibers to fibers. $\Phi $ can then be considered to be a bundle map induced by the identity. A similar diagram for $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{ \mathcal{\mathbf{P}}^c}$ shows that $\widetilde{\mathcal{\mathbf{P}}^{c}}$ is a trivial double cover of $ \widetilde{\mathcal{\mathbf{P}}}$ . The following diagram gives all possible relations among the transition functions and can be used to prove that $\widetilde{\mathcal{\mathbf{P}}^{c}}$ is a $ \Gamma $-structure on $\mathcal{\mathbf{P}}^{c}$. $$\begin{tabular}{lllll} $\widetilde{\mathcal{\mathbf{P}}^{c}}$ & & $\overset{\overline{\Phi }}{\rightarrow }$ & & $\mathcal{\mathbf{P}}^{c}$ \\ & $\widetilde{\overline{\pi }}\searrow $ & & $\swarrow $ $\overline{\pi }$ \ & \\ & & $\mathcal{\mathbf{B}}^c$ & & \\ $\overline{F}\downarrow $ & & $\downarrow $ & & $\downarrow F$ \\ & & $\mathcal{\mathbf{B}}$ & & \\ & $\nearrow $ & & $\nwarrow $ & \\ $\mathcal{\mathbf{P}}^{c}$ & & $\overset{\Phi }{\rightarrow }$ & & $\mathcal{\mathbf{P}}$% \end{tabular}$$ Considering $\mathcal{\mathbf{P}}^{c}$ and $\mathcal{\mathbf{P}} $ as base spaces, the map $\overline{\Phi }$ is induced by $\Phi $. Hence it is a bundle map. We need to check that it is equivariant, i.e., $$\overline{\Phi }\left( \widetilde{\overline{u}}\cdot \gamma \right) = \overline{\Phi }(\widetilde{\overline{u}})\cdot \rho (\gamma )\text{ \ \ \ \ \ for all }\gamma \in \Gamma .$$ We write $\overline{\Phi }$ explicitly. A map that satisfies all the properties of the above diagram is $$\overline{\Phi }(x^{\prime };\widetilde{\overline{u}})=\Phi (x^{\prime }; \widetilde{u}),$$ where $$p(x^{\prime }) = x$$ and $\widetilde{\overline{u}}$ ,$\widetilde{u}$ are the same pinor frames. Since multiplication by $\gamma $ leaves the fiber invariant, it is trivially true that $\overline{\Phi }$ is an equivariant map since $\Phi $ is itself equivariant.  Next we define the difference class of two structures. $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ are two $\Gamma $-structures on $\mathcal{\mathbf{P}}$, where the group actions differ by an isomorphism $\Psi $ . The [*difference class*]{} $\delta (\widetilde{\mathcal{\mathbf{P}}},\widetilde{\mathcal{\mathbf{P}}}^{\prime })$ is defined to be $$\delta _{ji}(x)=\gamma _{ji}(x)\Psi (\gamma _{ji}^{\prime -1}(x)),\text{ \ \ \ \ x}\in U_{ij} \: .$$ Similarly, we can define $\overline{\delta }$ for the respective double covers. The difference class $\delta (\widetilde{\mathcal{\mathbf{P}}},\widetilde{\mathcal{\mathbf{P}}}^{\prime })$ can be shown to be an element of H$ ^{1}(\mathcal{\mathbf{B}},\mathbf{Z}_{2}).$ Similarly, $\overline{\delta }(\widetilde{\mathcal{\mathbf{P}}^{c}}, \widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime })$ is an element of H$^{1}(\mathcal{\mathbf{B}}^c ,\mathbf{Z}_{2}).$   By definition, we have $$\delta _{ij}(x)=\gamma _{ij}(x)\Psi (\gamma _{ij}^{\prime }(x)^{-1}) \: ,$$ and $$\rho (\gamma _{ij})=\rho (\gamma _{ij}^{\prime })=g_{ij} \: .$$ This implies that $$\begin{aligned} \rho (\delta _{ij}) &=&1 \\ &\Longrightarrow &\delta _{ij}(x)\in \mathbf{Z}_{2}\end{aligned}$$ i.e., $\delta _{ij}$ is in the center of $\Gamma $ and $$\begin{aligned} (d\delta )(ijk) &=&\delta _{jk}\delta _{ik}^{-1}\delta _{ij}\nonumber \\ &=&\gamma _{ji}\Psi (\gamma _{jk}^{\prime -1})(\gamma _{ik}\Psi (\gamma _{ik}^{\prime -1}))\gamma _{ij}\Psi (\gamma _{ij}^{\prime -1}) \nonumber \\ &=&\gamma _{jk}\Psi (\gamma _{jk}^{\prime -1})(\Psi (\gamma _{ik}^{\prime })\gamma _{ik}^{-1})\gamma _{ji}\Psi (\gamma _{ij}^{\prime -1})\nonumber \\ &=&\gamma _{jk}(\gamma _{ij}\Psi (\gamma _{ij}^{\prime -1}))\Psi (\gamma _{jk}^{\prime -1})(\Psi (\gamma _{ik}^{\prime })\gamma _{ik}^{-1})\text{ since }\delta _{ij}\in C(\Gamma )\nonumber \\ &=&\gamma _{jk}\gamma _{ij}\Psi (\gamma _{ij}^{\prime -1}\gamma _{jk}^{\prime -1}\gamma _{ik}^{\prime })\gamma _{ik}^{-1} \nonumber \\ &=&1 .\end{aligned}$$ Hence $\delta _{ij}\in $ $H^{1}(\mathcal{\mathbf{B}},\mathbf{Z}_{2})$ . A similar proof works for $ \overline{\delta }_{ij}$.  The difference class can be used to define an equivalence relation among the $\Gamma -$ structures.  In fact, we have $\widetilde{\mathcal{\mathbf{P}}}\simeq _{W}\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ iff $\overline{\delta }(% \widetilde{\mathcal{\mathbf{P}}^{c}},\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime })=1$ .  To show this, suppose that $\widetilde{\mathcal{\mathbf{P}}}$ and $\widetilde{\mathcal{\mathbf{P}}}^{\prime }$ are weakly -equivalent, then $$\gamma _{ij}(x)=\Psi (\gamma _{ij}^{\prime }(x)) ,$$ with  $\Psi ^{2}=1$. Moreover we have $$\begin{aligned} \overline{\gamma }_{ij}(x) &=&\gamma _{ij}(p(x^{\prime })) \end{aligned}$$ since $$\overline{g}_{ij}( x^{\prime } ) \; =\; g_{ij}(p(x^{\prime })) ,$$ which implies that $$\overline{\gamma }_{ij}(x^{\prime })=\Psi (\overline{\gamma }_{ij}^{\prime }(x)) .$$ Hence the difference class becomes $$\begin{aligned} \overline{\delta }_{ij}(x) &=&\overline{\gamma }_{ij}(x^{\prime })\Psi ( \overline{\gamma }_{ij}^{\prime -1}(x^{\prime }))\nonumber \\ &=&\overline{\gamma }_{ij}(x^{\prime })(\overline{\gamma }_{ij}^{\prime -1}(x^{\prime })) \nonumber \\ &=&1 .\end{aligned}$$ Now suppose that $\overline{\delta }(\widetilde{\mathcal{\mathbf{P}}^{c}},\widetilde{ \mathcal{\mathbf{P}}^{c}}^{\prime })=1$. Hence there exists $\lambda _{i} : p^{-1}(U_{i})\rightarrow \Gamma $ such that $$d\lambda (ij)=\overline{\delta }(ij) \: .$$ This is a Čech-coboundary condition, therefore we have $$\overline{\gamma }_{ij}(x^{\prime })=\lambda _{i}^{-1}(x^{\prime })\Psi ( \overline{\gamma }_{ij}^{\prime }(x^{\prime }))\lambda _{j}(x^{\prime }) \: .$$ Now we try to construct locally the bundle isomorphism $$\Theta : \widetilde{\mathcal{\mathbf{P}}}\rightarrow \widetilde{\mathcal{\mathbf{P}}}^{\prime }$$ such that $$\begin{aligned} \Theta (\widetilde{u}\cdot \gamma ) &=&\Theta (\widetilde{u})\cdot \Psi (\gamma )\end{aligned}$$ and $$\begin{aligned} \Phi \circ \Theta &=&\Phi ^{\prime } \; .\end{aligned}$$ First we have the following commutative diagram: $$\begin{tabular}{lllll} $\widetilde{\mathcal{\mathbf{P}}^{c}}$ & & $\overset{\overline{\Theta }}{\rightarrow }$ & & $\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime }$ \\ & $\overline{\Phi }\searrow $ & & $\swarrow \overline{\Phi }^{\prime }$ & \\ & & $\mathcal{\mathbf{P}}^{c}$ & & \\ $\downarrow $ & & $\downarrow $ & & $\downarrow $ \\ & & $\mathcal{\mathbf{P}}$ & & \\ & $\Phi \nearrow $ & & $\nwarrow \Phi ^{\prime }$ & \\ $\widetilde{\mathcal{\mathbf{P}}}$ & & $\overset{\Theta }{\rightarrow }$ & & $\widetilde{% \mathcal{\mathbf{P}}^{\prime }}$ \end{tabular}$$ It should be clear from this diagram that locally $\Theta $ and $\overline{\Theta }$ are the same. Hence a construction of $\overline{\Theta }$ will immediately give one for $\Theta $ . Let $V_{i}=\widetilde{\overline{\pi }} (p^{-1}(U_{i}))$ and define $\overline{\Theta }$ locally by $$\overline{\Theta }_{i}:V_{i} \rightarrow \widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime }$$ $$\widetilde{\overline{u}}\rightarrow \sigma _{i}^{\prime }(\widetilde{ \overline{\pi }}(\widetilde{\overline{u}}))\cdot \lambda _{i}\Psi (\gamma _{i}(\widetilde{\overline{u}}))$$ where $\sigma _{i}^{\prime }$ is a local cross section of $\widetilde{ \mathcal{\mathbf{P}}^{c}}^{\prime }$ and $\gamma _{i}$ is an element of $\Gamma $ such that $$\widetilde{\overline{u}}=\sigma _{i}(\widetilde{\overline{\pi }}(\widetilde{ \overline{u}}))\cdot \gamma _{i}(\widetilde{\overline{u}})$$ If $\gamma \in \Gamma $ , then $$\begin{aligned} \overline{\Theta }_{i}(\widetilde{\overline{u}}\cdot \gamma ) &=&\sigma _{i}^{\prime }(\widetilde{\overline{\pi }}(\widetilde{\overline{u}}\cdot \gamma ))\cdot \lambda _{i}\Psi (\gamma _{i}(\widetilde{\overline{u}}\cdot \gamma )) \nonumber \\ &=&\overline{\Theta }_{i}(\widetilde{\overline{u}}).\Psi (\gamma ) \: .\end{aligned}$$ This map is well defined globally. On intersection $V_{i}\smallfrown V_{j}$ $$\begin{aligned} \sigma _{i}(\widetilde{\overline{\pi }}(\widetilde{\overline{u}}))\cdot \lambda _{i}\Psi (\gamma _{i}(\widetilde{\overline{u}}) &=&\sigma _{i}( \widetilde{\overline{\pi }}(\widetilde{\overline{u}}))\gamma _{ji}^{^{\prime }}\lambda _{i}\Psi (\gamma _{i}) \nonumber \\ &=&\sigma _{i}(\widetilde{\overline{\pi }}(\widetilde{\overline{u}}))\lambda _{j}\Psi (\gamma _{ji})\Psi (\gamma _{i}) \nonumber \\ &=&\sigma _{i}(\widetilde{\overline{\pi }}(\widetilde{\overline{u}}))\lambda _{j}\Psi (\gamma _{j}),\end{aligned}$$ as it should be. ### **Theorem IV** {#theorem-iv .unnumbered} [*[Let $\widetilde{\mathcal{\mathbf{P}}}$ be a $\Gamma$-structure on $\mathcal{\mathbf{P}}$. $\widetilde{\mathcal{\mathbf{P}}^{c}}$ is the corresponding double cover. Then for each element $\overline{\zeta }\in H^{1}(\mathcal{\mathbf{B}}^c,\mathbf{Z}_{2})$ there exist a non weakly-equivalent $\Gamma -structure$ $\widetilde{\mathcal{\mathbf{P}}}^{\prime }.$]{}*]{} Let $\overline{\zeta }_{ij}$ be a representation of $\overline{\zeta }$ and $\Psi $ is an isomorphism as above. Define the following functions: $$\overline{\gamma }_{ij}(x^{\prime })=\overline{\zeta }_{ij}^{-1}\cdot ( \overline{\gamma }_{ij}^{\prime }(x^{\prime })) \: .$$ They clearly satisfy the cocycle condition and hence they form the transition functions of a principal bundle which we call $(\widetilde{ \mathcal{\mathbf{P}}^{c}}^{\prime },\widetilde{\overline{\pi }}^{\prime },\mathcal{\mathbf{B}}^c ,\Gamma )$ . Note also that $$\overline{\delta }_{ij}=\overline{\gamma }_{ij}(\overline{\gamma } _{ij}^{^{\prime }})=\overline{\zeta }_{ij} \; ,$$ and $$\rho (\overline{\gamma }_{ij})=\rho (\overline{\gamma }_{ij}^{\prime })= \overline{g}_{ij}.$$ Now, to get the Principal bundle $(\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime }, \widetilde{\overline{\pi }}^{\prime },\mathcal{\mathbf{B}}^c,\Gamma )$ we use the fact that $(\mathcal{\mathbf{B}}^c,p,\mathcal{\mathbf{B}},\mathbf{Z}_{2})$ is a Principal bundle with transition $ \theta _{ij}(x)\in \mathbf{Z}_{2}$ . Hence the following diagram commutes: $$\begin{tabular}{lll} $\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime }$ & $\longrightarrow $ & $\mathcal{\mathbf{B}}^c$ \\ & & \\ & $p^{\prime }\searrow $ & $\downarrow $ \\ & & $\mathcal{\mathbf{B}}$ \end{tabular}$$ Therefore $(\widetilde{\mathcal{\mathbf{P}}^{c}}^{\prime },\widetilde{\overline{\pi }} ^{\prime },\mathcal{\mathbf{B}}^c,\Gamma )$ is a Principal bundle with transition functions $$\psi _{ij}(x)=\theta _{ij}(x)\cdot \overline{\gamma }_{ij}^{\prime }(x^{\prime }) \: .$$ Here, we have used the trivial extension of $\Gamma $ , i.e., $$1\rightarrow \mathbf{Z}_{2}\rightarrow \mathbf{Z}_{2}\otimes \Gamma \rightarrow \Gamma \rightarrow 1 \: .$$ The bundle $ \widetilde{\mathcal{\mathbf{P}}}^{\prime} $ is constructed with the same transition functions. Therefore we set $$\gamma _{ij}^{\prime }(x)=\theta _{ij}(x)\cdot \overline{\gamma } _{ij}^{\prime }(x^{\prime }) .$$  This ends our main section which relates the number of inequivalent pin structures to the first Cohomology group of the associated orientable cover of the underlying non-orientable space. APPLICATIONS ============  In this last section, we discuss two examples: the first deals with an electron in a nano-circuit.  The second deals with a non-orientable space. Transport in nano-Circuits -------------------------- In this section, we follow the notation of Negele and Orland [@negele]. The geometry of the circuit is nontrivial; it has a ‘hole’. The homotopy group of the torus is $\pi_{1}\left( T^{2}\right) =\mathbf{Z}\oplus \mathbf{Z}$. It is a two-dimensional surface. However the electrons are not only constrained to the surface, but they can be also inside. Therefore the geometry of the circuit is in fact homeomorphic to $D\times \mathbf{S}^{1}$ where $D$ is a disk in $\mathbb{R}^{2}$, i.e, a simply connected region. Moreover the manifold is orientable in this case and hence the nontrivial spinor is dictated by the circle around the hole (see figure 2).  According to our discussion in previous sections, any non-orientability will be factored out.  Hence, the following discussion will equally apply to a Mobius band.  For this manifold, there are two possibilities to define spinors since $H^{1}\left( \mathbf{M},\mathbf{Z}_{2}\right) =\mathbf{Z}_{2}$. The vector potential that corresponds to the non-trivial one differs by an element in the Cohomology class $\lambda$:[@petry] $$A_{\mu}\rightarrow A_{\mu}-\frac{i\hbar c}{2e}\lambda^{-1}\partial_{\mu }\lambda ,$$ with$$\oint\lambda^{-1}\partial_{\mu}\lambda\cdot dx=2\pi i.$$ The function $\lambda$ can always be chosen to be defined on the unit circle:$$\lambda:M\rightarrow\text{ \ }U(1)\subset\mathbb{C}\nonumber$$ Therefore the magnetic flux will change by a discrete value for each closed path traveled by an electron around the circuit$$\oint\left( A_{\mu}-\frac{i}{2e}\lambda^{-1}\partial_{\mu}\lambda\right) \cdot dx=\oint A_{\mu}\cdot dx^{\mu}+\frac{hc}{2e}%$$ It is interesting to observe that Magnus and Schoenmaker [@magnus] had to postulate the quantization of flux to be able to recover the Landauer-Buttiker formula for the conductivity. In our case the quantization is automatic for the non-trivial spin structure. It will be argued below that for this circuit, it is the configuration with non-trivial spin structure that must be adopted based on energy arguments. The Frohlich-Studer ($FS$) theory [@frohlich] is a non-relativistic theory that explicitly exhibits the spin degrees of freedom. This latter theory is $U(1)\times SU(2)$ gauge-invariant.  The $SU(2)$ symmetry comes from the spin degrees of freedom of the wave function of the electron. For a magnetic field $(\mathbf{A}=\frac{1}{2}\mathcal{\mathbf{B}}% \times\mathbf{r})$ and an electric field in the $z$-direction, the covariant derivatives in the $FS$ equation take the form: $$D_{t}=\partial_{t}+e\varphi-ig\mu S_{z}B$$ and the spatial derivatives are $$\begin{aligned} D_{1} & =\partial_{1}-ieA_{1}+i\left( -2g\mu+\frac{e\mu}{2m}\right) ES_{y},\\ D_{2} & =\partial_{2}-ieA_{2}+i\left( -2g\mu+\frac{e\mu}{2m}\right) ES_{x}, \nonumber\\ D_{3} & =\partial_{3} . \nonumber\end{aligned}$$ In two dimensions with $z=x+iy$, they acquire a simple form $$D_{+}=D_{1}+iD_{2}=\partial_{z}-ieA_{+}+g^{^{\prime}}ES_{+},$$ with$$A_{+}=\frac{1}{2}iBz,\qquad S_{+}=S_{x}+iS_{y},\qquad g^{\prime}=-2g\mu +\frac{e\mu}{2m}.\nonumber$$ For a one dimensional ring with radius $a$, $z=ae^{i\phi}$. Hence, we can simply set $x^{2}+y^{2}=a^{2}$ without loosing any essential spin-orbit type terms in the Hamiltonian as it is the case in the standard formulation [@meijer]. Next we comment on a procedure for obtaining the Green’s function and the effective action for a particle interacting with an electromagnetic field $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ based on the proper time method [@schwinger].  This is a relativistic method that starts with the Dirac equation. We just state the results since they exhibit explicit gauge invariance. For the one-particle Green’s function, we have $$\begin{aligned} G\left( x^{\prime},x^{\prime\prime}\right) & =i\int_{0}^{\infty }dse^{-im^{2}s}\\ & \times\left[ -\gamma_{\mu}\left( x(s)^{^{\prime}}\left| \Pi_{\mu }(s)\right| x(0)^{^{^{\prime\prime}}}\right) \right. \nonumber\\ & +\left. m\left( \left. x(s)^{^{\prime}}\right| x(0)^{^{^{\prime\prime}% }}\right) \right] , \nonumber\end{aligned}$$ where $$\begin{aligned} \left( \left. x(s)^{^{\prime}}\right| x(0)^{^{^{\prime\prime}}}\right) & =-\frac{i}{\left( 4\pi\right) ^{2}}\exp\left[ ie\int_{x"}^{x^{\prime}% }dx^{\mu}A_{\mu}(x)\right] \nonumber\\ & \frac{1}{s^{2}}e^{-L(s)}\exp\left[ i\frac{1}{4}\left( x^{^{\prime}% }-x^{^{\prime\prime}}\right) eF\coth\left( eFs\right) \left( x^{^{\prime}% }-x^{^{\prime\prime}}\right) \right] \nonumber\\ & \times\exp\left[ i\frac{1}{2}e\sigma Fs\right] ,\end{aligned}$$$$\begin{aligned} \left( x(s)^{^{\prime}}\left| \pi_{\mu}(s)\right| x(0)^{^{^{\prime\prime}}% }\right) & =\frac{1}{2}\left[ eF\coth\left( eFs\right) -eF\right] \nonumber\\ & \times\left( x^{^{\prime}}-x^{^{\prime\prime}}\right) \left( \left. x(s)^{^{\prime}}\right| x(0)^{^{^{\prime\prime}}}\right) ,\end{aligned}$$ and $$L(s)=\frac{1}{2}tr\log\left[ \left( eFs\right) ^{-1}\sinh\left( eFs\right) \right] .$$ The trace is over the Dirac matrices. These expressions are valid in Euclidean space. The phase factor is clearly isolated in the expression for the Green functions. Hence a non-trivial spin structure will clearly affect the Green’s function of the theory. In particular the energy will be different in both cases.  In the following, we will assume that there is only a magnetic field and no spin-orbit coupling. We calculate the energy in both cases. In terms of Green’s function with one-body potential, the energy is given by: $$\begin{aligned} E & =i\int dx\left( i\partial_{t}\right) \left. G\left( x,x^{^{\prime}% }\right) \right| _{x=x^{^{\prime}}}\\ & =\int dx\Psi^{+}\left( x^{^{\prime}}\right) i\partial_{t}\left. \Psi\left( x\right) \right| _{x=x^{^{\prime}}}\nonumber\end{aligned}$$ In Fourier space, we have $$G_{\alpha\alpha}\left( \omega,\mathbf{k}\right) =\frac{\theta\left( k_{F}-k\right) }{\omega-\in_{k}+\alpha\mu_{0}H-i\varepsilon},$$ where $\alpha=\pm1$, for spin up and spin down.  For \[ptb\] a periodic lattice with period $d = 2 \pi a$ in the $x$ direction, we have [@shulman] $$\begin{aligned} G_{\alpha}\left( x,x^{^{\prime}}+nd\widehat{x}\right) & =-i\theta \left( t^{^{\prime}}-t\right) \int\frac{d^{3}k}{\left( 2\pi\right) ^{3}}\theta\left( k_{F}-k\right) e^{i\mathbf{k\cdot}\left( \mathbf{x}-\mathbf{x}^{^{\prime}}\right) }\\ & e^{-ik_{x}nd}e^{-i\left( \in_{k}-\alpha\mu_{0}H\right) \left( t-t^{^{\prime}}\right) }. \nonumber\end{aligned}$$ For a regular periodic lattice in Euclidean space, the wave functions are periodic: this is the configuration that corresponds to the trivial spin structure. In this case the energy is given by $$\begin{aligned} E_{0} & =\sum_{n=-\infty}^{\infty}\sum_{\alpha}\partial_{t}G_{\alpha}\left. \left( x,x^{^{\prime}}+nd\widehat{x}\right) \right| _{x=x^{\prime}}\\ & =\frac{1}{2m}\int_{0}^{k_{F}}\frac{dk_{x}}{\left( 2\pi\right) ^{2}% }\left( k_{F}^{4}-k_{x}^{4}\right) \nonumber\\ & +\frac{1}{m}\sum_{n=1}^{\infty}\int_{0}^{k_{F}}\frac{dk_{x}}{\left( 2\pi\right) ^{2}}\cos\left( ndk_{x}\right) \left( k_{F}^{4}-k_{x}% ^{4}\right) .\nonumber\end{aligned}$$ For a twisted configuration, we have instead the energy: $$\begin{aligned} E_{t} & =\sum_{n=-\infty}^{\infty}\left( -1\right) ^{n}\sum_{\alpha }\partial_{t}G_{\alpha}\left. \left( x,x^{^{\prime}}+nd\widehat{x}\right) \right| _{x=x^{\prime}}\\ & =\frac{1}{2m}\int_{0}^{k_{F}}\frac{dk_{x}}{\left( 2\pi\right) ^{2}% }\left( k_{F}^{4}-k_{x}^{4}\right) \nonumber\\ & +\frac{1}{2m}\sum_{n=\pm2,\pm4,\pm6,..}\int_{0}^{k_{F}}\frac{dk_{x}% }{\left( 2\pi\right) ^{2}}\cos\left( ndk_{x}\right) \left( k_{F}% ^{4}-k_{x}^{4}\right) \nonumber\\ & -\frac{1}{2m}\sum_{n=\pm1,\pm3,\pm5,..}\int_{0}^{k_{F}}\frac{dk_{x}% }{\left( 2\pi\right) ^{2}}\cos\left( ndk_{x}\right) \left( k_{F}% ^{4}-k_{x}^{4}\right) \nonumber\end{aligned}$$ The difference in energy for typical values of $k_{F}=10^{8}cm^{-1}$ and $d=100nm$ $(d=10nm)$. In arbitrary units, we have: $% \begin{array} [c]{ccc}% k_{F} (cm^-1) & d (nm) & E_{t}-E_{0}\\ \\ 10^{8} & 100 & -561\\ 10^{8} & 10 & -19634 \end{array} $ Therefore as the size of the ring gets smaller, the nontrivial spin structure becomes lower in energy for a critical value of the radius. Hence from an energy point of view, the spin will choose to be in the lowest energy state possible that is compatible with the geometry of the circuit. In this case there will also be a flux quantization associated with changes in the current. Since we are in the ballistic regime, each electron travels in closed paths around the circuit. Any change in the number of particles that traveled around the torus will give rise to a flux or a vector potential. The current is not polarized at zero temperature and hence each pair of electrons with spin up and spin up will give a change in flux as that [*postulated*]{} in Ref. [@magnus] to recover the Landauer-Buttiker formula in non-simply connected circuits with one ’hole’. Therefore it seems the assumption can be proved if the nontrivial spin configuration is taken into account.  We also observe that having twisted leads in the circuit will not change the (s)pin structures in this calculation as shown in the previous sections. \[ptb\] Spin on a non-orientable space ------------------------------ In this section we treat non-orientable cases.  First we take a non-orientable manifold, $\mathcal{\mathbf{B}}=\mathbf{S}^{3}/\mathbf{Z}_{2}$, where $\mathbf{Z}_{2}=\left\{ (1,T),(1,I)\right\} $.  From our discussion above, it was found that inequivalent ${pin}$ structures were given by $H^{1}(R\times \mathbf{S}^{3},\mathbf{Z}_{2})$.  This latter result has been found by different methods in [@john].  Finally, we would like to say more about the new adopted definition for equivalence by going back to the example of the projective plane that we mentioned earlier. Here however, we take a more physically motivated approach. Let $\psi ^{a},a=1,2$, be a pinor field on $\mathbf{RP}_{2}$. The structure group of the frame bundle is $\mathbf{O}(2)$. Let $ \{e_{i}^{a}\}$ be a local frame on the open set $U_{i}.$ The sets $U_{i}$ cover $\mathbf{RP}_{2}$ and their intersections are contractible so local sections are always well defined. On intersections $U_{i}\smallfrown U_{j}$ we have $$e_{i}^{a}=(L_{ij})^{a}e_{i}^{a}\text{, with }L_{ij}\in \mathbf{O}(2) \: .$$ On the pinor frame level , we have $$\psi _{i}^{a}=(S_{ij})_{b}^{a}\psi _{j}^{a}\text{, with }S_{ij}\in \mathbf{Pin}(2) \; ,$$ and $\rho (S)=L$. We require that $\psi ^{\dagger }\psi $ and $\psi ^{\dagger }\gamma ^{a}\psi $ transform as a scalar and a vector respectively. The $\gamma ^{a}$ defined here are the Pauli matrices. From this we get the following conditions on $S$, $$\begin{aligned} S^{\dagger }S &=&1 \: , \\ S^{\dagger }\gamma ^{a}S &=&L_{b}^{a}\gamma ^{b} \: .\end{aligned}$$ To find explicit expressions, we need to choose a covering. $\mathbf{RP}_{2}$ is topologically equivalent to a disk with the boundary antipodally identified. Next we parametrize the boundary with an angle $\theta , 0\preccurlyeq \theta \prec 2\pi $ . Choose a simple cover for the strip adjacent to the boundary, we will need at least three open neighborhoods. Non-trivial transition functions will be needed only as we go along the boundary. They are of the form $$L=I\cdot e^{2i\theta } \: ,$$ where $I$ is a reflection about the first axis. Using this cover, we find that $S$ must have the form $e^{i\alpha }\gamma ^{1}e^{i\theta \gamma ^{3}}$ . Imposing boundary conditions on $\psi (\theta )$ , we find that $$e^{i\alpha }=\pm i \: .$$ Hence, the two $\mathbf{Pin}(2)$ structures predicted above. The phase factor $ e^{i\alpha }$ is clearly due to the reflection $I$. Moreover, it is physically irrelevant and ignoring it amounts to ignoring $I,$ i.e., the non-orientability of the space.  Therefore quantum mechanics should be studied first on the orientable cover and then projected on the configuration space. CONCLUSION ==========  In summary, we have given a definition to spin structures on non-orientable manifolds by going to the orientable double cover.  This allowed us to determine the number of inequivalent spin structures using our definition of equivalence.  We also showed that in the typical structure of a nano-circuit, the nontrivial spin configuration is probably more important than the trivial one at nanometer scale. This argument is supported indirectly by the work in ref. [@magnus]. A convincing proof of this statement will be to solve the problems with the constraints on the motion of the particle explicitly taken into account.  This is a very difficult problem. We believe the energy argument that we presented is compelling enough to continue looking into other aspects which can result from the nontrivial spin configuration.  Smaller non-orientable structures than those made by Tanda et al. should also be possible in the near future and provide an experimental test of the idea presented here.  Finally, there is one question that we did not discuss in this work and that is related to the nature of ’phase transition’ at the critical radius of ring between the two spin structures.  This is an interesting question mathematically and physically.  We are not the first to raise this question; Jarosczewicz asked a similar question regarding the spin of $\textbf{SU}(2)$ solitons [@jaros].  To avoid introducing one more flavor to quantize the spin, he introduced the idea of a rotating soliton which corresponds mathematically to the nontrivial paths in $\textbf{SO}(3)$.  A similar analysis to his may shed some light on the physics of our non-trivial spin configurations in a ring. The author is very grateful to R. Chantrell who made this work possible and thanks J. Friedman for initial discussions on this subject. [99]{} S. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya, and N. Hatakenaka, Nature **417**, 397 (2002). A. P. Balanchandran, G. Marmo, B. S. Skagerstman, and A. Stern, Classical Topology and Qunatum States, World Scientific (1991). D. J. Thouless, Phys. Rev. B **27**, 6083 (1983). A. Rebei and O. Heinonen, spin currents in the Rashba model, submitted to Phys. Rev. textbf[B]{}. E. Witten, Nucl. Phys. **B223**, 422 (1983). L. Alvarez-Gaume and P. Ginsparg, Ann. Phys. (NY) **161**, 423 (1985). E. Saitoh, S. Kasai, and H. Miyajima, J. Appl. Phys. **97**, 10J709 (2005). P. A. Dirac, Proc. Roy. Soc. (London) A 117, 610 (1928). H. B. Petry, J. Math. Phys. **20**, 231 (1979). L. S. Schulman, Techniques and Applications of Path integration. New York, Wiley 1981. L. Dabrowski, Group Actions on Spinors (Bibliopolis, 1988). M. Berg, C. DeWitt-Moretti, S. Gwo and E. Kramer, Rev. Math. Phys. **13**, 953 (2001). M. Atiyah, R. Bott and A. Shapiro, Topology **3**, Suppl. 1, 3 (1964). M. Karoubi, Ann. Scient. Ec. Norm.**1**, 161 (1968). D. Ebner, Gen. Rel. Grav. **8**, 15 (1977). T. Eguchi, P. B. Gilkey, A. J. Hanson, Phys. Rep. **66**, 213 (1980). J. Milnor,  Enseig. Math. **9**, 198 (1963). A. Chamblin, Commun.Math.Phys. **164**, 65 (1994). F. Hirzebruch, Topological Methods in Algebraic Geometry(Springer-Verlag, Berlin, 1966). E. Spanier, Algebraic Topology (Springer, New-York, 1966). W. Greub and H. Petry, On the Lifting of Structure Groups.  In:  Bleuler,K., Petry,H., Reetz,A. (eds) Differential Geometrical Methods in Mathematical Physics II. Proceedings, Bonn 1977, pp. 217-246. Berlin: (Springer-Verlag, Berlin, 1978). N. Steenrod, The Topology of Fiber Bundles (Princeton, 1951). J. Negele and H. Orland, Quantum Many-Partilce Systems, Addison-Wesley, 1988. W. Magnus and W. Schoenmaker, J. Math. Phys. **39**, 6715 (1998); Phys. Rev. B **61**, 10883 (2000). J. Frohlich and U. M. Studer, Rev. Mod. Phys. **65**, 733 (1993). F. E. Meijer, A. F. Morpurgo and T. M. Klapwijk, Phys. Rev. **B** 66, 033107 (2002). J. Schwinger, Phys. Rev. **82**, 664 (1951). L. S. Schulman, Phys. Rev. **188**, 1139 (1969). J. Friedman, Class.Quantum.Grav.**12**, 2231 (1995). T. Jaroszewicz, Phys. Rev. D **44**, 1311 (1991).

--- abstract: 'A ferromagnetic coupling between localized Mn spins was predicted in a series of ab initio and tight binding works and experimentally verified for the dilute magnetic semiconductor Ga$_{1-x}$Mn$_x$N. In the limit of small Mn concentration, x $ \lesssim$ 0.03, the paramagnetic properties of this material were successfully described using single ion crystal field model approach. However, there is still a need to investigate the effect of magnetic coupling on magnetic properties of dilute magnetic semiconductors using approach that goes beyond the classical approximation, e.g. Landau-Lifshitz-Gilbert one. Therefore, in order to obtain the magnetization $M(T,B)$ of (Ga,Mn)N in the presence of interacting magnetic centers, we extend the previous model of a single substitutional Mn$^{3+}$ ion in GaN by considering pairs, triplets and quartets of Mn$^{3+}$ ions coupled by a ferromagnetic superexchange interaction. Using this approach we investigate how the magnetic properties, particularly the magnitude of uniaxial anisotropy field change as the number of magnetic Mn$^{3+}$ ions in a given cluster increases from 1 to 4.' author: - 'D. Sztenkiel' title: 'Crystal field model simulations of magnetic response of pairs, triplets and quartets of Mn$^{3+}$ ions in GaN' --- Introduction ============ Magnetism in reduced dimensions, such as in magnetic nanostructures and magnetic clusters, has received a great research interest in the recent years due to its unexpected features and potential applications in high-density storage [@Sun:2000_Science], nanoelectronics and quantum computations [@Troiani:2005_PRL]. A major advantage with respect to analogous bulk based materials originates from an additional degrees of freedom of nanoparticles to tune the magnetic properties by modifications of their size, shape, number of magnetic ions and/or coupling with the substrate [@Spisak:2002_PRB]. For example, recently it has been shown that single cobalt atoms deposited onto platinum (111) surface pose a very large magnetic anisotropy energy (MAE) of about 9 meV [@Gambardella:2003_Science]. The single-ion MAE depends on the arrangements of atoms around the magnetic ion through the spin–orbit interaction and crystal field induced anisotropy of quantum orbital angular momentum ($L$). In the bulk materials the magnitude of $L$ is usually quenched or strongly diminished by electron delocalization, ligand fields and hybridization effects, what result in small values of MAE of the order of 0.01 meV/atom [@Evans:2014_JPhysCM]. However, it is possible to enhance the MAE by using low-coordination geometries, such as atoms deposited on the surface, 1D atomic chains, magnetic clusters or molecular complexes. The values of MAE of the order of $1 \div 10$ meV per atom have been routinely reported in such systems [@Gambardella:2003_Science; @Gambardella:2003_JPhysCM; @Tung:2010_PRB]. Experiments on small particles of iron, cobalt, and nickel revealed the strong dependence of per-atom magnetic moments on the cluster size [@Billas:1994_Science]. The ferromagnetism was present even for the clusters composed of about 30 atoms, with atom-like magnetization. These magnetic moments per one atom decreased with the number of ions in a given particle, approaching the bulk limit for about 500 atoms. Even in the material investigated here, the magnetic anisotropy strongly depends on the Mn ion concentration $x$, due to the dependence of lattice parameters $c$ and $a$ of Ga$_{1-x}$Mn$_x$N on $x$. The magnetic anisotropy is high in very diluted case[@Gosk:2005_PRB] and then decreases with x. [@Sztenkiel:2016_NatComm]. High MAE reduces the magnitude of the thermal fluctuations in superparamagnetic nanostructures and thus determines the potential applicability of these small-scale systems in high-density recording and magnetic memory operations. It is thus very relevant to investigate magnetic anisotropy properties of systems with reduced symmetry and/or coordination of magnetic aggregates. In this paper we numerically study how the MAE evolves from single isolated magnetic Mn$^{3+}$ impurity in GaN to very small magnetic clusters, composed of up to four Mn$^{3+}$ ions coupled by ferromagnetic superexchange interaction. Similar approach has been used to explain experimental results of singles and antiferromagnetically coupled pairs of Mn ions in InS-based dilute magnetic semiconductor (DMS) [@Tracy:2005_PRB]. In Ref the single ion crystal field model (CFM) was extended to simulate magnetic properties of pairs and triplets, however the basis functions for diagonalization of Hamiltonian were considerably restricted by taking only 10-fold degenerate functions of $^5$E symmetry. Here we use CFM approach to model small magnetic clusters with up to 4 ions, where all function of $^5$E and $^5$T symmetry are included in setting up the Hamiltonian with spin-orbit interaction and both trigonal and Jahn-Teller deformation taken into account. Model ===== The standard theoretical approach to tackle magnetic systems is to use density functional theory (DFT) calculations that can give insight into the values of MAE, exchange interactions and atomic moments. Due to numerical complexity of DFT, the simulations are generally limited to very small structures or bulk/2D periodic systems. In order to obtain macroscopic properties such the Curie temperature and total magnetization $M(T,B)$ of the large systems one can resort to classical approximations, namely atomistic spin models supplemented with Monte Carlo or Landau-Lifshitz-Gilbert (LLG) dynamics [@Evans:2015_PRB]. However, small magnetic nanostructures at very low temperature are fundamentally quantum mechanical systems due to the quantization of the relevant energy levels. For example, $M(T,B)$ characteristic of a single substitutional transition metal ion in a given semiconductor can be obtained using the full quantum-mechanical crystal field model approach. The CFM was developed by Vallin[@Vallin:1970_PRB; @Vallin:1974_PRB] for II-VI dilute magnetic semiconductors doped with Cr, and then successfully applied for other DMSs[@Mac:1994; @Twardowski:1993_JAP; @Herbich:1998; @Wolos:2004_PRB_b; @Gosk:2005_PRB; @Savoyant:2009_PRB; @Stefanowicz:2010_PRB; @Bonanni:2011_PRB; @Rudowicz:2019_JMMM]. Recently it was shown that CFM simulations can explain magnetic [@Gosk:2005_PRB; @Stefanowicz:2010_PRB; @Bonanni:2011_PRB], magnetooptic[@Wolos:2004_PRB_b] and even magnetoelectric[@Sztenkiel:2016_NatComm] properties in dilute Ga$_{1-x}$Mn$_x$N, with $x \leq 0.03$. Therefore it is a natural way to extend aforementioned model of a single substitutional Mn$^{3+}$ ion in GaN by considering pairs, triplets and quartets of Mn$^{3+}$ ions coupled by ferromagnetic superexchange interaction [@Bonanni:2011_PRB; @Sawicki:2012_PRB; @Stefanowicz:2013_PRB]. Due to the fact that the number of elementary operations and computer memory needed for calculations grow exponentially with the number of particles, the CFM simulations are restricted here to small magnetic clusters composed of up to four ions. The considered here cluster types are shown in Fig. \[Fig:ClusterTypes\]. Due to the short-ranged nature of spin-spin interactions, only the couplings between the nearest-neighbor (nn) Mn ions are taken into account. Each magnetic cluster is defined as a group of magnetic ions coupled by an nn ferromagnetic superexchange interaction, and decoupled from other more distant magnetic atoms. As Mn ions in Ga$_{1-x}$Mn$_x$N are randomly distributed over Ga cation sites [@Gas:2018_JALCOM], the distribution of the different types of clusters, which depends on the Mn concentration $x$, can be precisely estimated [@Shapira:2002_JAP]. The energy levels of a single Mn$^{3+}$ ion in wurtzite GaN can be obtained by numerical diagonalization of the following (25 x 25) Hamiltonian matrix $H_S$ (see also Ref. ) $$\begin{aligned} \label{eq:Hcf} H_S(j)=H_{\mathrm{CF}}+H_{\mathrm{JT}}(j)+H_{\mathrm{TR}}+H_{\mathrm{SO}}+H_{\mathrm{B}},\end{aligned}$$ where $H_{\mathrm{CF}}=-2/3B_4(\hat{O}_4^0-20\sqrt{2}\hat{O}_4^3)$ is the cubic field of tetrahedral $T_{d}$ symmetry, $H_{\mathrm{JT}}=\tilde{B}_2^0\hat{\Theta}_4^0+\tilde{B}_4^0\hat{\Theta}_4^2$ describes the static Jahn-Teller (J-T) distortion of the tetragonal symmetry, $H_{\mathrm{TR}}=B_2^0\hat{O}_4^0+B_4^0\hat{O}_4^2$ corresponds to the trigonal distortion along the GaN hexagonal $c$-axis, $H_{\mathrm{SO}}=\lambda\hat{\textbf{L}}\hat{\textbf{S}}$ represents the spin-orbit coupling. $H_B=\mu_{\mathrm{B}}(g_L\hat{\textbf{L}}+g_S\hat{\textbf{S}})\textbf{B}$ describes the Zeeman term where g-factors are $g_S=2$, $g_L=1$, $\mu_{\mathrm{B}}$ is the Bohr magneton and $B$ is the magnetic field. Here $\hat{\Theta}$ are Stevens equivalent operators for a tetragonal distortion along one of the three equivalent cubic $[100]$, $[010]$, $[001]$ directions denoted by $j=A, B, C$ respectivelly, and $\hat{O}$ are Stevens operators for a trigonal distortion along $[111]$ $\|$ $\textbf{c}$-axis of GaN. $B_i^k$, $\tilde{B}_i^k$, $\lambda_{TT}$, and $\lambda_{TE}$ denote parameters of the crystal field model, which are given in Tab. \[tab:Parameters\_CF\]. These parameters are taken from Ref. . We only modified values of $B_i^j$ corresponding to trigonal deformation, as they depend on the wurtzite lattice parameters $c$ and $a$ and can be controlled by strain engineering or electric field [@Sztenkiel:2016_NatComm]. All simulations are performed at temperature $T = 2$K. [ cccccccc]{} $B_4$ & $B_2^0$ & $B_4^0$ & $\tilde{B}_2^0$ & $\tilde{B}_4^0$ & $\lambda_{TT}$ & $\lambda_{TE}$ & $J$\ \ 11.44&0.99&-0.13&-5.85&-1.17&5.5&11.5&2.0\ \[tab:Parameters\_CF\] The basis of a single Mn$^{3+}$ ion (d$^4$ configuration, with $S=2$, $L=2$) consists of a total of W=$(2S+1)(2L+1)=25$ function $|m_L,m_S\rangle$ characterized by spin $-2 \leq m_S \leq 2$ and orbital $-2 \leq m_L \leq 2$ quantum numbers. Due to the presence of three different J-T centers ($j=A, B$ or $C$), the average magnetic moment $\textbf{M}$ of Mn ion (in $\mu_{\mathrm{B}}$ units) can be calculated according to the formula $$\label{eq:M_cf} <\textbf{M}>=Z^{-1}\sum_{j=A,B,C}Z_j\textbf{M}^j,$$ with $Z_j=\sum_{k=1}^W\mathrm{exp}(-E_k^j/k_{\mathrm{B}}T)$ representing the partition function of the j-th center, $Z=Z_A+Z_B+Z_C$ and $$\label{eq:M_cf_Center} \textbf{M}^j=\frac{-\sum_{k=1}^W<\varphi_{k}^j|g_L\hat{\textbf{L}}+g_S\hat{\textbf{S}}|\varphi_{k}^j>\mathrm{exp}(-E_k^j/k_{\mathrm{B}}T)}{Z_j},$$ where $E_k^j$, $\varphi_{k}^j$ are the k-th eigenenergy and the eigenfunction of the Mn$^{3+}$ ion being in $j$-th J-T center, respectively. ![\[Fig:ClusterTypes\] A magnetic cluster is defined here as a group of magnetic ions (red dots) coupled by a nearest-neighbor ferromagnetic superexchange interaction (green line), and decoupled from other more distant magnetic atoms. Only clusters with up to 4 ions are shown.](Cluster_types_names.pdf){width="8.6"} In this work, we consider singles, pairs, triplets and quartets of Mn$^{3+}$ ions coupled by a ferromagnetic superexchange interaction $H_{exch}(1,2) = J \hat{\textbf{S}}_1 \hat{\textbf{S}}_2$. The exact value of nearest neighbor superexchange coupling $J$ is not known. The magnitudes of $J$ obtained from first-principles methods[@Sato:2010_RMP] or tight binding approximations [@Simserides:2014_EPJ] are rather high $J>10$ meV. On the other hand, in our recent LLG simulations (not published yet) that described reasonably well ferromagnetic properties of Ga$_{1-x}$Mn$_x$N with $x=6\%$ we use $J$=1.4 meV. Therefore we assume $J$=2 meV here. Now, the relevant eigen-functions and eigen-values are obtained by a numerical diagonalization of the full (25×25), (25$^2$×25$^2$), (25$^3$×25$^3$), (25$^4$×25$^4$) Hamiltonian matrix, for a single ion, pair, triplet or quartet, respectively. Additionally, one should take into account that the number of different J-T configurations increases with the number of ions $N$ in given cluster, and equals $3$, $3^3$, $3^3$ and $3^4$ for $N$ = 1, 2, 3 and 4 respectively. For example, the hamiltonian for open triplet (c.f. Fig. \[Fig:ClusterTypes\]) reads $$\begin{gathered} \label{eq:Hcf_closedtriplet} H(j_1,j_2,j_3)=H_S(j_1,1) + H_S(j_2,2) + H_S(j_3,3) +\\+ H_{exch}(1,2) + H_{exch}(2,3),\end{gathered}$$ where $H_S(j,k)$ is the single ion hamiltonian for $k$-th ion being in the $j$-th J-T center and the base states are characterized by the set of quantum numbers $|m_{L_1},m_{S_1},m_{L_2},m_{S_2},m_{L_3},m_{S_3} \rangle$. Now, the magnetization of the cluster (in $\mu_{\mathrm{B}}$ units) is the thermodynamical and configurational average of the total magnetic moment operator $g_L(\hat{\textbf{L}}_1+\hat{\textbf{L}}_2+\hat{\textbf{L}}_3)+g_S(\hat{\textbf{S}}_1+\hat{\textbf{S}}_2+\hat{\textbf{S}}_3)$, and the sum $\sum_{j=A,B,C}$ in Eq. (\[eq:M\_cf\]) is replaced by $\sum_{j_1=A,B,C}\sum_{j_2=A,B,C}\sum_{j_3=A,B,C}$. In order to speed up the calculations the parallelization of the code is used. Additionally, due to the presence of the Boltzman factor in Eq. (\[eq:M\_cf\_Center\]), only the $k_{max} < W$ lowest energy levels and eigenfunctions need to be calculated in order to obtain a very good approximation of $M(B,T)$ curves. Here $W$ is equal to $25$, $25^2$, $25^3$, $25^4$ for a single ion, pair, triplet or quartet, respectively. In each case, the condition of $E_{k_{max}} - E_{k=0} \gg 30 k_{\mathrm{B}}T $ was fullfilled, ensuring that relative error in claculation of $M(B,T)$ is practically zero. $E_{k=0}$ is the ground state energy and $E_{k_{max}}$ is the maximal energy of the calculated excited states. Magnetic simulations ==================== ![\[Fig:Magnetization\] Magnetization per one ion as a function of the magnetic field $B$ of different Mn$^{3+}$ magnetic clusters in GaN at $T$=2 K obtained using crystal field model with ferromagnetic superexchange coupling $J$ = 2 meV. The magnetic easy axis $M_{\perp}$ (solid lines) is perpendicular to the $\textbf{c}$ axis of GaN, whereas the hard one $M_{||}$ (dash lines) is parallel to the $\textbf{c}$ axis. (a) The medium magnetic field, and (b) the high magnetic field region.](MagnetizUp4Ions.pdf){width="8.6"} ![\[Fig:OpenVsClosed\] Comparision of magnetization per one ion of magnetic clusters composed of three ions (open vs closed triplet) and four ions (string vs tetrahedron quartet)](OpenVsClosed.pdf){width="8.6"} In Fig. \[Fig:Magnetization\] we present the results of our simulations of magnetization per one ion as a function of magnetic field $B$ of different magnetic clusters at $T$=2 K. We see that $M(B)$ varies sharply with magnetic field and the saturation is observed for high $B$. The magnetic easy axis ($M_{\perp}$) is perpendicular to the $\textbf{c}$ axis of GaN, whereas the hard one ($M_{||}$) is parallel to $\textbf{c}$. Such uniaxial magnetic anisotropy is specific for Mn$^{3+}$ ions in wurtzite GaN [@Gosk:2005_PRB; @Stefanowicz:2010_PRB; @Sztenkiel:2016_NatComm]. As expected for ferromagnetic coupling between atoms, the magnetization per one ion increases with the size of the cluster $N$. In Fig. \[Fig:Magnetization\] only one representative example from clusters with $N$ = 1, 2, 3, 4 is presented. As the strength of superexchange interaction is much stronger than thermal energy $J \gg k_{\mathrm{B}}T \approx 0.17$ meV at $T = 2$ K, the $M(B,T=2~K)$ curve of cluster with given $N$ is practically independent of the cluster type and the number of ferromagnetic bonds $J\textbf{S}_i\textbf{S}_j$. It is exemplified in Fig \[Fig:OpenVsClosed\] where the same dependencies are observed for closed and open triplets as well as tetrahedron and string quartets. Therefore in the next figures only the results corresponding to singles, pairs, closed triplets and tetrahedron quartets will be shown. In Fig. \[Fig:Magnetization\] we see that $M(B)$ clearly depends on the orientation of the magnetic field $B$ and the cluster size. On the other hand other studies do not show any conclusive dependencies of the magnetic anisotropy on the number of ions in given clusers or 1D wires. For example, experimental results on cobalt atoms deposited on an atomically ordered platinum surface [@Gambardella:2003_Science], revealed that magnetic anisotropy decrease strongly with increasing Co coordination and number of Co particles. The first-principles DFT investigations of nanometric Co$_n$Ni$_m$ clusters (with size $N = n + m \leq 7$) were performed in Ref. . A strong enhancement of MAE of the clusters as compared with bulk-like values was found. However, MAE of clusters as a function of their composition exhibited a complex and a non-monotonous behavior. These features were related to Co–Ni (spd) hybridization processes as well as structural rearrangements of the atoms. ![\[Fig:MzMx\] (a) The hard axis magnetization $M_{||}$ as a function of easy axis magnetization $M_{\perp}$. The isotropic case with $M_{||}$=$M_{\perp}$ is represented by the dash line. (b) The $M_{||}/M_{\perp}$ ratio as a function of the magnetic field $B$. The uniaxial anisotropy filed ${B}_{a}$ corresponds to the field $B$ where $M_{||}/M_{\perp}=1$ (dash line).](MzMx.pdf){width="8.6"} The significant dependence of the magnetic anisotropy on the magnetic ion composition can also be observed in thin 2D films. For example, the variations of MAE with concentration of Mn in (Ga,Mn)As grown on GaAs is usually caused by the epitaxial strain originating from lattice mismatch [@Fedorych:2002_PRB; @Sawicki:2006_JMMM]. The similar effect takes place in the material investigated here, where the deviation of the $c$-lattice parameter of the Ga$_{1-x}$Mn$_x$N from that of GaN layer was observed as a function of the Mn content [@Kunert:2012_APL]. However, in this work, we have the advantage that all parameters of the single ion hamiltonian can remain unchanged during transition from singlet to quartets, and the only varying parameters are the the number and the geometry of ferromagnetic bonds $J\textbf{S}_i\textbf{S}_j$. [ c|cccc]{} & single & pair & triplet & quartet\ \ $B_a$ (T) & 14.6 & 11.5 & 9.6 & 8.7\ \ MAE (meV) & 0.204 & 0.217 & 0.223 & 0.227\ \[tab:MAE\] In order to quantify the strength of magnetic anisotropy we use two different approaches. Firstly, we plot in Fig. \[Fig:MzMx\](a) the hard axis magnetization $M_{||}$ as a function of easy axis magnetization $M_{\perp}$. Obviously, the deviation of $M_{||}$=f($M_{\perp}$) curve from the isotropic case $M_{||}$=$M_{\perp}$ (dash line in Fig. \[Fig:MzMx\](a)) indicates both the direction and the magnitude of the anisotropy. It is seen that the anisotropy increases with $N$, however the modifications are the most pronounced between $N$=1 (single) and $N$=2 (pair). Secondly, we calculate MAE in a standard way. The MAE is the energy needed to rotate the magnetization from its easy axis into the hard one and it can be obtained from the following formula: $$\label{eq:MAE_integral} MAE=\int_0^{{B}_{a}}(M_{\perp}-M_{||})dB,$$ where ${B}_{a}$ is the uniaxial anisotropy field (see also Fig. \[Fig:MzMx\](b)). The results of this procedure are summarized in Tab. \[tab:MAE\]. We observe a rather weak dependence of MAE per atom on cluster size, but still the MAE increases with $N$. It seems that the magnetic anisotropy at small magnetic fields increases with $N$ (see Fig. \[Fig:Magnetization\](a)) but at the same time the uniaxial anisotropy field ${B}_{a}$ decreases with $N$, as shown in Fig. \[Fig:MzMx\]b and Tab. \[tab:MAE\]. We even observe the reversal of magnetic anisotropy at very high magnetic fields $B > 12$ T for $N\geq$1: the low-field easy axis is perpendicular to the $\textbf{c}$ axis ($M_{\perp}>M_{||}$), whereas the magnetic anisotropy reverses (i.e., $M_{\perp}<M_{||}$) when B is sufficiently large . Conclusions {#conclusions .unnumbered} =========== In this paper we numerically compute magnetic response of small Mn$^{3+}$ magnetic clusters in GaN using quantum-mechanical crystal field approach. The calculations are performed for isolated ions, pairs, triples and quarters of Mn$^{3+}$ ions coupled by nearest neighbor ferromagnetic superexchange interaction. We show that magnetocrystalline anisotropy increases with number of ions N in given cluster, whereas the uniaxial anisotropy field ${B}_{a}$ decreases with $N$. Our simulations can be further exploited in explaining experimental magnetic properties of Ga$_{1-x}$Mn$_x$N in the dilute case ($x \leq 0.03$), where different small magnetic clusters play important role. Acknowledgments {#acknowledgments .unnumbered} =============== I would like to thank K. Gas and M. Sawicki for proofreading of the manuscript. The work is supported by the National Science Centre (Poland) through project OPUS 2018/31/B/ST3/03438 and by the Interdisciplinary Centre for Mathematical and Computational Modelling at the University of Warsaw through the access to the computing facilities. [33]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1126/science.287.5460.1989) [****,  ()](\doibase 10.1103/PhysRevLett.94.207208) [****,  ()](\doibase 10.1103/PhysRevB.65.235405) [****,  ()](\doibase 10.1126/science.1082857) [****,  ()](\doibase http://dx.doi.org/10.1088/0953-8984/26/10/103202) [****,  ()](\doibase https://doi.org/) [****,  ()](\doibase 10.1103/PhysRevB.81.094422) [****,  ()](\doibase 10.1126/science.265.5179.1682) [****,  ()](\doibase 10.1103/PhysRevB.71.094432) [****,  ()](\doibase http://dx.doi.org/10.1038/ncomms13232) [****,  ()](\doibase 10.1103/PhysRevB.72.165201) [****,  ()](\doibase https://doi.org/10.1016/S0304-8853(01)00460-7) [****,  ()](\doibase http://dx.doi.org/10.1103/PhysRevB.91.144425) [****,  ()](\doibase 10.1103/PhysRevB.2.4313) [****,  ()](\doibase 10.1103/PhysRevB.9.2051) [****,  ()](\doibase 10.1103/PhysRevB.50.14144) [****,  ()](\doibase 10.1063/1.353558) [****,  ()](\doibase 10.1103/PhysRevB.58.1912) [****,  ()](\doibase 10.1103/PhysRevB.70.245202) [****,  ()](\doibase 10.1103/PhysRevB.80.115203) [****,  ()](\doibase 10.1103/PhysRevB.81.235210) [****,  ()](\doibase 10.1103/PhysRevB.84.035206) [****,  ()](\doibase https://doi.org/10.1016/j.jmmm.2019.04.099) [****,  ()](\doibase 10.1103/PhysRevB.85.205204) [****,  ()](\doibase 10.1103/PhysRevB.88.081201) [****,  ()](\doibase https://doi.org/10.1016/j.jallcom.2018.03.056) [****, ()](\doibase 10.1063/1.1507808) [****,  ()](\doibase 10.1103/RevModPhys.82.1633) [****,  ()](\doibase http://dx.doi.org/10.1051/epjconf/20147501003) [****,  ()](\doibase 10.1039/C8CP01372A) [****,  ()](\doibase 10.1103/PhysRevB.66.045201) [****,  ()](\doibase 10.1016/j.jmmm.2005.10.020) [****,  ()](\doibase http://dx.doi.org/10.1063/1.4734761)

--- abstract: 'On the basis of the Berkovits pure spinor formalism of covariant quantization of supermembrane, we attempt to construct a M(atrix) theory which is covariant under $SO(1,10)$ Lorentz group. We first construct a bosonic M(atrix) theory by starting with the first-order formalism of bosonic membrane, which precisely gives us a bosonic sector of M(atrix) theory by BFSS. Next we generalize this method to the construction of M(atrix) theory of supermembranes. However, it seems to be difficult to obtain a covariant and supersymmetric M(atrix) theory from the Berkovits pure spinor formalism of supermembrane because of the matrix character of the BRST symmetry. Instead, in this paper, we construct a supersymmetric and covariant matrix model of 11D superparticle, which corresponds to a particle limit of covariant M(atrix) theory. By an explicit calculation, we show that the one-loop effective potential is trivial, thereby implying that this matrix model is a free theory at least at the one-loop level.' --- =cmr5 \#1[\#1\^[\^]{}]{} \#1 EDO-EP-45\ February, 2003\ hep-th/0302203\ **Covariant Matrix Model of Superparticle** in the Pure Spinor Formalism Ichiro Oda [^1]\ Edogawa University, 474 Komaki, Nagareyama City, Chiba 270-0198, JAPAN\ Introduction ============ Since the advent of M(atrix) theory by BFSS [@BFSS], there has been a strong desire to construct a manifestly Lorentz covariant M(atrix) theory, but no one has succeeded in constructing such a theory thus far. Although M(atrix) theory has been derived from the low-energy effective action of D-particles which is obtained via the dimensional reduction from the maximally supersymmetric Yang-Mills theory in ten dimensions, this theory can be also interpreted as a regularized supermembrane theory in the light-cone gauge [@de; @Witt]. Then, it is natural to start with the supermembrane action in eleven dimensions and quantize it in a covariant manner in order to obtain the covariant M(atrix) theory. However, as is well known, the fermionic kappa symmetry, which is used to reduce the number of fermionic degrees of freedom by half, has given us a difficulty in covariant quantization of the supermembrane action. Recently, there has been an interesting progress by Berkovits in the covariant quantization of the Green-Schwarz superstrings [@GS] using the pure spinors [@Ber1; @Ber2; @Ber3; @Ber4; @Ber5]. One of the key ingredients in the Berkovits approach is the existence of the BRST charge $Q_{BRST} = \oint \lambda^\alpha d_\alpha$ where $\lambda^\alpha$ are pure spinors satisfying the pure spinor equations $\lambda^\alpha \Gamma^m_{\alpha \beta} \lambda^\beta = 0$ and $d_\alpha \approx 0$ are the fermionic constraints associated with the kappa symmetry. It is remarkable that this approach provides us the same cohomology as the BRST charge of the Neveu-Schwarz-Ramond formalism [@Neveu] and the correct tree amplitudes of superstrings with keeping the Lorentz covariance of the theory. Afterwards, the Berkovits approach has been investigated from various different viewpoints [@Oda1; @Ber6; @Ber7; @Trivedi; @Oda2; @Grassi; @Kazama]. In particular, more recently, the generalization of this approach to supermembrane has be done by Berkovits [@Ber8]. Combining the above-mentioned two observations, we are naturally led to think that we could make use of the Berkovits pure spinor formalism to construct a covariant M(atrix) theory since the covariant quantization of supermembrane has been made and the difficulty of the quantization associated with the kappa symmetry has been resolved in the pure spinor formalism. Actually, Berkovits has proposed such an interesting idea in the conference of Strings 2002 [@Ber9], but it is a pity that this work has not been completed so far as long as I know. In this paper, we pursue this idea and attempt to construct a covariant M(atrix) theory by using the pure spinor formalism of supermembrane [@Ber8]. However, we will see that the construction of a covariant M(atrix) theory is rather difficult owing to the existence of the BRST invariance $Q_{BRST}$ which is now promoted to a matrix symmetry (like a local gauge symmetry) in the matrix model. In this paper, we will explain in detail why it is difficult to apply the Berkovits formalism to the construction of the covariant M(atrix) model. Thus, instead of constructing a covariant M(atrix) theory, we present how to construct a covariant matrix model of superparticle in eleven dimensions [@Brink] which in some sense corresponds to a particle limit of a covariant M(atrix) theory. This paper is organized as follows. In section 2, as a warmup, we construct a bosonic M(atrix) theory by starting with the first-order formalism of bosonic membrane. In section 3, we generalize the method to supermembrane and attempt to construct a covariant M(atrix) theory from the pure spinor formalism of supermembrane by Berkovits. Here we find a difficulty of constructing M(atrix) theory which is invariant under the BRST symmetry. Hence, instead we turn to the construction of a covariant matrix model of superparticle invariant under both the supersymmetry and the BRST symmetry. Furthermore, in section 4, we calculate the one loop effective potential and show that our matrix model is a free theory owing to the lack of the potential term. The final section is devoted to the conclusion. Bosonic M(atrix) theory ======================= In this section, we shall construct a bosonic M(atrix) theory since this construction gives us a good exercise in attempting to construct a M(atrix) theory of supermembrane based on the pure spinor formalism. In addition, we can clearly understand the difference of the construction of a matrix model between the bosonic theory and the supersymmetric one. A similar analysis has been thus far done from various different contexts [@Bergshoeff; @Santos; @Fujikawa; @Ishibashi; @Yoneya]. We begin with the well-known Nambu-Goto action of the bosonic membrane in eleven dimensions in a flat space-time: $$\begin{aligned} S_{NG} = - T \int d^3 \sigma \sqrt{-g}, \label{1}\end{aligned}$$ where $T$ is the membrane tension with dimension $(mass)^3$. And the induced metric and its determinant are respectively given by $g_{ij}= \partial_i x^a \partial_j x^b \eta_{ab}$, and $g = \det g_{ij}$. We take the Minkowskian metric signature $(-,+,+, \cdots,+)$. Moreover, the indices indicate $i, j = 0, 1, 2$ and $a, b, c = 1, 2, \cdots, 11$. We follow the notations and conventions of the Berkovits’ paper [@Ber8]. Let us perform the canonical quantization of the action (\[1\]). The canonical conjugate momenta of $x^a$ are derived as $$\begin{aligned} P_a &=& \frac{\partial S_{NG}}{\partial \dot{x}^a} {\nonumber}\\ &=& - T \sqrt{-g} g^{0j} \partial_j x_a {\nonumber}\\ &=& - T \sqrt{-g} (g^{00} \partial_0 x_a + g^{0I} \partial_I x_a), \label{2}\end{aligned}$$ where we have defined as $\dot{x}^a = \partial_0 x^a$ and $I, J = 1, 2$. From this expression, we have the primary constraints which generate the world-volume reparametrization invariance as follows: $$\begin{aligned} {\cal{H}}_0 &=& \frac{1}{2T} P_a P^a + \frac{T}{2} h \approx 0, {\nonumber}\\ {\cal{H}}_I &=& P_a \partial_I x^a \approx 0, \label{3}\end{aligned}$$ where $h_{IJ} = \partial_I x^a \partial_J x^b \eta_{ab}$ and $h = \det h_{IJ}$. Given the Poisson brackets $$\begin{aligned} \{P_a(\sigma^0, \vec{\sigma}), x^b (\sigma^0, \vec{\sigma}')\} = - \delta_a^b \delta^2(\vec{\sigma} - \vec{\sigma}'), \label{4}\end{aligned}$$ it is easy to show that the constraints constitute of the first-class constraints as required: $$\begin{aligned} \{ {\cal{H}}_0 (\sigma^0, \vec{\sigma}), {\cal{H}}_0 (\sigma^0, \vec{\sigma}') \} &=& \Bigl[ {\cal{H}}_I(\vec{\sigma}) h(\vec{\sigma}) h^{IJ}(\vec{\sigma}) + {\cal{H}}_I(\vec{\sigma}') h(\vec{\sigma}') h^{IJ}(\vec{\sigma}') \Bigr] \partial_J \delta(\vec{\sigma} - \vec{\sigma}'), {\nonumber}\\ \{ {\cal{H}}_0 (\sigma^0, \vec{\sigma}), {\cal{H}}_I (\sigma^0, \vec{\sigma}') \} &=& \Bigl[ {\cal{H}}_0(\vec{\sigma}) + {\cal{H}}_0(\vec{\sigma}') \Bigr] \partial_I \delta(\vec{\sigma} - \vec{\sigma}'), {\nonumber}\\ \{ {\cal{H}}_I (\sigma^0, \vec{\sigma}), {\cal{H}}_J (\sigma^0, \vec{\sigma}') \} &=& {\cal{H}}_J(\vec{\sigma}) \partial_I \delta(\vec{\sigma} - \vec{\sigma}') + {\cal{H}}_I(\vec{\sigma}') \partial_J \delta(\vec{\sigma} - \vec{\sigma}'). \label{5}\end{aligned}$$ Since the Hamiltonian vanishes weakly, we can introduce the extended Hamiltonian which is purely proportional to the constraints $$\begin{aligned} H &=& \int d^2 \vec{\sigma} \Bigl[ e^0 {\cal{H}}_0 + e^I {\cal{H}}_I \Bigr] {\nonumber}\\ &=& \int d^2 \vec{\sigma} \Bigl[ e^0 (\frac{1}{2T} P_a P^a + \frac{T}{2} h) + e^I P_a \partial_I x^a \Bigr], \label{6}\end{aligned}$$ where $e^0$ and $e^I$ are the Lagrange multiplier fields. Via the Legendre transformation, we can obtain the first-order action: $$\begin{aligned} S_0 &=& \int d^3 \sigma P_a \partial_0 x^a - \int d \sigma^0 H {\nonumber}\\ &=& \int d^3 \sigma \Bigl[ P_a \partial_0 x^a - e^0 \bigl(\frac{1}{2T} P_a P^a + \frac{T}{2} h \bigr) - e^I P_a \partial_I x^a \Bigr]. \label{7}\end{aligned}$$ Note that this action is very similar to the bosonic part of the Berkovits action of supermembrane [@Ber8] in that both the actions are in the first-order Hamiltonian form and invariant under only the world-volume reparametrizations as local symmetries, so it is worthwhile to construct a bosonic matrix model from this action. Actually, we will see that the construction of M(atrix) theory follows a very similar path to the present bosonic formalism. In order to construct a matrix model, we first perform the integration over $P_a$ whose result is given by $$\begin{aligned} S_0 &=& \frac{T}{2} \int d^3 \sigma \Bigl[ \frac{1}{e^0} \bigl( \partial_0 x^a - e^I \partial_I x^a \bigr)^2 - e^0 h \Bigr] {\nonumber}\\ &=& \frac{T}{2} \int d^3 \sigma \Bigl[ \frac{1}{e^0} \bigl( \partial_0 x^a - e^I \partial_I x^a \bigr)^2 - \frac{1}{2} e^0 \{ x^a, x^b \}^2 \Bigr], \label{8}\end{aligned}$$ where in the second equation we have introduced the Lie bracket defined as $$\begin{aligned} \{ X, Y \} = \varepsilon^{IJ} \partial_I X \partial_J Y. \label{9}\end{aligned}$$ Here let us try to understand the geometrical meaning of the Lagrange multiplier fields, which can be done by comparing the above action with the Polyakov action (which is at least classically equivalent to the Nambu-Goto action (\[1\])) $$\begin{aligned} S_P = T \int d^3 \sigma \Bigl( - \frac{1}{2} \sqrt{-g} g^{ij} \partial_i x^a \partial_j x^b \eta_{ab} + \frac{1}{2} \sqrt{-g} \Bigr). \label{10}\end{aligned}$$ Then we can express the metric tensor in terms of the Lagrange multiplier fields $$\begin{aligned} g_{ij} &=& \left( \begin{array}{cc} e^I e^J h_{IJ} - (e^0)^2 h & h_{JK} e^K \\ h_{IL} e^L & h_{IJ} \end{array} \right), {\nonumber}\\ g^{ij} &=& \left( \begin{array}{cc} - \frac{1}{(e^0)^2 h} & \frac{e^J}{(e^0)^2 h} \\ \frac{e^I}{(e^0)^2 h} & h^{IJ} - \frac{e^I e^J}{(e^0)^2 h} \end{array} \right). \label{11}\end{aligned}$$ Next we will fix the reparametrization invariance by two gauge conditions [^2]. The first choice of the gauge conditions is given by $$\begin{aligned} e^0 = \frac{1}{\sqrt{h}}, \ e^I = 0, \label{12}\end{aligned}$$ or equivalently, from (\[11\]), $$\begin{aligned} g_{ij} = \left( \begin{array}{cc} - 1 & 0 \\ 0 & h_{IJ} \end{array} \right). \label{13}\end{aligned}$$ With the gauge conditions (\[12\]), the action (\[8\]) reduces to $$\begin{aligned} S_0 = \frac{T}{2} \int d \sigma^0 \int d^2 \sigma \sqrt{h} \Bigl[ \bigl( \partial_0 x^a \bigr)^2 - \frac{1}{2 h} \{ x^a, x^b \}^2 \Bigr]. \label{14}\end{aligned}$$ Finally, we make the following replacements $$\begin{aligned} \int d^2 \sigma \sqrt{h} &\rightarrow& Tr, {\nonumber}\\ \frac{1}{\sqrt{h}} \{ x^a, x^b \} &\rightarrow& i [ x^a, x^b ]. \label{15}\end{aligned}$$ Consequently, we arrive at a matrix model of the bosonic membrane $$\begin{aligned} S_0 = \int d \tau Tr \Bigl\{ \frac{1}{2} \bigl( \partial_\tau x^a \bigr)^2 + \frac{1}{4} [ x^a, x^b ]^2 \Bigr\}, \label{16}\end{aligned}$$ where we have set $T = 1$ and $\sigma^0 = \tau$. This matrix model describes a matrix model of the bosonic membrane. We can also select another form of the gauge conditions $e^0 = \frac{1}{\sqrt{h}}$ and $\ e^I = \frac{1}{\sqrt{h}} \varepsilon^{IJ} \partial_J A_0$. Then, the action (\[8\]) takes the form $$\begin{aligned} S_0 = \frac{T}{2} \int d \sigma^0 \int d^2 \sigma \sqrt{h} \Bigl[ \bigl( \partial_0 x^a + \frac{1}{\sqrt{h}} \{ A_0, x^a \} \bigr)^2 - \frac{1}{2 h} \{ x^a, x^b \}^2 \Bigr]. \label{17}\end{aligned}$$ With the replacements (\[15\]), we have a matrix model $$\begin{aligned} S_0 = \int d \tau Tr \Bigl\{ \frac{1}{2} \bigl( D_\tau x^a \bigr)^2 + \frac{1}{4} [ x^a, x^b ]^2 \Bigr\}, \label{18}\end{aligned}$$ where $D_\tau x^a = \partial_\tau x^a + i [A_\tau, x^a]$ and $A_\tau \equiv A_0$. This matrix model is obviously invariant under the $SU(N)$ gauge symmetry $$\begin{aligned} x^a &\rightarrow& x^{\prime a} = U^{-1} x^a U, {\nonumber}\\ A_\tau &\rightarrow& A^{\prime}_\tau = U^{-1} A_\tau U - i U^{-1} \partial_\tau U. \label{19}\end{aligned}$$ With the gauge condition $A_\tau = 0$, this matrix model reduces to the previous matrix model (\[16\]). Note that if we selected the light-cone gauge, the matrix model (\[18\]) would become equivalent to the bosonic part of M(atrix) theory by BFSS except irrelevant dimensional factors and numerical constants [@BFSS] [^3]. In this way, we can obtain the bosonic M(atrix) theory by starting with the bosonic membrane action and utilizing the first-order Hamiltonian formalism. A covariant matrix model of 11D superparticle ============================================= We now turn our attention to an attempt of the construction of a covariant M(atrix) theory of supermembrane in the pure spinor formalism and point out a difficulty of it. Then we construct a new matrix model of superparticle in the pure spinor formalism. Before doing so, let us start by reviewing the pure spinor formalism of supermembrane [@Ber8]. From now on, we consider only the flat membrane such as toroidal membrane where the scalar density $\sqrt{h}$ can be set to unity. The first-order Hamiltonian action of supermembrane reads $$\begin{aligned} S = \int d^3 \sigma \Bigl[ P_c \Pi_0^c + L_{WZ} + e^0 \Bigl( P_c P^c + \det \bigl(\Pi_I^c \Pi_{J c} \bigr) \Bigr) + e^I P_c \Pi_I^c \Bigr], \label{20}\end{aligned}$$ where $\Pi_i^c = \partial_i x^c + \frac{i}{2} \theta \Gamma^c \partial_i \theta$ and $L_{WZ}$ denotes the Wess-Zumino term whose concrete expression takes the form $$\begin{aligned} L_{WZ} = \frac{i}{4} \varepsilon^{ijk} \theta \Gamma_{cd} \partial_i \theta \Bigl( \Pi_j^c \Pi_k^d - \frac{i}{2} \Pi_j^c \theta \Gamma^d \partial_k \theta - \frac{1}{12} \theta \Gamma^c \partial_j \theta \theta \Gamma^d \partial_k \theta \Bigr), \label{21}\end{aligned}$$ where we define as $\varepsilon_{012}= - \varepsilon^{012}= +1$ and $\varepsilon^{0IJ}= -\varepsilon^{IJ}$. This action is invariant under the kappa symmetry and the global space-time supersymmetry as well as the world-volume reparametrizations. The primary constraints consisting of 16 first-class and 16 second-class constraints appear when we evaluate the canonical conjugate momenta $p_\alpha$ of the spinor fields $\theta^\alpha$, which are given by $$\begin{aligned} d_\alpha &\equiv& p_\alpha - \frac{\partial^R S}{\partial \dot{\theta}^\alpha} {\nonumber}\\ &=& p_\alpha - \frac{i}{2} P^c (\Gamma_c \theta)_\alpha + \frac{i}{4} \varepsilon^{IJ} (\Gamma_{cd} \theta)_\alpha \Bigl( \Pi_I^c \Pi_J^d - \frac{i}{2} \Pi_I^c \theta \Gamma^d \partial_J \theta - \frac{1}{12} \theta \Gamma^c \partial_I \theta \theta \Gamma^d \partial_J \theta \Bigr) {\nonumber}\\ &+& \frac{1}{8} \varepsilon^{IJ} \theta \Gamma_{cd} \partial_I \theta \Bigl( \Pi_J^d - \frac{i}{6} \theta \Gamma^d \partial_J \theta \Bigr) (\Gamma^c \theta)_\alpha {\nonumber}\\ &\approx& 0, \label{22}\end{aligned}$$ where the superscript $R$ on $\frac{\partial^R S}{\partial \dot{\theta}^\alpha}$ denotes the right differentiation. These constraints satisfy the following Poisson bracket $$\begin{aligned} \{ d_\alpha(\sigma^0, \vec{\sigma}), d_\beta(\sigma^0, \vec{\sigma}')\} = \Bigl[-i P_c \Gamma_{\alpha\beta}^c + \frac{i}{2} \varepsilon^{IJ} \Pi_{I c} \Pi_{J d} \Gamma_{\alpha\beta}^{cd} \Bigr] \delta^2(\vec{\sigma} - \vec{\sigma}'). \label{23}\end{aligned}$$ In deriving this equation, we need to use the eleven dimensional Fierz identity $\Gamma_{(\alpha\beta}^b \Gamma_{\gamma\delta)}^{cd} \eta_{bc} = 0$ and the Poisson brackets $$\begin{aligned} \{\tilde{P}_c(\sigma^0, \vec{\sigma}), x^d (\sigma^0, \vec{\sigma}')\} &=& - \delta_c^d \delta^2(\vec{\sigma} - \vec{\sigma}'), {\nonumber}\\ \{ p_\alpha(\sigma^0, \vec{\sigma}), \theta^\beta(\sigma^0, \vec{\sigma}')\} &=& \delta_\alpha^\beta \delta^2(\vec{\sigma} - \vec{\sigma}'), \label{24}\end{aligned}$$ where $\tilde{P}_c$, the conjugate momenta of $x^c$, are defined as $$\begin{aligned} \tilde{P}_c &\equiv& \frac{\partial S}{\partial \dot{x}^c} {\nonumber}\\ &=& P_c + \frac{i}{2} \varepsilon^{IJ} \theta \Gamma_{cd} \partial_I \theta \Bigl( \Pi_J^d - \frac{i}{4} \theta \Gamma^d \partial_J \theta \Bigr). \label{25}\end{aligned}$$ Since we cannot quantize the action (\[20\]) covariantly owing to the kappa symmetry, Berkovits has proposed a pure spinor action, which is of form $$\begin{aligned} S &=& \int d^3 \sigma \Bigl[ P_c \Pi_0^c + L_{WZ} + d_\alpha \partial_0 \theta^\alpha + w_\alpha \partial_0 \lambda^\alpha - \frac{1}{2} \Bigl( P_c P^c + \det \bigl(\Pi_I^c \Pi_{J c} \bigr)\Bigr){\nonumber}\\ &+& (d \Gamma_c \partial_I \theta) \Pi_J^c \varepsilon^{IJ} + (w \Gamma_c \partial_I \lambda) \Pi_J^c \varepsilon^{IJ} - i \varepsilon^{IJ} (w \Gamma_c \partial_I \theta)(\lambda \Gamma^c \partial_J \theta) + i \varepsilon^{IJ}(w_\alpha \partial_I \theta^\alpha)(\lambda_\beta \partial_J \theta^\beta) {\nonumber}\\ &+& e^I (P_c \Pi_I^c + d_\alpha \partial_I \theta^\alpha + w_\alpha \partial_I \lambda^\alpha) \Bigr], \label{26}\end{aligned}$$ where $d_\alpha$ is defined as in (\[22\]). In this action, the kappa symmetry has been already gauge-fixed covariantly, whereas the shift symmetries of the world-volume reparametrizations are still remained. (The lapse symmetry is gauge-fixed to $e^0 = -\frac{1}{2}$.) This action is invariant under the BRST transformation $Q_B = \int d^2 \sigma \lambda^\alpha d_\alpha$. As a peculiar feature of supermembrane, additional constraints $$\begin{aligned} \lambda \Gamma^c \lambda = 0, \ (\lambda \Gamma^{cd} \lambda) \Pi_{Jc} =0, \ \lambda_\alpha \partial_J \lambda^\alpha = 0 \label{27}\end{aligned}$$ are required to guarantee the BRST invariance of the action and the nilpotence of the BRST transformation. Note that the constraints (\[27\]) break the covariance on the world-volume explicitly. Note that the bosonic part in the pure spinor action (\[26\]) of supermembrane shares the same form as in the bosonic membrane argued in the previous section, so as in the bosonic membrane, let us proceed to integrate over $P_c$ and choose the gauge conditions $e^I = - \varepsilon^{IJ} \partial_J A_0$ [^4]. As a result, the action (\[26\]) reduces to the form $$\begin{aligned} S &=& \int d^3 \sigma \Bigl[ \frac{1}{2} (D_0 x^c + \frac{i}{2} \theta \Gamma^c D_0 \theta)^2 + L_{WZ} + d_\alpha D_0 \theta^\alpha + w_\alpha D_0 \lambda^\alpha - \frac{1}{2} \det \bigl(\Pi_I^c \Pi_{J c} \bigr) {\nonumber}\\ &+& (d \Gamma_c \partial_I \theta) \Pi_J^c \varepsilon^{IJ} + (w \Gamma_c \partial_I \lambda) \Pi_J^c \varepsilon^{IJ} - i \varepsilon^{IJ} (w \Gamma_c \partial_I \theta)(\lambda \Gamma^c \partial_J \theta) {\nonumber}\\ &+& i \varepsilon^{IJ}(w_\alpha \partial_I \theta^\alpha)(\lambda_\beta \partial_J \theta^\beta) \Bigr], \label{28}\end{aligned}$$ where we have defined as $D_0 = \partial_0 - \varepsilon^{IJ} \partial_J A_0 \partial_I$. Via the replacements (\[15\]) (recall that we have set $h =1$) from the continuum theory to the matrix model, we obtain a covariant matrix model corresponding to the pure spinor action of supermembrane: $$\begin{aligned} S &=& \int d \tau Tr \Bigg\{ \frac{1}{2} \left(D_\tau x^c + \frac{i}{4} (\theta \Gamma^c D_\tau \theta - D_\tau \theta \Gamma^c \theta) \right)^2 + d_\alpha D_\tau \theta^\alpha + w_\alpha D_\tau \lambda^\alpha {\nonumber}\\ &-& (\Gamma_c d)_\alpha \left( i [x^c, \theta^\alpha] - \frac{1}{2}(\Gamma^c \theta)_\beta \{\theta^\alpha, \theta^\beta \} \right) - (\Gamma_c w)_\alpha \left( i [x^c, \lambda^\alpha] + \frac{1}{2} (\Gamma^c \theta)_\beta [\lambda^\alpha, \theta^\beta] \right) {\nonumber}\\ &-& \left(w_\alpha \lambda_\beta - (\Gamma_c w)_\alpha (\Gamma^c \lambda)_\beta \right) \{\theta^\alpha, \theta^\beta \} + L_{WZ} + L_{\det \pi^2} \Bigg\}, \label{29}\end{aligned}$$ where we have defined the covariant derivative as $D_\tau x^c = \partial_\tau x^c + i \left[A_\tau, x^c \right]$ as before, and the curly bracket $\{\ , \ \}$ denotes the anti-commutator whereas the square bracket $[\ , \ ]$ denotes the commutator. The last two terms $L_{WZ}$ and $L_{\det \pi^2}$ come from the Wess-Zumino term and $- \frac{1}{2} \det \bigl(\Pi_I^c \Pi_{J c} \bigr)$, respectively, and involve the complicated expression. Note that in moving the continuum theory to the matrix theory we must pay attention to how to order various terms (in particular, in $L_{WZ}$ and $L_{\det \pi^2}$). Our guiding principle is to order the terms in order to keep symmetries of the theory as much as possible. At this stage, compared with the bosonic membrane in the previous section, we further have to impose the requirements of the supersymmetry and the BRST invariance on the matrix model (\[29\]). First, let us consider the supersymmetry. This symmetry is a global symmetry, so the matrix extension can be given by $$\begin{aligned} \delta x^c = \frac{i}{2} \theta \Gamma^c \epsilon, \ \delta \theta^\alpha = \epsilon^\alpha, \label{30}\end{aligned}$$ where the parameter $\epsilon$ is not a matrix but a mere number. We have checked that under this supersymmetry the matrix model (\[29\]) is invariant except the Wess-Zumino term $L_{WZ}$. To do so, we need to define $L_{\det \pi^2}$ in an appropriately ordered form: $$\begin{aligned} L_{\det \pi^2} &=& \frac{1}{4} [x^a, x^b]^2 - \frac{i}{2}[x^a, x^b] (\Gamma_a \theta)_\alpha [x_b, \theta^\alpha] {\nonumber}\\ &+& \frac{1}{8}[x^a, x^b](\Gamma_a \theta)_\alpha (\Gamma_b \theta)_\beta \{\theta^\alpha, \theta^\beta\} - \frac{1}{8}(\Gamma_a \theta)_\alpha [x^b, \theta^\alpha] (\Gamma^a \theta)_\beta [x_b, \theta^\beta] {\nonumber}\\ &+& \frac{1}{8}(\Gamma_a \theta)_\alpha [x^b, \theta^\alpha] (\Gamma_b \theta)_\beta [x^a, \theta^\beta] + \frac{i}{8}(\Gamma^a \theta)_\alpha [x^b, \theta^\alpha] (\Gamma_{[b} \theta)_\beta (\Gamma_{a]} \theta)_\rho \{\theta^\beta, \theta^\rho \} {\nonumber}\\ &+& \frac{1}{64} (\Gamma_{[a} \theta)_\alpha (\Gamma_{b]} \theta)_\beta \{\theta^\alpha, \theta^\beta \} (\Gamma^a \theta)_\rho (\Gamma^b \theta)_\sigma \{\theta^\rho, \theta^\sigma\}, \label{31}\end{aligned}$$ where we have used the notation like $(\Gamma_{[a} \theta)_\alpha (\Gamma_{b]} \theta)_\beta \equiv \frac{1}{2} \left( (\Gamma_a \theta)_\alpha (\Gamma_b \theta)_\beta - (\Gamma_b \theta)_\alpha (\Gamma_a \theta)_\beta \right)$. The reason why the Wess-Zumino term is not invariant under the supersymmetry might be related to the fact that this term breaks the $SU(N)$ gauge symmetry since it includes not the covariant derivative $D_\tau$ but the ordinary derivative $\partial_\tau$. Since the supersymmetry is an essential ingredient of our formalism, we stick to keep this symmetry and drop the Wess-Zumino term $L_{WZ}$ from the matrix theory. Furthermore, in case of supermembrane in the pure spinor formalism, we must respect the BRST symmetry. The Berkovits’ BRST symmetry is simply given by $Q_B = \int d^2 \sigma \lambda^\alpha d_\alpha$. Thus, using Eq. (\[15\]) the matrix extension must take the form of $Q_B = Tr \lambda^\alpha d_\alpha$. Since we have dropped the Wess-Zumino term, $d_\alpha$ is now simply given by $$\begin{aligned} d_\alpha = p_\alpha - \frac{i}{4} \left( P_c (\Gamma^c \theta)_\alpha + (\Gamma^c \theta)_\alpha P_c \right). \label{32}\end{aligned}$$ Then we have the following BRST transformation $$\begin{aligned} Q_B \theta^\alpha &=& \lambda^\alpha, {\nonumber}\\ Q_B x^c &=& \frac{i}{4} \left( \theta \Gamma^c \lambda + \lambda \Gamma^c \theta \right), {\nonumber}\\ Q_B d_\alpha &=& - \frac{i}{2} \left( P_c (\Gamma^c \lambda)_\alpha + (\Gamma^c \lambda)_\alpha P_c \right), {\nonumber}\\ Q_B w_\alpha &=& d_\alpha. \label{33}\end{aligned}$$ Although this BRST symmetry has a rather simple form (essentially is of the same form as in superparticle), this symmetry constrains the form of the action severely since all the fields are now promoted to matrices. Actually we can check that the BRST-invariant matrix model must take the form $$\begin{aligned} S &=& \int d \tau Tr \Bigg\{ \frac{1}{2} \left(D_\tau x^c + \frac{i}{4} (\theta \Gamma^c D_\tau \theta - D_\tau \theta \Gamma^c \theta) \right)^2 + d_\alpha D_\tau \theta^\alpha + w_\alpha D_\tau \lambda^\alpha \Bigg\}. \label{34}\end{aligned}$$ It is also worthwhile to notice that the BRST transformation is nilpotent up to the ’gauge’ transformations $$\begin{aligned} \delta_G d_\alpha &=& \frac{1}{4} \left[ (D_\tau \theta \Gamma_c \lambda + \lambda \Gamma_c D_\tau \theta) (\Gamma^c \lambda)_\alpha + (\Gamma^c \lambda)_\alpha (D_\tau \theta \Gamma_c \lambda + \lambda \Gamma_c D_\tau \theta) \right] , {\nonumber}\\ \delta_G w_\alpha &=& - \frac{i}{2} \left[ P_c (\Gamma^c\lambda)_\alpha + (\Gamma^c \lambda)_\alpha P_c \right], \label{35}\end{aligned}$$ which are indeed symmetry of the matrix model (\[34\]). Let us notice that this matrix model is nothing but the matrix model which can be obtained from the pure spinor formalism of the 11D superparticle [@Brink] by generalizing all the local fields to matrices. We shall finally make comments on some features of matrix model (\[34\]). First of all, this matrix model is not only invariant under the space-time supersymmetry and the Berkovits’ BRST transformation but also manifestly covariant under $SO(1,10)$ Lorentz group, which is the most appealing point of the model at hand. However, the matrix model does not have the potential term given by $([x^a, x^b])^2$ (which exists in $L_{\det \pi^2}$) as in the BFSS M(atrix) model so the physical properties of the both models are quite different as shown in the next section. The one loop effective potential ================================ In this section, we wish to clarify the physical properties of our new matrix model of 11D superparticle. It is well known that the superparticle action in the continuum theory [@Brink] is the zero-slope limit of the superstring theory [@GS], so it might hopefully shed some light on the underlying structure of space-time. However, as shown below by evaluating the one-loop effective potential the matrix theory of superparticle in the pure spinor formalism is a free theory, so scattering amplitudes should be calculated by determining the vertex operators and inserting them in the path integral. In order to evaluate the one-loop effective potential, we take the gauge condition $A_\tau = 0$ and introduce the FP ghosts $(\bar{C}, C)$ [^5]. After integrating over $A_\tau$, we obtain the gauge-fixed, BRST-invariant action $$\begin{aligned} S &=& \int d \tau Tr \Bigg\{ \frac{1}{2} \left(\partial_\tau x^c + \frac{i}{4} (\theta \Gamma^c \partial_\tau \theta - \partial_\tau \theta \Gamma^c \theta) \right)^2 + d_\alpha \partial_\tau \theta^\alpha + w_\alpha \partial_\tau \lambda^\alpha - \bar{C} \partial_\tau C \Bigg\}. \label{36}\end{aligned}$$ As a background, we select a non-trivial classical solution $$\begin{aligned} x^1_{(0)} = \frac{1}{2} \left( \begin{array}{cc} v \tau & 0 \\ 0 & - v \tau \end{array} \right), \ x^2_{(0)} = \frac{1}{2} \left( \begin{array}{cc} b & 0 \\ 0 & - b \end{array} \right), \label{37}\end{aligned}$$ which describes two particles moving with velocities $v/2$ and $- v/2$ and separated by the distance $b$ along the $x^2$-th axis. Around this background, we expand $x^c$ by $x^c = x^c_{(0)} + y^c$ where the fluctuation $y^c$ takes the off-diagonal form $$\begin{aligned} y^c = \left( \begin{array}{cc} 0 & y^c \\ y^{\dagger c} & 0 \end{array} \right). \label{38}\end{aligned}$$ Similarly, $C$, $\bar{C}$, $w_\alpha$, $\lambda^\alpha$, $p_\alpha$ and $\theta^\alpha$ are expanded in the off-diagonal form like $y^c$. (For convenience, we have used the same letters as the original fields for expressing the off-diagonal matrix elements.) After inserting these equations into the action (\[34\]) and taking the quadratic terms with respect to the fluctuations, we obtain the following action: $$\begin{aligned} S_2 &=& \int d \tau Tr \ \Bigl( - y^\dagger_c \partial^2_\tau y^c + p_\alpha \partial_\tau \theta^{\dagger\alpha} + p^\dagger_\alpha \partial_\tau \theta^\alpha + w_\alpha \partial_\tau \lambda^{\dagger\alpha} + w^\dagger_\alpha \partial_\tau \lambda^\alpha {\nonumber}\\ &-& \bar{C} \partial_\tau C^\dagger - \bar{C}^\dagger \partial_\tau C \Bigr). \label{39}\end{aligned}$$ In deriving this quadratic action, we have used the fact that in the one-loop approximation, we can put $P^c = \partial_\tau x^c_{(0)}$. Then the partition function is given by $$\begin{aligned} Z &=& \int {\cal D}X \ {\mbox{\rm e}}^{- S_2} {\nonumber}\\ &=& (\det \partial_\tau^2)^{-11} (\det \partial_\tau)^{-46} (\det \partial_\tau)^{64} (\det \partial_\tau)^2 (\det \partial_\tau)^2 {\nonumber}\\ &=& (\det \partial_\tau)^{-22-46+64+2+2} {\nonumber}\\ &=& 1, \label{40}\end{aligned}$$ where we have symbolically denoted the integration measure by ${\cal D}X$ and taken account of the contribution from the missing ghosts $(b, c)$ [@Ber8] in the pure spinor formalism. The result shows that at least in the one-loop level the theory is trivial, in other words, two particles do not interact with each other [^6]. Recall that in M(atrix) theory by BFSS the similar calculation leads to the phase shift of D-particles in the eikonal approximation [@BFSS; @Douglas; @Becker; @Okawa]. Our matrix theory therefore seems to be a free theory owing to the lack of the potential term $([x^a, x^b])^2$. Thus, in order to have non-trivial physical scattering amplitudes we must evaluate the expectation values of the vertex operators even in the matrix theory. Finally, let us ask ourselves why we have obtained the matrix model (\[34\]) which is quite different from the BFSS matrix model. First, we should notice that the transformation law of the supersymmetry is completely different in both the formalisms. That is, our law (\[30\]) is purely from supermembrane whereas their law is from super Yang-Mills theory [@BFSS] $$\begin{aligned} \delta X^i &=& - 2 \epsilon^T \gamma^i \theta, {\nonumber}\\ \delta \theta &=& \frac{1}{2} \left( D_t X^i \gamma_i + \gamma_{-} + \frac{1}{2} [X^i, X^j] \gamma_{ij} \right) \epsilon + \epsilon' {\nonumber}\\ \delta A_0 &=& -2 \epsilon^T \theta, \label{41}\end{aligned}$$ where $\epsilon$ and $\epsilon'$ are two independent 16 component constant parameters. (Note that $A_0$ is needed to make the algebra of supersymmetry close.) Second, the Berkovits’ BRST invariance plays a role similar to a local symmetry in the matrix model, thereby strongly restricting the form of the action of the matrix model. In particular, the non-trivial potential $([x^a, x^b])^2$ in M(atrix) theory, which is also present in the $L_{\det \pi^2}$ in Eq. (\[31\]), is not allowed to satisfy the matrix version of the Berkovits’ BRST symmetry. In any case, since the symmetries in both the present matrix model and the BFSS M(atrix) theory are different so that the two theories belong to different universality classes, it is natural to obtain the different theories in the both approaches. Conclusion ========== In this article, we have investigated the possibility of making use of the Berkovits pure spinor formalism in order to make a Lorentz covariant M(atrix) theory. We have clarified that the naive expectation of it does not work well since symmetries in the Berkovits pure spinor formalism and the BFSS M(atrix) theory are different. Moreover, we have pointed out that the Berkovits’ BRST symmetry excludes the presence of the potential $([x^a, x^b])^2$ which not only leads to an interesting interpretation of space-time relevant to the non-commutative geometry but also produces the non-trivial interaction of 11D supergravitons in M(atrix) theory. Instead, we have constructed a matrix model of 11D superparticle which is in a sense a particle limit of M(atrix) theory. Obviously we have many remaining future works to be investigated. For instance, we have not constructed the vertex operators which should be also invariant under the Berkovits’ BRST transformation. Related to this work, there is a computation of scattering amplitude using superparticle in the continuum theory [@Green] where the amplitude leads to a divergent result and the coefficient is fixed by using the duality of superstring theory. We think that once the vertex operators are constructed in the present formalism, the scattering amplitude can be calculated and gives rise to a finite result. This study is under investigation and we wish to report the results in future publication. 1 [**Acknowledgements**]{} We are grateful to N. Berkovits and M. Tonin for valuable discussions and would like to thank Dipartimento di Fisica, Universita degli Studi di Padova for its kind hospitality. This work has been partially supported by the grant from the Japan Society for the Promotion of Science, No. 14540277. 1 [99]{} T. Banks, W. Fischler, S.H. Shenker and L. Susskind, [[Phys. Rev.]{} [**D55**]{} (1997) 5112, hep-th/9610043.]{} B. de Witt, J. Hoppe and H. Nicolai, [[Nucl. Phys.]{} [**B305**]{} (1988) 545 .]{} M.B. Green and J.H. Schwarz, [[Phys. Lett.]{} [**B136**]{} (1984) 367.]{} N. Berkovits, [[JHEP]{} [**0004**]{} (2000) 018, hep-th/0001035.]{} N. Berkovits and B.C. Vallilo, [[JHEP]{} [**0007**]{} (2000) 015, hep-th/0004171.]{} N. Berkovits, [[JHEP]{} [**0009**]{} (2000) 046, hep-th/0006003.]{} N. Berkovits, [[JHEP]{} [**0108**]{} (2000) 026, hep-th/0104247.]{} N. Berkovits, [[JHEP]{} [**0109**]{} (2000) 016, hep-th/0105050.]{} A. Neveu and J.H. Schwarz, [[Nucl. Phys.]{} [**B31**]{} (1971) 86]{}; P. Ramond, [[Phys. Rev.]{} [**D3**]{} (1971) 2415.]{} I. Oda and M. Tonin, [[Phys. Lett.]{} [**B520**]{} (2001) 398, hep-th/0109051.]{} N. Berkovits and P.S. Howe, [[Nucl. Phys.]{} [**635**]{} (2002) 75, hep-th/0112160.]{} N. Berkovits and V. Pershin, [hep-th/0205154.]{} G. Trivedi, [[Mod. Phys. Lett.]{} [**A17**]{} (2002) 2239, hep-th/0205217.]{} M. Matone, L. Mazzucato, I. Oda, D. Sorokin and M. Tonin, [[Nucl. Phys.]{} [**B639**]{} (2002) 182, hep-th/0206104.]{} P.A. Grassi, G. Policastro, M. Porrati and P. van Nieuwenhuizen, [[JHEP]{} [**0210**]{} (2002) 054, hep-th/0112162]{}; P.A. Grassi, G. Policastro and P. van Nieuwenhuizen, [[JHEP]{} [**0211**]{} (2002) 004, hep-th/0202123; hep-th/0206216, 0209026.]{} Y. Aisaka and Y. Kazama, [hep-th/0212316.]{} N. Berkovits, [[JHEP]{} [**0209**]{} (2002) 051, hep-th/0201151.]{} N. Berkovits, [talk presented at the Strings 2002 conference (Cambridge), July, 2002.]{} L. Brink and J.H. Schwarz, [[Phys. Lett.]{} [**B100**]{} (1981) 310.]{} E. Bergshoeff, E. Sezgin and P.K. Townsend, [[Phys. Lett.]{} [**B189**]{} (1987) 75.]{} J. Barcelos-Neto and D.S. Santos, [[Z. Phys.]{} [**C54**]{} (1992) 101.]{} K. Fujikawa and K. Okuyama, [[Phys. Lett.]{} [**B411**]{} (1997) 261, hep-th/9706027; [Nucl. Phys.]{} [**B510**]{} (1998) 175, hep-th/9709044.]{} M. Hayakawa and N. Ishibashi, [[Nucl. Phys.]{} [**B614**]{} (2001) 171, hep-th/0107103.]{} Y. Sekine and T. Yoneya, [[Nucl. Phys.]{} [**B619**]{} (2001) 22, hep-th/0108176.]{} I. Oda, [[Mod. Phys. Lett.]{} [**A13**]{} (1998) 203, hep-th/9709005; [Nucl. Phys.]{} [**B516**]{} (1998) 160, hep-th/9710030; [Phys. Lett.]{} [**B427**]{} (1998) 267, hep-th/9801051; Chaos, Solitons and Fractals on “Superstrings, M, F, S,...Theory”, Vol. 10, (1999) 483, hep-th/9806096.]{} M.D. Douglas, D. Kabat, P. Pouliot and S. Shenker, [[Nucl. Phys.]{} [**B485**]{} (1997) 85, hep-th/9608024.]{} K. Becker and M. Becker, [[Nucl. Phys.]{} [**B506**]{} (1997) 48, hep-th/9705091.]{} Y. Okawa and T. Yoneya, [[Nucl. Phys.]{} [**B538**]{} (1999) 67, hep-th/9806108.]{} M.B. Green, H. Kwon and M. Gutperle, [[JHEP]{} [**9908**]{} (1999) 012, hep-th/9907155.]{} [^1]: E-mail address: ioda@edogawa-u.ac.jp [^2]: For comparison with the case of supermembrane in the next section, we will not take the light-cone gauge explicitly in what follows. [^3]: M(atrix) theory manifestly depends on the background flat metric, so it is not a background independent formalism. See [@Oda3] for the pioneering works of the background independent matrix models. [^4]: Comparing (\[7\]) and (\[26\]), we notice that the definition of $e^0$ and $e^I$ in supermembrane differs from that in the bosonic membrane by the minus sign. [^5]: See [@Douglas; @Becker; @Okawa] for calculations of the effective action in M(atrix) theory. [^6]: One subtle point of the above calculation is that we have taken the ’axial’ gauge $A_\tau = 0$. It is known that the effective potential in general depends on the gauge conditions whereas the S-matrix does not depend on the gauge. To have the gauge-invariant effective action, we usually take the background field-dependent gauges as in [@Douglas; @Becker; @Okawa], which guarantees the gauge invariance at all the stage of calculations. Our result obtained above, however, is manifestly gauge-invariant so it is free from the problem of the gauge dependence.

--- author: - 'Timnit Gebru[^1]' - Jamie Morgenstern - Briana Vecchione - Jennifer Wortman Vaughan - Hanna Wallach - Hal Daumé III - Kate Crawford bibliography: - 'sample-base.bib' title: Datasheets for Datasets --- Introduction ============ \[sec:intro\] Objectives {#sec:objectives} ---------- Parallels in Other Contexts {#sec:standards} =========================== To contextualize and motivate our proposal, we first discuss parallels in two other contexts. #### Electronics. #### Medicine. Questions and Workflow {#sec:workflow_uses} ====================== Development Process {#sec:process} =================== Impact and Challenges {#sec:limitations} ===================== Acknowledgments {#sec:acknowledgments .unnumbered} --------------- Appendix ======== [^1]: Much of this research was conducted while Gebru, Morgenstern, and Vecchione were at Microsoft.

--- author: - 'Yosuke <span style="font-variant:small-caps;">Imamura</span>[^1]andKeisuke <span style="font-variant:small-caps;">Kimura</span>[^2]' title: 'On the Moduli Space of Elliptic Maxwell-Chern-Simons Theories' --- Introduction ============ Recently, there has been great interest in 3-dimensional superconformal field theories as theories for describing multiple M2-branes in various backgrounds. This was triggered by the proposal of a new class of 3-dimensional theories by Bagger and Lambert[@Bagger:2006sk; @Bagger:2007jr; @Bagger:2007vi], and Gusstavson[@Gustavsson:2007vu; @Gustavsson:2008dy]. The model (BLG model) possesses ${\cal N}_{(d=3)}=8$ superconformal symmetry and is based on Lie $3$-algebra. The action of the BLG model includes the structure constant $f^{abc}{}_d$ of a Lie $3$-algebra, which determines the form of the interactions, and a metric $h^{ab}$, which appears in the coefficients of the kinetic terms. These tensors must satisfy certain conditions required by the supersymmetry invariance of the action. If these tensors satisfy the conditions, we can write down the action of a BLG model. The constraint imposed on the structure constant is called a fundamental identity. It was soon realized that the identity is very restrictive[@Ho:2008bn], and it was proved that if we assume that the metric is positive definite and the algebra is finite dimensional, there is only one nontrivial Lie $3$-algebra[@Papadopoulos:2008sk; @Gauntlett:2008uf], which is called an $A_4$ algebra. The BLG model based on the $A_4$ algebra is a $SU(2)\times SU(2)$ Chern-Simons theory with levels $k$ and $-k$ for each $SU(2)$ factor. Analysis of this model showed that it describes a pair of M2-branes in certain orbifold backgrounds[@VanRaamsdonk:2008ft; @Lambert:2008et; @Distler:2008mk]. As a theory for an arbitrary number of M2-branes, a model based on an algebra with a Lorenzian metric was proposed in Refs. . Because of the indefinite metric, the model includes unwanted ghost modes. Although the ghost modes can be removed by treating them as background fields satisfying classical equations of motion[@Ho:2008ei; @Honma:2008un], or by gauging certain symmetries and fixing them[@Bandres:2008kj; @Gomis:2008be], this procedure breaks the conformal invariance, and the theory becomes D2-brane theory[@Ho:2008ei; @Bandres:2008kj; @Ezhuthachan:2008ch] by the mechanism proposed in Ref.  unless the parameter corresponding to the Yang-Mills coupling is sent to infinity or integrated over all values as a dynamical parameter[@Gomis:2008be]. There has also been some progress in 3-dimensional Chern-Simons theories with supersymmetries of less than $8$, which are closely related to M2-branes. Gaiotto and Witten[@Gaiotto:2008sd] proposed ${\cal N}_{(\rm d=3)}=4$ superconformal Chern-Simons theories, and Hosomichi et al.[@Hosomichi:2008jd] extended the theories by introducing twisted hypermultiplets. They derived the relation between their models and the BLG model, and showed that the $A_4$ BLG model is included as a special case of their ${\cal N}_{(d=3)}=4$ Chern-Simons theories. They also studied the M-crystal model [@Lee:2006hw; @Lee:2007kv; @Kim:2007ic], which is described by a circular quiver diagram with $2n$ vertices. The vertices represent Chern-Simons fields at level $\pm k$ with alternate signatures, and by analyzing the moduli space of the model they showed that it can be regarded as a theory describing M2-branes in the orbifold $({\mathbb C}^2/{\mathbb Z}_n)^2$.[^3] They also presented a realization of this model by using D3-, D5-, and NS5-branes, which give the model at level $\pm1$, and reproduce the orbifold as the M-theory dual of the brane system. Aharony et al. also proposed a similar model[@Aharony:2008ug] based on $U(N)\times U(N)$ Chern-Simons theory with levels $k$ and $-k$ for each $U(N)$ factor. They showed that the action possesses ${\cal N}_{(d=3)}=6$ superconformal symmetry, and describes $N$ M2-branes in the orbifold ${\mathbb C}^4/{\mathbb Z}_k$. Although ${\cal N}_{(d=3)}=8$ supersymmetry, which is expected when $k=1$ or $2$ is not manifest, the action does not have dimensionful parameters and the scale invariance is manifest. In Ref.  it is also shown that the theory can be realized as a theory on a brane system in type IIB string theory. The brane system consists of $N$ D3-, one NS5-, and one $(k,1)$5-branes. They showed that by T-duality and M-theory lift, M2-branes in the orbifold ${\mathbb C}^4/{\mathbb Z}_k$ are obtained. The purpose of this paper is to extend the models proposed in Refs.  and by generalizing the brane configurations in these references. In §\[brane.sec\] we consider a brane system with $n_A$ NS5-branes and $n_B$ $(k,1)$5-branes, and analyze the moduli space of the theory realized by the brane system. The theory is a $U(N)^{n_A+n_B}$ quiver gauge theory with nonvanishing Chern-Simons terms for some of the $U(N)$ factors. Some of the $U(N)$ fields are Yang-Mills fields without Chern-Simons coupling. The supersymmetry of this theory is ${\cal N}_{(d=3)}=3$, which is expected to be enhanced to ${\cal N}_{(d=3)}=4$ in the strong gauge-coupling limit. The reason for this is as follows. This theory can be obtained from the $U(N)\times U(N)$ theory proposed in Ref.  by combining two extensions. One is the inclusion of twisted hypermultiplets, as mentioned above, and the other is the inclusion of gauge groups with vanishing Chern-Simons couplings. The latter extension is discussed in Ref.  to describe general nonlinear sigma models of hypermultiplets. Both extensions are known to give ${\cal N}_{(d=3)}=4$ supersymmetric Chern-Simons theory, and it is plausible that the theory we discuss in this paper possesses ${\cal N}_{(d=3)}=4$ supersymmetry. In §\[moduli.sec\] we determine the moduli space of the theory. We focus only on the Higgs branch, which describes a mobile M2-brane. Under a certain assumption for flux quantization, we obtain a 4-dimensional orbifold ${\mathbb C}^4/\Gamma$, where $\Gamma$ is a discrete subgroup depending on $k$, $n_A$, and $n_B$. We reproduce the same orbifold in §\[dual.sec\] as an M-theory dual of the brane configuration. In §\[more.sec\] we consider models with more than two kinds of fivebranes. The moduli space is also a 4-dimensional manifold, but it is nontoric. The last section is devoted to discussion. Brane configuration and action {#brane.sec} ============================== The model proposed in Ref.  is a Chern-Simons theory with a $U(N)\times U(N)$ gauge group. It can be realized as a theory based on a brane system consisting of $N$ D3-branes, one NS5-brane, and one $(k,1)$5-brane. All these branes share the directions of 012, which are the coordinates of the 3-dimensional field theory. The $N$ D3-branes are wrapped on the compact direction 9. The NS5-brane and the $(k,1)$5-brane are spread along the 012345 and $012[36]_{\theta_1}[47]_{\theta_2}[58]_{\theta_3}$ directions, respectively, where $[ij]_\theta$ is the direction in the $i$-$j$ plane specified by the angle $\theta$. The angles $\theta_{1,2,3}$ are determined by the BPS conditions. ![Brane configuration for the $U(N)\times U(N)$ Chern-Simons model.[]{data-label="brane.eps"}](brane.eps) We refer to NS5- and $(k,1)$5-branes as A- and B-branes, respectively. The D3-brane worldvolume is divided into two parts by the intersecting fivebranes (Fig. \[brane.eps\]), and a $U(N)$ vector multiplet exists on each segment. Bifundamental chiral multiplets also arise at the intersections. This brane system is similar to the D4-NS5 system realizing the Klebanov-Witten theory[@Klebanov:1998hh], which is a 4-dimensional ${\cal N}_{(d=4)}=1$ superconformal field theory. In the D4-NS5 system, we have $N$ D4-branes wrapped on ${\mathbb S}^1$, instead of D3-branes, and the A- and B-branes in this case are NS5-branes along different directions. We generalize the D3-fivebrane system by introducing an arbitrary number of fivebranes. In the case of 4-dimensional ${\cal N}_{(d=4)}=1$ gauge theories, such a generalization is known as an elliptic model, and has been studied in detail[@Uranga:1998vf; @vonUnge:1999hc]. It is known that the moduli spaces of the theories are generalized conifolds. We here carry out a similar analysis in the 3-dimensional case. Let $n_A$ and $n_B$ be the numbers of A- and B-branes, respectively. We denote the total number of fivebranes by $n=n_A+n_B$. Let us label the fivebranes by $I=1,\ldots,n$ according to their order along ${\mathbb S}^1$. We identify $I=n+1$ with $I=1$. On the interval of D3-branes between two fivebranes $I$ and $I+1$, we have a $U(N)$ vector multiplet $V_I$ and an adjoint chiral multiplet $\Phi_I$. (We use the terminology of ${\cal N}_{(d=4)}=(1/2){\cal N}_{(d=3)}=1$ supersymmetry.) The kinetic terms of these multiplets are $$\begin{aligned} S_V&=&\int d^3x\sum_I\frac{1}{g_I^2}\tr \left[ -\frac{1}{4}(F^I_{\mu\nu})^2-\frac{1}{2}(D_\mu\sigma_I)^2+\frac{1}{2}D_I^2 +\mbox{fermions}\right], \label{sv}\\ S_\Phi&=&\int d^3xd^4\theta\sum_I \frac{1}{g_I^2}\tr (\Phi_I^*e^{V_I}\Phi_Ie^{-V_I}).\label{sphi}\end{aligned}$$ $\sigma_I$ is the real scalar field in the vector multiplet $V_I$. The adjoint chiral multiplets $\Phi_I$ describe the motion of the D-branes along the fivebranes. When two fivebranes $I$ and $I+1$ are not parallel, the chiral multiplet $\Phi_I$ becomes massive, and the mass term is described by the superpotential $$W=\frac{\mu}{2}\sum_I(q_{I+1}-q_I)\Phi_I^2, \label{massterm}$$ where $q_I=0$ for A-branes and $q_I=1$ for B-branes. The overall factor $\mu$ is related to the relative angle between A- and B-branes. We also have bifundamental chiral multiplets $X_I$ and $Y_I$, which arise from open strings stretched between two intervals of D-branes divided by the $I$th fivebrane. (See Fig. \[xyfv.eps\].) ![Brane system and fields.[]{data-label="xyfv.eps"}](xyfv.eps) These fields belong to the following representations of $U(N)_I\times U(N)_{I-1}$, where $U(N)_I$ is the gauge group associated with the vector multiplet $V_I$: $$X_I:(N,{\overline}N),\quad Y_I:({\overline}N,N).$$ The kinetic terms of these bifundamental fields are $$\begin{aligned} S_{XY} &=&\int d^3x d^4\theta\sum_{I=1}^n\tr\left[ X_I^*e^V_IX_Ie^{-V_{I-1}} +Y_Ie^{-V_I}Y^*_Ie^{V_{I-1}} \right] \nonumber\\ &=&\int d^3x \sum_{I=1}^n\tr\left[ -D_I(|X_I|^2-|Y_I|^2-|X_{I+1}|^2+|Y_{I+1}|^2) \right. \nonumber\\&& \left. -(|X_I|^2+|Y_I|^2)(\sigma_I-\sigma_{I-1})^2 +|F^X_I|^2 +|F^Y_I|^2 \right] +\cdots.\end{aligned}$$ In the component expression we show only the bosonic terms without derivatives. These bifundamental fields couple to the adjoint chiral multiplets through the superpotential $$W=\sum_{I=1}^n\tr\Phi_I(X_IY_I-Y_{I+1}X_{I+1}). \label{n2term}$$ The difference between the RR-charges of the A- and B-branes generates Chern-Simons terms[@Kitao:1998mf; @Bergman:1999na]. The bosonic part of the ${\cal N}_{d=3}=2$ completion of the Chern-Simons terms is $$S_{\rm CS} =\sum_{I=1}^n\frac{k_I}{2\pi}\int d^3x\tr\left[ \epsilon^{\mu\nu\rho} \left(\frac{1}{2}A^I_\mu\partial_\nu A^I_\rho +\frac{1}{3}A^I_\mu A^I_\nu A^I_\rho\right) +\sigma_I D_I \right], \label{csterm}$$ where the Chern-Simons coupling $k_I$ is given by $$k_I=k(q_{I+1}-q_I). \label{cscouplings}$$ We assume that $k$ is a positive integer. The Chern-Simons terms in (\[csterm\]) cause some of the vector multiplets to be massive. The masses $\sim k_Ig_I^2$ are proportional to the masses of adjoint chiral multiplets $\Phi_I$. We can promote the supersymmetry of this theory to ${\cal N}_{(d=3)}=3$ by matching the masses of $V_I$ and $\Phi_I$ by setting $\mu=k$. In $3$-dimensional field theories the coupling constants $g_I$ have mass dimension $1/2$, and taking the low-energy limit is equivalent to taking the strong-coupling limit $g_I\rightarrow\infty$. This makes the masses of $V_I$ and $\Phi_I$ infinity unless $k_I=0$, and we can integrate out the massive adjoint chiral multiplets. After this, the superpotential becomes[^4] $$W=\sum_{q_I=q_{I+1}}\tr\Phi_I(X_IY_I-Y_{I+1}X_{I+1}) +\sum_{q_I\neq q_{I+1}}(q_{I+1}-q_I)\tr(X_IY_IY_{I+1}X_{I+1}). \label{ellw}$$ Moduli space {#moduli.sec} ============ In this section we investigate the moduli space of the 3-dimensional field theory defined in the previous section. As we mentioned at the end of the previous section we need to take the strong-coupling limit $g_I\rightarrow\infty$ to obtain the conformal theory describing the low-energy limit of M2-branes. Although the dynamics in such a strong coupling region is highly nontrivial, we assume that the vacuum structure is not affected by quantum corrections, and we consider only the classical equations of motion derived from the action given in the previous section. In the strong-coupling limit, the kinetic terms (\[sv\]) and (\[sphi\]) vanish, and the fields $\phi_I$, the scalar components of $\Phi_I$, and $\sigma_I$ become auxiliary fields. The bifundamental chiral multiplets $X_I$ and $Y_I$ are still dynamical, and the moduli space is parameterized by the scalar components of these multiplets. We are interested in the moduli space for a single M2-brane, and we set $N=1$. Furthermore, we here focus only on the Higgs branch, which describes a mobile M2-brane, and assume $$X_I, Y_I\neq0. \label{coulomb}$$ F-term conditions ----------------- Let us first consider the F-term conditions derived from the superpotential (\[ellw\]). Because the superpotential is the same as the 4-dimensional elliptic model realized by the D4-NS5 brane system, the F-term conditions are also the same. Under the assumption (\[coulomb\]), the F-term conditions give the following solution: $$\Phi_{I\in A}=M_{I\in B}=u,\quad \Phi_{I\in B}=M_{I\in A}=v, \label{fcond}$$ where we define the mesonic operators as $M_I=X_IY_I$. $I\in A$ ($I\in B$) means that index $I$ is restricted to the values with $q_I=0$ ($q_I=1$). Although not directly related to our model, it may be instructive to demonstrate how we can obtain a Calabi-Yau $3$-fold as the moduli space of a 4-dimensional elliptic model in the case of the D4-NS5 system. In this case two complex numbers $u$ and $v$ can be interpreted as the coordinates of the D4-brane along B- and A-branes, respectively. The 4-dimensional theory possesses $U(1)^{n-1}$ gauge symmetry. In addition to the mesonic operators $M_I$, we can construct the gauge-invariant baryonic operators $$x=\prod_{I=1}^nX_I,\quad y=\prod_{I=1}^nY_I. \label{baryon}$$ By definition, these gauge-invariant operators are related by $$xy=u^{n_A} v^{n_B}.$$ This algebraic equation defines a Calabi-Yau 3-fold, which is often called a generalized conifold. The toric diagram of this generalized conifold is shown in Fig. \[gc.eps\]. ![Toric diagram of a generalized conifold.[]{data-label="gc.eps"}](gc.eps) D-term conditions {#dterm.sec} ----------------- In the strong-coupling limit $g_I\rightarrow\infty$, the vector multiplet $V_I$ includes two auxiliary fields $\sigma_I$ and $D_I$. The terms in the action including these auxiliary fields are $$\begin{aligned} S&=&\sum_{I=1}^n\left[ k_I\sigma_ID_I -D_I(|X_I|^2-|Y_I|^2-|X_{I+1}|^2+|Y_{I+1}|^2) \right. \nonumber\\&& \left. -(|X_I|^2+|Y_I|^2)(\sigma_I-\sigma_{I-1})^2 \right]. \label{daction}\end{aligned}$$ In this action, $D_I$ are Lagrange multipliers, and give the constraint $$k_I\sigma_I=|X_I|^2-|Y_I|^2-|X_{I+1}|^2+|Y_{I+1}|^2. \label{Dconst}$$ If we substitute this into the action (\[daction\]), the first line vanishes and the potential becomes $$V=\sum_{I=1}^n(|X_I|^2+|Y_I|^2)(\sigma_I-\sigma_{I-1})^2.$$ Because of the assumption (\[coulomb\]), vacua are given by $\sigma_I=\sigma_{I-1}$. Namely, all $\sigma_I$ are the same. Let $\sigma$ be the common value of $\sigma_I$. Then the constraint (\[Dconst\]) becomes $$q_I\sigma-(|X_I|^2-|Y_I|^2) =q_{I+1}\sigma-(|X_{I+1}|^2-|Y_{I+1}|^2).$$ This means that the left- and right-hand sides of this equation do not depend on the index $I$. Thus, we can write $$|X_I|^2-|Y_I|^2=q_I\sigma+c \label{dterm}$$ with a constant $c$. Although (\[dterm\]) is not the equation of motion of $D_I$, we can formally interpret it as an ordinary D-term condition associated with a certain symmetry. To rewrite (\[dterm\]) in the form of an ordinary D-term condition, let us define $U(1)$ transformation groups $G_I$ that act only on $X_I$ and $Y_I$ as $$G_I: X_I\rightarrow e^{i\lambda_I}X_I,\quad Y_I\rightarrow e^{-i\lambda_I}Y_I,$$ where $\lambda_I$ is a parameter of $G_I$. The groups $G_I$ are different from $U(1)_I$ defined in the previous section. The parameters $\alpha_I$ of $U(1)_I$ and $\lambda_I$ of $G_I$ are related by $$\lambda_I=\alpha_I-\alpha_{I-1}. \label{lambdaalpha}$$ Although each $G_I$ is not a symmetry of the theory, it is convenient to describe symmetry groups as subgroups of $\prod_IG_I$. For example, the gauge symmetry $G=U(1)^{n-1}$, which does not include the diagonal $U(1)$ decoupling from the theory, is the subgroup of $\prod_IG_I$ that does not rotate the baryonic operators (\[baryon\]). Let us rewrite (\[dterm\]) in the form of a D-term condition. Equation (\[dterm\]) is equivalent to the condition $$\sum_{I=1}^l\lambda_I(|X_I|^2-|Y_I|^2)=0, \label{dtermc}$$ for arbitrary $\lambda_I$ satisfying the constraints $$\sum_{I=1}^n\lambda_I=\sum_{I=1}^nq_I\lambda_I=0. \label{subg}$$ If we regard $\lambda_I$ as the parameters of $G_I$ transformations, the constraints (\[subg\]) imposed on $\lambda_I$ define a subgroup $H=U(1)^{n-2}$ of $\prod_I G_I$. Equation (\[dtermc\]) can be regarded as the D-term condition for $H$. We emphasize that we do not claim at this point that the gauge symmetry of the theory is $H$ or that relation (\[dterm\]) is obtained as the equations of motion of auxiliary fields in the vector multiplets associated with $H$. We only claim that the vacuum condition (\[dterm\]) is similar to the D-term condition of a gauge theory with the gauge symmetry $H$. In the next subsection, however, we will show that $H$ indeed emerges as the unbroken continuous gauge symmetry. It is convenient to define the subgroup $H$ in another way. Let us define the baryonic operators $$x_A=\prod_{I\in A}X_I,\quad y_A=\prod_{I\in A}Y_I,\quad x_B=\prod_{I\in B}X_I,\quad y_B=\prod_{I\in B}Y_I. \label{baryonicops}$$ The group $H$ can be defined as the subgroup of $\prod_IG_I$ that does not rotate these baryonic operators. Gauge symmetry -------------- To obtain the moduli space of a gauge theory, we need to remove unphysical degrees of freedom corresponding to gauge symmetries. In the case of Chern-Simons theories, we should carefully take account of symmetry breaking due to the existence of magnetic monopoles. Let us rewrite the abelian Chern-Simons terms in the form $$S_{\rm CS} =-\frac{k}{2\pi}\sum_{I=1}^nq_I(A^I-A^{I-1})\wedge {\widetilde}F +(\mbox{quadratic terms of $A^I-A^{I-1}$}), \label{csdiag}$$ where ${\widetilde}F$ is the field strength of the diagonal $U(1)$ gauge field ${\widetilde}A=(1/n)(A^1+A^2+\cdots+A^n)$. Equation (\[csdiag\]) is obtained by substituting $$A^I={\widetilde}A+(\mbox{linear combination of $A^I-A^{I-1}$})$$ into the Chern-Simons term in (\[csterm\]). The quadratic term of ${\widetilde}A$ vanishes because $\sum_Ik_I=0$. Because the diagonal gauge field ${\widetilde}A$ appears only in the first term of (\[csdiag\]), we can dualize it by adding the term $$\frac{1}{2\pi}\int d\tau\wedge{\widetilde}F,$$ and treating ${\widetilde}F$ as an unconstrained field. The equation of motion of ${\widetilde}F$ gives $$\sum_{I=1}^n k_IA_I=d\tau.$$ Upon the gauge transformation $\delta A_I=d\alpha_I$, the scalar field $\tau$ is transformed as $$\delta \tau=\sum_{I=1}^n k_I\alpha_I.$$ Let us assume that the period of $\tau$ is $2\pi$. This implies that the flux $\oint {\widetilde}F$ is quantized by $$\int{\widetilde}F\in 2\pi{\mathbb Z}. \label{feqf2}$$ Although we could not show this flux quantization on the field-theory side, we will later show that the moduli space obtained by assuming (\[feqf2\]) coincides with that obtained from the brane configuration by the T-duality and M-theory lift. If we adopt this assumption, the gauge fixing $\tau=0$ partially breaks the gauge symmetry and imposes the following constraint on the parameters $\lambda_I$ and $\alpha_I$: $$\sum_{I=1}^n k_I\alpha_I=k\sum_{I=1}^nq_I\lambda_I\in 2\pi{\mathbb Z}. \label{ubk}$$ (In the first equality we used (\[cscouplings\]) and (\[lambdaalpha\]).) Let us first focus on the continuous subgroup. It is generated by parameters satisfying $$\sum_{I=1}^n\lambda_I=\sum_{I=1}^nq_I\lambda_I=0. \label{sumqlambda}$$ The group defined by (\[sumqlambda\]) is simply group $H$ defined in §\[dterm.sec\]. Because of the emergence of the same group $H$ both in the equations of motion of auxiliary fields and in the unbroken gauge symmetry, we can obtain the moduli space as the coset ${\cal M}/H_{\mathbb C}$ or its orbifold, where ${\cal M}$ is the complex manifold defined by the F-term conditions and $H_{\mathbb C}$ is the complexification of the group $H$. This guarantees that the moduli space is a complex manifold. In addition to $H$, the group defined by (\[ubk\]) includes the discrete symmetry, which rotates the baryonic operators in (\[baryonicops\]) as $$x_A\rightarrow e^{\frac{2\pi i}{k}}x_A,\quad y_A\rightarrow e^{-\frac{2\pi i}{k}}y_A,\quad x_B\rightarrow e^{-\frac{2\pi i}{k}}x_B,\quad y_B\rightarrow e^{\frac{2\pi i}{k}}y_B. \label{discrete}$$ Moduli space {#moduli.ssec} ------------ Let us determine the moduli space. We first consider the $k=1$ case. In this case, the discrete gauge symmetry (\[discrete\]) becomes trivial, and we have the gauge-invariant operators $$u,\quad v,\quad x_A,\quad y_A,\quad x_B,\quad y_B. \label{giops}$$ By definition, these operators satisfy the following equations: $$x_Ay_A=u^{n_A},\quad x_By_B=v^{n_B}. \label{defeq}$$ These equations define the orbifold ${\mathbb C}^2/{\mathbb Z}_{n_A}\times{\mathbb C}^2/{\mathbb Z}_{n_B}$. Actually, the relation (\[defeq\]) can be solved as $$x_A=z_1^{n_A},\quad y_A=z_2^{n_A},\quad u=z_1z_2,\quad x_B=z_3^{n_B},\quad y_B=z_4^{n_B},\quad v=z_3z_4. \label{zdef}$$ We can identify $z_i$ as the coordinates of ${\mathbb C}^4$, the covering space of the orbifold. None of the variables in (\[giops\]) are changed by the transformations $$(z_1,z_2,z_3,z_4)\rightarrow (e^{2\pi i/n_A}z_1, e^{-2\pi i/n_A}z_2, z_3,z_4) \label{zna}$$ and $$(z_1,z_2,z_3,z_4)\rightarrow (z_1,z_2, e^{2\pi i/n_B}z_3, e^{-2\pi i/n_B}z_4). \label{znb}$$ Points in ${\mathbb C}^4$ mapped by these transformations should be identified with each other, and this identification defines the above orbifold. If $n_A=n_B$, the moduli space agrees with the result in Ref. , in which alternate A- and B-branes are considered. It is interesting that the moduli space does not depend on the order of the two kinds of fivebranes. Next, let us consider the case when $k>1$. In this case, we should take account of the discrete gauge transformation (\[discrete\]). The transformation of $z_i$ reproducing (\[discrete\]) is $$(z_1,z_2,z_3,z_4)\rightarrow (e^{2\pi i/kn_A}z_1,e^{-2\pi i/kn_A}z_2,e^{-2\pi i/kn_B}z_3,e^{2\pi i/kn_B}z_4). \label{znc}$$ The three transformations (\[zna\]), (\[znb\]), and (\[znc\]) generate a discrete subgroup of $U(1)^2$ with $kn_An_B$ elements. Let $\Gamma$ be this discrete group. The moduli space for general $k$ is the abelian orbifold ${\mathbb C}^4/\Gamma$. M-theory dual {#dual.sec} ============= In the previous section, we obtained the 4-dimensional orbifold ${\mathbb C}^4/\Gamma$ as the Higgs branch of the moduli space. The purpose of this section is to reproduce the same orbifold by the T-duality transformation and the M-theory lift from the D3-fivebrane system in type IIB string theory. For simplicity, we first consider a system in which the $(k,1)$5-branes are replaced by D5-branes. After determining the mapping from type IIB string theory to M-theory for NS5- and D5-branes, the dual object for the bound state of these two kinds of branes is easily obtained by superposing the objects for NS5- and D5-branes. Although the tilted angle of the branes should be appropriately chosen according to the charges of branes to preserve supersymmetry, we do not do this because the toric data do not change upon continuous deformations of the manifold and because we can determine the toric data of the dual geometry by using only the topological information. We start from the brane configuration for type IIB string theory in Table \[d5ns5.tbl\]. 0 1 2 3 4 5 6 7 8 9 ----- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- D3 $\circ$ $\circ$ $\circ$ $\circ$ D5 $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ NS5 $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ : Brane configuration in type IIB string theory[]{data-label="d5ns5.tbl"} Direction $9$ is compactified on ${\mathbb S}^1$. We replaced the $(k,1)$5-brane with the D5-brane and use a coordinate system in which the D5-brane is spread along 012678. In general, the $(k,1)$5-branes are not perpendicular to the NS5-brane, thus we use slanted coordinates. We first rearrange the coordinates in $4578$ space by using the Hopf fibration. We define $r_a$ ($a=1,2,3$) by $$r_a=u^\dagger \sigma_au,\quad u=\left(\begin{array}{c}x^4+ix^5\\x^7+ix^8\end{array}\right),$$ and we let $\psi$ be the coordinate of the ${\mathbb S}^1$ fiber. Then the NS5 and D5 worldvolumes are on the positive and negative parts of the $r_3$ axis, respectively, in the $r_a$ space. See Table \[d5ns52.tbl\]. The $\psi$ cycle shrinks at the center of the $r_a$ space, which is shown in the table as “KKM”. “s” in the table represents the shrinking cycle. 0 1 2 3 6 $r_3$ $r_1$ $r_2$ $\psi$ 9 ----- --------- --------- --------- --------- --------- ------- ------- ------- --------- --------- D3 $\circ$ $\circ$ $\circ$ $\circ$ D5 $\circ$ $\circ$ $\circ$ $\circ$ $-$ $\circ$ NS5 $\circ$ $\circ$ $\circ$ $\circ$ $+$ $\circ$ KKM $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ s $\circ$ : The same configuration as Table \[d5ns5.tbl\] with a different coordinate system. “s” represents the shrinking cycle and $+$ and $-$ mean that the branes are spread along the positive or negative part of the axis, respectively.[]{data-label="d5ns52.tbl"} Let us perform the T-duality transformation along direction $9$, and lift the configuration into M-theory. 0 1 2 3 6 $r_3$ $r_1$ $r_2$ $\psi$ 9 M ----------------------- --------- --------- --------- --------- --------- ------- ------- ------- --------- --------- --------- (D3$\rightarrow$)M2 $\circ$ $\circ$ $\circ$ (D5$\rightarrow$)KKM $\circ$ $\circ$ $\circ$ $\circ$ $-$ $\circ$ s $\circ$ (NS5$\rightarrow$)KKM $\circ$ $\circ$ $\circ$ $\circ$ $+$ $\circ$ $\circ$ s KKM $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ s $\circ$ $\circ$ : M-theory dual of the brane configuration.[]{data-label="d5ns53.tbl"} The D3-branes are mapped to M2-branes as shown in Table \[d5ns53.tbl\]. A single NS5-brane and a single D5-brane become KKM-branes associated with the $(1,0,0)$ and $(0,1,0)$ cycles, respectively, where the first, second, and last components correspond to the $M$, $9$, and $\psi$ coordinates, respectively. If we start with a $(k,1)$5-brane, which is the bound state of $k$ D5-branes and one NS5-brane, we obtain a single KKM-brane with $(0,1,k)$ cycle shrinking. In addition to these, we have one more KKM-brane, which originates from the special choice of the coordinates. The existence of the other KKM-branes make the shrinking cycle of the last KKM-brane ambiguous, and only the last component of the shrinking cycle has a definite value of $1$. The intersection with other branes changes the shrinking cycle, and the cycle should be determined according to the “charge conservation” of the KKM branes. See Fig. \[c4.eps\](a). ![(a) M-theory dual of the D3-fivebrane system in $36r_3$ space. This can be regarded as a webdiagram of the toric geometry. The corresponding toric diagram is shown in (b).[]{data-label="c4.eps"}](c4.eps) This system of KKM-branes in $36r_3$ space is simply a webdiagram describing a 4-dimensional toric manifold. We can easily obtain the toric diagram as a dual graph of the webdiagram. (Fig. \[c4.eps\](b)) The toric variety described by this diagram is in fact the orbifold we obtained in §\[moduli.ssec\], as we show in the rest of this section. The structure of a toric variety is mostly determined by the toric data, which are a set of generators of shrinking cycles. The generators are usually represented as vectors $\vec v_i$ in the lattice associated with the toric fiber. The toric data of ${\mathbb C}^4$ are given by $\vec v_i=\vec e_i$ ($i=1,2,3,4$), where $\vec e_i$ are the unit vectors in the 4-dimensional lattice. $$\vec e_1=(1,0,0,0),\quad \vec e_2=(0,1,0,0),\quad \vec e_3=(0,0,1,0),\quad \vec e_4=(0,0,0,1).$$ The orbifolding of a toric variety is realized by refining the lattice by adding new generators. In the case of the orbifold defined by (\[zna\])–(\[znc\]), we add three generators $$\begin{aligned} \vec e_5&=&\left(\frac{1}{n_A},-\frac{1}{n_A},0,0\right),\nonumber\\ \vec e_6&=&\left(0,0,\frac{1}{n_B},-\frac{1}{n_B}\right),\nonumber\\ \vec e_7&=&\left(\frac{1}{n_Ak},-\frac{1}{n_Ak},-\frac{1}{n_Bk},\frac{1}{n_Bk}\right).\end{aligned}$$ Of course, the seven vectors $\vec e_1,\ldots,\vec e_7$ are not linearly independent. Let us choose the following linearly independent basis: $$\begin{aligned} \vec f_1&=&-\vec e_5=\left(-\frac{1}{n_A},\frac{1}{n_A},0,0\right),\nonumber\\ \vec f_2&=&\vec e_7=\left(\frac{1}{n_Ak},-\frac{1}{n_Ak},-\frac{1}{n_Bk},\frac{1}{n_Bk}\right),\nonumber\\ \vec f_3&=&\vec e_3-\vec e_1=(-1,0,1,0),\nonumber\\ \vec f_4&=&\vec e_1=(1,0,0,0). \label{basisf}\end{aligned}$$ Using this basis, the toric data become $$\vec v_1=[0,0,0,1]_{\vec f},\quad \vec v_2=[n_A,0,0,1]_{\vec f},\quad \vec v_3=[0,0,1,1]_{\vec f},\quad \vec v_4=[n_B,kn_B,1,1]_{\vec f},$$ where $[a_1,\cdots,a_4]_{\vec f}=\sum_ia_i\vec f_i$. We have chosen basis (\[basisf\]) so that the toric data become the standard form in which the last components of the vectors are $1$. We can draw the toric diagram using the first three components of these vectors $\vec v_i$, which coincides with that in Fig. \[c4.eps\](b). Further generalization {#more.sec} ====================== Up to now we have considered a brane system with two kinds of fivebranes. It is also possible to introduce more than two kinds of fivebranes. To represent the types of branes we used $q_I=0$ and $1$. In this section we allow $q_I$ to be an arbitrary integer. In this case, we do not need to introduce the coefficient $k$ in (\[cscouplings\]) and we set $k=1$. This means that the $I$th fivebrane is a $(q_I,1)$5-brane, and the Chern-Simons couplings are given by $$\label{kInonzero} k_I=q_{I+1}-q_I.$$ For simplicity we assume that the Chern-Simons couplings do not vanish. This implies that all the adjoint chiral multiplets $\Phi_I$ and the vector multiplets $V_I$ become massive. It is easy to show that even if some of the $k_I$ vanish we obtain the same moduli space as derived below. By integrating out $\Phi_I$, we obtain the superpotential $$W=-\sum_{I=1}^n\frac{1}{2(q_{I+1}-q_I)}(X_IY_I-Y_{I+1}X_{I+1})^2.$$ (When we obtained the superpotential (\[ellw\]) we used $q_I=0$ and $1$, although we cannot use it here.) From the assumption (\[coulomb\]), the F-term conditions for $X_I$ and $Y_I$ give $$\frac{X_{I+1}Y_{I+1}-X_IY_I}{q_{I+1}-q_I} =\frac{X_IY_I-X_{I-1}Y_{I-1}}{q_I-q_{I-1}},$$ and this is solved as $$\label{XIYIaqIb} X_IY_I=a+q_Ib,$$ where $a$ and $b$ are arbitrary complex numbers. The equations of motion of the auxiliary fields $\sigma_I$ are solved by (\[dtermc\]) with the parameters $\lambda_I$ constrained by (\[subg\]). The constraint (\[subg\]) defines group $H$ and Eq. (\[dtermc\]) has the form of the D-term condition associated with group $H$. This group is identical to the continuous part of the unbroken gauge symmetry, which is given by (\[ubk\]) with $k=1$. The constraints imposed on the parameters are $$\sum_{I=1}^n\lambda_I=0,\quad \sum_{I=1}^nq_I\lambda_I\in\frac{1}{2\pi}{\mathbb Z}. \label{hp}$$ The following “baryonic operators” are invariant under gauge symmetries satisfying (\[hp\]), $$x=\prod_{I=1}^nX_I,\quad x_A=\prod_{I=1}^nX_I^{q_{\max}-q_I},\quad x_B=\prod_{I=1}^nX_I^{q_I-q_{\min}},$$ $$y=\prod_{I=1}^nY_I,\quad y_A=\prod_{I=1}^nY_I^{q_{\max}-q_I},\quad y_B=\prod_{I=1}^nY_I^{q_I-q_{\min}},$$ where $q_{\min}$ and $q_{\max}$ are the minimum and maximum of $q_I$, respectively. Any gauge-invariant monomial of $X_I$ and $Y_I$ can be represented as a monomial of the mesonic operators $M_I=X_IY_I$ and these baryonic operators. We now have the following $8$ gauge-invariant variables: $$a, b, x, x_A, x_B, y, y_A, y_B.$$ These operators satisfy $$\label{x0y0} xy=\prod_{I=1}^n(a+q_Ib),\quad x_Ay_A=\prod_{I=1}^n(a+q_Ib)^{q_{\max}-q_I},\quad x_By_B=\prod_{I=1}^n(a+q_Ib)^{q_I-q_{\min}},$$ $$\label{x0x1x} x_Ax_B=x^{q_{\max}-q_{\min}},\quad y_Ay_B=y^{q_{\max}-q_{\min}}.$$ Because the first equation in (\[x0y0\]) is not independent of the other two due to relations (\[x0x1x\]), these relations decrease the number of independent degrees of freedom by four, and the moduli space becomes a complex 4-dimensional space. If the $q_I$ take more than two different values, the equations in (\[x0y0\]) are not binary relations of monomials, and the moduli space is nontoric. Discussion ========== In this paper we studied the Higgs branch of Maxwell-Chern-Simons theories described by circular quiver diagrams. We first considered the model realized by the D3-NS5-$(k,1)$5-brane system with an arbitrary number of fivebranes, and showed that the moduli space is the orbifold ${\mathbb C}^4/\Gamma$, where $\Gamma$ is the discrete group generated by (\[zna\])–(\[znc\]). When we determined the orbifold group $\Gamma$, we made an assumption for the flux quantization (\[feqf2\]). Our result was confirmed by comparing it to the M-theory dual of the brane configuration. We also discussed the model realized by a brane system with more than two kinds of fivebranes, and we obtained a 4-dimensional nontoric moduli space. Note that our result is different from that expected from the orbifold method. In general, a quiver gauge theory obtained by the orbifold method introduced in Ref.  includes $n$ copies of fields of the parent theory, where $n$ is the order of the orbifolding group. Such analysis is carried out in Ref.  for the model proposed in Ref. , and a theory was obtained in which the number of $U(N)$ factors in the gauge group is proportional to the order of the corresponding orbifolding group. On the other hand, our construction gives $n=n_A+n_B$ copies of fields, whereas the order of the orbifolding group is proportional to the product $n_An_B$. Because the brane construction and orbifold method are both important methods for constructing field theories in string theory, it is very important to understand the reason for this discrepancy. The moduli spaces we obtained in this paper are completely determined by the number of fivebranes of each type. In the case of the brane system with A- and B-branes, the moduli space depends only on the level $k$ and the numbers of fivebranes $n_A$ and $n_B$. The orders of A- and B-branes along the compact direction do not affect the moduli space. This is also the case in the brane system with more than two types of fivebranes discussed in §\[more.sec\]. This may be interpreted as a duality similar to the Seiberg duality in the 4-dimensional ${\cal N}_{(d=4)}=1$ supersymmetric gauge theories[@Seiberg:1994pq]. In the 4-dimensional case, this duality can be understood as the exchange of the two types of branes[@Elitzur:1997fh]. In the brane system we consider in this paper, the exchange of A- and B-branes generates new D3-branes by the Hanany-Witten effect[@Hanany:1996ie]. It will be an interesting problem to clarify the relation among theories realized by brane systems with different orders of fivebranes. The models we considered in this paper are expected to flow to conformal fixed points in the low-energy limit, and thus the AdS/CFT correspondence is expected to be useful for studying low-energy dynamics. When we discuss the AdS/CFT correspondence, it is necessary to establish the correspondence between geometries and the UV description of quiver gauge theories. In the case of 4-dimensional ${\cal N}=1$ superconformal theories, brane tiling[@Hanany:2005ve; @Franco:2005rj; @Franco:2005sm] is a convenient tool for finding this correspondence in the toric Calabi-Yau case. Although the generalization of brane tiling to 3-dimensional gauge theories has been proposed[@Lee:2006hw; @Lee:2007kv; @Kim:2007ic], much less is known about the duality in the 4-dimensional case due to the small number of examples. We hope that the examples in this paper will be useful for investigating the general relation between four-manifolds and quiver Chern-Simons gauge theories. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank T. Eguchi for valuable discussions. We would also like to acknowledge the helpful comments of K. Ohta. Y. I. is partially supported by a Grant-in-Aid for Young Scientists (B) (\#19740122) from the Japan Ministry of Education, Culture, Sports, Science and Technology. [99]{} J. Bagger and N. Lambert, ; hep-th/0611108. J. Bagger and N. Lambert, ; arXiv:0711.0955. J. Bagger and N. Lambert, ; arXiv:0712.3738. A. Gustavsson, arXiv:0709.1260. A. Gustavsson, ; arXiv:0802.3456. P. M. Ho, R. C. Hou and Y. Matsuo, arXiv:0804.2110. G. Papadopoulos, ; arXiv:0804.2662. J. P. Gauntlett and J. B. Gutowski, arXiv:0804.3078. M. Van Raamsdonk, ; arXiv:0803.3803. N. Lambert and D. Tong, arXiv:0804.1114. J. Distler, S. Mukhi, C. Papageorgakis and M. Van Raamsdonk, ; arXiv:0804.1256. J. Gomis, G. Milanesi and J. G. Russo, arXiv:0805.1012. S. Benvenuti, D. Rodriguez-Gomez, E. Tonni and H. Verlinde, arXiv:0805.1087. P. M. Ho, Y. Imamura and Y. Matsuo, arXiv:0805.1202. Y. Honma, S. Iso, Y. Sumitomo and S. Zhang, arXiv:0805.1895. M. A. Bandres, A. E. Lipstein and J. H. Schwarz, arXiv:0806.0054. J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk and H. Verlinde, arXiv:0806.0738. B. Ezhuthachan, S. Mukhi and C. Papageorgakis, arXiv:0806.1639. S. Mukhi and C. Papageorgakis, ; arXiv:0803.3218. D. Gaiotto and E. Witten, arXiv:0804.2907. K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, arXiv:0805.3662. S. Lee, ; hep-th/0610204. S. Lee, S. Lee and J. Park, ; hep-th/0702120. S. Kim, S. Lee, S. Lee and J. Park, ; arXiv:0705.3540. O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, arXiv:0806.1218. I. R. Klebanov and E. Witten, ; hep-th/9807080. A. M. Uranga, ; hep-th/9811004. R. von Unge, ; hep-th/9901091. T. Kitao, K. Ohta and N. Ohta, ; hep-th/9808111. O. Bergman, A. Hanany, A. Karch and B. Kol, ; hep-th/9908075. M. R. Douglas and G. W. Moore, hep-th/9603167. M. Benna, I. Klebanov, T. Klose and M. Smedback, arXiv:0806.1519. N. Seiberg, ; hep-th/9411149. S. Elitzur, A. Giveon and D. Kutasov, ; hep-th/9702014. A. Hanany and E. Witten, ; hep-th/9611230. A. Hanany and K. D. Kennaway, hep-th/0503149. S. Franco, A. Hanany, K. D. Kennaway, D. Vegh and B. Wecht, ; hep-th/0504110. S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, ; hep-th/0505211. [^1]: E-mail: imamura@hep-th.phys.s.u-tokyo.ac.jp [^2]: E-mail: kimura@hep-th.phys.s.u-tokyo.ac.jp [^3]: The possibility that the existence of magnetic monopoles causes a discrete indentification in the orbifold is also mentioned. [^4]: We shift the field $\Phi_I$ by $(q_I-1/2)(X_IY_I+Y_{I+1}X_{I+1})$ and set $\mu=1$ to simplify the equations.

--- abstract: | Magnetically tunable scattering resonances have been used with great success for precise control of $s$-wave scattering lengths in ultracold atomic collisions. We describe relatively simple yet quite powerful analytic treatments of such resonances based on the analytic properties of the van der Waals long range potential. This theory can be used to characterize a number of properties of specific resonances that have been used successfully in various experiments with $^{87}$Rb, $^{85}$Rb, $^{40}$K, and $^{6}$Li. Optical Feshbach resonances are also possible and may be practical with narrow intercombination line photoassociative transitions in species like Sr and Yb. [**This will be published in [*Atomic Physics 20*]{}, the Proceedings of the 20th international Conference on Atomic Physics, Innsbruck, Austria, July 2006.**]{} author: - 'Paul S. Julienne' - Bo Gao title: Simple theoretical models for resonant cold atom interactions --- [ address=[Atomic Physics Division, NIST, Gaithersburg, Maryland 20899-8423 USA]{} ]{} [ address=[Department of Physics and Astronomy, University of Toledo, Toledo, Ohio 43606]{} ]{} Introduction ============ Tunable control of scattering resonances between interacting ultracold atoms has been widely used in many forefront experiments in atomic physics, many of which are described at this Conference. Such experiments may use bosonic or fermionic species and explore phenomena such as Bose-Einstein condensation (BEC), quantum degenerate Fermi gases, cold molecule formation, molecular BEC, few-body physics, or reduced dimensional structures in optical lattices. The control of such systems comes from the ability to tune the near-threshold scattering and bound state properties of interacting cold atoms. Threshold collisions are characterized by the scattering length $a$, which represents a quantum phase expressed in units of length: the scattering phase shift $\eta(E)\to -ka$ as $E\to 0$ at low collision energy $E$, where $\hbar k$ is the relative collision momentum. The ability to make $a$ vary over a wide range between $\pm \infty$ using a tunable scattering resonance is what allows for precise control of cold atomic gases. This talk will not survey the kinds of experiments that are being done, since these will be covered by many other speakers, but will instead concentrate on developing some simple models that provide understanding of these resonances, using as examples resonances that have been or are being exploited in key ultracold atom experiments. In general, scattering resonances are complex and require either detailed experimental studies in order to measure their properties or detailed knowledge of the molecular physics of the interacting system in order to calculate their properties. On the other hand, they remain amenable to simple models. The simplest is a [*universal*]{} model parameterized solely in terms of the scattering length $a$ and reduced mass $\mu$ of the pair of atoms. This will apply to scattering and bound states sufficiently close to threshold. However, the subject of this talk is a much more powerful model, extending over a much wider range of energy, based on the analytic properties of the van der Waals potential. We will primarily consider resonances tunable by a magnetic field $B$, but briefly consider resonances tunable by an optical field of frequency $\nu$. Resonant scattering =================== There is a long history of resonance scattering, going back to the earliest days of quantum physics [@Rice33; @Fano35], and widely developed in nuclear physics, especially the elegant formalism of Herman Feshbach [@Feshbach58], whose name has become attached to the resonances used in ultracold atom research. Tiesinga [*et al.*]{} [@Tiesinga93] pointed out the existence of sharp resonances versus $B$ in the scattering length of ultracold Cs atoms, and the quest for resonance control of ultracold collisions was underway. A recent review [@Kohler06] describes the near threshold properties of magnetically tunable Feshbach resonances in the context of cold molecule formation and atom paring. Whatever theory one uses, the colliding two-atom system is separated into an approximate bound state $|n\rangle$ with discrete energy $E_n$ and a scattering continuum $|E\rangle$ with a continuous spectrum of collision energies $E$ with some coupling $V$ between $| n \rangle$ and $|E\rangle$. The “bare” bound state $|n\rangle$ in the absence of coupling is the closed channel, or resonance, state, and the “bare” continuum $|E\rangle$ in the absence of coupling represents the open channel, or “background” scattering. Following Fano’s classic 1961 paper [@Fano61], the scattering phase shift $\eta(E)$ separates into background and resonance components: $\eta(E)=\eta_\mathrm{bg}+\eta_\mathrm{res}(E)$, where the weak energy dependence of $\eta_\mathrm{bg}$ over a narrow resonance can be ignored. The resonance contribution varies rapidly by $\pi$ as $E$ varies from below to above resonance at shifted energy $E_0=E_n+\delta E_n$: $$\eta_\mathrm{res}(E)=-\tan^{-1}\frac{\frac{1}{2}\Gamma_n}{E-E_n-\delta E_n}\,,$$ where $\Gamma_n=2\pi|\langle n|V|E_n\rangle|^2$ and $\delta E_n$ are the resonance width and shift respectively. This theory is modified for near-threshold resonances in that $\eta_\mathrm{bg}(E)$, $\Gamma_n(E)$, and $\delta E_n(E)$ become strongly $E$-dependent, following quantum threshold law behavior. For a magnetically tunable resonance $E_n = \delta \mu (B-B_n)$ crosses threshold ($E=0$) at $B=B_n$, and $\delta \mu$ gives the magnetic moment difference between the “bare” open channel atoms and the “bare” resonance state. The phase near threshold is: $$\eta_\mathrm{res}(E)=-\tan^{-1}\frac{\frac{1}{2}\Gamma_n(E)}{E-\delta \mu(B-B_n)-\delta E_n(E)}\,,$$ As $E\to0$ the threshold relations $\eta_\mathrm{bg}\to-ka_\mathrm{bg}$ and $\frac{1}{2}\Gamma_n(E) = (ka_\mathrm{bg}) \delta \mu\,\Delta_n$ [@Mies00] imply a tunable scattering length $$a(B) = a_\mathrm{bg}\left ( 1 - \frac{\Delta_n}{B-B_0} \right )\,,$$ where $\Delta_n$ is the width of the resonance in magnetic field units, and $a(B)$ is singular at the shifted resonance “position” $B_0=B_n+\delta B_n$, where $\delta B_n = \delta E_n/\delta\mu$. If $a(B)$ is positive and sufficiently large, there is a bound state with binding energy $E_b$ related to $a(B)$ by the “universal” relation $E_b=\hbar^2/(2 \mu a(B)^2)$. This universal relation applies for a number of known experimental resonances [@Kohler06]. Analytic van der Waals theory ============================= While universal resonance properties parameterized by $a(B)$ are certainly useful, a much more powerful theory is possible by introducing the analytic properties of the van der Waals potential, which varies at large interatomic separation $R$ as $-C_6/R^6$. The solutions for a single potential, depending solely on $C_6$, reduced mass $\mu$ and scattering length $a$ for that potential, have been worked out in a series of articles by B. Gao [@Gao98a; @Gao98b; @Gao99; @Gao00; @Gao01; @Gao04a]. The van der Waals theory is especially powerful for threshold resonances and bound states when the van der Waals solutions are used as the reference solutions for the specific form of the multichannel quantum defect theory (MQDT) developed by Mies and coworkers [@Mies84a; @Mies84b; @Julienne89; @Mies00a]. In particular, the MQDT is concerned with the analytic properties of single and coupled channels wave functions $\Psi(R,E)$ across thresholds as $E$ goes from positive to negative and between short and long range in $R$. The van der Waals and MQDT theories, when combined, give simple formulas for threshold resonance properties that have illuminating physical interpretations. For all cases where we have tested it numerically, the van der Waals MQDT gives scattering properties in excellent agreement with full coupled channels scattering calculations over a wide range of energies exceeding those normally encountered in cold atom experiments below 1 mK. It is especially good in the ultracold domain of a few $\mu$K and below. The key parameters are the scale length $R_\mathrm{vdw}=(1/2)\left (2\mu C_6/\hbar^2 \right)^{1/4}$ and corresponding energy $E_\mathrm{vdw}=\hbar^2/(2\mu R_\mathrm{vdw}^2)$ associated with the long range potential [@Jones06]. Table \[table1\] lists these for typical alkali species. When $|E| \ll E_\mathrm{vdw}$, bound and scattering wave functions approach their long range asymptotic form for $R \gg R_\mathrm{vdw}$ and oscillate rapidly on a length scale small compared to $R_\mathrm{vdw}$ for $R \ll R_\mathrm{vdw}$. When $E$ is much larger than $E_\mathrm{vdw}$, it is always a good approximation to make a semiclassical WKB connection between the short range and long range parts of the wave function, but when $E$ is small compared to $E_\mathrm{vdw}$, a quantum connection is necessary, even for $s$-waves. In such a case, $R_\mathrm{vdw}$ characterizes the distance range where the WKB connection fails [@Julienne89]. An alternative van der Waals length, $\bar{a}= 0.956 R_\mathrm{vdw}$, called the “mean scattering length” and which appears naturally in van der Waals theory, has been defined by Gribakin and Flambaum [@Gribakin93]. They also gave the the correction to universality in bound state binding energy due to the long range potential: $E_b=\hbar^2/(2 \mu (a-\bar{a})^2)$. -------------- ------------------ ---------------------- -------------------- -- -- -- Species $R_\mathrm{vdw}$ $E_\mathrm{vdw}/k_B$ $E_\mathrm{vdw}/h$ ($a_0$) (mK) (MHz) ${^6}$Li 31.3 29.5 614 ${^{23}}$Na 44.9 3.73 77.8 ${^{40}}$K 64.9 1.03 21.4 ${^{87}}$Rb 82.5 0.292 6.09 ${^{133}}$Cs 101 0.128 2.66 -------------- ------------------ ---------------------- -------------------- -- -- -- : Characteristic van der Waals length and energy scales ($a_0=0.0529$ nm). []{data-label="table1"} \ The key result from the MQDT analysis, derivable from formulas in the original papers [@Mies84a; @Mies84b], is that the energy-dependent threshold width and shift can be written in the following factored form: $$\frac{1}{2}\Gamma_n(E) =\frac{1}{2} \bar{\Gamma}_n C_\mathrm{bg}(E)^{-2}\hspace{2cm} \delta E_n(E) =-\frac{1}{2} \bar{\Gamma}_n \tan\lambda_\mathrm{bg}(E)\,,$$ where $\bar{\Gamma}_n$ is independent of $E$ and $B$ and depends on the short range physics that determines the strength of the resonance. The two functions $C_\mathrm{bg}(E)^{-2}$ and $\tan\lambda_\mathrm{bg}(E)$, as well as $\eta_\mathrm{bg}(E)$, are analytic functions of the entrance channel, or background, potential, and are completely given from the analytic van der Waals theory once $C_6$, reduced mass, and $a_\mathrm{bg}$ are specified. When $E \gg E_\mathrm{vdw}$ such that the semiclassical WKB approximation applies at all $R$, the two MQDT functions take on the following limiting behavior [@Mies84a; @Mies84b; @Julienne89]: $$\lim_{E \gg E_\mathrm{vdw}} C_\mathrm{bg}(E)^{-2} = 1\hspace{2cm} \lim_{E \gg E_\mathrm{vdw}} \tan\lambda_\mathrm{bg}(E) = 0\,.$$ On the other hand, when $E \ll E_\mathrm{vdw}$ so that the threshold law limiting behavior applies, then for the van der Waals potential [@Mies00a] $$\lim_{E\to 0} C_\mathrm{bg}(E)^{-2} = k\bar{a} \left ( 1+(1-r)^2\right ) \hspace{2cm} \lim_{E \to 0} \tan\lambda_\mathrm{bg}(E) = 1-r$$ $$\frac{1}{2}\bar{\Gamma}_n = \frac{r}{1+(1-r)^2} \delta \mu \Delta_n$$ where $r=a_\mathrm{bg}/\bar{a}$ represents the background scattering length in units of $\bar{a}$. With these results the phase shift due to a Feshbach resonance takes on a remarkably simple form: $$\eta(E,B)=\eta_\mathrm{bg}(E)-\tan^{-1}\left(\frac{\frac{1}{2} \bar{\Gamma}_n C_\mathrm{bg}(E)^{-2}}{E -\delta \mu(B-B_n)-\frac{1}{2} \bar{\Gamma}_n \tan\lambda_\mathrm{bg}(E)}\right ) \label{eta.mqdt}$$ The dependence on $B$ occurs only in the linear term in the denominator. The dependence on the entrance channel is contained solely in the $\eta_\mathrm{bg}(E)$, $C_\mathrm{bg}(E)^{-2}$, and $ \tan\lambda_\mathrm{bg}(E)$ functions. The strength of the resonant coupling is given by the single parameter $\bar{\Gamma}_n$. Numerical calculations show that phase shifts predicted by this formula are in superb agreement with full coupled channels calculations over wide ranges of $B$ and $E$ for typical resonances. One pleasing aspect of MQDT theory is that $C(E)^{-1}$ has a simple physical interpretation [@Mies84a; @Mies84b], described in the context of cold atom collisions by [@Julienne89]. The asymptotic entrance channel wave function for $R\gg\bar{a}$ is $f(R,E)=\sin(kR+\eta_\mathrm{bg}(E))/k^{1/2}$. At short range, $R\ll\bar{a}$, it is convenient to write the wave function with a different normalization, that of the WKB semiclassical approximation, $\hat{f}(R,E)=\sin b(R,E)/k(R,E)^{1/2}$, where $k(R)=\sqrt{2\mu(E-V(R))/\hbar^2}$ is the local wavenumber. Since the potential is quite large in magnitude at short range in comparison with $E\approx 0$, the $\hat{f}$ WKB function is essentially independent of energy and has a [*shape*]{} determined by the WKB phase $b(R)=\int k(R')dR'+\pi/4$, so that we can replace $\hat{f}(R,E)$ by $\hat{f}(R,0)$ when $R \ll \bar{a}$. The WKB-assisted MQDT theory shows that the relation between $f(R,E)$ and $\hat{f}(R,E)$ (at all $R$) is $f(R,E)=C(E)^{-1}\hat{f}(R,E)$ [@Mies84a; @Mies84b]. Thus, when $R\ll\bar{a}$, the background channel wave function can be replaced by $C(E)^{-1}\hat{f}(R,0)$, so that, in particular, the coupling matrix element that determines the $E$-dependent width of the resonance can be written in factored form: $V_n(E)=\langle n|V|E\rangle=C(E)^{-1}\hat{V}_n$, where $\hat{V}_n$ depends only on the short range physics and is independent of $E$ and $B$ near threshold. Thus, the short range physics, which depends on the large energy scale set by the deep short range potential, is separated from the long range physics and its very small energy scale near threshold. Consequently, given $a_\mathrm{bg}$, the threshold properties depend only on the long range part of the potential. ![Calculated threshold background channel functions $C(E)^{-2}$ and $-\tan\lambda(E)$ for collisions of $^{40}$K atoms in the $a=\{F=9/2,M=-9/2\}$ and $b=\{F=9/2,M=-7/2\}$ states, showing the limiting behavior for energies that are large or small compared to the van der Waals energy $E_\mathrm{vdw}$ (arrow). []{data-label="fig1"}](fig1.eps){height=".28\textheight"} ![Scattering ($\sin^2\eta(E,B)$ for $E>0)$ and bound states (lines for $E<0$) near the $B/B_G=$202 $^{40}$K $a+b$ resonance, where $B_G=10^{-4}$T $=$ 1 Gauss, with the resonant coupling $V$ turned off (bare) or on (dressed). Light-colored shading for $E>0$ implies scattering near the unitarity limit of the $S$-matrix. The energy zero is chosen to be the energy of two motionless separated atoms at each $B$. The energy scale of $E/h=\pm40$ MHz corresponds to a range of $E/k_B= \pm2$ mK, where $k_B$ is Boltzmann’s constant. The shift from $B_n$ to $B_0$, where the dressed (solid line) bound state crosses threshold, is evident due to the avoided crossing between the last bound state of the background channel at $E_{-1}$ and the bare resonant state with $E_n=\delta\mu(B-B_n)$ and $\delta \mu/h = 23.5$ GHz$/$T. The interference of the ramping bare resonance state with the background is evident for $E>0$. A “sharp” resonance ($\Gamma_n(E) \ll E$) only emerges for $E \gg E_\mathrm{vdw}$ [@Nygaard06]. []{data-label="fig2"}](fig2.eps){height=".28\textheight"} Figures  \[fig1\] and  \[fig2\] illustrate the threshold properties of a $^{40}$K resonance that has been used for experiments involving molecular Bose-Einstein condensation [@Greiner03], fermionic pairing [@Regal04], and reduced dimension [@Moritz05]. Figure  \[fig3\] shows similar results for a $^{85}$Rb resonance that has been used to make a stable condensate [@Cornish00] and exhibit atom-molecule interconversion [@Donley02]. The $a_\mathrm{bg}$ is positive (negative) for $^{40}$K ($^{85}$Rb). In both cases $|a_\mathrm{bg}| > \bar{a}$ and there is peak in $C(E)^{-2}$ near $E=\hbar^2/(2 \mu (a_{bg}-\bar{a})^2)$. Recall that $C(E)^{-2}$ represents the squared amplitude enhancement of the short range wave function relative to the WKB wave function. The peak value of $C(E)^{-2}$ is about 3.4 for the $^{85}$Rb case. Note also that the shift $\delta E_n(E) \propto \tan \lambda(E)$ vanishes as $E$ increases above $E_\mathrm{vdw}$. ![The left panel shows the $C(E)^{-2}$ and $\tan \lambda(E)$ functions for the background collision of two $^{85}$Rb atoms in the $e=\{F=2,M=-2\}$ state. The van der Waals energy is $E_\mathrm{vdw}/k_B=0.3$ mK or $E_\mathrm{vdw}/h=6$ MHz. The right panel is analogous to Fig. \[fig2\], with the same shading of $\sin^2\eta(E,B)$ for $E>0$ and with the bare and dressed bound states crossing threshold at $B_n$ and $B_0$ respectively ($B_G=10^{-4}$ T $= 1$ Gauss). Marcelis [*et al.*]{} [@Marcelis04] show a similar figure illustrating the large shift between $B_n$ and $B_0$. []{data-label="fig3"}](fig3.eps){height=".30\textheight"} The analytic van der Waals theory also gives the effective range of the potential [@Gao00; @Flambaum99], and can be put into an angular-momentum independent form that predicts the bound states for different partial waves, given $a_\mathrm{bg}$, $C_6$, and reduced mass [@Gao00; @Gao01; @Gao04a]. For example, if $|a_{bg}|$ becomes very large compared to $\bar{a}$, there will be a $g$-wave bound or quasibound state near threshold. This is the case for $^{133}$Cs and $^{85}$Rb, for example. On the other hand, if $a_{bg}$ is close to $\bar{a}$ there will be a $d$-wave bound or quasibound state very close to threshold. This is the case for $^{23}$Na, $^{87}$Rb, and $^{174}$Yb, all of which have $a$ slightly larger than $\bar{a}$ and all have quaisbound $d$-wave shape resonances close to threshold. The van der Waals theory also permits a criterion for classifying resonances according to their open or closed channel character [@Kohler06]. Let us define a dimensionless resonance strength parameter $S=(a_\mathrm{bg}/\bar{a})(\delta \mu \Delta_n/E_\mathrm{vdw})$. Let $Z$ be the norm of the closed channel component of the wave function of the near threshold dressed bound state ($1-Z$ is the norm of the open channel component). Open channel dominated resonances have $S\gg 1$ and closed channel dominated ones have $S \ll 1$. The bound states of the former are universal and have $Z \ll 1$ for a range of $|B-B_0|$ that is a significant fraction of $\Delta_n$; scattering states have $\Gamma_n(E) >E$ for $E < E_\mathrm{vdw}$, so that no resonance “feature” appears for $E <E_\mathrm{vdw}$. The bound states for closed channel dominated resonances have only a small domain of universality near $B_0$ over a range that is small compared to $\Delta_n$, and have $Z\approx 1$ except over this narrow range; scattering states can have $\Gamma_n(E) \ll E$ for $E < E_\mathrm{vdw}$, so that sharp resonance features can appear in this range very close to threshold. The $^{40}$K and $^{85}$Rb resonances described above, as well as the broad $^{6}$Li resonance near 834 G [@Bartenstein05], are open channel dominated. The very narrow $^{6}$Li resonance near 543 G [@Strecker03] and the $^{87}$Rb resonance near 1007 G [@Thalhammer06; @Volz06] are examples of closed channel dominated ones. A good description of the two $^{6}$Li resonances has been given by  [@Simonucci05]. One application of the expression in Eq.  is to provide an analytic form for the energy-dependent scattering length $a(E,B)=\tan^{-1}\eta(E,B)/k$. This quantity is defined so as to give the scattering length in the limit $E\to0$ but can be used at finite $E$ to define an energy-dependent pseudopotential for finding interaction energies in strongly confined systems with reduced dimension [@Blume02; @Bolda02; @Naidon06]. Finally, it is worth noting that optical Feshbach resonances [@Fedichev96; @Bohn99] follow a similar theory to the magnetic ones. An optical Feshbach is a photoassociation resonance [@Jones06] where a laser with frequency $h\nu$ and intensity $I$ couples colliding ground state atoms to an excited bound state at frequency $h\nu_0$ relative to $E=0$ ground state atoms. Both the resonance width $\Gamma_n(E)$ and shift (contained in $h\nu_0$) are proportional to $I$. The general expression for the resonant scattering length is $$a = a_\mathrm{bg} \left ( 1 - \frac{\Gamma_0}{E_0-i\frac{\gamma}{2}} \right ) = a_\mathrm{bg} -a_\mathrm{res} -ib_\mathrm{res} \,,$$ where, in the case of an optical resonance, $\Gamma_0=(l_\mathrm{opt}/a_\mathrm{bg})\gamma$, $E_0=-h(\nu-\nu_0)$, and $\gamma/\hbar$ is the spontaneous emission decay rate of the excited state. This compares with $\Gamma_0=\delta \mu \Delta_n$, $E_0=\delta \mu(B-B_0)$ and $\gamma/\hbar \approx 0$ for a typical magnetic resonance. The optical resonance strength is characterized by the optical length $l_\mathrm{opt}$ [@Ciurylo05], proportional to $I$ and the free-bound Franck-Condon factor $|\langle n|E\rangle|^2$. The two-body decay rate coefficient due to decay of the resonance is $k_2=2(h/\mu)b_\mathrm{res}$. Optical resonance are only practical when $|a_\mathrm{res}|\gg b_\mathrm{res}$, insuring that the decay rate is sufficiently small. Assuming the detuning $\Delta=h(\nu-\nu_0)$ to be large in magnitude compared to $\gamma$, then $a_\mathrm{res} =l_\mathrm{opt} (\gamma/\Delta)$ and $b_\mathrm{res} =(l_\mathrm{opt}/2) (\gamma/\Delta)^2$. Thus, the condition $\gamma/|\Delta| \ll 1$ ensures small decay, while the condition $l_\mathrm{opt} \gg |a_\mathrm{bg} -a_\mathrm{res}|$ is necessary in order to make a meaningful change in $a_\mathrm{bg}$. Satisfying the above conditions is difficult for strongly allowed optical transitions for which $\gamma$ is large, since very large detuning from atomic resonance is necessary in order to suppress losses, and then $l_\mathrm{opt}$ is too small for reasonable intersity $I$; alternatively changing $a_\mathrm{bg}$ is accompanied by large losses. These problems can be eliminated by working with narrow intercombination line transitions of species such as Ca, Sr, or Yb [@Ciurylo05; @Zelevinsky06; @Tojo06]. Quite large values of $l_\mathrm{opt}$ can be achieved for levels very close to the atomic dissociation limit of the molecular excited state, and it appears to be feasible to use such resonances to control scattering lengths [@Zelevinsky06]. [2]{} O. K. Rice, *J. Chem. Phys.* **1**, 375 (1933). U. Fano, *Nuovo Cimento* **12**, 154 (1935). H. Feshbach, *Ann. Phys. (NY)* **5**, 357 (1958); **19**, 287 (1962). E. Tiesinga, B. J. Verhaar, H. T. C. Stoof, *Phys. Rev.* A**47**, 4114 (1993). T. K[ö]{}hler, K. Goral, P. S. Julienne, *Rev. Mod. Phys.*, in press (2006); cond-mat/0601420. U. Fano, *Phys. Rev.* **124**, 1866 (1961). F. H. Mies, E. Tiesinga, and P. S. Julienne, *Phys. Rev.* A**61**, 022721 (2000). B. Gao, *Phys. Rev.* A [**58**]{}, 1728 (1998). B. Gao, *Phys. Rev.* A [**58**]{}, 4222 (1998). B. Gao, *Phys. Rev. Lett.* [**83**]{}, 4225 (1999). B. Gao, *Phys. Rev.* A [**62**]{}, 050702 (2000). B. Gao, *Phys. Rev.* A [**64**]{}, 010701 (2001). B. Gao, *J. Phys.* B [**37**]{}, 4273 (2004). F. H. Mies, *J. Chem. Phys.* [**80**]{}, 2514 (1984). F. H. Mies and P. S. Julienne, *J. Chem. Phys.* [**80**]{}, 2526 (1984). P. S. Julienne and F. H. Mies, *J. Opt. Soc. Am.* B[**6**]{}, 2257 (1989). F. H. Mies and M. Raoult, *Phys. Rev.* A [**62**]{}, 012708 (2000). K. Jones, P. D. Lett, E. Tiesinga, and P. S. Julienne, *Rev. Mod. Phys.* **78**, 483 (2006). G. F. Gribakin and V. V. Flambaum, *Phys. Rev.* A [**4**8]{}, 546 (1993). M. Greiner, C. A. Regal, and D. S. Jin, *Nature* (London) [**4**26]{}, 537 (2003). C. A. Regal, M. Greiner, and D. S. Jin, *Phys. Rev. Lett.* [**9**2]{}, 040403 (2004). H. Moritz, T. St[ö]{}ferle, K. G[ü]{}nter, M. K[ö]{}hl, and T. Esslinger, *Phys. Rev. Lett.* [**9**4]{}, 210401 (2005). N. Nygaard, B. I. Schneider, and P. S. Julienne, *Phys. Rev.* A [**7**3]{}, 042705 (2006). S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, *Phys. Rev. Lett.* [**8**5]{}, 1795 (2000). E. A. Donley, N. R. Claussen, S. T. Thompson, and C. E. Wieman, *Nature* (London) [**4**17]{}, 529 (2002). B. Marcelis, E. G. M. van Kempen, B. J. Verhaar, and S. J. J. M. F. Kokkelmans, *Phys. Rev.* A [**7**0]{}, 012701 (2004). V. V. Flambaum, G. F. Gribakin, and C. Harabati, Phys. Rev. A 59, 1998 (1999). M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin, J. H. Denschlag, R. Grimm, A. Simoni, E. Tiesinga, C. J. Williams, and P. S. Julienne, *Phys. Rev. Lett.* [**9**4]{}, 103201 (2005). K. E. Strecker, G. B. Partridge, and R. G. Hulet, *Phys. Rev. Lett.* [**9**1]{}, 080406 (2003). Thalhammer, G., K. Winkler, F. Lang, S. Schmid, R. Grimm, and J. H. Denschlag, *Phys. Rev. Lett.* [**9**6]{}, 050402 (2006). T. Volz, N. Syassen, D. M. Bauer, E. Hansis, S. D[ü]{}rr, and G. Rempe, cond-mat/0605184 (2006). D. Blume and C. H. Greene, *Phys. Rev.* A [**6**5]{}, 043613 (2002). E. L. Bolda, E. Tiesinga, and P. S. Julienne, *Phys. Rev.* A [**6**6]{}, 013403 (2002). P. Naidon, E. Tiesinga, W. F. Mitchell, and P. S. Julienne, physics/0607140 (2006). S. Simonucci, P. Pieri and G. C. Strinati, *Europhys. Lett.* [**6**9]{}, 713 (2005). P. O. Fedichev, Yu. Kagan, G. V. Shlyapnikov, J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996). J. L. Bohn, and P. S. Julienne, *Phys. Rev.* A[**6**0]{}, 414 (1999). R. Ciurylo, E. Tiesinga, and P. S. Julienne, *Phys. Rev.* A [**7**1]{}, 030701 (2005). T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciurylo, P. Naidon, and P. S. Julienne,*Phys. Rev. Lett.* [**9**6]{}, 203202 (2006). S. Tojo, M. Kitagawa, K. Enomoto, Y. Kato, Y. Takasu, M. Kumakura, and Y. Takahashi, *Phys. Rev. Lett.* [**9**6]{}, 153201 (2006).

--- author: - | Amir-Hossein Karimi $^{1,2}$ Julius von Kügelgen[^1] $^{1,3}$ Bernhard Schölkopf $^{1}$ Isabel Valera $^{1,4}$\ \ $^1$Max Planck Institute for Intelligent Systems, Tübingen, Germany\ $^2$Max Planck ETH Center for Learning Systems, Zürich, Switzerland\ $^3$Department of Engineering, University of Cambridge, United Kingdom\ $^4$Department of Computer Science, Saarland University, Saarbrücken, Germany\ `{amir, jvk, bs, ivalera}@tue.mpg.de` bibliography: - 'references.bib' title: 'Algorithmic recourse under imperfect causal knowledge: a probabilistic approach' --- Introduction {#sec:introduction} ============ Background and related work {#sec:background} =========================== Negative result: no recourse guarantees for unknown structural equations {#sec:motivating_example} ======================================================================== Individualised algorithmic recourse via (probabilistic) counterfactuals {#sec:recourse_via_probabilistic_counterfactuals} ======================================================================= Subpopulation-based algorithmic recourse via interventions and [<span style="font-variant:small-caps;">cate</span>]{}s {#sec:subpopulation_based_recourse_via_CATE} ====================================================================================================================== Solving the probabilistic-recourse optimisation problems {#sec:optimisation_problem} ======================================================== Experimental results {#sec:experiments} ==================== Discussion {#sec:discussion} ========== The authors would like to thank Adrian Weller, Floyd Kretschmar, Junhyung Park, Matthias Bauer, Miriam Rateike, Nicolo Ruggeri, Umang Bhatt, and Vidhi Lalchand for helpful feedback and discussions. Moreover, a special thanks to Adrià Garriga-Alonso for insightful input on some of the [<span style="font-variant:small-caps;">gp</span>]{}-derivations and to Adrián Javaloy Bornás for invaluable help with the [<span style="font-variant:small-caps;">cvae</span>]{}-training. Proofs {#app:proofs} ====== Further details on [<span style="font-variant:small-caps;">cvae</span>]{} training {#app:cvae_training_details} ================================================================================== (Non-)identifability of [<span style="font-variant:small-caps;">scm</span>]{}s under different assumptions {#app:nonidentifiability} ========================================================================================================== Experimental details, hyperparameter choices, and specification of [<span style="font-variant:small-caps;">scm</span>]{}s {#app:experimental_details} ========================================================================================================================= Additional results {#app:additional_results} ================== Derivation of a Monte-Carlo estimator for the gradient of the variance {#app:derivation_of_variance_grad} ====================================================================== [^1]: Equal contribution

   and We report preliminary results of a kinematical study for three Seyfert galaxies selected from a sample of nearby active galactic nuclei observed using 3D spectroscopy. The observations were performed at the prime focus of the SAO RAS 6-m telescope with the integral-field spectrograph MPFS and with a scanning Fabry-–Pérot interferometer, installed on the multimode device SCORPIO. Based on these data, the monochromatic maps and velocity fields in different emission lines were constructed. We have detected a nuclear outflow or ionized gas motions associated with a radio jet in all galaxies circumnuclear regions. galaxies: individual: interactions – galaxies: kinematics and dynamics – galaxies: Seyfert – galaxies: starburst We have studied the kinematics of the ionized gas and stellar component in three Seyfert galaxies using methods of 3D spectroscopy. The observations were performed at the prime focus of the SAO RAS 6-m telescope. The circumnuclear regions were observed with the integral-field spectrograph MPFS (Afanasiev et al. 2001). The large-scale kinematics of the ionized gas in the H$\alpha$ emission line was studied using the SCORPIO multimode focal reducer (Afanasiev & Moiseev 2005) operating in the mode of scanning Fabry-Pérot Interferometer (FPI). The integral-field spectrograph MPFS takes simultaneous spectra of 256 spatial elements arranged in the form of 16x16 square lenses array with a scale of 1 arcsec per spaxel. The wavelength interval included numerous emission lines of the ionized gas and absorption features of the stellar population. Based on these data, the monochromatic maps and velocity fields in different emission lines of the ionized gas were constructed. We used the ULySS software (Koleva et al. 2009) for fitting of the spectra of stellar population as well as for construction of the stellar velocity fields. The velocity fields were fitted by the model of a pure circular rotation using the modification of the ‘tilted-ring’ algorithm employed earlier to study NGC 6104 (Smirnova et al. 2006) and Mrk 334 (Smirnova & Moiseev 2010). Mrk 198 is a Sy2 spiral galaxy with a bright asymmetric spiral arm at North-East direction. Faint arc-like feature is seen at 60-70 arcsec northeast from the nucleus Mrk 198 in our deep image (Smirnova et al. 2010). This tidal envelope traces the event of probable past minor merging. The velocity fields in the Balmer emission lines seems to be in a good agreement with the model of a regular rotating thin disc (Figures 1, 2). The kinematics of stars shows the same behaviour. However, the motions of gas emitted in higher-excitation lines have significant deviations from the pure circular rotation. The map of residual velocities in the \[OIII\] emission line reveals the significant (from -70 to +130 km/s) non-circular motions along PA$\approx$30$^{\circ}$. The most probable explanation of the peculiar kinematics is a jet or an outflow penetrating through the surrounding interstellar medium. Our large-scale FPI data for the H$\alpha$ emission line demonstrate residual velocities in the central part of the galaxy, and also in the spiral arm. Significant non-circular motions seen only within the 3 arcsec, at larger distances (including bright asymmetric spiral arm) residual velocities are close to zero. Mrk 291 is a Sy2 galaxy, it has a bar and two spiral arms started at the end of the bar (Adams 1973). It is a strongly isolated object. The velocity field derived in the H$\alpha$ emission line demonstrates a regular rotation plus radial streaming motions along the stellar bar (two regions with non-zero residuals at r$>$5$''$, see Figure 3). In contrast to this picture, a significant excess of blueshifted velocities (about -100 km/s) in the \[OIII\] emission line is detected (Figure 4). Similar gas outflows associated with a nuclear starburst, or a jet-clouds interaction, or with a hot wind emerging from an active nucleus were already found in the integral field spectroscopy data for several Seyfert galaxies (for references see Smirnova & Moiseev 2010). Mrk 348 is a Sy2 spiral tidally-disrupted galaxy (Simkin et al. 1987). FPI data reveals extended spiral structure with a projected companion galaxy (Figure 5). Maps of the residual velocities in Balmer emission lines as well as in forbidden lines demonstrate significant non-circular motions ($\pm$50 km/s) at North and South direction from the nucleus (Figure 6). The radio image of the galaxy shows extended structure up to 0.2$''$ at P.A. =170$^{\circ}$ (Ulvestad & Wilson 1984). According to the HST images (Capetti 2002; Falcke et al. 1998) the extended (up to r=1.5$''$) \[OIII\] emission structure is elongated in the same direction. Thereby peculiar motions observed on the MPFS maps at the distances $r=1-4''$ are associated with a cocoon of the ionized gas surrounded the radio jet which is penetrate into interstellar medium. A similar combination (inner radio jet + velocities perturbations in the outer regions) was recently found in Mrk533 (Smirnova et al. 2007). [^1] Adams T. 1977, ApJS, v.33, p.19 Afanasiev V.L., Dodonov S.N., Moiseev A.V. 2001, [*in Stellar Dynamics: From Classic to Modern*]{}, eds. Ossipkov L. P. & Nikiforov I. I., Saint Petersburg, 103 Afanasiev V. L., Moiseev A. V. 2005, Astronomy Letters, 31, 193 Capetti A. 2002, RevMexAA (Serie de Conferencias), 13, 163 Falcke H., Wilson A.S., Simpson C. 1998, ApJ, 502, 199 Koleva M., Prugniel Ph., Bouchard A., Wu Y. 2009, A&A, 501, 1269 Simkin S., van Gorkom J., Hibbard J., Su H.-J. 1987, Science, 235, 1367 Smirnova A., Gavrilovic N., Moiseev A., Popovic L., Afanasiev V., Jovanovic P. 2007, MNRAS, 377, 480 Smirnova A. A., Moiseev A. V. 2010, MNRAS, 401, 30 Smirnova A. A., Moiseev A. V., Afanasiev V. L. 2006, Astronomy Letters, 32, 520 Smirnova A. A., Moiseev A. V., Afanasiev V. L. 2010, MNRAS, 408, 400 Ulvestad J., Wilson A. 1984, ApJ, 285, 439 [^1]: We thank the Organizing Committee for financial support and efficient organization of the conference. This work was supported by the Russian Foundation for Basic Research (project no. 09-02-00870) and by the Russian Federal Program ‘Kadry’ (contract no. 14.740.11.0800). A.M. is also grateful to the ‘Dynasty’ Fund.

--- abstract: 'The fireball concept of Rolf Hagedorn, developed in the 1960’s, is an alternative description of hadronic matter. Using a recently derived mass spectrum, we use the transport model GiBUU to calculate the shear viscosity of a gas of such Hagedorn states, applying the Green-Kubo method to Monte-Carlo calculations. Since the entropy density is rising ad infinitum near $T_H$, this leads to a very low shear viscosity to entropy density ratio near $T_H$. Further, by comparing our results with analytic expressions, we find a nice extrapolation behavior, indicating that a gas of Hagedorn states comes close or even below the boundary $1/4\pi$ from AdS-CFT.' author: - Jan Rais - Kai Gallmeister - Carsten Greiner bibliography: - 'paper.bib' title: The Shear Viscosity to Entropy Density Ratio of Hagedorn States --- Introduction {#sec:Intro} ============ The properties of hot and dense matter, created experimentally in heavy-ion collision performed at accelerators like RHIC or CERN, are usually extracted by applying relativistic hydrodynamics or kinetic transport theory. Doing hydrodynamics, transport coefficients like heat or electric conductivity, or shear- or bulk viscosity, are extrinsic inputs which should be calculated from an underlying field theory, as it is Quantum Chromodynamics (QCD) for the for the quark gluon plasma (QGP). The shear viscosity, as one of those transport coefficients, can be calculated employing two-particle scattering processes. Dealing with QGP, there is almost a perfect liquid characterized by a very small value for the shear viscosity to entropy density ratio, $\eta/s$. Nevertheless, this ratio never undergoes the value $1/4\pi$, which is derived within the anti-de Sitter/conformal field theory (AdS/CFT) [@Kovtun:2004de]. This boundary holds for all substances in nature. In [@Xu:2007ns] it was shown, within the BAMPS parton cascade, which includes inelastic gluonic $gg \leftrightarrow ggg$ reactions, that $\eta/s \sim 0.13$ in a pure gluon gas. This is as expected, because $\eta/s$ increases with decreasing $T$, which goes hand in hand with a decrease of the relevant hadronic cross section in the hadronic phase [@Gavin:1985ph; @Venugopalan:1994ux]. On the other hand, asymptotic freedom dictates that $\eta/s$ increases with $T$ in the deconfined phase. Here the coupling between quarks and gluons decreases logarithmically [@Arnold:2003zc]. There have been several efforts to study this transport coefficient $\eta/s$ in microscopic models using the Green-Kubo formalism, as e.g. in UrQMD [@Muronga:2003tb; @Demir:2008tr], in SMASH [@Rose:2017bjz] and in pHSD [@Ozvenchuk:2012kh]. On the partonic side, either pHSD [@Ozvenchuk:2012kh], PCM [@Fuini:2010xz], and BAMPS have been used [@Wesp:2011yy], while within the latter model also a critical test of the Green-Kubo method itself has been performed [@Reining:2011xn]. Very recently, there was an attempt using a $S$-matrix based Hadron Resonance Model via the Chapman-Enskog method [@Dash:2019zwq]. Before QCD made the calculation of phase transition and QGP possible, an alternative theory describing hadrons was devoloped by Rolf Hagedorn in the 1960’s [@Hagedorn:1965st]. He states a visual concept that reads “fireballs, consist of fireballs, which consist of fireballs …”. This yields a density of (hadronic) states as function of the mass as $$\begin{aligned} \label{eq:fitfunc} \rho(m) = \text{const.}\, m^{-a} \, \exp \left[m/T_H\right]\ ,\end{aligned}$$ with $T_H$ being the so-called “Hagedorn temperature”. Later, Frautschi invented a reformulation [@Frautschi:1971ij], yielding a bootstrap equation, $$\begin{aligned} \label{eq:bootstrap} \rho(m) =& \rho_0(m)\ +\ \sum_N\frac{1}{N!}\left[\frac{V_0}{(2\pi)^3}\right]^{N-1}\\ &\times \int \prod_{i=1}^{N}\left[{\mathrm{d}}m_i \rho(m_i) {\mathrm{d}}^3p_i\right] \delta^{(4)}\left(\sum_ip_i-p\right)\ .\nonumber\end{aligned}$$ Here $V_0$ is the volume of the Hagedorn states. In general, this equation can not be solved analytically. For the easiest inhomogeneity, $\rho_0(m)=\delta(m-m_0)$ with $m_0$ the mass of some initial state, Nahm [@Nahm:1972zc] found a solution with $a \approx 3$. With $V_0 \simeq (4\pi/3)m_\pi^{-3}$, $m_0 \simeq m_\pi$, one achieves a slope $T_H \simeq 150{{\ensuremath{\,{\rm MeV}}\xspace}}$. For more realistic inhomogenities of \[eq:bootstrap\], the solution has to be found numerically. Recently, our group developed a prescription with $N=2$, where the quantum numbers $B$ (baryon number), $S$ (strangeness) and $Q$ (electric charge) are conserved explicitly [@Beitel:2014kza; @Beitel:2016ghw; @Gallmeister:2017ths]. Summing over all quantum numbers, one gets for the prescription given in [@Gallmeister:2017ths] a Hagedorn spectrum, \[eq:fitfunc\], which is characterized by $T_H=165{{\ensuremath{\,{\rm MeV}}\xspace}}$ and $a=2.98$ (for a Hagedorn state radius $R=1.0{{\ensuremath{\,{\rm fm}}\xspace}}$)[^1]. As will be discussed below, these numbers can only be used for analytic estimates with restrictions. For real calculations the detailed, tabulated spectra are used. Nevertheless, the fitted distribution may be used to estimate some quantities in the vicinity of $T_H$. The aim of this paper is to study how the shear viscosity over entropy density of a Hagedorn gas behaves as a function of the temperature of the gas (cf. also [@NoronhaHostler:2008ju]). Thus analytical estimates are compared to Monte Carlo results obtained from box calculations based on the Green-Kubo formalism. In order to check the validity of the results, also results for a pion gas are analyzed, where three different charge states may interact elastically according an isotropic constant cross section $\sigma=30{{\ensuremath{\,{\rm mb}}\xspace}}$. Thus the interaction is direct comparable to that of the Hagedorn gas. The paper is organized as follows. In section II, analytic expressions for the thermodynamical quantities of the considered Hagedorns state gas are given. Also, an analytic expression for the shear viscosity of a gas of particles, which interpolates the non-relativistic regime to the relativistic regime necessary for pions is stated. Section III describes the numerical Green-Kubo method used in this analysis and shows intermediate results. The final results for $\eta$ and $\eta/s$ are presented in section IV and discussed in section V. Analytic Expressions {#sec:AnaExpr} ==================== Thermodynamical quantities {#sec:thermoQuantities} -------------------------- Knowing the general resonance gas partition function in Boltzmann approximation, $$\begin{aligned} \label{eq:partitionfunction} \ln \mathcal{Z}(T,V) = \frac{VT}{2\pi^2} \int {\mathrm{d}}m \, \rho(m)\,m^2 \operatorname{K}_2\left(\frac{m}{T}\right)\ ,\end{aligned}$$ with $\operatorname{K}_\nu$ being a modified Bessel function, one may derive all necessary thermodynamical quantities (cf. e.g. [@NoronhaHostler:2012ug]), as e.g. particle density $n$, energy density $e$, and entropy density $s$, as $$\begin{aligned} \label{eq:nDens} n(T)&=\frac{T}{2\pi^2}\int {\mathrm{d}}m \, \rho(m)\,m^2\operatorname{K}_2\left(\frac{m}{T}\right)\ ,\\ \label{eq:eDens} e(T)&=\frac{T}{2\pi^2}\int {\mathrm{d}}m \, \rho(m)\,m^3\left[3\frac{T}{m}\operatorname{K}_2\left(\frac{m}{T}\right)+\operatorname{K}_1\left(\frac{m}{T}\right)\right]\ ,\\ \label{eq:sDens} s(T)&=\frac{1}{2\pi^2}\int {\mathrm{d}}m \,\rho(m)\,m^3 \operatorname{K}_3\left(\frac{m}{T}\right)\ ,\end{aligned}$$ where all chemical potentials have been neglected, $\mu=0$. Since for Boltzmann statistics the pressure is given by $p=T\,n$, it is easily observed using the recurrence relations of the Bessel functions, that the well known Gibbs-Duhem relation, $$\begin{aligned} \label{eq:sHag} s &= \frac{e + p}{T} \ ,\end{aligned}$$ is fulfilled. It is important to note, that a simple minded insertion of \[eq:fitfunc\] into the thermodynamical integrals \[eq:nDens,eq:eDens,eq:sDens\] leads to faulty results: The Hagedorn spectrum fitted to a function according \[eq:fitfunc\] only describes the high mass ($m\gtrsim 2{{\ensuremath{\,{\rm GeV}}\xspace}}$) contribution, but totally fails below. Here the full Hagedorn spectrum is a sum of the known hadron states and the pure Hagedorn states, thus showing all the mass structures of the known hadrons, \[fig:HagSpectrum\]. ![The Hagedorn spectrum and a fit according \[eq:fitfunc\] ($T_H=0.165{{\ensuremath{\,{\rm GeV}}\xspace}}$, $a=2.98$) as function of the mass $m$.[]{data-label="fig:HagSpectrum"}](All1.eps){width="\columnwidth"} Therefore we will use the tabulated spectrum of hadrons and Hagedorn states instead of an analytic approximation in all what follows, except the extrapolations described below. The tabulation stops at Hagedorn state masses $m=10{{\ensuremath{\,{\rm GeV}}\xspace}}$. The resulting entropy as a function of temperature is shown in \[fig:s\_T3\]. ![The entropy density $s$, \[eq:sDens\], normalized to $T^3$ as function of the temperature $T$. Also the estimate for $m\to\infty$ (see text for details) is shown.[]{data-label="fig:s_T3"}](s_T3.eps){width="\columnwidth"} The entropy density for pions is simply calculated by replacing $\rho(m)$ by a properly scaled delta peak according the degeneracy at the pion mass. While the entropy density of the pion gas increases very slowly, the entropy of the Hagedorn gas increases exponentially and gets very steep for $T\gtrsim T_H$. Since the Hagedorn state tabulations only extends up to masses $m=10{{\ensuremath{\,{\rm GeV}}\xspace}}$, the (expected) divergence is weakened, showing only a constant increase on a logarithmic scale. This holds true for all thermodynamical quantities mentioned above, as e.g. energy and particle density. For some quantities, it is now possible to add the missing contribution by using the analytic fit function \[eq:fitfunc\] getting the real divergence. Inserting approximations for the Bessel function $K_\nu$ for large arguments, integrals like e.g. \[eq:nDens,eq:eDens,eq:sDens\] may be expressed in terms of the complementary incomplete gamma function. The corresponding result for the entropy density is also shown in \[fig:s\_T3\]. It is obvious that one has to abstain from this procedure, when directly comparing to the Monte Carlo simulations. We have to mention, that we consider the gas particles to be pointlike, such that there is no volume correction. Since the Hagedorn spectrum generates more and more particles, this also influences the space in a given box volume. Therefore it would be instructive to introduce volume corrections, as e.g. in [@NoronhaHostler:2012ug; @Rischke:1991ke; @Gorenstein:2007mw] in future studies. Shear viscosity {#sec:shearviscosity} --------------- To investigate the shear viscosity of pion or Hagedorn states gas, it is important to ensure that the underlying formulae are valid for the desired range of the variable $z=m/T$, while $m$ is the mass of the particle and $T$ the temperature of the system. For relevant temperatures $T=100-200{{\ensuremath{\,{\rm MeV}}\xspace}}$ and masses $m>138{{\ensuremath{\,{\rm MeV}}\xspace}}$, the covered range is $z=10^{-3}-1.5$. Thus one needs a non-relativistic prescription, which reaches till $m\sim T$. For this the expression valid for all masses and all temperatures is selected as [@DeGroot:1980dk] $$\begin{aligned} \label{eq:eta_general} \eta=\frac{15}{16}\,\frac{T}{\sigma} \frac{z^4\,\operatorname{K}_3^2(z)}{(az^2+b)\operatorname{K}_2(2z)+(cz^3+dz)\operatorname{K}_3(2z)}\ .\end{aligned}$$ Choosing the values of the constants as $$\begin{aligned} a=15\,,\ b=2\,,\ c=3\,,\ d=49\,,\end{aligned}$$ yields the well known first order approximations [@DeGroot:1980dk][^2]. By slightly adjusting these constants to $$\begin{aligned} \label{eq:eta_FaksMod} a=14.55\,,\ b=1.13\,,\ c=2.95\,,\ d=46.85\,,\end{aligned}$$ \[eq:eta\_general\] gives a nice interpolation of numerical results [@KOX1976155] and yields the also well known higher order limiting formulae [@Huovinen:2008te; @Wiranata:2012br] $$\begin{aligned} \eta\ \underset{m\ll T}{\simeq}\ & 1.2654\,\frac{T}{\sigma}\ ,\\ \eta\ \underset{m\gg T}{\simeq}\ & 0.3175\sqrt{\pi}\,\frac{T}{\sigma}\,\sqrt{\frac{m}{T}}\left(1+1.6349\frac{T}{m}\right)\ .\end{aligned}$$ Thus, \[eq:eta\_general\] with the modified factors \[eq:eta\_FaksMod\] will be used further-on in this work. Another expression covering all values of $z$ may be found in [@Gorenstein:2007mw]. This expression differs from the given one by more than 20for $z>0.1$ and is therefore not covered here. The overall shear viscosity of a mixture of particles is given by the weighted sum of the viscosities of each particle species [@Reif:1987], which in the given case of the Hagedorn gas converts into a integration over all masses, $$\begin{aligned} \label{eq:etachapens} \eta(T)&=\frac{T}{2\pi^2\,n(T)}\int {\mathrm{d}}m \, \rho(m)\,m^2\operatorname{K}_2\left(\frac{m}{T}\right)\,\eta(m)\end{aligned}$$ with $\eta(m)$ given by \[eq:eta\_general,eq:eta\_FaksMod\]. Numerical Contemplation {#sec:NumCalc} ======================= Implementation into GiBUU {#sec:GiBUU} ------------------------- The Gießen Boltzmann-Uehling-Uhlenbeck (GiBUU) project [@Buss:2011mx] simulates nuclear reactions as $e+A$, $\gamma + A$, $\nu+A$, $\text{hadron} + A$ (i.e. $p+A$, $\pi+A$) or $A+A$ at energies of $10{{\ensuremath{\,{\rm MeV}}\xspace}}$ to $100{{\ensuremath{\,{\rm GeV}}\xspace}}$. Here the BUU equation $$\left[\partial_t \left(\nabla_p \mathcal{H}_i \right) \nabla_r - \left(\nabla_r \mathcal{H}_i\right)\nabla_p\right] f_i(r,p,t) = C\left[f_i, f_j,...\right]$$ is solved, where $i = N, \Delta, \pi, \rho,...$. The collision term $C$ conventionally involves the decay and scattering of 1-, 2- and 3- body processes, $C=C_{1\rightarrow x}+C_{2\rightarrow x}+C_{3\rightarrow x}$, which splits into a resonance model for low energies and the string model for high energies. In the actual implementation [@Gallmeister:2017ths], all interactions (even elastic scattering) are replaced by Hagedorn state creation and decay processes, i.e. by $2\to1$ and $1\to2$ processes alone. The Hagedorn spectrum tabulation limits the available energy range to be below $10{{\ensuremath{\,{\rm GeV}}\xspace}}$. In the simulations, the particles are thermally initialized in a box (non-reflecting boundaries) with fixed volume of $(10{{\ensuremath{\,{\rm fm}}\xspace}})^3$. The interaction is according a constant cross section of $\sigma = 30{{\ensuremath{\,{\rm mb}}\xspace}}$ for the pion gas and $\sigma=\pi R^2=31.4{{\ensuremath{\,{\rm mb}}\xspace}}$ for the Hagedorn gas. For the pion gas a time step size of $\Delta t=0.1{{\ensuremath{\,{\rm fm}}\xspace}}$ and $N_t=30000$ timesteps was chosen, while the Hagedorn gas where calculated at a lower time step size of $\Delta t=0.01{{\ensuremath{\,{\rm fm}}\xspace}}$ and $N_t=25000$ timesteps, which is justified because of the less steady correlation function at higher time, however bypassing too long calculation times. These values are extracted from the error estimation via the colored noise studies described below. It is checked, that detailed balance is fully respected and the mass and quantum number distributions are constant over the full simulation time. Green-Kubo formalism {#sec:Green-Kubo} -------------------- The Green-Kubo method is the common method to compute transport coefficients like shear viscosity, electric or heat conductivity etc. assuming, that the probability distribution of the time-averaged dissipative flux is Gaussian [@Searles:2000]. It can be derived from the dissipation-fluctuation theorem [@Kubo:1966; @Nyquist:1928zz] and reads (see e.g. [@Wesp:2011yy]) $$\eta = \frac{1}{T} \int_V{\mathrm{d}}^3r\int_{0}^{\infty} {\mathrm{d}}t \left\langle \pi^{xy}(\vec r,t)\pi^{xy}(0,0)\right\rangle\ .$$ Here $\pi^{xy}$ indicates a fixed spatial component of the volume averaged shear tensor[^3] and $\left\langle...\right\rangle$ denotes the ensemble average of the argument. The shear stress component, defined as $$\begin{aligned} \label{eq:pimunucont} \pi^{xy}(\vec r,t) =\int\frac{g{\mathrm{d}}^3 p}{(2\pi)^3\,E}\,p^xp^y\,f(\vec r,t;\vec p\,)\ ,\end{aligned}$$ is in the simulation replaced by a discretized version, $$\label{eq:pimunu} \overline{\pi}^{xy} = \frac{1}{V} \sum_{i=1}^{N_{\text{part}}}\frac{p_i^x p_i^y}{p_i^0}\ ,$$ summing up all particles in the box with volume $V$ at a given time $t$. The correlator is obtained by the time and ensemble average in the limit $t_{\rm corr} \rightarrow \infty$, $$\begin{aligned} \label{eq:C0_Fourier} C^{xy}(t) &=\left\langle \overline{\pi}^{xy}(t)\,\overline{\pi}^{xy}(0)\right\rangle\nonumber\\ &=\left\langle \frac{1}{t_{\rm corr}} \int_{0}^{t_{\rm corr}} \overline{\pi}^{xy}(t+t')\,\overline{\pi}^{xy}(t')\,{\mathrm{d}}t'\right\rangle\nonumber\\ &=\left\langle \frac{1}{N_{\rm corr}} \sum_{j=0}^{N_{\rm corr}-1} \overline{\pi}^{xy}(i\Delta t + j\Delta t)\,\overline{\pi}^{xy}(j\Delta t)\right\rangle\nonumber\\ &=\mathcal{F}_\omega \left[\vert \overline{\pi}_\omega^{xy}\vert^2\right](t)\end{aligned}$$ where $N_{\rm corr} = t_{\rm corr}/\Delta t$ and $i = t/\Delta t$ and $ \mathcal{F}_\omega$ denotes the Fourier-transformed of its argument, applying the Wiener-Khinchin theorem. Here, $\overline\pi_\omega$ stands for the Fourier-transformed of $\overline\pi$. If the system fluctuates around the equilibrium state, one finds [@Muronga:2003tb] $$C^{xy}(t) = C^{xy}(0)\, {\mathrm{e}}^{-\frac{t}{\tau}}.$$ Therefore one obtains $$\begin{aligned} \label{eq:eta} \eta = \frac{V}{T} \int_{0}^{\infty} {\mathrm{d}}t \, C^{xy}(t) = \frac{C^{xy}(0) V \tau}{T}\ .\end{aligned}$$ This procedure is illustrated in \[fig:Tmunu\], showing an example of the oscillating $\pi^{xy}(t)$, and in \[fig:slopes\], where the exponential decaying slopes are cleary visible. ![Example of $\pi^{xy}(t)$ as function of time $t$.[]{data-label="fig:Tmunu"}](Tmunu.eps){width="\columnwidth"} ![The correlation function $C^{xy}$ for different temperatures[]{data-label="fig:slopes"}](slopes.eps){width="\columnwidth"} The value $C^{xy}(0)$ is of special interest because the analytic expression can be calculated easily noticing, that $C^{xy}(0) = \text{Var} \left[\overline{\pi}^{xy}\right]$. Thus, using the continuous formulation \[eq:pimunucont\], one obtains for one single particle species with mass $m$ and degeneracy $g$ [@Wesp:2011yy; @Rose:2017bjz] $$\begin{aligned} C^{xy}_m(0) &=\frac{g}{30\pi^2 V}\int_0^\infty {\mathrm{d}}p \frac{p^6}{E^2} \exp\left(-\frac{E}{T}\right)\end{aligned}$$ with $E = \sqrt{m^2+p^2}$. This integral has to be performed numerically. Finally, to get a result for the Hagedorn gas, one has to sum over all masses, $$\begin{aligned} \label{eq:C0_ana} C^{xy}(0) &= \frac{1}{30\pi^2 V}\int{\mathrm{d}}m\,\rho(m)\int_0^\infty {\mathrm{d}}p \frac{p^6}{E^2} \exp\left(-\frac{E}{T}\right)\ .\end{aligned}$$ Irrespective of the numerical integrations, we will call these results still ’analytical’ to contrast them from the results obtained via the Monte Carlo calculations. One observes a very nice agreement of analytical, \[eq:C0\_ana\], and numerical results, \[eq:C0\_Fourier\], as shown in \[fig:c0\]. ![A comparison of analytical, \[eq:C0\_ana\], and numerical results, \[eq:C0\_Fourier\], of $C^{xy}(0)$.[]{data-label="fig:c0"}](C0.eps){width="\columnwidth"} If $C^{xy}(0)$ is one value of interest one gets out of the Green-Kubo formalism, the other one is the relaxation time $\tau$, the slope of the correlator, shown in \[fig:tau\]. ![Results for the slope parameter $\tau$ from the fitting procedure for varying temperature $T$. The error bars indicate the statistical error (see text for details). []{data-label="fig:tau"}](tau.eps){width="\columnwidth"} One observes, that the $\tau$ parameter of the pion gas decreases smoothly and less rapid than that of the Hagedorn gas. Here, no analytic estimator is available at the moment. While during the fitting procedure, $C^{xy}(0)$ varies only little and agrees nearly perfectly with the analytic estimate, the results of the fits for the relaxation time $\tau$ vary drastically between different runs. Therefore also the statistical error of this quantity as obtained by calculating the Jackknife variance (for a review see [@Miller:1974]) is shown in \[fig:tau\]. Nevertheless, considering a relaxation time as the inverse of an interaction rate, one may express (in a low density approximation) $\tau=1/\Gamma=1/\langle n\sigma v_{\rm rel}\rangle$. Thus, the exponential behavior of $\tau$ as function of the temperature $T$ is mainly dictated by the increase of the particle density $n$. It may be matter of debate, if the factor $\langle \sigma v_{\rm rel}\rangle$ really directly translates into the transport cross section $\sigma_{\rm tr}=2/3\sigma_{\rm tot}$. Here further investigations are at order. It is very instructive to check the Green-Kubo method against some known input. For this, an implementation of the algorithm of generating random numbers with memory [@Schmidt:2014zpa] enables to dial in specific values for the correlation and compare with the results of the Green-Kubo method. Error estimates according a Jacknife method show clearly, that the error scales as usual with $1/\sqrt{N_{\rm run}}$, if $N_{\rm run}$ independent runs are performed, but with $1/N_{\rm timestep}$ in a single run. Thus it is more preferable, to perform long runs, than doing multiple short runs. In addition, having an estimate for the correlation time $\tau$, the effect of the timestep size may be estimated correctly. Results ======= The final results for the shear viscosity using the Green-Kubo method \[eq:eta\] are shown in \[fig:eta\] and compared to analytic estimates. ![A comparison of analytical, \[eq:etachapens\], and numerical values of the shear viscosity $\eta$. The error bars shown emerge from the error bars of $\tau$ shown in \[fig:tau\]. []{data-label="fig:eta"}](eta.eps){width="\columnwidth"} The agreement is very well; while there is some tiny underestimation for $T<100{{\ensuremath{\,{\rm MeV}}\xspace}}$, Monte Carlo results coincide very well with the analytic estimates for higher temperatures. Here one can also see, that $\eta$ stays more or less the same for both species at lower temperature and starts to diverge the more particles are created in the box in the Hagedorn case. The values explode, if the particle number density increases ad infinitum near $T_H$. Nevertheless, it increases less rapidly than the entropy density as shown in \[fig:s\_T3\]. It is interesting to observe, that the competing differences in the intermediate result of $C(0)$ and $\tau$ cancel each other at low temperatures and only for $T\gtrsim 140{{\ensuremath{\,{\rm MeV}}\xspace}}$, a different behavior between the pion gas and the Hagedorn state gas my be observable. Combining both the results of the thermodynamical quantities (the entropy density), and the shear viscosity, \[fig:eta\_s\] shows the final result, the shear viscosity to entropy density ratio. ![The final result is $\eta$ normalized to the entropy density for numerical and analytical estimates. Error bars are as in \[fig:eta\]. The KSS bound $1/4\pi$ is indicated. The Hagedorn extrapolation $M\to\infty$ contains both separate $\eta$ and $s$ extrapolations. []{data-label="fig:eta_s"}](eta_sloglog.eps){width="\columnwidth"} As expected, at low temperatures the results for the pion gas and the Hagedorn gas coincide. Since the entropy density $s$ very rapidly starts to diverge with increasing temperature, also the fraction $\eta/s$ diverges. Finally, all the calculated results via the Monte Carlo/Green Kubo approach a stop at values above the KSS bound of $1/4\pi$. The analytic estimates indicate, that the results drop below this boundary and go to zero when temperature increases further. Using the statistical error for $\tau$, one can compute the errors for $\eta$ and $\eta/s$. In \[fig:eta\] and \[fig:eta\_s\] one observes, that the numerical results including the errorbars do not match the analytical curve. This leads us to the finding, that there are some systematic error in the Green-Kubo formalism, which are underestimated in the current work. Conclusions {#sec:Conclusions} =========== In the present work, the transport coefficient $\eta/s$ has been calculated for a gas of Hagedorn resonances. Using the usual way of doing Monte Carlo simulations with a Green-Kubo analysis, it has been shown, that these results coincide very well with some analytic estimates. In addition, the same analysis has been performed for a single pion gas with elastic interactions. This, on one hand side allows to check the used analysis routines and also on the second hand side indicates the differences of the interactions. Here, while the MC calculations only consider Hagedorn states with masses $m<10{{\ensuremath{\,{\rm GeV}}\xspace}}$, the analytic estimates allow to extrapolate to a Hagedorn spectrum up to infinite masses. Interestingly, the influence of high mass Hagedorn states with $m>10{{\ensuremath{\,{\rm GeV}}\xspace}}$ is only visible in the present analysis at temperatures $T>160{{\ensuremath{\,{\rm MeV}}\xspace}}$, which are very close to the underlying Hagedorn temperature $T_H=165{{\ensuremath{\,{\rm MeV}}\xspace}}$. Finally, the main result of this study is the finding, that the fraction $\eta/s$ drops while approaching the limiting Hagedorn temperature. While $\eta$ itself increases with increasing temperature, the growth of $s$ overwhelms it and dominates the overall behavior. The KSS bound is violated at $T_H$. This singular behavior may be cured by a phase transition to some other phase with increasing $\eta/s$, being beyond the Hagedorn picture, since the Hagedorn temperature is a limiting temperature. The authors thank Harri Niemi for useful discussions. This work was supported by the Bundesministerium für Bildung und Forschung (BMBF), grant No. 3313040033. [^1]: Non-vanishing chemical potentials disallow the independent summation over quantum numbers in the Hagedorn spectrum and the thermal distribution [@Gallmeister:2017ths] [^2]: Please note the typo concerning $a$ in the original references [@ANDERSON1977408; @DeGroot:1980dk]. The prefactor is chosen here such that $\sigma=\sigma_{\rm tot}$. [^3]: We use internally the three combinations $\pi^{xy}$,$\pi^{xz}$,$\pi^{yz}$ as an additional possibility to estimate the statistical error.

--- address: - 'Institute of Nuclear and Particle Physics, National Centre for Scientific Research “Demokritos”, GR-15310 Aghia Paraskevi, Attiki, Greece' - 'Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigrad Road, 1784 Sofia, Bulgaria' - 'Department of Physics, University of Istanbul, 34134 Istanbul, Turkey' - 'Wright Laboratory, Yale University, New Haven, Connecticut 06520, USA' - 'Facility for Rare Isotope Beams, 640 South Shaw Lane, Michigan State University, East Lansing, MI 48824 USA' - 'Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany' author: - 'S. Sarantopoulou' title: 'Proxy-SU(3) symmetry in heavy nuclei: Prolate dominance and prolate-oblate shape transition' --- [1]{}, , , , , , , [1]{} [2]{} [3]{} [4]{} [5]{} [6]{} Using a new approximate analytic parameter-free proxy-SU(3) scheme, simple predictions for the global feature of prolate dominance and for the locus of the prolate-oblate shape transition have been made and compared with empirical data. Emphasis is placed on the mechanism leading to the breaking of the particle-hole symmetry, which is instrumental in shaping up these predictions. It turns out that this mechanism is based on the SU(3) symmetry and the Pauli principle alone, without reference to any specific Hamiltonian. Intoduction =========== The recent introduction of the proxy-SU(3) scheme [@proxy1] has led to parameter independent predictions of the deformation parameters $\beta$ and $\gamma$ [@proxy2], as well as to predictions of the locus of the prolate to oblate shape transition on the nuclear chart and the prolate over oblate dominance in deformed rare earth nuclei [@proxy2]. In Section 2 of the present work we extend the study of the prolate to oblate transition to the actinides and superheavy elements, while in Section 3 we study in detail the particle-hole symmetry breaking, which leads to the prolate over oblate dominance and to the determination of the border of the prolate to oblate transition. Prolate to oblate transition in actinides and superheavy elements ================================================================= In order to determine in proxy-SU(3) the SU(3) irreducible representation (irrep) corresponding to a given nucleus, one needs the highest weight (h.w.) irreps of the relevant U(n) algebras for protons and neutrons, given in Table I. This is an extension of Table [I]{} of Ref. [@proxy2], to which the U(28) and U(36) results have been added. These results can be obtained through the UNTOU3 code [@code]. For the h.w. irreps, in particular, an analytic formula exists [@Kota1; @Kota2; @Kota3]. The extra columns in Table I can be used for the determination of the h.w. irreps in the actinides, as well as in superheavy elements (SHE) [@Ring; @Skalski]. For the illustrative and pedagogical purposes of this work, including the text and Table II, we take the relevant shells for the actinides and super heavy nuclei as $Z = 82$-126 and $N = 126$-184 although the upper bounds are by no means certain and microscopic calculations give many varying scenarios. Consider $^{260}_{100}$Fm$_{160}$ as an example. This nucleus possesses $100-82=18$ valence protons in the 82-126 shell, which is approximated by the proxy-SU(3) pfh shell with U(21) symmetry. Therefore the h.w. irrep for the valence protons is (36,6). It also has 160-126=34 valence neutrons in the 126-184 shell, approximated by the sdgi proxy-SU(3) shell with U(28) symmetry. Therefore the h.w. irrep for the valence neutrons is (46,16). The whole nucleus is then described by the stretched [@DW2] irrep (82,22). Results for nuclei with protons in the 82-126 shell and neutrons in the 126-184 shell are summarized in Table II. Prolate nuclei are characterized by $\lambda>\mu$, while oblate nuclei have $\lambda<\mu$ [@proxy2]. In Table II we see that a small region of oblate nuclei (underlined) appears in the lower right corner of the table, i.e., just below the proton shell closure and the neutron shell closure. This result is similar to what is seen in Tables II and III of Ref. [@proxy2] for rare earth nuclei. Beta and gamma deformation parameters for the nuclei appearing in Table II can be found in another paper in this Workshop [@BonatsosSDANCA17]. Breaking of the particle-hole symmetry {#ph} ====================================== The breaking of the particle-hole (p-h) symmetry plays a key role in the determination of the place of the prolate to oblate transition within a given shell, as seen in Ref. [@proxy2] (see especially Fig. 1 in this reference). This topic has been briefly discussed in Ref. [@EPJA], but it does deserve a more detailed discussion. In a simple way, one can say that the highest weight (h.w.) irrep corresponds to the most probable distribution of a given number of particles over the available levels in a given harmonic oscillator (h.o.) shell. This is explained in a clear way in Fig. 1 of Ref. [@code]. In more detail, let us consider the $n$-th shell of the harmonic oscillator, which possesses and overall U((n+1)(n+2)/2) symmetry. For example, the sd shell has $n=2$ and possesses the U(6) symmetry, the pf shell has $n=3$ and possesses the U(10) symmetry, and so on. Let us consider the U(6) case in more detail. If we have a system of bosons with U(6) symmetry, as in the Interacting Boson Model (IBM) [@IA], one boson will give the (2,0) h.w. irrep, 2 bosons will give (4,0), 3 bosons will give (6,0), 4 bosons will give (8,0), N bosons will give (2N,0), as in IBM. This happens because bosons will crowd the most symmetric irrep of U(6), characterized by a Young diagram with only one line (fully symmetric). For 1, 2, 3, 4 bosons the U(6) irreps are \[1\], \[2\], \[3\], \[4\] respectively. If we now consider a system of (like) fermions with U(6) symmetry, one fermion will give (2,0), 2 fermions will give (4,0), 3 fermions will give (4,1), 4 fermions will give (4,2), as in Table I. This is a consequence of the Pauli principle. Because of antisymmetrization, the U(6) irreps in this case can have only 2 columns, thus for 1, 2, 3, 4 fermions the U(6) irreps will be \[1\], \[2\], \[21\], \[22\], as in Table I. We see that the differences between the boson irreps and the fermion irreps, both in U(6) and in its SU(3) subalgebra, arise solely from the Pauli principle, without reference to any specific Hamiltonian. In a similar way, the particle-hole asymmetry arises because of the restrictions imposed by the Pauli principle, irrespectively of any specific Hamiltonian. Let us now consider a pf shell and start filling it with fermions. One fermion will have the (3,0) h.w. irrep, 2 fermions will have the (6,0) irrep, 3 fermions will have the (7,1) irrep, 4 fermions will have the (8,2) irrep, 5 fermions will have the (10,1) irrep, 6 fermions will have the (12,0) irrep, as in Table 1, the corresponding U(10) irreps being \[1\], \[2\], \[21\], \[22\], \[221\], \[222\] respectively. By the way, if we had bosons, the U(10) irreps would have been \[1\], \[2\], \[3\], \[4\], \[5\], \[6\], and the SU(3) irreps (3,0), (6,0), (9,0), (12,0), (15,0), (18,0) respectively. The consequences of the Pauli principle are again vividly present. We go on gradually filling the pf shell and we end up with the results of Table I. These indicate that up to 4 particles or 4 holes, there is a particle-hole symmetry, but the symmetry is broken beyond this point. Actually this p-h symmetry is seen up to 4 particles or 4 holes in all U(n) symmetries of the h.o., irrespectively of n. Full results of the irreps occurring for 2, 4, 6, 8, 12, 14, 16, 18 particles are given in Table III, produced by the code of Ref. [@code]. Several comments apply. 1\) In the case of 2 and 18 particles, 2 irreps appear in each case. The h.w. irreps are conjugate to each other, as it should be expected with p-h symmetry present. 2\) In the case of 4 and 16 particles, 11 different irreps appear (some of them more than once, as indicated by the exponents, which stand for multiplicity numbers). The irreps appearing for 16 particles are the conjugates of the irreps appearing for 4 particles, but they do not appear in the same order. However, the h.w. irrep (2,8) in the case of 16 particles is the conjugate of the (8,2) irrep, which is the h.w. irrep in the case of 4 particles. Thus the p-h symmetry is preserved, but only as far as the h.w. irrep is concerned. 3\) In the case of 6 and 14 particles, 24 different irreps appear (some of them more than once, as indicated by the exponents in Table 3, which indicate multiplicities of irreps). The irreps appearing for 14 particles are the conjugates of the irreps appearing for 6 particles, but they do not appear in the same order. Furthermore, while the (12,0) irrep is the h.w. irrep for 6 particles, its conjugate, (0,12), appears in the 8th place in the case of 14 particles. In addition, the (6,6) irrep, which is the h.w. irrep for 14 particles, appears in the 3rd place in the case of 6 particles. In other words, while the (12,0) irrep is the most probable one for 6 particles, for 14 particles the (6,6) irrep is the most probable one. This is a consequence of the Pauli principle alone, irrespectively of any Hamiltonian. The restrictions imposed by the Pauli principle in the lower half of the shell, where the shell is relatively empty and the particles have more choices at their disposal, leads to a result different from the one in the upper half of the shell, where particles get crowded and have fewer choices at their disposal. 4\) The same asymmetry is seen in the case of 8 and 12 particles. The sets of 32 different irreps appearing in these two cases are still conjugate to each other, but the order in which they appear and the h.w. irrep are different for each particle number. Similar conclusions can be drawn from the lists of SU(3) irreps occurring in the sdg shell with U(15) symmetry, given in Table IV. From the above it becomes clear that a particle-hole symmetry holds as far as the set of irreps appearing in each case is concerned. However, the p-h symmetry is broken as far as the order of appearance of the irreps according to their weight is concerned. As a result, the h.w. irrep is different if more than 4 particles or holes are present. This is a result imposed by the Pauli principle alone, without involving any specific Hamiltonian. In earlier work, the argument was used that the irrep with the highest eigenvalue of the second order Casimir operator of SU(3) should be used, because it corresponds to the highest value of the quadrupole-quadrupole interaction, through the well-known relation [@Draayer] $$C_2= {1\over 4} Q\cdot Q + {3\over 4} L^2,$$ where $C_2$ stands for the second order Casimir operator of SU(3), $Q$ is the quadrupole operator and $L$ denotes the angular momentum. This choice gives identical results with the h.w. choice up to the middle of the shell, as one can see in Table I, in which the irreps corresponding to the highest eigenvalue of $C_2$ are given in the columns labelled by $C$. Beyond the middle of the shell the $C_2$ choice gives results corresponding to the p-h symmetry, since obviously the irrep with the highest $C_2$ eigenvalue will be the same one both in the upper and in the lower part of the shell, since the expression for the eigenvalues is symmetric in $\lambda$ and $\mu$, namely [@Draayer] $$C_2(\lambda,\mu) = (\lambda+\mu+3)(\lambda +\mu) -\lambda \mu.$$ As an example, consider the case of 8 and 12 particles in the pf shell. For 8 particles, the h.w. irrep is the (10,4) one. One can easily see that this is the irrep with the highest $C_2$ eigenvalue among all the irreps appearing for 8 particles. In the upper part of the shell, the irrep (4,10) still has the highest $C_2$ eigenvalue, but the Pauli principle has pushed it down to 4th place as far as the weight, i.e. the probability of appearance, is concerned. The h.w. irrep in this case is (12,0), despite the fact that it possesses a lower $C_2$ eigenvalue. From the above it becomes clear that the p-h symmetry breaking appearing in the proxy approach is imposed by the Pauli principle alone, without reference to any particular Hamiltonian. In contrast, the choice of the irrep with the highest $C_2$ eigenvalue is based on a particular choice of the Hamiltonian. Fortunately, most of the applications in earlier work have been carried out in the lower half of various shells, thus the bulk of the relevant results remains perfectly valid. It is well known [@code] that the h.w. irrep has to be a unique one, i.e., it has to occur only once. From Table 3 we see that the percentage of different irreps which qualify as candidates for the h.w. place is reduced with increasing number of particles within the lower half of the shell. For 2 particles there are 2 unique irreps out of 2 (100%), for 4 particles there are 9 unique irreps out of 11 (81.2%), for 6 particles there are 9 unique irreps out of 24 (37.5%), for 8 particles there are 8 unique irreps out of 32 (25%). Once more, similar conclusions can be drawn from the lists of SU(3) irreps occurring in the sdg shell with U(15) symmetry, given in Table IV. Conclusions =========== The particle-hole symmetry breaking, which is instrumental in causing the prolate over oblate dominance in deformed nuclei and in defining the locus of the prolate to oblate shape transition, is found to be a consequence of the SU(3) symmetry and the Pauli principle, without reference to any specific Hamiltonian. Acknowledgements {#acknowledgements .unnumbered} ================ Work partly supported by the Bulgarian National Science Fund (BNSF) under Contract No. DFNI-E02/6, by the US DOE under Grant No. DE-FG02- 91ER-40609, and by the MSU-FRIB laboratory, by the Max Planck Partner group, TUBA-GEBIP, and by the Istanbul University Scientific Research Project No. 54135. [99]{} D. Bonatsos, I. E. Assimakis , N. Minkov, A. Martinou, R. B. Cakirli, R. F. Casten, and K. Blaum, Proxy SU(3) symmetry in heavy deformed nuclei, Phys. Rev. C **95**, 064325 (2017). D. Bonatsos, I. E. Assimakis, N. Minkov, A. Martinou, S. Sarantopoulou, R. B. Cakirli, R. F. , and K. Blaum, Analytic predictions for nuclear shapes, prolate dominance and the prolate-oblate shape transition in the proxy-SU(3) model, Phys. Rev. C **95**, 064326 (2017). J. P. Draayer, Y. Leschber, S. C. Park, and R. Lopez, Representations of U(3) in U(N), Comput. Phys. Commun. **56**, 279 (1989). V. K. B. Kota, Reduction of oscillator orbital symmetry partitions into irreducible representations of SU(3), Physical Research Laboratory (Ahmedabad, India) Technical Report PRL-TN-97-78 (1978). V. K. B. Kota, Single particle SU(3) parentage coefficients, Pramana [**9**]{}, 129 (1977). V. K. B. Kota, private communication. S. E. Agbemava, A. V. Afanasjev, T. Nakatsukasa, and P. Ring, Covariant density functional theory: Reexamining the structure of superheavy nuclei, Phys. Rev. C [**92**]{}, 054310 (2015). M. Kowal, P. Jachimowicz, and J. Skalski, Ground state and saddle point: masses and deformations for even-even superheavy nuclei with $98 \leq Z \leq 126$ and $134\leq N \leq 192$, arXiv:1203.5013 \[nucl-th\]. J. P. Draayer and K. J. Weeks, Towards a shell model description of the low-energy structure of deformed nuclei I. Even-even systems, Ann. Phys. (N.Y.) **156**, 41 (1984). D. Bonatsos, I.E. Assimakis, N. Minkov, A. Martinou, S. Peroulis, S. Sarantopoulou, R.B. Cakirli, R.F. Casten, and K. Blaum, Proxy-SU(3): A symmetry for heavy nuclei, Proceedings of the International Workshop on Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects (SDANCA17), Bulg. J. Phys. (2017) in press. D. Bonatsos, Prolate over oblate dominance in deformed nuclei as a consequence of the SU(3) symmetry and the Pauli principle, Eur. Phys. J. A **53**, 148 (2017). F. Iachello and A. Arima, [*The Interacting Boson Model*]{} (Cambridge University Press, Cambridge, 1987). J. P. Draayer, Fermion models, in [*Algebraic Approaches to Nuclear Structure: Interacting Boson and Fermion Models*]{}, ed. R. F. Casten (Harwood, Chur, 1993) p. 423.

--- abstract: 'We investigate Co nanostructures on Bi$_{2}$Se$_{3}$ by means of scanning tunneling microscopy and spectroscopy \[STM/STS\], X-ray absorption spectroscopy \[XAS\], X-ray magnetic dichroism \[XMCD\] and calculations using the density functional theory \[DFT\]. In the single adatom regime we find two different adsorption sites by STM. Our calculations reveal these to be the fcc and hcp hollow sites of the substrate. STS shows a pronounced peak for only one species of the Co adatoms indicating different electronic properties of both types. These are explained on the basis of our DFT calculations by different hybridizations with the substrate. Using XMCD we find a coverage dependent spin reorientation transition from easy-plane toward out-of-plane. We suggest clustering to be the predominant cause for this observation.' address: - '$^1$ Institute of Applied Physics, University of Hamburg, Jungiusstra[ß]{}e 11, 20355 Hamburg, Germany' - '$^2$ Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, 30 Mickiewicza Av., 30-059 Krakow, Poland' - '$^3$ Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52428 Jülich, Germany' - '$^4$ Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN, USA' author: - 'T Eelbo$^1$, M Sikora$^2$, G Bihlmayer$^3$, M Dobrzański$^2$, A Kozłowski$^2$, I Miotkowski$^4$ and R Wiesendanger$^1$' title: 'Co atoms on Bi$_{2}$Se$_{3}$ revealing a coverage dependent spin reorientation transition' --- With decreasing dimensionality and size of nanostructures, there is an increasing importance of the electronic interactions with the supporting substrate. This may lead to new electronic and magnetic properties, e.g. on substrates with a large spin-orbit interaction \[SOI\] giant magnetic anisotropies of isolated Co atoms were discovered [@Gambardella2003] while on alkali metals these atoms behave as being quasi-free with vanishing magnetic anisotropy energies [@Gambardella2002]. The interaction of nanostructures with the supporting substrate might also induce spin reorientation transitions \[SRTs\]; for example, a monolayer of Fe on W(110) reveals an in-plane anisotropy while the anisotropy changes to out-of-plane if a second layer is grown on top [@Pietzsch2000]. The adsorption of individual magnetic adatoms and nanostructures on exotic surfaces, like three dimensional topological insulators is currently of high interest both from a fundamental point-of-view as well as in view of potential applications. Topological insulators \[TI\] are a new class of materials characterized by a strong SOI which leads to gapless surface states with an odd number of Fermi-level crossings within the bulk band gap [@Fu2007]. In the simplest case, a single Dirac cone consists of two spin-polarized linear dispersing branches and the crossing point, i.e. the Dirac point \[DP\], is protected by time reversal symmetry [@Kane2005]. The time reversal symmetry is broken if magnetic impurities are introduced and the Dirac point gets massive if species with a net out-of-plane magnetic moment are added [@Hor2010; @Chen2010; @Wray2011; @Okada2011; @Xu2012; @Chang2013]. For this reason, we investigated the properties of Co atoms after their adsorption on the surface of Bi$_{2}$Se$_{3}$. By means of scanning tunneling microscopy and spectroscopy \[STM/STS\], X-ray absorption spectroscopy \[XAS\] and X-ray magnetic circular dichroism \[XMCD\] we explore the electronic and magnetic properties of the transition metal \[TM\] adatoms. We explain our findings based on density functional theory \[DFT\] calculations performed in the generalized gradient approximation [@Perdew1996]. The experiments have been carried out in two separate ultrahigh vacuum systems. Scanning tunneling microscopy and spectroscopy experiments were performed at 5K on Bi$_2$Se$_3$ single crystals *in situ* cleaved at low temperatures. Using electron beam evaporators Co was directly deposited onto the cold sample at 12K to obtain well-isolated Co adatoms on the surface. To gain information about the local density of states \[LDOS\] electronic conductance \[d$I$/d$U$\] spectra were acquired by means of a lock-in technique using a modulation voltage $U_{\rm{mod}}=20$mV and a frequency $f=5$kHz. The XAS and XMCD experiments have been carried out at the ID08 beamline at the European Synchrotron Radiation Facility. While the measurement temperature was about $T\approx 8$K, the single crystals had to be cleaved at room temperature and immediately cooled down afterward. Co atoms have been deposited by using an electron beam evaporator with the substrate remaining in the measurement stage at a temperature of $T\approx 10$K. X-ray absorption spectra were obtained in the total-electron-yield mode using almost $100 \%$ polarized light. Magnetic fields of up to 5T were applied collinear to the incident beam. In addition, the sample was rotated from normal \[$0^\circ$\] to steep \[$70^\circ$\] incidence angle to obtain information about the in- and out-of-plane magnetic properties. All spectra have been normalized with respect to the incident beam intensity and the Co $L_3$ pre-edge intensity. ![\[Fig1\] (a) STM topography of two different isolated Co adatoms on Bi$_{2}$Se$_{3}$. The tunneling parameters are $U=0.2$V and $I=0.75$nA. The inset shows a theoretical simulation of the bare surface. The white lines are guide lines to the eyes representing Se top sites. (b) and (d) Magnified views on both types of adsorbates showing the affected surface in their vicinity as well. (c) and (e) Simulations of Co atoms adsorbed in the fcc and hcp hollow site for a bias voltage of $U=0.1$V.](Fig1a_1e.eps){width="50.00000%"} We used STM to address the local properties of the Co adatoms. Contrary to a recent work on Co monomers on Bi$_{2}$Se$_{3}$, where the authors concluded Co atoms to adsorb on Se top sites [@Ye2012], we find two different types of Co atoms on Bi$_{2}$Se$_{3}$. Figure \[Fig1\](a) proves that both species \[in the following Co$_{\rm A/B}$\] occupy different adsorption sites which we illustrate by the white guidelines to the eyes. Regarding their apparent heights and shapes as well as the influence on the surrounding substrate further differences appear among both species. On the one hand, for the stabilization voltage chosen, Co$_{\rm A}$ shows a larger apparent height \[$\approx0.7$Å\] than Co$_{\rm B}$ \[$\approx0.3$Å\]. On the other hand, in case of Co$_{\rm A}$, the substrate shows an almost sixfold symmetric pattern in its vicinity whereas Co$_{\rm B}$ induces a threefold symmetric pattern, compare figures \[Fig1\](b) and (d). Importantly, at no bias voltage, the Co adatoms appear as dark triangular depressions which generally would hint toward a substitution of Bi at its lattice site, which e.g. has been found for Fe adatoms after room temperature annealing [@Schlenk2013]. For a coverage of 0.01 monolayer equivalent \[MLE\] the relative abundance of both species is approximately three to one with a predominance of Co$_{\rm A}$ type adatoms indicating that the adsorption site of Co$_{\rm A}$ is energetically favorable. We note that depending on the tunneling parameters \[e.g. during STS\] Co$_{\rm B}$ type atoms can be manipulated and afterward appear as Co$_{\rm A}$. A change from Co$_{\rm A}$ toward Co$_{\rm B}$ has never been observed. To elucidate our STM/STS observations, DFT calculations using the full-potential linearized augmented planewave method as implemented in the [Fleur]{}-code [@fleur] have been performed. The model comprises a $\left(\sqrt{3} \times \sqrt{3}\right)\rm{R}30^{\circ}$ unit cell of four quintuple layers of Bi$_{2}$Se$_{3}$. We find the fcc hollow position to be the energetically most favorable adsorption site. Nevertheless, the hcp hollow position is unfavorable by only $\approx90$meV per atom. For a comparison with the experimental data, STM topographies of the bare surface have been simulated using the vacuum density of states up to +100mV, compare the inset of figure \[Fig1\](a). At this voltage the Se atoms show up as bright triangles. Guidelines which cross at these positions reveal that Co$_{\rm A/B}$ do occupy different hollow sites. Further simulations of Co atoms adsorbed in the fcc and hcp hollow sites reveal different appearances of both species, shown in figures \[Fig1\](c) and (e). The good agreement between the simulated topographies and the experimental data let us conclude that Co$_{\rm A}$ is adsorbed in the fcc hollow site while Co$_{\rm B}$ is adsorbed in the hcp hollow site. This assignment is further supported by the relative abundance if the energy difference between both adsorption sites is taken into consideration. ![\[Fig2\] STS of pristine Bi$_{2}$Se$_{3}$ \[right y-axis\] and 0.01 MLE Co/Bi$_{2}$Se$_{3}$ \[left y-axis\]. The off-dopant spectrum has been acquired after Co deposition far away from any Co adatom. The blue and black arrows indicate the energetic position of the DP before and after Co deposition and, hence, reveal a shift of $\approx-50$mV. Tunneling parameters are $U=0.2$V and $I=0.1$nA.](Fig2.eps){width="55.00000%"} ![\[Fig3\] Fat band analysis and vacuum DOS of Co adsorbed in the fcc hollow site (a) and in the hcp hollow site (b). The red and blue lines indicate spin up and spin down states with respect to the easy axes while the black lines depict the total DOS. The size of the circles gives the spin-polarization of the states in a region above the surface. The gray lines depict the computed Dirac points at -0.55eV and -0.25eV for fcc and hcp occupation, respectively.](Fig3a_3b.eps "fig:"){width="80.00000%"}\ In addition, the simulations indicate a relaxation of the Co atoms into the surface of Bi$_{2}$Se$_{3}$ in both cases by $\approx0.2$Å. Opposite to theory, the experimentally resolved apparent heights differ significantly in case of fcc and hcp occupation. Thus, we relate the difference of the observed apparent heights to different hybridizations with the surrounding atoms resulting in different electronic properties for both species \[adsorption sites\]. Therefore, the adatoms have been investigated by means of STS. Figure \[Fig2\] shows STS spectra of pristine Bi$_{2}$Se$_{3}$ and Co/Bi$_{2}$Se$_{3}$. While for the pristine crystal the onset of the bulk valence \[conduction\] band is detected at $\approx-450$mV \[$\approx0$mV\], we find a global minimum \[blue arrow\] which we assign to the Dirac point \[DP\] at $-300$mV. The shift with respect to the Fermi level indicates that the crystal is naturally electron doped, being in agreement with previous observations [@Hor2009; @Bianchi2010] and attributed to the existence of Se$_{\text{Bi}}$ antisite defects [@Urazhdin2002]. Upon Co deposition of 0.01 MLE the off-dopant spectrum exhibits a global minimum \[black arrow\] at $\approx-350$mV, which hence depicts an additional shift of $\approx-50$mV of the DP with respect to the Fermi level. We conclude that Co further n-dopes the substrate and acts as a donor. In contrast to recent predictions [@Liu2009; @Schmidt2011], no indication of a global surface band gap has been found after the deposition of Co adatoms. Regarding the STS spectra of the adatoms, no distinct resonances have been found in case of Co$_{\rm A}$, whereas Co$_{\rm B}$ type adatoms reveal a pronounced peak at $-400$mV. According to our theoretical calculations, the different electronic properties and the appearance of a resonance only in the hcp case are caused by different hybridizations of the Co $3d$ electrons with Bi$_{2}$Se$_{3}$, as is illustrated in figure \[Fig3\]. The fat band analysis of Co adatoms in the fcc \[figure \[Fig3\](a)\] and hcp hollow site \[figure \[Fig3\](b)\] reveal the Dirac points to be located at -0.55eV and -0.25eV, respectively. Spin up and spin down weights are defined with respect to the easy axes of both configurations, which are in-plane for fcc and out-of-plane for the hcp adsorption. While in case of hcp occupation the band structure shows two additional Co bands at $\approx-0.1$eV below the DP, these bands are shifted toward $\approx0.3$eV above the DP in case of fcc. We also notice the different dispersion of these bands in the fcc and hcp geometry: while in the latter case the hybridization with the topmost valence band (mainly Bi $p_z$ states [@Zhang2009]) leads to an increase of the binding energy at the center of the Brillouin-zone, in the fcc-case the dispersion of the band is inverted. As a result, the Co-induced peak in the vacuum DOS is energetically higher in the hcp than in the fcc case. Although the calculations were performed for a coverage of 0.33 MLE and, hence, the absolute positions of the peaks cannot be directly compared to the STS data, the differences between the fcc and hcp occupation are significant and can be related to the different STS observations, in particular the pronounced peak observed for Co$_{\rm B}$. ![\[Fig4\] (a) Sketch of the experimental setup. The magnetic field can be applied parallel to the incident beam direction whereas the sample can be inclined with respect to this direction. (b) XAS and XMCD spectra for 0.01 MLE of Co/Bi$_{2}$Se$_{3}$ for normal (upper panel) and grazing (lower panel) incidence angle. The inset shows the XMCD signal strength normalized by the XAS $L_3$ peak height. (c) Spectra for an increased coverage of 0.08 MLE. (d) Normalized XMCD signals for a series of coverages ranging between (b) and (c) for normal (left panel) and grazing (right panel) incidence angle. The colored lines are guidelines to the eyes indicating differences between both angles.](Fig4a_4d.eps "fig:"){width="70.00000%"}\ The electronic and magnetic properties have been further tested by XAS and XMCD measurements, summarized in figure \[Fig4\]. A sketch of the experimental setup is depicted in figure \[Fig4\](a). Different coverages ranging from 0.01 MLE to 0.08 MLE have been investigated. Independent of the coverage, the shapes of the XAS spectra show no distinct multipeak structures besides a slight shoulder on the high-energy side of the Co $L_3$ edge at approximately $780.2$eV, compare figures \[Fig4\](b) and (c). The XAS line shape suggests the Co atoms to be in the electronic configuration of $3d^7$ [@Laan1992]. This result is in agreement to recent works on Fe/Bi$_{2}$Se$_{3}$ [@Honolka2012] as well as Fe and Co/Bi$_{2}$Te$_{3}$ [@Shelford2012] where the TM adatoms were found to be in their pristine configuration as well. Furthermore, the XAS spectra can be used to estimate the branching ratio \[$BR$\] which serves as an indicator of the spin character of the ground state of the adatoms [@branching; @Thole1988]. Within the coverage range investigated, we find a constant value of $BR=0.84 \pm 0.01$ which suggests a high-spin ground state of the Co adsorbates. The inset in figure \[Fig4\](b) shows the angular dependence of the XMCD signal normalized with respect to the $L_3$ XAS intensity, which can be used as an indicator for the easy axis of the Co adatoms. For the low coverage regime we find the signal strength for normal incidence angle to be enhanced by about 20% compared to the signal at grazing incidence angle. This suggests the easy axis to reside in the surface plane similar to Fe/Bi$_{2}$Se$_{3}$ [@Honolka2012] and contrary to predictions of an out-of-plane anisotropy for Co/Bi$_{2}$Se$_{3}$ in [@Ye2012; @Schmidt2011]. The experimentally indicated easy-plane easy axis is particularly in line with our calculations, which predict an easy-plane magnetocrystalline anisotropy energy \[MAE\] in case of fcc hollow site occupation \[Co$_{\rm A}$; $K_{fcc}=-6$meV\] and an out-of-plane MAE in case of the hcp hollow site \[Co$_{\rm B}$; $K_{hcp}=+3$meV\], where $K$ denotes the MAE per adatom. Taking the relative abundance into account \[low coverage: $n_{\rm{Co}_{\rm A}} / n_{\rm{Co}_{\rm B}} = 3/1$\] an easy-plane anisotropy is theoretically expected. The agreement further supports the assignment of the adsorption sites to the different types of Co adatoms. We note, that a site-dependent anisotropy has been reported before on metal substrates [@Blonski2010; @Khajetoorians2013] and as usual depends on the layer thickness, which – for our calculations – has been 0.33 MLE. Moreover, we investigated the anisotropy using the normalized $L_3$ XMCD intensity as a function of the coverage, see figure \[Fig4\](d). Based on the accuracy level given for the estimation of the coverage, the relative $L_3$ XMCD/XAS intensities acquired at both angles suggest an in-plane easy axis at low coverages \[$0.01$ MLE – $0.04$ MLE\] and an out-of-plane easy axis for 0.08 MLE. Unfortunately, the magnetic moments of the Co adatoms have not been saturated at the maximum magnetic field available \[5T\] and, thus, deducing the orbital and effective spin moments would be unreliable. However, the ratio \[$R$\] of orbital to effective spin moment [@ratio] is independent of the saturation and can be used for conclusions about the investigated sample. In the regime of single atoms, we find $R=0.33 \pm 0.02$, which is in good agreement with [@Ye2012] and significantly larger than the Co bulk value [@Chen1995]. The increase of the Co coverage toward 0.08 MLE goes hand in hand with a gain of the ratio until $R=0.49 \pm 0.03$ which indicates relevant changes in the orbital and effective spin moments. Such a trend was observed before [@Ye2012] although the orbital moment is expected to diminish upon increasing the mean cluster size [@Gambardella2003]. This contradiction can be understood taking the SRT into consideration. In case of the low coverage \[0.01 MLE\] we find an easy-plane anisotropy, i.e. $m_L^{\text{x}}>m_L^{\text{z}}$ for z denoting the direction parallel to the surface normal. Hence, the ratio is given by $R^{0.01}=m_L^{\text{x}}/m_S^{\text{x}}=0.33 \pm 0.02$. For the high coverage regime \[0.08 MLE\] we deduce an out-of-plane easy axis which, according to Bruno [@Bruno1989], means that $m_L^{\text{x}}<m_L^{\text{z}}$ in this case. Furthermore, in good agreement with our theoretical model, we can assume the magnetic spin moment to be independent of the orientation, i.e. $m_S^{\text{x}}\approx m_S^{\text{z}}$ \[$1.17 \mu_{\rm B}/\text{atom}\approx 1.15 \mu_{\rm B}/\text{atom}$\]. If the ratio of in-plane orbital moment and in-plane spin moment additionally remains constant while increasing the coverage, then an increase of the ratio given by $R^{0.08}=m_L^{\text{z}}/m_S^{\text{z}}=0.49 \pm 0.03$ is possible and essentially driven by the SRT. We note, that these conclusions are based on the assumption of a vanishing spin dipole moment $m_D$ and that the magnetic spin moments have been calculated for a coverage of 0.33 MLE. In order to understand the spin reorientation transition in more detail, we performed a series of STM/STS experiments at elevated coverages \[up to 0.1 MLE\] and determined the ratio of adatoms showing a resonance at -400mV, i.e. Co$_{\rm B}$, to those not exhibiting this resonance. Although we observe slight changes of this ratio, no drastic modifications were found. Therefore, we experimentally rule out the possibility that a change of the relative population of fcc and hcp hollow sites upon increasing the total Co coverage might cause the observed SRT. Instead, we assign the transition to the decrease of the mean distance between the adatoms upon increasing the Co coverage. From this an emerging interaction as well as the growth of clusters seems plausible, which consequently causes serious modifications of the electronic properties as well as the anisotropy of the magnetic moments. This conclusion is supported by the XAS line shape showing a decrease of the high-energy shoulder at the Co $L_3$ edge upon increasing the coverage, compare figures \[Fig4\](b) and (c). Furthermore, the series of coverage dependent XMCD spectra \[figure \[Fig4\](d)\] shows that the peak position monotonically shifts downward in energy while increasing the coverage. The exact shape of the XAS spectra is determined by the average chemical state and the average crystal field of the ensemble probed by the X-ray beam, since the symmetry and the splitting of the crystal field as well as the chemical state might be slightly different for hcp and fcc occupation. Hence, we relate the variations of the XAS line shape and the shift of the $L_3$ peak position to changes of these quantities. The upcoming growth of clusters while the coverage is raised certainly influences both and, therefore, is a likely explanation for the mutation of the XAS/XMCD spectra. Note, that in the high coverage regime, similar to the low coverage regime, no gap opening has been detected for spectra acquired as far away from any impurity as possible, although the out-of-plane easy axis is evidenced by XMCD. A possible reason might be given in view of the clusters being the predominant cause for the variation of the anisotropy, since spectra in their vicinity have not been measurable because the tunneling conditions have not been sufficiently stable at these locations. In contrast, by the global technique of XMCD the clusters’ influence can be easily detected. In general, our conclusions are supported by a recent study on bulk-doped Mn-Bi$_2$Se$_3$ where excess Mn clusters have been observed on the crystal’s surface and suggested to significantly influence the surface magnetization of the sample [@Zhang2012]. In conclusion, by means of scanning tunneling microscopy we find Co adatoms to occupy fcc and hcp hollow sites on Bi$_{2}$Se$_{3}$ after low temperature deposition. The irreversible switching from Co$_{\rm B}$ into Co$_{\rm A}$ as well as the employed DFT calculations let us assign the Co$_{\rm A}$ type adatoms to be adsorbed in fcc hollow sites. The DFT calculations further indicate different hybridization effects to be the cause for different electronic properties of both species found within the STS spectra. Using XMCD, we determine an easy axis of the magnetization in the surface plane for the low coverage regime, whereas it changes to out-of-plane upon increasing the Co coverage. Cluster formation is suspected to be responsible for the observed spin reorientation transition. We thank M. Gyamfi for fruitful discussions. Financial support from the ERC Advanced Grant FURORE and the DFG-Sonderforschungsbereich 668 is gratefully acknowledged. We gratefully acknowledge computing time from the Jülich supercomputing centre (JSC). References {#references .unnumbered} ========== [23]{} Gambardella P 2003 [*Science*]{} [**300**]{} 1130 Gambardella P, Dhesi S S, Gardonio S, Grazioli C, Ohresser P and Carbone C 2002 [*Phys. Rev. Lett.*]{} [**88**]{} 047202 Pietzsch O, Kubetzka A, Bode M and Wiesendanger R 2000 [*Phys. Rev. Lett.*]{} [**84**]{} 5212 Fu L, Kane C L and Mele E J 2007 [*Phys. Rev. Lett.*]{} [**98**]{} 106803 Kane C L and Mele E J 2005 [*Phys. Rev. Lett.*]{} [**95**]{} 226801 Hor Y S 2010 [*Phys. Rev. B*]{} [**81**]{} 195203 Chen Y L 2010 [*Science*]{} [**329**]{} 659 Wray L A, Xu S Y, Xia Y, Hsieh D, Fedorov A V, Hor Y S, Cava R J, Bansil A, Lin H and Hasan M Z 2011 [*Nat. Phys.*]{} [**7**]{} 32 Okada Y 2011 [*Phys. Rev. Lett.*]{} [**106**]{} 206805 Xu S-Y 2012 [*Nat. Phys.*]{} [**8**]{} 616 Chang C-Z 2013 [*Science*]{} [**340**]{} 167 Perdew J P, Burke K and Ernzerhof M 1996 [*Phys. Rev. Lett.*]{} [**77**]{} 3865 Ye M 2012 [*Phys. Rev. B*]{} [**85**]{} 205317 Schlenk T 2013 [*Phys. Rev. Lett.*]{} [**110**]{} 126804 For a detailed description, see http://www.flapw.de Hor Y S, Richardella A, Roushan P, Xia Y, Checkelsky J G, Yazdani A, Hasan M Z, Ong N P and Cava R J 2009 [*Phys. Rev. B*]{} [**79**]{} 195208 Bianchi M, Guan D, Bao S, Mi J, [Brummerstedt Iversen]{} B, King P D C and Hofmann P 2010 [*Nat. Commun.*]{} [**1**]{} 128 Urazhdin S, Bilc D, Tessmer S H, Mahanti S D, Kyratsi T and Kanatzidis M G 2002 [*Phys. Rev. B*]{} [**66**]{} 161306(R) Liu Q, Liu C X, Xu C, Qi X L, and Zhang S C 2009 [*Phys. Rev. Lett.*]{} [**102**]{} 156603 Schmidt T M, Miwa R H and Fazzio A 2011 [*Phys. Rev. B*]{} [**84**]{} 245418 Zhang H, Liu C X, Qi X L, Dai X, Fang Z and Zhang S C 2009 [*Nat. Phys.*]{} [**5**]{} 438 G and Kirkman I W 1992 [*J. Phys.: Condens. Matter*]{} [**4**]{} 4189 Honolka J 2012 [*Phys. Rev. Lett.*]{} [**108**]{} 256811 Shelford L R, Hesjedal T, [Collins-McIntyre]{} L, Dhesi S S, Maccherozzi F and [van der Laan]{} G 2012 [*Phys. Rev. B*]{} [**86**]{} 081304(R) $BR$: Ratio of the integrated XAS of the $L_3$ edge divided by the sum of the integrated XAS of the $L_3$ and $L_2$ edges. Thole B T and van der Laan G 1988 [*Phys. Rev. B*]{} [**38**]{} 3158 B[ł]{}oński P, Lehnert A, Dennler S, Rusponi S, Etzkorn M, Moulas G, Bencok P, Gambardella P, Brune H and Hafner J 2010 [*Phys. Rev. B*]{} [**81**]{} 104426 Khajetoorians A A, Schlenk T, Schweflinghaus B, dos Santos Dias M, Steinbrecher M, Bouhassoune M, Lounis S, Wiebe J and Wiesendanger R 2013 [*Phys. Rev. Lett.*]{} [**111**]{} 157204 $R$: Ratio of orbital magnetic moment ($m_{L}$) to the sum of spin moment ($m_{S}$) and spin dipole moment ($m_{D}$) according to $R$ = $m_{L}$ / ($m_{S}$ + 7 $m_{D}$), where the moments are given in $\mu_{\rm B}$/atom and are projected along the incident beam direction. Chen C T, Idzerda Y U, Lin H J, Smith N V, Meigs G, Chaban E, Ho G H, Pellegrin E and Sette F 1995 [*Phys. Rev. Lett.*]{} [**75**]{} 152 Bruno P 1989 [*Phys. Rev. B*]{} [**39**]{} 865 Zhang D 2012 [*Phys. Rev. B*]{} [**86**]{} 205127

--- abstract: 'The nonequilibrium dynamics of generic quantum systems denies a fully understanding up to now, even if the thermalization in the long-time asymptotic state has been explained by the eigenstate thermalization hypothesis which proposes a universal form of the observable matrix elements in the eigenbasis of the Hamiltonian. In this paper, we study the form of the quantum state, i.e., of the density matrix elements. We propose that the density matrix has also a universal form in chaotic systems, which is used to understand the nonequilibrium dynamics in the whole time scale, from the transient regime to the long-time steady limit, and then extends the applicability of eigenstate thermalization hypothesis to true nonequilibrium phenomena such as nonequilibrium steady states. Our assumption is numerically tested in various models, and its intimate relation to the eigenstate thermalization hypothesis is discussed.' author: - Xinxin Yang - Pei Wang bibliography: - 'a.bib' title: Nonequilibrium dynamics of the density matrix in chaotic quantum systems --- \[sec:level 1\]Introduction =========================== The nonequilibrium dynamics of quantum many-body systems keeps on attracting attention of both experimentalists [@Bloch08] and theorists [@Polkovnikov11; @Eisert15]. For integrable systems, the case-by-case study of exact solutions revealed exotic properties of the quantum states driven out of equilibrium. The long-time asymptotic state is far from thermal equilibrium, but should be described by the generalized Gibbs ensemble [@rigol2007relaxation]. On the other hand, for systems whose classical counterparts are chaotic, it is widely believed that they will finally thermalize in the long time limit [@rigol2008thermalization]. But the dynamics in the transient and intermediate time scale is still hard to explore, due to the lack of a reliable analytical or numerical method. The study of the dynamics in quantum chaotic systems dated back to the early days of quantum mechanics, when the question has been raised as to how the statistical properties of equilibrium ensembles arise from the linear dynamics of Schrödinger equation in a complex system [@Neumann29; @Goldstein10]. A breakthrough was made in 1950s by Wigner [@Wigner55; @Wigner57; @Wigner58], who stated that the statistics of the eigenenergies of a chaotic system should be as same as that of a random matrix, that is the level spacing follows the Wigner-Dyson distribution. This statement was verified by both experiments and numerical simulations [@rosenzweig1960repulsion; @brody1981random; @bohigas1984characterization; @schroeder1987statistical; @guhr1998random]. But for an integrable system, the level spacing satisfies a Poisson distribution according to Berry and Tabor [@Berry77]. In the random matrix theory (RMT), the eigenstates of the Hamiltonian are considered to be random vectors in the Hilbert space. This oversimplified picture ignores the dependence of the structure of the eigenstate on the eigenenergy, and then fails to explain why the observables are in fact a function of the energy or temperature of the system. A further step was made in the eigenstate thermalization hypothesis (ETH) [@deutsch1991quantum; @srednicki1994chaos; @srednicki1999approach], which proposed a generic form of the matrix elements of observable operators in the eigenbasis of the Hamiltonian: $$\label{eth} O_{\alpha\beta}={O}\left(\bar{E}\right)\delta_{\alpha\beta} +D^{-\frac{1}{2}}\left(\bar{E}\right)f_O\left(\omega,\bar{E}\right)R^O_{\alpha\beta},$$ where $\alpha$ and $\beta$ are the eigenstates of the Hamiltonian with $E_{\alpha}$ and $E_{\beta}$ being their eigenenergies, respectively. $\bar{E}= (E_{\alpha}+E_{\beta})/2$ and $\omega=E_{\alpha}-E_{\beta}$ denote the average energy and the energy difference of $\alpha$ and $\beta$, respectively. $D\left(\bar{E}\right)$ is the density of many-body states, which increases exponentially with the system size (or the total number of particles). ${O}\left(\bar{E}\right)$ and $f_O\left(\omega,\bar{E}\right)$ are both smooth functions, with the former describing how the expectation value of the observable changes with energy. The randomness of the eigenstates is reflected in Eq. (\[eth\]) by the random number $R^O_{\alpha\beta}$, which has zero mean and unit variance according to definition. When a chaotic system is driven out of equilibrium, its density matrix evolves according to the quantum Liouville equation. In the asymptotic long-time state, the off-diagonal elements of the density matrix obtain completely randomized phases, therefore, only the diagonal elements, which construct the so-called diagonal ensemble, have a contribution to the expectation value of observables. ETH builds the equivalence between the microcanonical ensemble and the diagonal ensemble, and then explains thermalization successfully. Its correctness has been verified in plenty of numerical simulations [@d2016quantum], while its limitation was also noticed. ETH has to be modified for the order parameter in the presence of spontaneous symmetry breaking [@PhysRevE.92.040103], and it fails in a many-body localized system [@Gornyi05; @basko2006metal] which cannot thermalize. In spite of the success of ETH, it cannot explain how an observable relaxes towards steady value, because it says nothing about the off-diagonal elements of density matrix which are important in the transient and intermediate time scale. The off-diagonal elements of density matrix are even the key of describing an asymptotic long-time state, in the case that the thermodynamic limit and the long-time limit are noncommutative. This noncommutativity defines an important class of nonequilibrium states - the nonequilibrium steady states, which is the basis of understanding mesoscopic transport phenomena. A stationary current flows through a central region which is coupled to multiple thermal reservoirs at different temperatures and chemical potentials. The description of such a quantum state goes beyond the ability of diagonal ensemble, Gibbs ensemble or generalized Gibbs ensemble, but requires the knowledge of the off-diagonal elements. This motivated one of the authors to propose the nonequilibrium steady state hypothesis (NESSH) [@wang2017theory]. In this paper, we make NESSH complete by proposing the form of both the diagonal and off-diagonal elements in the density matrix, which is $$\begin{aligned} \label{NESSHt} \rho_{\alpha\beta}&=&D^{-1}\left(\bar{E}\right)\Big(\frac{1}{\sqrt{2\pi\sigma^2_s}} e^{-\frac{\left(\bar{E}-\mu_s\right)^2}{2\sigma^2_s}}\delta_{\alpha\beta}\nonumber\\ &&+D^{-\frac{1}{2}}\left(\bar{E}\right)f\left(\omega,\bar{E}\right)R^s_{\alpha\beta}\Big),\end{aligned}$$ where $\mu_s$ and $\sigma^2_s$ denote the mean and variance of the system’s energy, respectively. Note that the first term in Eq. (\[NESSHt\]) is absent in the previous paper [@wang2017theory]. $f\left(\omega,\bar{E}\right)$ is the dynamical characteristic function, which is determined by the initial state and contains all the information for understanding the real-time dynamics of a chaotic system from the transient regime to the long-time steady limit. It is worth emphasizing that Eq. (\[NESSHt\]) stands for the density matrix of arbitrary chaotic system whether it thermalizes or evolves into a nonequilibrium steady state. In this paper, we will carry out numerical simulations in different models of spins in different dimensions to support our assumption (\[NESSHt\]), supplement to the numerical simulations of fermionic models in the previous study. Furthermore, we will show how to derive ETH by using Eq. (\[NESSHt\]), and then build an intimate connection between NESSH and ETH, which both stand in quantum chaotic systems. The rest of the paper is organized as follows. The physical meaning of Eq. (\[NESSHt\]) will be discussed in Sec. \[sec:level 2\], in which we also derive a generic expression for the real-time dynamics of an observable based on our assumption and ETH. The numerical evidence of our assumption is presented in Sec. \[sec:level 3\] and \[sec:level 4\]. The connection between NESSH and ETH is the content of Sec. \[sec:level 5\]. Sec. \[sec:level 6\] summarizes our results. \[sec:level 2\]Nonequilibrium dynamics of the density matrix ============================================================ Let us consider an isolated system with the Hamiltonian $\hat H$. Without loss of generality, we suppose $t=0$ as the initial time, at which the quantum state of the system is denoted by $\ket{s}$. According to quantum mechanics, the expectation value of an arbitrary observable evolves as $$\label{quench} O(t)=\sum_{\alpha,\beta} e^{-i\omega t}\rho_{\alpha\beta}O_{\beta\alpha},$$ where $\alpha$ and ${\beta}$ denote the eigenstates of $\hat H$, and $\omega=E_{\alpha}-E_{\beta}$ is the difference between their eigenenergies. $\rho_{\alpha\beta}=\bra{\alpha}\hat{\rho}\ket{\beta}$ with $\hat \rho =\ket{s}\bra{s}$ denotes the element of the initial density matrix in the eigenbasis of $\hat H$, and $O_{\beta\alpha}=\bra{\beta}\hat{O}\ket{\alpha}$ denotes the matrix element of the observable operator. In the case that the initial state is not an eigenstate of $\hat H$, the system is out of equilibrium at $t>0$. To study the nonequilibrium dynamics of a system is equivalent to calculate $O(t)$. For this purpose, we need to know the eigenenergies, the initial density matrix and the observable matrix. For integrable systems, $E_\alpha$, $\rho_{\alpha\beta}$ and $O_{\alpha\beta}$ differ from model to model. There is no common way of understanding nonequilibrium dynamics of integrable systems. But it is not the case for chaotic systems, which are “similar” to each other. RMT tells us that the eigenenergies of chaotic systems all follow the Wigner-Dyson distribution [@kravtsov2012random] $$\label{WDdis} P(E_1,E_2,\cdots) = \frac{1}{\mathcal{N}} \displaystyle e^{-\frac{E_1^2+E_2^2+\cdots}{2\sigma^2}} \left| \prod_{\alpha>\beta}\left(E_\alpha-E_\beta\right) \right|,$$ where $\sigma$ is connected to the energy bandwidth and $\mathcal{N}$ is a normalization constant. And according to ETH, $O_{\alpha\beta}$ has the universal form (\[eth\]), independent of the model being of fermions, bosons or spins, or in which dimensions. Once if we know the form of $\rho_{\alpha\beta}$, $O(t)$ can be calculated, even if the exact solution of any specific chaotic model is inaccessible. Different from integrable models, our knowledge of $E_\alpha$, $O_{\alpha\beta}$ and $\rho_{\alpha\beta}$ in chaotic models is not precise, but only statistical. Eq. (\[WDdis\]) only gives the statistics of the eigenenergies, and Eq. (\[eth\]) contains a random number $R^O_{\alpha\beta}$. This is what we have to pay for not really solving the model. But it does not prevent us from obtaining the information that we are interested in, i.e. $O(t)$. Before discussing the form of $\rho_{\alpha\beta}$, we need to make clear which kind of initial states are interesting to us. The initial state $\ket{s}$ should be some quantum state that we can prepare in a laboratory. Preparing a quantum state is usually equivalent to measuring the state which inevitably causes the wave function collapsing into an eigenstate of the observable operator. Therefore, it is natural to choose $\ket{s}$ as an eigenstate of a complete set of observable operators. For example, in a spin lattice model, we can choose $\ket{s}$ to be a configuration of spin eigenstates in the $z$-direction on each lattice site, or we choose $\ket{s}$ to the spin eigenstates in the $x$- or $y$-directions. Such kind of initial states will be called natural states in next. Of course, $\ket{s}$ cannot be an eigenstate of $\hat H$, otherwise, the system is already thermalized at the initial time according to ETH. $\ket{s}$ is also not a fine-tuned state, such as the superposition of a few eigenstates of $\hat H$. Such kind of fine-tuned states are diffcult to creat in experiments for a many-body chaotic system. On the other hand, $\ket{s}$ can be an eigenstate of a Hamiltonian $\hat H_0$ which includes no interaction between particles and is then integrable, e.g., a spin model without interaction between spins at different sites. Usually, this kind of $\hat H_0$ is commutative with some observable operators so that they have common eigenstates. $\ket{s}$ can also be the eigenstate of a chaotic Hamiltonian $\hat H'$ which is noncommutative with $\hat H$. For example, $\hat H'$ and $\hat H$ describe interacting spins with different interaction strength. In this case, the eigenstate of $\hat H'$ looks like a random vector in the eigenbasis of $\hat H$, which is the foundation of our assumption (\[NESSHt\]). Starting from a natural state, the dynamics of the density matrix follows the quantum Liouville equation. We propose that $\rho_{\alpha\beta}=\braket{\alpha|s}\braket{s|\beta}$ has a universal form which can be expressed as $$\label{NESSH} \rho_{\alpha\beta}=D^{-1}\left(\bar{E}\right)\left(\rho\left(\bar{E}\right) \delta_{\alpha\beta}+D^{-\frac{1}{2}}\left(\bar{E}\right) f\left(\omega,\bar{E}\right)R^s_{\alpha\beta}\right),$$ where the first term in the bracket denotes the diagonal element, while the second term denotes the off-diagonal element. $\bar{E}=\left(E_\alpha+E_\beta\right)/2$ and $\omega=E_\alpha - E_\beta$ are the energy average and the energy difference, respectively. The density of states $D\left(\bar{E}\right)$ appears in Eq. (\[NESSH\]) to indicate how $\rho_{\alpha\beta}$ scales with increasing system’s size $N$. Note that both ETH and NESSH should be treated as assumptions in the thermodynamic limit $N\to \infty$. But $\rho_{\alpha\beta}$ vanishes as $N\to \infty$, therefore, we have to start from a finite $N$ and separate the diverging factor in $\rho_{\alpha\beta}$, which is $D$. $D$ increases exponentially with $N$ and diverges in thermodynamic limit. The exponents of $D$ in the diagonal and off-diagonal terms can be deduced from the fact that a local observable $O(t)$ must be convergent in thermodynamic limit. Let us consider the diagonal term in Eq. (\[NESSH\]). $\rho_{\alpha\alpha}= \braket{\alpha|s}\braket{s|\alpha}$ is in fact the probability of measuring the energy of state $\ket{s}$ and finding it is $E_\alpha$. This probability distribution should be centered around the mean energy of $\ket{s}$. It is natural to think that this distribution is a Gaussian distribution, which is also supported by our numerics. $\rho\left(\bar{E}\right)$ can then be expressed as $$\label{NESSHd} \rho\left(\bar{E}\right)=\frac{1}{\sqrt{2\pi\sigma^2_s}} e^{-\frac{(\bar{E}-\mu_s)^2}{2\sigma^2_s}}+C_sR^s_{\alpha\alpha},$$ where $\mu_s$ and $\sigma_s^2$ denote the mean energy and the energy fluctuation of the state $\ket{s}$. An additional term $C_sR^s_{\alpha\alpha}$ is added to fit the numerics into Eq. (\[NESSHd\]). $C_sR^s_{\alpha\alpha}$ describes the deviation from the Gaussian distribution with $C_s$ being a constant and $R^s_{\alpha\alpha}$ being an independent random number with zero mean and unit variance. The properties of $C_s$ and $R^s_{\alpha\alpha}$ will be further discussed in Sec. \[sec:level 3\]. They indeed vanish in thermodynamic limit. Anyway, $C_sR^s_{\alpha\alpha}$ has no effect on the value of $O(t)$. The parameters $\mu_s$ and $\sigma^2_s$ in Eq. (\[NESSHd\]) can be determined. We have $$\label{mean} \begin{split} \mu_s =&\int\bar{E}\rho\left(\bar{E}\right)\mathrm{d}\bar{E} \\ =&\sum_{\alpha}E_{\alpha}\braket{\alpha|s}\braket{s|\alpha} \\ =& \bra{s} \hat H \ket{s}, \end{split}$$ where we have used $\int\mathrm{d}E_{\alpha}D(E_{\alpha})= \sum_{\alpha}$ and $E_{\alpha}=\bar{E}$ for the diagonal elements. Note $\int C_sR^s_{\alpha\alpha} \bar{E}\mathrm{d}\bar{E}=0$, because $R^s_{\alpha\alpha}$ at different $\alpha$ are independent random numbers with zero mean. Similarly, we obtain the variance of energy $$\begin{aligned} \label{var} \sigma_s^2&=&\int\bar{E}^2\rho\left(\bar{E}\right)\mathrm{d} \bar{E}-\left(\int\bar{E}\rho\left(\bar{E}\right)\mathrm{d}\bar{E}\right)^2\nonumber\\ &=&\bra{s}\hat{H}^2\ket{s}-\bra{s}\hat{H}\ket{s}^2 \nonumber\\ &=&\sum_{s'\ne s}H^2_{ss'}.\end{aligned}$$ Here the sum with respect to $s'$ is over the natural states, i.e., the common eigenstates of the complete set of observable operators, which form a natural basis of the Hilbert space. Different from $\mu_s$, the variance is determined by the off-diagonal elements of the Hamiltonian in the natural basis. Next we consider the off-diagonal elements in Eq. (\[NESSH\]). The factor $D^{-1/2}$ indicates that the off-diagonal elements are exponentially smaller than the diagonal elements. RMT says that the eigenstates are random vectors in the Hilbert space, therefore, $\braket{s|\alpha}$ and $\braket{s|\beta}$ are both random numbers, so is $\rho_{\alpha\beta}$. The randomness of $\rho_{\alpha\beta}$ in Eq. (\[NESSH\]) is reflected by the random number $R^s_{\alpha\beta}$, which by definition has zero mean and unit variance. And $R^s_{\alpha\beta}$ at different $(\alpha,\beta)$ are independent to each other. But the variance of $\rho_{\alpha\beta}$ is not a constant, but depends on the energies $E_\alpha$ and $E_\beta$. In order to describe the structure of $\rho_{\alpha\beta}$ which is ignored by RMT, we introduce the dynamical characteristic function $f\left(\omega,\bar{E}\right)$ with $\bar{E}=(E_{\alpha} +E_{\beta})/2$ and $\omega=E_{\alpha}-E_{\beta}$. One can understand the off-diagonal term of Eq. (\[NESSH\]) as follows. $\rho_{\alpha\beta}$ fluctuates heavily as $E_\alpha$ or $E_\beta$ changes. But if we integrate out the fluctuation, the average of $\left| \rho_{\alpha\beta} \right|^2$ changes smoothly with $E_\alpha$ and $E_\beta$, or with $\bar{E}$ and $\omega$. This is the central idea of NESSH and is highly nontrivial, being true only for chaotic systems. Our numerics in the previous study [@wang2017theory] already showed its breakdown in integrable models. By substituting Eqs. (\[eth\]) and (\[NESSH\]) into Eq. (\[quench\]), we obtain $$\begin{aligned} \label{de2q} O(t)&=&\int_{-\infty}^{\infty}\mathrm{d}\bar{E}\rho \left(\bar{E}\right)O\left(\bar{E}\right)\\ &&+\mathcal{C}_{so}\int_{-\infty}^{\infty}\mathrm{d}\bar{E} \int_{-\infty}^{\infty}\mathrm{d}\omega e^{-i\omega t}f\left(\omega,\bar{E}\right) f_O\left(-\omega,\bar{E}\right),\nonumber\end{aligned}$$ where we have used $\sum_{\alpha}\to\int\mathrm{d}E_{\alpha}D(E_{\alpha})$ and $D(\bar{E}\pm\omega/2)\approx D\left(\bar{E}\right)$. The latter approximation comes from the factor that the integration function $ff_O $ decays to zero quickly with increasing $\omega$. $\mathcal{C}_{so}= \overline{R^s_{\alpha\beta}R^O_{\beta\alpha}}$ denotes the correlation between the random numbers $R^s_{\alpha\beta}$ and $R^O_{\beta\alpha}$, which can be estimated numerically by averaging $R^s_{\alpha\beta}R^O_{\beta\alpha}$ over a small energy box centered at a specific value of $(\bar{E},\omega)$. $\mathcal{C}_{so}$ changes slowly with $\bar{E}$ or $\omega$, and can be treated as a constant and taken out of the integral. Note that the density of states disappears in Eq. ([\[de2q\]]{}), therefore, $O(t)$ converges in thermodynamic limit, as we expect. The first term of Eq. (\[de2q\]) is independent of time, being exactly the expectation value of the observable with respect to the diagonal ensemble. Since $\rho(\bar{E})$ is a Gaussian distribution (the second term of Eq. (\[NESSHd\]) cancels out in the integral), the first term of Eq. (\[de2q\]) evaluates $O(\mu_s)$, once if $O(\bar{E})$ changes slowly in the range $\left( \mu_s -\sigma_s, \mu_s+\sigma_s\right)$. $O(\mu_s)$ is indeed the value of the observable after thermalization. Imagine the evolution processes starting from two different microscopic states $\ket{s_1}$ and $\ket{s_2}$ that have the same mean energy $\mu_s$. In the long-time steady limit, one cannot distinguish $\ket{s_1}$ from $\ket{s_2}$ by any measurement. This is exactly what thermalization means - the memory of initial state is lost and the properties of the system is only determined by few parameters such as the mean energy. The second term of Eq. (\[de2q\]) is more interesting. It is time-dependent and displays how $O(t)$ relaxes to its stationary limit. Note that the second term is a Fourier transformation of $ff_O$ with respect to the variable $\omega$. The transient dynamics of $O(t)$ is determined by the asymptotic behavior of $ff_O$ in the large-$\omega$ limit, while the long-time asymptotic behavior of $O(t)$ is determined by the asymptote of $ff_O$ at small $\omega$. In this way, $f$ influences the dynamics of an arbitrary observable in the whole time scale, which is the reason why $f$ is called the dynamical characteristic function. In thermodynamic limit, two different situations can be distinguished in the long-time asymptotic behavior of $O(t)$. First, since numerics already shows that $f_O$ always develops a plateau at small $\omega$, if $f$ converges in the limit $\omega\to 0$, according to Riemann-Lebesgue lemma, the Fourier transformation of $ff_O$ must decay to zero in the limit $t\to\infty$. The second term of Eq. (\[de2q\]) then vanishes in the long-time limit, and $\lim_{t\to\infty} O(t)$ coincides with the value of $O$ in the diagonal ensemble. In this case, the system thermalizes. Second, if $f$ asymptotes to $1/\omega$ in the small-$\omega$ limit, the second term goes to a nonzero value in the limit $t\to\infty$. In this case, the system does not thermalize but evolves into a nonequilibrium steady state, in which the values of observables are different from their equilibrium counterparts. The failure of thermalization is attributed to the infinite imbalance in the initial state $\hat \rho$, which cannot be removed for thermalization to happen [@wang2017theory]. Moreover, the first situation (thermalization) can be further classified according to whether $O(t)$ relaxes in an exponential way or in a power-law way, etc., by supposing different asymptotic behavior of $f(\omega)$ in the limit $\omega \to 0$. Therefore, Eq. (\[de2q\]) serves as a benchmark for understanding the nonequilibrium dynamics of chaotic quantum systems. \[sec:level 3\]Numerical simulation of the diagonal part of density matrix ========================================================================== Next, we test NESSH (Eqs. (\[NESSH\]) and (\[NESSHd\])) in spin lattice models. We consider the two-dimensional (2D) transverse field Ising model (TFIM) and the one-dimensional (1D) disordered XXZ model. The Hamiltonian of TFIM is $$\label{isingh} \hat{H}_{Ising}=-J\sum_{<i,j>}\hat{\sigma}^z_i\hat{\sigma}^z_j+g\sum_i\hat{\sigma}^x_i,$$ where $\hat{\sigma}^z_i$ and $\hat{\sigma}^x_i$ are the Pauli matrices. We consider only the interaction between nearest-neighbor sites. The ferromagnetic coupling $J$ is set to the energy unit and $g$ denotes the transverse field. The total number of lattice sites in numerical simulation is set to $N$. This model has already been studied for testing ETH, and found to be chaotic [@mondaini2017eigenstate; @mondaini2016eigenstate]. We choose the natural states $\ket{s}$ to be the eigenstates of $\{\hat \sigma_i^z\}$. After a straightforward calculation, we obtain $\sigma^2_s=Ng^2$ which is a constant, thereafter, the fluctuation of energy density is $\frac{\sigma^2_s}{N^2}=\frac{g^2}{N}$, which goes to zero in the thermodynamic limit $N\to\infty$, as we expect. The second model we study is the one-dimensional XXZ model: $$\label{xxzm} \hat{H}_{XXZ}=-J\sum_i\left(\hat{\sigma}_i^x\hat{\sigma}_{i+1}^x +\hat{\sigma}_i^y\hat{\sigma}_{i+1}^y+\hat{\sigma}_i^z \hat{\sigma}_{i+1}^z\right)+\sum_ih_i\hat{\sigma}_i^z,$$ where $h_i\in\left[-h, h\right]$ is a random number with uniform distribution and $h$ is the disorder strength. Again $J$ is set to unity. The XXZ model without disorder is integrable, but infinitesimal disorder destroys integrability. As is well known, the XXZ model is in the many-body localized phase in the case of strong disorder. In our study, we control $h$ to be small enough for avoiding localization. ![(Color online) The plot of $\left(\overline{\rho_{\alpha\alpha}} D\right)$ as a function of $\bar{E}$ for (a) 2D-TFIM with $g=2$, $\Delta\bar{E}=1$ and $N=12$, and (b) the disordered XXZ chain with $h=0.05$, $\Delta\bar{E}=1$ and $N=16$. The black solid lines are the Gaussian functions of $\mu_s=4.58\times10^{-6}$, $\sigma^2_s=61.3$, and $\mu_s=-0.18$, $\sigma^2_s=3.73$, respectively.[]{data-label="fig1"}](fig1.eps){width="45.00000%"} In order to obtain the density matrix elements, we first diagonalize the model Hamiltonians. For the 2D-TFIM, we choose a lattice of specific shape that breaks the geometric symmetries (see Ref. \[\] for detail). Similarly, there is a symmetry in the XXZ model. Following previous literatures [@bertrand2016anomalous], we focus on a subspace of the Hilbert space associated with the operator $\hat{\sigma}^z=\sum_i\hat{\sigma}_i^z$. Only the subspace $\sigma^z=0$ is considered. After diagonalization, we calculate the density matrix elements $\rho_{\alpha\beta} =\braket{\alpha | s}\braket{s| \beta}$. ![(Color online) Panels (a) and (b) plot $\kappa_s$ of the different natural states $\ket{s}$ for 2D-TFIM with $g=2$ and the XXZ model with $h=0.05$, respectively. The natural states are sorted with $s$ denoting their sequence number, and $s_{max}$ denotes the total number of natural states which equals the dimension of the Hilbert space. Different types of dots with different colors represent different $N$. Panels (c) and (d) plot $\bar{\kappa}$ as a function of $1/N$ for 2D-TFIM and the XXZ model, respectively. The dashed lines are the fitting functions with the form $\bar{\kappa} \propto \left(1/N\right)^z$.[]{data-label="fig2"}](fig2.eps){width="45.00000%"} According to our assumption, the diagonal element $\rho_{\alpha\alpha}$ is a Gaussian function blurred by the fluctuation $C_s R^s_{\alpha\alpha}$. Since the mean of $R^s_{\alpha\alpha}$ is zero, we calculate the average of $\rho_{\alpha\alpha}$ over a thin energy shell centered at $\bar{E}$, which should give $$\overline{\rho_{\alpha\alpha}}=D^{-1}\left( \bar{E} \right) \frac{1}{\sqrt{2\pi \sigma^2_s}} e^{-\frac{(\bar{E}-\mu_s)^2}{2\sigma_s^2}}.$$ The width of the energy shell is set to $2\Delta\bar{E}=2$ in practice to contain enough number of eigenstates. But it is worth mentioning that $\Delta\bar{E}$ can be made smaller and smaller as the system’s size increases, since the density of states increases. And in thermodynamic limit, $\Delta\bar{E} $ can be made arbitrarily small, while there are still infinite number of states in the shell. Fig. \[fig1\] shows $\left(\overline{\rho_{\alpha\alpha}} \cdot D\right)$ as a function of $\bar{E}$. It fits well into a Gaussian function (the solid line). The deviation should be attributed to the finite system’s size in numerical simulation. We expect that the deviation vanishes in thermodynamic limit. $\rho_{\alpha\alpha}$ fluctuates around its average $\overline{\rho_{\alpha\alpha}}$. According to our assumption, their difference should be $\rho_{\alpha\alpha}- \overline{\rho_{\alpha\alpha}}=D^{-1} C_s R^s_{\alpha\alpha}$. The amplitude of the fluctuation is $\kappa_s = D^{-1} C_s$, which can be computed according to $$\label{eq:kappadef} \kappa_s= \sqrt{\displaystyle \overline{\left(\rho_{\alpha\alpha}- \overline{\rho_{\alpha\alpha}}\right)^2} },$$ where the overline denotes the average over $\alpha$. To obtain Eq. (\[eq:kappadef\]), we have used the definition $\overline{\left(R^s_{\alpha\alpha}\right)^2}=1$. Fig. \[fig2\](a) and (b) plot $\kappa_s$ for different states $\ket{s}$ in the natural basis. It is clear that the values of $\kappa_s$ concentrate. As the system’s size increases, the change of $\kappa_s$ with $s$ becomes even smaller. We then guess that $\kappa_s$ should be independent of the initial state $\ket{s}$ for sufficiently large $N$. We study the average of $\kappa_s$ over ${s}$, which is denoted by $\bar{\kappa}$. Fig. \[fig2\](c) and (d) display how $\bar{\kappa}$ changes with the system’s size. For both TFIM and XXZ models, $\bar{\kappa}$ decays with increasing $N$. Our numerics indicates $\kappa_s \to 0$ in thermodynamic limit, which means that the fluctuation vanishes and the diagonal element $\rho_{\alpha\alpha}$ approaches $\overline{\rho_{\alpha\alpha}}$, while the latter is a Gaussian function of $E_\alpha$ according to Fig. \[fig1\]. \[sec:level 4\]Numerical simulation of the off-diagonal part ============================================================ ![(Color online) The distribution of $\rho_{\alpha\beta}$ for (a) 2D-TFIM at $g=2$ and $N=12$ and (b) the XXZ model at $h=0.05$ and $N=14$. We choose $\bar{E}=0$ and $\omega=1$ as the center of the energy box whose sides are set to $\Delta{\bar{E}}=0.1$ and $\Delta\omega=0.1$. The red lines are the stable distributions with the parameters $a=0.99$, $b=0.05$, $c=5.78\times 10^{-5}$ and $\delta=-2.37\times10^{-6}$ for panel (a), and $a=0.51$, $b=-5.51\times10^{-4}$, $c=1.67\times 10^{-5}$ and $\delta=-1.96\times10^{-9}$ for panel (b).[]{data-label="fig3"}](fig3.eps){width="45.00000%"} Next let us study $\rho_{\alpha\beta}$ for $\alpha\neq\beta$, which should be expressed as $D^{-3/2} f R^s_{\alpha\beta}$ according to our assumption. $R_{\alpha\beta}^s$ is a random number of zero mean and unit variance, therefore, $\rho_{\alpha\beta}$ should be a random number of zero mean and the variance $D^{-3}f^2$. We consider the set of $\rho_{\alpha\beta}$ within a small rectangular energy box centered at $(\bar{E},\omega)$ with the sides $2\Delta{\bar{E}}$ and $2\Delta{\omega}$, that is $\alpha$ and $\beta$ satisfy $\bar{E}-\Delta{\bar{E}}<(E_\alpha +E_\beta)/2 < \bar{E}+\Delta{\bar{E}}$ and $\omega-\Delta{\omega}<E_\alpha -E_\beta< \omega+\Delta{\omega}$. We choose small $\Delta{\bar{E}}$ and $\Delta{\omega}$ so that $f$ and $D$ are approximately constants within the energy box, and then obtain the statistics of $\rho_{\alpha\beta}$. Fig. $\ref{fig3}$ plots the distribution of $\rho_{\alpha\beta}$. It is clear that the distribution is symmetric with respect to zero, indicating that the mean of $\rho_{\alpha\beta}$ is zero. And the distribution function has a similar shape for the TFIM and XXZ models. It is also quite similar to that in the fermionic models studied previously [@wang2017theory]. Since the distribution of $\rho_{\alpha\beta}$ is indeed determined by the random number $R^s_{\alpha\beta}$, our finding suggests that $R^s_{\alpha\beta}$ has a universal distribution in arbitrary chaotic system. We fit the histogram of $\rho_{\alpha\beta}$ to the stable distribution (the red lines in Fig. $\ref{fig3}$), which is defined as the Fourier transformation $$\begin{split} P\left( x \right)=& \frac{1}{2\pi} \int dp \, e^{-ip(x-\delta)} \\ & \times e^{-c^a \left|p \right|^a \left[1+i b\, \text{sign}\left(p\right) \tan(\pi a/2) \left( \left(c\left|p \right| \right)^{1-a} - 1\right)\right]} \end{split}$$ with the parameters $a$, $b$, $c$ and $\delta$. $\text{sign}(p)$ denotes the sign of $p$. $\delta$ is the location parameter, which is almost zero, indicating that the distribution is symmetric to zero. $c$ is the scale parameter, which is also small. The shape parameters $a$ and $b$ measure the concentration and the asymmetry of the distribution, respectively. ![(Color online) The variance of $\rho_{\alpha\beta}$ as a function of $\omega$ is plotted for TFIM. (a) $\Sigma(\omega)$ at different system’s size. (b) The plateau of $\Sigma(\omega)$ at small $\omega$ is plotted as a function of the system’s size. The dotted line is the function $0.21 e^{-1.31 N}$. (c) The functions $\Sigma(\omega)$ for different initial states $\ket{s_1}$ and $\ket{s_2}$, whose spin configurations are depicted in panel (d) with the circles and squares representing the spins up and down, respectively.[]{data-label="fig4"}](fig4.eps){width="45.00000%"} We study the variance of $\rho_{\alpha\beta}$, denoted by $\Sigma(\bar{E},\omega)= D^{-3}(\bar{E})f^2(\bar{E},\omega)$. $\Sigma$ is no more than the squared dynamical characteristic function weighted by a factor $D^{-3}$, which is exponentially small as the system’s size increases. We average $\Sigma$ over $\bar{E}$, and then obtain a function of $\omega$. The averaged $\Sigma$ reflects how the dynamical characteristic function changes with $\omega$. The results are plotted in Fig. \[fig4\]. $\Sigma$ (or $f$) is a smooth function of $\omega$, as we expect. And it develops a plateau at small $\omega$, indicating that the system will thermalize in the long-time limit (see the discussion in the previous section). The thermalizing consequence agrees with previous studies [@mondaini2017eigenstate; @mondaini2016eigenstate]. For large $\omega$, the dynamical characteristic function decays exponentially to zero. This is believed to be a typical feature of $f$. For $N=11$, $\Sigma(\omega)$ displays a peak before the exponential decay (see Fig. \[fig4\](a)), which should be attributed to the small value that $N$ takes. This peak vanishes as we choose $N=12$. We also see that the value of $\Sigma$ decreases with increasing $N$. This is due to the factor $D^{-3}$ in the expression of $\Sigma$, which decays exponentially as $N$ increases. We denote the value of $\Sigma$ at the plateau as $S_d$ and display it as a function of $N$ in Fig. \[fig4\](b). As we expect, $S_d$ does decay exponentially. In general, the dynamical characteristic function should be dependent on the initial state $\ket{s}$. Fig. \[fig4\](c) shows $\Sigma(\omega)$ for two different initial states $\ket{s_1}$ and $\ket{s_2}$, which are depicted in Fig. \[fig4\](d). As we see, their dynamical characteristic functions differ from each other, but the difference is not significant, indicating that the real-time dynamics starting from $\ket{s_1}$ and $\ket{s_2}$ has similar properties. In this section, we focus on the dependence of $\Sigma$ or $f$ on $\omega$, but neglect their dependence on $\bar{E}$. The former is more important, since the time-dependent observable $O(t)$ is a Fourier transformation with respect to the variable $\omega$ (see Eq. (\[de2q\])). The real-time dynamics is then sensitive to the dependence of $f$ on $\omega$. \[sec:level 5\]The connection between NESSH and ETH =================================================== In this section, we try to connect ETH, i.e. the assumption about the matrix elements of an observable operator $\hat O$, to our assumption about the density matrix $\hat \rho=\ket{s}\bra{s}$. For this purpose, we notice that the natural state $\ket{s}$ can be chosen to the eigenstate of some observable operators. Without loss of generality, we suppose that $\ket{s}$ is the eigenstate of $\hat O$. And all the natural states form a complete basis of the Hilbert space, satisfying $\sum_s\ket{s}\bra{s}=1$. In this natural basis, $O_{ss'}=\bra{s}\hat{O}\ket{s'}$ is a diagonal matrix. The matrix elements of the observable in the eigenbasis of the Hamiltonian can then be expressed as $$\label{ob} O_{\alpha\beta}=\sum_s O_{ss} \rho_{\alpha\beta},$$ where $\rho_{\alpha\beta}=\braket{\alpha |s} \braket{s|\beta}$ denotes the element of the density matrix. NESSH states that $\rho_{\alpha\beta}$ has a universal expression in chaotic systems (see Eqs. (\[NESSH\]) and (\[NESSHd\])). Therefore, our target is to use Eqs. (\[NESSH\]) and (\[NESSHd\]) to prove that $O_{\alpha\alpha}$ changes smoothly with $E_\alpha$, which is the central idea of ETH for explaining thermalization. Substituting Eqs. (\[NESSH\]) and (\[NESSHd\]) into Eq. (\[ob\]), we obtain $$\label{NESSH2eth} \begin{split} O_{\alpha\alpha}=& \sum_s D^{-1}\left({E_\alpha}\right) O_{ss} \rho\left({E_\alpha}\right) \\ = & \sum_s D^{-1} O_{ss} \frac{1}{\sqrt{2\pi\sigma^2_s}} e^{-\frac{({E}_\alpha-\mu_s)^2}{2\sigma^2_s}} \\ & + \sum_s O_{ss} D^{-1}C_sR^s_{\alpha\alpha}. \end{split}$$ Our numerics has shown that $D^{-1}C_sR^s_{\alpha\alpha}$ goes to zero in thermodynamic limit (see Fig. \[fig2\] and the corresponding discussion). And since $R^s_{\alpha\alpha}$ is a random number of zero mean independent of $O_{ss}$, we have sufficient reason to believe that $\sum_s O_{ss} D^{-1}C_sR^s_{\alpha\alpha} \to 0$ in thermodynamic limit. ![${O}'_{max}$ is plotted as a function of $1/N$.[]{data-label="fig6"}](fig5.eps){width="45.00000%"} Next we use 2D-TFIM as an example to show that the first term in Eq. (\[NESSH2eth\]) changes smoothly with $E_\alpha$ in the limit $N\to \infty$. The observable operator is chosen to be $\hat O=\hat \sigma_i^z$ with $i$ denoting the bottom site in Fig. \[fig4\](d). The first term of Eq. (\[NESSH2eth\]) is the sum of a series of Gaussian functions weighted by $O_{ss}$. $O_{ss}$ is usually bounded, and the sum of finite number of Gaussian functions must changes smoothly. In order to extend this conclusion to the limit $N\to \infty$, we compute the derivative $${O}'_{\alpha\alpha} = \sum_s O_{ss} \frac{d}{dE_\alpha} \left( D^{-1} \frac{1}{\sqrt{2\pi\sigma^2_s}} e^{-\frac{({E}_\alpha-\mu_s)^2}{2\sigma^2_s} } \right).$$ And we define $O'_{max}$ as the maximum of $\left|{O}'_{\alpha\alpha}\right|$ over $\alpha$. A scaling analysis of $O'_{max}$ is given in Fig. \[fig6\], which clearly shows that ${O}'_{max}$ converges in the limit $N\to\infty$. This means that $O_{\alpha\alpha}$ has a finite derivative at arbitrary energy, i.e., $O_{\alpha\alpha}$ changes smoothly with $E_\alpha$. We then reach the central idea of ETH and also the basis of thermalization - $O_{\alpha\alpha}$ does not fluctuate infinitely with $\alpha$ even in thermodynamic limit. \[sec:level 6\]summary ====================== We summarize our results. NESSH assumes a universal form of the density matrix in the eigenbasis of the Hamiltonian of a quantum chaotic system. The main assumption of NESSH is given in Eq. (\[NESSHt\]). The diagonal element of the density matrix $\rho_{\alpha\alpha}$ is a Gaussian function, with the mean $\mu_s$ and the variance $\sigma^2_s$ determined by the initial state. The off-diagonal elements $\rho_{\alpha\beta}$ are random numbers with a universal distribution. Its standard deviation is dubbed the dynamical characteristic function, which governs the real-time dynamics of the system according to Eq. (\[de2q\]). For a typical initial state that thermalizes in the long-time limit, the dynamical characteristic function exhibits a plateau at low frequencies but an exponential decay at high frequencies. We provide the numerical evidence of NESSH in two chaotic spin models - the 2D transverse field Ising model and the 1D disordered XXZ model. The numerics for these two models is consistent with the prediction of NESSH, both for the diagonal and off-diagonal elements. Furthermore, we show how to reach ETH from the assumptions of NESSH by factorizing the observable matrix elements into the density matrix elements and the expectation value of observable in the natural basis. By using the assumptions of NESSH, we show that the diagonal element of the observable matrix changes smoothly with energy, which explains why thermalization happens. acknowledgements {#acknowledgements .unnumbered} ================ This work is supported by NSF of China under Grant Nos. 11774315 and 11304280. Pei Wang is also supported by the Junior Associates programm of the Abdus Salam International Center for Theoretical Physics.

--- address: - | Department of Mathematics\ University of Illinois\ Urbana\ IL 61801\ USA - | Departamento de Matem[á]{}ticas\ Facultad de Ciencias\ Universidad Aut[ó]{}noma de Madrid\ 28049 Madrid\ Spain - | Departamento de Matemáticas\ Facultad de Ciencias\ Universidad Autónoma de Madrid\ 28049 Madrid\ Spain - | Department of Mathematical Sciences\ University of Liverpool\ Peach Street\ Liverpool L69 7ZL\ UK author: - 'S.B. Bradlow' - 'O. García-Prada' - 'V. Muñoz' - 'P.E. Newstead' date: 29 May 2002 title: 'Coherent systems and Brill-Noether theory' --- [^1] Suplico a vuesa merced, señor don Quijote, que mire bien y especule con cien ojos lo que hay allá dentro: quizá habrá cosas que las ponga yo en el libro de mis [*Transformaciones*]{} (El ingenioso hidalgo don Quijote de la Mancha, Book 2, Chapter XXII) I beg you, don Quixote sir: look carefully, inspect with a hundred eyes what you see down there. Who knows, maybe you will find something that I can put in my book on [*Transformations*]{}. Introduction {#sec:intro} ============ Augmented algebraic vector bundles often have moduli spaces which depend not only on the topological type of the augmented bundle, but also on an additional parameter. The result is that the moduli spaces occur in discrete families. First exploited by Thaddeus in a proof of the Verlinde formula, this phenomenon has been responsible for several interesting applications (cf. [@BeDW], [@BGG], [@BG2]). In this paper we examine the augmented bundles known as coherent systems and discuss the use of their moduli spaces as a tool in Brill-Noether theory. By a [*coherent system*]{} on an algebraic variety (or scheme) we mean an algebraic vector bundle together with a linear subspace of prescribed dimension of its space of sections. As such it is an example of an augmented bundle. Introduced in [@KN], [@RV] and [@LeP], there is a notion of stability which permits the construction of moduli spaces. This notion depends on a real parameter, and thus leads to a family of moduli spaces. That these moduli spaces are related to Brill-Noether loci follows almost immediately from the definitions. The Brill-Noether loci are natural subvarieties within the moduli spaces of stable bundles over an algebraic curve, defined by the condition that the bundles have at least a prescribed number of linearly independent sections. A similar condition defines Brill-Noether loci in the moduli spaces of (S-equivalence classes of) semistable bundles. But any bundle which occurs as part of a coherent system must evidently have at least a prescribed number of linearly independent sections. Conversely, a bundle with a prescribed number of linearly independent sections determines, in a natural way, a coherent system. In order to convert these observations into a precise relationship between the coherent systems moduli spaces and the Brill-Noether loci, one extra tool is needed, namely a precise relationship between bundle stability and coherent system stability. In general, for an arbitrary choice of the coherent system stability parameter, no such relationship exists. However, for values of the parameter close to $0$, the required relationship holds and there is a map from the coherent systems moduli space to the semistable Brill-Noether locus. While this map is not necessarily surjective, it does include the entire stable Brill-Noether locus in its image. It is via this map that information about the coherent systems moduli spaces can be applied to answer questions about higher rank Brill-Noether theory. In [@BG2] the first two authors initiated a programme to do just this, i.e. to use coherent systems moduli spaces to study higher rank Brill-Noether theory. There the goals were limited to explaining results of [@BGN] from the perspective of coherent systems. This turned out to require only a limited understanding of the coherent systems moduli spaces. In this paper we build on the foundation laid in [@BG2]. We study the moduli spaces for coherent systems $(E,V)$consisting of an algebraic vector bundle $E$ together with a linear subspace $V$ of its space of sections. While not required by the definitions, we consider only the case of bundles over a smooth irreducible projective algebraic curve $X$ of genus $g$. The [*type*]{} of the coherent system is defined by a triple of integers $(n,d,k)$ giving the rank of $E$, the degree of $E$, and the dimension of the subspace $V$. The infinitesimal study of coherent systems follows a standard pattern and is summarised in section \[sec:infinitesimal\]. This allows us in particular to identify the Zariski tangent space to each moduli space at any point, and to show that every irreducible component of every moduli space has dimension at least equal to a certain number ${\beta}(n,d,k)$, called the [*Brill-Noether number*]{} and often referred to as the [*expected dimension*]{}. For each type $(n,d,k)$, there is a family of moduli spaces. While each such family of moduli spaces has some properties which depend on ${(n,d,k)}$, there are some features that are common to all types. In particular: - [*The families have only a finite number of distinct members.*]{} The different members in the family correspond to different values for the real parameter ${\alpha}$ in the definition of stability. As ${\alpha}$ varies, the stability condition changes only as ${\alpha}$ passes through one of a discrete set of points in the real line. In some cases (if $k<n$) the range for the parameter is a finite interval, in which case it follows automatically that the family has only finitely many distinct members. However, even in the cases for which the range of the parameter is infinite, it turns out that there can be only a finite number of distinct moduli spaces. This is a consequence of the [*stabilisation theorem*]{} Proposition \[prop:flips\]. - [*The families are ordered, and the coherent systems in the terminal member are as simple as possible.*]{} The ordering comes from the fact that the moduli spaces are labelled by intervals on the real line. By the terminal member of the family we mean the moduli space corresponding to the last of these intervals (in the natural ordering of ${{\mathbb R}}$). In section \[sec:alpha-large\] we analyse the coherent systems corresponding to the points in this terminal moduli space. While the specifics depend on the type of the coherent system, in all cases we find that the structure of these coherent systems is (in a suitable sense) the best possible. The most obvious type-dependent feature is the description of the terminal moduli space. This divides naturally into distinct cases, according to whether $k<n$, $k=n$ or $k> n$. The case $k<n$ was discussed in some detail in [@BG2]. The results are summarized in section \[subsec:alpha-large.k&lt;n\]. When $k\ge n$, we show (in section \[subsec:quot\]) how to relate the terminal moduli space to a Grothendieck Quot scheme of quotients of the trivial bundle of rank $k$. Denoting the terminal moduli space of stable (respectively semistable) objects by $G_L$ (respectively ${\widetilde}{G_L}$), we prove \[thm:main1\] [**\[Theorem \[G\_L(k=n)\]\]**]{} Let $k=n\ge2$. If ${\widetilde}{G}_L$ is non-empty then $d=0$ or $d\geq n$. For $d>n$, ${\widetilde}{G}_L$ is irreducible and $G_L$ is smooth of the expected dimension $dn-n^2+1$. For $d=n$, $G_L$ is empty and ${\widetilde}{G}_L$ is irreducible of dimension $n$ (not of the expected dimension). For $d=0$, $G_L$ is empty and ${\widetilde}{G}_L$ consists of a single point. The non-emptiness of $G_L$ for the case $k>n$ is not so obvious and is related to the non-emptiness of Quot schemes. However, in this case there is a duality construction that relates coherent systems of types $(n,d,k)$ and $(k-n,d,k)$. If the parameter is large these “dual” moduli spaces are birationally equivalent. A similar idea has been considered for small ${\alpha}$ by Butler [@Bu2], but the construction seems to be more natural for large ${\alpha}$ and turns out to be an important tool to prove non-emptiness for large values of the parameter. For instance if $k=n+1$, the non-emptiness is given by the classical rank 1 Brill-Noether theory, i.e. we get \[thm:main2\] [**\[Theorem \[thm:dual-span\]\]**]{} Suppose that the curve $X$ is generic and that $k=n+1$. Then $G_L$ is non-empty if and only if ${\beta}=g-(n+1)(n-d+g)\ge0$. Moreover $G_L$ has dimension ${\beta}$ and it is irreducible whenever ${\beta}>0$. In addition to these ‘absolute’ results about the terminal moduli spaces, we also give ‘relative’ results which characterize the differences between moduli spaces within a given family. We compare the moduli spaces and identify subvarieties within which the differences are localised. In section \[sec:critical\] we give some general results estimating the codimensions of these subvarieties. With a view to applications, we examine a number of special cases in which either $k$ or $n$ (or both) are small. In these cases, discussed in sections \[sec:k=1\] - \[sec:k=3\], we can get more detailed results, especially for the codimension estimates on the difference loci between moduli spaces within a family. Having amassed all this information about the coherent systems moduli spaces, we end with some applications to Brill-Noether theory in section \[sec:applications\]. In all cases the strategy is the same: starting with information about $G_L$, and using our results about the relation between different moduli spaces within a given family, we deduce properties of the moduli space corresponding to the smallest values of the stability parameter. This is then passed down to the Brill-Noether loci using the morphism from the coherent systems moduli space to the moduli space of semistable bundles. This allows us, for example, - [**\[Theorems \[thm:bn2\] and \[thm:bn3\]\]**]{} to prove the irreducibility of the Brill-Noether loci for $k=1, 2, 3$, and - [**\[Theorem \[thm:bn4\]\]**]{} to compute the Picard group of the smooth part of the Brill-Noether locus for $k=1$. While the irreducibility result was previously known for $k=1$ and any $d$, and for $k=2,3$, $k<n$ and $d<\min\{2n,n+g\}$, our theorems have no such restrictions. These results should be regarded as a sample of what can be done. Our methods are certainly applicable more widely and we propose to pursue this in future papers. Throughout the paper $X$ denotes a fixed smooth irreducible projective algebraic curve of genus $g$ defined over the complex numbers. Unless otherwise stated, we make no assumption about $g$. For simplicity we shall write ${{\mathcal O}}$ for ${{\mathcal O}}_X$ and $H^0(E)$ for $H^0(X,E)$. We shall consistently denote the ranks of bundles $E,E',E_1\ldots$ by $n,n',n_1\ldots$, their degrees by $d,d',d_1\ldots$ and the dimensions of subspaces $V,V',V_1\ldots$ of their spaces of sections by $k,k',k_1\ldots$. Definitions and basic facts {#sec:definitions} =========================== Coherent systems and their moduli spaces {#subsec:coh} ---------------------------------------- Recall (cf. [@LeP], [@KN]) that a coherent system $( E, V)$ on $X$ of type $(n,d,k)$ consists of an algebraic vector bundle $ E$ over $X$ of rank $n$ and degree $d$, and a linear subspace $V\subseteq H^0(E)$ of dimension $k$. Strictly speaking, it is better to consider triples $( E,{{\mathbb V}},{\phi})$where ${{\mathbb V}}$ is a dimension $k$ vector space and ${\phi}:{{\mathbb V}}{\otimes}{{\mathcal O}}{\rightarrow}E$ is a sheaf map such that the induced map $H^0({\phi}):{{\mathbb V}}{\rightarrow}H^0(E)$ is injective. The linear space $V\subseteq H^0(E)$ is then the image $H^0({\phi})({{\mathbb V}})$. Under the natural concepts of isomorphism, isomorphism classes of such triples are in bijective correspondence with isomorphism classes of coherent systems. We will usually use the simpler notation $( E, V)$, but occasionally it is helpful to use the longer one. For a summary of basic results about coherent systems (and other related augmented bundles) we refer the reader to [@BDGW]. By introducing a suitable definition of stability, one can construct moduli spaces of coherent systems. The correct notion (i.e. the one dictated by Geometric Invariant Theory) depends on a real parameter ${\alpha}$, which a posteriori must be non-negative (cf. [@KN]). In the situation under consideration (i.e. where $E$is a vector bundle over a smooth algebraic curve), the definition may be given as follows. \[def:stable\] Fix ${\alpha}\in{{\mathbb R}}$. Let $( E,V)$ be a coherent system of type ${(n,d,k)}$. The ${\alpha}$-[*slope*]{} $\mu_{{\alpha}}(E,V)$ is defined by $$\mu_{{\alpha}}( E,V)=\frac{d}{n}+{\alpha}\frac{k}{n}\ .$$ We say $( E,V)$ is ${\alpha}$-[*stable*]{} if $$\mu_{{\alpha}}(E',V')<\mu_{{\alpha}}(E,V)$$ for all proper subsystems $(E',V')$ i.e. for every non-zero subbundle $E'$ of $E$ and every subspace $V'\subseteq V\cap H^0(E')$ with $(E',V')\ne(E,V)$. We define ${\alpha}$-[*semistability*]{} by replacing the above strict inequality with a weak inequality. A coherent system is called ${\alpha}$-[*polystable*]{} if it is the direct sum of ${\alpha}$-stable coherent systems of the same ${\alpha}$-slope. Sometimes it is necessary to consider a larger class of objects than coherent systems in which one replaces $E$ by a general coherent sheaf and $H^0({\phi}):{{\mathbb V}}{\rightarrow}H^0(E)$ is not necessarily injective. By doing so one obtains an abelian category [@KN]. One can easily extend the definition of ${\alpha}$-stability to this category. It turns out, however, that ${\alpha}$-semistability forces $E$ to be locally free and $H^0({\phi})$ to be injective, and hence ${\alpha}$-semistable objects in this category can be identified with ${\alpha}$-semistable coherent systems up to an appropriate definition of isomorphism. One has the following result. \[prop:filtration\]([@KN Corollary 2.5.1]) The ${\alpha}$-semistable coherent systems of any fixed ${\alpha}$-slope form a noetherian and artinian abelian category in which the simple objects are precisely the ${\alpha}$-stable systems. In particular the following statements hold. 1. [*(Jordan-Hölder Theorem)*]{} For any ${\alpha}$-semistable coherent system $(E,V)$, there exists a filtration by ${\alpha}$-semistable coherent systems $(E_j,V_j)$, $$0=(E_0,V_0)\subset (E_1,V_1)\subset ...\subset (E_m,V_m)=(E,V),$$ with $(E_j,V_j)/(E_{j-1},V_{j-1})$ an ${\alpha}$-stable coherent system and $$\mu_{\alpha}((E_j,V_j)/(E_{j-1},V_{j-1}))=\mu_{\alpha}(E,V) \;\;\; \mbox{for}\;\; 1\leq j\leq m.$$ 2. If $(E,V)$ is an ${\alpha}$-stable coherent system, then ${\operatorname{End}}(E,V)\cong{{\mathbb C}}$. Any filtration as in (i) is called a [*Jordan-Hölder*]{} filtration of $(E,V)$. It is not necessarily unique, but the associated graded object is uniquely determined by $(E,V)$. \[def:polystable\] We define the [*graduation*]{} of $(E,V)$ to be the ${\alpha}$-polystable coherent system $${\operatorname{gr}}(E,V)=\bigoplus_j (E_j,V_j)/(E_{j-1},V_{j-1}).$$ Two ${\alpha}$-semistable coherent systems $(E,V)$ and $(E',V')$ are said to be $S$-equivalent if ${\operatorname{gr}}(E,V)\cong{\operatorname{gr}}(E',V')$. We shall denote the moduli space of ${\alpha}$-stable coherent systems of type $(n,d,k)$ by $G({\alpha})=G({\alpha};n,d,k)$, and the moduli space of S-equivalence classes of ${\alpha}$-semistable coherent systems of type $(n,d,k)$ by ${\widetilde}{G}({\alpha})= {\widetilde}{G}({\alpha};n,d,k)$. The moduli space ${\widetilde}{G}({\alpha})$ is a projective variety which contains $G({\alpha})$ as an open set. Now suppose that $k\geq 1$. By applying the ${\alpha}$-semistability condition for $(E,V)$ to the subsystem $(E,0)$ one obtains that ${\alpha}\geq 0$. This means that there are no semistable coherent systems for negative values of ${\alpha}$. For ${\alpha}=0$, $(E,V)$ is $0$-semistable if and only if $E$ is semistable. For $k\geq 1$ there are no 0-stable coherent systems. \[def:cv\] We say that ${\alpha}>0$  is a [*virtual critical value*]{} if it is numerically possible to have a proper subsystem $(E',V')$ such that $\frac{k'}{n'}\ne\frac{k}{n}$ but $\mu_{{\alpha}}( E',V')=\mu_{{\alpha}}( E,V)$. We also regard $0$ as a virtual critical value. If there is a coherent system $(E,V)$ and a subsystem $(E',V')$ such that this actually holds, we say that ${\alpha}$ is an [*actual critical value*]{}. It follows from this (cf. [@BDGW]) that, for coherent systems of type $(n,d,k)$, the non-zero virtual critical values of ${\alpha}$ all lie in the set $$\{\ \frac{nd'-n'd}{n'k-nk'}\ |\ 0\le k'\le k\ ,\ 0<n'< n\ ,\ n'k\ne nk'\ \}\cap (0,\infty)\ .$$ We say that ${\alpha}$ is [*generic*]{} if it is not critical. Note that, if ${\operatorname{GCD}}(n,d,k)=1$ and ${\alpha}$ is generic, then ${\alpha}$-semistability is equivalent to ${\alpha}$-stability. If we label the critical values of ${\alpha}$ by ${\alpha}_i$, starting with ${\alpha}_0=0$, we get a partition of the ${\alpha}$-range into a set of intervals $({\alpha}_{i},{\alpha}_{i+1})$. For numerical reasons it is clear that within the interval $({\alpha}_{i},{\alpha}_{i+1})$ the property of ${\alpha}$-stability is independent of ${\alpha}$, that is if ${\alpha},{\alpha}'\in ({\alpha}_{i},{\alpha}_{i+1})$, $G({\alpha})= G({\alpha}')$. We shall denote this moduli space by $G_i=G_i(n,d,k)$. The construction of moduli spaces thus yields one moduli space $G_i$ for the interval $({\alpha}_i,{\alpha}_{i+1})$. If ${\operatorname{GCD}}(n,d,k)\ne1$, one can define similarly the moduli spaces ${\widetilde}{G}_i$ of semistable coherent systems. The GIT construction of these moduli spaces has been given in [@LeP] and [@KN]. A previous construction for $G_0$ had been given in [@RV] and in [@Be] for big degrees. When $k=1$ the moduli space of coherent systems is equivalent to the moduli space of vortex pairs studied in [@B; @BD1; @BD2; @G; @HL1; @HL2; @Th]. The relationship between the semistability of a coherent system and the underlying vector bundle is given by the following (cf. [@BDGW], [@KN]). \[small-alpha\] Let ${\alpha}_1$ be the first critical value after $0$ and let $0<{\alpha}<{\alpha}_1$. Then 1. $(E,V)$ ${\alpha}$-stable implies $E$ semistable; 2. $E$ stable implies $(E,V)$ ${\alpha}$-stable. Brill-Noether loci {#subsec:basicBN} ------------------- \[def:BN\] Let $X$ be an algebraic curve, and let $M(n,d)$ be the moduli space of stable bundles of rank $n$ and degree $d$. Let $k\geq 0$. The [*Brill-Noether loci*]{} of stable bundles are defined by $$B(n,d,k):=\{E\in M(n,d)\; |\; \dim H^0(E)\geq k\}.$$ Similarly one defines the Brill-Noether loci of semistable bundles $${\widetilde}{B}(n,d,k):=\{[E]\in {\widetilde}{M}(n,d)\; |\; \dim H^0({\operatorname{gr}}(E))\geq k\},$$ where ${\widetilde}{M}(n,d)$ is the moduli space of S-equivalence classes of semistable bundles, $[E]$ is the S-equivalence class of $E$ and ${\operatorname{gr}}(E)$ is the polystable bundle defined by a Jordan-Hölder filtration of $E$. The spaces $B(n,d,k)$ and ${\widetilde}{B}(n,d,k)$ have previously been denoted by $W_{n,d}^{k-1}$ and ${\widetilde}{W}_{n,d}^{k-1}$, but we have chosen to change the classical notation in an attempt to get rid of the $k-1$, which in the arbitrary rank case does not make much sense. (In fact the same loci have also been denoted by $W_{n,d}^k$ and ${\widetilde}{W}_{n,d}^k$, but, while logical, this seems a little confusing!) By semicontinuity, the Brill-Noether loci are closed subschemes of the appropriate moduli spaces. The main object of Brill-Noether theory is the study of these subschemes, in particular questions related to their non-emptiness, connectedness, irreducibility, dimension, and topological and geometric structure. It is in particular not difficult to describe them as determinantal loci, from which one obtains the following general result. We begin with a definition. \[def:bnnumber\] For any $(n,d,k)$, the [*Brill-Noether number*]{} ${\beta}(n,d,k)$ is defined by $$\label{bnnumber} {\beta}(n,d,k)=n^2(g-1)+1-k(k-d+n(g-1)).$$ \[thm:BN\] If $B(n,d,k)$ is non-empty and $B(n,d,k)\neq M(n,d)$, then - every irreducible component $B$ of $B(n,d,k)$ has dimension $$\dim B\geq {\beta}(n,d,k),$$ - $B(n,d,k+1)\subset Sing B(n,d,k)$, - the tangent space of $B(n,d,k)$ at a point $E$ with $\dim H^0(E)=k$ can be identified with the dual of the cokernel of the [*Petri map*]{} $$\label{petri} H^0(E){\otimes}H^0(E^*{\otimes}K){\longrightarrow}H^0({\operatorname{End}}E {\otimes}K)$$ (given by multiplication of sections), - $B(n,d,k)$ is smooth of dimension ${\beta}(n,d,k)$ at $E$ if and only if the Petri map is injective. For details, see for example [@M2]. When $n=1$, $M(n,d)$ is just $J^d$, the Jacobian of $X$ consisting of line bundles of degree $d$, and the Brill-Noether loci are the classical ones for which a thorough modern presentation is given in [@ACGH]. In particular we have the following results. - If ${\beta}(1,d,k)\ge0$, then $B(1,d,k)$ is non-empty. - If ${\beta}(1,d,k)>0$, then $B(1,d,k)$ is connected. - For a generic curve $X$ and $n=1$, the Petri map is always injective. Hence - $B(1,d,k)$ is smooth outside $B(1,d,k+1)$. - $B(1,d,k)$ has dimension ${\beta}(1,d,k)$ whenever it is non-empty and not equal to $M(n,d)$. - $B(1,d,k)$ is irreducible if ${\beta}(1,d,k)>0$. None of these statements is true for $n\ge2$ (see, for example, [@T1; @T2; @BGN; @BeF; @Mu]). Rather than referring repeatedly to a [*generic*]{} curve, we prefer to use the following more precise term. \[def:Petri\] A curve $X$ is called a [*Petri curve*]{} if the Petri map $$H^0(L){\otimes}H^0(L^*{\otimes}K){\longrightarrow}H^0(K)$$ is injective for every line bundle $L$ over $X$. One may note that any curve of genus $g\le2$ is Petri, the simplest examples of non-Petri curves being hyperelliptic curves with $g\ge3$. There is currently no sensible generalisation of Definition \[def:Petri\] to higher rank. Indeed, at least for $g\ge6$, there exist stable bundles $E$ on Petri curves for which the Petri map (\[petri\]) is not injective (see [@T1 §5]). Moreover the condition of Definition \[def:Petri\] is not sufficient to determine even the non-emptiness of Brill-Noether loci in higher rank (see [@M2; @Mu; @V]). Relationship between $B(n,d,k)$ and $G_0$ {#subsec:alpha-small} ----------------------------------------- The relevance of the moduli spaces of coherent systems in relation to Brill-Noether theory is given by Proposition \[small-alpha\]. The assignment $(E,V)\mapsto E$ defines a map $$\label{bnmap} G_0(n,d,k){\longrightarrow}{\widetilde}{B}(n,d,k),$$ which is one-to-one over $B(n,d,k)-B(n,d,k+1)$ and whose image contains $B(n,d,k)$. When ${\operatorname{GCD}}(n,d,k)\ne1$, this map can be extended to $$\label{bnmap2} {\widetilde}{G}_0(n,d,k){\longrightarrow}{\widetilde}{B}(n,d,k).$$ Even (\[bnmap2\]) may fail to be surjective. This happens for example (as observed in [@BG2]) when $d=0$, $0<k<n$. In this case ${\widetilde}{G}_0=\emptyset$ but ${\widetilde}{B}$ is non-empty. On the other hand, if $(n,d)=1$, the loci $B$ and ${\widetilde}{B}$ coincide and (\[bnmap\]) is surjective. Even when $(n,d)\ne1$, we may be able to obtain information about $B$ from properties of $G_0$. For example, if $G_0(n,d,k)$ is non-empty, then certainly ${\widetilde}{B}(n,d,k)$ is non-empty. Moreover, if $B(n,d,k)$ is non-empty and $G_0(n,d,k)$ is irreducible, then $B(n,d,k)$ is also irreducible. We are therefore interested in studying $G_0=G_0(n,d,k)$. Our approach to this consists of having - a detailed description of at least one (usually large ${\alpha}$) moduli space, - a thorough understanding of the “flips” to go from $G_i$ to $G_{i-1}$, until we get to $G_0$. The meaning of ‘thorough’ can vary, depending on the application. For instance, for non-emptiness questions all we require are the codimensions of the flip loci, or at least sufficiently good estimates thereof. In the case $n=1$, everything is much simpler. The concept of stability is vacuous and independent of ${\alpha}\in (0,\infty)$. We shall therefore denote the moduli space of coherent systems by $G(1,d,k)=G({\alpha};1,d,k)$. It consists of coherent systems $(L,V)$ such that $L$ is a line bundle of degree $d$ and $V\subset H^0(L)$ is any subspace of dimension $k$. These spaces have been studied classically (see for example [@ACGH], where $G(1,d,k)$ is denoted by $G_d^{k-1}$). The map (\[bnmap\]) becomes $$\label{bnmap3} G(1,d,k){\longrightarrow}B(1,d,k)$$ and is always surjective. The fibre of (\[bnmap3\]) over $L$ can be identified with the Grassmannian ${\operatorname{Gr}}(k,h^0(L))$. When $X$ is a Petri curve, we have - $G(1,d,k)$ is non-empty if and only if ${\beta}=g-k(k-d+g-1)\ge0$, - If ${\beta}\ge0$, $G(1,d,k)$ is smooth of dimension ${\beta}$, - If ${\beta}>0$, $G(1,d,k)$ is irreducible. Comparing this with section \[subsec:basicBN\], we see that, for $X$ a Petri curve, $G(1,d,k)$ provides a desingularisation of $B(1,d,k)$ whenever $B(1,d,k)\ne J^d$. In higher rank, if ${\operatorname{GCD}}(n,d,k)=1$, ${\beta}(n,d,k)\le n^2(g-1)$ and $G_0(n,d,k)$ is smooth and irreducible, and $B(n,d,k)$ is non-empty, then the map (\[bnmap\]) is a desingularisation of the closure of $B(n,d,k)$. We shall see that, for $k=1$, all these conditions hold (see section \[sec:applications\]). Note that it was proved in [@RV] that $G_0(n,n(g-1),1)$ is a desingularisation of ${\widetilde}{B}(n,n(g-1),1)$, which coincides with the generalised theta-divisor in $M(n,n(g-1))$. Infinitesimal study and extensions {#sec:infinitesimal} ================================== The infinitesimal study of the moduli space of coherent systems as well as the study of extensions of coherent systems is carried out in [@He; @LeP] (see also [@Th; @RV]). We review here the main results and refer to these papers, in particular for omitted proofs. Given two coherent systems $(E,V)$ and $(E',V')$ one defines the groups $${\operatorname{Ext}}^q((E',V'),(E,V)),$$ and considers the long exact sequence ([@He Corollaire 1.6]) $$\label{long-exact} \begin{array}{ccccccc} 0& {\longrightarrow}& {\operatorname{Hom}}((E',V'),(E,V)) & {\longrightarrow}& {\operatorname{Hom}}(E',E) & {\longrightarrow}& {\operatorname{Hom}}(V', H^0(E)/V) \\ & {\longrightarrow}& {\operatorname{Ext}}^1((E',V'),(E,V)) & {\longrightarrow}& {\operatorname{Ext}}^1(E',E) & {\longrightarrow}& {\operatorname{Hom}}(V', H^1(E))\\ & {\longrightarrow}& {\operatorname{Ext}}^2((E',V'),(E,V)) & {\longrightarrow}& 0. & & \end{array}$$ Notice that since we are on a curve ${\operatorname{Ext}}^2(E',E)=0$. Also, since $E'$ is a vector bundle, $${\operatorname{Ext}}^1(E',E)=H^1({E'}^*{\otimes}E).$$ We can now apply this to the study of infinitesimal deformations of the moduli space of coherent systems as well as to the study of extensions of coherent systems. Extensions {#subsec:extensions} ---------- We will have to deal later with extensions of coherent systems arising from the one-step Jordan-Hölder filtration of a semistable coherent system. By standard results on abelian categories, we have \[prop:extensions\] Let $(E_1,V_1)$ and $(E_2,V_2)$ be two coherent systems on $X$. The space of equivalence classes of extensions $$0{\longrightarrow}(E_1,V_1){\longrightarrow}(E,V) {\longrightarrow}(E_2,V_2){\longrightarrow}0$$ is isomorphic to ${\operatorname{Ext}}^1((E_2,V_2),(E_1,V_1))$. Hence the quotient of the space of non-trivial extensions by the natural action of ${{\mathbb C}}^*$ can be identified with the projective space ${{\mathbb P}}({\operatorname{Ext}}^1((E_2,V_2),(E_1,V_1)))$. \[prop:C21\] Let $(E_1,V_1)$ and $(E_2,V_2)$ be two coherent systems on $X$ of types $ (n_1,d_1,k_1)$ and $(n_2,d_2,k_2)$ respectively. Let ${{{\mathbb H}}}_{21}^0:={\operatorname{Hom}}((E_2,V_2),(E_1,V_1))$ and ${{\mathbb H}}^2_{21}:={\operatorname{Ext}}^2((E_2,V_2),(E_1,V_1))$. Then $$\label{C21} \dim {\operatorname{Ext}}^1((E_2,V_2),(E_1,V_1))= C_{21}+\dim {{{\mathbb H}}}^0_{21}+ \dim {{{\mathbb H}}}^2_{21},$$ where $$\begin{aligned} C_{21} &:= & k_2\chi (E_1)-\chi(E_2^\ast{\otimes}E_1)- k_1k_2 \nonumber\\ & = & n_1n_2(g-1)-d_1n_2+d_2n_1+k_2d_1-k_2n_1(g-1)-k_1k_2. \label{dim-ext}\end{aligned}$$ Moreover, $$\label{obs-ext} {{{\mathbb H}}}_{21}^2={\operatorname{Ker}}(H^0(E_1^*{\otimes}K){\otimes}V_2 \to H^0(E_1^* {\otimes}E_2{\otimes}K))^*.$$ Finally, if $N_2$ is the kernel of the natural map $V_2{\otimes}{{\mathcal O}}\to E_2$ then $$\label{obs-ext2} {{{\mathbb H}}}_{21}^2=H^0(E_1^*{\otimes}N_2{\otimes}K)^*.$$ [*Proof.* ]{} This follows from (\[long-exact\]) applied to $(E_1,V_1)=(E,V)$ and $(E_2,V_2)=(E',V')$, together with Serre duality for the last part. $\Box$ In order to use this result we will need to be able to estimate the dimension of ${{{\mathbb H}}}_{21}^2$. \[lem:hopf\] Suppose that $k_2>0$ and $h^0(E_1^*{\otimes}K)\neq 0$. Then the dimension of ${{\mathbb H}}_{21}^2$ is bounded above by $(k_2-1)(h^0(E_1^*{\otimes}K)-1)$. [*Proof.*]{} Use the Hopf lemma which states that, if ${\phi}:A{\otimes}B \to C$ is a bilinear map between finite-dimensional complex vector spaces such that, for any $a\in A-\{0\}$, ${\phi}(a,\cdot)$ is injective and, for any $b\in B-\{0\}$, ${\phi}(\cdot,b)$ is injective, then the image of ${\phi}$ has dimension at least $\dim A+\dim B-1$. The result follows from this and (\[obs-ext\]). $\Box$ Infinitesimal deformations {#subsec:infinitesimal} -------------------------- By standard arguments in deformation theory we have (see [@He Théorème 3.12]) \[prop:tangent\] Let $(E,V)$ be an ${\alpha}$-stable coherent system. 1. If ${\operatorname{Ext}}^2((E,V),(E,V))=0$, then the moduli space of ${\alpha}$-stable coherent systems is smooth in a neighbourhood of the point defined by $(E,V)$. This condition is satisfied if and only if the homomorphism ${\operatorname{Ext}}^1(E,E){\rightarrow}{\operatorname{Hom}}(V,H^1(E))$ is surjective. 2. The Zariski tangent space to the moduli space at the point defined by $(E,V)$ is isomorphic to $${\operatorname{Ext}}^1((E,V),(E,V)).$$ \[lem:BN-number\] Let $(E,V)$ be an ${\alpha}$-stable coherent system of type $(n,d,k)$. Then $$\dim {\operatorname{Ext}}^1((E,V),(E,V))= {\beta}(n,d,k) + \dim {\operatorname{Ext}}^2((E,V),(E,V)),$$ where ${\beta}(n,d,k)$ is the Brill-Noether number defined in Definition \[def:bnnumber\]. [*Proof.* ]{} By considering the long exact sequence (\[long-exact\]) for $(E',V')=(E,V)$, we see that $$\begin{aligned} \dim {\operatorname{Ext}}^1((E,V),(E,V)) &=& k\chi (E)-\chi({\operatorname{End}}E)- k^2 + \dim{\operatorname{End}}(E,V)\nonumber\\ & &\quad\quad \mbox{} +\dim {\operatorname{Ext}}^2((E,V),(E,V))\nonumber \\ &=& k(d+n(1-g))-n^2(1-g)- k^2 +1\nonumber\\ & &\quad\quad\mbox{}+ \dim {\operatorname{Ext}}^2((E,V),(E,V))\nonumber\\ &=& n^2(g-1)+1 -k(k-d +n(g-1))\label{dim-mod}\\ & &\quad\quad\mbox{} + \dim {\operatorname{Ext}}^2((E,V),(E,V)),\nonumber\end{aligned}$$ since ${\operatorname{End}}(E,V)\cong{{\mathbb C}}$ by Proposition \[prop:filtration\](ii). $\Box$ \[cor:exp\] Every irreducible component $G$ of every moduli space $G_i(n,d,k)$ has dimension $$\dim G\ge{\beta}(n,d,k).$$ [*Proof.* ]{} See [@He Corollaire 3.14].$\Box$ The following further corollary of Lemma \[lem:BN-number\] will be useful. \[cor:euler\] Let $C_{21}$ be defined by [*(\[dim-ext\])*]{} and $C_{12}$ by interchanging indices in [*(\[dim-ext\])*]{}. Then $${\beta}(n,d,k)={\beta}(n_1,d_1,k_1)+{\beta}(n_2,d_2,k_2)+C_{12}+C_{21}-1.$$ [*Proof.*]{} This follows from (\[dim-ext\]) and (\[dim-mod\]) using the facts that $\chi(E)=\chi(E_1)+\chi(E_2)$ and $$\chi({\operatorname{End}}E)=\chi({\operatorname{End}}E_1)+\chi({\operatorname{End}}E_2)+\chi(E_1^*{\otimes}E_2) +\chi(E_2^*{\otimes}E_1).$$ $\Box$ \[rem:duality\] Notice that if $k>n$ then ${\beta}(n,d,k)={\beta}(k-n,d,k)$. This is easily seen by writing $${\beta}(n,d,k)=n(g-1)(n-k)-k(k-d)+1.$$ We will come back to this symmetry later when studying the dual span of a coherent system (see section \[subsec:alpha-large.k&gt;n\]). We are now ready to extend to coherent systems the standard fact about smoothness of Brill-Noether loci. First we extend the definition of Petri map. \[def:petri\] Let $(E,V)$ be a coherent system. The Petri map of $(E,V)$ is the map $$V{\otimes}H^0(E^*{\otimes}K){\longrightarrow}H^0({\operatorname{End}}E{\otimes}K)$$ given by multiplication of sections. \[prop:Petri\] Let $(E,V)$ be an ${\alpha}$-stable coherent system of type $(n,d,k)$. Then the moduli space $G({\alpha};n,d,k)$ is smooth of dimension ${\beta}(n,d,k)$ at the point corresponding to $(E,V)$ if and only if the Petri map is injective. [*Proof.* ]{} By Proposition \[prop:tangent\] and Lemma \[lem:BN-number\], the moduli space is smooth of the correct dimension at $(E,V)$ if and only if ${\operatorname{Ext}}^2((E,V),(E,V))=0$. The result is now a special case of (\[obs-ext\]). $\Box$ \[rem:Petri\] This is a strengthening of the result for Brill-Noether loci (Theorem \[thm:BN\]), and it justifies the idea that the spaces of coherent systems provide partial desingularisations of the Brill-Noether loci (see sections \[subsec:alpha-small\] and \[sec:applications\]). In view of Proposition \[prop:Petri\] and Corollary \[cor:exp\], we often refer to ${\beta}(n,d,k)$ as the [*expected dimension*]{} of $G_i(n,d,k)$. There is a special case in which it is easy to check the injectivity of the Petri map. \[prop:smooth\] Let $(E,V)$ be an ${\alpha}$-stable coherent system such that $k\leq n$. If $V{\otimes}{{\mathcal O}}\to E$ is injective then the moduli space is smooth of dimension ${\beta}(n,d,k)$ at the point corresponding to $(E,V)$. This happens in particular when $k=1$. [*Proof.* ]{} We have an exact sequence $$0{\longrightarrow}V{\otimes}{{\mathcal O}}{\longrightarrow}E{\longrightarrow}F{\longrightarrow}0,$$ where $F$ is a coherent sheaf. Tensoring with $E^*{\otimes}K$ gives $$0{\longrightarrow}V{\otimes}E^*{\otimes}K{\longrightarrow}{\operatorname{End}}E{\otimes}K{\longrightarrow}F{\otimes}E^*{\otimes}K{\longrightarrow}0.$$ Taking sections, we see that the Petri map is injective.$\Box$ When the hypotheses of Proposition \[prop:smooth\] fail, we will need to estimate the dimension of the kernel of the Petri map. In fact Lemma \[lem:hopf\] gives us such an estimate. Range for ${\alpha}$ {#sec:range} ==================== \[lem:range\] If $k<n$ then the moduli space of ${\alpha}$-semistable coherent systems of type $(n,d,k)$ is empty for ${\alpha}> \frac{d}{n-k}$. In particular, we must have $d\ge 0$ in order for ${\alpha}$-semistable coherent systems to exist. Also we must have $d>0$ in order for ${\alpha}$-stable coherent systems to exist. [*Proof.* ]{} Suppose that $(E,V)$ is an ${\alpha}$-semistable coherent system of type $(n,d,k)$. By applying the ${\alpha}$-semistability condition to the subsystem $(E',V)$, where $E'={\operatorname{Im}}(V{\otimes}{{\mathcal O}}{\rightarrow}E)$, one obtains the upper bound ${\alpha}\leq \frac{d}{n-k}$, which in particular implies that $d\geq 0$ in order to have non-empty moduli spaces. The final assertion is similar. $\Box$ From this lemma and the considerations of section \[subsec:coh\], we deduce at once the following proposition. \[prop:last\] Let $k<n$ and let ${\alpha}_L$ be the biggest critical value smaller than $\frac{d}{n-k}$. The ${\alpha}$-range is then divided in a finite set of intervals determined by $$0={\alpha}_0<{\alpha}_1<\dots <{\alpha}_L<\frac{d}{n-k}\ .$$ Moreover, if ${\alpha}_i$ and ${\alpha}_{i+1}$ are two consecutive critical values, the moduli spaces for two different values of ${\alpha}$ in the interval $({\alpha}_i,{\alpha}_{i+1})$ coincide, and if ${\alpha}> \frac{d}{n-k}$ the moduli spaces are empty. \[lem:degree\] Let $k\geq n$. We must have $d\ge 0$ in order for ${\alpha}$-semistable coherent systems of type $(n,d,k)$ to exist. Also we must have $d>0$ in order for ${\alpha}$-stable coherent systems to exist except in the case $(n,d,k)=(1,0,1)$. [*Proof.* ]{} The first assertion is clear if the map ${{\mathcal O}}{\otimes}V{\rightarrow}E$ is generically surjective, otherwise one has to apply the ${\alpha}$-semistability condition to the subsystem $(E',V)$, where $E'={\operatorname{Im}}({{\mathcal O}}{\otimes}V{\rightarrow}E)$. For the second assertion, suppose $d=0$ and apply the ${\alpha}$-stability condition to $(E',V)$, where $E'={\operatorname{Im}}({{\mathcal O}}{\otimes}V{\rightarrow}E)$, to get that ${{\mathcal O}}{\otimes}V{\rightarrow}E$ is generically surjective. Therefore $E\cong {{\mathcal O}}^n$ and ${\alpha}$-stability forces $k=n=1$. $\Box$ Although in the case $k\geq n$ the stability condition does not provide us with a bound for ${\alpha}$, we will show that in fact after a certain finite value of ${\alpha}$ the moduli spaces do not change. We show first that for ${\alpha}$ big enough the vector bundle $E$ for an ${\alpha}$-semistable coherent system is generically generated by the sections in $V$. More precisely \[prop:BGN-last\] Suppose $k\geq n$. Then there exists ${\alpha}_{gg}>0$ such that for ${\alpha}\geq{\alpha}_{gg}$ if $(E,V)$ is ${\alpha}$-semistable then the map ${\phi}:V{\otimes}{{\mathcal O}}{\rightarrow}E$ is generically surjective, i.e. we have an exact sequence $$0{\longrightarrow}N{\longrightarrow}{{\mathcal O}}^{\oplus k}\stackrel{{\phi}}{{\longrightarrow}} E{\longrightarrow}T{\longrightarrow}0$$ where 1. $T$ is a torsion sheaf (possibly $0$), 2. ${\operatorname{rk}}N=k-n$, 3. $H^0(N)= 0$. In fact, $${\alpha}_{gg}\leq \frac{d(n-1)}{k}.$$ [*Proof.* ]{} Let $N={\operatorname{Ker}}{\phi}$ and $I={\operatorname{Im}}{\phi}$, and suppose ${\operatorname{rk}}I=n-l<n$. One has the exact sequence $$0{\longrightarrow}N{\longrightarrow}{{\mathcal O}}^{\oplus k}{\longrightarrow}E{\longrightarrow}E/I{\longrightarrow}0.$$ Consider the subsystem $(I,V)$. One has $d_I:=\deg I\geq 0$ since $I$ is generated by global sections. Now ${\alpha}$-semistability implies that $\mu_{\alpha}(I,V)\leq\mu_{\alpha}(E,V)$, which means that $$\frac{d_I}{n-l}+{\alpha}\frac{k}{n-l} \leq\frac{d}{n} +{\alpha}\frac{k}{n},$$ and hence $${\alpha}\leq \frac{d(n-l)}{kl}\leq\frac{d(n-1)}{k}.$$ We conclude that if ${\alpha}>\frac{d(n-1)}{k}$ then ${\operatorname{rk}}I=n$ so that $E/I=T$ is pure torsion. Finally $H^0(N)=0$ since $H^0({\phi})$ is injective by definition of coherent system. $\Box$ Our next object is to show that the ${\alpha}$-stability condition is independent of ${\alpha}$ for ${\alpha}>d(n-1)$. More precisely \[stabilization\] 1. If there exists a subsystem $(E',V')$ of $(E,V)$ with $\displaystyle{\frac{k'}{n'}>\frac{k}{n}}$, then $(E,V)$ is not ${\alpha}$-semistable for ${\alpha}>d(n-1)$. 2. If there exists a subsystem $(E',V')$ of $(E,V)$ with $\displaystyle{\frac{k'}{n'}=\frac{k}{n}\hbox{ and }\frac{d'}{n'}\ge\frac{d}{n}}$, then $(E,V)$ is not ${\alpha}$-stable for any ${\alpha}$. 3. If neither [*(i)*]{} nor [*(ii)*]{} holds, and $E$ is generically generated by its sections, then $(E,V)$ is ${\alpha}$-stable for ${\alpha}>d(n-1)$. [*Proof*]{}. (i) Suppose $(E,V)$ is ${\alpha}$-semistable. Replacing $E'$ by a subbundle if necessary, we can suppose that $E'$ is generically generated by its sections and hence $d'\ge0$. Then we have $${\alpha}\frac{k'}{n'}\le \frac{d}{n}+{\alpha}\frac{k}{n},$$ i.e. $${\alpha}\le\frac{n'd}{nk'-n'k}\le d(n-1).$$ \(ii) is obvious. \(iii) If neither (i) nor (ii) holds and $(E',V')$ contradicts the ${\alpha}$-stability of $(E,V)$, then we must have $\displaystyle{\frac{k'}{n'}<\frac{k}{n}}$. If $E$ is generically generated by its sections, then so is $E/E'$; hence $\deg(E/E')\ge0$ and $d'=\deg E'\le d$. Thus we have $$\frac{d}{n}+{\alpha}\frac{k}{n}\le \frac{d'}{n'}+{\alpha}\frac{k'}{n'}\le \frac{d}{n'}+{\alpha}\frac{k'}{n'},$$ i.e. $${\alpha}\le\frac{d(n-n')}{n'k-nk'}\le d(n-1).$$ $\Box$ We have thus proved the following. \[prop:flips\] Let $k\geq n$. Then there is a critical value, denoted by ${\alpha}_L$, after which the moduli spaces stabilise, i.e. $G({\alpha})=G_L$ if ${\alpha}>{\alpha}_L$. The ${\alpha}$-range is thus divided into a finite set of intervals bounded by critical values $$0={\alpha}_0<{\alpha}_1<\dots<{\alpha}_L<\infty$$ and such that 1. if ${\alpha}_i$ and ${\alpha}_{i+1}$ are two consecutive critical values, the moduli spaces for any two different values of ${\alpha}$ in the interval $({\alpha}_i,{\alpha}_{i+1})$ coincide, 2. for any two different values of ${\alpha}$ in the range $({\alpha}_L,\infty)$, the moduli spaces coincide. Moduli for ${\alpha}$ large {#sec:alpha-large} =========================== The moduli space $G_L$ for $k<n$ {#subsec:alpha-large.k<n} -------------------------------- Recall that, when $k<n$, $G_L$ denotes the moduli space of coherent systems for ${\alpha}$ large, i.e. ${\alpha}_L<{\alpha}<\frac{d}{n-k}$. The description of $G_L$ in this case has been carried out in [@BG2], where we refer for details. We summarise here the main results. \[def:BGN\] A [*BGN extension*]{} [@BGN] is an extension of vector bundles $$0{\longrightarrow}{{\mathcal O}}^{\oplus k}{\longrightarrow}E{\longrightarrow}F{\longrightarrow}0$$ which satisfies the following conditions: - ${\operatorname{rk}}E=n>k$, - $\deg E=d>0$, - $H^0(F^*)=0$, - if $\vec{e}=(e_1,\dots,e_k)\in H^1(F^\ast{\otimes}{{\mathcal O}}^{\oplus k})=H^1(F^\ast)^{\oplus k}$ denotes the class of the extension, then $e_1,\dots,e_k$ are linearly independent as vectors in $H^1(F^\ast)$. The BGN extensions which differ only by an automorphism of ${{\mathcal O}}^{\oplus k}$, i.e. by the action of an element in ${{\operatorname{GL}}}(k)$, comprise a BGN extension class of type $(n,d,k)$. \[prop:BGN\] Suppose that $0<k<n$ and $d>0$. Let ${\alpha}_L<{\alpha}<\frac{d}{n-k}$. Let $(E,V)$ be an ${\alpha}$-semistable coherent system of type $(n,d,k)$. Then $(E,V)$ defines a BGN extension class represented by an extension $$\label{bgn} 0{\longrightarrow}{{\mathcal O}}^{\oplus k}{\longrightarrow}E{\longrightarrow}F{\longrightarrow}0,$$ with $F$ semistable. In the converse direction, any BGN extension of type $(n,d,k)$ in which the quotient $F$ is stable gives rise to an ${\alpha}$-stable coherent system of the same type. In the last part of Proposition \[prop:BGN\], it is essential to have $F$ stable. If $F$ is only semistable, the coherent system can fail to be ${\alpha}$-semistable. \[G\_L(k&lt;n)\] Let $0<k<n$ and $d>0$. If $g\ge2$, the moduli space $G_L(n,d,k)$ of ${\alpha}$-stable coherent systems of type $(n,d,k)$ is birationally equivalent to a fibration over the moduli space $M(n-k,d)$ of stable bundles of rank $n-k$ and degree $d$ with fibre the Grassmannian ${{\operatorname{Gr}}}(k,d+(n-k)(g-1))$. In particular $G_L$ is non-empty if and only if $k\leq d+(n-k)(g-1)$, and it is then always irreducible and smooth of dimension ${\beta}(n,d,k)$. If $(n-k,d)=1$ then the birational equivalence is an isomorphism. [*Proof.* ]{} This follows directly from Proposition \[prop:BGN\] and the remark immediately preceding it. $\Box$ \[rmk:g=0,1\] Proposition \[prop:BGN\] remains true when $g=0$ or $1$, but Theorem \[G\_L(k&lt;n)\] can fail because $M(n-k,d)$ may be empty. In fact, if $g=0$, $M(n-k,d)=\emptyset$ unless $n-k=1$. Furthermore, if $n-k$ does not divide $d$, then ${\widetilde}{M}(n-k,d)=\emptyset$, and it follows from Proposition \[prop:BGN\] that $G_L(n,d,k)=\emptyset$. If $d=(n-k)a$ with $a\in{{\mathbb Z}}$, then ${\widetilde}{M}(n-k,d)$ consists of a single point corresponding to the bundle $$F={\mathcal O}(a)\oplus\ldots\oplus{\mathcal O}(a).$$ It is not clear from the results of [@BG2] whether there exist $\alpha$-stable coherent systems as in (\[bgn\]). Thus we conclude, for $g=0$, - $G_L(n,d,n-1)\ne\emptyset$ if and only if $d\ge n$, and it is then isomorphic to the Grassmannian ${{\operatorname{Gr}}}(n-1,d-1)$, - $G_L(n,d,k)=\emptyset$ if $k\le n-2$ and $d$ is not divisible by $n-k$, - if $k\le n-2$ and $d=(n-k)a$ with $a\in{{\mathbb Z}}$, then $G_L(n,d,k)=\emptyset$ if $d<n$; if $d\ge n$, a more detailed analysis is required. Turning now to $g=1$, we know that ${\widetilde}{M}(n-k,d)$ is always non-empty and that $M(n-k,d)\ne\emptyset$ if and only if $(n-k,d)=1$; moreover in this case $M(n-k,d)$ is isomorphic to the curve $X$. (All this follows essentially from [@At].) We conclude, for $g=1$, - if $(n-k,d)=1$, then $G_L(n,d,k)\ne\emptyset$ if and only if $d\ge k$, and it is then isomorphic to a fibration over $X$ with fibre ${{\operatorname{Gr}}}(k,d)$, - if $(n-k,d)\ne1$, a more detailed analysis is required. Quot schemes {#subsec:quot} ------------ When $k\ge n$, we can follow [@BeDW] and relate $G_L$ to a Grothendieck Quot scheme. In fact, by Proposition \[prop:BGN-last\], any element of ${\widetilde}{G}_L$ can be represented in the form $$\label{quot} 0{\longrightarrow}N{\longrightarrow}{{\mathbb V}}{\otimes}{{\mathcal O}}\stackrel{{\phi}}{{\longrightarrow}}E,$$ where ${\phi}$ is generically surjective. Dualising (\[quot\]), we obtain $$\label{quot2} 0{\longrightarrow}E^*{\longrightarrow}{{\mathbb V}}^*{\otimes}{{\mathcal O}}{\longrightarrow}F{\longrightarrow}0,$$ where $F$ is a coherent sheaf but is not torsion-free (unless ${\phi}$ is surjective). Conversely, given (\[quot2\]), one can recover (\[quot\]) (in fact $N\cong F^*$). It follows that there is a bijective correspondence between triples $(E,{{\mathbb V}},{\phi})$ and points of $Q=\hbox{Quot}_{k-n,d}({{\mathcal O}}^{\oplus k})$, the Quot scheme of quotients of ${{\mathcal O}}^{\oplus k}$ of rank $k-n$ and degree $d$. In order to obtain ${\widetilde}{G}_L$, we therefore need to construct a GIT quotient of $Q$ by the natural action of ${{\operatorname{GL}}}(k)$ with respect to a stability condition corresponding to the ${\alpha}$-stability of coherent systems for large ${\alpha}$. This situation requires detailed analysis, but even if we complete the construction, it may still be difficult to obtain information about $G_L$, since even basic information about $Q$ is often lacking, e.g., when it is non-empty, irreducible etc. However, potentially this would be a useful source of information about $G_L$. In sections \[subsec:alpha-large.k=n\] and \[subsec:alpha-large.k&gt;n\], we shall use the sequences (\[quot\]) and (\[quot2\]) to obtain information about $G_L$ in the cases $k=n$ and $k>n$. The moduli space $G_L$ for $k=n$ {#subsec:alpha-large.k=n} -------------------------------- We are now able to prove Theorem \[thm:main1\] in a stronger form which covers ${\widetilde}{G}_L$ as well as $G_L$. \[G\_L(k=n)\] Let $k=n\ge2$. If ${\widetilde}{G}_L$ is non-empty then $d=0$ or $d\geq n$. For $d>n$, ${\widetilde}{G}_L$ is irreducible and $G_L$ is smooth of the expected dimension $dn-n^2+1$. For $d=n$, $G_L$ is empty and ${\widetilde}{G}_L$ is irreducible of dimension $n$ (not of the expected dimension). For $d=0$, $G_L$ is empty and ${\widetilde}{G}_L$ consists of a single point. [*Proof.* ]{} Let $(E,V)$ be an ${\alpha}$-semistable coherent system for any ${\alpha}$, represented by ${\phi}:{{\mathcal O}}^{\oplus n}{\rightarrow}E$. If ${\phi}$ is an isomorphism, then $d=0$ and $(E,V)\cong({{\mathcal O}}^{\oplus n},{{\mathbb C}}^n)$, which is clearly ${\alpha}$-semistable but not ${\alpha}$-stable. Otherwise there exists ${{\mathcal O}}\subset{{\mathcal O}}^{\oplus n}$ which defines a section of $E$ with a zero. It follows that this section is contained in a subbundle of $E$ of rank $1$ with degree $>0$. This subbundle together with the section defines a subsystem which contradicts ${\alpha}$-semistability if $d<n$ and ${\alpha}$-stability if $d=n$. Now suppose $d\ge n$. Let $(E,V)$ be any ${\alpha}$-semistable coherent system for ${\alpha}$ large. By Proposition \[prop:BGN-last\] with $k=n$, we have an extension $$0 \to {{{\mathcal O}}}^{\oplus n} \to E \to T \to 0,$$ where $T$ is torsion. The generic torsion sheaf is of the form $T={{{\mathcal O}}}_D$ for a divisor $D$ consisting of $d$ distinct points. For such $T$, $E$ is given by an extension class $\xi \in {\operatorname{Ext}}^1(T,{{{\mathcal O}}}^{\oplus n}) \cong {\operatorname{Hom}}(T, {{{\mathbb C}}}^n)$, which is equivalent to a collection of $d$ vectors $\xi_i\in {{{\mathbb C}}}^n$, one for each point $P_i$ in the support of $D$. We claim that all the coherent systems defined by extensions in $$U=\{ ({{{\mathcal O}}}_D, \xi)| \text{any subset of $n$ vectors of $\xi_1,\ldots, \xi_d$ is linearly independent}\}$$ are ${\alpha}$-stable (for $d>n$) or ${\alpha}$-semistable (for $d=n$). Suppose for the moment that the claim holds. Let $$U^{ss}=\{ (T, \xi)\;|\; \xi\in {\operatorname{Ext}}^1(T,{{\mathcal O}}^{\oplus n}) \text{ and determines an ${\alpha}$-semistable coherent system}\}.$$ Then $U$ is dense and open in $U^{ss}$. Also $U^{ss}$ dominates the moduli space of ${\alpha}$-semistable coherent systems. Thus ${\widetilde}{G}_L$ is irreducible. The fact that $G_L$ is smooth of the expected dimension follows at once from Proposition \[prop:smooth\]. We can also compute directly the dimension of the space of coherent systems determined by $U$. The space ${\operatorname{Ext}}^1(T,{{{\mathcal O}}}^{\oplus n})$ has dimension $dn$, and we have to quotient out by the automorphisms ${\operatorname{Aut}}_{{{\mathcal O}}} T= {\operatorname{GL}}(1)^d$, for $T={{{\mathcal O}}}_D$, and by ${\operatorname{Aut}}_{{{\mathcal O}}} {{{\mathcal O}}}^{\oplus n}={\operatorname{GL}}(n)$. For $d>n$, the centraliser of the action of the product on $({{\mathcal O}}_D,\xi) \in U$ is ${{{\mathbb C}}}^*$. So the dimension of the space of coherent systems determined by $U$ is $d+dn-d-n^2+1$, which is in agreement with our previous answer. For $d=n$, ${\operatorname{GL}}(n)$ acts freely on any collections of $n$ linearly independent vectors in ${{{\mathbb C}}}^n$. So the dimension of the space of coherent systems determined by $U$ is $d+dn-n^2=n$. It is possible that different elements of $U$ give rise to S-equivalent systems, thus reducing $\dim {\widetilde}{G}_L$. However, if $g\ge1$, the coherent systems $$\left({{\mathcal O}}(P_1),H^0({{\mathcal O}}(P_1))\right)\oplus\ldots \oplus\left({{\mathcal O}}(P_n),H^0({{\mathcal O}}(P_n))\right),$$ where $P_1,\ldots,P_n\in X$, are clearly ${\alpha}$-semistable and no two of them are S-equivalent, so $\dim{\widetilde}{G}_L\ge n$, which completes the computation. If $g=0$, there is a unique line bundle ${{\mathcal O}}(1)$ of degree $1$, and $h^0({{\mathcal O}}(1))=2$; in this case the coherent systems $$({{\mathcal O}}(1),V_1)\oplus\ldots\oplus({{\mathcal O}}(1),V_n),$$ where $V_1,\ldots,V_n$ are subspaces of dimension $1$ of $H^0({{\mathcal O}}(1))$, are ${\alpha}$-semistable and form a family of dimension $n$. This gives the same conclusion. Now we prove the claim, i.e. every $(E,V)$ in the image of $U$ is ${\alpha}$-stable for ${\alpha}$ large. Let $(E_1,V_1)$ be a coherent subsystem of $(E,V)$. As $E_1 \subset E$ we must have $k_1 \leq n_1$. If $k_1<n_1$ then $(E_1,V_1)$ cannot violate ${\alpha}$-stability. If $k_1=n_1$ then we have a diagram $$\begin{array}{ccccc} {{{\mathcal O}}}^{\oplus n_1} & \to & E_1 &\to & T_1 \\ \downarrow & & \downarrow & & \downarrow \\ {{{\mathcal O}}}^{\oplus n} & \to & E &\to & T \end{array}$$ where $d_1=\deg T_1$. Then the image of $\xi \in {\operatorname{Ext}}^1(T, {{{\mathcal O}}}^{\oplus n})$ in ${\operatorname{Ext}}^1(T_1, {{{\mathcal O}}}^{\oplus n})$ lies in the subspace ${\operatorname{Ext}}^1(T_1, {{{\mathcal O}}}^{\oplus n_1})$. This is equivalent to $\xi_i \in {{{\mathbb C}}}^{n_1}$ for any $P_i$ in the support of $T_1$. $E$ is ${\alpha}$-semistable if $d_1/n_1 \leq d/n$ for all possible choices of diagrams as above. Now any subcollection of $n$ vectors of the $\xi_i$ is linearly independent, so for $d_1\geq n$ we have $n_1=n$ and $E_1=E$. For $d_1<n$ we have $n_1\ge d_1$ and $d_1/n_1\le 1$. Hence, for $d>n$ the coherent systems are ${\alpha}$-stable, while for $d=n$ they are ${\alpha}$-semistable.$\Box$ \[rem:d=n\] For $d\le n$, the proof shows that $(E,V)$ cannot be ${\alpha}$-stable for any ${\alpha}$; moreover, if $0<d<n$, $(E,V)$ cannot be ${\alpha}$-semistable. For $d=0$, any ${\alpha}$-semistable coherent system is isomorphic to $({{\mathcal O}}^{\oplus n},{{\mathbb C}}^n)$. For $d=n$, one can show that ${\widetilde}{G}({\alpha})$ is independent of ${\alpha}$ and that ${\widetilde}{G}({\alpha})\cong S^nX$ see [@BGN Theorem 8.3] for the case ${\alpha}=0$. The moduli space $G_L$ for $k> n$. The dual span construction {#subsec:alpha-large.k>n} ------------------------------------------------------------- We can represent a coherent system by a sequence (\[quot\]), where we now suppose that $k>n$ and that ${\phi}$ is surjective; so we have $$0{\longrightarrow}N{\longrightarrow}{{\mathbb V}}{\otimes}{{\mathcal O}}\stackrel{{\phi}}{{\longrightarrow}}E{\longrightarrow}0$$ and $$0{\longrightarrow}E^*{\longrightarrow}{{\mathbb V}}^*{\otimes}{{\mathcal O}}\stackrel{\psi}{{\longrightarrow}} N^*{\longrightarrow}0.$$ In the case where $V=H^0(E)$ and $E$ is generated by its sections, this construction has been used by a number of authors (see for example [@Bu1; @Bu2; @EL; @M1; @PR]), the main question being to determine conditions under which the stability of $E$ implies that of $N$. Recently Butler noted that the construction belongs more naturally to the theory of coherent systems and began to investigate it using ${\alpha}$-stability. However he restricted attention to small ${\alpha}$. Our purpose in this section is to show that the construction works better if we consider large ${\alpha}$. It is convenient here to make partial use of the wider notion of coherent system, introduced in [@KN] and mentioned in section \[subsec:coh\], by dropping the assumption that $H^0({\phi})$ is injective. This makes no essential difference as $(E,V)$ cannot be ${\alpha}$-semistable unless $H^0({\phi})$ is injective. It does however mean that $(N^*,{{\mathbb V}}^*,\psi)$ always determines a coherent system, which we may call the [*dual span*]{} of $(E,{{\mathbb V}},{\phi})$ and denote by $D(E,{{\mathbb V}},{\phi})$ (or $D(E,V)$ in the case where $H^0({\phi})=0$). \[def:strongly-unstable\] $(E,{{\mathbb V}},{\phi})$ is [*strongly unstable*]{} if there exists a proper coherent subsystem $(E',{{\mathbb V}}',{\phi}')$ such that $${\frac{k'}{n'}>\frac{k}{n}}\, .$$ \[prop:dual-strongly\] Suppose that $E$ is generated by ${{\mathbb V}}$. Then $(E,{{\mathbb V}},{\phi})$ is strongly unstable if and only if $D(E,{{\mathbb V}},{\phi})$ is strongly unstable. [*Proof*]{}. Suppose that $(E,{{\mathbb V}},{\phi})$ is strongly unstable and that $(E',{{\mathbb V}}',{\phi}')$ is a subsystem as in the definition above. Replacing $E'$ by the (sheaf) image of ${{\mathbb V}}'$ in $E$ if necessary, we have a short exact sequence $$0\to N'\to{{\mathbb V}}'{\otimes}{{\mathcal O}}\to E'\to0.$$ $D(E',{{\mathbb V}}',{\phi}')$ is then a quotient system of $D(E,{{\mathbb V}},{\phi})$. The corresponding subsystem has rank $(k-n)-(k'-n')$ and dimension $k-k'$ and $$(k-n)(k-k')-((k-n)-(k'-n'))k=nk'-n'k>0.$$ So $D(E,{{\mathbb V}},{\phi})$ is strongly unstable. The converse is similar, which completes the proof. $\Box$ By Proposition \[stabilization\], any strongly unstable coherent system fails to be ${\alpha}$-semistable for ${\alpha}>d(n-1)$. The converse may fail because we have to take account of subsystems with $\displaystyle{\frac{k'}{n'}=\frac{k}{n}}$. However, if $(n,k)=1$, there are no such subsystems and we have \[cor:dual-span\] Suppose that $E$ is generated by $V$. If $(n,k)=1$, then $(E,V)$ is ${\alpha}$-stable for large ${\alpha}$ if and only if $D(E,V)$ is ${\alpha}$-stable for large ${\alpha}$. By Proposition \[stabilization\] it is sufficient to take ${\alpha}>\max\{d(n-1),d(k-n-1)\}$. \[thm:dual-span\] Suppose that $X$ is a Petri curve and that $k=n+1$. Then $G_L$ is non-empty if and only if ${\beta}=g-(n+1)(n-d+g)\ge0$. Moreover $G_L$ has dimension ${\beta}$ and it is irreducible whenever ${\beta}>0$. [*Proof. *]{} If $(E,V)\in G_L$, then by Proposition \[prop:BGN-last\], $E$ is generically generated by $V$. If we suppose further that $E$ is generated by $V$, then $D(E,V)\in G(1,d,n+1)$. Since $X$ is Petri, $G(1,d,n+1)$ is non-empty if and only if the Brill-Noether number $${\beta}={\beta}(1,d,n+1)=g-(n+1)(n-d+g)\ge0.$$ Moreover, if this holds, $G(1,d,n+1)$ has dimension ${\beta}$, and it is irreducible whenever ${\beta}>0$. Note also that the dimension of the subvariety consisting of systems $(L,W)$ for which $L$ is not generated by $W$ has dimension at most $$g-(n+1)(n-(d-1)+g)+1<{\beta}.$$ So $G(1,d,n+1)$ has a dense open subset in which $L$ is generated by $W$. The Brill-Noether number ${\beta}(n,d,n+1)={\beta}$ by Remark \[rem:duality\] so the systems $(E,V)$ which are ${\alpha}$-stable for large ${\alpha}$ and for which $V$ generates $E$ are parametrised by a variety of the expected dimension. If $E$ is only generically generated by $V$ and $E'$ is the subsheaf generated by $V$, we can put $\deg E'=d-t$ with $t>0$. Then, by the argument above, the variety parametrising the systems $(E',V)$ has the expected dimension, which is ${\beta}-(n+1)t$. On the other hand, the variety parametrising the extensions $$0\to E'\to E\to T\to 0,$$ where $T$ is a torsion sheaf of length $t$, has dimension $nt$ (after factoring out by the action of ${\operatorname{Aut}}T$). So the variety parametrising all the corresponding systems $(E,V)$ has dimension $<{\beta}$. Since every component of $G_L$ has dimension $\ge{\beta}$, this completes the proof. $\Box$ Crossing critical values {#sec:critical} ======================== In this section we analyse the differences between consecutive moduli spaces in the family $\{G_0,G_1,\dots,G_L\}$. Recall that $G_i$ denotes the moduli space of ${\alpha}$-stable coherent systems, where ${\alpha}$ is (anywhere) in the interval bounded by the critical values ${\alpha}_i$ and ${\alpha}_{i+1}$. The differences between $G_{i-1}$ and $G_i$ are thus due to the differences between the ${\alpha}$-stability conditions for ${\alpha}<{\alpha}_i$ and ${\alpha}>{\alpha}_i$. The basic mechanism {#subsec:mechanism} ------------------- The following lemma describes the basic mechanism responsible for a change in the stability property of a coherent system. Let $(E,V)$ be a coherent system of type $(n,d,k)$ and let $(E',V')$ be a subsystem of type $(n',d',k')$. Then $\mu_{{\alpha}}(E',V')-\mu_{{\alpha}}(E,V)$ is a linear function of ${\alpha}$ which is - monotonically increasing if $\frac{k'}{n'}- \frac{k}{n}>0$, - monotonically decreasing if $\frac{k'}{n'}- \frac{k}{n}<0$, - constant if $\frac{k'}{n'}- \frac{k}{n}=0$. In particular, if ${\alpha}_i$ is a critical value and $\mu_{{\alpha}_i}(E',V')=\mu_{{\alpha}_i}(E,V)$, then - $(\mu_{{\alpha}}(E',V')-\mu_{{\alpha}}(E,V))({\alpha}-{\alpha}_i)>0$, for all ${\alpha}\ne{\alpha}_i$, if $\frac{k'}{n'}- \frac{k}{n}>0$, - $(\mu_{{\alpha}}(E',V')-\mu_{{\alpha}}(E,V))({\alpha}-{\alpha}_i)<0$, for all ${\alpha}\ne{\alpha}_i$, if $\frac{k'}{n'}- \frac{k}{n}<0$, - $\mu_{{\alpha}}(E',V')-\mu_{{\alpha}}(E,V)=0$ for all ${\alpha}$ if $\frac{k'}{n'}- \frac{k}{n}=0$. [*Proof.* ]{} This follows easily from $$\mu_{{\alpha}}(E',V')-\mu_{{\alpha}}(E,V)=\frac{d'}{n'}-\frac{d}{n}+ {\alpha}\left(\frac{k'}{n'}- \frac{k}{n}\right).$$ $\Box$ In particular, we have Let $( E,V)$ be a coherent system of type ${(n,d,k)}$. Suppose that it is ${\alpha}$-stable for ${\alpha}>{\alpha}_i$, but is strictly ${\alpha}$-semistable for ${\alpha}={\alpha}_i$. Then $( E,V)$ is unstable for all ${\alpha}<{\alpha}_i$. [*Proof.* ]{} Any such coherent system must have a subsystem, say $(E',V')$, for which $\mu_{{\alpha}_i}(E',V')=\mu_{{\alpha}_i}(E,V)$ but such that $\mu_{{\alpha}}(E',V')<\mu_{{\alpha}}(E,V)$ if ${\alpha}>{\alpha}_i$. It follows from the previous lemma that $\mu_{{\alpha}}(E',V')>\mu_{{\alpha}}(E,V)$ for all ${\alpha}<{\alpha}_i$, i.e. the subsystem $(E',V')$ is destabilising for all ${\alpha}<{\alpha}_i$. Thus, if we study the effect on $G_L{(n,d,k)}$ of monotonically reducing ${\alpha}$, we see that “once a coherent system is removed it can never return”. In contrast to this, it can happen that “a coherent system once added may have to be later removed” [@BG2]. We define $G_i^+\subseteq G_i=G_i(n,d,k)$ to be the set of all $(E,V)$ in $G_i$ which are not ${\alpha}$-stable if ${\alpha}<{\alpha}_i$. Similarly, we define $G_i^-\subseteq G_{i-1}$ to be the set of all $(E,V)$ in $G_{i-1}$ which are not ${\alpha}$-stable if ${\alpha}>{\alpha}_i$. We can identify the sets $G_i-G_i^+ = G_{i-1}-G_i^- $ and hence (set theoretically) we get - $G_{i+1}=G_i-G_{i+1}^- +G_{i+1}^+$, - $G_{i-1}=G_i-G_i^+ +G_i^-$. In fact, we can be more precise. The subset $G_i^+$ consists of the points in $G_i$ corresponding to coherent systems which are not ${\alpha}_i$-stable; they therefore form a closed subscheme of $G_i$. Similarly $G_i^-$ is a closed subscheme of $G_{i-1}$. Hence $G_i-G_i^+$ and $G_{i-1}-G_i^-$ have natural scheme structures, and as such are isomorphic. Destabilising patterns {#subsec:destab} ---------------------- The following lemma allows us to describe the sets $G_i^+$ and $G_i^-$, and also to estimate their codimensions in the moduli spaces $G_i$. It is important to note that, unlike the Jordan-Hölder filtrations for semistable objects, the descriptions we obtain are always as extensions, i.e. 1-step filtrations. This simplification results from a careful exploitation of the stability parameter. For convenience, we denote values of ${\alpha}$ in the intervals on either side of ${\alpha}_i$ by ${\alpha}_i^-$ and ${\alpha}_i^+$ respectively. \[lem:vicente\] Let ${\alpha}_i$ be a critical value of ${\alpha}$ with $1\le i\le L$. Let $(E,V)$ be a coherent system of type ${(n,d,k)}$. 1. Suppose that $(E,V)$ is ${\alpha}_i^+$-stable but ${\alpha}^-_i$-unstable. Then $(E,V)$ appears as the middle term in an extension $$\label{destab} 0\to (E_1,V_1)\to (E,V)\to (E_2,V_2) \to 0$$ in which 1. $(E_1,V_1)$ and $(E_2,V_2)$ are both ${\alpha}^+_i$-stable, with $\mu_{{\alpha}_i^+}(E_1,V_1)<\mu_{{\alpha}_i^+}(E_2,V_2)$, 2. $(E_1,V_1)$ and $(E_2,V_2)$ are both ${\alpha}_i$-semistable, with $\mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)$, 3. $\frac{k_1}{n_1}$ is a maximum among all proper subsystems $(E_1,V_1)\subset (E,V)$ which satisfy [(b)]{}, 4. $n_1$ is a minimum among all subsystems which satisfy [(c)]{}. 2. Similarly, if $(E,V)$ is ${\alpha}_i^-$-stable but ${\alpha}^+_i$-unstable, then $(E,V)$ appears as the middle term in an extension $\mathrm{(\ref{destab})}$ in which 1. $(E_1,V_1)$ and $(E_2,V_2)$ are both ${\alpha}^-_i$-stable, with $\mu_{{\alpha}_i^-}(E_1,V_1)<\mu_{{\alpha}_i^-}(E_2,V_2)$, 2. $(E_1,V_1)$ and $(E_2,V_2)$ are both ${\alpha}_i$-semistable, with $\mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)$, 3. $\frac{k_1}{n_1}$ is a minimum among all proper subsystems $(E_1,V_1)\subset (E,V)$ which satisfy [(b)]{}, 4. $n_1$ is a minimum among all subsystems which satisfy [(c)]{}. [*Proof.*]{} Since its stability property changes at ${\alpha}_i$, the coherent system $(E,V)$ must be strictly ${\alpha}_i$-semistable, i.e. it must have a proper subsystem $(E',V')$ with $\mu_{{\alpha}_i}(E',V')=\mu_{{\alpha}_i}(E,V)$. Consider the (non-empty) set $${{{\mathcal F}}}_1=\{(E_1,V_1)\subsetneq (E,V)\ |\ \mu_{{\alpha}_i}(E_1,V_1)= \mu_{{\alpha}_i}(E,V)\ \}.$$ Any such subsystem $(E_1,V_1)$ must have $n_1<n$ and $V_1=V\cap H^0(E_1)$ (otherwise replacing $V_1$ by $V\cap H^0(E_1)$ would contradict the ${\alpha}_i$-semistability of $(E,V)$). [*Proof of (i).*]{} Suppose first that $(E,V)$ is ${\alpha}_i^+$-stable but ${\alpha}^-_i$-unstable. We observe that if $(E_1,V_1)\in {{{\mathcal F}}}_1$, then $\frac{k_1}{n_1}< \frac{k}{n}$, since otherwise $(E,V)$ could not be ${\alpha}_i^+$-stable. But the allowed values for $\frac{k_1}{n_1}$ are limited by the constraints $0<n_1< n$ and $0\le k_1\le k$. We can thus define $${\lambda}_0=\max\left\{\frac{k_1}{n_1}\ \biggm|\ (E_1,V_1)\in {{{\mathcal F}}}_1\ \right\}$$ and set $${{{\mathcal F}}}_2=\left\{(E_1,V_1)\subset {{{\mathcal F}}}_1\ \biggm|\\ \frac{k_1}{n_1}={\lambda}_0\ \right\}.$$ Let $(E_1,V_1)$ be any coherent system in ${{\mathcal F}}_2$. Since $V_1=V\cap H^0(E_1)$, we can write $$0\to (E_1,V_1)\to (E,V)\to (E_2,V_2) \to 0 $$ for some coherent system $(E_2,V_2)$. Since $\mu_{{\alpha}_i}(E_1,V_1)= \mu_{{\alpha}_i}(E,V)=\mu_{{\alpha}_i}(E_2,V_2)$ and $(E,V)$ is ${\alpha}_i$-semistable, it follows that both $(E_1,V_1)$ and $(E_2,V_2)$ are ${\alpha}_i$-semistable. We now show that $(E_2,V_2)$ is ${\alpha}_i^+$-stable. Suppose not. Then there is a proper subsystem $(E_2',V_2')\subset (E_2,V_2)$ with - $\mu_{{\alpha}_i}(E_2',V_2')=\mu_{{\alpha}_i}(E_2,V_2)$, - $\frac{k'_2}{n'_2}\ge\frac{k_2}{n_2}$. Consider now the subsystem $(E',V')\subset (E,V)$ defined by the pull-back diagram $$0\to (E_1,V_1)\to (E',V')\to (E'_2,V'_2) \to 0 \ .$$ This has $\mu_{{\alpha}_i}(E',V')=\mu_{{\alpha}_i}(E,V)$ and thus satisfies $\frac{k'_2+k_1}{n'_2+n_1}\le\frac{k_1}{n_1}$. It follows that $$\frac{k'_2}{n'_2}\le\frac{k_1}{n_1}<\frac{k_2}{n_2}\ ,$$ which is a contradiction. Now consider $(E_1,V_1)\in{{\mathcal F}}_2$ with [*minimum rank in*]{} ${{\mathcal F}}_2$. If $(E_1,V_1)$ is not ${\alpha}_i^+$-stable, then it must have a proper subsystem $(E'_1,V'_1)$ with - $\mu_{{\alpha}_i}(E_1',V'_1)=\mu_{{\alpha}_i}(E_1,V_1)$, - $\frac{k'_1}{n'_1}\ge\frac{k_1}{n_1}$. But then $n'_1<n_1$, which contradicts the minimality of $n_1$. Finally, notice that since $(E,V)$ is ${\alpha}_i^+$-stable, we must have $\mu_{{\alpha}_i^+}(E_1,V_1)<\mu_{{\alpha}_i^+}(E,V)<\mu_{{\alpha}_i^+}(E_2,V_2)$. [*Proof of (ii).*]{} If $(E,V)$ is ${\alpha}_i^-$-stable but ${\alpha}^+_i$-unstable, then $\frac{k_1}{n_1}> \frac{k}{n}$ for all $(E_1,V_1)\in{{\mathcal F}}_1$. The proof of (i) must thus be modified as follows. With $${\lambda}_0=\min\left\{\frac{k_1}{n_1}\ \biggm|\ (E_1,V_1)\in {{{\mathcal F}}}_1\ \right\}$$ we can define $${{{\mathcal F}}}_2=\left\{(E_1,V_1)\subset {{{\mathcal F}}}_1\ \biggm|\\ \frac{k_1}{n_1}={\lambda}_0\ \right\}$$ and select $(E_1,V_1)\in{{\mathcal F}}_2$ such that $E_1$ has minimal rank in ${{\mathcal F}}_2$. It follows in a similar fashion to that above that $(E,V)$ has a description as $$0\to (E_1,V_1)\to (E,V)\to (E_2,V_2) \to 0$$ in which both $(E_1,V_1)$ and $(E_2,V_2)$ are ${\alpha}_i^-$-stable. We refer to the extensions of the form (\[destab\]) with the properties of Lemma \[lem:vicente\] as the [*destabilising patterns*]{} of the coherent systems. Codimension estimates for $G_i^-$ and $G_i^+$ --------------------------------------------- \[def:wi\] Let $W^+({\alpha}_i, {\lambda}, n_1; n,d,k)$ (abbreviated to $W^+_i({\lambda},n_1)$ whenever possible) denote the set of all destabilising patterns $$0\to (E_1,V_1)\to (E,V)\to (E_2,V_2) \to 0$$ in which - $(E,V)$ is ${\alpha}^+_i$-stable and of type ${(n,d,k)}$, - ${{\operatorname{rk}}}(E_1)=n_1$ and $\dim(V_1)={\lambda}n_1$, - $\mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)=\mu_{{\alpha}_i}(E,V)$, - $(E_1,V_1)$ and $(E_2,V_2)$ are both ${\alpha}_i^+$-stable, - $\dim(V_1)$ and ${{\operatorname{rk}}}(E_1)$ satisfy the min-max criteria given in [(c)]{} and [(d)]{} of Lemma \[lem:vicente\](i). Similarly, let $W^-({\alpha}_i, {\lambda}, n_1; n,d,k)$ (abbreviated to $W^-_i({\lambda},n_1)$ whenever possible) denote the set of all destabilising patterns $$0\to (E_1,V_1)\to (E,V)\to (E_2,V_2) \to 0$$ in which - $(E,V)$ is ${\alpha}^-_i$-stable and of type ${(n,d,k)}$, - ${{\operatorname{rk}}}(E_1)=n_1$ and $\dim(V_1)={\lambda}n_1$, - $\mu_{{\alpha}_i}(E_1,V_1)=\mu_{{\alpha}_i}(E_2,V_2)=\mu_{{\alpha}_i}(E,V)$, - $(E_1,V_1)$ and $(E_2,V_2)$ are both ${\alpha}_i^-$-stable, - $\dim(V_1)$ and ${{\operatorname{rk}}}(E_1)$ satisfy the min-min criteria given in [(c)]{} and [(d)]{} of Lemma \[lem:vicente\](ii). Define $$W^+({\alpha}_i,n,d,k)=\bigsqcup_{{\lambda}<\frac{k}{n},\, n_1<n} W^+({\alpha}_i, {\lambda}, n_1; n,d,k),$$ $$W^-({\alpha}_i,n,d,k)=\bigsqcup_{{\lambda}>\frac{k}{n},\, n_1<n} W^-({\alpha}_i, {\lambda}, n_1; n,d,k).$$ We abbreviate these to $W^+_i$ and $W^-_i$ whenever possible. \[lem:wi\] Fix ${(n,d,k)}$ and also ${\alpha}_i$. Then each set $W^{\pm}_i({\lambda},n_1)$ is contained in a family of dimension bounded above by $$\begin{aligned} w^{\pm}_i({\lambda},n_1)&= \dim G({\alpha}_i^{\pm};n_1,d_1,k_1)&\mbox{} + \dim G({\alpha}_i^{\pm};n_2,d_2,k_2)\\ & & \mbox{} + \max{\dim {\operatorname{Ext}}^1((E_2,V_2),(E_1,V_1))}-1. \end{aligned}$$ Here $n=n_1+n_2$, $d=d_1+d_2$ and $k=k_1+k_2$, and the maximum is taken over all $(E_1,V_1)$, $(E_2,V_2)$ which satisfy the relevant part of Definition \[def:wi\]. Thus the set $W^{+}_i$ is contained in a family whose dimension is bounded above by the maximum of $w^{+}_i({\lambda},n_1)$ for all ${\lambda}<\frac{k}{n} $ and $\ n_1<n$. Similarly, the set $W^{-}_i$ is contained in a family whose dimension is bounded above by the maximum of $w^{-}_i({\lambda},n_1)$ for all ${\lambda}>\frac{k}{n} $ and $\ n_1<n$. [*Proof.*]{} In general the coherent systems moduli spaces do not support universal objects. In order to obtain families in the strict sense of the term, it is necessary to lift back from the moduli spaces to a level (for example, that of Quot schemes) on which families can be constructed. One can then do a dimensional calculation. In fact this gives the same answer is if we simply assumed that the moduli spaces support genuine families (for a similar calculation, see, for example, [@BGN Lemma 4.1]). Given this, the lemma follows at once from the definitions and Lemma \[lem:vicente\].$\Box$ Note that $G({\alpha}_i^+;n_1,d_1,k_1)=G_i(n_1,d_1,k_1)$ and $G({\alpha}_i^-;n_1,d_1,k_1)=G_{i-1}(n_1,d_1,k_1)$; the version used in the lemma appears more natural in this context. There are clearly surjective maps $$W^{\pm}_i\twoheadrightarrow G_i^{\pm}\ .$$ The maps may fail to be injective because a coherent system in $G_i^{\pm}$ may have more than one subsystem which satisfies the criteria on $(E_1,V_1)$ in Lemma \[lem:vicente\]. Nevertheless, we can use the dimension estimates on $W^{\pm}_i$ to estimate the codimension of $G_i^{\pm}$ in $G({\alpha}_i^{\pm};n,d,k)$ by $${\operatorname{codim}}G_i^{+} \ge \dim G({\alpha}^{+}_i;n,d,k) - \max\left\{ w^+_i({\lambda},n_1)\ \biggm|\ {\lambda}<\frac{k}{n}\ ,\ \ n_1<n \right\}$$ and $${\operatorname{codim}}G_i^{-} \ge \dim G({\alpha}^{-}_i;n,d,k) - \max\left\{ w^-_i({\lambda},n_1)\ \biggm|\ {\lambda}>\frac{k}{n}\ ,\ \ n_1<n \right\}.$$ It follows from (\[destab\]) and Proposition \[prop:filtration\](ii) that in our situation $${{\mathbb H}}^0_{21}={{\operatorname{Hom}}}((E_2,V_2),(E_1,V_1))=0.$$ When $G({\alpha}_i^{\pm};n,d,k)$, $G({\alpha}_i^{\pm};n_1,d_1,k_1)$ and $G({\alpha}_i^{\pm};n_2,d_2,k_2)$ have their expected dimensions, and $\mathbb H^2_{21}$ is zero for all relevant $(E_1,V_1)$ and $(E_2,V_2)$, we have $$\begin{aligned} {\operatorname{codim}}G_i^{+}&\geq&{\beta}(n,d, k)\nonumber\\ & &\mbox{}-\max\left\{ ({\beta}(n_1,d_1,k_1) + {\beta}(n_2,d_2,k_2) + C_{21}-1)\, \biggm|\, \frac{k_1}{n_1}<\frac{k}{n}\ ,\ \ n_1<n \right\}\nonumber \\ &=& \min\left\{\ C_{12}\ \biggm|\ \frac{k_1}{n_1}<\frac{k}{n}\ ,\ \ n_1<n \right\}\label{codim+}\end{aligned}$$ by Corollary \[cor:euler\]. Similarly $$\label{codim-} {\operatorname{codim}}G_i^- \geq \min\left\{\ C_{12}\ \biggm|\ \frac{k_1}{n_1}>\frac{k}{n}\ ,\ \ n_1<n \right\}.$$ Of course in general we have to allow for the fact that the moduli spaces may have dimensions greater than the expected ones and take into account the contribution from ${{\mathbb H}}^2$ in the computations of the actual dimensions. For later use, we state a very general result and then we particularise to a result that covers the cases considered in this paper. In general, we shall describe the process of going from $G({\alpha}^+_i;n,d,k)$ to $G({\alpha}^-_i;n,d,k)$ (or vice versa) as a [*flip*]{}, although it is not necessarily a flip in any technical sense. For all allowable values of $({\lambda},n_1)$, we denote the image of $W^+_i({\lambda},n_1)$ in $G^+_i$ by $G^+_i({\lambda},n_1)$. For any irreducible component $G$ of $G({\alpha}^+_i;n,d,k)$, we shall say that the flip is $({\lambda},n_1)$-[*good on*]{} $G$ if $G^+_i({\lambda},n_1)\cap G$ has positive codimension in $G$. A similar definition applies to irreducible components of $G({\alpha}^-_i;n,d,k)$. If a flip is $({\lambda},n_1)$-good on all irreducible components of both $G({\alpha}^+_i;n,d,k)$ and $G({\alpha}^-_i;n,d,k)$ and for all allowable values of $(n_1,{\lambda})$, we shall call it a [*good flip*]{}. \[lem:flip-good\] Let ${\alpha}_i$ be a critical value and suppose that - $n_1+n_2=n$, $d_1+d_2=d$, $k_1+k_2=k$, - $\frac{d_1}{n_1}+{\alpha}_i\frac{k_1}{n_1}= \frac{d_2}{n_2}+{\alpha}_i\frac{k_2}{n_2} =\frac{d}{n}+{\alpha}_i\frac{k}{n}$, - ${\lambda}=\frac{k_1}{n_1}<\frac{k}{n}$. Let $G$ be an irreducible component of $G({\alpha}^{+}_i;n,d,k)$ of excess dimension $e\ge0$. Let $\{S_t\}$ be a stratification of $$G({\alpha}^{+}_i;n_1,d_1,k_1){\times}G({\alpha}^{+}_i;n_2,d_2,k_2)$$ such that $\dim{{\mathbb H}}^2_{21}$ is constant on each $S_t$. Write $e_1$, $e_2$ for the excess dimensions of irreducible components $G^1$ of $G({\alpha}^{+}_i;n_1,d_1,k_1)$ and $G^2$ of $G({\alpha}^{+}_i;n_2,d_2,k_2)$. Then the flip at ${\alpha}_i$ is $({\lambda},n_1)$-good if $$\label{flipeq} C_{12}> \dim{{\mathbb H}}^2_{21}+e_1+e_2-e-{\operatorname{codim}}_{G^1{\times}G^2} (S_t\cap(G^1{\times}G^2))$$ for all $G^1$, $G^2$ and all $S_t$ such that there exist extensions (\[destab\]) satisfying the conditions of Lemma \[lem:vicente\](i) with $(E,V)\in G$ and $((E_1,V_1),(E_2,V_2))\in S_t\cap(G^1{\times}G^2)$. A similar result holds for $G({\alpha}^{-}_i;n,d,k)$ if we replace the condition ${\lambda}<\frac{k}{n}$ by ${\lambda}>\frac{k}{n}$ and Lemma \[lem:vicente\](i) by Lemma \[lem:vicente\](ii). [*Proof.*]{} We need to adjust the formulae (\[codim+\]) and (\[codim-\]) by allowing for all the obstructions. For this we use (\[C21\]) and recall that we have already noted that ${{\mathbb H}}^0_{21}=0$. $\Box$ \[cor:flip\] Suppose that, for every allowable choice of $(n_1,d_1,k_1)$ with $\frac{k_1}{n_1}<\frac{k}{n}$, $G({\alpha}_i^{\pm};n_1,d_1,k_1)$ and $G({\alpha}_i^{\pm};n_2,d_2,k_2)$ have the expected dimensions, and that stratifications $\{S_t^+\}$, $\{S_t^-\}$ of $$G({\alpha}^{+}_i;n_1,d_1,k_1){\times}G({\alpha}^{+}_i;n_2,d_2,k_2),\quad G({\alpha}^{-}_i;n_1,d_1,k_1){\times}G({\alpha}^{-}_i;n_2,d_2,k_2)$$ exist such that $\dim{{\mathbb H}}^2_{21}$ is constant on every stratum $S_t^+$ and $\dim{{\mathbb H}}^2_{12}$ is constant on every stratum $S_t^-$. Suppose further that $$\label{criteria} C_{12}>\dim{{\mathbb H}}^2_{21}-{\operatorname{codim}}S_t^+,\quad\mathrm{and}\quad C_{21}>\dim{{\mathbb H}}^2_{12}-{\operatorname{codim}}S_t^-$$ for every $(n_1,d_1,k_1)$ and every stratum $S_t^{\pm}$. Then the flip at ${\alpha}_i$ is good. [*Proof.*]{} The hypotheses give $e_1=e_2=0$ for every choice of $G^1$, $G^2$. The flip is therefore $({\lambda},n_1)$-good for ${\lambda}< \frac{k}{n}$ by Lemma \[lem:flip-good\]. Now note that interchanging the indices $12$ changes a destabilising pattern with ${\lambda}=\frac{k_1}{n_1}<\frac{k}{n}$ into one with ${\lambda}=\frac{k_2}{n_2}>\frac{k}{n}$ and vice-versa. So the second inequality in the statement shows that the flip is good for ${\lambda}>\frac{k}{n}$.$\Box$ Of course, one needs to prove (\[criteria\]) only for non-empty strata. Moreover, if the extension (\[destab\]) is trivial, $(E,V)$ cannot be ${\alpha}$-stable for any ${\alpha}$. So, for proving the first inequality, we may also assume that $\dim{\operatorname{Ext}}^1((E_2,V_2),(E_1,V_1))>0$, i.e. by (\[C21\]) $$C_{21}+\dim{{\mathbb H}}^2_{21}>0.$$ Similarly, for the second inequality, we may assume $$C_{12}+\dim{{\mathbb H}}^2_{12}>0.$$ Coherent systems with $k=1$ {#sec:k=1} =========================== We want to deal with applications of the theory developed so far to the case of coherent systems with few sections and also to the case of small rank. We devote the following sections to this task. We start by analysing the case $k=1$ and $n\geq 2$. The moduli space of coherent systems in this case coincides with the moduli space of pairs $(E,{\phi})$ which are ${\alpha}$-stable (see [@BG1]). The particular case $n=2$, $k=1$, $d>0$ has been studied thoroughly by Thaddeus [@Th], showing in particular that the spaces $G({\alpha};2,d,1)$ are irreducible and of the expected dimension $2g +d-2$. We assume that $g\ge2$ partly because of the complications of Remark \[rmk:g=0,1\] and partly because the proof fails for $g=0$. \[thm:k=1\] Let $g\ge2$. For $n>1$, the moduli spaces $G_i(n,d,1)$ are non-empty, smooth, irreducible and of the expected dimension ${\beta}=(n^2-n)(g-1)+d$. They are birationally equivalent for different values of $i$. The critical values are all of the form $\frac{s}{m}\in (0,\frac{d}{n-1})$ with $0<m<n$ and $0<s<d$. [*Proof.*]{} The smoothness property follows from Proposition \[prop:smooth\]. Theorem \[G\_L(k&lt;n)\] shows that the large ${\alpha}$ moduli space $G_L$ is irreducible and of the expected dimension. So it only remains to prove that all the moduli spaces are birationally equivalent for different values of ${\alpha}$. This follows at once when we check that the flips are good. By Corollary \[cor:flip\] we need only to verify the inequalities (\[criteria\]) for $k_1=0$, $k_2=1$, but we do need to know that all non-empty $G({\alpha}_i^{\pm};n_1,d_1,k_1)$ with $n_1<n$ and $k_1=0,1$ have the expected dimensions. For $k_1=0$, these spaces are the full moduli spaces, for which we know the result to be true. We can therefore proceed by induction on $n$. For the base case, we take the equivalent theorem for $n=1$, namely that $G(1,d,1)$ has dimension $d$. This is clear since $G(1,d,1)=S^dX$. We can therefore proceed to the inductive step. Note first that ${{\mathbb H}}^2_{21}=0$ by Lemma \[lem:hopf\] and ${{\mathbb H}}^2_{12}=0$ since $V_1=0$. The critical value ${\alpha}_i$ is given by $$\frac{d_1}{n_1}=\frac dn +\frac{{\alpha}_i}{n} =\frac{d_2}{n_2}+\frac{{\alpha}_i}{n_2},$$ i.e. $$\label{eqn:critical} {\alpha}_i=\frac{1}{n_1}(d_1n_2-d_2n_1)$$ We have by (\[dim-ext\]) $$C_{12}=n_1n_2(g-1)-d_2n_1+d_1n_2=n_1n_2(g-1)+n_1{\alpha}_i>0.$$ On the other hand $$C_{21}=n_1n_2(g-1)-d_1n_2+d_2n_1+d_1-n_1(g-1).$$ Now $$d_1n_2-d_2n_1=n_1{\alpha}_i<\frac{n_1d}{n-1},$$ which gives $d_1n_2-d_2n_1<d_1$. So $$C_{21}>n_1(n_2-1)(g-1)\ge0.$$ $\Box$ Coherent systems with $k=2$ {#sec:k=2} =========================== We look next at the case $k=2$. \[thm:k=2\] Let $X$ be a Petri curve of genus $g\geq 2$. Then we have - For $n=2$ the moduli spaces $G_i(2,d,2)$ are non-empty if and only if $d> 2$. They are irreducible and of the expected dimension $2d-3$. - For $n>2$ the moduli spaces $G_i(n,d,2)$ are non-empty if and only if $d>0$. They are always irreducible and of the expected dimension $(n^2-2n)(g-1)+2d-3$. [*Proof.*]{} We start by considering the moduli space $G_L$. Here the result follows from Theorem \[G\_L(k=n)\] when $n=2$ and from Theorem \[G\_L(k&lt;n)\] when $n>2$. It remains to prove that all the flips are good. Again we proceed by induction on $n$, noting that we already know that the moduli spaces for $k=0,1$ do have the expected dimensions. For the base case, we take the statement that the moduli spaces $G(1,d,2)$ have the expected dimensions. This is true by section \[subsec:alpha-small\] since we are assuming that the curve is Petri. Note incidentally that these spaces are not necessarily irreducible, but irreducibility is not needed for the argument. We now proceed to the inductive step. According to Corollary \[cor:flip\], we can restrict attention to the two cases $k_1=0$, $k_2=2$ and $k_1=k_2=1$, $n_1>n_2$. In each case we need to prove the inequalities (\[criteria\]). 1. $k_1=0$, $k_2=2$. The critical value is given by $$\frac{d_1}{n_1}=\frac{d_2}{n_2}+\frac{2{\alpha}_i}{n_2},$$ i.e. $${\alpha}_i=\frac{1}{2n_1}(d_1n_2-d_2n_1).$$ So $$C_{12}=n_1n_2(g-1)-d_2n_1+d_1n_2=n_1n_2(g-1)+2n_1{\alpha}_i>0.$$ By Proposition \[prop:C21\], ${{\mathbb H}}^2_{21}={\operatorname{Ext}}^2((E_2,V_2),(E_1,0))=H^0(E_1^*{\otimes}N_2 {\otimes}K)^*$, where $N_2$ is the kernel of ${{{\mathcal O}}}^2\to E_2$. If $N_2=0$ we have finished as we have already proved that $C_{12}>0$. When $N_2$ is non-zero we have an exact sequence $N_2\to {{{\mathcal O}}}^2\twoheadrightarrow L$ onto some line bundle $L$ with at least two sections. Therefore $\deg N_2=-\deg L \leq -\frac{g+2}2$, by section \[subsec:basicBN\], since the curve is Petri. So $$\deg (E_1^*{\otimes}N_2{\otimes}K) \leq -d_1+n_1(2g-2-\frac{g+2}2) <n_1(2g-2).$$ Then by Clifford’s theorem [@BGN] applied to the semistable bundle $E_1^*{\otimes}N_2{\otimes}K$, if $h^0(E_1^*{\otimes}N_2{\otimes}K)>0$ then $$\begin{aligned} \dim {{\mathbb H}}^2_{21} &\leq & \frac{-d_1+n_1(2g-2-\frac{g+2}2)}2 +n_1= -\frac{d_1}{2} + \frac{n_1}{4}(3g-2) \\ & <& \frac34 n_1(g-2) + n_1 < n_1n_2(g-2)+ n_1n_2+ 2n_1{\alpha}_i=C_{12}. \end{aligned}$$ On the other hand, ${{\mathbb H}}^2_{12}=0$ since $k_1=0$. Therefore we only need to prove that $C_{21}>0$. Now $$C_{21}=n_1n_2(g-1)+n_1 d_2-n_2d_1-2n_1(g-1)+2d_1.$$ If $n_2>2$ then we use the bound on the ${\alpha}$-range given by ${\alpha}_i<\frac{d_2}{n_2-2}$. Hence ${\alpha}_i< \frac{d_2}{n_2}+\frac{2}{n_2}{\alpha}_i=\frac{d_1}{n_1}$ and $$C_{21}=n_1(n_2-2)(g-1)+2d_1-2n_1{\alpha}_i>0.$$ If $n_2=2$ then $d_2> 2$ by induction hypothesis, and so $\frac{d_1}{n_1}= \frac{d_2}{2}+{\alpha}_i > {\alpha}_i$, whence $C_{21}=2d_1-2n_1{\alpha}_i >0$. If $n_2=1$ then $d_2\geq \frac{g+2}2$ since $E_2$ is a line bundle with at least two sections on a Petri curve. As $\frac{d_1}{n_1}>d_2$, we have $$\begin{aligned} C_{21} &=& -n_1(g-1)+n_1d_2+d_1 > 2n_1d_2-n_1(g-1)\\ &\geq & n_1(g+2-g+1)>0. \end{aligned}$$ So in all the cases $C_{21}>0$, as required. 2. $k_1=k_2=1$, $n_1>n_2$. The critical value is given by $$\frac{d_1}{n_1}+\frac{{\alpha}_i}{n_1}=\frac{d_2}{n_2}+\frac{{\alpha}_i}{n_2}= \frac{d}{n}+\frac{2}{n}{\alpha}_i.$$ i.e. $${\alpha}_i=\frac{1}{n_1-n_2}(d_1n_2-n_1d_2).$$ By Lemma \[lem:hopf\], we have ${{\mathbb H}}^2_{21}=0$ and ${{\mathbb H}}^2_{12}=0$. We compute $$\begin{aligned} C_{12}&=&n_1n_2(g-1)-n_1d_2+n_2d_1-n_2(g-1)+d_2-1=\\ &=&(n_1-1)n_2(g-1)+{\alpha}_i(n_1-n_2)+d_2-1>0, \\ C_{21} &=& n_1n_2(g-1)+n_1d_2-n_2d_1 -n_1(g-1)+d_1-1= \\ &=& (n_2-1)n_1(g-1)+ d_1-{\alpha}_i(n_1-n_2)-1. \end{aligned}$$ For $n_2>1$ we use the ${\alpha}$-range condition to get ${\alpha}_i<\frac{d_1}{n_1-1}$ and so $d_1-{\alpha}_i(n_1-n_2)>d_1 - \frac{d_1}{n_1-1}(n_1-n_2)\geq 0$ and thus $C_{21}>0$. In the case $n_2=1$, we have $C_{21}=n_1d_2-1>0$. $\Box$ \[rem:G(2,2,2)\] In the case $n=d=k=2$, ${\widetilde}{G}_L$ consists only of reducible coherent systems and it is irreducible and of dimension $2$ by Theorem \[G\_L(k=n)\]. It is easy to see that in this case there are no flips. Coherent systems with $n=2$ {#sec:n=2} =========================== Now we are going to deal with coherent systems of rank $2$. Our results in this case are partial. This is due to two reasons. On the one hand our understanding of the moduli space $G_L$ of coherent systems for large values of the parameter ${\alpha}$ for $k\geq 4$ is very limited, in particular we do not know whether these spaces are irreducible and of the expected dimension. On the other hand we only manage to check that the flips are good for $k\leq 4$. We need a preliminary result on rank $1$ coherent systems. \[lem:St\] Let $X$ be a Petri curve of genus $g\geq 2$. Consider in $G(1,d,k)$ the stratification given by the sets $S_t=\{ (L,V) \in G(1,d,k)\; |\; h^0(L)=t\}$. Then - If $d \leq g-1+k$ then the number of sections $h^0(L)$ of a generic $(L,V) \in G(1,d,k)$ is $k$, and ${\operatorname{codim}}S_{k+j}=j(g-d-1+k+j)$, when non-empty. - If $d \geq g-1+k$ then the number of sections $h^0(L)$ of a generic $(L,V) \in G(1,d,k)$ is $p=d-g+1$, and ${\operatorname{codim}}S_{p+j}=j(d-g+1-k+j)$, when non-empty. [*Proof.* ]{} Let $p$ be the number of sections of a generic $(L,V) \in G(1,d,k)$. Then it must be $\dim G(1,d,k)=\dim G(1,d,p)+\dim {\operatorname{Gr}}(k,p)$. By an easy computation it follows that either $p=d-g+1$ or $p=k$. If $d<g-1+k$ then it must be $p=k$ and ${\operatorname{codim}}S_{k+j}=\dim G(1,d,k)-\dim G(1,d,k+j)- \dim {\operatorname{Gr}}(k,k+j)$. If $d \geq g-1+k$ then ${\operatorname{codim}}S_{d-g+1}=0$ so $p=d-g+1$. The computation of ${\operatorname{codim}}S_{p+j}$ is left to the reader. $\Box$ Now we focus on the study of $G_i(2,d,k)$ for $k>0$. The expected dimension is ${\beta}(2,d,k)=(4-2k)g + k d -k^2+2k- 3$. For $k=1$ this has been treated in section \[sec:k=1\] and for $k=2$ in section \[sec:k=2\]. So we may restrict to the case $k>2$. By Lemma \[lem:degree\] it must be $d>0$ for stable objects to exist. \[thm:n=2\] Let $X$ be a Petri curve of genus $g\geq 2$. Then - For $k=2$ the moduli spaces $G_i(2,d,2)$ are non-empty if and only if $d> 2$. They are irreducible and of the expected dimension ${\beta}=2d-3$. - For $k=3$ the moduli spaces $G_i(2,d,3)$ are non-empty if and only if $d\geq \frac{2g+6}3$. They are always of the expected dimension ${\beta}=3d-2g-6$ and irreducible when ${\beta}>0$. - For $k=4$ the moduli spaces $G_i(2,d,4)$ are birational to each other. [*Proof.*]{} We start by considering the moduli space $G_L$. Here the result follows from Theorem \[G\_L(k=n)\] for $k=2$ and from Theorem \[thm:dual-span\] for $k=3$. Let now $k=2$, $3$ or $4$ and we will prove that the flips are good. By Corollary \[cor:flip\] we have to prove the inequalities for $n_1=n_2=1$ and all possible choices of $k_1<\frac{k}{2}$, since the moduli spaces of coherent systems of type $(1,d',k')$ have the expected dimension for a Petri curve, by section \[subsec:alpha-small\]. As $k\leq 4$ we have that $k_1=0$ or $1$. More in general, let $k\geq 2$ be an integer, and consider extensions as in of the form $(L_1,V_1) \to (E,V) \to (L_2,V_2)$ where $n_1=n_2=1$ and $k_1<\frac{k}{2}$ satisfying $k_1\leq 1$. Then we are going to prove that the inequalities are satisfied. By Lemma \[lem:flip-good\] this implies that the flip is $({\lambda}, 1)$-good on $G({\alpha}^+_i;2,d,k)$ for ${\lambda}=0,1$ and $({\lambda},1)$-good on $G({\alpha}_i^-;2,d,k)$ for ${\lambda}=k,k-1$. The critical value ${\alpha}_i$ is given by $$d_1+k_1\, {\alpha}_i=\frac d2+\frac k2{\alpha}_i= d_2 +k_2\, {\alpha}_i,$$ i.e. $${\alpha}_i=\frac{d_1-d_2}{k_2-k_1}.$$ We start by proving the second inequality in . In this case Lemma \[lem:hopf\] implies that ${{\mathbb H}}_{12}^2=0$ since $k_1\leq 1$. By Theorem \[thm:BN\], in order for coherent systems of type $(1,d_2,k_2)$ to exist we must have $$\label{eqn:T} d_1>d_2 \geq \frac{k_2-1}{k_2}g +k_2-1.$$ We compute $$\begin{aligned} C_{21}&=&g-1+d_2-d_1+k_2(d_1-g+1-k_1) \\ &=&d_2 +(k_2-1)(d_1-g+1-k_1)-k_1 \\ &\geq& k_2\left(\frac{k_2-1}{k_2}g +k_2-1\right) +(k_2-1)(-g+2-k_1)-k_1\\ &=& (k_2-k_1-1)k_2+2(k_2-1) \geq 2k_1 > 0.\end{aligned}$$ Now we prove the first inequality in . We have $$\begin{aligned} C_{12}&=&g-1+d_1-d_2+k_1(d_2-g+1-k_2)\\ &\geq& g-1+1 +k_1\left( \frac{k_2-1}{k_2}g -g\right)=\frac{k_2-k_1}{k_2}g>0.\end{aligned}$$ If ${{\mathbb H}}^2_{21}=0$ then we have finished. Otherwise, Lemma \[lem:hopf\] gives the bound $$\dim {{\mathbb H}}^2_{21}\leq (k_2-1)(h^0(L_1^*{\otimes}K)-1).$$ We stratify $G(1,d_1,k_1)$ by using the subsets defined in Lemma \[lem:St\]. Let $S_t$ be the subspace of those $(L_1,V_1)\in G(1,d_1,k_1)$ with $h^0(L_1)=t$. It only remains to check that $C_{12}>\dim {{\mathbb H}}^2_{21}-{\operatorname{codim}}S_t$ at the points in $S_t$. Suppose first that $d_1> g-1+k_1$. Lemma \[lem:St\] says that the generic number of sections $h^0(L_1)$ of an element $(L_1,V_1)\in G(1,d_1,k_1)$ is $p=d_1-g+1$ and that ${\operatorname{codim}}S_{p+t}=t(d_1-g+1-k_1+t)$. Also $h^0(L_1^*{\otimes}K)=t$ at a point in $S_{p+t}$. Suppose that $\dim {{\mathbb H}}^2_{21}-{\operatorname{codim}}S_{p+t}>0$ since otherwise there is nothing to prove. So $$\begin{aligned} \label{eqn:n=2.case1} \dim {{{\mathbb H}}}^2_{12}-{\operatorname{codim}}S_{p+t} &\leq& (k_2-1)(t-1)- t(d_1-g+1-k_1+t) \nonumber \\ &\leq & t(g-d_1-2+k-t) \\ &\leq& (k_2-1)(g-d_1+k-3), \nonumber\end{aligned}$$ since it must be $1\leq t \leq k_2-1$ for the second line to be non-negative. In the other case, $d_1\leq g-1+k_1$, the generic number of sections of $L_1$ is $p=k_1$ and ${\operatorname{codim}}S_{p+t}=t(g-1-d_1+k_1+t)$. Since $h^0(L_1^*{\otimes}K)=g-1-d_1+k_1+t$ at a point in $S_{p+t}$, we have $$\begin{aligned} \label{eqn:n=2.case2} \dim {{{\mathbb H}}}^2_{12}-{\operatorname{codim}}S_{p+t} &\leq& (k_2-1)(g-1-d_1+k_1+t-1)- t(g-1-d_1+k_1+t) \nonumber \\ &\leq& (k_2-1-t)(g-1-d_1+k_1+t) \\ &\leq& (k_2-1)(g-d_1+k-3). \nonumber\end{aligned}$$ So using either or it only remains to prove that $$g-1+d_1-d_2+k_1(d_2-g+1-k_1k_2)>(k_2-1)(g-d_1+k-3).$$ Rearranging terms this is equivalent to $$k_2d_1+(k_1-1)d_2>(k-2)g+(k-2)(k_2-2)+k_1k_2.$$ Using it suffices to show that $$k_2+(k-1)\left(\frac{k_2-1}{k_2}g +k_2-1\right)>(k-2)g+(k-2)(k_2-2)+k_1k_2.$$ This holds for $k_1=0$ or $1$ and $k_2=k-k_1$. $\Box$ \[rem:noflips\] In order to have any flips, imposes the condition $$d\geq 2\left( \frac{k_2-1}{k_2}g +k_2-1\right) +1,$$ for some $k_2>\frac k2$. This implies that $d \geq 2(\frac g2 +1)+1=g+3$. So when $d\leq g+2$ there are no flips for $G({\alpha};2,d,k)$. Checking whether the flips are good when $k_1>1$ is difficult in general. Nonetheless we have the following positive result for the case $k_1=2$. \[thm:k=5,6\] Let $X$ be a Petri curve of genus $g\geq 2$. Consider the moduli spaces of coherent systems of type $(2,d,k)$ with $k>4$, and let ${\alpha}_i$ be a critical value corresponding to coherent subsystems with $n_1=1$ and $k_1=2$. Then the flip at ${\alpha}_i$ is $({\lambda}=k-2,n_1=1)$-good on $G({\alpha}_i^-;2,d,k)$. In particular, when $k=5$ or $k=6$, if the moduli space $G_0(2,d,k)$ is non-empty then $G_L(2,d,k)$ is non-empty also. [*Proof.*]{} By Lemma \[lem:flip-good\] we need to check that for $n_1=n_2=1$ and $k_1=2$ we have the inequality $C_{21}>\dim {{\mathbb H}}^2_{12}-{\operatorname{codim}}S_t$ at the points of $S_t$, for a suitable stratification $\{S_t\}$ of $G({\alpha}^{-}_i;1,d_1,2){\times}G({\alpha}^{-}_i;1,d_2,k-2)$. By the proof of Theorem \[thm:n=2\], we already know that $C_{21}\geq 2k_1=4>0$. We distinguish two cases. First suppose that $d_2 \geq g+k-3$. We consider the stratification of $G(1,d_2,k-2)$ given by $S_t=\{ (L_2,V_2)\; |\; h^0(L_2)=t\}$. By Lemma \[lem:hopf\] we know that $\dim {{\mathbb H}}^2_{12} \leq h^0(L_2^* {\otimes}K)-1$. The generic number of sections of $L_2$ for an element $(L_2,V_2) \in G(1,d_2,k-2)$ is $p=d_2-g+1$. Using Lemma \[lem:St\] we have that for any $t\geq 0$, at a point in $S_{p+t}$, $$\dim {{\mathbb H}}^2_{12} -{\operatorname{codim}}S_{p+t} \leq t-1 -t(d_2-g+1-k+2+t) \leq 0<C_{21},$$ as required. The other case is $d_2 <g+k-3$. Then the generic number of sections of $L_2$ for an element $(L_2,V_2)$ is $p=k-2$. So for any $t\geq 1$ we have at a point in $S_{p+t}$, $$\dim {{\mathbb H}}^2_{12} -{\operatorname{codim}}S_{p+t} \leq (1-t)(g-1-d_2+k-2+t) \leq 0<C_{21}.$$ This means that we may restrict to the case where $(L_2,V_2)$ lies in the open subset $S_{k-2}\subset G(1,d_2,k-2)$. A coherent system $(L_2,V_2)\in S_{k-2}$ is determined by its underlying line bundle $L_2$. Now consider the exact sequence $N_1 \to {{{\mathcal O}}}^2\to L_1$, where $N_1$ is the kernel. Then $N_1$ is a line bundle of degree $-l$, say. One clearly has $l\leq d_1$. Define the stratification of $S_{k-2}$ given by the subsets $$T_t=\{ (L_2,V_2) \in S_{k-2} \; |\; h^0(N^*_1{\otimes}L_2)=t\}.$$ Clearly $\dim T_t \leq \dim G(1,d_2+l,t)$. Also we stratify $G(1,d_1,2)$ by the subsets $W_l$ of those coherent systems $(L_1,V_1)$ such that the image of the map ${{{\mathcal O}}}^2\to L_1$ is a line bundle of degree $l$. Generically this map is surjective, so $W_{d_1}$ is an open dense subset. We start by considering the stratum $W_{d_1}\subset G(1,d_1,2)$. An easy calculation using that $d_1>d_2\geq \frac{k-3}{k-2}g +k-3$ (see ) and $k\geq 5$ shows that $$\dim G(1,d,d-g+4)< \dim G(1,d_2,k-2).$$ Therefore the generic number of sections of the line bundle $N^*_1{\otimes}L_2$, for $(L_2,V_2) \in S_{k-2}$ and $(L_1,V_1) \in W_{d_1}$, is $p\leq d-g+3$. Note that in particular $d-g+3 \geq 0$. At a point of $T_t\subset S_{k-2}$ with $t\leq d-g+3$ we have $$\dim {{\mathbb H}}^2_{12}=\dim H^0(L_2^*{\otimes}N_1{\otimes}K)=g-1-d+t \leq 2< C_{21}.$$ At a point of $T_{d-g+4+t}$ with $t\geq 0$ we have, $$\begin{aligned} \dim {{\mathbb H}}^2_{12} &- & {\operatorname{codim}}T_{d-g+4+t}=g-1-d+d-g+4+t - {\operatorname{codim}}T_{d-g+4+t} \\ &\leq & 3+t- \dim G(1,d_2,k-2) +\dim G(1,d,d-g+4+t) \\ &<& 3+t-t(d-g+7+t) \leq 3 \leq C_{21}. \end{aligned}$$ For the stratum $W_{d_1-1}$ we use that $\dim G(1,d-1,d-g+3)< \dim G(1,d_2,k-2)$ to prove that the generic number of sections of $N^*_1{\otimes}L_2$, for $(L_2,V_2) \in S_{k-2}$ and $(L_1,V_1) \in W_{d_1-1}$, is $p\leq d-g+2$. Working as before we get that $$\dim {{\mathbb H}}^2_{12}- {\operatorname{codim}}T_{d-g+2+t} \leq 2<C_{21},$$ for $t\geq 0$. Finally consider the strata $W_l\subset G(1,d_1,2)$ where $l\leq d_1-2$. It is easy to check that ${\operatorname{codim}}W_l=d_1-l \geq 2$. We have that $l\geq \frac{g+2}2$, since $N^*_1$ has two sections and $X$ is a Petri curve. Now an easy calculation shows that $$\dim G(1,d_2+l, d_2+l-g+7)<\dim G(1,d_2,k-2).$$ Therefore the generic number of sections of $N^*_1{\otimes}L_2$ is $p\leq d_2+l-g+6$. At a point $((L_1,V_1),(L_2,V_2)) \in W_l{\times}T_t \subset G(1,d_1,2) {\times}S_{k-2}$ with $t\leq d_2+l-g+6$ we have $$\dim {{\mathbb H}}^2_{12} -{\operatorname{codim}}W_l \leq g-1-(d_2+l) +t -2 \leq 3<C_{21}.$$ At a point of $W_l{\times}T_{d_2+l-g+7+t}$ with $t\geq 0$, we have $$\dim {{\mathbb H}}^2_{12}-{\operatorname{codim}}W_l-{\operatorname{codim}}T_{d_2+l-g+7+t} < 6+t-t(d_2+l-g+13+t)-2 \leq 4\leq C_{21},$$ concluding that in all cases the flip is $(k-2,1)$-good. $\Box$ Coherent systems with $k=3$ {#sec:k=3} =========================== Now we shall work out the case of the moduli spaces $G_i(n,d,3)$ of coherent systems with $k=3$ sections and rank $n>1$. Note that the case $n=2$ follows from section \[sec:n=2\]. We need a preliminary result, similar in spirit to Lemma \[lem:St\] but for the case of bundles of higher rank. This result is somewhat restricted as the only input is information on coherent systems with at most $2$ sections. \[lem:bound\] Let $d\leq n(g-1)$. Stratify the moduli space $M(n,d)$ by $S_t=\{F\in M(n,d)\; | \; h^0(F)=t\}$. Then $2h^0(F^*{\otimes}K)-{\operatorname{codim}}S_t\leq 2(n(g-1)-d)+1$ at a point in $S_t$. [*Proof.* ]{} For $F\in S_0$ we have $2h^0(F^*{\otimes}K)= 2(n(g-1)-d)$. For $t=1$ we have, by Proposition \[small-alpha\], $\dim S_1\leq \dim G({\alpha};n,d,1)= (n^2-n)(g-1)+d$, where ${\alpha}>0$ is a small number. Hence ${\operatorname{codim}}S_1\geq n^2(g-1)+1- (n^2-n)(g-1)+d=n(g-1)-d+1$ and $$2h^0(F^*{\otimes}K)-{\operatorname{codim}}S_1 \leq 2(n(g-1)-d+1)-n(g-1)-d+1=n(g-1)-d+1.$$ For $t\geq 2$ we have that $\dim S_t +\dim {\operatorname{Gr}}(2,t) \leq \dim G({\alpha};n,d,2)=(n^2-2n)(g-1)+2d-3$, using Theorem \[thm:k=2\]. So we deduce that $$\begin{aligned} & &2h^0(F^*{\otimes}K)- {\operatorname{codim}}S_t \leq \\ & & \leq 2(n(g-1)-d+t)-n^2(g-1)-1+(n^2-2n)(g-1)+2d-3-2(t-2)=0. \end{aligned}$$ The statement follows. $\Box$ Now we obtain Clifford bounds type results for coherent systems. The following results are not sharp, but they are good enough for our purposes in this section. In the next two Lemmas, $X$ is [*any*]{} curve of genus $g\geq 2$. \[clif-lem\] Suppose $(E,V)$ is an ${\alpha}$-semistable coherent system with $\mu(E) \geq 2g-2$ and $h^1(E) >0$. Then $$h^0(E) \leq \frac{d}{2} +n + (n-1)k{\alpha}.$$ [*Proof.* ]{} We want to bound $h^0(E)=h^1(E)+ d+n(1-g)$. Put $N=h^1(E)=h^0(E^*{\otimes}K)$. Then there are $N$ linearly independent maps $E \to K$. For any divisor $D$ on $X$ of degree $[\frac{N-1}{n}]$ we may find a non-zero map $E\to K(-D)$. The ${\alpha}$-semistability implies then $$\begin{aligned} & & \frac{d}{n}+{\alpha}\frac{k}{n} \leq 2g-2-\deg D +{\alpha}k, \\ & & \left[\frac{N-1}{n}\right]\leq 2g-2-\frac{d}{n}+{\alpha}k\frac{n-1}{n}, \\ & & N \leq n(2g-2)-d+{\alpha}k (n-1)+n, \\ & & h^0(E) \leq n(g-1)+n +{\alpha}k (n-1) \leq \frac{d}{2}+n+{\alpha}k(n-1).\end{aligned}$$ $\Box$ \[lem:clif-cs\] Let $(E,V)$ be an ${\alpha}$-semistable coherent system with $0\leq\mu(E)<2g-2$. Then $$h^0(E) \leq \frac{d}{2} +n +(n-1)k{\alpha}.$$ [*Proof.* ]{} For $n=1$ the last term is dropped and the result is the usual Clifford theorem for line bundles. Also for ${\alpha}>0$ very small, $E$ is a semistable bundle and the result follows by the Clifford theorem in [@BGN]. We also may suppose that $k>0$. Note that the bound weakens as we increase ${\alpha}$, so it is enough to check what happens when we cross a critical value ${\alpha}_i$ to the coherent systems $(E,V)$ that are ${\alpha}_i$-semistable but not ${\alpha}_i^-$-semistable. Then there is a pattern $$0\to (E_1,V_1) \to (E,V) \to (E_2, V_2) \to 0,$$ with $k_1/n_1 < k/n < k_2/n_2$, $\mu_{{\alpha}_i}(E_1, V_1)= \mu_{{\alpha}_i}(E_2, V_2)=\mu_{{\alpha}_i}(E, V)$ and where $(E_1,V_1)$, $(E_2, V_2)$ are ${\alpha}_i$-semistable. Therefore $k_2>0$, and by Lemma \[lem:degree\] we have $d_2 \geq 0$. Hence $0\leq d_2/n_2< d/n<2g-2$, and by induction, $$\label{clif2} h^0(E_2) \leq \frac{d_2}{2} +n_2 +(n_2-1) k_2{\alpha}_i.$$ There are three cases to consider: - $d_1/n_1 <2g-2$. As $d_1/n_1 > d/n \geq 0$, we apply induction to get $$h^0(E_1) \leq \frac{d_1}{2} +n_1 +(n_1-1)k_1{\alpha}_i\, ,$$ which together with gives the result using that $h^0(E)\leq h^0(E_1)+h^0(E_2)$. Note that $(n_1-1)k_1 + (n_2-1)k_2 \leq (n-1)k$, whenever $n=n_1+n_2$, $0<n_1,n_2<n$ and $k=k_1+k_2$, $k_1,k_2\geq 0$. - $d_1/n_1 \geq 2g-2$ and $h^1(E_1) \neq 0$. We use Lemma \[clif-lem\] to conclude $$h^0(E_1) \leq \frac{d_1}{2} +n_1 +(n_1-1)k_1{\alpha}_i\, ,$$ and the result follows as in the previous case. - $d_1/n_1 \geq 2g-2$ and $h^1(E_1) =0$. Then $h^0(E_1)=d_1+n_1(1-g)$. We have $$h^0(E_1) =\frac{d_1}{2} + \frac{n_1}{2}\left[ \frac{d}{n}+ {\alpha}_i\left(\frac{k}{n}-\frac{k_1}{n_1}\right)\right]+ n_1(1-g) <$$ $$<\frac{d_1}{2}+ \frac{n_1}{2}(2g-2) +n_1(1-g) + {\alpha}_i \frac{n_1k}{2n}< \frac{d_1}{2}+n_1+{\alpha}_i\frac{(n-1)k}{2n},$$ from which we get again the result since $\frac{(n-1)k}{2n} +(n_2-1)k_2 \leq (n-1)k$. $\Box$ \[thm:k=3\] Let $X$ be a Petri curve of genus $g\geq 2$. Then we have - For $n=2$ the moduli spaces $G_i(2,d,3)$ are non-empty if and only if $d\geq \frac{2g+6}3$. They are always of the expected dimension ${\beta}=3d-2g-6$ and irreducible when ${\beta}>0$. - For $n=3$ the moduli spaces $G_i(3,d,3)$ are non-empty if and only if $d>3$. They are irreducible and of the expected dimension ${\beta}=3d-8$. - For $n>3$ the moduli spaces $G_i(n,d,3)$ are non-empty if and only if $d>0$ and $d\geq n-(n-3)g$. They are always irreducible and of the expected dimension ${\beta}=(n^2-3n)(g-1)+3d-8$. [*Proof.*]{} The case $n=2$ follows from Theorem \[thm:n=2\], so we may restrict to the case $n\geq 3$. The moduli space $G_L$ for the largest possible values of the parameter satisfies the statement of the Theorem, using Theorem \[G\_L(k=n)\] for the case $k=n=3$ and Theorem \[G\_L(k&lt;n)\] for the case $n>k=3$. It remains to check that the flips are good. We proceed by induction on $n$, noting that we already know that the moduli spaces for $k=0,1,2$ have the expected dimensions for a Petri curve. According to Corollary \[cor:flip\], we have two cases: $k_1=0$, $k_2=3$ and $k_1=1$, $k_2=2$. 1. $k_1=0$, $k_2=3$. The critical value ${\alpha}_i$ is given by $$\frac{d_1}{n_1}=\frac{d_2}{n_2}+\frac{3}{n_2}{\alpha}_i= \frac{d}{n}+\frac{3}{n}{\alpha}_i\, .$$ i.e. $${\alpha}_i=\frac{d_1n_2-d_2n_1}{3n_1}\, .$$ We start by proving the first inequality in . We have $$C_{12}=n_1n_2(g-1)-n_1d_2+n_2d_1=n_1n_2(g-1)+3n_1{\alpha}_i>0.$$ Now Lemma \[lem:hopf\] implies $\dim {{\mathbb H}}^2_{21} \leq 2(h^0(E_1^*{\otimes}K)-1)$ or else ${{\mathbb H}}^2_{21}=0$. There are two cases: 1. If $d_1 \leq n_1(g-1)$ then we use Lemma \[lem:bound\]. Define the stratification given by $S_t=\{E_1\in M(n_1,d_1)\; |\; h^0(E_1)=t\}$. Then $$2h^0(E_1^*{\otimes}K)-{\operatorname{codim}}S_t\leq 2(n_1(g-1)-d_1)+1.$$ Hence $C_{12}> \dim {{\mathbb H}}^2_{21}-{\operatorname{codim}}S_t$ is implied by $$n_1(n_2-2)(g-1)+2d_1+3n_1{\alpha}_i>-1.$$ For $n_2\geq 2$ this obviously holds. For $n_2=1$ we have that $$\qquad -n_1(g-1)+2d_1+3n_1{\alpha}_i=-n_1(g-1)+2d_1+d_1-n_1d_2=$$ $$\qquad =3d_1-n_1(g-1+d_2)\geq n_1(2d_2-g+1)>-1,$$ using that $\frac{d_1}{n_1}>d_2\geq \frac{2g+6}3$, the last inequality being necessary for the existence of coherent systems of type $(1,d_2,3)$ on a Petri curve. 2. If $d_1 > n_1(g-1)$ then we use Clifford theorem for the stable bundle $E_1^*{\otimes}K$. So either $h^0(E_1^*{\otimes}K)=0$ in which case there is nothing to prove, or $$h^0(E_1^*{\otimes}K)\leq \frac{n_1(2g-2)-d_1}{2}+n_1,$$ whence $\dim {{\mathbb H}}^2_{21}\leq 2n_1g-d_1-2$. The inequality $C_{12} >2n_1g-d_1-2$ is equivalent to $$n_1(n_2-2)(g-2) + n_1(n_2-4) +d_1+3n_1{\alpha}_i >-2.$$ For $n_2\geq 4$ this is obviously true. For $n_2=3$ it must be $d_2>3$ by induction hypothesis, so $\frac{d_1}{n_1} > 1+{\alpha}_i$ and $d_1 -n_1>0$, which yields the result. For $n_2=2$ we have $\frac{d_1}{n_1} > \frac{d_2}2 \geq 2$ as $d_2\geq \frac{2g+6}3$, by induction hypothesis. So $-2n_1+d_1 >0$ and we are done. For $n_2=1$ and $g\leq 5$ we have that $\frac{d_1}{n_1} >d_2\geq \frac{2g+6}3$ implies $\frac{d_1}{n_1}>d_2 \geq g+1$ and hence $$-n_1(g-2)-3n_1+d_1+3n_1{\alpha}_i> d_1-n_1(g+1)>-2,$$ as required. The same argument covers the case $n_2=1$ and $d_1\geq n_1(g+1)$. Finally the case $n_2=1$, $n_1(g-1)<d_1<n_1(g+1)$ and $g\geq 6$ requires a special treatment. We use the improvement of Clifford theorem given in [@M3 Theorem 1]. Since $2+\frac{2}{g-4} \leq g-3 <2g-2-\frac{d_1}{n_1}<g-1$ and the curve is Petri, we have $$h^0(E_1^*{\otimes}K)\leq \frac{n_1(2g-2)-d_1}{2},$$ which gives $\dim {{\mathbb H}}^2_{21}\leq 2n_1(g-1)-d_1-2$, and hence $$C_{12}=n_1(g-1)+3n_1{\alpha}_i > \dim {{\mathbb H}}^2_{21}.$$ Now we pass on to prove the second inequality in . In this case ${{\mathbb H}}^2_{12}=0$. We compute $$\begin{aligned} C_{21}&=&n_1n_2(g-1)-3n_1 {\alpha}_i+3(d_1-n_1(g-1)) \\ &=&n_1(n_2-3)(g-1)+3d_1-3n_1{\alpha}_i. \end{aligned}$$ We have the following cases: 1. If $n_2>3$ then ${\alpha}_i<\frac{d_2}{n_2-3}$. Computing we obtain that ${\alpha}_i<\frac{d_2}{n_2}+\frac{3}{n_2}{\alpha}_i=\frac{d_1}{n_1}$ and thus $C_{21}>0$. 2. If $n_2=3$ then $C_{12}=3d_1-3n_1{\alpha}_i =d_2n_1 >0$. 3. If $n_2=2$ then $\frac{d_1}{n_1}>\frac{d_2}2$. As $d_2\geq \frac{2g+6}{3}$ by induction hypothesis, we have $$\begin{aligned} \qquad\qquad C_{21}&=&-n_1(g-1)+3d_1-2d_1+d_2n_1=n_1(d_2-g+1)+d_1 \\ &>&n_1 \left(\frac32 d_2-g+1\right) \geq n_1(g+3-g+1)>0. \end{aligned}$$ 4. If $n_2=1$ then $d_2\geq \frac{2g+6}{3}$ in order to have stable coherent systems of type $(1,d_2,3)$ on a Petri curve. Also $\frac{d_1}{n_1}>d_2$, so $$\begin{aligned} \qquad\qquad C_{21}&=&-2n_1(g-1)+3d_1-d_1+d_2n_1=n_1(d_2-2g+2)+2d_1 \\ &>& n_1(3d_2-2g+2) \geq n_1(2g+6-2g+2)>0. \end{aligned}$$ 2. $k_1=1$, $k_2=2$. The critical value is given by $$\frac{d_1}{n_1}+\frac{1}{n_1}{\alpha}_i=\frac{d_2}{n_2}+\frac{2}{n_2}{\alpha}_i= \frac{d}{n}+\frac{3}{n}{\alpha}_i\, ,$$ i.e. $${\alpha}_i=\frac{d_2n_1-d_1n_2}{n_2-2n_1}\, .$$ It must be $n_2-2n_1 \neq 0$. We start proving the second inequality in . We have ${{\mathbb H}}^2_{12}=0$ and $$C_{21}=n_1n_2(g-1)+n_1d_2-n_2d_1+2(d_1-n_1(g-1)-1).$$ We have the following cases: 1. $n_2-2n_1>0$. Then $C_{21}=n_1(n_2-2)(g-1) +{\alpha}_i(n_2-2n_1)+2d_1-2>0$ since $d_1>0$ and $n_2>2$. 2. $n_2-2n_1<0$ and $n_2\geq 2$. Then ${\alpha}_i=\frac{d_1n_2-d_2n_1}{2n_1-n_2} <\frac{d_1}{n_1-1}$ implies that ${\alpha}_i(2n_1-n_2)<2d_1-d_2$. So $$\begin{aligned} \qquad C_{21} & =& n_1(n_2-2)(g-1)-{\alpha}_i(2n_1-n_2)+2d_1-2 \\ &> & n_1(n_2-2)(g-1)+d_2-2 \geq 0. \end{aligned}$$ (Recall that in the particular case $n_2=2$ we must have $d_2> 2$.) 3. $n_2-2n_1<0$ and $n_2=1$. Using that $\frac{d_1}{n_1}>d_2$ and $d_2\geq \frac{g+2}{2}$, we have $$\begin{aligned} \qquad C_{21}&=& n_1(d_2-g+1)+d_1-2>n_1(2d_2-g+1)-2 \\ &\geq & n_1(g+2-g+1)-2=3n_1-2>0. \end{aligned}$$ Now we pass on to prove the first inequality in . We have $$C_{12}=n_1n_2(g-1)-n_1d_2+n_2d_1+(d_2-n_2(g-1)-2).$$ On the other hand, either ${{\mathbb H}}^2_{21}=0$ or else Lemma \[lem:hopf\] and Lemma \[clif-lem\] or Lemma \[lem:clif-cs\] imply that $$\label{eqn:k=3.last} \dim {{\mathbb H}}^2_{21}\leq h^0(E_1^*{\otimes}K)-1 \leq n_1g -\frac{d_1}{2} +(n_1-1){\alpha}_i-1.$$ We have the following cases: - $2n_1-n_2>0$. As we are supposing $n\geq 3$ it follows that $n_1>1$. Then $$\qquad \qquad C_{12}=(n_1-1)n_2(g-1) +{\alpha}_i(2n_1-n_2)+d_2-2\geq 1+1+1-2 >0.$$ We need to prove that $C_{12}>\dim {{\mathbb H}}^2_{21}$ using . 1. $n_2=1$, $n_1\geq 2$. Then $C_{12}> n_1g-\frac{d_1}2+(n_1-1){\alpha}_i-1$ is equivalent to $n_1{\alpha}_i +d_2+\frac{d_1}2>n_1+g$. In order for coherent systems of type $(1,d_2,2)$ to exist on a Petri curve, it is necessary that $d_2\geq \frac{g+2}2$. Also $$d_1>n_1d_2 \geq n_1\frac{g+2}2\geq g+2n_1-2.$$ Easily we get the result. 2. $n_2=2$, $n_1\geq 2$. Then $C_{12}>n_1g-\frac{d_1}2+(n_1-1){\alpha}_i-1$ is equivalent to $$(n_1-2)(g-2)+{\alpha}_i(n_1-1)+d_2 +\frac{d_1}2>3.$$ This holds since $d_2\geq 3$, for a stable coherent system of type $(2,d_2,2)$ to exist. 3. $n_2>2$, $n_1\geq 2$. Then generically the map ${{{\mathcal O}}}^2\to E_2$ has no kernel, for $(E_2,V_2)\in G_i(n_2,d_2,2)$. This happens since in $G_L(n_2,d_2,2)$ all coherent systems have this property by Proposition \[prop:BGN\], and because all $G_i(n_2,d_2,2)$ are birational to each other, by induction hypothesis. Therefore the subset $S \subset G_i(n_2,d_2,2)$ of those coherent systems $(E_2,V_2)$ such that ${{\mathcal O}}^2\to E_2$ is not injective is of positive codimension. For $(E_2,V_2)\notin S$ we have ${{\mathbb H}}^2_{21}=0$ by . So it is enough to prove $C_{12}>\dim {{\mathbb H}}^2_{21}-1$, i.e. $$\qquad \qquad\qquad (n_1-2)(g-2)+(n_2-2)(n_1-1)(g-1)+(d_2-{\alpha}_i(n_2-2))+$$ $$\qquad \qquad \qquad \qquad\qquad \qquad \qquad +{\alpha}_i(n_1-1)+\frac{d_1}2>2.$$ This holds clearly. The only case to be considered separately is $g=2$, $n_1=2$, $n_2=3$, $d_1=1$. But in this case ${\alpha}_i \in {{\mathbb Z}}$, hence ${\alpha}_i\geq 1$ and the result follows easily. - $2n_1-n_2<0$. Then $$C_{12}=(n_1-1)n_2(g-1) -{\alpha}_i(n_2-2n_1)+d_2-2.$$ Now ${\alpha}_i=\frac{d_2n_1-d_1n_2}{n_2-2n_1}<\frac{d_2}{n_2-2}$ gives ${\alpha}_i(n_2-2n_1)<d_2-2d_1$. Thus $$C_{12}>(n_1-1)n_2(g-1)+2d_1-2\geq 0.$$ We have the following cases: 1. $n_1\geq 2$. We use the bound . Then $C_{12}>n_1g -\frac{d_1}2 +(n_1-1){\alpha}_i-1$ is equivalent to $$(n_1-2)(g-2)+(n_2-2)(n_1-1)(g-1)+$$ $$\qquad \qquad\qquad \qquad \qquad \qquad +(d_2-{\alpha}_i(n_2-2n_1))-{\alpha}_i(n_1-1)+\frac{d_1}2>3.$$ Use that ${\alpha}_i(n_1-1)<d_1$ and $d_2-{\alpha}_i(n_2-2n_1)>2d_1$ to get that the left hand side is bigger or equal than $3+2d_1-d_1+\frac{d_1}2>3$. 2. $n_1=1$, $n_2>2$. By Proposition \[prop:C21\], ${{\mathbb H}}^2_{21}=H^0(E_1^*{\otimes}N_2{\otimes}K)^*$, where $N_2\hookrightarrow {{{\mathcal O}}}^2\to E_2$. Hence if $N_2=0$ then ${{\mathbb H}}^2_{21}=0$ and we have finished. So we may suppose that $N_2\neq 0$. By it is enough to prove that $C_{12}=d_1n_2-2>g-\frac{d_1}2-1$. Let $L$ be the image of ${{{\mathcal O}}}^2\to E_2$, which is a line bundle of degree $l\geq \frac{g+2}2$. We may write an inclusion of coherent systems $(L,V_2) \subset (E_2,V_2)$. By ${\alpha}_i$-semistability, $l+2{\alpha}_i\leq \frac{d_2+2{\alpha}_i}{n_2}=d_1+{\alpha}_i$. So $d_1 \geq {\alpha}_i+ \frac{g+2}2$ and then $$d_1n_2-2>n_2\frac{g+2}{2}-2>g-\frac{d_1}2-1.$$ Applications of coherent systems to Brill-Noether theory {#sec:applications} ======================================================== In this section, we shall describe in more detail the relationship between $G_0(n,d,k)$ and $B(n,d,k)$ introduced in section \[subsec:alpha-small\], and give some applications of our results to Brill-Noether theory. Although a good deal is known about non-emptiness of Brill-Noether loci, even quite simple geometrical properties (for example, irreducibility) have been established only in a very few cases. The results given here begin to fill these gaps in our knowledge, and should be regarded as a sample of what is possible. We plan to return to these questions in future papers and obtain more extensive and comprehensive results. Although many of the proofs are valid for all $g$, one may as well assume in this section that $g\ge2$, since Brill-Noether theory itself is trivial for $g=0,1$. General remarks. {#subsec:bngen} ---------------- \[lem:bn1\] If $\beta(n,d,k)\ge n^2(g-1)+1$, then $B(n,d,k)=M(n,d)$. [*Proof.* ]{} By (\[bnnumber\]), $$\label{eq:bn1} \beta(n,d,k)\ge n^2(g-1)+1\Leftrightarrow d-n(g-1)\ge k.$$ When these equivalent conditions hold, it follows from the Riemann-Roch Theorem that, for any $E\in M(n,d)$, $$\dim H^0(E)\ge k.$$ So $B(n,d,k)=M(n,d)$. $\Box$ On the other hand, we have \[lem:bn2\] If $\beta(n,d,k)\le n^2(g-1)$, then every irreducible component $B$ of $B(n,d,k)$ contains a point outside $B(n,d,k+1)$. [*Proof.* ]{} (This is [@Lau Lemma 2.6]; for the convenience of the reader, we include a proof.) The content of the statement is that there exists $E\in B$ such that $\dim H^0(E)=k$. To see this, note first that, if $\dim H^0(E')\ge1$ and $P$ is a point of $X$ such that the sections of $E'$ generate a non-zero subspace of the fibre $E'_P$, we can find an extension $$0\to F\to E'\to{\mathcal O}_P\to 0$$ such that the map $H^0(E')\to {\mathcal O}_P$ is non-zero and hence $\dim H^0(F)=\dim H^0(E')-1$. Now let $E'$ be a point of $B$ not contained in any other irreducible component of $B(n,d,k)$ and suppose that $H^0(E')=k+r$ with $r\ge1$. By iterating the above construction, we can find points $P_1,\ldots,P_r$ of $X$ and an exact sequence $$0\to F\to E'\to{\mathcal O}_{P_1}\oplus\ldots\oplus{\mathcal O}_{P_r}\to 0$$ such that $\dim H^0(F)=\dim H^0(E')-r=k$. Now consider the extensions $$\label{eq:bn2} 0\to F\to E\to{\mathcal O}_{Q_1}\oplus\ldots\oplus{\mathcal O}_{Q_r}\to 0,$$ where $Q_1,\ldots,Q_r\in X$. These form an irreducible family of bundles with $\dim H^0(E)\ge k$, whose generic member is stable (since $E'$ is stable). It follows that the generic extension (\[eq:bn2\]) belongs to $B$. Moreover, by the Riemann-Roch Theorem and (\[eq:bn2\]), $$\begin{aligned} \dim H^1(F)&=&\dim H^0(F)-(d-r)+n(g-1)\\ &>& k-k+r=r. \end{aligned}$$ By considering the dual sequence $$0\to E^*\otimes K\to F^*\otimes K\to{\mathcal O}_{Q_1}\oplus\ldots\oplus{\mathcal O}_{Q_r}\to 0,$$ in which $\dim H^0(F^*\otimes K)>r$, we can choose $Q_1,\ldots,Q_r$ and $E$ so that $$\dim H^0(E^*\otimes K)=\dim H^0(F^*\otimes K)-r;$$ hence (again by Riemann-Roch) $$\dim H^0(E)=\dim H^0(F).$$ It follows that the generic extension (\[eq:bn2\]) satisfies $\dim H^0(E)=k$. Since we already know that $E\in B$, this completes the proof.$\Box$ As envisaged at the end of section \[subsec:alpha-small\], we introduce \[cond\] [ ]{} - $\beta(n,d,k)\le n^2(g-1)$, - $G_0(n,d,k)$ is irreducible, - $B(n,d,k)\ne\emptyset$. For the moment we do not assume that ${{\operatorname{GCD}}}(n,d,k)=1$ or that $G_0(n,d,k)$ is smooth. We denote by $$\psi:G_0(n,d,k)\to\widetilde{B}(n,d,k)$$ the map given by assigning to every $(E,V)\in G_0(n,d,k)$ the underlying bundle $E$ (see (\[bnmap\])). \[thm:bn1\] Suppose Conditions \[cond\] hold. Then 1. $B(n,d,k)$ is irreducible, 2. $\psi$ is one-to-one over $B(n,d,k)-B(n,d,k+1)$, 3. $\dim B(n,d,k)=\dim G_0(n,d,k)$, 4. for any $E\in B(n,d,k)-B(n,d,k+1)$, the linear map $${\rm d}{\psi}:T_{(E,H^0(E))}G_0(n,d,k)\longrightarrow T_EB(n,d,k)$$ of Zariski tangent spaces is an isomorphism. [*Proof.* ]{} (i) If $E\in B(n,d,k)$, then $(E,V)\in G_0(n,d,k)$ for any $k$-dimensional subspace of $H^0(E)$. It follows that the image of $\psi$ contains $B(n,d,k)$ as a non-empty Zariski-open subset. Since $G_0(n,d,k)$ is irreducible, it follows that $B(n,d,k)$ is irreducible. \(ii) If $E\in B(n,d,k)-B(n,d,k+1)$, then $\psi^{-1}(E)=\{(E,H^0(E)\}$. \(iii) follows from (i), (ii) and Lemma \[lem:bn2\]. \(iv) Taking $(E',V')=(E,V)$ in (\[long-exact\]) and putting $V=H^0(E)$, we get a map $${{\operatorname{Ext}}}^1((E,H^0(V)),(E,H^0(V)))\to {{\operatorname{Ext}}}^1(E,E)$$ which can be identified with the map $$T_{(E,H^0(E))}G_0(n,d,k)\to T_EM(n,d)$$ induced by $\psi$. By (\[long-exact\]) this map is injective and its image is $${{\operatorname{Ker}}}({{\operatorname{Ext}}}^1(E,E)\to {{\operatorname{Hom}}}(H^0(E),H^1(E))).$$ By standard Brill-Noether theory, this image becomes identified with the subspace $T_EB(n,d,k)$ of $T_EM(n,d)$.$\Box$ \[cor:smooth\] Suppose Conditions \[cond\] hold and $G_0(n,d,k)$ is smooth. Then $\psi$ is an isomorphism over $B(n,d,k)-B(n,d,k+1)$. Moreover, if ${{\operatorname{GCD}}}(n,d,k)=1$, then $G_0(n,d,k)$ is a desingularisation of the closure $\overline{B(n,d,k)}$ of $B(n,d,k)$ in the projective variety $\widetilde{M}(n,d)$. [*Proof.* ]{} The first part follows from (ii) and (iv). For the second part, recall that, when ${{\operatorname{GCD}}}(n,d,k)=1$, $G_0(n,d,k)$ is projective; hence the image of $\psi$ is precisely $\overline{B(n,d,k)}$.$\Box$ \[cor:coprime\] Suppose Conditions \[cond\] hold, $G_0(n,d,k)$ is smooth and $(n,d)=1$. Then $B(n,d,k)$ is projective and $G_0(n,d,k)$ is a desingularisation of $B(n,d,k)$. [*Proof.* ]{} In this case $M(n,d)=\widetilde{M}(n,d)$.$\Box$ Irreducibility and dimension of Brill-Noether loci. {#sec:irred} --------------------------------------------------- In many cases our methods yield information about the irreducibility and dimension of $B(n,d,k)$, and more precisely about its birational structure. We illustrate this with results for $k=1,2,3$, where we have good estimates for the codimensions of the flips. The main respect in which our results improve those previously known is that they impose no restriction on $d$ other than that required for the Brill-Noether locus to be non-empty and not equal to $M(n,d)$. We begin with $k=1$. \[thm:bn2\] Suppose $0<d\le n(g-1)$. Then 1. $G_0(n,d,1)$ is a desingularisation of $\overline{B(n,d,1)}$, 2. $B(n,d,1)$ is irreducible of dimension $\beta(n,d,1)$, smooth outside $B(n,d,2)$, 3. $B(n,d,1)$ is birationally equivalent to a fibration over $M(n-1,d)$ with fibre ${\mathbb P}^{d+(n-1)(g-1)-1}$, 4. if $(n-1,d)=1$, $B(n,d,1)$ is birationally equivalent to $$M(n-1,d)\times{\mathbb P}^{d+(n-1)(g-1)-1}.$$ \[bn:rem1\] In the case $d=n(g-1)$, a stronger form of (i) is proved in [@RV]. Part (ii) is proved in [@Su]. Parts (iii) and (iv) are implicit in [@Su]. We have chosen to prove the complete theorem to illustrate our methods. [*Proof.* ]{} We first check Conditions \[cond\]. The first follows at once from (\[eq:bn1\]), the second from Theorem \[thm:k=1\] and the third is elementary and well known (see for example [@Su]). Moreover $G_0(n,d,1)$ is smooth of dimension $\beta(n,d,1)$ by Theorem \[thm:k=1\] (or Proposition \[prop:smooth\]). Parts (i) and (ii) now follow from Theorem \[thm:bn1\] and Corollary \[cor:smooth\]. Part (iii) follows from Theorem \[G\_L(k&lt;n)\] and Theorem \[thm:k=1\], as does part (iv) if we note that in this case the existence of a universal bundle over $X\times M(n-1,d)$ implies that the fibration of Theorem \[G\_L(k&lt;n)\] is locally trivial in the Zariski topology.$\Box$ For $k=2,3$, we need a lemma. \[lem:bn3\] Suppose $k=2$ or $3$, $n\ge2$ and that $X$ is a Petri curve of genus $g\ge2$. Then $B(n,d,k)$ is non-empty precisely in the following cases: 1. $k=2$, $n=2$, $d\ge3$, 2. $k=2$, $n\ge3$, $d\ge1$, 3. $k=3$, $n=2$, $d\ge\frac{2g+6}{3}$, 4. $k=3$, $n=3$, $d\ge4$, 5. $k=3$, $n=4$, either $g=2$ and $d\ge2$ or $g\ge3$ and $d\ge1$, 6. $k=3$, $n\ge5$, $d\ge1$. The Petri condition is required only for case (c). [*Proof.* ]{} All parts except (c) follow from [@BGN] and either [@T1] or [@M5]. For (c), see [@Bu2] or [@Tan].$\Box$ \[thm:bn3\] Let $X$ be a Petri curve of genus $g\ge2$, $k=2$ or $3$, $d<n(g-1)+k$. Suppose further that one of the conditions of Lemma \[lem:bn3\] holds. Then [(i)]{} $B(n,d,k)$ is irreducible of dimension $\beta(n,d,k)$. If in addition $k<n$ (i.e. in cases [*(b), (e), (f)*]{} of Lemma \[lem:bn3\]), then [(ii)]{} $B(n,d,k)$ is birationally equivalent to a fibration over $M(n-k,d)$ with fibre ${{\operatorname{Gr}}}(k,d+(n-k)(g-1))$; [(iii)]{} if $(n-k,d)=1$, $B(n,d,k)$ is birationally equivalent to $$M(n-k,d)\times{{\operatorname{Gr}}}(k,d+(n-k)(g-1)).$$ [*Proof.* ]{} (i) Note that the conditions of Lemma \[lem:bn3\] for the non-emptiness of $B(n,d,k)$ are exactly the same as those of Theorems \[thm:k=2\] and \[thm:k=3\] for the non-emptiness of $G_0(n,d,k)$. Conditions \[cond\] follow from (\[eq:bn1\]) and Theorems \[thm:k=2\] and \[thm:k=3\]. The result now follows from Theorem \[thm:bn1\]. \(ii) and (iii) follow from Theorem \[G\_L(k&lt;n)\] in the same way as the corresponding parts of Theorem \[thm:bn2\].$\Box$ When $k<n$ (cases (b), (e), (f)), the irreducibility of $B(n,d,k)$ for $d<\min\{2n,n+g\}$ and the fact that $B(n,d,k)$ has the expected dimension for $d\le 2n$ have been proved previously [@BGN; @M1]. For $k\le n$ (all cases except (c)), it was proved in [@T1] that $B(n,d,k)$ has a component of the expected dimension. Parts (ii) and (iii) are known for $d<\min\{2n,n+g\}$ [@M1]. Picard group. {#sec:pic} ------------- Our methods become potentially even more useful in computing cohomological information about Brill-Noether loci. In general, the calculations will be complicated and we restrict attention here to computing the Picard group in the case $k=1$. \[thm:bn4\] Let $X$ be a Petri curve of genus $g\ge2$. Suppose $0<d\le n(g-1)$, $n\ge3$, $(n-1,d)=1$ and $(n,d)=1$. Then $${{\operatorname{Pic}}}(B(n,d,1)-B(n,d,2))\cong{{\operatorname{Pic}}}(M(n-1,d))\times{\mathbb Z}.$$ [*Proof.* ]{} Note first that, by Theorem \[G\_L(k&lt;n)\], $G_L(n,d,1)$ is a projective bundle over $M(n-1,d)$, so $${{\operatorname{Pic}}}(G_L(n,d,1))={{\operatorname{Pic}}}(M(n-1,d))\times{\mathbb Z}.$$ From the proof of Theorem \[thm:k=1\], we see that the codimensions $C_{12}$, $C_{21}$ are both at least $2$ (we need $n\ge3$ here since otherwise we could have $n_1=n_2=1$, $d_2=1$, giving $C_{21}=1$). Hence $${{\operatorname{Pic}}}(G_0(n,d,1))={{\operatorname{Pic}}}(M(n-1,d))\times{\mathbb Z}.$$ To complete the proof, we need to show that $\psi^{-1}(B(n,d,2))$ has codimension at least $2$ in $G_0(n,d,1)$. Now the fibre of $\psi$ over a point of $B(n,d,k)-B(n,d,k+1)$ is a projective space of dimension $k-1$. It is therefore sufficient to prove that $B(n,d,k)$ has codimension at least $k+1$ in $B(n,d,1)$ for all $k\ge2$. In view of Lemma \[lem:bn2\], it is enough to prove this for $k=2$, i.e. to prove $${{\operatorname{codim}}}_{B(n,d,1)}B(n,d,2)\ge3.$$ For this, note that $$\begin{aligned} \beta(n,d,2)&=&\beta(n,d,1)-n(g-1)+d-3\\ &\le&\beta(n,d,1)-3\end{aligned}$$ since $d\le n(g-1)$. The result now follows from Theorem \[thm:bn3\]. \[rem:bn2\] Note that we need Theorem \[thm:bn3\] here to show that $B(n,d,2)$ always has the expected dimension. This is the only point in the proof where the Petri condition is used. It may be that this condition is not essential. [AAAA]{} E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, [*Geometry of Algebraic Curves*]{}, Vol. 1, Springer-Verlag, New York, 1985. M. F. Atiyah, Vector bundles over an elliptic curve, [*Proc. London Math. Soc.*]{} (3) [**7**]{} (1957) 414–452. A. Bertram, Stable pairs and stable parabolic pairs, [*J. Alg. Geom.*]{} [**3**]{} (1994) 703–724. A. Bertram, G. Daskalopoulos and R. Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, [*J. Amer. Math. Soc.*]{} [**9**]{} (1996) 529–571. A. Bertram and B. Feinberg, On stable rank 2 bundles with canonical determinant and many sections. [*Algebraic Geometry: papers presented for the EUROPROJ conferences in Catania and Barcelona*]{}, ed. P. E. Newstead, Lecture Notes in Pure and Applied Mathematics [**200**]{}, 259–269, Marcel Dekker, New York, 1998. S.B. Bradlow, Special metrics and stability for holomorphic bundles with global sections, [*J. Diff. Geom.* ]{} [**33**]{} (1991) 169–214. S.B. Bradlow and G. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces, [*Internat. J. Math.*]{} [**2**]{} (1991) 477–513. S.B. Bradlow and G. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces II, [*Internat. J. Math.*]{} [**4**]{} (1993) 903–925. S.B. Bradlow, G. Daskalopoulos and R. Wentworth, Birational equivalences of vortex moduli, [*Topology*]{}, [**35**]{} (1996) 731–748. S.B. Bradlow, G. Daskalopoulos, O. García-Prada and R. Wentworth, Stable augmented bundles over Riemann surfaces. [*Vector Bundles in Algebraic Geometry, Durham 1993*]{}, ed. N.J. Hitchin, P.E. Newstead and W.M. Oxbury, LMS Lecture Notes Series [**208**]{}, 15–67, Cambridge University Press, 1995. S.B. Bradlow and O. García-Prada, A Hitchin-Kobayashi correspondence for coherent systems on Riemann surfaces, [*J. London Math. Soc.*]{} [**(2) 60**]{} (1999) 155–170. S.B. Bradlow and O. García-Prada, An application of coherent systems to a Brill-Noether problem, [*J. Reine Angew. Math.*]{}, to appear. S.B. Bradlow, O. García-Prada and P. Gothen, Representations of the fundamental group of a surface in $PU(p,q)$ and holomorphic triples, [*C. R. Acad. Sci. Paris*]{}, [**333**]{}, Série I (2001), 347–352. L. Brambila-Paz, I. Grzegorczyk and P. Newstead, Geography of Brill-Noether loci for small slope. [*J. Alg. Geom.*]{} [**6**]{} (1997) 645–669. L. Brambila-Paz, V. Mercat, P. Newstead and F. Ongay, Nonemptiness of Brill-Noether loci, [*Internat. J. Math.*]{} [**11**]{} (2000) 737–760. D. Butler, Normal generation of vector bundles over a curve, [*J. Diff. Geom.*]{} [**39**]{} (1994) 1–34. D. Butler, Birational maps of moduli of Brill-Noether pairs, preprint. L. Ein and R.K. Lazarsfeld, Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves, [ *Complex Projective Geometry*]{}, ed. G. Ellingsrud, C. Peskine, G. Sacchiero and S. A. Stromme, LMS Lecture Notes Series [**179**]{}, Cambridge University Press, 1992. O. García-Prada, Dimensional reduction of stable bundles, vortices and stable pairs, [*Internat. J. Math.*]{} [**5**]{} (1994) 1–52. M. He, Espaces de modules de systèmes cohérents, [*Internat. J. Math.*]{} [**9**]{} (1998) 545–598. D. Huybrechts and M. Lehn, Stable pairs on curves and surfaces, [*J. Alg. Geom.*]{} [**4**]{} (1995) 67–104. D. Huybrechts and M. Lehn, Framed modules and their moduli, [*Internat. J. Math.*]{} [**6**]{} (1995) 297–324. A. King and P. Newstead, Moduli of Brill-Noether pairs on algebraic curves, [*Internat. J. Math.*]{} [**6**]{} (1995) 733–748. G. Laumon, Fibrés vectoriels spéciaux, [*Bull. Soc. Math. France*]{} [**119**]{} (1991) 97–119. J. Le Potier, Faisceaux semi-stables et systèmes cohérents, [*Vector Bundles in Algebraic Geometry, Durham 1993*]{}, ed. N.J. Hitchin, P.E. Newstead and W.M. Oxbury, LMS Lecture Notes Series [**208**]{}, 179–239, Cambridge University Press, 1995. V. Mercat, Le problème de Brill-Noether pour des fibrés stables de petite pente, [*J. reine angew. Math.*]{} [**56**]{} (1999) 1–41. V. Mercat, Le problème de Brill-Noether et le théorème de Teixidor, [*Manuscripta Math.*]{} [**98**]{} (1999) 75–85. V. Mercat, Le problème de Brill-Noether: Présentation, [http://www.bnt.math.jussieu.fr/]{} V. Mercat, Clifford’s Theorem and higher rank vector bundles, [*Internat. J. Math.*]{}, to appear. S. Mukai, Vector bundles and Brill-Noether theory, [*Current Topics in Algebraic Geometry*]{}, Math. Sci. res. Inst. Publ. [**28**]{}, 145–158, Cambridge University Press, 1995. K. Paranjape and S. Ramanan, On the canonical ring of a curve, [*Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata*]{} (1987) 503–516. N. Raghavendra and P.A. Vishwanath, Moduli of pairs and generalized theta divisors, [*Tôhoku Math. J.*]{} [**46**]{} (1994) 321–340. N. Sundaram, Special divisors and vector bundles, [*Tôhoku Math. J.*]{} [**39**]{} (1987) 175–213. Tan X.-J., Some results on the existence of rank 2 special stable bundles, [*Manuscripta Math.*]{} [**75**]{} (1992) 365–373. M. Teixidor i Bigas, Brill-Noether theory for stable vector bundles, [*Duke Math. J.*]{} [**62**]{} (1991) 385–400. M. Teixidor i Bigas, On the Gieseker-Petri map for rank 2 vector bundles, [*Manuscripta Math.*]{} [**75**]{} (1992) 375–382. M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, [*Invent. Math.*]{} [**117**]{} (1994) 317–353. C. Voisin, Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, [*Acta Math.*]{} [**168**]{} (1992) 249–272. [^1]: All authors are members of the research group VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). Support was also received from the Acciones Integradas Programme (HB 1998-0006). The first author was partially supported by the National Science Foundation under grant DMS-0072073. The fourth author would like to thank the Isaac Newton Institute, Cambridge and the organisers of the HDG programme for their hospitality during the completion of work on this paper.

--- author: - 'I. Negueruela' - 'J. S. Clark' date: 'Received / Accepted ' title: ' Further Wolf-Rayet stars in the starburst cluster [^1]' --- Introduction ============ Wolf-Rayet (WR) stars represent the last stage in the evolution of massive stars and are characterised by very high temperatures and exhaustive mass loss [see @hucht for a recent review]. In spite of their interest, our current understanding of their evolutionary paths is still rather limited. The evolutionary links between different WR subtypes are not well established and, more importantly, the correspondence between observed characteristics and position on theoretical tracks is still unclear [cf. @mm03]. Observation of WR stars in open clusters and comparison with other massive members have yielded most of the constraints on which understanding of these objects is based, providing ages and progenitor masses [e.g., @mas01]. Unfortunately, the number of WR stars in clusters is relatively small and generally each cluster contains only one or two WR stars, resulting in rather poor evolutionary constraints. The young open cluster (henceforth Wd 1; @west61) offers the possibility of studying an important population of WR stars of a given age and chemical composition within the context of a large homogeneous population of massive stars. This highly reddened cluster is found at a distance of between 2 and 5 kpc (most likely close to the upper limit; see @main) and contains a large number of evolved massive stars. Spectra of 11 WR stars obtained with the ESO 1.5-m telescope were presented by Clark & Negueruela ([-@one]; henceforth ). Because of the small size of the telescope and the low spatial resolution of the configuration used (as well as the lack of appropriate finding charts for the field), the identification of some of these objects was problematic, as many WR spectra appeared to arise from what were obviously unresolved blends of stars. Moreover the Signal-to-Noise Ratio (SNR) of several of the spectra was low, allowing only very approximate spectral classifications. The objects observed comprised 5 late WC stars (WCL), 5 late WN stars (WNL) and one broad-lined, presumably early, WN star. @main reported the identification of three further objects with WR-like characteristics. Two of them had been observed serendipitously at intermediate resolution as they fell on the slit when observations of brighter objects were taken. Their spectra were therefore of very low SNR, but were suggestive of a WCL and a transitional Ofpe object, belonging to a class recently rechristened as very late WN stars (WNVL). The third object had only been observed at very low resolution and it appeared as an OB supergiant with very strong emission lines, also suggestive of a transitional object. Here we present higher quality spectra of the majority of these objects, taken under exceptional seeing conditions, allowing a much better characterisation of the WR population in . We also present spectra of 5 new WR stars found in the field of . This brings the total number of WR stars known in to 17 + two transitional objects. In what follows, we will adopt the naming convention of @main, but will drop the word “candidate” from the name of those Wolf-Rayet stars whose identification has been secured. Observations & data reduction ============================= Observations of were obtained with the ESO Multi-Mode Instrument (EMMI) on the 3.5-m New Technology Telescope (NTT) at La Silla, Chile. They were taken during a run on 2003 June 5th-8th, though only the 6th and the 7th were useful because of cloud cover. On the night of the 6th, though some high cirrus were present, the seeing was exceptionally good, staying below 06 for most of the night and reaching $<$04 at times. Imaging of the cluster area was obtained using the $R$ and H$\alpha$ (\#654) filters. \[fig:south\] Due to the varied science goals of the observations and the serendipitous nature of many of the detections, spectra of a number of sources were obtained using a varied set of instrumental configurations. For intermediate resolution spectra, we used the red arm with gratings \#6 and \#7. On the night of June 6th, grating \#6 covered the $\lambda\lambda6440-7140$[Å]{} range. On June 7th, grating \#6 covered $\lambda\lambda8225-8900$[Å]{} and grating \#7, $\lambda\lambda6310-7835$[Å]{}. For low resolution, we used the red arm with grisms \#1 and \#4. Grism \#1 covers the $\lambda\lambda3850-10000$[Å]{} range, with a resolution $R=263$. Grism \#4 covers the $\lambda\lambda5550-10000$[Å]{} range, with a resolution $R=613$. Note, however, that because of the high reddening, the signal-to-noise ratio rapidly decreases in the blue end of the spectra - effectively limiting the spectra to $\lambda > 6500$Å for grism \#4 and $\lambda > 5500$Å for grism \#1. An observation of the field to the South of the cluster was also conducted in slitless spectroscopy mode. This technique, based on the use of a low dispersion grism (in our case, \#1) coupled with a broad-band filter (Bessel $R$) resulting in an “objective prism-like” spectrogram of all the objects in the field, has been used by @bp01 to search for emission line stars in open clusters. Image pre-processing was carried out with [*MIDAS*]{} software, while data reduction was achieved with the [*Starlink*]{} packages [ccdpack]{} and [@draper] and [figaro]{} [@shortridge]. Analysis was carried out using [figaro]{} and [dipso]{} [@howarth]. The results of this inevitably somewhat varied observational approach was the identification of a further 5 WR stars within Wd 1. Inspection of the slitless image led to the location of three obvious candidates lying in the outskirts of the cluster, which were later confirmed as WR stars by long-slit spectroscopy. A fourth WR star was found serendipitously while obtaining spectra of objects in the Southern part of the cluster. Following the notation used in , these are WR stars N, O, P and Q. Finally, as part of a dedicated investigation into the apparent Ofpe star W14, we identified a final WR, designated R. Object Wavelength range (Å) -------- ---------------------- WR A 5550$-$10000 WR B 5550$-$10000 WR C 5550$-$10000 WR K 5550$-$10000 6440$-$7140 WR L 5550$-$10000 WR M 3850$-$10000 5550$-$10000 6440$-$7140 WR N 3850$-$10000 WR O 3850$-$10000 WR P 3850$-$10000 WR Q 5550$-$10000 WR R 5550$-$10000 W5 5550$-$10000 6440$-$7140 6310$-$7835 : Observation log, presenting all the spectroscopic observations for each target. See text for the configurations resulting in each spectral range. \[fig:cluster\] For aesthetic reasons, the finder in Fig. 2 is based on an $R$-band image obtained with VLT UT1/FORS2 on June 10th 2004, in service mode. Results {#sec:res} ======= Due to crowding and the low spatial resolution of the Boller & Chivens spectrograph, the exact identifications of some of the WR stars presented in and @main were uncertain. The WR features were observed in spectra attributable to a blend of several objects, an example of this being the identification of the continuum of an O-type supergiant and emission lines typical of a WR star in the low-resolution spectrum of W14 [@main]. This situation has been somewhat alleviated with the current dataset. We now can confirm the exact identification for all but 5 of the currently identified WR stars. For the purposes of this paper, we consider that a WR identification is sufficiently secure when either several spectra of the same star exist, all displaying the WR features, or a spectrum with sufficiently high spatial resolution allows the identification of a single candidate. This is the case for WR star N (Fig. 1), and all the objects circled in Fig. 2 (also marked by their corresponding letters). For those objects without such secure identification, the word “cand” is shown in front of the corresponding letter, with an arrow pointing to the most likely identification. Of the previously identified WRs, the new observations have confirmed the positions of WRs A, B, C, E, F, L and M, while the most probable candidates for WRs D, G, I and J remain unchanged. However, we present a new identification for WR K and suggest a different candidate for WR H. Moving to the new WRs, star N is an outlier some $4\farcm5$ South of the cluster. This object is listed in the USNO catalogue as USNO-B1.0 0440-0523445, with no measured blue magnitudes, red magnitudes $r_{1}=16.6$ and $r_{2}=16.9$ and infrared magnitude $i=13.0$. These magnitudes and colour make it a very likely cluster member, further supported by its late-WC classification (see Section \[sec:wcs\]). A finder for this object is shown in Fig. 1. Another nearby object, USNO-B1.0 0440-0523458, was found to display emission lines, but its spectrum shows it to be a foreground Be star. Its foreground character is confirmed by the relatively low reddening ($b_{2}=14.08$, $r_{2}=13.04$). WR stars O and P are within the area of the cluster previously explored and their positions are identified in Fig. 2; no photometry is available for either object. Their spectra are shown in Fig. 3. WR star Q lies on the Southwestern reaches of the cluster and it is one of the westernmost likely members identified by @main, who give the following magnitudes $B=23.7$, $V=20.3$, $R=17.5$, $I=14.7$. Its spectrum is shown in Fig. 4 and discussed in Section \[sec:wns\]. Its location with respect to the main body of the cluster can be seen in Fig. 2. Finally, in our new images, W14 is clearly resolved into three objects of similar brightness, which we designate W14a, b and c. (see Fig. 2 for identifications). Individual spectra of all three objects have been obtained and WR features are unmistakably associated with the Easternmost object, W14c which we designate as WR R; its spectrum is displayed in Fig. 4, but no photometry is available for the individual components of W14. Spectral Classification ----------------------- \[fig:lowres\] Our new spectroscopic observations – listed in Table 1 – allow the classification of the newly identified WRs (WRs N-Q), while permitting a more accurate analysis of stars previously observed at lower resolution and/or S/N (WRs A, B, C, K–M). The results are summarised in Table 2. ### WC stars {#sec:wcs} \[fig:gr4\] Red spectra of WC stars are displayed in Fig. 4. The lower-resolution spectrum of WR star N can be seen in Fig. 3. Spectroscopic classification criteria for WCL stars between 6-10000[Å]{} are discussed in . We employ the ratio between the Equivalent Width (EW) of  9900Å  and  9710Å as a key diagnostic, found to be $>0.14$ for WC9 stars. WR M was preliminarily classified WC9 by @main based on a rather noisy $R$-band spectrum. Our new spectrum, shown in Fig. 4, for which EW()/EW() $=0.17\pm0.03$, corroborates this classification, showing an obvious resemblance to the spectra of the WC9 stars WR E and WR F displayed in . The new spectrum of WR C - which classified as WC8 - is also presented in Fig. 4. It is clearly very similar to that of WR M, the main difference being the slightly weaker features. The EW()/EW() ratio is $0.12\pm0.03$, on the borderline between WC8 and WC9. The new spectrum of WR K, also seen in Fig. 4, is clearly earlier than those of WR M and WR C. The features are clearly much weaker and the  7726Å  line is very obviously present. The EW()/EW() ratio is only $0.03\pm0.01$, suggesting that WR K is WC7. This is confirmed by the much higher resolution spectrum shown in Fig. 5. The blend around the position of H$\alpha$ is dominated by  6581Åin WC9 spectra [@vreux83], but in WR K, it peaks at $\lambda$6568Å as is typical of earlier-type WC stars, where it is dominated by  6560Å. The blend around $\lambda$7065Åis likely dominated by  7062Å. Finally, we examine the newly discovered WR N, for which we only have the low-resolution grism\#1 spectrum shown in Fig. 3. The ratio between  5696Å and  5812Å is suggestive of a WC9 spectral type. However, the absence of  5875Åsupports an earlier type. As the S/N ratio is rather low in the $\lambda\lambda5500-6000$Å region, we prefer to use the red-end classification criteria. The EW()/EW() ratio is $0.07\pm0.02$, typical of WC8 and indeed the strength of features appears intermediate between that of WR M and WR K. We therefore adopt WC8 for WR N. \[fig:highres\] ### WN stars {#sec:wns} Spectra of four WN stars observed with grism \#4 are displayed in Fig. 4, while two other objects observed at lower resolution are shown in Fig. 3. Of the four newly discovered WN stars, the best S/N ratio has been achieved for WR Q, which we therefore choose to discuss first. We find its spectrum to be extremely similar to that of the WN6 star WR 85 displayed by @vreux83. Indeed the weakness of  7065Å and  6678Å +  6683Å argues against a spectral type later than WN7. Unfortunately, the region covered by our spectra is not very sensitive to the spectral type of WN stars and so a spectral type WN7 cannot be ruled out. The spectrum of WR star R (also in Fig. 4) has a much lower S/N ratio, but does not appear to differ in any important respect from that of WR star Q, leading to the same classification. For WR star O, we have a low-resolution spectrum reaching $\lambda$5500Å (Fig. 3). The general aspect is very similar to those of WR stars R and Q. Both  5875Å and  5808Åappear to be absent (within the – rather large – uncertainty allowed by the limited S/N ratio), supporting a WN6 classification. The spectrum of WR star P (also in Fig. 3) is very different. The strength of all the lines supports a WN8 spectral type. Specifically, only the WN8 stars in the catalogues of @vreux83 and @vreux89 fulfil the condition  6678Å(+ 6683Å) $\la$ H$\alpha$ ((+), as happens in WR star P. WR star B was previously unclassifiable, due to the extremely low S/N ratio of the available spectrum. Our new spectrum, presented in Fig. 4, reveals the emission lines of this star to be very weak, strongly suggesting the presence of an OB companion in the spectrum. Within the very limited S/N ratio,  7065Å seems to be rather strong compared to  7114Å, suggesting a spectral type WN8, but an exact classification cannot be given. Finally, we present a new higher S/N spectrum of WR star A, the only broad-lined object in our sample (Fig. 4). The spectrum does not offer any clues for classification, except for the fact that it should be earlier than WN7. ### The WNVL candidates {#sec:wnvls} Clark et al. (2004) identified the first example of a WNVL star within Wd 1, the WN9 object W44 (= WR star L), while speculating that the emission line star W5 may possess an even later classification. New low resolution spectra (Table 1; not presented here) confirm the line identifications previously reported for both stars – with the addition of weak He[i]{} 7282[Å]{} emission in W44 – the emission lines appearing to be narrow and single peaked. In Fig. 6, we present a higher resolution spectrum of W5 which reveals a wealth of new details, including P Cygni He[i]{} emission lines and emission from low excitation metallic species such as N[ii]{} and C[ii]{}. In particular we are able to confirm the presence of the strong emission feature at $\sim$7235[Å]{}, which we attribute to a C[ii]{} doublet. \[fig:w5\] As with W44, the lack of N[iv]{} 7116[Å]{} precludes a classification earlier than WN9 for W5. The H[i]{} and He[i]{} emission spectrum is similar to that of the WN11 star , although the presence and strength of the C[ii]{} emission suggests a temperature lower than the $24\:$kK inferred for (Paul Crowther, private communication 2004). If W5 is cooler than $24\:$kK, it is not expected to display emission. Observationally, this is difficult to check, as  4686Åis outside our spectral range and lines in the red would be expected to be very weak. A lack of emission would prevent a WR classification, making W5 an early B supergiant. In this case, the strength of the emission lines and the lack of a P-Cygni profile in H$\alpha$ would indicate a very extreme early B0–0.5Ia$^{+}$ classification (Paul Crowther, private communication 2004). Given this uncertainty, we choose to denote W5 as [*Candidate*]{} WR star S and adopt a provisional classification of WNVL/early BIa$^{+}$, apparently intermediate between the [*bona fide*]{} WN9 star W44 and the extreme B5Ia$^{+}$ stars W7 and W33 [@main]. ----------- ------------- --------------------------- -------------------------------------------------------------------- -------------- -------------- -------------- ---------------- -- WR star Alternative $\alpha$ $\delta$ V R I Spectral Names Type A WR77s, W72 16h47m08.32s $-45\degr50\arcmin45\farcs5$ 19.69 16.59 13.68 [**$<$WN7**]{} B WR77n 16h47m05.35s $-45\degr51\arcmin05\farcs0$ 20.99 17.50 14.37 [**WN8?**]{} C WR77l 16h47m04.40s $-45\degr51\arcmin03\farcs8$ - - - [**WC8.5**]{} D WR77q [*16h47m06.24s*]{} [*$-$45$\degr$51$\arcmin$26$\farcs$5*]{} - - - WN6-8 E WR77p,W241 16h47m06.06s $-45\degr52\arcmin08\farcs3$ - - - WC9 F WR77m, W239 16h47m05.21s $-45\degr52\arcmin25\farcs0$ 17.86 15.39 12.90 WC9 G WR77i [*16h47m04.02s*]{} [*$-$45$\degr$51$\arcmin$25$\farcs$2*]{} 20.87 17.75 14.68 WN6-8 H WR77k [**[*16h47m04.1s*]{}**]{} [**$-$[*45*]{}$\degr$[*51*]{}$\arcmin$[*20*]{}$\farcs$[*0*]{}**]{} - - - WC9 I WR77e [*16h47m01.67s*]{} [*$-$45$\degr$51$\arcmin$19$\farcs$9*]{} - - - WN6-8 J WR77c [*16h47m00.89s*]{} [*$-$45$\degr$51$\arcmin$20$\farcs$9*]{} - - - WNL K WR77g [**16h47m03.1s**]{} [$-45\degr50\arcmin43\arcsec$]{} - - - [**WC7**]{} L WR77j, W44 16h47m04.20s $-45\degr51\arcmin07\farcs0$ 18.86 15.61 12.52 [**WN9**]{} M WR77h, W66 16h47m04.0s $-45\degr51\arcmin37\farcs5$ 19.79 16.85 13.96 [**WC9**]{} [**N**]{} WR77b [**16h46m59.9s**]{} [$-45\degr55\arcmin26\arcsec$]{} - [**16.9**]{} [**13.0**]{} [**WC8**]{} [**O**]{} WR77r [**16h47m07.6s**]{} [$-45\degr52\arcmin36\arcsec$]{} - - - [**WN6**]{} [**P**]{} WR77d, W57c [**16h47m01.5s**]{} [$-45\degr51\arcmin45\arcsec$]{} - - - [**WN8**]{} [**Q**]{} WR77a [**16h46m55.4s**]{} [$-45\degr51\arcmin34\arcsec$]{} [**20.3**]{} [**17.5**]{} [**14.7**]{} [**WN6-7**]{} [**R**]{} WR77o, W14c [**16h47m06.0s**]{} [$-45\degr50\arcmin22\arcsec$]{} - - - [**WN6-7**]{} [*S*]{} WR77f, W5 16h47m02.97s $-45\degr50\arcmin19\farcs5$ 17.49 14.98 12.48 [**WNVL**]{} ----------- ------------- --------------------------- -------------------------------------------------------------------- -------------- -------------- -------------- ---------------- -- Discussion & Concluding Remarks =============================== Our current census for the WRs within Wd1 consists of 2 transitional/WNVL(9-11), 9 WNL(6-8), one indeterminate WN (the broad-lined WR star A) and 7 WCL (7-9) stars. As such, we appear to be lacking early WR stars of both flavours. This may be most simply explained by the intrinsic faintness of such objects. Given a median value of $V-M_{V} \sim25.3$ for Wd 1 [@main], assuming the absolute visual magnitudes for early WR stars presented by @hucht yields apparent $V$-band magnitudes $\ga22$, clearly beyond the reach of our current photometry and spectroscopy. Theoretical models [@mm03] also predict that WNE stars will be rare compared to both WNL and WCL stars, due to their shorter lifetimes. With the above considerations in mind, it is of interest to compare the currently identified WR population of Wd 1 to those of other young massive clusters in the Galaxy. First, we would like to note that the resolution of W14 into three components, including a WNL, and the consequent refutation of a possible early O supergiant classification removes the sole observational datum that might suggest non-coevality for Wd 1. As discussed in @main, an age of 3.5-5 Myrs may safely be estimated for Wd 1. Therefore we expect its population to differ from those of the younger NGC3603 and Arches clusters, which are found to be dominated by WN stars [e.g., @crowther; @figer02]. A more revealing comparison may be established with the Quintuplet cluster [@figer] and the concentration of massive stars within the central parsec of the Galactic Centre [e.g., @horrobin; @genzel], which have estimated ages similar to those of Wd 1. Excluding the five Quintuplet Proper Members, which may be exceptionally dusty WCLs, the Quintuplet hosts 6 WN and 5 WCL stars [@figer; @homeier]. While the population of WC stars in both the the Quintuplet and Wd 1 consists exclusively of WC7-9 stars, five of the six Quintuplet WN stars are classified as WN9 or later, with only one earlier WN6 object (@figer, and refs. therein); by comparison, we have only found two $\geq$WN9 stars within Wd 1, with 9 of the remaining 10 objects being WN6-8. Inevitably, this comparison is prone to concerns due to completeness and selection effects. Given that the majority of the currently identified non-dusty WRs are amongst the faintest spectroscopically surveyed stars within the Quintuplet, it is possible that a further population of earlier WN stars lies below current detection thresholds. Moreover, @figer infer very high intrinsic luminosities for the WNL stars within the Quintuplet when compared to those within Wd 1. Further observational and analytic efforts – to arrive at a common detection threshold and to determine if the stellar parameters of the WNLs in both clusters systematically differ – are required before meaningful conclusions as to the relative WR populations of both clusters may be drawn. In contrast, the evolved population of the central parsec – as currently determined – is remarkably similar to that of Wd 1. To date, it consists of 6 Ofpe/LBV[^2], 8 WNL (7-9), 4 WNE (5/6), 10 WCL (8/9)[^3] and 1 WCE (5/6) stars, of which only the latter spectral type is not represented in Wd 1 (@genzel; Genzel, priv. comm. 2004; @maillard). In spite of this similarity, the Galactic Centre population may not represent a good analogue for Wd 1, as the origin of these massive evolved stars in the central parsec is at present unclear, with @genzel finding that they occupy two thin coeval discs, each with similarly-sized populations of evolved stars, about half the size of the currently identified population of Wd 1. One possibility for the formation of such a distribution is the spiral in of two clusters – indeed a possible remnant of such a disrupted cluster, IRS13, is found to be associated with one ring (Genzel, priv comm. 2004; @maillard). Alternatively, @genzel suggest that a collision between two interstellar clouds resulted in two counter-rotating discs of gas which then formed the (evolved) massive stellar population. In any event, the stars found within the central parsec do not appear to form a stellar cluster in the same manner as apparently ‘monolithic’ clusters such as Wd 1 and the Arches. We therefore find that, at present, Wd 1 possesses the largest WR population of any known galactic cluster. Indeed, given the combination of distance and high reddening to Wd 1, the intrinsic faintness of even WNL and WCL objects [@hucht], crowding in the central regions of the cluster and the somewhat [*ad hoc*]{} nature of the current observations, we suggest that our current census likely remains significantly incomplete. Further investigation is needed in order to achieve a full characterisation of the WR population of Wd 1, but the results presented here already show that this cluster can provide us with a unique laboratory to study the evolutionary paths followed by massive stars. IN is a researcher of the programme [*Ramón y Cajal*]{}, funded by the Spanish Ministerio de Educación y Ciencia and the University of Alicante, with partial support from the Generalitat Valenciana and the European Regional Development Fund (ERDF/FEDER). This research is partially supported by the Spanish MEC under grant AYA2002-00814. We are very grateful to Dr. Amparo Marco for her help during this observing run. The NTT team have provided excellent support for this project since its start. We would like to specifically thank John Willis and Emanuella Pompei for their dedicated support. We also thank Paul Crowther for many informative discussions on the nature and classification of WR stars, and Karel van der Hucht for comments on the manuscript . Bernabei, S., & Polcaro, V.F. 2001, A&A, 371, 123 Clark, J. S., Negueruela, I. 2002, A&A 396, L25 Clark, J. S., Negueruela, I., Crowther, P.A., & Goodwin, S.P. 2005, A&A, in press Crowther P. A., Dessart L. 1998, MNRAS, 296, 622 Draper, P.W., Taylor, M. , & Allan, A. 2000, Starlink User Note 139.12, R.A.L. Figer D. F., McLean I. S., Morris M., 1999, ApJ, 514, 202 Figer D. F., Najarro F., Gilmore D., et al. 2002, ApJ, 581, 258 Genzel R., Schoel R., Ott T., et al., 2003, ApJ, 594, 812 Howarth, I., Murray, J., Mills, D., & Berry, D.S. 1997, Starlink User Note 50.20, R.A.L. Horrobin M., Eisenhauer F., Tecza M., et al., 2004, AN, 325, 88 van der Hucht, K.A. 2001, New Ast. Rev. 45, 135 Homeier N. L., Blum R. D., Pasquali A., Conti P. S., & Damineli A., 2003, A&A, 408, 153 Maillard J. P., Paumard T., Stolovy S. R., & Rigaut F., 2004, A&A, 423, 155 Massey, P., DeGioia-Eastwood, K.,& Waterhouse, E. 2001, AJ 121, 1050 Meynet, G., & Maeder, A. 2003, A&A 404, 975 Paumard T., Genzel R., Maillard J. P., et al. 2005, to appear in ‘Young Local Universe’, Proceedings of XXXIXth Recontres de Moriond, La Thuile, Aosta Valley, Italie, March 21-28, 2004, Eds: Chalabaev, A., Fukui, T., Montmerle, T. & Tran-Thanh-Van J., Editions Frontieres, Paris, (astro-ph/0407189) Shortridge, K., Meyerdicks, H., Currie, M., et al. 1997, Starlink User Note 86.15, R.A.L Smith N., Vink J. S., & de Koter A., 2004, ApJ 615, 475 Vreux J. M., Dennefeld M., & Andrillat Y., 1983, A&AS, 54, 437 Vreux J. M., Dennefeld M., Andrillat Y., & Rochowicz, K. 1989, A&AS, 81, 353 Westerlund, B.E. 1961, PASP 73, 51 [^1]: Based on observations collected at the European Southern Observatory, La Silla, Chile (ESO 71.D-0151) [^2]: Compared to the WNVL/early BIa$^{+}$ star W5, the LBV W243, the B5Ia$^{+}$ stars W7, 33 & 42 and arguably the six YHGs within Wd 1 – @smith claim that YHGs occupy a closely related [ *evolutionary*]{} state to LBVs. [^3]: Additional dusty objects such as IRS13 E3A & B may also host WCL stars [@maillard].

--- abstract: | We present the temperature dependence of the specific heat, without external magnetic field and with [*9 T*]{}, for LaMnO$_{3}$, La$_{1.35}$Sr$_{1.65}$Mn$% _{2}$O$_{7}$, La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ single crystals. We found that spin-wave excitations in the ferromagnetic and bilayer-structure La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ were suppressed by the [*9 T*]{} magnetic field. On the other hand, the external magnetic field had no effect in the specific heat of the other three antiferromagnetic samples. Also, the electronic part of the interactions were removed at very low temperatures in the La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ single crystal, even with a zero applied magnetic field. Below [*4 K*]{}, we found that the specific heat data for La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ crystals could be fitted to an exponential decay law. Detailed magnetization measurements in this low temperature interval showed the existence of a peak close to 2 K. Both results, magnetizations and specific heat suggest the existence of an anisotropy gap in the energy spectrum of La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$% _{1.5} $Sr$_{0.5}$NiO$_{4}$ compounds. address: - | Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas,\ UNICAMP, 13083-970, Campinas, SP, Brazil - | Depto. de Física, Universidade Federal de São Carlos, CP-676,\ São Carlos, SP, 13565-905, Brazil - 'Department of Physics, Oxford University, Oxford OX1 3PU, United Kingdom' author: - 'J. López and O. F. de Lima' - 'C. A. Cardoso and F. M. Araujo-Moreira' - 'D. Prabhakaran' title: | Comparative study of specific heat measurements in LaMnO$_{3}$, La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$, La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$% _{0.5}$CoO$_{4}$ --- \[theorem\][Acknowledgement]{} \[theorem\][Algorithm]{} \[theorem\][Axiom]{} \[theorem\][Claim]{} \[theorem\][Conclusion]{} \[theorem\][Condition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Criterion]{} \[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Exercise]{} \[theorem\][Lemma]{} \[theorem\][Notation]{} \[theorem\][Problem]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Solution]{} \[theorem\][Summary]{} Introduction ============ The combined ordering of charge (CO) and spin (SO) is proving to be a common phenomenon in transition-metal oxides like LaMnO$_{3}$, La$_{1.35}$Sr$% _{1.65} $Mn$_{2}$O$_{7}$, La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$% _{0.5}$CoO$_{4}$. The magnetic properties of Mn, Co and Ni perovskites are considered to arise from the strong competition involving ferromagnetic and antiferromagnetic interactions and the spin-phonon coupling[@Radaelli]$-$[@J.López2]. The dimensionality of the relevant structure involving the transition metal ions, three (3D) or quasi-two-dimensional (2D), also plays an important role. For instance, the bilayer-structure compounds La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$, in which MnO$_{2}$ and (La,Sr)$_{2}$O$_{2}$ layers are stacked alternatively have 2D electronic and magnetic properties [@Okuda]. The well known LaMnO$_{3}$ is an antiferromagnetic insulator with 3D characteristics. The specific heat at low temperature, for LaMnO$_{3+\delta } $ samples, was found by L. Ghivelder et al.[@Ghivelder] to be very sensitive to small variations of $\delta $. Previous specific heat measurements in the related Nd$_{0.5}$Sr$_{0.5}$MnO$_{3}$, Nd$_{0.5}$Ca$% _{0.5}$MnO$_{3}$, Sm$_{0.5}$Ca$_{0.5}$MnO$_{3}$, Dy$_{0.5}$Ca$_{0.5}$MnO$% _{3} $ and Ho$_{0.5}$Ca$_{0.5}$MnO$_{3}$ samples revealed a Schottky-like anomaly at low temperatures[@JLópez3]$^{,}$[@JLópez4]. Differently from the LaMnO$_{3}$ crystal, the La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$% _{1.5} $Sr$_{0.5}$CoO$_{4}$ compounds have their magnetic ions (Ni and Co) confined in planes which are insulated by (La,Sr)$_{2}$O$_{2}$ layers. I. A. Zaliznyak et al.[@Zaliznyak] presented elastic and quasielastic neutron scattering measurements characterizing peculiar short-range charge-orbital and spin order in the layered perovskite La$_{1.5}$Sr$_{0.5}$CoO$_{4}$. They found that, below [*T*]{}$_{c}$[* = 750 K*]{}, holes introduced by Sr doping lose mobility and enter into a statically ordered charge-glass-phase with loosely correlated checkerboard arrangement of empty and occupied d$% _{3z^{2}-r^{2}}$ orbitals (Co$^{3+}$ and Co$^{2+}$). La$_{1.5}$Sr$_{0.5}$NiO$% _{4}$, like its parent compound La$_{2}$NiO$_{4}$, is an insulator, contrary to the related compound La$_{2}$CuO$_{4}$, where the antiferromagnetic insulator phase is rapidly destroyed by doping, leading to a metallic superconductor phase at moderate hole concentration[@Sachan], La$_{2}$NiO$_{4}$ remains nonmetallic up to quite large hole concentrations[@Wochner]. R. Kajimoto et al. [@Kajimoto] studied the CO in La$_{1.5}$Sr$% _{0.5}$NiO$_{4}$ with neutron diffraction technique. They found a rearrangement of CO from checkerboard-type to stripe-type as a function of temperature. In their measurements the stripe phase persisted up to x = 0.7 for highly hole-doped samples of Nd$_{2-x}$Sr$_{x}$NiO$_{4}$ with 0.45 $\leq $ x $\leq $ 0.7. A large number of papers discuss the properties of Mn, Co and Ni perovskite compounds treated separately. However, to our knowledge, a comparative study of the low temperature specific heat for these perovskite families is missing. Here, we present the temperature dependence of the specific heat, without external magnetic field ([*H*]{}) and with [*H=9 T*]{}, for LaMnO$% _{3}$, La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$, La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ single crystals. We found that the specific heat data for La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ crystals could be fitted to an exponential decay law below [*4 K*]{}. Detailed magnetization measurements in this low temperature interval showed the existence of a peak close to 2 K. Both results, magnetization and specific heat suggest the existence of an anisotropy gap in the energy spectrum of La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$% _{1.5} $Sr$_{0.5}$NiO$_{4}$ compounds. Our macroscopical measurements confirm the complex magnetic excitation and electronic band structure due to charge ordering and the quasi-2D confinement of the magnetic ions. Experimental methods ==================== Large single crystals of LaMnO$_{3}$, La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$, La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ were grown by the floating zone method described elsewhere[@Prabhakaran]. The magnetization measurements were done with a Quantum Design MPMS-5S SQUID magnetometer. Specific heat measurements were made with a Quantum Design PPMS calorimeter that uses a [*two relaxation times*]{} technique, and data was always collected during sample cooling. The intensity of the heat pulse applied to the sample was calculated to produce a variation in the temperature bath of [*0.5 %*]{}. Experimental errors during the specific heat and magnetization measurements were lower than [*1 %*]{} for all temperatures and samples. Results and Discussion ====================== Specific heat measurements at low temperatures give valuable information about the ground state excitations. In contrast to magnetization, which has a vector character, the specific heat is an scalar property. Figures 1a, 1b, 1c and 1d show the dependence of the specific heat measurements with temperature, between 2 and 30 K, for LaMnO$_{3}$, La$_{1.35}$Sr$_{1.65}$Mn$% _{2}$O$_{7}$, La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ single crystals, studied with [*H=0*]{} and [*9 T*]{}. The data is plotted as [*C/T*]{} vs. [*T*]{}$^{2}$ to facilitate the interpretation. The LaMnO$% _{3}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ samples showed both an almost linear behavior and no magnetic field dependence. On the other hand, the magnetic field strongly affected the low temperature ([*T 5 K*]{}) behavior of the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ compound. The specific heat curves for the La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ sample did not change with the applied field, but were not linear at temperatures below [*5 K*]{}. Continuous lines in figure 1 indicate the fitting of the experimental data between 2 and 30 K by the following expression[@Gordon]: $$C=\sum \beta _{2n+1}\,T^{\text{\/}2n+1}{\em +\ }\beta _{3/2}\,T^{\text{\/}% 3/2} \label{5}$$ The whole temperature interval, from [*2 K*]{} to [*30 K*]{}, was possible to be fitted with natural values of $n$ from [*0*]{} to [*4*]{}. ${\em C}$ is the specific heat, ${\em T}$ is the temperature and $\beta $ parameters represent the contributions of electron interactions ($\beta _{1}$), ferromagnetic spin waves ($\beta _{3/2}$) and phonon modes ($\beta _{3}$, $% \beta _{5}$, $\beta _{7}$ and $\beta _{9}$). The coefficient $\beta _{1}$ ([*n=0*]{}) is also known as $\gamma $, and $\beta _{3}$ ([*n=1*]{}) as $% \beta $. The results of the specific heat data fitting are shown in Table 1. It is worth stressing that, differently from the LaMnO$_{3}$ crystal, the La$% _{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ compounds have their magnetic ions (Ni and Co) confined in planes which are insulated by (La,Sr)$_{2}$O$_{2}$ layers. However, these latter two compounds show a different magnetic behavior compared to the quasi-2D La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ sample: the Mn ions order ferromagnetically, while the corresponding Ni and Co ions order antiferromagnetically. Therefore, as seen in figures 1c and 1d, there is not much contribution of the magnetic field into the measured specific heat. &gt;From resistivity measurements the LaMnO$_{3}$, La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ samples are electrical insulators at low temperatures, and applied magnetic fields up to 9 T are not strong enough to destroy this characteristic[@Zaliznyak],[@Wochner],[@Myron]. Therefore, we should not expect for the previous three crystals the linear contribution from free electrons to the specific heat. However, other kind of many-body excitations could also lead to a linear contribution[@Smolyaninova1]. On the other hand, the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ is metallic below about 100 K[@Okuda]. Our fitting shows that $\beta _{1} $ values are big for La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$_{1.5} $Sr$_{0.5}$NiO$_{4}$. To facilitate even more the comparison, we have re-plotted in figure 2 the data for the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ crystals at temperatures below 10 K. Closed symbols represent the measurements with [*H=0*]{} and open ones with [*9 T*]{}. Okuda et al.[@Okuda] found that the decrease of specific heat at low temperatures, due to an applied magnetic field of 9 T, was about ten times larger in a La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ sample than the observed values in La$_{1-x}$Sr$_{x}$MnO$_{3}$ samples (with x=0.3 and 0.4). They also calculated the theoretical reduction in specific heat upon application of a magnetic field for the ideal simple-cubic (3D) and simple-square (2D) lattices and concluded that the observed change in specific heat for the bilayered manganite was large, but still less than that for an ideal 2D ferromagnetism. Besides, Okuda et al.[@Okuda] reported values of $\beta _{1}$=3 mJ/mole-K$^{2}$ for a La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ sample, an order of magnitude smaller than the one found by us. Because their $\beta _{1}$ value was similar to the one found in three dimensional perovskites, like La$_{0.7}$Sr$_{0.3}$MnO$_{3}$[@Okuda0], they concluded that dimensionality did not affect the value of the electron-electron interaction constant. However, our result shows that the quasi-2D confinement of the electrons in La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ do increase the electron-electron interaction constant in comparison to the 3D counterpart. As expected, a term of the type T$^{3/2}$ appears only for the La$_{1.35}$Sr$% _{1.65}$Mn$_{2}$O$_{7}$ sample due to its ferromagnetic interactions. B. F. Woodfield et al.[@Woodfield] studied the specific heat in La$_{1-x}$Sr$% _{x}$MnO$_{3}$ with [*x*]{} between 0.1 and 0.3 and found $\beta _{3/2}$ values in the interval 0.9 to 3.7 mJ/mole-K$^{5/2}$. Our $\beta _{3/2}$ values, for La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$, are approximately equals to the 3D counterpart and roughly duplicate upon the application of a 9 T magnetic field. Figure 1c (also in fig 2) shows that the La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ curves make a downward turn at low temperatures in both [*H=0*]{} and [*9 T*]{} and this region is not well fitted by equation 1. The values of $\beta _{1}$ for this sample (see table 1) apply only for temperatures higher than 4 K, and reveal a very high electronic interactions. Martinho et al. [@Martinho] interpreted the specific heat of La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$ samples ([*0.29x 0.51*]{}) as thermal excitations of a two-dimensional gas of ferromagnetic magnons. However, they did not discuss the very low temperature interval (below 4 K). In figure 3 we re-scale the specific heat data in the temperature interval below [*4 K*]{} for the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$% _{7}$ and La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ crystals. The x-axis is now equal to the inverse of temperature and the y-axis in presented in logarithmic scale to facilitate the comparison with an exponential decay law. In both, [*H=0*]{} and [*9 T*]{}, the data points are well fitted by straight lines. The estimated energy gap ([*E*]{}$_{gap}$) in the La$_{2-2x}$Sr$_{1+2x}$Mn$% _{2}$O$_{7}$ crystal was [*0.30 meV*]{} and [*0.57 meV*]{} for zero and 9 T, respectively. On the other hand, the estimated [*E*]{}$_{gap}$ in the La$% _{1.5}$Sr$_{0.5}$NiO$_{4}$ crystal was [*0.63 meV*]{} and [*0.65 meV*]{} for zero and [*9 T*]{}, respectively. The graph for La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ remind us the structure-related superconductor La$_{1.5}$Sr$_{0.5}$CuO$_{4}$. In a superconductor an exponential decay in the specific heat is interpreted as the opening of a gap in the electronic structure[@Kittel]. However, our sample, differently from superconductors, did not show a noticeable dependence with a magnetic field up to [*9 T*]{}. If a BCS-like theory[@Kittel] were to be valid in this crystal ([*E*]{}$_{gap}$[*= 7/2 k*]{}$_{B}$[* T*]{}$_{c}$) the sample should have a corresponding critical temperature ([*T*]{}$_{c}$) at about [*2 K*]{}. Figure 4 shows a detailed measurement of the zero field cooling ([*ZFC*]{}) magnetization in the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ crystal, between 1.8 and 10 K, with applied magnetic field of 20 Oe (a) and 50 Oe (b). Measurements were done with the magnetic field parallel (triangles) and perpendicular (squares) to the [*c*]{} axes and temperature steps of [*0.1 K*]{}. The results reveal a clear anisotropy due to the quasi-bidimensional distribution of the magnetic ions. The magnetization shows three features: a minimum close to 2 K, a small maximum close to 4 K and a plateau close to 7 K. Figure 5 shows [*ZFC* ]{}magnetization measurement in the La$_{1.5}$Sr$% _{0.5}$NiO$_{4}$ crystal between 1.8 and 10 K with applied magnetic field of 1 T (a) and 3 T (b). The magnetic field was applied parallel (triangles) and perpendicular (squares) to the [*c*]{} axes and the temperature steps were of [*0.1 K*]{}. These graphs also display an anisotropy behavior due to the quasi-bidimensional distribution of the magnetic ions. The magnetization here shows two features: a maximum close to 2 K and a minimum close to 4 K in the orientation of the applied magnetic field parallel to the [*c*]{} axes. The absolute value of magnetization is higher in La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$, due to its ferromagnetic alignment, in comparison with the antiferromagnetic alignment in La$_{1.5}$Sr$_{0.5}$NiO$_{4}$. In both cases the position of the peaks close to 2 K in the magnetization curves (figures 4 and 5) seems to be correlated to the exponential decay in the specific heat (figure 3). Recently, Boothroyd[@Boothroyd] et al. studied a single crystal of La$% _{1.5}$Sr$_{0.5}$NiO$_{4}$ with polarized neutrons at 10 K. They made neutron energy scans with the direction of the incident beam fixed and aligned parallel, as well as perpendicular, to the NiO layers. They found inter-plane correlations at low energies (negligible for E $\geq $ 5 meV) and a reduction in intensity below an energy of 4 meV. Given that neutrons scatter from spin fluctuations perpendicular to the wave vector, these observations indicated that the intensity reduction below 4 meV was due to the freezing out of the c-axis component of the spin fluctuations. They also pointed out the existence of a 4 meV energy gap due to single-ion out-of-plane anisotropy. Although we found in the La$_{1.5}$Sr$_{0.5}$NiO$% _{4}$ crystal, using specific heat measurements, a gap value smaller than the one reported by Boothroyd et al.[@Boothroyd], both results, one microscopically and the other macroscopically, seem to confirm the complex magnetic excitation and band structure due to charge ordering and the quasi-2D confinement of the Ni ions. Further studies are clearly necessary to elucidate better these points. Conclusions =========== Single crystals of LaMnO$_{3}$, La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$, La$% _{1.5}$Sr$_{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ were characterized by magnetization and specific heat measurements. The bilayer compound La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ presents a ferromagnetic transition, while the other studied compositions show an antiferromagnetic order. Spin-wave excitations in the bilayer-structure La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ are suppressed by a 9 T magnetic field as indicated by specific heat measurements. This is attributed to the reduced magnetic dimensionality. The effect is larger than in the case of 3D compounds, but not as large as expected in an ideal 2D system. Unlike a previous report[@Okuda], we found that the quasi-2D confinement of the electrons in La$% _{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ do increase the electron-electron interaction constant in comparison to its 3D counterpart. The specific heat does not change with a 9 T magnetic field in LaMnO$_{3}$, La$_{1.5}$Sr$% _{0.5} $NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$. However, electronic excitations are drastically removed at very low temperatures in the La$% _{1.5} $Sr$_{0.5}$NiO$_{4}$ single crystal, as revealed by the downward turn in the specific heat. Below [*4 K*]{}, we also found that the specific heat data for the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ crystals could be fitted by an exponential decay law. From the fittings we were able to estimate characteristic energy gaps for both compounds. Detailed magnetization measurements in this low temperature interval showed the existence, close to 2 K, of a maximum for La$_{1.5}$Sr$% _{0.5}$NiO$_{4}$ and a minimum for La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$. Our measurements of magnetizations and specific heat, combined with a previous report on neutron diffraction[@Boothroyd], suggest the existence of an anisotropy gap in the energy spectrum of the La$_{1.35}$Sr$% _{1.65}$Mn$_{2}$O$_{7}$ and La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ compounds. Acknowledgments =============== We thank the Brazilian science agencies FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for financial support. Table 1. Results of the fitting to the law $C=\sum \beta _{2n+1}\,T^{\text{\/% }2n+1}$[* +* ]{}$\beta _{3/2}\,T^{\text{\/}3/2}$, with [*n*]{} from 0 to 4, for the four studied single crystals and [*H= 0*]{} and [*9 T*]{}. The units are mJ/mole-K$^{j}$, where [*j*]{} is the subscript of the coefficient. --------------------------------------------------------------------------------------------------------------------------------------- Sample H ( T ) $\beta _{1}$ $\beta _{3/2}$ $\beta _{3}$ $\beta $\beta _{7}$ $\beta _{9}$ _{5} $ --------------------------------------- --------- -------------- ---------------- -------------- -------- -------------- -------------- LaMnO$_{3}$ 0 1.1 0 0.20 0.31 -0.48 2.0 9 1.1 0 0.20 0.31 -0.48 2.0 La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ 0 24 2.4 0.12 1.3 -1.4 5.3 9 14 4.4 0.20 0.98 -1.0 3.5 La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ 0 16 0 0.18 0.67 -0.79 2.8 9 16 0 0.17 0.71 -0.82 2.9 La$_{1.5}$Sr$_{0.5}$CoO$_{4}$ 0 1.4 0 0.12 0.61 -0.76 2.9 9 1.9 0 0.12 0.61 -0.76 2.9 --------------------------------------------------------------------------------------------------------------------------------------- Figure Captions Figure 1. Specific heat measurements between 2 and 30 K in the single crystals LaMnO$_{3}$, La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$, La$_{1.5}$Sr$% _{0.5}$NiO$_{4}$ and La$_{1.5}$Sr$_{0.5}$CoO$_{4}$, with [*H=0*]{} and [*9 T*]{}. Continuous lines represent the fitting of the law: $C=\sum \beta _{2n+1}\,T^{\text{\/}2n+1}$[* +* ]{}$\beta _{3/2}\,T^{\text{\/}3/2}$. The data is plotted as [*C/T*]{} vs. [*T*]{}$^{2}$ to facilitate the interpretation. Figure 2. Specific heat of La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$% _{1.5}$Sr$_{0.5}$NiO$_{4}$ crystals below 10 K. Close symbols represent the measurements with [*H=0*]{} and open ones with [*9 T*]{}. Figure 3. Re-scaled specific heat data in the low temperature interval (below [*4 K*]{}) for the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$_{7}$ and La$% _{1.5}$Sr$_{0.5}$NiO$_{4}$ crystals. The x-axis is equal to the inverse of temperature and the y-axis in presented in logarithmic scale to facilitate the test of an exponential decay law. Figure 4. Magnetization measurements in the La$_{1.35}$Sr$_{1.65}$Mn$_{2}$O$% _{7}$ crystal between 1.8 and 10 K with applied magnetic field of 20 Oe (a) and 50 Oe (b). Measurements were done with the magnetic field parallel (triangles) and perpendicular (squares) to the [*c*]{} axes. The results reveal an anisotropy due to the quasi-bidimensional distribution of the magnetic ions. Figure 5. Magnetization measurements in the La$_{1.5}$Sr$_{0.5}$NiO$_{4}$ crystal between 1.8 and 10 K with applied magnetic field of 1 T (a) and 3 T (b). Magnetic field was applied parallel (triangles) and perpendicular (squares) to the [*c*]{} axes. The small peak close to 2 K, in the parallel orientation to the [*c*]{} axes, might be correlated with a gap in the electronic structure. P. G. Radaelli, D. E. Cox, M. Marezio and S-W. Cheong, Phys. Rev. B 55 (5) 3015 (1997) Guo-meng Zhao, K. Ghosh and R. L. Greene, J. Phys.: Condens. Matter 10, L737 (1998) Y. Moritomo, Phys. Rev. B 60 (14) 10374 (1999) Gang Xiao, G. Q. Gong, C. L. Canedy, E. J. McNiff, Jr. and A. Gupta, J. Appl. Phys. 81 (8) 5324 (1997) S. Mori, C. H. Chen and S.-W. Cheong, Nature 392, 473 (1998) J. López, P. N. Lisboa-Filho, W. A. C. Passos, W. A. Ortiz and F. M. Araujo-Moreira, Journal of Magnetism and Magnetic Materials 226-230, 507-508 (2001) J. López, P. N. Lisboa-Filho, W. A. C. Passos, W. A. Ortiz, F. M. Araujo-Moreira, O. F. de Lima, D. Schaniel and Kartik Ghosh, Phys. Rev. B 63 (22) 224422 (2001) T. Okuda, T. Kimura and Y. Tokura, Phys. Rev. B 60, 3370 (1999) L. Ghivelder, I. A. Castillo, M. A. Gusmão, J. A. Alonso and L. F. Cohen, Phys. Rev. B 60 (17) 12184 (1999) J. López, P. N. Lisboa-Filho, O. F. de Lima and F. M. Araujo-Moreira, Journal of Magnetism and Magnetic Materials (242-245P2) 683-685 (2002) J. López, O. F. de Lima, P. N. Lisboa-Filho and F. M. Araujo-Moreira, Phys. Rev. B 66, 214402 (2002) I. A. Zaliznyak, J. P. Hill, J. M. Tranquada, R. Erwin, Y. Moritomo, Preprint cond-mat/0006115 V. Sachan, D. J. Buttrey, J. M. Tranquada, J. E. Lorenzo, and G. Shirane, Phys. Rev. B 51, 12742 (1995) P. Wochner, J. M. Tranquada, D. J. Buttrey and V. Sachan, Phys. Rev. B 57, 1066 (1998) R. Kajimoto, K. Ishizaka, H. Yoshizawa and Y. Tokura, Phys. Rev. B 67, 014511 (2003) D. Prabhakaran, P. Islab, A.T. Boothroyd, J. Crystal Growth 237, 815 (2002). J. E. Gordon, R. A. Fisher, Y. X. Jia, N. E. Phillips, S. F. Reklis, D. A. Wright and A. Zettl, Phys. Rev. B 59 (1) 127 (1999) Myron B. Salamon and Marcelo Jaime, Reviews of Modern Physics 73, 583 (2001) V. N. Smolyaninova, K. Ghosh and R. L. Greene, Phys. Rev. B 58 (22) R14725 (1998) T. Okuda, A. Asamitsu, Y. Tomioka, T. Kimura, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. 81, 3203 (1998) B. F. Woodfield, M. L. Wilson and J. M. Byers, Phys. Rev. Lett. 78 (16) 3201 (1997) H. Martinho, C. Rettori, D. L Huber, J. F. Mitchell and S. B. Oseroff, Phys. Rev. B 67, 214428 (2003) Kittel, Charles, Introduction to Solid State Physics, 7th Ed., Wiley, (1996) A.T. Boothroyd, P.G. Freeman, D. Prabhakaran, M. Enderle, J. Kulda, PHYSICA B 345 (1-4): 1 (2004)

--- author: - | A.P.Kharel$^1$, K.H.Soon$^1$, J.R.Powell$^1$, A. Porch$^1$, M.J.Lancaster$^1$, A. V. Velichko$^{1,2}$ and R.G.Humphreys$^3$\ [$^1$School of Electronics & Electrical Engineering, University of Birmingham, UK]{}\ [$^2$Institute of Radiophysics & Electronics of NAS, Ukraine]{}\ [$^3$DERA, Malvern WR14 3PS, UK]{} title: | Non-linear Microwave Surface Impedance of Epitaxial\ HTS Thin Films in Low DC Magnetic Fields --- 11.5 pt Introduction ============ Understanding the mechanisms of the non-linearity of high-$T_c$ superconductors (HTS) at microwave frequencies is very important from the point of view of application of the materials in both passive and active microwave devices [@Heinrev]. Recently, unusual features such as decrease of the surface resistance $R_s$ and reactance $X_s$ of HTS thin films with microwave field $H_{r\!f}$ have been reported [@Hein97; @Khar98]. Similar observations were made in weak ($\leq 20$ mT) static fields $H_{dc}$ [@Hein97; @Choudh97], which have shown that a small dc magnetic field can cause a decrease of $R_s$ and $X_s$ in both the linear and nonlinear regimes. In the present paper we report measurements of the microwave field dependences of $R_s$ and $X_s$ of high-quality epitaxial YBaCuO thin films in zero and finite ($\leq 12$ mT) applied dc magnetic fields. All the samples have rather different functional form of $R_s(H_{r\!f})$, but $X_s(H_{r\!f})$ is universal and nearly temperature-independent. At the same time, $H_{dc}$ applied parallel to the c-axis of the films has a qualitatively similar effect on both $R_s(H_{r\!f})$ and $X_s(H_{r\!f})$, giving evidence of non-monotonic behavior of $R_s$ and $X_s$ as a function of $H_{dc}$ both in the linear and nonlinear regimes. An even more striking feature is that for some of the samples the dc field can decrease $R_s$ below its low-power zero-field value, thereby offering a possible way of reducing the microwave losses of HTS thin films. Experimental Results {#exp} ==================== The films are deposited by e-beam co-evaporation onto polished (001)-oriented MgO single crystal $10\times10$ mm$^2$ substrates. The films are 350 nm thick. The c-axis misalignment of the films is typically less than 1$\%$, and the $dc$ critical current density $J_c$ at 77 K is around $2\cdot10^6$ A/cm$^2$ [@Chew]. The films were patterned into linear coplanar transmission line resonators with resonance frequency of $\sim 8$ GHz using the technique described in [@Porch]. The nonlinear measurements were performed using a vector network analyzer with a microwave amplifier providing CW output power up to 0.3 W. The low-power values of $R_s$ and $\lambda$ at 15 K are 60, 35, 50 $\mu\Omega$ and 260, 210, 135 nm for samples TF1, TF2 and TF3, respectively. \#1\#2[0.45\#1]{} Changes in $R_s$ and $X_s$ with $H_{r\!f}$, $\Delta R_s=R_s(H_{r\!f})-R_s(0)$ and $\Delta X_s=X_s(H_{r\!f})-X_s(0)$, are plotted in Fig. \[fig1\] for all three samples. It is seen that the $H_{r\!f}$-dependence of $\Delta R_s$ is rather different for different samples, whereas $\Delta X_s(H_{r\!f})$ is universal. For sample TF1, $\Delta R_s\sim H_{r\!f}^2$ from the lowest $H_{r\!f}$. For sample TF2, a decrease in $R_s$ is observed with increased $H_{r\!f}$, and the absolute value of $R_s$ falls below the corresponding low-power value. Finally, for sample TF3, $\Delta R_s$ is rather independent of $H_{r\!f}$ up to sufficiently high fields ($\sim$60 kA/m), after which a skewing of the resonance curve is observed. The surface reactance, $X_s$, for all three samples is a sublinear function of $H_{r\!f}$ ($\sim H_{r\!f}^n$, $n<1$) at low powers, then has a kink, followed by a superlinear functional dependence ($\sim H_{r\!f}^n$, $n>1$). The effect of dc magnetic fields ($\leq 12$ mT) on the microwave power dependence of $R_s$ and $X_s$ for all the samples is illustrated in Fig. \[fig2\] and Fig. \[fig3\]. The common feature for all three samples is that the dependences of $R_s(H_{r\!f})$ and $X_s(H_{r\!f})$ upon $H_{dc}$ are non-monotonic. For samples TF1 and TF2 (for particular $H_{r\!f}$-range and $H_{dc}$-values), the static field leads to a decrease in $R_s$ compared to the low-power zero-field value. This means that both dc and rf fields can cause a [*reduction of the microwave losses*]{} in YBaCuO (see Fig. \[fig1\]a and Fig. \[fig2\]a,b). A possible mechanism of such a behavior is discussed later. \#1\#2[0.5\#1]{} \#1\#2[0.5\#1]{} \[fig3\] One can see that for sample TF1, a dc field of a certain strength (10 mT) can cause a decrease in $R_s$, whereas $X_s$ is always enhanced by a dc field. Similarly, for sample TF3, the behavior of $R_s(H_{r\!f})$ and $X_s(H_{r\!f})$ in $H_{dc}$ is also uncorrelated. However, for TF1 we observe a reduction of $R_s$ without an accompanying decrease in $X_s$, whereas for TF3 the effect is opposite; for particular values of $H_{dc}$ (5, 10 mT) the in-field ($H_{dc}\neq 0$) value of $X_s(H_{r\!f})$ is lower than the corresponding value for $H_{dc}=0$ (Fig. \[fig2\]c), while the in-field value of $R_s(H_{r\!f})$ is always higher than corresponding zero-field value (Fig. \[fig2\]c). Here, the most pronounced decrease in $X_s$ for TF3 is observed at $H_{dc}=5$ mT. Finally, for sample TF2 there is a well pronounced correlated behavior of $R_s(H_{r\!f})$ and $X_s(H_{r\!f})$ in a dc field; $H_{dc}$ of any value from 5 to 12 mT decreases both $R_s$ and $X_s$ (see Fig. \[fig2\]b and Fig. \[fig3\]b). Discussion and Conclusions ========================== A powerful approach in distinguishing between various non-linear mechanisms is a parametric representation of the data in terms of the $r$ parameter, where $r=\Delta R_s/\Delta X_s$ [@Halb97; @Golos95]. In Fig. \[fig4\] we plotted the $H_{r\!f}$-dependence of the $r$ parameter for all three samples in different dc magnetic fields from 0 to 12 mT. One can see that for sample TF1, all the in-field $r(H_{r\!f})$ curves almost collapse over the entire range of $H_{r\!f}$, whereas the zero-field $r(H_{r\!f})$ data are clearly different from the in-field ones. This is especially noticeable at low $H_{r\!f}$ (between 3-7 kA/m), where the $r$ values differ by up to a factor of 10 between the zero-field and in-field $r(H_{r\!f})$ dependences. At the same time, at the lowest $H_{r\!f}$ (2-3 kA/m), the zero-field $r$-values match very well with the in-field ones (see Fig. \[fig4\]a). Therefore, the low-power nonlinearity for sample TF1 appears to have the same origin for zero-field and in-field regimes, whereas the high-power range mechanisms are likely to be different. For sample TF2 at $H_{dc}=5$ mT and 10 mT, $r(H_{r\!f})$ is rather noisy, which clearly correlates with the noisy dependence of $X_s(H_{r\!f})$ for this sample at the relevant dc fields (see inset in Fig. \[fig3\]b). The $r$ parameter oscillates between $-$4 and 6 with an average values close to 0.3–0.4 and 0.2–0.3 for 5 and 10 mT, respectively. For zero field and 12 mT, $r(H_{r\!f})$ are quite consistent, starting to increase from large negative values $\sim -1$ at low powers, and saturating to the level of $-$0.2 to $-$0.1 at higher $H_{r\!f}$. \#1\#2[0.5\#1]{} Finally, for sample TF3 at zero dc field, $r$ increases with $H_{r\!f}$ at low powers, whereas the 10 and 12 mT $r$-values decrease, but all three curves level off for $H_{r\!f}\geq 10$ kA/m to a value of $\sim 0.1$. However, $r(H_{r\!f})$ at 12 mT appears to tend to small negative values at high $H_{r\!f}$, consistent with the decrease in $R_s(H_{r\!f})$ at 12 mT in the relevant $H_{r\!f}$ range (Fig. \[fig2\]c). Standing apart from other dependences is $r(H_{r\!f})$ at 5 mT, which shows very high values ($\sim 2$) at low $H_{r\!f}$, saturating at a level of $\sim 0.4$ at higher $H_{r\!f}$. Note that this value is about a factor of 4 larger than the saturation level for other curves. This seems to imply that $H_{dc}=5$ mT causes a switching of the mechanism of the nonlinearity in the film, as compared to the mechanism at other fields, including zero-field results. Recently Ma et al. [@Ma] have found that YBaCuO thin films deposited by the same method exhibit correlation of $R_s(H_{r\!f})$ with the values of low-power residual $\lambda_{res}$ and the normal-fluid conductivity $\sigma_n$. At the same time, they failed to note any correlation between the power dependence and $R_{res}$. A similar conclusion can be drawn from our results (see Fig. \[fig1\]a). One can see that $R_s$ is almost independent of $H_{r\!f}$ for sample TF3, which has the lowest $\lambda(15~K)=135$ nm, whereas sample TF1 with the largest $\lambda(15~K)=260$ nm exhibits the strongest $H_{r\!f}$-dependence. On the other hand, there is no strict correlation between $R_s(H_{r\!f})$ and low-power $R_s$ (see Sec. \[exp\] and Fig. \[fig1\]a), which is also consistent with the results of Ma et al. However, the strongest power dependence, $R_s\sim H_{r\!f}^2$, is observed for sample TF1 with both the highest $R_{res}$ and $\lambda_{res}$, in agreement with recent results on YBaCuO thin films with different low-power characteristics [@Velich]. There are two further distinctive features for samples TF1, when compared with the two other samples. The functional form of $R_s(H_{r\!f})$ is noticeably changed by a dc field ($R_s\sim H_{r\!f}^n$, where $n=$2, 1.12, 0.8 and 1.24 for 0, 5, 10 and 12 mT, respectively), whereas $X_s(H_{r\!f})$ is not affected by $H_{dc}$. In addition, for TF1, a dc magnetic field changes not only the power dependence of $R_s$, but the absolute value of the low-power $X_s$ (see Fig. \[fig2\]a), while for TF2 and TF3 no such effect is observed (Fig. \[fig2\]b,c). The effect of a dc field on $R_s(H_{r\!f})$ is also seen for sample TF3 at $H_{dc}=5$ mT which, as will be argued later, may switch the mechanism of nonlinearity for this sample. Recently Habib et al. [@Habib] have found that for a stripline resonator with a weak link in the middle, $R_s(H_{r\!f})$ is strongly affected by the junction, whereas $X_s(H_{r\!f})$ was found to be insensitive to the presence of the weak link. Based on this finding, we can suggest that the difference between $R_s(H_{r\!f})$ for our samples may originate from different microstructure (type, dimension and number of defects) of the samples, whereas the similar form of $X_s(H_{r\!f})$ appears to reflect the intrinsic behavior of each film, mostly exhibited by grains. This assumption is further supported by the strong effect of a small dc field on $R_s(H_{r\!f})$ for samples TF1 (Fig. \[fig2\]a) and TF3 (at 5 mT, Fig. \[fig2\]c), whereas the functional form of $X_s(H_{r\!f})$ is unchanged by $H_{dc}$. Analysis of Possible Mechanisms ------------------------------- As we have shown earlier [@Khar98], such uncorrelated behavior of $R_s(H_{r\!f})$ and $X_s(H_{r\!f})$, as we observed for our samples (Fig. \[fig1\]), cannot be explained by any of the known theoretical models, including Josephson vortices [@Halb97] (where $r_{JF}<1$, $\Delta R_s$,$\Delta X_s\sim H_{r\!f}^n$, $0.5<n<2$), heating of weak links [@Golos95] ($r_{HE}<1$, $\Delta R_s,\Delta X_s\sim H^2$) and the RSJ model [@Halb97] ($r_{RSJ}<1$, $\Delta R_s$ increasing in a stepwise manner and $\Delta X_s$ oscillating with $H_{r\!f}$), intrinsic pair breaking or uniform heating [@Golos95] (for both mechanisms $r<10^{-2}$, and $\Delta R_s,\Delta X_s\sim H^2$). We can also rule out the mechanism of the superconductivity stimulation by microwave radiation [@Eliash], recently claimed by us [@Khar98] and Choudhury et al. [@Choudh97]. In this mechanism, the dc magnetic field decreases the order parameter, increasing both $R_s$ and $X_s$, which we do not observe for any of our samples. Moreover, we see that even in the low-power regime $H_{dc}$ can cause reduction of both $\Delta R_s$ and $\Delta X_s$, which is not explained by the above model at all. The most plausible mechanism responsible for the decrease in $R_s$ and $X_s$ with both $H_{r\!f}$ and $H_{dc}$ fields seems to be field-induced alignment of the spins of magnetic impurities, which are likely to be present in most HTS (particularly in YBaCuO) [@Kres1]. This mechanism was recently claimed by Hein et al. [@Hein97] to explain their results on non-monotonic behavior of $R_s$ and $X_s$ in $H_{dc}$ and $H_{r\!f}$ for YBaCuO thin films. However, because our non-linear results for $\Delta R_s$ and $\Delta X_s$ are not correlated, and moreover, exhibit different $R_s(H_{r\!f})$ dependences, we suppose that other strong nonlinear mechanism(s) may interfere with the spin-alignment mechanism. We suggest that this mechanism might be Cooper pair breaking at low powers, and nucleation and motion of rf-vortices [@Halb97] at higher powers. Heating effects at high powers may also play an important role. However, additional investigations are necessary to answer this question unambiguously. In conclusion, we have presented here the results on non-monotonic microwave power dependence of $R_s$ and $X_s$ in both zero and weak ($\leq 12$ mT) dc magnetic field for very high-quality epitaxial YBaCuO thin films. Since this unusual behavior has come into being only owing to a significant progress in the thin films fabrication for the past few years, we conclude that the features observed by us seem to originate from the intrinsic properties of superconductors. However, different functional form of $R_s(H_{r\!f})$ for different samples and universal $X_s(H_{r\!f})$ behavior seem to imply that the microstructure still plays a significant role in the macroscopic properties of the samples. In addition, the observed decreases in $R_s$ and $X_s$ below their zero-field low-power values means that there is still room for improvement of the microwave properties of the thin films. This can be realized upon adequate understanding of the mechanisms responsible for the unusual behavior observed, and can lead to improved characteristics of HTS-based microwave devices. M. A. Hein, “Microwave properties of high-temperature superconductors: surface impedance, circuits and systems” in [*Studies of High-Temperature Superconductors*]{}, vol. 18, A. Narlikar Editor, Eds. New York: Nova Sci. Publ. , 1996, pp. 141–216. M.A. Hein, C. Bauer, W. Diete, S. Hensen, T. Kaiser, V. Z. Kresin, G. Müller, “Magnetic field-induced recovery of superconductivity in epitaxial $Y\!Ba_2Cu_3O_{7-\delta}$ films at microwave frequencies, [*J.Supercond.*]{}, vol. 10, pp. 485-493, May 1997. A. P. Kharel, A. V. Velichko, J. R. Powell, A. Porch, M. J. Lancaster, and R.G. Humphreys, “Unusual features in the nonlinear microwave surface impedance of $Y\!BaCuO$ thin films”, [*Phys. Rev. B*]{}, in press. D. P. Choudhury, B. A. Willemsen, J. S. Derov and S. Sridhar, “Nonlinear response of HTSC thin film microwave resonators in an applied DC magnetic field”, [*IEEE Trans. Appl. Supercond.*]{}, vol. 7, pp. 1260-1263, June 1997. N.G. Chew, S. W. Goodyear, J. A. Edwards, J. S. Satchell, S. E. Blenkinsop, R. G. Humphreys, “Effect of small changes in composition on the electrical and structural properties of $Y\!BaCuO$ thin-films”, [*Appl. Phys. Lett.*]{}, vol. 57, pp. 2016-2018, October 1990. A. Porch, M.J. Lancaster, R.G. Humphreys, “The coplanar resonator technique for determining the surface impedance $Y\!Ba_2Cu_3O_{7-\delta}$ thin-films”, [*IEEE Trans. MTT*]{}, vol. 43, pp. 306-314, February 1995. J. Halbritter, “Nonlinear surface impedance of “low” and “high-T$_c$” superconductors”, [*J. Supercond.*]{}, vol. 10, pp. 91-96, February 1997. M. A. Golosovsky, H. J. Snortland, M. R. Beasley, “Nonlinear microwave properties of superconducting Nb microstrip resonators”, [*Phys. Rev. B*]{}, vol. 51, pp. 6462-6469, May 1997. Z. Ma, E. de Obaldia, G. Hampel, P. Polakos, P. Mankiewich et al. , “RF power dependence study of large area YBCO thin films” [*IEEE Trans. Appl. Supercond.*]{}, vol. 7, pp. 1911-1916, June 1997. A. V. Velichko, E. V. Izhyk, A. Ya. Kirichenko, N. T. Cherpak, and A. V. Moroz, “Microwave-power dependent surface resistance of YBaCuO superconductors. Material quality linkage,” [*Supercond. Sci. Technol.*]{}, vol. 11, pp. 716-729, August 1997. Y. M. Habib, D. E. Oates, G. Dresselhaus and M. S. Dresselhaus, “Power dependence of microwave $Z_s$ in high-T$_c$ Josephson junctions: measurements and modeling”, [*IEEE Trans. Appl. Supercond.*]{}, vol. 7, pp. 2553-2557, June 1997. G. M. Eliashberg, “Superconductivity of films stimulated by high-frequency field”, [*Sov. JETP Lett.*]{}, vol. 11, pp. 186-188, March 1970. Yu.N. Ovchinnikov and V.Z. Kresin, “Recovery of superconductivity and critical field in layered superconductors”, [*Phys. Rev. B*]{}, vol. 54, pp. 1251-1263, January 1996.

--- abstract: 'We construct an epidemic model for the transmission of dengue fever with an early-life stage in the vector dynamics and age-structure within hosts. The early-life stage of the vector is modeled via a general function that supports multiple vector densities. The [*basic reproductive number*]{} and [*vector demographic threshold*]{} are computed to study the local and global stability of the infection-free state. A numerical framework is implemented and simulations are performed.' author: - 'Fabio Sanchez[^1] and Juan G. Calvo[^2]' title: 'Dengue model with early-life stage of vectors and age-structure within host' --- Introduction {#sec:intro} ============ Dengue fever has been a burden to public health officials in the tropics and subtropics since the 20th century [@cdc; @harris2000]. Dengue virus, of the genus *Flavivirus* of the family Flaviviridae, is an infectious disease transmitted by the mosquitoes [*Aedes aegypti*]{} and [*Aedes albopictus*]{} [@gubler1998]. There are four serotypes of the dengue virus, called DEN-1, DEN-2, DEN-3, and DEN-4. After infection of one serotype, the infected person acquires lifelong immunity for that specific serotype and short-term immunity to other serotypes [@cdc]. There are two stages in the transmission cycle of dengue that have been reported. There is an enzootic transmission cycle between primates mostly in forests, with transmission between vectors through feeding from infected animals [@gubler1998]. These infected mosquitoes rarely wander far from the forest, so the infection to human populations comes from humans or livestock who visit a forest with presence of the virus, encounter an infected vector, become infected, and then infect the mosquitoes in their population center, which then can spread the disease to the rest of the population. In the case of rural, small populations, since the population rapidly gets saturated with the infection and subsequently immunized, the epidemic usually is short-termed. The other stage is between vectors and humans, where vectors bite an infected human and can potentially become infected. In this work we will focus on the interaction between vectors and humans. The model in [@sanchez2006] focuses on the early-life stage of the vector and explores the effect of multiple vector densities on dengue outbreaks. Previous work on dengue models mostly focus on the adult vector-host interactions; see, e.g., [@feng1997; @esteva1998; @esteva1999; @sanchez2012; @manore2014; @murillo2014; @brauer2016; @sanchez2018]. The model we present here incorporates age-structure within the host population, as well as the early-life stage of the vector as in [@sanchez2006]. The inclusion of age-structure in the human/host population can help to determine prevention and control strategies based on specific population age groups and other social factors inherent to a subgroup of the population at risk. This article is organized as follows. In Section \[sec:model\], we outline the mathematical model. In Section \[sec:R0\], we compute the [*vector demographic threshold*]{}, the [*basic reproductive number*]{}, and determine the stability of the system. Section \[sec:num\] includes the numerical scheme and numerical simulations, confirming the theoretical results. Finally, in Section \[sec:disc\] we present some relevant conclusions and final thoughts. Mathematical model {#sec:model} ================== We consider a compartmental model with age-structure within the host population, with susceptible, infectious and recovered hosts, denoted by $S_h(t,a)$, $I_h(t,a)$ and $R_h(t,a)$, respectively. Vectors are described by three states: $E(t)$ (egg/larvae at time $t$), $S_v(t)$ (number of non-infected vectors) and $I_v(t)$ (number of infected vectors). Hosts and vectors are coupled via a transmission process, where susceptible hosts can become infected at rate $\beta(a) \frac{I_v(t)}{N_v(t)}$, where $\beta(a)$ represents the age-dependent contact rate (vector-human) and $N_v(t)=S_v(t)+I_v(t)$ is the total number of vectors in the system. The number of new hosts coming into the system, $\Lambda$, is assumed to be constant. Infected individuals can recover at rate $\gamma(a)$, and all hosts exit the system at rate $\mu_h(a)$. We will restrict ourselves to the case of proportional mixing: $$p(t,a) = \frac{c(a)n(t,a)}{\int_0^\infty c(a) n(t,a)\ da},$$ with $c(a)$ the age-specific per-capita contact/activity rate. We then define the force of infection $$B(t) = \int_0^\infty \frac{I_h(t,a)}{n(t,a)} p(t,a)da,$$ where $n(t,a)= S_h(t,a)+I_h(t,a)+R_h(t,a)$ is the population density and $\int_{0}^{\infty}n(t,a)da$ is the total population. In the vector classes, we consider that eggs enter the system at rate $f(N_v)$, where $f(N_v)$ is assumed to be a Kolmogorov type function, $f(N_v)=N_v g(N_v)$, with $g:\mathbb{R}^+\rightarrow \mathbb{R}^+$ a differentiable function such that $g(0)>0$, $g(\infty)=0$. They also exit the system at rate $\mu_e$ and become mosquitoes at rate $\delta$. Mosquitoes become infectious at rate $\beta_v B(t)$, where $\beta_v$ is the transmission rate from humans to mosquitoes and $B(t)$ is the force of infection. Mosquitoes also exit the system at rate $\mu_v$. In our analysis, we assume that parameters $\delta, \mu_e, \mu_v, \beta_v$ are constant and $\beta, \mu_h, \gamma$ are continuous functions on age. The model we just described is given by the system of differential equations: \[eq:system\] $$\begin{aligned} \begin{split} \frac{dE}{dt} &= f(N_v)-(\delta+\mu_e)E, \end{split}\\ \begin{split} \frac{dS_v}{dt} &= \delta E-\beta_v S_v B(t) -\mu_v S_v, \end{split}\\ \begin{split} \frac{dI_v}{dt} &= \beta_v S_v B(t)-\mu_v I_v, \end{split}\\ \begin{split} \Big(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\Big)S_h(t,a) &= -\beta(a)S_h(t,a)\frac{I_v}{N_v}-\mu_h(a)S_h(t,a), \end{split}\\ \begin{split} \Big(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\Big)I_h(t,a) &= \beta(a)S_h(t,a)\frac{I_v}{N_v}-(\mu_h(a)+\gamma(a))I_h(t,a), \end{split}\\ \begin{split} \Big(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\Big)R_h(t,a) &= \gamma(a)I_h(t,a)-\mu_h(a)R_h(t,a), \end{split}\end{aligned}$$ along with initial conditions given by $$\begin{array}{rlrlrl} E(0) &= E_0, & S_v(0) &=S_{v_0}, & I_v(0) &= I_{v_0} \\ S_h(t,0) &= \Lambda, & I_v(t,0) &=0, & R_h(t,0)&=0,\\ S_h(0,a) &= S_{h_0}(a), & I_h(0,a) &= I_{h_0}(a), & R_h(0,a)&=R_{h_0}(a). \end{array}$$ The total host population satisfies $$\Big(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\Big)n(t,a) = -\mu_h(a) n(t,a),$$ and we then can compute explicitly that $$\label{eq_nta} n(t,a) = \left\lbrace \begin{array}{cl} n_0(a-t) \dfrac{\mathcal{F}(a)}{\mathcal{F}(a-t)} & {\rm if }\ a\geq t, \\ \Lambda \mathcal{F}(a) & {\rm if }\ a<t, \\ \end{array} \right.$$ where $$\mathcal{F}(a) = e^{-\int_0^a \mu_h(s)\ ds}$$ is the proportion of individuals that survive at age $a$. Therefore, we define $$\begin{aligned} n^*(a) &:= \lim_{t\rightarrow \infty} n(t,a) = \Lambda \mathcal{F}(a),\\ p_\infty(a) &:= \lim_{t\rightarrow \infty} p(t,a) = \frac{c(a)\mathcal{F}(a)}{\int_0^\infty c(a) \mathcal{F}(a)\ da}.\end{aligned}$$ Model analysis {#sec:R0} ============== In this section we explore the conditions for multiple vector demographic steady states and determine their stability. We also compute the [*basic reproductive number*]{} and analyze local and global stability for the solutions of System . Vector demographic number, $\mathcal{R}_v$ ------------------------------------------ Since $n(t,a)$ is given explicitly in , we first rescale variables $$s_h(t,a) = \frac{S_h(t,a)}{n(t,a)},\quad i_h(t,a) = \frac{I_h(t,a)}{n(t,a)},\quad r_h(t,a) = \frac{R_h(t,a)}{n(t,a)},$$ to obtain the equivalent system \[age\_vsys\_2\] $$\begin{aligned} \begin{split} \frac{dE}{dt} &= f(N_v)-(\delta+\mu_e)E , \end{split}\\ \begin{split} \frac{dS_v}{dt} &= \delta E-\beta_v B(t) S_v -\mu_v S_v, \end{split}\\ \begin{split} \label{eq_Iv} \frac{dI_v}{dt} &= \beta_v B(t) S_v-\mu_v I_v, \end{split}\\ \begin{split} N_v &= S_v+I_v, \end{split}\\ \begin{split} \Big(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\Big)s_h(t,a) &= -\beta(a)s_h(t,a)\frac{I_v}{N_v}, \end{split}\\ \begin{split} \label{eq_ih} \Big(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\Big)i_h(t,a) &= \beta(a)s_h(t,a)\frac{I_v}{N_v}-\gamma(a)i_h(t,a), \end{split}\\ \begin{split} \Big(\frac{\partial}{\partial t}+\frac{\partial}{\partial a}\Big)r_h(t,a) &= \gamma(a)i_h(t,a), \end{split}\\ \begin{split} B(t) &= \int_0^{\infty}p(t,a)i_h(t,a)da. \end{split}\end{aligned}$$ For a given equilibrium state $(E^*, S_{v}^{*}, I_{v}^{*},N_{v}^{*}, s_{h}^{*}(a), i_{h}^{*}(a), r_{h}^{*}(a),B^*)$ of System , we study its local stability by using the perturbations $$\begin{aligned} E(t) &=& E^*+e^{\psi t}\widetilde{E},\\ S_v(t) &=& S_{v}^{*}+e^{\psi t}\widetilde{S_v},\\ I_v(t) &=& I_{v}^{*}+e^{\psi t}\widetilde{I_v},\\N_v(t) &=& N_{v}^{*}+e^{\psi t}\widetilde{N_v},\\ s_h(t,a) &=& s_{h}^{*}(a)+e^{\psi t}\widetilde{s_h}(a),\\ i_h(t,a) &=& i_{h}^{*}(a)+e^{\psi t}\widetilde{i_h}(a),\\ r_h(t,a) &=& r_{h}^{*}(a)+e^{\psi t}\widetilde{r_h}(a),\\ B(t) &=& B^* + e^{\psi t}\widetilde{B},\end{aligned}$$ where $$B^* = \int_0^\infty p_\infty(a) i_{h}^{*}(a)\ da, \quad \widetilde{B} = \int_0^\infty p_\infty(a) \widetilde{i_h}(a)\ da.$$ Linearization leads to the eigenvalue problem $$\begin{aligned} \psi \widetilde{E} &= f^\prime(N_{v}^{*})\widetilde{N_v}-(\mu_e+\delta)\widetilde{E} \label{eqE},\\ \psi \widetilde{N_v} &= \delta \widetilde{E} - \mu_v \widetilde{N_v} \label{eqL},\\ \psi \widetilde{I_v} &= \beta_v (\widetilde{S_v}B^* + S_{v}^{*} \widetilde{B} ) - \mu_v \widetilde{I_v} \label{eqJ},\\ \psi \widetilde{S_v} &=\delta \widetilde{E}-\beta_v ( B^* \widetilde{S_v}+ S_{v}^{*} \widetilde{B})-\mu_v \widetilde{S_v},\\ \dfrac{d}{da}\widetilde{s_h}(a) + \psi \widetilde{s_h}(a) &= -\beta(a)\left( \frac{I_{v}^{*}}{N_{v^*}} \widetilde{s_h}(a) + \frac{\widetilde{I_v}}{N_{v}^{*}}s_{h}^{*}(a) -\frac{I_{v}^{*}}{N_{v}^{*}} \frac{\widetilde{N_v}}{N_{v}^{*}} s_{h}^{*}(a)\right),\\ \dfrac{d}{da}\widetilde{i_h}(a) + \psi \widetilde{i_h}(a) &=\beta(a)\left( \frac{I_{v}^{*}}{N_{v}^{*}} \widetilde{s_h}(a) + \frac{\widetilde{I_v}}{N_{v}^{*}}s_{h}^{*}(a) -\frac{I_{v}^{*}}{N_{v}^{*}} \frac{\widetilde{N_v}}{N_{v}^{*}} s_{h}^{*}(a)\right)-\gamma(a)\widetilde{i_h}(a) \label{eqI},\\ \dfrac{d}{da}\widetilde{r_h}(a) + \psi \widetilde{r_h}(a) &=\gamma(a)\widetilde{i_h}(a).\end{aligned}$$ For $\widetilde{E}, \widetilde{N_v} \neq 0$, equations and imply that $$\dfrac{\delta f'(N_{v}^*)}{(\mu_e+\delta+\psi)(\psi+\mu_v )} = 1.$$ Let $$\mathcal{H}_v(\psi) = \dfrac{\delta f'(N_{v}^*)}{(\mu_e+\delta+\psi)(\psi+\mu_v )}.$$ We then define the [*demographic vector*]{} number $$\mathcal{R}_v = {\cal H}_v(0) = \dfrac{f'(N_{v}^*)}{\phi},$$ where $$\phi = \dfrac{(\delta+\mu_e) \mu_v}{\delta}$$ represents the proportion of eggs that survive to the adult stage. Recall that the rate that eggs enters the system is given by $f(N_v)=N_v g(N_v)$. We can then establish the following result: \[lem:stabNv\] Suppose that the set $g^{-1}(\phi) = \lbrace N_v\in (0,+\infty):g(N_v) = \phi \rbrace$ is non-empty. For each $N_v \in g^{-1}(\phi)$, there exists a positive vector state $$\label{eq_eqState} (E^*, N_v^*)=\left(\dfrac{\mu_v}{\delta} N_v, N_v \right),$$ which is locally asymptotically stable if $\mathcal{R}_v<1$ and unstable otherwise. We have that $(E,N_v)$ satisfies the system $$\begin{aligned} \label{eq:pde_E_Sv} \frac{dE}{dt} &= f(N_v)-(\delta+\mu_e)E,\nonumber \\ \frac{dN_v}{dt} &= \delta E -\mu_v N_v,\end{aligned}$$ with appropriate initial conditions. Therefore, fixed points satisfy $$\begin{aligned} f(N_v^*) &= (\delta+\mu_e)E^*,\nonumber \\ \delta E^* &=\mu_v N_v^*.\end{aligned}$$ Multiplying both equations and using the fact that $f(N_v^*)=N_v^* g(N_v^*)$, we then deduce that $g(N_v^*)= \phi$ (for $E^*N_v^*\neq 0$). Thus, for each $N_v \in g^{-1}(\phi)$ there exists the positive state given in . Moreover, for a fixed state , the associated Jacobian to System is given by $$\left[ \begin{array}{cc} -(\delta +\mu_e) & f'(N_v^*) \\ \delta & -\mu_v \end{array} \right],$$ which eigenvalues are given by $$\frac{1}{2} \left( -(\delta+\mu_e+\mu_v) \pm \sqrt{(\delta+\mu_e+\mu_v)^2-4\mu_v(\mu_e+\delta)\left(1-\mathcal{R}_v \right)} \right).$$ If $R_v<1$, we then conclude that both eigenvalues have negative real part and is locally stable. [\[rem\_gNv\] Since we assume that $f(N_v)=N_vg(N_v)$, it is straightforward to verify that $$\mathcal{R}_v = 1+g'(N_v^*) \dfrac{N_v^*}{\phi}.$$ Thus, equilibrium points given by are locally stable if $g'(N_v^*)<0$, and unstable otherwise; see Figure \[fig:stabNv\*\]. ]{} ![An example for $g(N_v)$ as a function of $N_v$ for which multiple steady states exist. The dashed line corresponds to the value of $\phi$. Each intersection of both curves corresponds to an endemic state $ N_v^* \in g^{-1}(\phi)$. Black filled dots correspond to stable points since $g'(N_v^*)<0$ and circles correspond to unstable fixed points; see Lemma \[lem:stabNv\] and Remark \[rem\_gNv\]. In this case, $N_v^*=0$ is unstable; see Lemma \[lem:Rv0\]. \[fig:stabNv\*\]](fig_g_N){width=".7\textwidth"} \[lem:Rv0\] The vector-free state $(E^*,S_v^*,I_v^*) = (0,0,0)$ is locally asymptotically stable if $g(0)<\phi$, and unstable otherwise. Suppose that $g(0)<\phi$. For $N_v^*=0$, $\mathcal{R}_v$ simplifies to $\mathcal{R}_v = \dfrac{g(0)}{\phi}<1$. Since ${\cal H}_v: [0,\infty)$ is a decreasing function of $\psi$, the equation $\mathcal{H}_v(\psi)=1$ can have only solutions with negative real part, and $(E^*,N_v^*)$ is locally stable. The result then follows since $S_v^*, I_v^*$ are non negative and $N_v^* = S_v^* + I_v^*$. If $g(0)>\phi$, then $\mathcal{H}_v(\psi)=1$ has one positive solution and the result holds. Basic reproductive number, $\mathcal{R}_0$ ------------------------------------------ Consider now the disease-free state $$(E^*, S_{v}^{*}, I_{v}^{*},N_{v}^{*}, s_{h}^{*}(a), i_{h}^{*}(a), r_{h}^{*}(a),B^*) = (E^*,S_v^*,0,S_v^*,1,0,0,0)$$ for System . From we get $$\dfrac{\widetilde{I_v}}{N_{v}^*} = \dfrac{\beta_v \widetilde{B}}{\psi + \mu_v}.$$ Substituting in and solving, we obtain that $$\widetilde{{i}(a)} = \dfrac{\beta_v \widetilde{B}}{\psi + \mu_v} \int_0^a \beta(\tau) e^{-\int_\tau^a (\psi + \gamma(h))\ dh}\ d\tau.$$ Multiplying by $p_\infty(a)$ and integrating with respect to $a$, we deduce that $$\widetilde{B} = \dfrac{\beta_v \widetilde{B}}{\psi + \mu_v} \int_0^\infty \int_0^a p_\infty(a) \beta(\tau) e^{-\int_\tau^a (\psi + \gamma(h))\ dh}\ d\tau\ da.$$ For $\widetilde{B}\neq 0$, we obtain that $$\dfrac{\beta_v}{\psi+\mu_v} \int_0^\infty \int_0^a p_\infty(a) \beta(\tau) e^{-\int_\tau^a (\psi + \gamma(h))\ dh}\ d\tau\ da = 1.$$ Let $$G(\psi) = \dfrac{\beta_v}{\psi+\mu_v} \int_0^\infty \int_0^a p_\infty(a) \beta(\tau) e^{-\int_\tau^a (\psi + \gamma(h))\ dh}\ d\tau\ da.$$ We then define the [*basic reproductive number*]{} $$\label{eq_defR0} \mathcal{R}_0= G(0) = \dfrac{\beta_v}{\mu_v} \int_0^\infty \int_0^a p_\infty(a) \beta(\tau) e^{-\int_\tau^a \gamma(h)\ dh}\ d\tau\ da.$$ In the particular case of constant parameters, it reduces to $$\mathcal{R}_0 = \dfrac{\beta_v \beta}{\mu_v (\mu_h+\gamma)}.$$ We then have the following results: \[th:R0\] Assume that $\mathcal{R}_0 < 1$. Then, the disease-free solution of System is globally asymptotically stable. From we have that $$i_h(t,a)=\int_0^a \beta(\tau) e^{-\int_\tau^a \gamma(h)\ dh} s_h(\tau+t-a,\tau) \dfrac{I_v(\tau+t-a)}{N_v(\tau+t-a}\ d\tau$$ for $t>a$. Multiplying by $p(t,a)$ and integrating with respect to $a$ we get $$B(t) =\int_0^\infty \int_0^a \beta(\tau) p(t,a) e^{-\int_\tau^a \gamma(h)\ dh} s_h(\tau+t-a,\tau) \dfrac{I_v(\tau+t-a)}{N_v(\tau+t-a)}\ d\tau.$$ Since $s(t,a)\leq 1$, $$B(t) \leq \int_0^\infty \int_0^a \beta(\tau) p(t,a) e^{-\int_\tau^a \gamma(h)\ dh} \dfrac{I_v(\tau+t-a)}{N_v(\tau+t-a)}\ d\tau,$$ and therefore $$\label{eq2} B^* \leq \dfrac{I_v^*}{N_v^*} \int_0^\infty \int_0^a \beta(\tau) p_\infty(a) e^{-\int_\tau^a \gamma(h)\ dh} \ d\tau.$$ From , if $S_v^* = 0$ then $I_v^*=0$. Otherwise, it holds that $$\label{eq3} B^* = \dfrac{\mu_v}{\beta_v} \dfrac{I_v^*}{S_v^*}.$$ Combining , , and using , we obtain that $$0\leq \dfrac{I_v^*}{N_v^*} \leq \dfrac{I_v^*}{S_v^*}\leq \dfrac{I_v^*}{N_v^*} \mathcal{R}_0,$$ since $S_v^*\leq N_v^*$. By assumption, $\mathcal{R}_0<1$, and therefore $I_v^*=0$. Hence, $B^*=0$ and $i_h^*(a)=0$. \[th:endemic\] Assume that there exists $N \in \lbrace N\in (0,+\infty):g(N) = \phi \rbrace$ with $g'(N)<0$. If $\mathcal{R}_0>1$, there exists one endemic non-uniform stable steady state for System . It is straightforward to verify that there exists the endemic state $$\begin{aligned} N_v^* &= N,\\ E^* &= N \dfrac{\mu_v}{\delta}, \\ S_v^* &= N \frac{\mu_v}{(\mu_v + \beta_v B^*)},\\ I_v^* &= N \frac{\beta_v B^* }{(\mu_v + \beta_v B^*)}.\end{aligned}$$ By hypothesis and Remark \[rem\_gNv\], it holds that $(E^*,N_v^*)$ is a positive local stable fixed point. We will prove that $B^*>0$ which implies that $I_v^*>0$. A non-uniform steady state for System is a solution of the nonlinear system $$\begin{aligned} \label{eq:systB*} \dfrac{I_{v}^{*}}{N_{v}^{*}} &= \frac{\beta_v B^*}{\mu_v + \beta_v B^*},\nonumber\\ \dfrac{d s_{h}^{*}(a)}{da} &= -\beta(a)s_{h}^{*}(a)\dfrac{I_{v}^{*}}{N_{v}^{*}},\nonumber\\ \dfrac{d i_{h}^{*}(a)}{da} &= \beta(a)s_{h}^{*}(a)\dfrac{I_{v}^{*}}{N_{v}^{*}}-\gamma(a) i_{h}^{*}(a),\\ \dfrac{d r_{h}^{*}(a)}{da} &= \gamma(a) i_{h}^{*}(a),\nonumber\\ B^* &= \int_0^\infty p_\infty(a) i_{h}^{*}(a)\ da,\nonumber\end{aligned}$$ for $a>0$, with initial conditions $$s_{h}^{*}(0)=1,\ i_{h}^{*}(0)=0,\ r_{h}^{*}(0)=0.$$ Consider the linear system of equations with parameter $B$ given by $$\begin{aligned} \label{eq:systB} \dfrac{d s^*_{h_B(a)}}{da} &= -\beta(a)s_{h}^{*}(a)\frac{\beta_v B}{(\mu_v + \beta_v B)},\nonumber\\ \dfrac{d i^*_{h_B(a)}}{da} &= \beta(a)s_{h}^{*}(a)\frac{\beta_v B}{(\mu_v + \beta_v B)}-\gamma(a) i_{h}^{*}(a),\\ \dfrac{d r^*_{h_B(a)}}{da} &= \gamma(a) i_{h}^{*}(a),\nonumber\end{aligned}$$ for $a>0$, with initial conditions $$s_{h_B}^{*}(0)=1,\ i_{h_B}^{*}(0)=0,\ r_{h_B}^{*}(0)=0.$$ Given the solution $(s_B^*(a),i_B^*(a),r_B^*(a))$ of system , define $$H(B)= \int_0^\infty i_B^*(a) p_\infty(a)\ da.$$ It holds that $(s_B^*(a),i_B^*(a),r_B^*(a))$ satisfies system if and only if $B$ is a fixed point of $H$; i.e., $H(B) = B$. Moreover, if $B=0$ then $i_B^*(a)=0$ and $H(0)=0$. Thus, in order to guarantee existence of at least one non-trivial solution to , it is just necessary to prove that $H(B)$ has a positive fixed point. Solving for $s_{h_B}^{*}(a)$ and $i_{h_B}^{*}(a)$, it follows that $$\begin{aligned} s_{h_B}^{*}(a) &= e^{\frac{- \beta_v B}{\mu_v + \beta_v B} \int_0^a \beta(h)\ dh},\\ i_{h_B}^{*}(a) &= \frac{\beta_v B}{\mu_v + \beta_v B}\int_0^a e^{-\int_\tau^a \gamma(h)\ dh} \beta(\tau)s_{h_B}^*(\tau)\ d\tau.\end{aligned}$$ Define $$G(B):=\dfrac{H(B)}{B}\quad {\rm for\ } B\neq 0.$$ The function $G:[0,1]\rightarrow\mathbb{R}$ is continuous by defining $G(0) = H'(0)$. We have that $$G(0) = \frac{\beta_v}{\mu_v} \int_0^\infty \int_0^a e^{-\int_\tau^a \gamma(h)\ dh} \beta(\tau) p_\infty(a)\ d\tau\ da = \mathcal{R}_0 > 1$$ and $G(1)=H(1)<1$. Therefore, there exists $B^* \in (0,1)$ such that $G(B^*)=1$, i.e., $H(B^*)=B^*$ and there exists at least one endemic state for $i_h^*(a)$. Moreover, $B^*$ is unique since $G(B)$ is strictly decreasing. In particular, $B^*>0$ implies that $I_v^*>0$, reaching an endemic steady state, both for vectors and humans. Numerical implementation {#sec:num} ======================== For simplicity, we discretize the system of partial differential equations with a first-order upwind finite difference scheme. We approximate the solution on the physical domain of interest given by the rectangle $\lbrace(t,a):t\in [0,T], a\in [0,A]\rbrace$. We divide the intervals $[0,T]$ and $[0,A]$ in $N_T$ and $N_A$ subintervals, respectively, and consider the grid given by the nodes $$(t_j,a_k) = \left( j\Delta t, k\Delta a\right),$$ for $j\in\lbrace 0,1,\ldots,N_T\rbrace$, $k\in\lbrace 0,1,\ldots,N_A\rbrace$, where $$\Delta t := \frac{T}{N_T},\quad \Delta a := \frac{A}{N_A}$$ are the corresponding step sizes. For any function $x$ and a grid point $(t_j,a_k)$, we denote the approximation of $x(t_j,a_k)$ by $x_k^j$. If the function depends only on age or time, it is denoted simply by $x_k$ or $x^j$. We approximate the force of infection $B(t_j)$ by the composite trapezium rule $$B^j = \Delta a \left( \sum_{k=1}^{N_A-1} p_j^k\ (i_h)_j^k\ da + \frac{1}{2}p_j^{N_A} (i_h)_j^{N_A} \right).$$ We then have the explicit scheme given by the equations $$\begin{aligned} (N_v)^{j} &= (S_v)^{j} + (I_v)^{j},\\ E^{j+1} &= E^{j}+\Delta t\left[f\left((N_v)^{j}\right)-(\delta + \mu_e)E^{j}\right],\\ (S_v)^{j+1} &= E^{j}+\Delta t\left[\delta E^{j} - \beta_v B^j (S_v)^j - \mu_v (S_v)^j\right],\\ (I_v)^{j+1} &= (I_v)^{j}+\Delta t\left[ \beta_v B^j (S_v)^j - \mu_v (I_v)^j\right],\\ (s_h)^{j+1}_k &= (s_h)^{j}_k + \Delta t \left[- \beta_k (s_h)^j_k \dfrac{(I_v)^j}{(N_v)^j} - \frac{(s_h)^j_k - (s_h)^{j}_{k-1}}{\Delta a}\right],\\ (i_h)^{j+1}_k &= (i_h)^{j}_k + \Delta t\left[ \beta_k (s_h)^j_k \dfrac{(I_v)^j}{(N_v)^j} - \gamma_k (i_h)^j_k - \frac{(i_h)^j_k - (i_h)^{j}_{k-1}}{\Delta a} \right],\\ (r_h)^{j+1}_k &= (r_h)^{j}_k+ \Delta t\left[ \gamma_k (i_h)^j_k - \frac{(r_h)^j_k - (r_h)^{j}_{k-1}}{\Delta a} \right],\\\end{aligned}$$ for $1\leq k\leq N_A$ and $0\leq j\leq N_T-1$. Thus, given the initial conditions $E^0$, $(S_v)^0 $, $(I_v)^0$, $s_0^j$, $s_k^0$, $i_0^j$, $i_k^0$, $r_0^j$, $r_k^0$, we can compute the values of the unknowns on the grid points in successive time steps. We present different scenarios where we confirm the results from Theorems \[th:R0\] and \[th:endemic\]. For the age dependent parameters, we show the distributions used in Figure \[fig:ex1\_gamma\_c\]. We consider three different functions $f(N_v) = N_v g(N_v)$ in the following sections: a logistic-type function, a case with multiple vector steady-states, and a seasonal example. Logistic growth {#sec:numexp} --------------- We first consider $$g(N_v) = r\left (1-\dfrac{N_v}{N_{\max}}\right)$$ for given constants $r$ (mosquito growth rate) and $N_{\max}$ (maximum number of mosquitoes that the system can hold); see Figure \[fig\_g\]. Recall that for a positive steady-state on vectors we require $\phi=g(N_v)$. In this case: 1. If $\phi>r$, there exists only the trivial state $N_v^*=0$, which is stable since $\mathcal{R}_v = \dfrac{g(0)}{\phi} <1$; see Figure \[fig\_g2\]. 2. For $\phi < r$, besides the unstable state $N_v^*=0$, we have the non-zero state $$N_v^*= N_{\max}\left(1-\dfrac{\phi}{r}\right),$$ which is stable since $g'(N_v^*)<0$; see Remark \[rem\_gNv\] and Figure \[fig\_g1\]. \[ex1\] We first confirm that the condition $\mathcal{R}_0>1$ could lead to an endemic state, as long as there is a stable positive steady-state for vectors. We consider a set of parameters for which $\phi \approx 3.39$ and $\mathcal{R}_0 \approx 1.60$. First, if $r=0.20$, the only stable fixed point is $(E^*,S_v^*,I_v^*)=(0,0,0)$ for which $i^*(a)=0$; see Figures \[fig:ex1a\], \[fig:ex1b\]. Second, if $r=5$, we have the stable fixed point $(E^*,S_v^*,I_v^*) \approx (9633,7505,88824)$. In this case, we have an endemic state as shown in Figures \[fig:ex1c\], \[fig:ex1d\], according to Theorem \[th:endemic\]. \[ex2\] In this example we confirm that the condition $\mathcal{R}_0<1$ is sufficient to guarantee a disease-free steady state. We take $r=5$ and reduce $\beta$ such that $\mathcal{R}_0<1$. The infected class reaches a disease-free state as shown in Figure \[fig:ex2a\], according to Theorem \[th:R0\]. Even though there exists a positive state for vectors $(E^*,N_v^*)$ as shown in Figure \[fig:ex2b\], we observe that $I_v^*=0$. \[ex3\] We now consider the case $\mathcal{R}_0>1$ with initial conditions $E_0=0$, $S_{v_0} = 10$, $I_{v_0} = 1$, $i_h(0,a) = r_h(0,a) =0$ (no infected or immune humans at time $t=0$); see results for the infected class in Figure \[fig:ex3\]. It is clear that $\mathcal{R}_0>1$ guarantees an endemic state as long as vectors can survive. Multiple vector demographic states ---------------------------------- In a second set of experiments we use $$\label{eq:g2} g(N_v) = r e^{-N_v/c_1} (\sin(c_2 N_v)+1),$$ for given constants $r$, $c_1$, $c_2$. Here, $r$ is the vector per-capita fertility rate, $c_1$ is a form of vector control and $c_2$ represents the variations in vector densities; for a particular choice of parameters see Figure \[fig:stabNv\*\]. Equation represents the different growth rates of vectors for the wet and dry seasons. In this way, we simulate variations based on vector control efforts, obtaining multiple vector demographic steady states for $(E^*,N_v^*)$. For the particular choice of parameters we have used, we obtain eight positive fixed points. Numerically we confirm that four of them are locally stable. \[ex4\] We first confirm the result proved in Lemma \[lem:Rv0\]. We observe that if $r<\phi$, the infection-free steady state is stable and unstable otherwise; results for $(E_0,S_{v0},I_{v0})=(10,10,10)$ are shown in Figure \[fig:ex4a\]. We then obtain different solutions for different initial conditions for the vector classes for which different positive steady-states are reached; see Figure \[fig:g2\_difI\]. Despite having multiple vector densities the outbreaks are similar in severity. This implies that even when vector density is low an outbreak is possible. \[ex5\] Similarly as Example \[ex2\], we confirm that $\mathcal{R}_0<1$ is sufficient for the disease to die out, even in the presence of a positive population of vectors. In this case, $(E^*,N_v^*) = (7430, 74305)$, but $I_v^*=0$; see Figure \[fig:ex5\]. Even when the [*demographic vector number*]{} is bigger than one the disease can be under control when $\mathcal{R}_0<1$. \[ex6\] Similarly as Example \[ex3\], we confirm that $\mathcal{R}_0>1$ implies the existence of an endemic state, as long as a positive equilibrium state exists for the vectors to survive; see Figure \[fig:ex6\]. Effect of seasonality on dengue dynamics {#sec:seasons} ---------------------------------------- In most places where dengue is endemic, seasonal variations in vector populations play a major role in disease transmission. Moreover, it determines the distribution of resources allocated for preventive/control measures. Typically, dengue incidence is correlated with the rainy season. The importance of understanding seasonal variations per location could potentially help public health officials to allocate resources, as well as having better preventive/control measures to reduce dengue incidence (focused primarily towards the reduction of vector breeding sites). We include some numerical results where $g(N_v)$, $\beta_v$ and $\delta$ depend periodically on time, simulating high and low seasons in the dynamics of vectors. Here we use parameters for the vector classes as in [@sanchez2006]. We consider a population with only susceptible humans. In the vector classes, we include one infected vector in order to observe the propagation of the disease; see results in Figure \[fig:ex8\]. For different values of the transmission rate ($\beta$(a)) the infected host distribution distinctly affects the younger and senior age groups. Discussion {#sec:disc} ========== We have constructed a model with age-structure within host and early-life stage of the vector with a general function $f(N_v)$ that represents the new vectors from the egg/larvae stage in the vector system. This gives the possibility of multiple demographic steady states for the vector population and its stability depends on the [*vector demographic number*]{}, $\mathcal{R}_v$. The local stability of the vector-free state when $R_v<1$ was established. The [*basic reproductive number*]{} was computed and the local and global asymptotic stability of the disease-free equilibrium was determined when $\mathcal{R}_0<1$. When $\mathcal{R}_0>1$ and we have a stable vector demographic steady state ($\mathcal{R}_v>1$), the disease is then endemic. Control measures on the early-life stage of the vector can guarantee an adult vector-free state and hence, the disease dies out. Vector control measures such that $\phi > \max g(N_v)$ implies that vectors will die out independently on the value of $\mathcal{R}_0$. There are important public health implications when we are able to include host age distribution, which can determine better strategies for hospitalized individuals. Furthermore, control measures on the early-life stage of the vector can effectively change the landscape on how public health officials lead prevention efforts before the onset of a dengue outbreak. Acknowledgements ================ We thank the Research Center in Pure and Applied Mathematics and the Mathematics Department at Universidad de Costa Rica for their support during the preparation of this manuscript. The authors gratefully acknowledge institutional support for project B8747 from an UCREA grant from the Vice Rectory for Research at Universidad de Costa Rica. [99]{} ”Some models for epidemics of vector-transmitted diseases”, , [**1**]{}:79–87. , 2019. Available from: <https://www.cdc.gov/Dengue/>. ”Analysis of a dengue disease transmission model”, , [**150**]{}:131–151. ”A model for dengue disease with variable human population”, , [**38**]{}:220–240. ”Competitive exclusion in a vector-host model for the dengue fever”, , [**35**]{}:423–544. ”Resurgent vector-borne diseases as a global health problem”, , [**4**]{}:442–450. ”Clinical, epidemiologic, and virologic features of dengue in the 1998 epidemic in Nicaragua”, , [**63**]{}(1):5–11. ”Comparing dengue and chikungunya emergence and endemic transmission in [*A. aegypti*]{} and [*A. albopictus*]{}”, , [**356**]{}:174–191. “Vertical Transmission in a Two-Strain Model of Dengue Fever”, , [**1**]{}(2):249–271. “Comparative estimation of parameters for dengue and chikungunya in Costa Rica from weekly reported data”, , [**67**]{}(1):163–174. “Models for Dengue Transmission and Control”, . AMS Contemporary Mathematics Series. Gumel A. (Chief Editor), Castillo-Chavez, C., Clemence, D.P. and R.E. Mickens. ”Change in host behavior and its impact on the transmission dynamics of dengue”, in [*International Symposium on Mathematical and Computational Biology*]{}, (Eds. R.P. Mondaini), BIOMAT 2011:191–203. [^1]: Centro de Investigación en Matemática Pura y Aplicada (CIMPA), Escuela de Matemática, Universidad de Costa Rica. San Pedro de Montes de Oca, San José, Costa Rica, 11501. Email: [fabio.sanchez@ucr.ac.cr](fabio.sanchez@ucr.ac.cr) [^2]: Centro de Investigación en Matemática Pura y Aplicada (CIMPA), Escuela de Matemática, Universidad de Costa Rica. San Pedro de Montes de Oca, San José, Costa Rica, 11501Email: [juan.calvo@ucr.ac.cr](juan.calvo@ucr.ac.cr)

--- abstract: 'We propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace–Beltrami operator and thus favor low-frequency eigenfunctions, we aim for a spectrum that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem which we exploit to improve the accuracy of our computed correspondences. To effectively search in this high dimensional space of solutions, we incorporate into our minimization state-of-the-art regularizers. We solve the resulting highly non-linear and non-convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester-type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.' author: - Omri Azencot - Rongjie Lai bibliography: - 'basis-search.bib' title: Shape Analysis via Functional Map Construction and Bases Pursuit --- {width="\textwidth"} Introduction ============ Functional maps (FM) [@ovsjanikov2012functional] were recently introduced in the geometry processing community in the context of shape matching. During the last few years, FM were quickly adopted by many, serving as the key building block in a range of shape analysis frameworks. Applicative instances include mesh quadrangulation or fluid simulation tasks, in addition to the original shape correspondence problem. The goal of this paper is to propose an efficient and easy to code framework for computing improved functional map matrices. The key idea behind FM is that instead of aligning points as in Iterative Closest Point (ICP) approaches [@besl1992method], it is often simpler to align scalar functions defined on the input shapes. Thus, a typical pipeline for computing functional maps is composed of three steps. Given a pair of shapes, one first collects a set of corresponding descriptors, such as the Wave Kernel Signature [@aubry2011wave]. Second, one performs dimensionality reduction by projecting the descriptors onto a spanning subspace of basis functions. Finally, one solves an optimization problem, seeking a matrix that best aligns the projected features, possibly while minimizing additional regularizing terms. [faust\_pod\_vis]{} (4,21) [LB]{} (7,20) [$\mathfrak{b}_1$]{} (16,20) [$\mathfrak{b}_5$]{} (25,20) [$\mathfrak{b}_9$]{} (34,20) [$\mathfrak{b}_{11}$]{} (43,20) [$\mathfrak{b}_{26}$]{} (52,21) [POD]{} (56,20) [$b_1$]{} (65,20) [$b_5$]{} (74,20) [$b_9$]{} (83,20) [$b_{11}$]{} (92,20) [$b_{26}$]{} Numerous extensions to the original pipeline [@ovsjanikov2012functional] were proposed in the literature. These extensions can be generally classified into two research avenues. On the one hand, recent works focus on the formulation of novel regularization terms that can be incorporated into the functional map computation phase. For instance, cycle-consistency is promoted in [@huang2014functional], whereas [@nogneng2017informative] favor the preservation of the given descriptors. On the other hand, some papers aim to improve the functional subspaces onto which the features are being projected. For example, [@kovnatsky2013coupled] design basis elements to account for sign or ordering ambiguities. In this context, our work contributes in that it *combines* the tasks of functional map computation and basis design into a single unified framework. Our formulation allows to harness the advancements in functional map regularization as well as to benefit from the increase in the search space of solutions when the bases are allowed to change during the optimization. Choosing a good basis set is crucial in FM applications. In the original work [@ovsjanikov2012functional], the authors propose to utilize the spectrum of the Laplace–Beltrami (LB) operator as the spanning subspace in the second step of the pipeline. More generally, over the last few years, LB bases became the prevailing choice for function representation in many geometry processing tasks such as computing descriptors [@rustamov2007laplace], distances [@solomon2014earth], and generating shape segments [@reuter2009discrete], just to name a few. While this choice can be optimal under certain conditions [@aflalo2016best], it may sometimes lead to subpar results. To this end, Kovnatsky and others [@kovnatsky2013coupled; @kovnatsky2015functional; @kovnatsky2016madmm; @litany2017fully] optimize for joint diagonalizable (JD) basis elements to improve shape matching tasks. Solving for the bases increases the solution space, allowing to produce better correspondences. In this paper, we address two limitations that appear in most existing work on functional maps. The first shortcoming is related to the common choice of the LB spectrum. While LB encodes the geometry of the surface, it is completely independent of the selected features, potentially introducing large representation errors. Indeed, high frequency signals such as locally supported functions will exhibit poor spectral representations [@nogneng2018improved]. Thus and in contrast to previous work, our approach is based on designing basis elements that are tailored to a given collection of descriptors. In practice, we observe that employing LB-based representations often leads to the elimination of many degrees of freedom that could be re-introduced into the problem. Instead of using LB, we utilize the Proper Orthogonal Decomposition (POD) modes for dimensionality reduction purposes. POD subspaces share many of the advantageous properties of LB—they are orthonormal and have a natural ordering. However, POD modes are superior to LB in capturing high frequency data and thus improve descriptors’ transfer between shapes. The second limitation we alleviate deals with the split between the tasks of basis design and functional map computation. Indeed, most existing work focus only on one of these tasks: facilitating a fixed basis or alternatively using a closed-form solution for the functional map. Instead, we propose to merge these objectives into one larger problem. In practice, this modification leads to an increase in the search space, allowing to find better solutions for a given problem. In addition, we can independently regularize and constrain the bases and the functional map to the specific requirements of the application at hand, leading to a flexible yet effective framework. The main contribution in this work is an effective minimization framework for computing functional basis sets on a pair of shapes and a corresponding functional map. The resulting optimization is unfortunately highly non-linear and non-convex. Nevertheless, we construct a novel and highly efficient Alternating Direction of Multipliers Method (ADMM) scheme. Specifically, our unique choice of auxiliary variables allows to naturally incorporate state-of-the-art complex regularizers promoting e.g., cycle-consistency or metric preservation. Moreover, our scheme converges empirically and we additionally show that our method is similar to a *provably convergent* procedure. We evaluate our approach on several shape analysis tasks including shape matching, joint quadrangulation and function transfer. Our comparison to previous work indicates that our method achieves beyond state-of-the-art results in shape correspondences on challenging scenarios where the shapes do not share the connectivity and or only approximately isometric. Related Work ============ Functional maps [@ovsjanikov2012functional] have recently gained a lot of attention in geometry processing and related fields. Some of the applications in which functional maps were found useful include shape exploration [@rustamov2013map], fluid simulation [@azencot2014functional] and function transfer [@nogneng2018improved]. We refer the interested reader to a recent course discussing the functional map framework and a few related applications [@ovsjanikov2016computing]. One of the main scenarios in which functional maps are employed is for computing shape correspondences between a given pair or collection of shapes. In this context, many works extend the original approach [@ovsjanikov2012functional] to include various regularization terms. For instance, Nogneng and Ovsjanikov  show that minimizing commutativity with descriptor operators leads to better functional maps. In [@huang2014functional], the authors promote consistency with respect to the inverse mapping, and recently, [@ren2018continuous] formulate orientation preserving terms into the functional maps pipeline. In addition to improving the accuracy of functional map matrices, the approaches for extracting point-to-point maps are also under development. [@rodola2015point] cast this problem as a probability density function estimation, whereas [@ezuz2017deblurring] propose to minimize the error from projecting delta functions onto the basis and its orthogonal complement. Finally, [@ren2018continuous] iteratively alternate between improving the map in its spectral and spatial domains. In all of the above works, while the functional map could be computed in any scalar basis, the eigenfunctions of the Laplace–Beltrami operator are typically used. This choice is natural given the wealth of theoretical results related to the LB spectrum, but, on the other hand, it is completely independent of the input descriptors. Recently, [@schonsheck2018nonisometric] proposed to design a basis via a conformal deformation while the other basis set is fixed. Probably closest to our approach is the line of work of [@kovnatsky2013coupled; @kovnatsky2015functional; @kovnatsky2016madmm; @litany2017fully] where the authors look for spectral coefficients such that the resulting basis elements are as close as possible to the LB eigenfunctions while preserving the given constraints. In contrast to their perspective, we advocate the use of a linear domain in which the descriptors are better represented, in addition to the incorporation of different regularizers. We provide a detailed comparison between our method and theirs in \[subsec:cmp\_to\_ajd\]. Motivation and Background ========================= To motivate our approach, we will need the following notation. Let ${\mathcal{M}}_1 = ({\mathcal{V}}_1,{\mathcal{F}}_1)$ and ${\mathcal{M}}_2 = ({\mathcal{V}}_2,{\mathcal{F}}_2)$ be a pair of manifold triangle meshes, where ${\mathcal{V}}_1,{\mathcal{V}}_2$ are their vertex sets and ${\mathcal{F}}_1,{\mathcal{F}}_2$ are their triangle sets. We represent scalar functions using real values on vertices, i.e., $f_1:{\mathcal{V}}_1 \rightarrow {\mathbb{R}}$ is a scalar function on ${\mathcal{M}}_1$, and similarly, $f_2:{\mathcal{V}}_2 \rightarrow {\mathbb{R}}$ is a function on ${\mathcal{M}}_2$. Thus, $f_1$ and $f_2$ are real-valued vectors of sizes $|{\mathcal{V}}_1|=m_1$ and $|{\mathcal{V}}_2|=m_2$, respectively. We define the inner product on ${\mathcal{M}}_1$ to be $$\langle f_1, g_1 \rangle_{{\mathcal{M}}_1} := f_1^T G_1 g_1 \ ,$$ where $G_1 \in {\mathbb{R}}^{m_1 \times m_1}$ is the diagonal (lumped) mass matrix of the nodes of ${\mathcal{M}}_1$ (see e.g., [@botsch2010polygon Chap. 3]), and similarly, we have $\langle f_2, g_2 \rangle_{{\mathcal{M}}_2} = f_2^T G_2 g_2 $. The input to our problem is a collection of functional constraints $\{f_{1\,j}\}_{j=1}^n$ and $\{f_{2\,j}\}_{j=1}^n$ such that $f_{1\,j}$ and $f_{2\,j}$ encode the same information but on different meshes, for every $j$. Finally, we arrange the given constraints in matrices, $$\tilde{F}_1 = [ f_{1 1} \; f_{1 2} \; ... f_{1 n} ] \in {\mathbb{R}}^{m_1 \times n}, \quad \tilde{F}_2 = [ f_{2 1} \; f_{2 2} \; ... f_{2 n} ] \in {\mathbb{R}}^{m_2 \times n} \ .$$ In its most simple form, the task of computing functional maps consists of finding a matrix $C$ that aligns the descriptors, i.e., $$C \, B_1^T G_1 \, \tilde{F}_1 \approx B_2^T G_2 \, \tilde{F}_2 \ ,$$ where $B_j$ is a basis of scalar functions on ${\mathcal{M}}_j$ for $j=1,2$. Typically, $C$ is a moderately sized $k\times k$ matrix with $k<300$. If we assume that the $G_j$ and $\tilde{F}_j$ matrices are fixed, it is natural to ask whether optimizing for $C$ *and* for the $B_j$ matrices will yield improved feature matching. We show in Fig. \[fig:teaser\] an example of a functional map with its bases obtained in this way (left), leading to a high quality map between non-isometric shapes (right). Solving for the bases and map significantly increases the parameters from $k^2$ to $k^2\times k \cdot m_1\times k \cdot m_2$, resulting in a challenging to solve problem as $m_j$ are very large. To deal with this issue, we can consider a subspace of solutions of size $k^2\times k \cdot r_1\times k \cdot r_2$, where $r_j$ represent the spectral dimensions of some fixed bases. That is, instead of finding spatial bases, we look for spectral coefficient matrices onto predefined linear subspaces. ![We plot the error distributions of feature matching when LB and POD bases are used. The above results show that designing POD modes is beneficial both in the spectral and spatial domains.[]{data-label="fig:feature_err"}](feature_err){width="\linewidth"} In practice, most existing work utilize the subspace spanned by the LB eigenfunctions. In this work, we propose to change this common choice and take subspaces that better fit the given descriptors. There are several approaches in the machine learning community that could be investigated to achieve this objective. In this work, we advocate the utilization of the Proper Orthogonal Decomposition (POD) modes [@berkooz1993proper], which can be classified as a linear manifold learning method. Given a set of descriptors, the POD can be easily computed using the Singular Value Decomposition, see Sec. \[sec:imp\]. One of the main reasons for preferring POD modes over other bases is due to the Karhunen–Loéve theorem, stating that these modes best approximate the input data under many choices of norms [@xiu2010numerical]. Therefore, incorporating POD modes instead of the LB spectrum may be considered as a data-driven approach for representing and manipulating signals. We show in Fig. \[fig:faust\_pod\_vis\] a few modes related to the LB operator (left) and resulting from POD computation (right). One significant difference between these bases is that POD modes allow for higher frequencies when compared to a similar truncation of LB. For example, the LB $\mathfrak{b}_5$ is significantly smoother than its related POD $b_5$. Moreover, encoding descriptors in a POD subspace induces less information loss in comparison to LB representations in the context of designing bases and functional maps, as we show below. To quantify the difference between the LB and POD subspaces, we measure the average matching error distributions per mode and per vertex, namely $$\begin{aligned} e_1 = \frac{1}{n} \sum_{j=1}^n \left( C \, B_1^T G_1 \, f_{1\,j} - B_2^T G_2 \, f_{2\,j} \right)^2 \ , \\ e_2 = \frac{1}{n} \sum_{j=1}^n \left( B_2 \, C \, B_1^T G_1 \, f_{1\,j} - f_{2\,j} \right)^2 \ ,\end{aligned}$$ where the squares are taken pointwise, i.e., $e_1 \in {\mathbb{R}}^{k}$ and $e_2 \in {\mathbb{R}}^{m_2}$. In Fig. \[fig:feature\_err\] we compare these errors when the bases are fixed as well as designed. Our results indicate that the fixed POD subspaces are extremely accurate for matching in the spectral domain, but yield the most error spatially. Moreover, LB bases produce poor results when designed, e.g., using joint diagonalization methods [@kovnatsky2013coupled] for both error measures. Finally, designed POD modes give the most accurate estimation in the spatial domain and is second best in the spectral regime. We remark that for the POD case these errors naturally depend on the particular descriptors in use. In our applications, we employ a mixture of features such as the Wave Kernel Signature [@aubry2011wave] or landmarks provided by the user. Functional Map and Basis Search (FMBS) ====================================== Our main goal is to find basis matrices $B_1 \in {\mathbb{R}}^{m_1 \times k}$ and $B_2 \in {\mathbb{R}}^{m_2 \times k}$ and a functional map $C \in {\mathbb{R}}^{k \times k}$ such that these objects best align the constraints $\tilde{F}_1$ and $\tilde{F}_2$. To reduce clutter, we scale each $\tilde{F}_j$ by its corresponding $G_j$ and denote $F_j = G_j \tilde{F_j}, j=1,2$. Formally, we consider the problem $$\label{eq:fmbs} \begin{aligned} & \operatorname*{minimize}_{B_1,B_2,C} & & \frac{1}{2} \left| C \, B_1^T F_1 - B_2^T F_2 \right|_F^2 \\ & \text{subject to} & & B_1^T G_1 \, B_1 = I, \; B_2^T G_2 \, B_2 = I \end{aligned}$$ where the terms $B_j^T F_j$ can be viewed as projecting the constraints onto the basis matrices. The equality conditions given by $B_j^T G_j \, B_j = I$ constrain the bases to be orthogonal with respect to the mass matrix. Unfortunately, the minimization problem  is highly non-linear and non-convex, and thus practical solvers are challenging to construct. To alleviate these difficulties, we propose in the next section a splitting scheme that is based on the Alternating Direction Method of Multipliers (ADMM) [@gabay1975dual; @glowinski1975approximation]. An ADMM Approach to FMBS ======================== The basic idea behind ADMM depends on splitting the original complex optimization task into simpler subproblems that can be solved efficiently. Under certain conditions on the objective function and constraints, it can be shown that ADMM converges. Therefore, ADMM is often the optimization framework of choice, arguably due to its computational complexity and theoretical guarantees. To allow splitting in our problem  above, we introduce the auxiliary variables $B_1'$ and $B_2'$ and arrive at the following optimization $$\label{eq:fmbs_admm} \begin{aligned} & \operatorname*{minimize}& & {\mathcal{E}}_{{\texttt{fid}}}(B_1,B_2,C) \\ & \text{subject to} & & B_1^T G_1 \, B_1' = I, \; B_2^T G_2 \, B_2' = I, \; B_1 = B_1', \; B_2 = B_2' \end{aligned}$$ where ${\mathcal{E}}_{{\texttt{fid}}}(B_1,B_2,C) = \frac{1}{2} \left| C \, B_1^T F_1 - B_2^T F_2 \right|_F^2$ is the *data fidelity* term. We stress that while ADMM can be viewed as a standard optimization technique, the choice of auxiliary variables is highly dependent on the problem and there is no general rule for how to “correctly” set these variables. In particular, our unique choice leads to an empirically converging scheme for a large range of parameters, and it further allows for a natural incorporation of novel regularizers . Finally, we mention that the auxiliary variables linearize the difficult orthogonality constraints which may lead to non-orthogonal bases in practice. However, this issue can be solved in a post-processing step. To minimize  we facilitate an iterative scheme $k=0,1,...$ where at each step, the unknowns are updated in an alternating style. Namely, all the variables are kept fixed except for the one which is being updated. In our case, the update order for the primal variables is $( B_1, B_2, B_1', B_2', C)$, followed by the update of the dual variables $(P_1, P_2, Q_1', Q_2')$. We note that each of the subproblems is at most quadratic in the unknown, and thus can be solved efficiently. In what follows, we discuss in detail each of the update tasks including their formulation and solution. To shorten the mathematical formulations below, we omit the step $k$ with the understanding that the variables are updated in a sequential fashion as shown in Alg. \[alg:fmbs\]. In addition, we denote by ${\mathcal{L}}_j(B_j,B_j',P_j,Q_j')$ the *scaled* Lagrangian terms, i.e., $${\mathcal{L}}_j(B_j,B_j',P_j,Q_j') = \frac{\rho}{2} \left| B_j^T G_j B_j' - I + P_j \right|_F^2 + \frac{\rho}{2} \left| B_j - B_j' + Q_j' \right|_{{\mathcal{M}}_j}^2 \ ,$$ where $j = 1, 2$, and $\rho \in {\mathbb{R}}^{+}$ is a penalty parameter provided by the user and it may be updated during the optimization. Updating the bases, $B_1$ and $B_2$ ----------------------------------- The variable $B_1$ is being updated first, using the estimations of the other variables from the previous step. Specifically, we have $$\begin{aligned} \label{eq:B_pbm} B_1^{k+1} = \operatorname*{arg\,min}_{B_1} \; {\mathcal{E}}_{{\texttt{fid}}}(B_1,B_2,C) + {\mathcal{L}}_1(B_1,B_1',P_1,Q_1') \ .\end{aligned}$$ Computing the first order optimality conditions of  lead to a *Sylvester Equation* of the form $$\label{eq:B1_update} \begin{gathered} F_1 F_1^T \, B_1 \, C^T C + \left( \rho G_1 B_1' B_1'^T G_1 + \rho G_1 \right) B_1 = \\ F_1 F_2^T B_2 C + \rho G_1 B_1' (I-P_1)^T + \rho G_1 (B_1'-Q_1') \ , \end{gathered}$$ which can be efficiently solved with numerical algorithms such as [@golub1979hessenberg] implemented via e.g., [`dlyap`]{} in MATLAB. We emphasize that the dimensionality of Eq.  introduces a practical challenge, as it involves dense matrices of size $m_1 \times m_1$. These concerns, along with other implementation aspects, are considered in Section \[sec:imp\]. The update for $B_2$ is carried after the update of $B_1$, but before the other variables. Therefore, we use the estimate of $B_1$ at step $k+1$, whereas the rest of the variables are taken from the $k$th step. The minimization takes the following form $$\begin{aligned} \label{eq:BT_pbm} B_2^{k+1} = \operatorname*{arg\,min}_{B_2} {\mathcal{E}}_{{\texttt{fid}}}(B_1,B_2,C) + {\mathcal{L}}_2(B_2,B_2',P_2,Q_2') \ .\end{aligned}$$ Problem  is quadratic in $B_2$, and its solution can be computed through the following linear system $$\label{eq:B2_update} \begin{gathered} \left( F_2 F_2^T + \rho G_2 + \rho G_2 B_2' B_2'^T G_2 \right) B_2 = \\ F_2 F_1^T B_1 C^T + \rho G_2 B_2' (I-P_2)^T + \rho G_2 (B_2'-Q_2') \ . \end{gathered}$$ Updating the auxiliary variables, $B_1'$ and $B_2'$ --------------------------------------------------- The minimization problems associated with the unknowns $B_1'$ and $B_2'$ are similar. These optimization problems take the form $$\begin{aligned} B_j'^{k+1} = \operatorname*{arg\,min}_{B_j'} \frac{\rho}{2} \left| B_j^T G_j B_j' - I + P_j \right|_{{\mathcal{M}}_j}^2 + \frac{\rho}{2} \left| B_j - B_j' + Q_j' \right|_{{\mathcal{M}}_j}^2 \ ,\end{aligned}$$ for $j=1,2$. The solution is given via the linear system $$\begin{aligned} \label{eq:BP_update} \left( \rho G_j + \rho G_j B_j B_j^T G_j \right) B_j' = \rho G_j B_j (I-P_j) + \rho G_j (B_j + Q_j') \ .\end{aligned}$$ Updating the functional map, $C$ -------------------------------- Given the basis matrices $B_1$ and $B_2$, finding the best functional map that aligns the constraints in a least squares sense has a closed-form solution. Namely, we want to minimize the term ${\mathcal{E}}_{{\texttt{fid}}}(B_1,B_2,C)$ with respect to $C$, and the solution is given by $$\begin{aligned} \label{eq:C_update} C^{k+1} = \left( B_2^T F_2 \right) \left( B_1^T F_1 \right)^+ \ ,\end{aligned}$$ where $A^+$ is the pseudo-inverse of the matrix $A$. Updating the dual variables, $P_j$ and $Q_j'$ --------------------------------------------- The last step of our scheme is trivial and for $j=1,2,$ it is given by $$\label{eq:dual_update} \begin{aligned} P_j &= P_j + B_j^T G_j \, B_j' - I \ , \\ Q_j' &= Q_j' + B_j - B_j' \ . \end{aligned}$$ We summarize the above steps in pseudocode in Alg. \[alg:fmbs\]. We note that generating ${\mathcal{O}}_1^{-1}$ is computationally prohibitive as ${\mathcal{O}}$ is a *large* and *dense* matrix. However, we significantly reduce the computation costs by representing $B_j$ in a spectral subspace, as we discuss in Sec. \[sec:imp\]. ![Our [`fmbs`]{} algorithm shows good empirical behavior for a large range of parameters. We compute the normalized energy and primal/dual residuals per normalized minimization step, across all our FAUST tests. The graphs above show the averaged and standard deviation of the energy (left), primal residual (middle) and dual residual (right), where all exhibit decay.[]{data-label="fig:admm_stats"}](admm_stats){width="\linewidth"} Provably convergent FMBS ------------------------ Most convergence results related to ADMM handle problems with convex objective functions and linear constraints. Recently, [@WangYinZeng2015_global] and [@gao2018admm] extended the convergence analysis of ADMM to a significantly larger class of problems including non-convex objective terms and non-linear constraints. In particular, in the latter work, the authors investigate the case where *biaffine* constraints are given, namely, constraints involving two variables which become linear when one variable is kept fixed. For instance, our orthogonality conditions $B_j^T G_j B_j' = I$ are exactly of this form. Moreover, [@gao2018admm] relax the convexity requirements on the objective function and allow to include differentiable terms instead. In practice, Alg. \[alg:fmbs\] behaves well and it exhibits energy decrease for many choices of parameters as we show in Fig. \[fig:admm\_stats\] and in Sec. \[sec:eval\], however, it does not satisfy the conditions given in [@gao2018admm]. To show convergence, we consider in App. \[app:prop\_proof\] a different minimization  for which we can show the following result. Under some mild conditions, problem  satisfies all the requirements in [@gao2018admm] and thus its ADMM converges to a constrained stationary point. That is, the sequence of variable updates $\left\{ {\mathcal{X}}^k, {\mathcal{Z}}^k \right\}_{k=0}^\infty$ is bounded and every limit point $({\mathcal{X}}^*,{\mathcal{Z}}^*)$ is a constrained stationary point. {width="\textwidth"} Regularized FMBS ---------------- One of the key aspects of our minimization  is that it introduces many degrees of freedom via the unknowns $B_1, B_2$ and $C$. While in general it is a positive feature of our approach, the associated optimization requires a significant amount of descriptors $n$. To relax this dependency, we propose to incorporate regularization terms into our problem. In particular, we add a consistency regularizer that takes into account the inverse functional map $D$. Moreover, we add an isometry promoting term which is given by commutativity with the LB operator [@ovsjanikov2012functional] and Dirichlet energies that favor smooth basis elements. We note that other regularizers such as descriptor commutativity [@nogneng2017informative] or orientation preservation [@ren2018continuous] may be also considered. Formally, we propose the following objective function $$\begin{aligned} \label{eq:fmbs_regs} & {\mathcal{E}} = {\mathcal{E}}_{{\texttt{fid}}} + \mu_{{\texttt{cfid}}} {\mathcal{E}}_{{\texttt{cfid}}} + \mu_{{\texttt{iso}}} {\mathcal{E}}_{{\texttt{iso}}} + \mu_{{\texttt{dir}}}{\mathcal{E}}_{{\texttt{dir}}} \ , \\ & {\mathcal{E}}_{{\texttt{cfid}}} = \frac{1}{2} | B_1^T F_1 - D \, B_2^T F_2 |_F^2\ , \\ & {\mathcal{E}}_{{\texttt{iso}}} = \frac{1}{2} | C \, B_1^T W_1 \, B_1' - B_2^T W_2 \, B_2' \, C |_F^2\ , \\ & {\mathcal{E}}_{{\texttt{dir}}} = \frac{1}{2} \operatorname{Tr}\left( B_1^T W_1 B_1' \right) + \frac{1}{2} \operatorname{Tr}\left( B_2^T W_2 B_2' \right) \ , \end{aligned}$$ where $\mu_{{\texttt{cfid}}}, \mu_{{\texttt{iso}}}, \mu_{{\texttt{dir}}} \in {\mathbb{R}}^{+}$ are penalty scalars, $\operatorname{Tr}$ yields the trace of a matrix, and $W_j$ is the cotangent weights matrix [@pinkall1993computing] of shape ${\mathcal{M}}_j$ for $j=1,2$. One of the key aspects of our framework resulting from our ADMM formulation  is that it allows to combine challenging regularizers  in a straightforward way. Thus, the formulation of the minimization that uses ${\mathcal{E}}$ and its associated ADMM is somewhat technical, and we defer the derivation to the supplementary material. [cmp\_map\_tex\_lr]{} (12,22) [GT]{} (24,22) [FMAPS]{} (41,22) [AJD]{} (55,22) [CFM]{} (70,22) [DPC]{} (84,22) [Ours]{} Implementation Details {#sec:imp} ====================== In what follows we describe a few technical aspects related to our method including dimensionality reduction of problem , variable initialization, stopping condition and the development platform. #### Dimensionality reduction Solving  directly is computationally prohibitive when the shapes consist many vertices. To overcome this difficulty, we propose to reduce the spatial dimension and use a spectral domain instead, allowing for fast computation times while retaining a significant amount of degrees of freedom. Specifically, we take the left singular vectors obtained by computing the Singular Value Decomposition (SVD) of the given constraints. Namely, $$\tilde{U}_j \tilde{S}_j \tilde{V}_j^T= {\texttt{SVD}} \left( \tilde{F}_j \right) \ , \quad j=1,2 \ ,$$ where we denote $U_j = \tilde{U}_j(: \,,1:r)$ the $r$ most significant modes. In our experiments we choose $r$ such that $U_j$ covers at least $90\%$ of the spectrum. We note that other spectral bases could be considered, e.g., the LB basis itself [@kovnatsky2013coupled]. However, each choice leads to a different optimization having its own set of assumptions and challenges. In Sec. \[sec:eval\], we compare our approach to other methods. To incorporate $U_j$ into our optimization, we denote the changes in boldface and perform the following modifications, $$\begin{aligned} \label{eq:dim_red} \bm{B}_j = U_j^T B_j \ , \quad \bm{F}_j = U_j^T F_j \ , \quad \bm{G}_j = U_j^T G_j \, U_j \ , \quad j=1,2 \ ,\end{aligned}$$ yielding matrices of sizes $r_j \times k$, $r_j \times n$ and $r_j \times r_j$, respectively. Substituting the above components with their high dimensional counterparts is the only change needed to obtain a spectral version of Alg. \[alg:fmbs\]. Finally, given $\bm{B}_j$, we reconstruct $B_j$ via $B_j = U_j \, \bm{B}_j$. We note that while our approach strongly depends on the input features for deriving the low-dimensional subspaces, in our experiments we observed that it works quite well with a variety of descriptors such as WKS and segmentation information. #### Variable initialization and stopping rule In our tests we noticed that our method is robust to the choice of initial values. Nevertheless, we describe the particular values we used in our experiments. To initialize the primal variables $B_j$ and $B_j'$, we take the first $k$ singular vectors of the respective descriptors, $\tilde{F}_j$. This computation is denoted by ${\texttt{SVD}}(\tilde{F}_j,k)$ in Alg. \[alg:fmbs\]. Using these bases, we can solve Eq.  to obtain an initial $C$. The dual variables $P_j$ and $Q_j'$ are set to zero matrices of an appropriate size. The stopping condition we used is based on the primal and dual residuals and is detailed in [@boyd2011distributed Sec. 3.3], where the maximum number of steps is $10,000$. Dataset r $\mu_{{\texttt{cfid}}}$ $\mu_{{\texttt{iso}}}$ $\mu_{{\texttt{dir}}}$ ---------------------- ------ ------------------------- ------------------------ ------------------------ FAUST intra 0.9 FAUST inter 0.9 SCAPE 0.9 Remeshed FAUST intra 0.99 Remeshed FAUST inter 0.99 Remeshed SCAPE 0.99 : The parameter values used in our tests for each dataset.[]{data-label="tab:param"} #### Development platform and parameters We implemented our method in MATLAB, using its built-in optimization tools such as [`dlyap`]{} and [`mldivide`]{}. Our approach was tested on an Intel Core i7 2.6GHz processor with 16GB RAM. We show in Fig. \[fig:timings\_graph\] a runtime comparison to AJD [@kovnatsky2013coupled] and OPC [@ren2018continuous] on meshes of sizes $1k-500k$ vertices. The parameters of our method include the penalty scalars $\mu_{\texttt{cfid}}, \mu_{\texttt{iso}}$ and $\mu_{\texttt{dir}}$ for the different energy terms . We list our choices in Tab. \[tab:param\], which also shows how much of the spectrum we employ, given by the $r$ parameter. Finally, the size of the functional map and the associated bases was $k=20$ unless noted otherwise. ![We compare the total pre-processing and computation times of the above methods on a pair of shapes for a large range of vertex counts, $m$. Our method is significantly faster than AJD and OPC for high vertex counts, where for low number of vertices OPC is more efficient than our approach.[]{data-label="fig:timings_graph"}](timings_graph_mac){width="\linewidth"} Evaluation and Results {#sec:eval} ====================== To evaluate our method, we consider several applications in which functional maps are useful such as extraction of point-to-point maps [@ovsjanikov2012functional], function transfer [@nogneng2018improved], and consistent quadrangulation [@azencot2017consistent]. We test our approach on a variety of datasets including SCAPE [@anguelov2005correlated], TOSCA [@bronstein2008numerical] and FAUST [@bogo2014faust], and SHREC07 [@giorgi2007shape]. In our comparison, we consider other methods for computing functional maps such as functional maps (FMAPS) [@ovsjanikov2012functional], approximate joint diagonalization (AJD) [@kovnatsky2013coupled], coupled functional maps (CFM) [@eynard2016coupled], descriptor preservation via commutativity (DPC) [@nogneng2017informative] and orientation preserving correspondences (OPC) [@ren2018continuous]. In our comparison, we only use the functional map matrices as computed using the above techniques, and we discard any other improvements related to a specific application. In all cases, we used the authors’ recommended parameters or we searched for the best ones. Extracting point-to-point maps ------------------------------ One of the main applications of functional maps is the computation of point-to-point correspondences between pairs of shapes. In our comparison, we consider two different scenarios. The first includes the original FAUST and SCAPE shapes using $20$ landmarks and $100$ Wave Kernel Signature (WKS) [@aubry2011wave] features. In the second case, we remesh the shapes and use consistent segmentation data [@kleiman2018robust] with WKS descriptors. We emphasize that the latter scenario is extremely challenging as it is completely automatic, it involves approximate features, and the meshes have different connectivities. The pairs we use appeared previously in [@kim2011blended; @chen2015robust]. For map extraction we employ the ICP method proposed in [@ovsjanikov2012functional] and the recent BCICP approach [@ren2018continuous], although other methods [@rodola2015point; @ezuz2017deblurring] could be used. Our evaluation metrics include the computation of cumulative geodesic errors [@kim2011blended] and visualization of transferred scalar functions or textures. In Fig. \[fig:lmk\_wks\_cull\_err\] we show the average cumulative geodesic errors of the first scenario. We note that our approach achieves a significant improvement over all the other competing methods. In particular, when ICP extraction is facilitated, our method yields very good results on FAUST intra which involves pairs of different poses of the same people. Interestingly, our method benefits the most from recent advances in map extraction techniques [@ren2018continuous] as can be seen in the second row. Specifically, using BCICP increases the gap between our results vs. others on FAUST inter (different people, different pose) and SCAPE. This hints that our functional map and associated bases introduce more degrees of freedom which could be exploited in elaborated methods such as [@ren2018continuous]. This behavior can be additionally seen in AJD [@kovnatsky2013coupled] BCICP results which surpass most methods even though their ICP measures were lower than others in general. We do not compare to OPC [@ren2018continuous] in this setup as we use non-symmetric landmarks and thus there is no advantage in using their orientation preserving regularization over, e.g., DPC [@nogneng2017informative]. [cmp\_energy\_terms]{} (34,45) [${\mathcal{E}}_{{\texttt{fid}}}$]{} (55,45) [+${\mathcal{E}}_{{\texttt{iso}}}$]{} (78,45) [+${\mathcal{E}}_{{\texttt{dir}}}$]{} We demonstrate the error measures of Scenario $2$ in Fig. \[fig:seg\_wks\_cul\_err\]. We stress that this setup is particularly challenging as the shapes do not share the connectivity and we use automatically computed features. Nevertheless, our method exhibits the best results on FAUST both for the isometric and non-isometric cases when ICP map extraction is applied. Moreover, when we utilize BCICP on FAUST our method and OPC yield the best scores compared to the alternative methods. Finally, the remeshed SCAPE was an extremely difficult test case, leading to mappings of poor quality in general for most methods (notice the $y$-axis gets to $0.6$ instead of $1$). For this dataset, CFM and DPC produced good measures for ICP, and our method and OPC were the highest with BCICP refinement. The point-to-point correspondence allows to map information from the target to the source. In Fig. \[fig:cmp\_map\_tex\], we compare the mappings generated in Scenario $1$ on a single pair of FAUST intra using texture transfer. The meshes in this dataset are in $1-1$ correspondence and thus we can use the ground-truth (GT) map for comparison. Overall, the performance of the tested methods was generally good. However, small parts of the body such as hands and legs were less accurate for FMAPS, AJD and DPC. Moreover, other methods exhibit large errors in the head, whereas ours correctly finds the symmetry line (see the zoom below). Effect of regularization ------------------------ To evaluate the benefits of utilizing regularizing terms, we visualize the map quality via coordinate function transfer in Fig. \[fig:cmp\_energy\_terms\]. Indeed, there is a clear improvement when ${\mathcal{E}}_{{\texttt{iso}}}$ is introduced (middle right) vs. using ${\mathcal{E}}_{{\texttt{fid}}}$ alone (middle left) as can be seen on the chest and head. Adding ${\mathcal{E}}_{{\texttt{dir}}}$ (right) is not beneficial in this case as it is visually indistinguishable from the ${\mathcal{E}}_{{\texttt{iso}}}$ (middle right) case. In addition to this visualization, we also run our algorithm on FAUST and SCAPE in scenario 1, using different regularization configurations. We show in Fig. \[fig:cmp\_energy\_terms\_graph\] the cumulative geodesic error of these tests. For each dataset the solid line represents using only ${\mathcal{E}}_{{\texttt{fid}}}$, the dashed is the result when we incorporate ${\mathcal{E}}_{{\texttt{iso}}}$, the dotted line is produced by adding ${\mathcal{E}}_{{\texttt{dir}}}$, and we get the dash-dot line by minimizing the full ${\mathcal{E}}$. On average, we observe a significant gain when regularization is used (see also the zoomed plots in Fig. \[fig:cmp\_energy\_terms\_graph\]). In particular, the consistency term (dash-dot line) helps both with respect to the accuracy of the results and the empirical convergence of the problem. We note that the Dirichlet penalization (dotted line) improves the results of FAUST, whereas for SCAPE its contribution is less apparent. ![We plot the cumulative geodesic error for maps computed using various regularization settings. Our results indicate that the regularized problems yield better correspondences. See the text for additional details.[]{data-label="fig:cmp_energy_terms_graph"}](cmp_energy_terms_graph){width="\linewidth"} [cmp\_ajd\_lr]{} (5,45) [AJD]{} (5,20) [Ours]{} Comparison with AJD [@kovnatsky2013coupled] {#subsec:cmp_to_ajd} ------------------------------------------- Perhaps closest to our approach is the method that finds approximate joint diagonalized bases of Kovnatsky et al. . In this work, the authors explore an optimization problem which is conceptually similar to ours. However, there are several key differences between our technique and theirs as we detail below. In terms of the energy functional, our technique is fundamentally different from theirs. Their approach favors basis elements which diagonalize the LB operator, leading to smooth functions. However, the disadvantage in this point of view is that one implicitly assumes that smooth basis functions span the descriptors subspace. Unfortunately, many practical descriptors that are currently used in functional map pipelines do not fit into this assumption. Indeed, any high frequency signal such as segment information will undergo a low pass filter which may lead to data loss in practice, as we show in Fig. \[fig:feature\_err\]. In contrast, our method does not favor smooth basis elements and may output high frequency functions, see e.g., Fig. \[fig:faust\_pod\_vis\]. Finally, even when we include Dirichlet terms in our minimization, they are weighted weakly. Another significant difference is in the data fidelity term. The formulation in [@kovnatsky2013coupled] and others [@litany2017fully] fixes the associated functional map $C$ to attain a *particular structure*. Namely, they include a term that takes the following form $$\begin{aligned} \label{eq:ajd_fid} \tilde{{\mathcal{E}}}_{{\texttt{fid}}} = \frac{1}{2} |B_1^T F_1 - B_2^T F_2 |_F^2 \ ,\end{aligned}$$ which can be interpreted as setting $C$ to be $C \approx B_2^T B_1$. There are two disadvantages to formulation  which our approach overcomes. First, regularizing the functional map $C$ is not straightforward as in our formulation , and may lead to *quartic* expressions in the unknowns $B_j$. Indeed, our formulation allows to independently constrain the bases or the functional map and its inverse, manifesting greater flexibility alongside the natural utilization of state-of-the-art regularizers. Second, our method allows for *general* functional map matrices and thus it increases the search space of solutions when compared with AJD frameworks. To summarize, our approach generalizes AJD methods in that it combines work on joint diagonalization and functional map optimization in a unified framework. There are three key differences in our technique. First, we consider a much larger search space of solutions as we utilize the Proper Orthogonal Decomposition (POD) modes which are better suited to the given features, and we further allow for general functional map matrices. Second, since we jointly optimize for the functional map and the bases, we can naturally incorporate regularization terms. Finally, on the algorithmic side, AJD approaches facilitate a constrained minimization tool which is inefficient in practice as can be seen in Fig. \[fig:timings\_graph\] and its convergence is not guaranteed. In contrast, we analyze our approach and show that it is similar to a provably convergent problem. [cmp\_ajd\_tex]{} (23,45) [GT]{} (47.5,45) [AJD]{} (72.5,45) [Ours]{} ![Switching to POD-based design with AJD (blue dash line) yields an improvement over the LB subspaces (blue line). Still, our framework generates correspondences that are more accurate.[]{data-label="fig:cmp_ajd_pod"}](cmp_ajd_pod_err){width="\linewidth"} In addition to this qualitative comparison, we show in Figs. \[fig:cmp\_ajd\] and \[fig:teaser\] the differences between the designed basis elements. Indeed, AJD (top row) produces highly consistent basis functions compared to ours (bottom row). However, we believe that this behavior limits the design process significantly, which may lead to less accurate matching results as can be seen in Figs. \[fig:lmk\_wks\_cull\_err\] and \[fig:seg\_wks\_cul\_err\]. Specifically, we select a pair of shapes from TOSCA and visualize the correspondence differences via texture transfer in Fig. \[fig:cmp\_map\_tex\]. Overall, AJD produces reasonable results as compared to the ground-truth (GT). However, various parts of the shape such as head, legs and tail, display large errors. In contrast, our technique was able to accurately match most areas of the shapes including the challenging parts. To conclude our qualitative comparison, we modify AJD to use POD modes in their design process instead of the LB eigenfunctions and we plot the cumulative geodesic error that was obtained for FAUST in Fig. \[fig:cmp\_ajd\_pod\]. Indeed, switching to POD modes (blue dashed line) yields a large improvement compared to LB-based AJD (blue line). However, our method (red line) is still significantly more accurate, which can be attributed in part to the state-of-the-art regularization terms we include in our optimization. ![Our technique is particularly suited to methods whose input is a functional map with its associated bases such as [@azencot2017consistent]. We demonstrate the consistent quadrangulations obtained by using their method with input generated by our approach.[]{data-label="fig:qremesh"}](qremesh_lr){width="\linewidth"} Consistent quadrangulation and function transfer ------------------------------------------------ The increasing interest in the functional map approach over the last few years lead to the development of techniques which can utilize a given functional map directly, without the need to convert it to a point-to-point map. For instance, [@azencot2017consistent] proposed an optimization framework for designing consistent cross fields on a pair of shapes for the purpose of generating approximately consistent quadrangular remeshings of the input shapes. Our computed functional map and bases can be directly used within their method to produce quad meshes. In Fig. \[fig:qremesh\], we show an example of the remeshing results of two pairs of shapes having different connectivities (left and right) and genus (right). Still, we obtain highly consistent results as can be seen in the matching singularity points (red spheres). We provide an additional instance of this pipeline in Fig. \[fig:cmp\_qremesh\] comparing the quadrangulation achieved with fixed LB bases (left) vs. our technique with designed POD modes (right). Indeed, we observe a much better alignment of isolines and singularity points with our approach compared to Fixed LB. The last application we consider involves the transfer of scalar valued information between shapes. Recently, [@nogneng2018improved] showed that by extending the usual functional basis to include basis products, an improved function transfer can be performed. In Fig. \[fig:func\_transfer\], we utilize this pipeline using our functional map and bases to transfer an extremely challenging data given by a localized Gaussian function. Indeed the transfer is improved using the extended basis as the noise is less severe and the maximum is more localized. Limitations =========== One limitation of our framework is related to the dependencies between the given constraints and our choice of dimensionality reducing subspaces $U_j$. Indeed, one can always add the standard LB spectrum to these subspaces. However, we observe that in general, the results may change depending on the particular subspace in use and its size. For instance, while increasing $r$ allows for greater flexibility for representing scalar functions, it also requires more regularization, otherwise unwanted solutions may potentially become local minimizers. Another shortcoming of our approach is that it tends to produce maps that are less smooth compared to those generated with LB bases. This behavior is somewhat expected, as our bases are designed to potentially transfer high frequency information which in turn leads to less uniform correspondences. We leave further investigation of these aspects to future work. [cmp\_qremesh\_lr]{} (19,38) [Fixed LB]{} (68,38) [Designed POD]{} [func\_transfer]{} (5,3) [Standard transfer]{} (40,3) [Product transfer]{} {width="\linewidth"} Conclusions and Future Work =========================== In this paper, we proposed a method for designing basis elements on a pair of triangle meshes along with an associated functional map. Unlike most existing work which utilize the spectrum of the Laplace–Beltrami operator, our technique adopts the Proper Orthogonal Decomposition (POD) modes to reduce the dimensionality of the problem. This choice introduces many degrees of freedom and it significantly extends the space of potential solutions. To effectively solve the problem, we incorporate state-of-the-art regularization terms which promote consistency, isometry and smoothness. Our optimization scheme is based on the Alternating Direction Method of Multipliers (ADMM) and it consists of easy-to-solve linear or Sylvester-type equations. We show that in practice our method behaves well in terms of convergence, and we additionally prove that a similar problem to ours is guaranteed to converge. We evaluate our machinery in the context of shape matching, function transfer and consistent quadrangulation, and we demonstrate that our results yield a significant improvement over state-of-the-art approaches for computing functional maps. In the future, we would like to characterize the dependencies between the subspaces spanned by the bases to the given constraints and the relation to the functional map. Moreover, we believe that many applications may benefit from the proposed pipeline on a single shape. Namely, generate a self functional map with a set of basis elements defined on the shape. Examples include symmetry detection, fluid simulation and data interpolation, among many other possibilities. On the other hand, extending our framework to handle multiple shapes is an interesting direction as well. Finally, we believe that many of the questions that we consider in our work could benefit from the recent advancements in machine learning with deep neural networks. We plan to investigate how the task of designing a basis and a functional map can be solved using deep learning approaches. Proof of Proposition 1 {#app:prop_proof} ====================== We consider a modified version of our problem  given by $$\label{eq:fmbs_admm2} \begin{aligned} & \operatorname*{minimize}& & {\mathcal{F}}({\mathcal{X}},{\mathcal{Z}}) \\ & \text{subject to} & & {\mathcal{P}}({\mathcal{X}}) + {\mathcal{Q}}({\mathcal{Z}}) = 0 \end{aligned}$$ where ${\mathcal{X}} = (B_1,B_1',\tilde{B}_1',B_2,B_2',\tilde{B}_2', C)$ and ${\mathcal{Z}} = (Z,B_1'',\tilde{B}_1'',B_2'',\tilde{B}_2'')$ are blocks of variables. Further, the objective function is given by $$\begin{aligned} {\mathcal{F}}({\mathcal{X}},{\mathcal{Z}}) &= {\mathcal{G}}({\mathcal{X}}) + {\mathcal{H}}({\mathcal{Z}}) \ , \\ {\mathcal{G}}({\mathcal{X}}) &= \frac{1}{2} | C \tilde{B}_1'^T F_1 - \tilde{B}_2'^T F_2 |_F^2 \ , \\ {\mathcal{H}}({\mathcal{Z}}) &= \frac{\nu}{2} | Z - I |_F^2 \\ &+ \frac{\mu}{2} |B_1''|_{{\mathcal{M}}_1}^2 + \frac{\mu}{2} |\tilde{B}_1''|_{{\mathcal{M}}_1}^2 \\ &+ \frac{\mu}{2} |B_2''|_{{\mathcal{M}}_2}^2 + \frac{\mu}{2} |\tilde{B}_2''|_{{\mathcal{M}}_2}^2 \ .\end{aligned}$$ Finally, the constraints are formed via $$\begin{aligned} {\mathcal{P}}({\mathcal{X}}) = \begin{pmatrix} B_1^T G_1 \, B_1' \\ B_2^T G_2 \, B_2' \\ B_1-B_1' \\ B_2 - B_2' \\ B_1 - \tilde{B}_1' \\ B_2 - \tilde{B}_2' \end{pmatrix} \ , \quad {\mathcal{Q}}({\mathcal{Z}}) = \begin{pmatrix} -Z \\ -Z \\ -B_1'' \\ -B_2'' \\ -\tilde{B}_1'' \\ -\tilde{B}_2'' \end{pmatrix} \ .\end{aligned}$$ Under some mild conditions, problem  satisfies all the requirements in [@gao2018admm] and thus its ADMM converges. #### Proof We need to show that the requirements in Assumption 1 and Assumption 2 in [@gao2018admm] hold. Our variables are updated sequentially in the order $B_1,B_1',\tilde{B}_1',B_2,B_2',\tilde{B}_2', C$ and a single block of $(Z,B_1'',\tilde{B}_1'',B_2'',\tilde{B}_2'')$. We have that $\operatorname{Im}({\mathcal{Q}}) \supseteq \operatorname{Im}({\mathcal{P}})$ since the image of ${\mathcal{Q}}$ is spanned by the identity matrix in each of the components. The objective function ${\mathcal{F}}({\mathcal{X}},{\mathcal{Z}})$ is coercive on the feasible set $\{ ({\mathcal{X}},{\mathcal{Z}}) | {\mathcal{P}}({\mathcal{X}}) + {\mathcal{Q}}({\mathcal{Z}}) = 0 \}$ since for every variable in ${\mathcal{Z}}$ the function behaves as $|x|^2$. This also holds for the variables in ${\mathcal{X}}$ because of the constraints. Moreover, the function ${\mathcal{H}}({\mathcal{Z}})$ is a strongly convex function because its Hessian is positive definite. Also, every subproblem in the ADMM of  is a trivial, linear or Sylvester-type equation and thus it attains its optimal value when $\rho$ is sufficiently large. Finally, our objective term ${\mathcal{G}}({\mathcal{X}})$ is differentiable and, in particular, it is lower semi-continuous.

--- abstract: 'Quantum discord is the minimal bipartite resource which is needed for a secure quantum key distribution, being a cryptographic primitive equivalent to non-orthogonality. Its role becomes crucial in device-dependent quantum cryptography, where the presence of preparation and detection noise (inaccessible to all parties) may be so strong to prevent the distribution and distillation of entanglement. The necessity of entanglement is re-affirmed in the stronger scenario of device-independent quantum cryptography, where all sources of noise are ascribed to the eavesdropper.' author: - Stefano Pirandola title: Quantum discord as a resource for quantum cryptography --- *Introduction*.– One of the hot topics in the quantum information theory is the quest for the most appropriate measure and quantification of quantum correlations. For pure quantum states, this quantification is provided by quantum entanglement [@Nielsen] which is the physical resource at the basis of the most powerful protocols of quantum communication and computation [@Tele; @Ekert; @Shor]. However, we have recently understood that the characterization of quantum correlations is much more subtle in the general case of mixed quantum states [Qdiscord,Qdiscord2]{}. There are in fact mixed states which, despite being separable, have correlations so strong to be irreproducible by any classical probability distribution. These residual quantum correlations are today quantified by quantum discord [@VedralRMP], a new quantity which has been studied in several contexts with various operational interpretations and applications, including work extraction [@Demon], quantum state merging [Merging1,Merging2]{}, remote state preparation [@Remote], quantum metrology [@Qmetrology], discrimination of unitaries [@Gu] and quantum channel discrimination [@Qillumination]. In this paper, we identify the basic role of quantum discord in one of the most practical tasks of quantum information, i.e., quantum key distribution (QKD) [@Gisin]. The claim that quantum discord must be non-zero to implement QKD is intuitive. In fact, quantum discord and its geometric formulation are connected with the concept of non-orthogonality, which is the essential ingredient for quantum cryptography. That said, it is still very important to characterize the general framework where discord remains the only available resource for QKD. Necessarily, this must be a scenario where key distribution is possible despite entanglement being absent. Here we show that this general scenario corresponds to device-dependent (or trusted-device) QKD, which encompasses all realistic protocols where the noise affecting the devices and apparata of the honest parties is assumed to be trusted, i.e., not coming from an eavesdropper but from the action of a genuine environment. This can be preparation noise (e.g., due to imperfections in the optical switches/modulators or coming from the natural thermal background at lower frequencies [Filip,Usenko,Weedbrook2010,Weedbrook2012,Weedbrook2014]{}) as well as measurement noise and inefficiencies affecting the detectors (which could be genuine or even added by the honest parties [@Renner2005; @Pirandola2009]). Such trusted noise may be high enough to prevent the distribution and distillation of entanglement, but still a secure key can be extracted due to the presence of non-zero discord. By contrast, if the extra noise in the apparata is not trusted but considered to be the effect of side-channel attacks [@SideCH], then we have to enforce device-independent QKD [@Ekert]. In this more demanding scenario, quantum discord is still necessary for security but more simply becomes an upper bound to the coherent information. This means that secure key distribution becomes just a consequence of entanglement distillation. *Quantum discord.*– Discord comes from different quantum extensions of the classical mutual information. The first is quantum mutual information, measuring the total correlations between two systems, $A$ and $% B $, and defined as $I(A,B):=S(A)-S(A|B)$, where $S(A)$ is the entropy of system $A$, and $S(A|B):=S(AB)-S(B)$ its conditional entropy. The second extension is $C(A|B):=S(A)-S_{\min}(A|B)$, where $S_{\min}(A|B)$ is the entropy of system $A$ minimized over an arbitrary measurement on $B$. This local measurement is generally described by a positive operator valued measure (POVM) $\{M_{y}\}$, defining a random outcome variable $% Y=\{y,p_{y}\} $ and collapsing system $A$ into conditional states $% \rho_{A|y}~$[@Nota2]. Thus, we have$$S_{\min}(A|B):=\inf_{\{M_{y}\}}S(A|Y),~S(A|Y)=\sum_{y}p_{y}S(\rho_{A|y}),$$ where the minimization can be restricted to rank-1 POVMs [@VedralRMP]. The quantity $C(A|B)$ quantifies the classical correlations between the two systems, corresponding to the maximal common randomness achievable by local measurements and one-way classical communication (CC) [@Winter]. Thus, quantum discord is defined as the difference between total and classical correlations [@Qdiscord; @Qdiscord2; @VedralRMP]$$D(A|B):=I(A,B)-C(A|B)=S_{\min}(A|B)-S(A|B)\geq0.$$ An equivalent formula can be written by noticing that $I_{c}(A\rangle B):=-S(A|B)$ is the coherent information [@Qcap; @Qcap2]. Then, introducing an ancillary system $E$ which purifies $\rho_{AB}$, we can apply the Koashi-Winter relation [@Koashi] and write $S_{\min}(A|B)=E_{f}(A,E)$, where the latter is the entanglement of formation between $A$ and $E$. Therefore$$D(A|B)=I_{c}(A\rangle B)+E_{f}(A,E)\geq\max\{0,I_{c}(A\rangle B)\}. \label{KWvar}$$ It is important to note that $D(A|B)$ is different from $D(B|A)$, where system $A$ is measured. For instance, in classical-quantum states $% \rho_{AB}=\sum _{x}p_{x}|x\rangle_{A}\langle x|\otimes\rho_{B}(x)$, where $A$ embeds a classical variable via the orthonormal set $\{|x\rangle\}$ and $B$ is prepared in non-orthogonal states $\{\rho_{B}(x)\}$, we have $D(B|A)=0$ while $D(A|B)>0$. By contrast, for quantum-classical states ($B$ embedding a classical variable), we have the opposite situation, i.e., $D(A|B)=0$ and $% D(B|A)>0$. *Device-dependent QKD protocols.– *Any QKD protocol can be recast into a measurement-based scheme, where Alice sends Bob part of a bipartite state, then subject to local detections. Adopting this representation, we consider a device-dependent protocol where extra noise affects Alice’s state preparation, as in Fig. \[picU\] (this is generalized later). In her private space, Alice prepares two systems, $A$ and $a$, in a generally mixed state $\rho_{Aa}$. This state is purified into a 3-partite state $\Phi_{PAa}$ with the ancillary system $P$ being inaccessible to Alice, Bob or Eve. System $a$ is then sent to Bob, who gets the output system $B$. From the shared state $\rho_{AB}$, Alice and Bob extract two correlated variables: System $A$ is detected by a rank-1 POVM $\{M_{x}\}$, providing Alice with variable $X=\{x,p_{x}\}$, while $B$ is detected by another rank-1 POVM $% \{M_{y}\}$, providing Bob with variable $Y=\{y,p_{y}\}$, whose correlations with $X$ are quantified by the classical mutual information $I(X,Y)$. ![Device-dependent protocol with preparation noise. Alice prepares a generally-mixed input state $\protect\rho_{Aa}$, which is purified into a state $\Phi_{PAa}$ by adding an extra system $P$ inaccessible to all parties. System $a$ is sent through an insecure line, so that Alice and Bob share an output state $\protect\rho_{AB}$. By applying rank-1 POVMs on their local systems, $A$ and $B$, they derive two correlated random variables, $X$ and $Y$, which are processed into a secret key. In the middle, Eve attacks the line using a unitary $U$ which couples system $a$ with a pure-state ancilla $e$. The output ancilla $E$ is then stored in a quantum memory, which is coherently detected at the end of the protocol.[]{data-label="picU"}](picU.eps){width="50.00000%"} After the previous process has been repeated many times, Alice and Bob publicly compare a subset of their data. If the error rate is below a certain threshold, they apply classical procedures of error correction and privacy amplification with the help of one-way CC, which can be either forward from Alice to Bob (direct reconciliation), or backward from Bob to Alice (reverse reconciliation). Thus, they finally extract a secret key at a rate $K\leq I(X,Y)$, which is denoted by $K(Y|X)$ in direct reconciliation and $K(X|Y)$ in reverse reconciliation. To quantify these rates, we need to model Eve’s attack. The most general attack is greatly reduced if Alice and Bob perform random permutations on their classical data [@Renner; @Renner2]. As a result, Eve’s attack collapses into a collective attack, where each travelling system is probed by an independent ancilla. This means that Eve’s interaction can be represented by a two-system unitary $U_{ae}$ coupling system $a$ with an ancillary system $e $ prepared in a pure state [@Noteiso]. The output ancilla $E$ is then stored in a quantum memory which is coherently measured at the end of the protocol (see Fig. \[picU\]). In this attack, the maximum information which is stolen on $X$ or $Y$ cannot exceed the Holevo bound. *Non-zero discord is necessary*.– Before analyzing the secret-key rates, we briefly clarify why discord is a necessary resource for QKD. Suppose that Alice prepares a quantum-classical state $\rho_{Aa}=% \sum_{k}p_{k}\rho _{A}(k)\otimes|k\rangle_{a}\langle k|$ with $\{|k\rangle\}$ orthogonal, so that $D(A|a)=0$. Classical system $a$ is perfectly clonable by Eve. This implies that the three parties will share the state [@Add]$$\rho_{ABE}=\sum_{k}p_{k}\rho_{A}(k)\otimes|k\rangle_{B}\langle k|\otimes |k\rangle_{E}\langle k|,$$ with Eve fully invisible, since her action is equivalent to an identity channel for Alice and Bob ($\rho_{AB}=\rho_{Aa}$). Direct reconciliation fails since $\rho_{ABE}$ is symmetric under $B$-$E$ permutation, which means that Eve decodes Alice’s variable with the same accuracy of Bob. Reverse reconciliation also fails. Bob encodes $Y$ in the joint state $\rho_{AE|y}=\sum_{k}p_{k|y}\rho_{A}(k)\otimes|k\rangle_{E}% \langle k|$, where $p_{k|y}:=\langle k|M_{y}|k\rangle$. Then, Eve retrieves $% K=\{k,p_{k|y}\}$ by a projective POVM, while Alice decodes a variable $X$ with distribution $$p_{x|y}=\mathrm{Tr}(M_{x}\rho_{A|y})=\sum_{k}p_{x|k}p_{k|y},~p_{x|k}:=% \mathrm{Tr}[M_{x}\rho_{A}(k)].$$ This equation defines a Markov chain $Y\rightarrow K\rightarrow X$, so that $% I(Y,K)\geq I(Y,X)$ by data processing inequality, i.e., Eve gets more information than Alice [@cohe]. As expected, system $a$ sent through the channel must be quantum $D(A|a)>0$ in order to have a secure QKD. Indeed, this is equivalent to sending an ensemble of non-orthogonal states. By contrast, the classicality of the private system $A$ is still acceptable, i.e., we can have $D(a|A)=0$. In fact, we may build QKD protocols with preparation noise using classical-quantum states $$\rho_{Aa}=\sum_{x}p_{x}|x\rangle_{A}\langle x|\otimes\rho_{a}(x), \label{cqSTATE}$$ whose local detection (on system $A$) prepares any desired ensemble of non-orthogonal signal states $\{\rho_{a}(x),p_{x}\}$. For instance, the classical-quantum state of two qubits $\rho_{Aa}=(|0,0\rangle_{Aa}% \langle0,0|+|1,\varphi\rangle_{Aa}\langle1,\varphi|)/2$, with $\{|0\rangle ,|1\rangle\}$ orthonormal and $\langle0|\varphi\rangle\neq0$, realizes the B92 protocol [@B92]. *Secret-key rates*.– Once we have clarified that non-zero input discord $D(A|a)>0$ is a necessary condition for QKD, we now study the secret-key rates which can be achieved by device-dependent protocols. Our next derivation refers to the protocol of Fig. \[picU\] and, more generally, to the scheme of Fig. \[PicD\], where Alice and Bob share an output state $\rho_{AB}$, where only part of the purification is accessible to Eve (system $E$), while the inaccessible part $P$ accounts for all possible forms of extra noise in Alice’s and Bob’s apparata, including preparation noise and detection noise (quantum inefficiencies, etc...) ![Output state from a device-dependent QKD protocol. Alice and Bob extract a secret-key by applying rank-1 POVMs on their local systems $A$ and $B$. Eve steals information from system $E$, while the extra system $P$ is inaccessible and completes the purification of the global state $\Psi_{ABEP}$.[]{data-label="PicD"}](picD.eps){width="50.00000%"} In direct reconciliation, Alice’s variable $X$ is the encoding to guess. The key rate is then given by $K(Y|X)=I(X,Y)-I(E,X)$, where $I(E,X)=S(E)-S(E|X)$ is the Holevo bound quantifying the maximal information that Eve can steal on Alice’s variable [@Notation1]. We can write an achievable upper bound if we allow Bob to use a quantum memory and a coherent detector. In this case, $I(X,Y)$ must be replaced by the Holevo quantity $I(B,X)=S(B)-S(B|X)$ and we get the forward Devetak-Winter (DW) rate [@DW1; @DW2; @DWnote]$$K(Y|X)\leq K(B|X):=I(B,X)-I(E,X). \label{DWdr}$$ The optimal forward-rate is defined by optimizing on Alice’s individual detections $K(\blacktriangleright):=\sup_{\{M_{x}\}}K(B|X)$. We can write similar quantities in reverse reconciliation, where Bob’s variable $Y$ is the encoding to infer. The secret key rate is given by $% K(X|Y)=I(X,Y)-I(E,Y)$, where $I(E,Y)=S(E)-S(E|Y)$ is Eve’s Holevo information on $Y$. Assuming a coherent detector for Alice, this rate is bounded by the backward DW rate$$K(X|Y)\leq K(A|Y):=I(A,Y)-I(E,Y), \label{DWrevEQ}$$ which gives the optimal backward-rate $K(\blacktriangleleft)$ by maximizing on Bob’s individual detections $\{M_{y}\}$. Playing with system $P$, we can easily derive upper and lower bounds for the two optimal rates. Clearly, we get lower bounds $K_{\ast}\leq K$ if we assume $P$ to be accessible to Eve, which means to extend $E$ to the joint system $\mathbf{E}=EP$ in previous Eqs. (\[DWdr\]) and (\[DWrevEQ\]). By exploiting the purity of the global state $\Psi_{AB\mathbf{E}}$ and the fact that the encoding detections are rank-1 POVMs (therefore collapsing pure states into pure states), we can write the entropic equalities $S(AB)=S(% \mathbf{E})$, $S(B|X)=S(\mathbf{E}|X)$ and $S(A|Y)=S(\mathbf{E}|Y)$. Then we easily derive$$K_{\ast}(\blacktriangleright)=I_{c}(A\rangle B),~K_{\ast}(\blacktriangleleft )=I_{c}(B\rangle A),$$ where the coherent information $I_{c}(A\rangle B)$ and its reverse counterpart [@RevCOH; @RMP] $I_{c}(B\rangle A)$ quantify the maximal entanglement which is distillable by local operations and one-way CC, forward and backward, respectively. It is also clear that we get upperbounds $K^{\ast}\geq K$ by assuming $P$ to be accessible to the decoding party, Alice or Bob, depending on the reconciliation. In direct reconciliation, we assume $P$ to be accessible to Bob, which means extending his system $B$ to $\mathbf{B}=BP$ in Eq. ([DWdr]{}). Using the equalities $S(A\mathbf{B})=S(E)$ and $S(\mathbf{B}% |X)=S(E|X)$, we get $$K^{\ast}(\blacktriangleright)=I_{c}(A\rangle\mathbf{B})=I_{c}(A\rangle B)+I(A,P|B),$$ where $I(A,P|B)\geq0$ is the conditional quantum mutual information. Then, in reverse reconciliation, we assume $P$ to be accessible to Alice, so that $% A$ becomes $\mathbf{A}=AP$ in Eq. (\[DWrevEQ\]). Using $S(\mathbf{A}% B)=S(E) $ and $S(\mathbf{A}|Y)=S(E|Y)$, we get $K^{\ast}(\blacktriangleleft )=I_{c}(B\rangle A)+I(B,P|A)$. Thus, the optimal key rates satisfy the inequalities$$\begin{aligned} I_{c}(A\rangle B)~ & \leq~K(\blacktriangleright)~\leq~I_{c}(A\rangle B)+I(A,P|B), \label{eq1} \\ I_{c}(B\rangle A)~ & \leq~K(\blacktriangleleft)~\leq~I_{c}(B\rangle A)+I(B,P|A), \label{eq2}\end{aligned}$$ where the right hand sides can be bounded using$$I(A,P|B),I(B,P|A)\leq I(AB,P)\leq2\min\{S(P),S(AB)\}.$$ According to Eqs. (\[eq1\]) and (\[eq2\]), key distribution can occur ($% K\geq0$) even in the absence of distillable entanglement ($I_{c}=0$). It is now important to note the following facts: (i) Device-dependent QKD is the only scenario where this is possible. In fact, only in the presence of trusted noise, i.e., $S(P)>0$, we can have $% K\geq I_{c}$ in the previous equations. Therefore device-dependent QKD is the only scenario where security may be achieved in the absence of distillable entanglement and, more strongly, in the complete absence of entanglement. (ii) In device-dependent QKD, we can indeed build protocols which are secure ($K>0$) despite entanglement being completely absent (in any form, distillable or bound). Clearly, this is possible as long as the minimal condition $D(A|a)>0$ is satisfied. A secure protocol based on separable Gaussian states is explictly shown in the supplementary material. In general, there is an easy way to design device-dependent protocols which are secure and free of entanglement. Any prepare and measure protocol whose security is based on the transmission of non-orthogonal states $\{\rho _{a}(x),p_{x}\}$ can be recast into a device-dependent protocol, which is based on a classical-quantum state $\rho_{Aa}$ as in Eq. (\[cqSTATE\]), whose classical part $A$ is detected while the quantum part $a$ is sent through the channel. This is as secure as the original one as long as the purification of the classical-quantum state is inaccessible to Eve. Thus in such assumption of trusted noise, any prepare and measure protocol has an equivalent discord-based representation, where non-zero discord guarantees security in the place of non-orthogonality. *Side-channels and device-independent QKD*.– Let us consider the more demanding scenario where all sources of noise are untrusted. This means that the extra noise in Alice’s and Bob’s apparata comes from side-channel attacks, i.e., system $P$ in Fig. \[PicD\] is controlled by Eve. In this case, the secret-key rates are given by$$K(\blacktriangleright)=I_{c}(A\rangle B),~K(\blacktriangleleft)=I_{c}(B\rangle A), \label{standard}$$ so that QKD is equivalent to entanglement distillation. It is easy to check that quantum discord upperbounds these key rates. Applying Eq. (\[KWvar\]) to Eq. (\[standard\]), we obtain the cryptographic relations$$\begin{aligned} K(\blacktriangleright) & =D(A|B)-E_{f}(A,E)\leq D(A|B), \label{B1} \\ K(\blacktriangleleft) & =D(B|A)-E_{f}(B,E)\leq D(B|A). \label{B2}\end{aligned}$$ The optimal forward rate $K(\blacktriangleright)$, where Alice’s variable must be inferred, equals the difference between the output discord $D(A|B)$, based on Bob’s detections, and the entanglement of formation $E_{f}(A,E)$ between Alice and Eve. Situation is reversed for the other rate $% K(\blacktriangleleft )$. Note that quantum discord not only provides an upper bound to the key rates, but its aymmetric definition, $D(A|B)$ or $% D(B|A)$, is closely connected with the reconciliation direction (direct $% \blacktriangleright$ or reverse $\blacktriangleleft$). *Ideal QKD protocols*.– In practical quantum cryptography, extra noise is always present, and we distinguish between device-dependent and device-independent QKD on the basis of Eve’s accessibility of the extra system $P$. In theoretical studies of quantum cryptography, it is however common to design and assess new protocols by assuming no-extra noise in Alice’s and Bob’s apparata (perfect state preparation and perfect detections). This is an ideal scenario where system $P$ of Fig. \[PicD\] is simply absent. For such ideal QKD protocols, the secret-key rates satisfy again Eqs. (\[standard\]), (\[B1\]), and (\[B2\]), computed on the corresponding output states. Remarkably, the discord bound can be found to be tight in reverse reconciliation. In fact, as we show in the supplementary material, we can have $K(\blacktriangleleft)=D(B|A)$ in an ideal protocol of continuous-variable QKD, where Alice transmits part of an Einstein-Podolsky-Rosen (EPR) state over a pure-loss channel, such as an optical fiber. *Conclusion and discussion*.– Quantum discord can be regarded as a bipartite formulation of non-orthogonality, therefore capturing the minimal requisite for QKD. In this paper we have identified the general framework, device-dependent QKD, where discord remains the ultimate cryptographic primitive able to guarantee security in the place of quantum entanglement. We have considered a general form of device-dependent protocol, where Alice and Bob share a bipartite state which can be purified by two systems: One system ($E$) is accessible to Eve, while the other ($P$) is inaccessible and accounts from the presence of trusted noise, e.g., coming from imperfections in the state preparation and/or the quantum detections. This is a scenario where the optimal key rate may outperform the coherent information and key distribution may occur in the complete absence of entanglement (in any form, distillable or bound) as long as discord is non-zero. As a matter of fact, any prepare and measure QKD protocol whose security is based on non-orthogonal quantum states can be recast into an entanglement-free device-dependent form which is based on a classical-quantum state, with non-zero discord transmitted through the channel. This discord-based representation is secure as long as the extra system $P$ is truly inaccessible to Eve, i.e., Alice’s and Bob’s private spaces cannot be accessed. Such a condition fails assuming side-channel attacks, where no noise can be trusted and $P$ becomes part of Eve’s systems. In this case, the secret-key rates are again dominated by the coherent information, which means that entanglement remains the crucial resource for device-independent QKD. For both device-independent QKD and ideal QKD (where system $P$ is absent), discord still represents an upper bound to the optimal secret-key rates achievable in direct or reverse reconciliation, with non trivial cases where this bound becomes tight. In conclusion, quantum discord is a necessary resource for secure QKD. This is particularly evident in device-dependent QKD where entanglement is a sufficient but not a necessary resource. Entanglement becomes necessary in device-independent and ideal QKD, where discord still provides an upper bound to the secret-key rates. Future work may involve the derivation of a direct mathematical relation between the amount of quantum discord in Alice and Bob’s output state and the optimal secret-key rates which are achievable in device-dependent QKD. *Acknowledgments*.– S.P. was supported by a Leverhulme Trust research fellowship and EPSRC (grants EP/J00796X/1 and EP/L011298/1). [99]{} M.M. Wilde,* Quantum Information Theory* (Cambridge University Press, Cambridge, 2013). A.K. Ekert, Phys. Rev. Lett. **67**, 661 (1991). C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. **70**, 1895 (1993). P. Shor, SIAM J. Comput. **26**, 1484 (1997). H. Ollivier and W.H. Zurek, Phys. Rev. Lett. **88**, 017901 (2001). L. Henderson and V. Vedral, J. Phys. A **34**, 6899 (2001). K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, Rev. Mod. Phys. **84**, 1655-1707 (2012). W.H. Zurek, Phys. Rev. A **67**, 012320 (2003). D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, A. Winter, Phys. Rev. A **83**, 032324 (2011). V. Madhok and A. Datta, Phys. Rev. A **83**, 032323 (2011). B. Dakic, Y.O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, and P. Walther, Nature Phys. **8**, 666-670 (2012). D. Girolami, T. Tufarelli, and G. Adesso, Phys. Rev. Lett. 110, 240402 (2013). M. Gu, H.M. Chrzanowski, S.M. Assad, T. Symul, K. Modi, T.C. Ralph, V. Vedral, and P. Koy Lam, Nature Phys. **8**, 671-675 (2012). C. Weedbrook, S. Pirandola, J. Thompson, V. Vedral, and M. Gu, *Discord Empowered Quantum Illumination*, preprint arXiv:1312.3332. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. **74**, 145 (2002). R. Filip, Phys. Rev. A **77**, 022310 (2008). V.C. Usenko and R. Filip, Phys. Rev. A **81**, 022318 (2010). C. Weedbrook, S. Pirandola, S. Lloyd, and T.C. Ralph, Phys. Rev. Lett. **105**, 110501 (2010). C. Weedbrook, S. Pirandola, and T.C. Ralph, Phys. Rev. A **86**, 022318 (2012). C. Weedbrook, C. Ottaviani, and S. Pirandola, Phys. Rev. A **89**, 012309 (2014). R. Renner, N. Gisin, and B. Kraus, Phys. Rev. A **72**, 012332 (2005). S. Pirandola, R. García-Patrón, S.L. Braunstein, and S. Lloyd, Phys. Rev. Lett. **102**, 050503 (2009). C. Weedbrook, S. Pirandola, R. Garcia-Patron, N.J. Cerf, T.C. Ralph, J.H. Shapiro, and S. Lloyd, Rev. Mod. Phys. **84**, 621 (2012). S.L. Braunstein and S. Pirandola, Phys. Rev. Lett. **108**, 130502 (2012). Variables can be discrete or continuous. In the latter case, sums become integrals and probability distributions become densities. I. Devetak and A. Winter, IEEE Trans. Inform. Theory. **50**, 3183 (2004). B. Schumacher and M.A. Nielsen, Phys. Rev. A **54**, 2629 (1996). S. Lloyd, Phys. Rev. A **55**, 1613 (1997). M. Koashi and A. Winter, Phys. Rev. A **69**, 022309 (2004). R. Renner, Nature Phys. **3**, 645 (2007). R. Renner, and J.I. Cirac, Phys. Rev. Lett. **102**, 110504 (2009). Up to isometries which do not increase Eve’s information. Note that we can always add another ancilla $E^{\prime}$ to make the output $\Psi_{ABEE^{\prime}P}$ globally pure. This reasoning can easily be extended to consider coherent detections for Eve and Alice. C.H. Bennett, Phys. Rev. Lett. **68**, 3121-3124 (1992). Note that, for $A$,$B$ quantum systems and $X,Y$ classical variables, we use $I(X,Y)$ for the classical mutual information, $% I(A,B)$ for the quantum mutual information, and $I(A,X)$ for the Holevo information. I. Devetak and A. Winter, Phys. Rev. Lett. **93**, 080501 (2004). I. Devetak and A. Winter, Proc. R. Soc. Lond. A **461**, 207-235 (2005). In general, the DW rate satisfies Eq. (\[DWdr\]) with $% \geq$, the separation ($>$) being possible for higher-rank POVMs [DW1,DW2]{}. These POVMs are considered here by including their detection noise in system $P$ (e.g., as in Fig. \[PicD\]). What remain are rank-1 POVMs applied to $A $ and $B$. R. García-Patrón, S. Pirandola, S. Lloyd, J.H. Shapiro, Phys. Rev. Lett. **102**, 210501 (2009). [Supplementary Material]{} Keys from separable Gaussian states =================================== Here we provide a simple example of device-dependent QKD protocol which is based on the distribution of a bipartite Gaussian state which is mixed and separable (not in a tensor-product, therefore having non-zero discord). We show that the key rates can be positive despite no entanglement being present. The reader not familiar with the formalism of bosonic systems and Gaussian states can find these concepts in Ref. [@RMP2], whose notation is here adopted ($\hslash=2$ and unit vacuum noise). Let us consider a continuous variable QKD protocol where Alice prepares two bosonic modes, $A$ and $a$, in a separable Gaussian state $\rho_{Aa}$, with zero mean and covariance matrix (CM)$$\mathbf{V}_{Aa}=\left( \begin{array}{cc} \mu\mathbf{I} & \mathbf{G} \\ \mathbf{G} & \mu\mathbf{I}\end{array} \right) ,$$ where $\mu\geq1$ and $\mathbf{G}$ is a diagonal correlation block which can be in one of the following forms$$\mathbf{G}=\left( \begin{array}{cc} g & \\ & g\end{array} \right) :=g\mathbf{I,~G}=\left( \begin{array}{cc} g & \\ & -g\end{array} \right) :=g\mathbf{Z}.$$ Here the parameter $g$ must satisfy $|g|\leq\mu-1$, so that $\mathbf{V}_{Aa}$ is both physical and separable [@twomodes]. Apart from the singular case $g=0$, this symmetric Gaussian state has always non-zero discord, i.e., $D(A|a)=D(a|A)>0$ [@VeriDISCO]. Mode $a$ is sent through the channel, where Eve performs a collective Gaussian attack, whose most general description can be found in Ref [colleGAUSS]{}. Assuming random permutations (so that quantum de Finetti applies), this is the most powerful attack against Gaussian protocols [RMP2]{}. One of the canonical forms of this attack is the so-called ‘entangling cloner’ attack [@RMP2], where Eve uses a beam splitter with transmissivity $\tau$ to mix the incoming mode $a$ with one mode $e$ of an EPR state $\rho_{eE^{\prime}}$ with CM $$\mathbf{V}_{eE^{\prime}}=\left( \begin{array}{cc} \omega\mathbf{I} & \sqrt{\omega^{2}-1}\mathbf{Z} \\ \sqrt{\omega^{2}-1}\mathbf{Z} & \omega\mathbf{I}\end{array} \right) :=\mathbf{V}(\omega), \label{Vomega}$$ where $\omega\geq1$. One output mode $B$ is sent to Bob, while the other output mode $E$ is stored in a quantum memory together with the retained mode $E^{\prime}$. Such memory will be coherently detected at the end of the protocol. In order to extract two correlated (complex) variables, $X$ and $Y$, Alice and Bob heterodyne their local modes $A$ and $B$. (Note that other protocols involving homodyne detection for one of the parties or even two homodynes may be considered as well.) One can easily check that Alice remotely prepares thermal states on mode $a$. In fact, by heterodyning mode $A$, the other mode $a$ is collapsed in a Gaussian state $\rho_{a|X}$ with CM $\mathbf{V}_{a|X}=(1+\varepsilon)\mathbf{I}$, where$$\varepsilon:=\mu-1-\frac{g^{2}}{\mu+1}\geq0$$ quantifies the thermalization above the coherent state. This conditional thermal state is randomly displaced in the phase space according to a bivariate Gaussian distribution with variance $\mu-1-\varepsilon$ (so that the average input state on mode $a$ is thermal with the correct CM $\mu\mathbf{I}$). At the output of the channel, Bob’s average state is thermal with CM $\nu _{B}\mathbf{I}$, where $\nu_{B}:=\tau\mu+(1-\tau)\omega$. By propagating the conditional thermal state $\rho_{a|X}$, we also get Bob’s conditional state $\rho_{B|X}$, which is randomly displaced and has CM $\nu_{B|X}\mathbf{I}$, where$$\nu_{B|X}:=\tau(1+\varepsilon)+(1-\tau)\omega=\nu_{B}-\frac{\tau g^{2}}{\mu +1}.$$ Therefore, we can easily compute Alice and Bob’s mutual information, which is equal to $$I(X,Y)=\log_{2}\frac{\nu_{B}+1}{\nu_{B|X}+1}.$$ The next step is the calculation of Eve’s Holevo information on Alice’s and Bob’s variables. We derive the global state of Alice, Bob and Eve, which is pure Gaussian with zero mean and CM$$\mathbf{V}_{ABEE^{\prime}}=\left( \begin{array}{cccc} \mu\mathbf{I} & \sqrt{\tau}\mathbf{G} & -\sqrt{1-\tau}\mathbf{G} & \mathbf{0} \\ \sqrt{\tau}\mathbf{G} & \nu_{B}\mathbf{I} & \gamma\mathbf{I} & \delta \mathbf{Z} \\ -\sqrt{1-\tau}\mathbf{G} & \gamma\mathbf{I} & \nu_{E}\mathbf{I} & \kappa\mathbf{Z} \\ \mathbf{0} & \delta\mathbf{Z} & \kappa\mathbf{Z} & \omega\mathbf{I}\end{array} \right) ,$$ where $\mathbf{0}$ is the $2\times2$ zero matrix, and$$\begin{aligned} \nu_{E} & :=\tau\omega+(1-\tau)\mu,~\gamma:=\sqrt{\tau(1-\tau)}(\omega -\mu), \\ \delta & :=\sqrt{1-\tau}\sqrt{\omega^{2}-1},~\kappa:=\sqrt{\tau(\omega ^{2}-1)}.\end{aligned}$$ From this global CM, we extract Eve’s reduced CM $\mathbf{V}_{EE^{\prime}}:=\mathbf{V}_{\mathbf{E}}$ describing the two output modes $\mathbf{E}=EE^{\prime}$ of the entangling cloner. This reduced CM has symplectic spectrum [@RMP2]$$\nu_{\mathbf{E}}^{\pm}=\frac{\sqrt{\alpha^{2}+4\beta}\pm\alpha}{2},$$ where $\alpha:=(1-\tau)(\mu-\omega)$ and $\beta:=\tau+(1-\tau)\mu\omega$. The von Neumann entropy of Eve’s average state is then given by $S(\mathbf{E})=h(\nu_{\mathbf{E}}^{+})+h(\nu_{\mathbf{E}}^{-})$, where$$h(x):=\frac{x+1}{2}\log_{2}\frac{x+1}{2}-\frac{x-1}{2}\log_{2}\frac{x-1}{2}.$$ By transforming the global CM under heterodyne detection [@RMP2], we compute Eve’s conditional CMs. First, we derive Eve’s CM conditioned to Bob’s detection $$\mathbf{V}_{\mathbf{E}|Y}=\mathbf{V}_{\mathbf{E}}-\frac{1}{\nu_{B}+1}\left( \begin{array}{cc} \gamma^{2}\mathbf{I} & \gamma\delta\mathbf{Z} \\ \gamma\delta\mathbf{Z} & \delta^{2}\mathbf{I}\end{array} \right) ,$$ which has symplectic spectrum$$\nu_{\mathbf{E}|Y}^{-}=1,~\nu_{\mathbf{E}|Y}^{+}=\frac{\mu+\beta}{1+\mu \tau+(1-\tau)\omega}.$$ Then, Eve’s CM conditioned to Alice’s detection is $$\mathbf{V}_{\mathbf{E}|X}=\mathbf{V}_{\mathbf{E}}-\frac{(1-\tau)g^{2}}{\mu +1}\left( \begin{array}{cc} \mathbf{I} & \mathbf{0} \\ \mathbf{0} & \mathbf{0}\end{array} \right) ,$$ and has symplectic spectrum$$\nu_{\mathbf{E}|X}^{\pm}=\frac{\sqrt{\theta^{2}+4(\mu+1)\phi}\pm\theta}{2(\mu+1)},$$ where$$\theta:=(1-\tau)g^{2}-(\mu+1)\alpha,~\phi:=(\mu+1)\beta-(1-\tau)\omega g^{2}.$$ From the previous conditional spectra, we compute Eve’s conditional entropies$$S(\mathbf{E}|X)=h(\nu_{\mathbf{E}|X}^{+})+h(\nu_{\mathbf{E}|X}^{-}),~S(\mathbf{E}|Y)=h(\nu_{\mathbf{E}|Y}^{+}),$$ and, therefore, we can derive the two Holevo quantities $I(\mathbf{E},X)=S(\mathbf{E})-S(\mathbf{E}|X)$ and $I(\mathbf{E},Y)=S(\mathbf{E})-S(\mathbf{E}|Y)$. By subtracting these from Alice and Bob’s mutual information $I(X,Y)$, we finally get the two key rates in direct and reverse reconciliation, i.e., $K(Y|X)$ and $K(X|Y)$. It is easy to check the existence of wide range of parameters for which these two rates are strictly positive, so that Alice and Bob can extract a secret key despite the absence of entanglement (at the input state $\rho_{Aa} $ and, therefore, also at the output state $\rho_{AB}$). As an example, we may consider the maximum correlation value $g=\mu-1$ for the separable Gaussian state $\rho_{Aa}$, and we may take the large modulation limit $\mu \rightarrow+\infty$, as typical in continuous variable QKD. In this case, we get the following asymptotical expression for Alice and Bob’s mutual information$$I(X,Y)\rightarrow\log_{2}\frac{\tau\mu}{1+3\tau+(1-\tau)\omega}+O(\mu^{-1}),$$ and the following asymptotical spectra$$\begin{aligned} \nu_{\mathbf{E}}^{-} & \rightarrow(1-\tau)\mu+\tau\omega+O(\mu^{-1}), \\ \nu_{\mathbf{E}}^{+} & \rightarrow\omega+O(\mu^{-1}), \\ \nu_{\mathbf{E}|Y}^{+} & \rightarrow\frac{1+(1-\tau)\omega}{\tau}+O(\mu ^{-1}), \\ \nu_{\mathbf{E}|X}^{\pm} & \rightarrow\xi_{\pm}+O(\mu^{-1}),\end{aligned}$$ where $$\begin{aligned} \xi_{\pm} & :=\frac{\sqrt{(\omega+3)^{2}+\tau^{2}(\omega-3)^{2}-2\tau (\omega^{2}+7)}}{2} \\ & \pm\frac{(1-\tau)(\omega-3)}{2}.\end{aligned}$$ Then, using the expansion $h(x)\simeq\log_{2}(ex/2)+O(1/x)$ for large $x$, we can write the two asymptotical rates$$\begin{aligned} K(Y|X) & =R(\tau,\omega)+h(\xi_{+})+h(\xi_{-}), \\ K(X|Y) & =R(\tau,\omega)+h\left[ \frac{1+(1-\tau)\omega}{\tau}\right] ,\end{aligned}$$ where we have introduced the common term$$R(\tau,\omega):=\log_{2}\frac{2\tau}{e(1-\tau)[1+3\tau+(1-\tau)\omega ]}-h(\omega).$$ As we can see from Fig. \[soglie\], there are wide regions of positivity for these rates. ![*Left panel*. Rate $K(Y|X)$ in direct reconciliation, as a function of channel transmissivity $\protect\tau$ and thermal variance $\protect\omega$. $K$ is positive in the white area, while it is zero in the black area. *Right panel*. Rate $K(X|Y)$ in reverse reconciliation, as function of $\protect\tau$ and $\protect\omega$. White area ($K>0$) is wider at low $\protect\omega$.[]{data-label="soglie"}](Soglie.eps){width="45.00000%"} In particular, for a pure loss channel ($\omega=1$), the previous asymptotical rates simplify to the following$$K(Y|X)=\log_{2}\frac{\tau}{e(1-\tau^{2})}+h(3-2\tau),$$ which is positive for any $\tau>0.693$, and$$K(X|Y)=\log_{2}\frac{\tau}{e(1-\tau^{2})}+h\left( \frac{2}{\tau}-1\right) ,$$ which is positive for any $\tau>0.532$. Discord bound can be tight ========================== Here we discuss a typical scenario where the optimal backward rate $K(\blacktriangleleft)$ of an ideal QKD protocol is exactly equal to the output discord $D(B|A)$ shared by Alice and Bob. This happens in continuous variable QKD, where reverse reconciliation is important for its ability to beat the 3dB loss-limit affecting direct reconciliation [@RMP2]. Consider an ideal QKD protocol which is based on the distribution of an EPRstate $\rho_{Aa}$, with CM $\mathbf{V}_{Aa}=\mathbf{V}(\mu)$ defined according to Eq. (\[Vomega\]) with $\mu\geq1$. By performing a rank-1 Gaussian POVM on mode $A$, Alice remotely prepares an ensemble of Gaussianly-modulated pure Gaussian states on the other mode $a$. For instance, heterodyne prepares coherent states, while homodyne prepares squeezed states. On average, mode $a$ is described by a thermal state with CM $\mu\mathbf{I}$. Suppose that signal mode $a$ is subject to a pure-loss channel. This means that Eve is using a beam splitter of transmissivity $\tau$ mixing the signal mode with a vacuum mode $e$. At the output of the beam splitter, mode $B$ is detected by Bob, while mode $E$ is stored in a quantum memory coherently detected by Eve (this is a collective entangling cloner attack with $\omega=1 $). Since the average state of mode $a$ is thermal and mode $e$ is in the vacuum, no entanglement can be present between the two output ports $B$ and $E$ of the beam splitter. This implies that their entanglement of formation must be zero $E_{f}(B,E)=0$ and, therefore, the optimal backward rate $K(\blacktriangleleft )$ must be equal to the discord $D(B|A)$ of the Gaussian state $\rho_{AB}$. Since this output state has CM$$\mathbf{V}_{AB}=\left( \begin{array}{cc} \mu\mathbf{I} & \sqrt{\tau(\mu^{2}-1)}\mathbf{Z} \\ \sqrt{\tau(\mu^{2}-1)}\mathbf{Z} & (\tau\mu+1-\tau)\mathbf{I}\end{array} \right) ,$$ its discord is easy to compute and is equal to [@Bconj] $$D(B|A)=h(\mu)-h[\tau+(1-\tau)\mu].$$ For large modulation ($\mu\rightarrow+\infty$), we have the asymptotic expression $$K(\blacktriangleleft)=D(B|A)=\log_{2}\left( \frac{1}{1-\tau}\right) ,$$ which is positive for any $0<\tau<1$. One can check that this rate can be achieved by heterodyne detections at Bob’s side (and coherent detection at Alice’s side). C. Weedbrook, S. Pirandola, R. Garcia-Patron, N.J. Cerf, T.C. Ralph, J.H. Shapiro, and S. Lloyd, Rev. Mod. Phys. **84**, 621 (2012). S. Pirandola, A. Serafini, and S. Lloyd, Phys. Rev. A **79**, 052327 (2009). S. Rahimi-Keshari, C.M. Caves, and T.C. Ralph, Phys. Rev. A **87**, 012119 (2013). S. Pirandola, S.L. Braunstein, and S. Lloyd, Phys. Rev. Lett. **101**, 200504 (2008). S. Pirandola *et al*., preprint arXiv:1309.2215.

--- abstract: 'In Newtonian theory, gravity inside a constant density static sphere is independent of spacetime dimension. Interestingly this general result is also carried over to Einsteinian as well as higher order Einstein-Gauss-Bonnet (Lovelock) gravity notwithstanding their nonlinearity. We prove that the necessary and sufficient condition for universality of the Schwarzschild interior solution describing a uniform density sphere for all $n\geq4$ is that its density is constant.' author: - Naresh Dadhich - Alfred Molina - Avas Khugaev title: 'Uniform density static fluid sphere in Einstein-Gauss-Bonnet gravity and its universality' --- Introduction ============ In Newtonian gravity, the gravitational potential at any point inside a fluid sphere is given by $-M(r)/r^{n-3}$ for $n\geq4$ dimensional spacetime. Now $M(r)= \int \rho r^{n-2}dr$ which for constant density will go as $\rho r^{n-1}$ and then the potential will go as $\rho r^{n-1}/r^{n-3} = \rho r^2$ and is therefore independent of the dimension. This is an interesting general result: for the uniform density sphere, gravity has the universal character that it is independent of the dimension of spacetime. It is then a natural question to ask, Does this result carry over to Einsteinian gravity? In general relativistic language it is equivalent to ask, Does Schwarzschild interior solution that describes the uniform density sphere in four dimensions remain good for all $n\geq4$? The main purpose of this paper is to show that it is indeed the case not only for Einstein gravity but also for higher order Einstein-Gauss-Bonnet (Lovelock) gravity. It is remarkable that this general feature holds true notwithstanding the highly nonlinear character of the theory. In static spherically symmetric fluid spacetime, we have two equations to handle: one is for density which easily integrates to give $g_{rr}$,and the other is the pressure isotropy equation determining $g_{tt}$. So long as density remains constant,the former equation will always integrate to give $g_{rr}$ in all dimensions with constant density redefined. Then we just need to make the latter equation free of dimension $n$ so that the constant density Schwarzschild interior solution becomes universally true for all $n$. In particular it turns out that the universality condition indeed implies constant density. Thus constant density is a necessary and sufficient condition for universality of the Schwarzschild interior solution for $n\geq4$ not only for Einstein but also for Einstein-Gauss-Bonnet (EGB) theory. Higher dimension is a natural playground for string theory and string inspired investigations (see a comprehensive review [@emparan]). The most popular studies have been of higher dimensional black holes [@tang-mayper-chong] with a view to gain greater and deeper insight into quantum phenomena, black hole entropy and the well-known AdS/CFT correspondence [@stro-mal-witt]. There have also been studies of fluid spheres in higher dimensions [@krori-shen-ponce]. We shall, however, focus on the universal character of constant density solution in Einstein and EGB theory and its matching with the corresponding exterior solution. The paper is organized as follows. In the next section, we establish the universality of the uniform density solution for Einstein and EGB theories and demonstrate the matching with an exterior solution for the five-dimensional Gauss-Bonnet black hole. We conclude with a discussion. Uniform density sphere ====================== Einstein case ------------- We begin with the general static spherically symmetric metric given by $$ds^2= e^\nu dt^2 - e^\lambda dr^2 - r^2d\Omega_{n-2}^2$$ where $d\Omega_{n-2}^2$ is the metric on a unit $(n-2)$-sphere. For the Einstein equation in the natural units ($8\pi G=c=1$), $$G_{AB} = R_{AB} - \frac{1}{2} R g_{AB} = - T_{AB}$$ and for perfect fluid, $T_A^B = diag(\rho, -p, -p, ..., -p)$, we write $$e^{-\lambda}(\frac{\lambda^{\prime}}{r} - \frac{n-3}{r^2}) + \frac{n-3}{r^2} = \frac{2}{n-2}\rho \label{density}$$ $$e^{-\lambda}(\frac{\nu^{\prime}}{r} + \frac{n-3}{r^2}) - \frac{n-3}{r^2} = \frac{2}{n-2}p$$ and the pressure isotropy is given by $$\begin{aligned} e^{-\lambda}(2\nu^{\prime\prime} + \nu^{\prime^2} - \lambda^{\prime}\nu^{\prime} - 2\frac{\nu^{\prime}}{r}) \nonumber \\ - 2(n-3)(\frac{e^{-\lambda}\lambda^{\prime}}{r} + 2\frac{e^{-\lambda}}{r^2} - \frac{2}{r^2}) = 0 \label{isotropy}. \end{aligned}$$ Let us rewrite this equation in a form that readily yields the universal character of the Schwarzschild interior solution for all $n\geq4$, $$\begin{aligned} e^{-\lambda}(2\nu^{\prime\prime} + \nu^{\prime^2} - \lambda^{\prime}\nu^{\prime} - 2\frac{\nu^{\prime} + \lambda^{\prime}}{r} - \frac{4}{r^2}) + \frac{4}{r^2} \nonumber \\ - 2(n-4) \Bigl((n-1)(\frac{e^{-\lambda}}{r^2} - \frac{1}{r^2}) + \frac{2\rho}{n-2} \Bigr) = 0. \label{iso}\end{aligned}$$ We now set the coefficient of $(n-4)$ to zero so that the equation remains the same for all $n\geq4$. This then straightway determines $e^{-\lambda}$ without integration and it is given by $$e^{-\lambda} = 1 - \rho_0 r^2 \label{sol}$$ where $\rho_0=2\rho/{(n-1)(n-2)}$. This when put in Eq. (\[density\]) implies constant density. We thus obtain $\rho=const.$ as the neceessary condition for universality of the isotropy equation for all $n\geq4$. The sufficiency of constant density is obvious from the integration of Eq. (\[density\]) for $\rho=const$, giving the same solution as above where a constant of integration is set to zero for regularity at the center. Thus constant density is a necessary and sufficient condition for universality of field inside a fluid sphere, i.e. independent of spacetime dimension. An alternative identification of constant density is that the gravitational field inside a fluid sphere is independent of spacetime dimension $\geq4$. This universal property is therefore true if and only if density is constant. As is well known, Eq. (\[iso\]) on substituting Eq. (\[sol\]) admits the general solution as given by $$e^{\nu/2} = A + Be^{-\lambda/2} \label{sol2}$$ where $A$ and $B$ are constants of integration to be determined by matching to the exterior solution. This is the Schwarzschild interior solution for a constant density sphere that is independent of the dimension except for a redefinition of the constant density as $\rho_0$. This proves the universality of the Schwarzschild interior solution for all $n\geq4$. The Newtonian result that gravity inside a uniform density sphere is independent of spacetime dimension is thus carried over to general relativity as well despite nonlinearity of the equations. That is, Schwarzschild interior solution is valid for all $n\geq4$. Since there exist more general actions like Lovelock polynomial and $f(R)$ than the linear Einstein-Hilbert, it would be interesting to see whether this result would carry through there as well. That is what we take up next. Gauss-Bonnet(Lovelock) case --------------------------- There is a natural generalization of Einstein action to Lovelock action that is a homogeneous polynomial in Riemann curvature with Einstein being the linear order. It has the remarkable property that on variation it still gives the second order quasilinear equation that is its distinguishing feature. The higher order terms make a nonzero contribution in the equation only for dimension $\geq5$. The quadratic term in the polynomial is known as Gauss-Bonnet, and for that we write the action as $$\label{action} S=\int d^nx\sqrt{-g}\biggl[\frac{1}{2}(R-2\Lambda+\alpha{L}_{GB}) \biggr]+S_{\rm matter},$$ where $\alpha$ is the GB coupling constant and all other symbols have their usual meaning. The GB Lagrangian is the specific combination of Ricci scalar, Ricci, and Riemann curvatures, and it is given by $${L}_{GB}=R^2-4R_{AB}R^{AB}+R_{ABCD}R^{ABCD}.$$ This form of action is known also to follow from the low-energy limit of heterotic superstring theory [@gross]. In that case, $\alpha$ is identified with the inverse string tension and is positive definite, which is also required for the stability of Minkowski spacetime. The gravitational equation following from the action (\[action\]) is given by $$G^A_B +\alpha H^A_B = - T^A_B, \label{beq}$$ where $$\begin{aligned} && H_{AB}\equiv 2\Bigl[RR_{AB}-2R_{AC}R^C_B-2R^{CD}R_{ACBD} \nonumber\\ && \hspace*{4em} +R_{A}^{~CDE}R_{BCDE}\Bigr] -{\frac12}g_{AB}{L}_{GB}.\label{def-H}\end{aligned}$$ Now density and pressure would read as follows: $$\begin{aligned} \rho = \frac{(n-2)e^{-\lambda}}{2r^2} \Bigl(r\lambda^\prime - (n-3)(1-e^{\lambda})\Bigr)+\qquad\qquad\nonumber \\ +\frac{(n-2) e^{-2\lambda}\tilde\alpha}{2r^4}(1-e^{\lambda}) \Bigl(-2r\lambda^\prime + (n-5)(1-e^{\lambda})\Bigr) \label{rho-gb} \end{aligned}$$ $$\begin{aligned} p = \frac{(n-2)e^{-\lambda}}{2r^2} \Bigl(r\nu^\prime + (n-3)(1-e^{\lambda}) \Bigr)-\qquad\qquad \nonumber \\ - \frac{(n-2) e^{-2\lambda}\tilde\alpha}{2r^4}(1-e^{\lambda}) \Bigl(2r\nu^\prime + (n-5)(1-e^{\lambda})\Bigr). \label{p-gb}\end{aligned}$$ The analogue of the isotropy Eq. (\[iso\]) takes the form $$I_{GB} \equiv \biggl(1+\frac{2\tilde\alpha f}{r^2}\biggr){I_E}+ \frac{2\tilde\alpha}{r}\biggl(\frac{f}{r^2}\biggr)^{\prime} \biggl[r\psi^\prime + \frac{f}{1-f}\psi\biggr] = 0 \label{iso-gb}$$ where $\psi=e^{\nu/2}, e^{-\lambda}=1-f, \tilde\alpha=(n-3)(n-4)\alpha$ and $I_E$ is given by the left-hand side (LHS) of Eq. (\[isotropy\]), $$\begin{aligned} I_E \equiv \frac{(1-f)}{\psi}\biggl\{\psi^{\prime\prime} - \biggl(\frac{f^{\prime}}{2(1-f)}+\frac{1}{r}\biggr)\psi^{\prime} - \qquad\qquad\nonumber \\ - \frac{(n-3)}{2r^2(1-f)}(rf^{\prime} - 2f)\psi\biggr\}. \label{isop}\end{aligned}$$ From Eq. (\[rho-gb\]), we write $$(\tilde\alpha r^{n-5}f^2 + r^{n-3}f)^{\prime} = \frac {2}{n-2}\rho r^{n-2}$$ which integrates for $\rho=const.$ to give $$\tilde\alpha r^{n-5}f^2 + r^{n-3}f = \rho_0r^{n-1} + k$$ where $k$ is a constant of integration that should be set to zero for regularity at the center and $2\rho/(n-1)(n-2)=\rho_0$ as defined earlier. Solving for $f$, we get $$e^{-\lambda} = 1 - f = 1 - \rho_{0GB}r^2 \label{sol-gb}$$ where $$\rho_{0GB} = \frac{\sqrt{1 + 4\tilde\alpha\rho_0} - 1}{2\tilde\alpha}.$$ So the solution is the same as in the Einstein case and the appropriate choice of sign is made so as to admit the limit $\alpha\rightarrow 0$ yielding the Einstein $\rho_0$ (the other choice would imply $\rho_{0GB}<0$ for positive $\alpha$). This, when substituted in the pressure isotropy Eq. (\[iso-gb\]), would lead to $I_E=0$ in Eq. (\[isop\]) yielding the solution (\[sol2\]) as before. This establishes sufficient condition for universality. For the necessary condition, we have from Eqn (\[iso-gb\]) that either $$\Bigl(\frac{f}{r^2}\Bigr)^\prime = 0$$ or $$r\psi^\prime + \frac{f}{1-f}\psi = 0$$ The former straightway leads with the use of Eq. (17) to the same constant density solution (\[sol-gb\]) and $I_E=0$ integrates to Eq. (\[sol2\]) as before. This shows that universality implies constant density as the necessary condition. For the latter case, when Eq. (22) is substituted in Eq. (\[isop\]) and $I_E = 0$ is now solved for $\lambda$, we again obtain the same solution (\[sol-gb\]). Equation (17) again implies $\rho=const$ as the necessary condition. Now $\psi$ is determined by Eq. (22), which means the constant $A$ in solution (\[sol2\]) must vanish. Then the solution turns into de Sitter spacetime with $\rho=-p=const.$ which is a particular case of Schwarzschild solution. This is, however, not a bounded finite distribution. Thus universality and finiteness of a fluid sphere uniquely characterize the Schwarzschild interior solution for Einstein as well as for Einstein-Gauss-Bonnet gravity. That is, gravity inside a fluid sphere of finite radius is universal, i.e it is true for all $n\geq4$ if and only if the density is constant and it is described by the Schwarzschild interior solution. It is only the constant density that gets redefined in terms of $\rho_0$ and $\rho_{0GB}$. If we relax the condition of finiteness, it is de Sitter spacetime with $\rho=-p=const$. Our entire analysis is based on the two equations (\[iso-gb\]) and (17). Let us look at GB contributions in them. In the former, there is a multiplying factor to the Einstein second order differential operator $I_E$ and another term with the factor $\tilde\alpha (f/r^2)^\prime$. This indicates that the contributions of higher orders in Lovelock polynomial will obey this pattern to respect quasilinearity of the equation. The higher orders will simply mean inclusion of the corresponding couplings in the multiplying factor as well as in the second term appropriately while the crucial entities, $I_E$ and $(f/r^2)^\prime$ on which the proof of the universality of Schwarzschild solution hinges remain intact. On the other hand, Eq. (17) is quadratic in $f$ for the quadratic GB action, which means the degree of $f$ is tied to the order of the Lovelock polynomial. It essentially indicates that as $\rho_{0GB}$ is obtained from a quadratic algebraic relation, similarly in higher order its analogue will be determined by the higher degree algebraic relation. The solution will always be given by Eq. (\[sol\]). Thus what we have shown explicitly for EGB will go through for the general Einstein-Lovelock gravity. Since Eqs. (15-17) owe their form and character to quasilinearity of the EGB equation, hence the carrying through of the Newtonian result of universality of gravity inside a uniform density fluid sphere critically hinges on quasilinearity. Thus this general result will not go through in theories like $f(R)$ gravity which do not in general respect quasilinearity. It could in a sense be thought of as yet another identifying feature of Einstein-Lovelock gravity. Let us now also indicate an itriguing and unusual feature of GB(Lovelock) gravity. What happens if the multiplying factor, $1+2\tilde\alpha f/r^2 = 0$ in Eq. (\[iso-gb\])? Then the entire equation becomes vacuous, leaving $\psi$ completely free and undetermined while $e^{-\lambda} = 1 + r^2/2\tilde\alpha$. This leads to $p = -\rho = (n-1)(n-2)/8\tilde\alpha$, which is an anti-de Sitter distribution for $\alpha\ge0$. This is a special prescription where density is given by GB coupling $\alpha$. There is no way to determine $\psi$, and so we have a case of genuine indeterminacy of the metric. It is because GB(Lovelock) contributes such a multiplying factor involving ($\alpha,~ r,~ f$) to sceond order quasilinear operator, which could be set to zero and thereby annul the equation altogether. Such a situation has been studied in the Kaluza-Klein split-up of six-dimensional spacetime into the usual $M^4$ and $2$-space of constant curvature in EGB theory  [@mada]. It gave rise to a black hole from pure curvature where the equations split-up into a four-dimensional part and a scalar constraint from an extra- dimensional part. As here by fine-tuning $\alpha, \Lambda$, and the constant curvature of the $2$-space, the four-dimensional part was turned vacuous, and then the metric was, however, determined by the remaning single scalar equation. This was because for vacuum (the null energy condition implies $\nu+\lambda =0$ in our notation), there was only one free parameter to be determined for which there was still a scalar constraint equation. The solution of that gave the black hole without matter support on $M^4$  [@mada]. In contrast, here we have two metric functions to be determined and there is only one equation remaining after the fine-tuning of density with $\alpha$. Thus one metric function will have to remain undetermined. As argued above, the form of Eq. (\[iso-gb\]) will be generic for the Lovelock system, and hence this kind of indeterminacy under the fine-tuning of parameters will also be generic. Matching with the exterior -------------------------- Now we would like to demonstrate matching of the interior with the corresponding exterior five-dimensional Gauss-Bonnet black hole solution [@boul]. In the interior, pressure is given by $$\begin{aligned} &&\hspace*{-3em} p=\frac{3}{4\alpha}(1-\mu ) \biggl[ 1- \frac{\mu}{1+\frac{2A\sqrt\alpha}{B\sqrt{r^2(1-\mu )+4\alpha}}} \biggr] \end{aligned}$$ where $$\mu = \sqrt{1 + 8\alpha \rho_{0GB}}.$$ At the boundary, $r=r_\Sigma$, pressure vanishes, which is equivalent to the continuity of $g_{tt}^\prime$, and that is what we shall employ. Besides this, the metric should be continuous across $r_\Sigma$. The five-dimensional Gauss-Bonnet black hole is given by the metric [@boul], $$ds^2= F(r)dt^2 - \frac{dr^2}{F(r)} - r^2(d\theta^2+\sin^2(\theta)( d\varphi^2+\sin^2(\psi) d\psi^2))$$ where $$F(r) = 1 + \frac{r^2}{4\alpha}(1 -\sqrt{1 + 8M\alpha/r^4}).$$ Now matching $g_{rr}$ means $[g_{rr}]_\Sigma=0$ which after appropriate substitutions determines the mass enclosed inside the radius $r_\Sigma$, $$M= \frac16\rho_{0GB} r_\Sigma^4.$$ Further $[g_{tt}]_\Sigma=0$ and $[g_{tt}^\prime]_\Sigma=0$ determine the constants, $$A=(1-B)\sqrt{1-\rho_{0GB}r_\Sigma^2}$$ and $$B=-(1 + \frac{8\alpha M}{r_\Sigma^4})^{-1/2}.$$ This completes the matching of the interior and exterior solutions. Discussion ========== We have established that the gravitational field inside a constant density fluid sphere has a universal character for spacetime dimensions $\geq4$. This is true not only for Einstein-Hilbert action but also for the more general Lovelock action which is a homogeneous polynomial in Riemann curvature. We have explicitly shown this for the linear Einstein and the quadratic Gauss-Bonnet cases and have argued that the proof would go through for the general Lovelock polynomial. That is, the Schwarzschild interior solution describing the gravitational field of the constant density sphere is true for all spacetime dimensions $\geq4$ for Einstein as well as for higher order Einstein-Lovelock polynomial gravity. It turns out that the necessary and sufficient condition for the universality of fluid sphere is that its density must be constant. Equivalently, universality uniquely characterizes the Schwazschild interior solution for a fluid sphere of finite radius. This result is obvious but perhaps not much noticed in Newtonian gravity as argued in the opening of the paper. It is, however, not so for Einstein-Lovelock gravity because of its highly nonlinear character. Yet it is carried through because the equation of motion still remains second order quasilinear. It is this feature that carries the general character of the solution into higher order gravity. Clearly it would not in general be carried along for non-quasilinear theory like $f(R)$ gravity. Apart from Lovelock’s original derivation of the action [@lov], there are two other characterizations of Lovelock action [@dad; @irani]. In [@dad], the identifying feature is the existence of the homogeneous polynomial in curvatures analogus to the Riemann curvature whose trace of the Bianchi derivative yields the corresponding analogue of the Einstein tensor in the equation while for [@irani] it is the requirement that both metric and Palitini variations give the same equation of motion. Here we have yet another identifying property of Einstein-Lovelock gravity. Also it exhibits that the obvious Newtonian result is carried through in higher order nonlinear theories. Universality characterizes uniform density for the static fluid sphere. The main aim of such investigations is essentially to probe and identify universal features of gravity for greater understanding and insight. Such universal features also provide discerning criteria for competing genralizations of Einstein gravity. This work was initiated by ND and AM during their visit to the University of Kwa-Zulu Natal, Durban, and they would like to warmly thank Professor Sunil Maharaj for the wonderful hospitality. AK gratefully acknowledges TWAS for support while visiting IUCAA and also thanks IUCAA for warm hospitality which facilitated this collaboration. AM also acknowledges financial suport from the Ministerio de Educación, Grant No. FIS2007/63 and from the Generalitat de Catalunya, 2009SGR0417 (DURSI). Finally we also thank the anonymous referee for the constructive criticism that has substantially improved clarity and precision. [9]{} R Emparan and HS Reall, Living Rewiev Relativity, [**11:6**]{} (2008). FR Tangherlini, Nuovo Cimento [**27**]{}, 636 (1963), RC Myers and MJ Perry, Ann. Phys. (NY) [**172**]{}, 304 (1986), ZW Chong, M Cvetic, H Lu and CN Pope, Phys. Rev. Lett. [**95**]{} 161301 (2005). A Strominger and C Vafa, Phys. Lett. [**B379**]{}, 99104 (1986), J Maldacena, Adv. Theor. Math. phys. [**2**]{}, 231 (1998), E Witten, Adv. Theor. Math. Phys. [**2**]{}, 253 (1998). KD Krori, P Borgohan and K Das, Phys. Lett [**A132**]{}, 321 (1988), YG Shen and ZQ Tan, Phys. Lett. [**A142**]{}, 341 (1988), JP de Leon, Gen.Rel.Grav. [**32**]{}, 1207(2000), A Das and A DeBenedictis, Prog. Theor. Phys. [**108**]{}, 119 (2002). D Gross, Nucl. Phys. Proc. Suppl. [**74**]{}, 426, (1999). H Maeda and N Dadhich, Phys. Rev. [**74**]{}, 021501 (2006), [**75**]{}, 044007 (2007); N Dadhich and H Maeda, Int. J. Mod. Phys., [**D17**]{}, 513 (2008). DG Boulware and S Deser, Phys. Rev. Lett. [**55**]{}, 2656 (1985). D Lovelock, J. Math. Phys. [**12**]{}, 498 (1971). N Dadhich, On gravitational dynamics, arxiv:08023034 (2008). O. Exirifard and M. M. Sheikh-Jabbari (2008) Phys.Lett. [**B661**]{}, 158 (2008).

--- abstract: 'Tracking by detection is a common approach to solving the Multiple Object Tracking problem. In this paper we show how learning a deep similarity metric can improve three key aspects of pedestrian tracking on a multiple object tracking benchmark. We train a convolutional neural network to learn an embedding function in a Siamese configuration on a large person re-identification dataset. The offline-trained embedding network is integrated in to the tracking formulation to improve performance while retaining real-time performance. The proposed tracker stores appearance metrics while detections are strong, using this appearance information to: prevent ID switches, associate tracklets through occlusion, and propose new detections where detector confidence is low. This method achieves competitive results in evaluation, especially among online, real-time approaches. We present an ablative study showing the impact of each of the three uses of our deep appearance metric.' author: - Michael Thoreau and Navinda Kottege bibliography: - 'ostab.bib' title: 'Deep Similarity Metric Learning for Real-Time Pedestrian Tracking' --- Introduction ============ Accurately tracking objects of interest such as pedestrians and vehicles in video streams is an important problem with applications in many fields such as surveillance, robotics and autonomous vehicles. The problem of Multiple Object Tracking (MOT) in video has mostly been addressed in recent literature using the ‘tracking by detection’ framework. In this formulation, detections are combined to estimate the trajectories of tracked objects. Solutions can generally be grouped in to online and batch processes. The difference being, online solutions use measurements only as they arrive while a batch process may build globally optimal trajectories by considering measurements at all times. In this paper we present an online approach to solving the MOT problem for pedestrian tracking and evaluate it on the MOTChallenge dataset [@Leal-TaixeMOTChallenge2015Benchmark2015; @MilanMOT16BenchmarkMultiObject2016]. Motivated by the large amounts of labelled data now available for pedestrian re-identification problems, the proposed method uses a deep-learning approach to appearance modelling. We present a convolutional neural network, trained in a Siamese configuration to produce a discriminative appearance similarity metric for pedestrians. We present three ways in which this deep appearance metric learning can be used in MOT and show how using two of these components together can achieve competitive performance on a tracking benchmark. We compare our results to those of other methods and evaluate each use of the proposed appearance metric independently in an ablative study. First we show how a learned appearance metric can be used to improve the *assignment* of candidate detections to form short tracks (tracklets) as the first step in creating longer optimal tracks. Next we show how the same metric learner can perform *detection boosting* to reduce false negatives where detections are missing within a person’s track. Lastly the deep appearance metric is used to perform iterative appearance based merging of tracklets to form longer tracks, a process we call *tracklet association*. We accomplish this as an online process, with a playback delay of only a few seconds, at a frame rate suitable for real-time applications. The rest of the paper is organised as follows; section \[sec:relatedwork\] describes the related approaches in the literature, section \[sec:SMDT\] introduces the proposed Siamese Deep Metric Tracker, section \[sec:evaluation\] evaluates the proposed method on the publicly available MOT16 dataset of the MOTChallenge, section \[sec:discussion\] discusses the evaluation results and section \[sec:conclusions\] concludes the paper. Related Work {#sec:relatedwork} ============ Solutions to the multiple object tracking problem fall in to two distinct categories; batch and online processing. In batch processing, detections are combined in a global sense, rather than frame by frame, to form optimal tracks[@RenFasterRCNNRealTime2015; @LiuSSDSingleShot2016; @Tychsen-SmithDeNetScalableRealtime2017]. Despite the apparent performance advantages of batch methods as described by Luo et al. in their extensive literature review[@LuoMultipleObjectTracking2014a], we consider only online methods in this work, where filtered tracks are available with little to no delay, motivated by potential real-time applications in surveillance, robotics and industrial safety. In some online approaches, tracklet states are estimated by a probabalistic model such as a Kalman filter [@BochinskiHighSpeedtrackingbydetectionusing2017; @BewleySimpleOnlineRealtime2016]. Others have used deep learning to learn to estimate the motion of tracked objects from data, including estimating the birth and death of tracks[@MilanOnlineMultiTargetTracking2016]. A difficult aspect of tracking by detection is solving the data association problem present when grouping detections or merging tracklets. Some works is present in the literature that use confidence estimation to aid in the data association problem by prioritising high confidence tracks[@BaeConfidenceBasedDataAssociation2017; @BaeRobustOnlineMultiobject2014]. Others leverage image information, where even simple appearance modelling has been shown to make data association more robust[@takala2007multi]. Appearance modelling plays a larger role in single object tracking, where only appearance is used to track objects given a prior[@BolmeVisualobjecttracking2010]. More recently, the availability of labelled data has motivated methods utilising deep learning for appearance modelling. Siamese networks have been used for single object tracking to great effect by Feichtenhofer et al. and Liu et al., where the deep appearance model is used to search successive frames[@FeichtenhoferDetectTrackTrack2017; @TaoSiameseInstanceSearch2016]. In multiple object tracking, online learning has been used to discriminate between tracked objects based on appearance albeit at limited speed due to computational complexity[@BaeConfidenceBasedDataAssociation2017]. Some methods using deep learning have achieved outstanding results on the MOT Challenge [@LiuEndtoEndComparativeAttention2017; @YuPOIMultipleObject2016]. For example, [@Leal-TaixeLearningtrackingSiamese2016] achieve good results with a siamese network, however with a requirement for a gallery of images to be stored from past tracks to do re-identification [@Leal-TaixeLearningtrackingSiamese2016]. Wojke et al. solves this by using a deep similarity metric learning network to store a gallery of metrics [@WojkeSimpleOnlineRealtime2017]. He et al. goes one step further and uses a deep recurrent network to compute an appearance metric which incorporates temporal information from the tracked object, to good effect [@HeSOTMOT2017]. Informed by the literature, we have developed our ‘Siamese Deep Metric Tracker’; with competitive tracking accuracy and real-time performance, crucial for applications in robotics. ![The proposed process where *Assignment*, *boosting*, and *tracklet association* components benefit from the use of deep appearance modelling.[]{data-label="fig:OSTAB"}](figures/OSTAB_new.pdf){width="0.8\columnwidth"} Siamese Deep Metric Tracker {#sec:SMDT} =========================== Here we present our proposed Siamese Deep Metric Tracker to perform online multiple object tracking. A strong appearance model is central to this proposed method. We use a single deep neural network, detailed in section \[sec:deep\_metric\], to enable or assist three components of our object tracking algorithm shown at a high level in figure \[fig:OSTAB\]. We solve the problem in multiple stages; firstly, detections are *Assigned* to tracklets, as detailed in section \[sec:assignment\]; detections are then *Boosted* as described in section \[sec:boosting\]; and finally tracklets are *Associated* as described in section \[sec:Tracklet\_Association\]. Notation -------- We use the following notation in all equations, explanations and algorithm listings in this paper. Let the set of estimated tracklets be $\mathcal{T}$, containing $J$ tracklets $T_j$. Let the estimated state of tracklet $j$ at time $t$ be $T_j^t$ and the predicted state of tracklet be ${T_j^t}^{\prime}$. Let a set of detections at time $t$ be $\mathcal{D}^t$ containing $I$ detections $D_i$. Deep Similarity Metric {#sec:deep_metric} ---------------------- A robust appearance model can improve simple object tracking by preventing tracks from drifting to false positive detections, and by enabling objects to be tracked through occlusion. At each time step we compute a feature vector $f \in \mathbb{R}^{128}$ for each candidate detection in a single batch. Computing all features in a batch is an efficient use of GPU resources, taking only $\approx 20\text{\,ms}$ for a typical batch of 40 image patches. The network, with layers listed in table \[tab:nn\_structure\], uses pre-trained convolutional layers from VGG-16[@SimonyanVeryDeepConvolutional2014], followed by two fully connected layers with batch and $l_2$ normalisation on the output layer. The use of pre-trained networks as feature extractors in Siamese/triplet networks has been shown to reduce the number of iterations required for convergence and improve accuracy[@HermansDefenseTripletLoss2017]. Euclidean distance between feature vectors lying within a unit hypersphere measures the distance $d_a = || f_1-f_2 ||$ between two input patches in the appearance similarity space. The appearance affinity $A_a$ between two patches is $A_a = 1-d_a$. We use an appearance affinity threshold $\tau_a = 0.895$, determined offline, to separate similar and dissimilar pairs. Two implementations of Siamese networks are shown in figure \[fig:twofigs\]. Figure \[fig:newmethod\] is the proposed implementation, using margin contrastive loss with a fixed margin of $0.2$. Figure \[fig:oldmethod\] is an alternative implementation, using a learned softmax classifier to give a similarity score between the input images. In our approach, we compute the feature vector for a detection and store it with the state of the tracklet at the time of the detection, meaning that we don’t have to store a gallery of images for each tracked object. We assume that the appearance metric computed from the detection will closely match the appearance metric of the true bounding box of the subject. This assumption appears to hold during testing, as bounding boxes are usually well regressed to the true bounding box of the detected object. Layer Output shape --------------------- --------------------------- Input $128 \times 64 \times 3$ VGG-16 $32 \times 16 \times 256$ Fully connected $128$ Fully connected $128$ Batch normalisation $128$ $l_2$ normalisation $128$ : Similarity Network Structure[]{data-label="tab:nn_structure"} Training {#sec:training} -------- The deep similarity network was trained on the Market-1501 pedestrian re-identification dataset[@ZhengScalablePersonReidentification2015], containing $\approx$ 32,000 annotated images of 1501 unique pedestrians in six camera views. Triplet loss has recently been used to good effect in training networks for pedestrian re-identification[@HermansDefenseTripletLoss2017]. Networks using triplet loss have been known to be difficult to train, due to a stagnating training loss. Batch-hard example mining has been shown to improve convergence when training with triplet loss[@HermansDefenseTripletLoss2017]. Our approach uses batch-hard sampling to train our network in a Siamese fashion using margin contrastive loss in a large batch. We sample 4 images each from 32 identities, compute their feature vectors in a forward pass and select the hardest pairings, maximising Euclidean distance between feature vectors for positive pairs and minimising distance for negative pairs, for each of the 128 images. {width="100.00000%"} Detection Assignment {#sec:assignment} -------------------- Detections are combined across time to estimate the trajectory of a tracked object. This algorithm is shown in listing \[lst:assignment\] and detailed below. The motion of small segments, tracklets, are estimated via a Kalman filter with a constant velocity constraint. Tracklet states are predicted at each time step, but are considered inactive after two predictions without being assigned a detection. A tracklet’s state is predicted for another 90 steps for tracklet association, discussed in section \[sec:Tracklet\_Association\]. The association of new detections to the set of active tracklets is solved as a data association problem using the Hungarian algorithm[@Hungarianmethodassignment]. The Hungarian algorithm maximises the affinity between tracklets and assigned detections, provided in the affinity matrix $\tilde{A}$, and creates entries in the matching matrix $\tilde{M}$. The affinity used to assign candidate detections to tracklets is a combination of motion affinity, preferencing detections close to the predicted position of the tracklet, and appearance affinity which attempts to match the tracklet with a detection whose appearance is closest to stored appearance information. Motion affinity is implemented as the Intersection over Union (IoU)[@YuUnitBoxAdvancedObject2016] between a candidate detection $D_i^{t}$ and the predicted bounding box of the tracklet ${T_j^{t}}^{\prime}$, as shown in equation \[eq:motionAffinity\]. Motion affinity is constrained to be strictly greater than a motion affinity threshold $\tau_m = 0.3$ for assignment. $$A_m(T_j, D_i^t) = IoU({T_j^t}^{\prime}, D_i^t),\;\; \textit{IoU}(b_1, b_2) = \frac{b_1 \cap b_2}{b_1 \cup b_2} \label{eq:motionAffinity}$$ Appearance affinity is computed as the mean affinity between a candidate detection’s feature vector and the stored feature vectors for a tracklet, shown in equation \[eq:appearance\_affinity\] with $t_0$ denoting the first state of the tracklet. An example of appearance affinity degrading as a track drifts to an overlapping detection is shown in figure \[fig:drifting\_track\]. A subset of N past states of the tracklet is used for computational tractability, in practice $N \leq 20$. $$A_a(T_j, D_i^{t}) = \frac{1}{N}\sum_{n = n_0}^{N} \> f(T_j^n, D_i^{t}),\;\; n \in \{t_0,t-1\} \label{eq:appearance_affinity}$$ $$f(T_j^n, D_i^{t}) = 1 - || f_1-f_2 || \label{eq:euclidean}$$ The total affinity, shown in equation \[eq:total\_Affinity\], is a combination of appearance and motion affinity, balanced by the parameter $\lambda$, typically between $0.3$ and $0.7$. A value of $0$ may be used to ignore appearance entirely when computing assignments, potentially improving frame rate under some implementations. $$A(T_j^t, D_i^t) = \lambda A_a(T_j, D_i^t) + (1-\lambda)A_m(T_j, D_i^t) \label{eq:total_Affinity}$$ $\tilde{A}_{j,i} \gets A(T_j, D_i^t)$ $\tilde{A}_{j,i} \gets 0$ $\tilde{M} \gets H(\tilde{A})$ update state of tracklet $T_j$ with detection $D_i^t$ Tracklet Confidence ------------------- A minimum length requirement $\tau_l = 6$ is imposed on tracklets for them to be considered positive. Tracks containing less than six states are considered negative and therefore are not reported. The mean confidence of detections assigned to a given tracklet is also used to filter out low confidence tracklets, with a minimum mean confidence of $\tau_c = 0.2$ used in practice. The average cost of assigning detections to a tracklet is used to estimate confidence in it being positive. A tracklet with a high mean assignment cost is likely to be varying in appearance or in motion and is considered negative. Tracklet association and boosting considers only positive tracks to avoid joining false positives with true positives. Detection Boosting {#sec:boosting} ------------------ In the case that in a given frame, there exists no detection which matches to a tracklet, but the tracked object is not occluded or out of frame, we wish to re-identify that person. Using the predicted location of the object as a prior, we perform dense sampling around the prediction and select the candidate bounding box which maximises appearance affinity and satisfies the appearance affinity constraint $\tau_a = 0.895$. This detection is added to the detection set and association is performed again, as shown in figure \[fig:OSTAB\]. In order to prevent track drift, boosting is limited to no more than once per two frames per track. To stop partial detections from drifting to a true person via boosting and therefore adding false positives, Non Maximum Suppression (NMS) is performed on the detections with a NMS-IoU threshold of $0.5$. Method MOTA $\uparrow$ FP $\downarrow$ FN $\downarrow$ IDs $\downarrow$ FPS $\uparrow$ -------------------------- ----------------- ----------------- ----------------- ------------------ ---------------- MOTDT[@long2018tracking] **47.6** 9,253 **85,431** 792 20.6 TestUnsup 41.5 12,596 93,404 643 19.7 PMPTracker 40.3 10,071 97,524 1,343 148.0 **SDMT (ours)** 39.6 11,130 98,343 **602** 19.8 RNN\_A\_P 34.0 8,562 109,269 2,479 19.7 cppSORT[@murray2017real] 31.5 3,048 120,278 1,587 **687.1** DCOR 28.3 **1,618** 128,345 849 32.9 Tracklet Association {#sec:Tracklet_Association} -------------------- Targets may be tracked through occlusion by matching tracklets across time using their appearance. Our association algorithm is shown in listing \[lst:association\] and described below. Due to uncertainties in camera and target motion, a much looser motion constraint is used to associate tracklets, requiring only a small overlap between the predicted bounding box of the older tracklet and the first bounding box of the newer track i.e. $IoU({T_j^t}^{\prime}, T_k^t) > 0$. $$A_a(T_j, T_k) = \frac{1}{N} \frac{1}{M} \sum_{n = n_0}^{N} \sum_{m = m_0}^{M}\> f(T_j^n, T_k^m) \label{eq:tracklet_appearance}$$ As tracking is done in the image plane, changes in camera motion may frequently violate the constant velocity constraint imposed by our Kalman filter based tracking. By building small tracklets with a stricter motion constraint and linking high confidence tracklets in to longer tracks with a looser motion constraint, intuitively our tracking may be robust to changes in camera motion. Tracklet association need not run at every time step, once every 20 time steps is sufficient to not impact performance, resulting in a higher refresh rate. After tracklets have been merged, temporal gaps are filled by interpolation with a constant velocity, giving a reasonable estimate for the state of the object while it is occluded. \[ln:beginning\] $\mathcal{C} \gets \mathcal{C} \cup T_k$ select best match\ $T_o = \underset{l} {\mathrm{\text{arg\,max}}} \; A_a(T_j, T_l) \quad \forall \> T_l \in \mathcal{C} $ merge tracklets $T_j \gets T_j \cup T_o$ **go to** \[ln:beginning\] Method MOTA $\uparrow$ FP $\downarrow$ FN $\downarrow$ IDs $\downarrow$ FPS $\uparrow$ --------------------------------- ----------------- ----------------- ----------------- ------------------ ---------------- SDMT ($\lambda = 0.5$) **34.6** 6,014 65,863 317 29.8 SDMT ($\lambda = 0$) 34.3 6,541 65,651 **295** 29.6 SDMT ($\lambda = 1$) 33.9 6,512 66,040 373 28.7 SDMT (w/ boosting) 34.2 6,743 **65,533** 334 25.9 SDMT (w/o association) 32.4 **3,968** 69,965 686 31.6 SDMT (w/o appearance modelling) 32.9 4,069 69,468 587 **96.8** Evaluation {#sec:evaluation} ========== The Siamese deep metric network was validated on a subset of Market-1501 dataset not used for training. The network achieved an area under the receiver operator characteristic curve of $0.98$ after $90,000$ training iterations, with an equal mix of positive and negative pairs and distractors sampled from the background. This validates the training of the deep similarity network. CLEAR MOT Metrics ----------------- The CLEAR MOT[@StiefelhagenCLEAR2006Evaluation2006] metrics are used here to compare our performance to others, as well as compare the benefits of each of the uses of our deep appearance model. The specific metrics we use ($\uparrow$ denotes metrics in which a higher score is better, $\downarrow$ denotes metrics in which a lower score is better): - MOTA $\uparrow$, combines FP, FN and IDs to give a single metric to summarise accuracy. - FP $\downarrow$, is the number of false positive bounding boxes. - FN $\downarrow$, is the number of false negative bounding boxes. - IDs $\downarrow$, is the number of times tracked targets swap ID’s. - FPS $\uparrow$, is the update frequency, an important metric for real-time applications. MOT16 results ------------- A selection of methods suitable for real-time applications was made for comparison. Online approaches that achieve an update rate of greater than 15Hz on the MOT16 test set using the public detections are shown in table \[tab:MOT16\] compared to our approach. Among the comparable approaches, our method achieves a competitive tracking accuracy (MOTA) for a relatively simple method, but crucially the lowest number of ID switches, thanks to robust tracklet association and detection assignment. We performed repeated testing while enabling/disabling certain aspects of our algorithm, presented in table \[tab:ablative\]. The best performing method from this ablative testing was used in testing presented in table \[tab:MOT16\]. The best method did not include boosting and used a lambda value of $\lambda = 0.5$. Changing $\lambda$ to 0 or 1 reduced accuracy on the training set. Adding boosting to the optimal method reduced false negatives but significantly increased false positives. Removing tracklet association, or appearance modelling entirely significantly reduced tracking accuracy. The method without any appearance modelling removed the need to compute feature vectors for each detection, significantly increasing the update rate. Discussion {#sec:discussion} ========== We found that using our deep appearance metric for detection assignment and tracklet association improved the overall performance of multiple object tracking. Only ‘detection boosting’ was found to hurt the accuracy of our tracking on this dataset, despite reducing the number of false negatives as intended. This was likely due to the high recall rate of 43% (with significant occlusion) but relatively low precision of the DPM v5 detections provided with the test sequences[@MilanMOT16BenchmarkMultiObject2016]. Boosting is most useful when there exists no detection for a given target, yet the target is not completely occluded or out of frame, however we suggest this case does not occur often in the MOT16 dataset. Tracklets built from false positive detections that contain some part of a true object, may be boosted, causing drift towards the true object. This may lead to the tracks being merged, explaining the increase in false positives. We found that the addition of deep metric learning significantly reduced the number of ID switches. The ablative study suggests that the largest change in ID switches was due to tracklet association. This reflects the major benefit of a tracking formulation with a strong appearance model, estimating the position of objects while they are occluded. e.g. A pedestrian walking behind a bus. Conclusions {#sec:conclusions} =========== We presented three uses of deep appearance metric learning for improving multiple object tracking, and demonstrated how two of these uses significantly improved tracking accuracy on the MOT16 dataset. Our method achieved competitive results for online methods suitable for real-time applications, with the lowest number of ID switches. Our ablative testing may be used to inform further use of deep appearance metrics in multiple object tracking.

--- abstract: 'A generalized Strangeness-incorporating Statistical Bootstrap Model (SSBM) is constructed so as to include indepedent fugacities for up and down quarks. Such an extension is crucial for the confrontation of multiparticle data emerging from heavy ion collisions, wherein isospin symmetry is not satisfied. Two constraints, in addition to the presence of a critical surface which sets the boundaries of the hadronic world, enter the extended model. An analysis pertaining to produced particle multiplicities and ratios is performed for the $S+Ag$ interaction at 200 GeV/nucleon. The resulting evaluation, concerning the location of the source of the produced system, is slightly in favor the source being outside the hadronic domain.' --- 6.5in -0.2in -0.5in UA/NPPS-7-1999 \ \ PACS numbers: 25.75.Dw, 12.40.Ee, 12.38.Mh, 05.70.Ce [**1. Introduction**]{} Multiparticle production in high energy collisions is a subject of intense research interest, whose history goes almost as far back as that of the strong interaction itself. Indeed, it registers as one of the key features entering the analysis of collision processes, involving the strong force, at both the experimental and the theoretical fronts. With specific reference to relativistic heavy ion collisions, the task of accounting for the produced multiparticle system is by far the most important issue to consider for extracting information of physical interest. A notably successful theoretical approach, through which experimentally observed particle multiplicities have been confronted, is based on the idea of thermalization. Within such a context, one views the multiparticle system, emerging from a given high energy collision, as being comprised of a large enough number of particles to be describable in terms of a thermodynamical set of variables. Relevant, standard treatments appearing in the literature \[1-5\] adopt an “Ideal Hadron Gas” (IHG) scheme, wherein any notion of interaction is totally absent[^1]. The fact that such analyses, ranging from $e^+e^-$ to $A+A$ collisions, produce very satisfactory results simply verifies, a posteriori, that the thermalization assumption is justifiable. Beyond this realization, however, no fundamental insight and/or information is gained with respect to the [ *dynamics*]{} operating during the process, which produced the multiparticle system in the first place. Given, in particular, that the object of true interest, in the case of relativistic heavy ion collisions, is whether the original thermal source of the multiparticle system is traceable to a region that belongs, or not, to the hadronic phase, an IHG-type of analysis renders itself totally inadequate. Clearly, only if interactions are taken into consideration does it become relevant to ask whether or not a change of phase has taken place during the dynamical development of the system. In a recent series of papers \[6-8\], we have pursued a line of investigation which, on the one hand, approaches the study of multiparticle systems from the hadronic side (just as the IHG case) while, on the other, incorporates the effects of interactions in a self-consistent way. We are referring to the employment of a scheme, known as Statistical Bootstrap Model (SBM), which was originally introduced by Hagedorn \[9-11\], much before QCD was conceived and was subsequently developed via notable contributions by a number of authors. Excellent reviews articles on the SBM can be found in Refs \[12\]. The crucial feature of the SBM is that it adopts a statistical-thermodynamical mode of description, which admits interactions among its relativistic constituent particles via a bootstrap logic. According to the SBM, the constitution of the system is viewed at different levels of organization (fireballs) with each given level being generated as a result of interactions operating at the preceeding one. The remarkable feature of the SBM is that the so-called bootstrap equation (BE), which results from the aformentioned reasoning, defines a critical surface in the space of the thermodynamical variables, which sets an upper bound to the world of hadrons and implies, under precisely specified conditions, the existence of a new phase of matter beyond. Let us briefly review this construction while giving, at the same time, an overview account of the bootstrap scheme itself. We start by displaying the generic bootstrap equation, whose final form reads $$\varphi(T,\{\lambda\})=2G(T,\{\lambda\})-\exp(T,\{\lambda\})+1\;,$$ where $\varphi$ is the so-called [*input function*]{}, since its specification involves an input from all observable hadrons and $G$ incorporates, via the bootstrap logic, the mass-spectrum of the system given in terms of fireballs of increasing complexity. Note that, $\varphi$ and $G$ as functions depend on a thermodynamic set of variables (temperature and fugacities). The key feature of the BE is that it exhibits a square root branch point at $$\varphi(T,\{\lambda\})=\ln 4-1\;,$$ which defines a critical surface in the space of thermodynamical variables that sets the limits of the hadronic phase, in the sense that eq. (1) does not posses physically meaningful solutions beyond this surface. This is not to say that the BE is thermodynamically consistent with the existence of a different phase on the other side. In this connection, the deciding factor is the form of the so-called spectrum function $\rho(m^2)$ entering the definition of $G$ and, in particular, the way it factorizes into a kinematical and a dynamical part (see following section). The final ingredient of the SBM is the employment of a grand canonical partition function $Z(V,T,\{\lambda\})$, which accounts for thermodynamical properties. In combination with the BE, it furnishes a thermal description of a system comprised of relativistic entities (hadrons/ fireballs) [*interacting with each other*]{}. Recognizing the importance of the role played by the quantum number of strangeness in providing possible signals for a presumed QGP phase, we have extended the SBM by introducing a fugacity variable for strangeness into the scheme \[6,7\][^2]. We shall be referring to the resulting extended construction as the “SSBM”. Imposing the condition $<S>=0$ we proceeded to study thermodynamical properties of the SSBM. Central emphasis was placed on the choice of the spectrum function, in order to acheive an acceptable thermodynamical description, consistent with the existence of a phase beyond the hadronic one. The end result is encoded into the following relation expressing the partition function in terms of the “bootstrap function” $G$ \[6,7\] $$\ln Z(V,T,\{\lambda\})=\frac{VT^3}{4\pi^3 H_0} \int_0^T \frac{1}{y^5}G(y,\{\lambda\})dy\;,$$ where $H_0\equiv\frac{2}{(2\pi)^3 4B}$ with $B$ the MIT bag constant. In combination with the critical surface condition furnished by the BE, one is able to relate the critical temperature $T_0$ at vanishing chemical potential with $B$. This occurence provides a direct connection between QCD-inspired phenomenology and critical temperature for the hadronic state of matter. Our numerical studies have been based on the choice $T_0=183$ MeV, which corresponds to the maximum acceptable value for $B^{1/4}$, namely 235 MeV. Such a choice is consistent with the strangeness chemical potential $\mu_s$ remaining positive definite throughout the hadronic phase while maximally extending the region of the hadronic phase and thereby rendering our appraisal of the proximity of the source of the multiparticle system to the critical surface (or beyond) as conservative as possible. Subsequently, we generalized the SSBM \[8\] by introducing a further “fugacity” variable $\gamma_s$, which allows for [*partial*]{} strangeness chemical equilibrium. This extension of the model enables us to confront the data with an open perspective on strange particle production, as we let the observed particle multiplicities and ratios determine whether strangeness saturation has taken place or not. In \[8\] we also conducted a systematic study of multiparticle states (particle multiplicities and particle ratios) produced in $S+S$ as well as in $p+\bar{p}$ collisions at CERN (experiments NA35 and UA5, respectively), the latter considered more as a test case. Our results yield an excellent account of particle multiplicities and ratios (equally good, if not slightly better than IHG results \[5,14,15\]). More importantly, we have identified a region in the space of thermodynamical parameters where the source of the produced multiparticle state is expected to lie and appraised its location with respect to the limiting surface of the hadronic phase. For the $S+S$ interaction we summarize the highlights of our findings as follows: \(a) The quality of our results were similar to that given by IHG analysis. This further justifies the thermalization hypothesis. \(b) Almost full saturation of strangeness was observed, which accounts for an enhanced production of strange particles relative to non-strange ones. \(c) The source of the multiparticle system was found to lie just outside the limits of the hadronic phase, as established by the SSBM. \(d) An excess of pions (SSBM/experimental = 0.73) is observed, which is not fully compatible with the theoretical prediction of a purely hadronic phase. At the same time, entropy considerations also give SSBM/QGP $=0.71-0.78$, pointing, together with (c), to a source being in the doorway of a deconfined phase. These findings strongly suggest that in the $S+S$ interaction at 200 GeV/nucleon the thermalized, strangeness-saturated source of the multiparticle system has exceeded the hadronic sector and has entered the lower limits of the QGP phase. Now, the $S+S$ colliding system is symmetric under isospin transformations, hence consistent with the simplification $\lambda_u=\lambda_d$ adopted in our previous work on the SSBM. In the present paper we shall further extend the model so as to accomodate isospin non-symmetric systems. Such a step will enable us to confront multiparticle data for the $S+Ag$ collision experiment (NA35), at CERN. As we shall see, this further extension imposes a new constraint on the system which relates charge and baryon numbers. It follows that the SSBM extension we shall be discussing amounts, at the hadronic level, to introducing a fugacity variable pertaining to total charge. The presentation of the new extension of the SSBM, accomodating isospin non-symmetric systems, will be accomplished in Section 2. The profile of the relevant construction, accentuated by the presence of the critical surface in the space of thermodynamical parameters as well as the two surfaces resulting by the imposition of the two physical constraints, will be discussed. Our confrontation of the data (particle multiplicities and ratios) for the $S+Ag$ experiment (NA35 at CERN) is presented in Section 3. Our concluding remarks are made in Section 4. Two appendices are devoted to corresponding discussion of a more specialized nature. In Appendix A we establish that the value for the critical temperature for zero chemical potentials corresponds to a maximum on the critical surface. Appendix B discusses the subtle points involved in the minimization of the $\chi^2$-variable given the presence of constraints and the critical surface beyond which the SSBM has no analytical validity. In this section we shall realize the construction of a maximally extended SSBM, accomodating both partial strangeness saturation and isospin asymmetry. In this way we shall be in position to perform thermal analyses pertaining to (non)strange particle production in nucleus-nucleus collisions in which the total number of participating protons differs from that of neutrons. The set of variables in terms of which the initial quantification of the bootstrap scheme is accomplished naturally associates itself with input particle (and fireball) attributes. These are number densities and four-momenta pertaining to particle/fireball species. As hinted to in the introduction the situation we wish to consider in this paper involves, the following number densities: Baryon number $b$, net strangeness $s$, overall strangeness $|s|$ and “net charge” $q$. The employed sequence respects “historical” order in the following sense. In the original SBM only $b$ enters, the SSBM construction of Ref. \[6,7\] includes $s$(=strangeness minus anti-strangeness number) while the extension of Ref \[8\] has added $|s|$(=strangeness plus anti-strangeness number) to the list. Our present effort amounts to a further extension of the SSBM through which we incorporate a “net charge” number density into the bootstrap scheme. To quantify our considerations regarding this new variable let us focus on the initial states entering a nucleus-nucleus collision process and consider the ratio $\frac{N_p^{in}}{N_n^{in}}$, where $N_p^{in}$($N_n^{in}$) denotes the total number of protons(neutrons) participating in the collision. Suppose this ratio is equal to unity. It then, follows that $$\frac{N_p^{in}}{N_n^{in}}=\frac{<Q>^{in}}{<B>^{in}-<Q>^{in}}=1\;,$$ where $<Q>^{in}$ and $<B>^{in}$ are the incoming total charge and baryon numbers, respectively. Equivalently, the above condition reads $$<B>=2<Q>\;.$$ By introducing a “net charge” particle density into the bootstrap scheme we declare our intention to confront $A+A$ collision processes which do not, a priori, respect the condition given by (5). It is not hard to see that the latter corresponds to an isospin non-symmetric system, at least as far as its initial (incoming) composition is concerned. With reference to (5) the quantification of isospin asymmetry can be parametrized as follows $$<B>=\beta 2<Q>\;,$$ where $$\beta=\frac{N_p^{in}+N_n^{in}}{2N_p^{in}}.$$ Our actual preoccupation, of course, is with the description of the produced, final states. Accordingly, we shall eventually impose (6) as a [*constraint*]{} on the system. We close this general exposition with a brief discussion of the particular version of the SSBM we have adopted throughout our work as far as the issue of dynamics vs kinematics is concerned. Generically speaking, the SSBM construction involves a mass spectrum function $\tau$ whose dependence is on the set of variables $\{p^2,b,s,|s|,q\}$. A kinematic factor $\tilde{B}(p^2)$ enters the equation (see the BE in the following subsection), the specific choice of which classifies different versions of bootstrap models, according to asymptotic behaviour, as the fireball mass goes to infinity. Our specific commitment to the form $\tilde{B}(p^2)$ has been discussed at great lenght in Refs. \[6,7\]. We have argued that there are desicive physical advantages in favor of the choice $$\tilde{B}(p^2)=B(p^2)=\frac{2V^{\mu}p_{\mu}}{(2\pi)^3}\;,$$ where $V^{\mu}$ is the (boosted) four-volume associated with a given particle/fireball and $p_{\mu}$ the corresponding four-momentum. The two four-vectors being parallel to each other imply a relation of the form $$V_{\mu}=\frac{V}{m} p_{\mu}\;,$$ $V$ being the rest frame volume. We, therefore, have $$B(p^2)\rightarrow B(m^2)=\frac{2Vm}{(2\pi)^3}\;.$$ The mass mass spectrum aquires the asymptotic form $$\tilde{\tau}(m^2,\{\lambda\})\stackrel{m\rightarrow\infty} {\longrightarrow} C'(\{\lambda\})m^{-1-\alpha} \exp [m/T^*(\{\lambda\})]\;\;\;.$$ The above relations determine the version ($\alpha=4$) of the bootstrap model we have found to be physically relevant. It should be pointed out that the bulk of the work surrounding the bootstrap model, prior to the introduction of strangeness, was based on the choice $\alpha=2$ \[13\]. The initial form of the BE reads $$\tilde{B}(p^2)\tilde{\tau}(p^2,b,q,s,|s|)= \underbrace{g_{bqs|s|}\tilde{B}(p^2)\delta_0(p^2-m^2_{bqs|s|})}_ {input\;term}+\sum_{n=2}^{\infty} \frac{1}{n!} \int \delta^4 (p-\sum_{i=1}^n p_i)\cdot$$ $$\cdot \sum_{b_i} \delta_K (b-\sum_{i=1}^n b_i) \sum_{b_i} \delta_K (q-\sum_{i=1}^n q_i) \sum_{s_i} \delta_K (s-\sum_{i=1}^n s_i) \sum_{|s|_i} \delta_K (b-\sum_{i=1}^n |s|_i)$$ $$\prod_{i=1}^n \tilde{B}(p^2_i)\tilde{\tau}(p^2_i,b_i,q_i,s_i,|s|_i)d^4p_i\;.$$ The new feature, with respect to our previous extensions of the bootstrap model, is the introduction of electric charge $Q$ as an additional variable[^3]. Performing five Laplace transforms (one continuous and four discrete) leads to the following replacement of variables $$(p^2,b,s,|s|,q)\rightarrow(T,\lambda_B,\lambda_S,\lambda_{|S|},\lambda_Q)\;,$$ where the $\lambda$’s represent fugacity variables corresponding to number densities and $T$ is the temperature, as recorded in the center of mass frame. As the final states are composed of hadrons, rather than just baryons, we find it more convenient to pass from the original set of fugacities into one given in terms of valence quark fugacities. The transcription is made according to the relations $$\lambda_B=\lambda_u \lambda_d^2,\;\lambda_Q=\lambda_u \lambda_d^{-1},\; \lambda_{|S|}=\gamma_s,\;\lambda_S=\lambda_d \lambda_s^{-1}\;.$$ The important implication of the above relations is that they facilitate a thermodynamical description of the system in terms of (valence) quark fugacities, thereby enabling us to accomodate the presence of any kind of hadronic particle in the final system. Specifically, the form of the functions $\varphi$ and $G$ entering the bootstrap scheme is given by $$\varphi(T,\lambda_u,\lambda_d,\lambda_s,\gamma_s;H_0)= 2\pi H_0 T \sum_{\rm a} \lambda_{\rm a}(\lambda_u,\lambda_d,\lambda_s,\gamma_s) \sum_i g_{{\rm a}i}m_{{\rm a}i}^3 K_1 \left( \frac{m_{{\rm a}i}}{T}\right)$$ and $$G(T,\lambda_u,\lambda_d,\lambda_s,\gamma_s;H_0)= 2\pi H_0 T \int_0^{\infty} m^3 \tau_0(m^2,\lambda_u,\lambda_d,\lambda_s, \gamma_s) K_1 (m/T) dm^2\;,$$ where $K_1$ denotes the modified Bessel function of the second kind and where the general form of the fugacities $\lambda_{\rm a}$, pertaining to the [*totality*]{} of hadronic families, is $$\lambda_{\rm a}(\{\lambda\})=\lambda_u^{n_u-n_{\bar{u}}} \lambda_d^{n_d-n_{\bar{d}}} \lambda_s^{n_s-n_{\bar{s}}} \gamma_s^{n_s+n_{\bar{s}}}\;,$$ where $n_i$ is the number of the $i$ quarks contained in the hadron of the ${\rm a}$ family. For the particular case of the fugacities of light unflavored mesons, one can employ the parametrizacion $c_1(u\bar{u}+d\bar{d})+c_2s\bar{s}$, with $c_1+c_2=1$, see Ref \[8\], whereupon the corresponding variables assume the form $$\lambda_{\rm a}(\{\lambda\})=c_1+c_2\gamma_s^2\;.$$ The bootstrap equation (1), written analytically for the case in hand, reads $$\varphi(T,\lambda_u,\lambda_d,\lambda_s,\gamma_s)= 2G(T,\lambda_u,\lambda_d,\lambda_s,\gamma_s)- \exp[G(T,\lambda_u,\lambda_d,\lambda_s,\gamma_s)]+1\;,$$ while the critical surface is determined by (either one of) the relations $$\varphi(T_{cr},\mu_{u\;cr},\mu_{d\;cr},\mu_{s\;cr},\gamma_{s\;cr};H_0)= \ln4-1$$ and $$G(T_{cr},\mu_{u\;cr},\mu_{d\;cr},\mu_{s\;cr},\gamma_{s\;cr};H_0)=\ln2$$ Clearly, the critical surface corresponds to a 4-dimensional surface immersed in the space of the 5 thermodynamical variables. The constant parameter $H_0$, related directly to the MIT-bag constant (see remark following eq (3) and Refs \[6,7\]), can also be linked to the critical temperature at vanishing chemical potentials by $$\varphi(T_0,\mu_u =0,\mu_d =0,\mu_s =0,\gamma_s;H_0)=\ln4-1\;.$$ Through this relation $H_0$ can be directly related to $T_0$, for a fixed value of the “fugacity” $\gamma_s$. At the same time we demonstrate, in Appendix A, that $T_0$ corresponds to the maximum value for the temperature on the critical surface, irrespective of the value $\gamma_s$. In our previous work this feature was simply assumed. In order to acquire a concrete sense concerning the profile of the critical surface we have conducted a number of numerical studies which are displayed in Figs. 1-3. In these figures we present various sections of the critical surface, having chosen $H_0$ such that $T_0=183$ MeV for $\gamma_s=1$. Fig. 1 depicts projections of the critical surface on the $(\mu_u,T)$-plane for three representative values of $\mu_d$ and three for $\mu_s$. One observes that the critical surface “shrinks” (equivalently, “narrows”) as $\mu_s$ reaches higher positive values, starting from zero. This “shrinkage” is more pronounced in the vicinity of vanishing $\mu_u$. Fig. 2 displays critical surface projections on the $(\mu_u,\mu_d)$-plane for three different values of $T$ and $\mu_s$. One notices that the projections are (approximately) symmetric with respect to the line $\mu_u=\mu_d$. A second point is that the lowering of $\mu_s$ causes an expansion of the region occupied by the hadronic phase in the $(\mu_u,\mu_d)$ plane. Finally, Fig. 3 shows projections on the $(T,\mu_s)$ plane of the critical surface for fixed values of $\mu_u$ and of $\mu_d$. We notice that for fixed $\mu_d$ and $\mu_s$ the critical temperature falls with increasing (absolute) values of $\mu_u$. The same holds true under the exchange $\mu_u\leftrightarrow\mu_d$. Given the constitution of the initial colliding states, we must impose the constraints $<S>=0$ and $<B>-\beta2<Q>=0$ on the system as a whole. To this end we must refer to the partition function for our chosen version of the bootstrap scheme, as given by eq. (3). The constraints have the generic form $$H_k(T,\{\lambda\})\equiv\int_0^T \frac{1}{y^5} \frac{F_k(y,\{\lambda\})}{2-\exp[G(y,\{\lambda\})]}dy=0\;,\;k=1,2\;,$$ with $$F_1(y,\{\lambda\})=\lambda_s \frac{\partial \varphi(y,\{\lambda\})}{\partial \lambda_s}$$ for the imposition of $<S>=0$ and $$F_2(y,\{\lambda\})= \frac{1-4\beta}{3}\lambda_u \frac{\partial \varphi(y,\{\lambda\})}{\partial \lambda_u}+ \frac{1+2\beta}{3}\lambda_d \frac{\partial \varphi(y,\{\lambda\})}{\partial \lambda_d}\;,$$ for securing the constraint $<B>-\beta2<Q>=0$. These conditions constitute a system of two equations whose solution yields a 3-dimensional hypersurface in the space of thermodynamical variables on which the given system is constrained to exist. Let us denote this surface by $H_{ph}$, where “$ph$” stands for physical. Clearly, the intersection between $H_{ph}$ and the critical surface defines the limits of the hadronic world for the system with the given constraints. This intersection comprises a two-dimensional surface whose numerical study is presented in Figs. 6-8. Figures 4 and 5 give corresponding perspectives of the profile of $H_{ph}$ whose basic aim is to display its variation with $\beta$. We have considered the cases $\beta=1$ $(N_p^{in}=N_n^{in})$, $\beta=2$ $(N_p^{in}<N_n^{in})$ and $\beta=1/2$ $(N_p^{in}>N_n^{in})$[^4]. Fig. 4 depicts projections of $H_{ph}$ in the $(T,\mu_s)$-plane for fixed values of $\lambda_u$ and $\gamma_s$, while Fig. 5 shows corresponding projections in the $(\mu_u,\mu_d)$ plane. From the first figure we record the tendency of $\mu_s$ to increase with $\beta$, for fixed values of $(T,\lambda_u,\gamma_s)$. From the second we witness the (expected) behavior $\mu_u=\mu_d$ for $\beta=1$, $\mu_u<\mu_d$ for $\beta>1$ and $\mu_d<\mu_u$ for $\beta<1$. Finally, in Figs. 6-8 we present results of numerical studies pertaining to the intersection between $H_{ph}$ and the critical surface. In Fig. 6 the 2-dimensional intersection is projected on the $(\mu_u,T)$-plane, for our three representative values of $\beta$. As one might expect, an increase of $\beta$ induces a decrease of $\mu_{u\;cr}$ for constant temperature. Fig. 7 shows the corresponding projections on the $(\mu_u,\mu_d)$-plane exhibiting similar connections between $\beta$-values and the relation among $\mu_{u\;cr}$ and $\mu_{d\;cr}$. In Fig. 8 we consider projections in the $(\mu_u,\mu_s)$-plane. Here we surmise that for fixed value of $\mu_{u\;cr}$ an upward move of $\beta$ with respect to 1 $(N_p^{in}<N_n^{in})$ induces an increase in the (critical) chemical potential of the strange quark. This concludes our discussion of the isospin non-symmetric SSBM. We shall proceed, in the next section, to confront experimental data encoded in the multiparticle system produced in $A+A$ collisions, in which we do not have isospin symmetry. In this section we shall perform a data analysis referring to particle multiplicities recorded in the NA35 $S+Ag$ experiment at 200 GeV/nucleon at CERN. The method we shall use is similar to the one presented in \[8\]. The main differences are that our space is described by the set of the six thermodynamical variables $(VT^3/4\pi^3,T,\{\lambda\})$, i.e. one more variable is present and that the system is subject to two constaints, namely $<S>=0$ and $<B>=2\beta<Q>$, instead of one. The latter will be enforced via the introduction of corresponding Lagrange multipliers. The theoretical values of the thermodynamical parameters are adjusted via a $\chi^2$-fit by minimizing the function $$\chi^2(VT^3/4\pi^3,T,\{\lambda\},\{l\})= \sum_{i=1}^N\left[\frac{N_i^{exp}-N_i^{theory} (VT^3/4\pi^3,T,\{\lambda\})} {\sigma_i}\right]^2$$ $$+\sum_{k=1}^2 l_k H_k(T,\{\lambda\})\;\;.$$ where $l_k$ are Lagrange multipliers accompanying the corresponding constraints as given by (23) and the $N_i^{theory}$ are given by $$N_i^{theory}=\left.\left(\lambda_i\frac {\partial \ln Z(VT^3/4\pi^3,T,\{\lambda\},\ldots,\lambda_i,\ldots)} {\partial \lambda_i}\right)\right|_{\ldots=\lambda_i=\ldots=1} \;\;.$$ The minimization of $\chi^2$ ammounts to solving the following system of eight equations $$\frac{\partial \chi^2(x_1,\ldots,x_8)}{\partial x_i}=0 \;\;(i=1,\ldots,8)\;,$$ with $\{x_i\}=(VT^3/4\pi^3,T,\{\lambda\},\{l\})$. An outline of the procedure involved in realizing a numerical solution of the minimization problem has been given in Ref. 10. There, we have also discussed the methodology by which we determine correction factors for Bose/Fermi statistics. We shall not repeat the general argumentation here, nevertheless we do present in Appendix B a discussion of some technical aspects involved in the relevant procedure for the case in hand. Turning our attention to the $S+Ag$ collision at an energy of 200 GeV/nucleon, using the methodology that has just been described, we set as our first task to specify the value of the $\beta$-parameter appropriate for the process under study. As far as the $^{32}S$ nucleus is concerned, the single isotope with nucleon number 32 ($Z=16$) is employed, whereas for silver there are two stable isotopes with nucleon numbers 107 and 109 ($Z=47$) entering, respectively, a mixture composed of $51.84\%$ and $48.16\%$ fractions. This accounts for an average nucleon number of 107.96. It turns out that it makes little difference whether one assumes that all the nucleons entering the $S+Ag$ system participate in the collision process, or that the “active” part of the $Ag$ nucleus is determined by some “realistically” assumed geometrical configuration. For our numerical applications we shall fix the value of $\beta$ at 1.10. The emerging results are displayed in a series of Tables and Figures. In table 1 we present adjusted sets of values for the thermodynamical parameters with a corresponding estimation for $\chi^2 /dof$[^5]. The presented numbers correspond to evaluations where all particle multiplicities are taken into account (1st row) and where, in turn, one of the particle species is excepted. One notices a decive improvement when $h^-$ (mostly pions) are excluded from the fit. This occurence makes meaningful the separate treatment of the full multiplicity analysis from the one(s) where pions are excluded. The experimental data pertaining to particle multiplicities have been taken from \[16-20,14,5\] and are entered in the first column of Table 2. The second column gives the theoretical estimates of populations based on the corressponding adjusted set of thermodynamical parameters with all particle species included. The third column pertains to the adjusted set with the absence of pions. The last column corresponds to the same situation but with the critical surface pushed slightly outwards by setting $T_0=183.5$ MeV at $\gamma_s=1$. Table 3 exhibits the correction factors due to Bose/Fermi statistics for each particle species. We have covered each of the three cases entering the previous table: All particle species, exclusion of pions with $T_0=183$ MeV at $\gamma_s=1$ and $T_0=183.5$ MeV at $\gamma_s=1$, respectively. Table 4 summarizes the adjustment of the thermodynamical parameters according to the $\chi^2$ fit (along with the estimate for $\chi^2/dof$) for each of the three aformentioned cases. Finally, Table 5 presents particle ratios (used only for the case where all the multiplicities are included), taken with respect to negative hadron population which has the smallest experimental uncertainty (see Ref \[7\] for a relevant comment). Pictorial representation of results with physical significance is given in Figs. 9-12. In the first of these figures we display bands, per particle ratio, in the ($\mu_u,T$)-plane with $\gamma_s$ fixed at 0.67 (see Table 4). These bands are determined by the experimental uncertainty per ratio. The bold solid line marks the boundary of the hadronic world, beyond which the SSBM does not present analytic solutions. All particle ratios are used (as per Table 5), i.e. pions have not been excluded in the plot. No overlap region of the various bands is observed, nevertheless we [*have*]{} marked with a cross the center of a region of “optimum overlap” which lies inside the hadronic world. Fig. 10 considers corresponding bands of particle populations. Since the variable $VT^3/4\pi^3$ also enters our considerations we fix it according to its adjusted value of 1.23 (see second column of Table 4). It should be pointed out that the upper limit for the experimental $K_s^0$ population (17) as well as the whole band of the negative hadron population (175-197) correspond to fitted values for the thermodynamical parameters that are outside the hadronic domain (bold solid line). The dotted line marks the smallest value of $\chi^2$ on the boundary surface. We have determined that $51.6\%$ of particle multiplicities are compatible with a source [*outside*]{} the hadronic domain. A zoom around the vicinity of the minimal $\chi^2$-value on the critical surface is presented in Fig. 11. A comparison between measured and theoretically determined, according to our $\chi^2$ fit, multiplicities is summarized in Fig. 12. Experimental points, with error bars, are represented by heavy dots. Theoretically determined points are marked according to the three cases studied throughout, i.e. all particle species inclusion and exclusion of pions with $T_0$ set, respectively, at 183 MeV and 183.5 MeV for $\gamma_s=1$. Once again we notice a dramatic improvement of the fits when pions ($h^-$) are excluded. As we have argued in Ref. 10, this occurence seems to signify an excess of pion production, incompatible with a pure hadronic phase (SSBM/experimental= $0.69\pm0.04$). In this work we have applied the SSBM to analyze experimental data from the $S+Ag$ collision at 200 GeV per nucleon pertaining to produced particle multiplicities ($4\pi$ projection) recorded by the NA35 collaboration at CERN. The quintessential aspect of the model is that it accomodates interactions in a self-consistent way within the framework of a thermal description of the (relativistic) multiparticle system. Moreover, it designates a precicely defined boundary for its applicability, a feature which plays a central role in the assesment of our results. It should finally be reminded that for the construction of this bounbary we have chosen the largest possible, physically meaningful values of $T_0(B)$ so as to enlarge the hadron gas domain and avoid over-optimistic interpretations, regarding the location of the source with respect to this boundary. Our primary objective has been to locate the [*source*]{} of the multipaticle system in the space of the relevant thermodynamical set of parameters. In this connection, we have found that the data point towards a thermal source that lies just outside the hadronic phase. The situation is not as pronounced as in the previously analysed case of $S+S$ collisions \[8\] at 200 GeV per nucleon, where a much larger weight in favor of the source being outside the hadronic boundaries was determined. The overall situation resulting from our data analyses ($p+\bar{p}$, $S+S$, $S+Ag$) is depicted in Fig. 13. One notices the proximity of the source location for the two nucleus-nucleus collision proccess just beyond the hadron phase as well as the (expected) placement of the source for the $p+\bar{p}$ collision well inside the hadronic domain. The experimental data give a substantial excess of pion (entropy) production compared to theoretical predictions. This strongly hints that the source of the emerging multiparticle system from the $S+Ag$ collision is in the doorway of the QGP phase. As already pointed out in the Introduction, an estimate of the entropy associated with the pionic component of the produced system for the $S+S$ collision gives \[8\] a theoretical to experimental ratio which is not compatible with hadronic physics and necessitates a location of the source outside the hadronic domain. A similar behavior persists in the present case as well. A final result of interest was the observance of a tendency towards strangeness saturation. This indicates that the source has acheived almost full thermal and chemical equilibrium, as expected and required for a phase transition to QGP. We thus conclude that in the $S+S$ and $S+Ag$ interactions at 200 GeV/nucleon we have witnessed for the first time the appearance of definite signals linking these interactions with the QGP phase. The fully extended SSBM is now in position to confront multiparticle data emerging from any $A+A$ collision experiment, including strange particles. In this respect, the methodology can be applied to other ongoing experiments, e.g. $Pb+Pb$, as data becomes available and, more importantly, on the future ones from RHIC and LHC. On the theoretical side, it would be extremely interesting to connect a scheme such as the SSBM coming from the hadronic side to corresponding microscopic-oriented accounts of QGP physics \[21\]. [**Appendix A**]{} We shall show that the critical temperature value $T_0$, as defined in the text, corresponds to its maximum value on the critical surface. We start by re-expressing eq. (20) in the form $$T=f(\{\lambda\},\gamma_s)\;,$$ where we ignore critical value indications on each variable for notational simplicity. A maximum for $T$ corresponds to an extremum $$\left.\frac{\partial f (\{\lambda\},\gamma_s)}{\partial \lambda_i} \right|_{\textstyle \varphi} =0\;,\;\;\;i=1,\ldots,4\;.$$ As long as one remains on the critical surface the above condition can be easily transcribed to $$\left.\frac{\partial f (\{\lambda\},\gamma_s)}{\partial \lambda_i} \right|_{\textstyle \varphi}= -\frac{\partial\varphi/\partial\lambda_i}{\partial\varphi/\partial T}= -\frac{\lambda_i\partial\varphi/\partial\lambda_i} {\lambda_i\partial\varphi/\partial T}\;,\;\;\;i=1,\ldots,4$$ and since $\lambda_i\partial\varphi/\partial T \neq 0$ it must be that $$\lambda_i\frac{\partial\varphi}{\partial\lambda_i}=0\;,\;\;\;i=1,\ldots,4\;.$$ Now, for the hadronic fugacities we may write $$\lambda_{\rm a}=c_1+\lambda_b\{\lambda\}\gamma_s^{N_{s{\rm a}}}\;,$$ where $\lambda_b$, $c_1$ and $N_{s{\rm a}}$ can be read from (17) and (18). (For examble, in the case of the $\Lambda$ Baryons we have $c_1=0$, $\lambda_b=\lambda_u\lambda_d\lambda_s$ and $N_{s{\rm a}}=1$.) Therefore $$\lambda_i\frac{\partial\lambda_{\rm a}}{\partial\lambda_i}= N_{i{\rm a}}\lambda_b\{\lambda\}\gamma_s^{N_{s{\rm a}}}\;,\;i=1,2,3\;$$ and $$\lambda_i\frac{\partial\lambda_{\rm a}}{\partial\lambda_i}= N_{s{\rm a}}\lambda_b\{\lambda\}\gamma_s^{N_{s{\rm a}}}\;,\;i=4\;.$$ Given the above set of equations there will be a corresponding family of antiparticles for which $$\lambda_i\frac{\partial\lambda_{\rm a}}{\partial\lambda_i}= -N_{i{\rm a}}\lambda_b\{\lambda\}^{-1}\gamma_s^{N_{s{\rm a}}}\;,\;i=1,2,3\;$$ and $$\lambda_i\frac{\partial\lambda_{\rm a}}{\partial\lambda_i}= N_{s{\rm a}}\lambda_b\{\lambda\}^{-1}\gamma_s^{N_{s{\rm a}}}\;,\;i=4\;.$$ will hold true. The last four equations applied to (32) give $$\sum_{\rm a}(\lambda_b\{\lambda\}-\lambda_b\{\lambda\}^{-1}) N_{i{\rm a}}\gamma_s^{N_{s{\rm a}}}F_{\rm a}(V,T)=0\;,\;i=1,2,3\;,$$ $$\sum_{\rm a}(\lambda_b\{\lambda\}+\lambda_b\{\lambda\}^{-1}) N_{s{\rm a}}\gamma_s^{N_{s{\rm a}}}F_{\rm a}(V,T)=0\;,\;i=4\;,$$ where the index “${\rm a}$” runs solely over particles. Next we see that the equation (39) is not possible to hold true because the left part is always positive (the numbers $N_{s{\rm a}}$ are physical). Therefore an extremum of the temperature with respect to $\gamma_s$ does not exist. Turning to (38) we observe that a solution could be found if for all $\{\lambda\}$ we had $$\lambda_b\{\lambda\}-\lambda_b\{\lambda\}^{-1}=0,\;\forall b \Leftrightarrow \lambda_b\{\lambda\}=1,\;\forall b\;.$$ An obvious solution for (40) is $$\lambda_u=\lambda_d=\lambda_s=1 \Leftrightarrow \mu_u=\mu_d=\mu_s=1\;.$$ The last equation defines an extremum for the critical temperature with constant $\gamma_s$. On the other hand we have $$\frac{\partial}{\partial \lambda_j}\left(\lambda_i \frac{\partial \varphi}{\partial \lambda_i}\right)=\frac{1}{\lambda_j} \sum_{\rm a}(\lambda_b\{\lambda\}+\lambda_b\{\lambda\}^{-1}) N_{i{\rm a}}N_{j{\rm a}}\gamma_s^{N_{s{\rm a}}}F_{\rm a}(V,T)>0\;,\;i=1,2,3\;.$$ That is, for every value of $i=1,2,3$ each one of the above equations, once two values among the $\{\lambda\}$ are fixed, will have a unique solution. That happens, because from (42), one can infer that $\lambda_i\frac{\partial \varphi}{\partial \lambda_i}$ is a genuine rising function with respect to $\lambda_j$ and so it has a unique solution. By extension the simultaneous solution of the three equations will be unique. So the extremum we have calculated is unique. This extremum cannot correspond to minimum, since the critical surface has zero critical temperature for non zero chemical potentials (e.g. see Figs. 1,3). Since always $T\geq 0$, if the point which corresponds to (41) was a local minimum, then we should have another extremum somewhere else, which is imposible, since the extremum is unique. Therefore (41) corresponds to a [*total*]{} maximum for a given value of $\gamma_s$. In our study we have to calculate $r$ constraints ($r=1$ for isospin symmetry and $r=2$ for isospin non-symmetry) and different particle multiplicities as functions of the thermodynamical variables $(T,\{\lambda\})$. In general all these quatities can be written as $$R_j(T,\{\lambda\})\equiv \int_0^T \frac{1}{y^5} \frac{Q_j(y,\{\lambda\})}{2-\exp [G(y,\{\lambda\})]} dy\;,$$ where $$R_j\equiv H_j\;,\;j\leq r\;\;\;R_j\equiv N_{j-r}^{theory}\;,\;j > r\;,$$ where $H$ and $N^{theory}$ are given from (23) and (27), respectively, and $$Q_j\equiv F_j\;,\;j\leq r\;\;\;,\;\;\; Q_j\equiv \frac{VT^3}{4\pi^3 H_0}\left.\left[ \frac{\partial\varphi(y,\{\lambda\},\cdots,\lambda_{j-r},\cdots)} {\partial\lambda_{j-r}}\right]\right|_{\cdots=\lambda_{j-r}=\cdots=1} \;,\;j > r \;,$$ with $F_j$ given from (24) and (25). In order to evaluate the optimized set of variables $(T,\{\lambda\})$ in our large working space we have to turn to the use of the generalized Newton-Raphson method which converges quickly but requires the knowledge of the derivatives of (43). If we try to calculate these derivatives with respect to a fugacity $\lambda_i$ from (43) we find $$\left.\frac{\partial R_j(T,\{\lambda\})}{\partial \lambda_i}\right|_T= \int_0^T\frac{dy}{y^5}\left\{ \frac{\exp [G(y,\{\lambda\})]} {\{2-\exp [G(y,\{\lambda\}]\}^3} \frac{\partial \varphi (y,\{\lambda\})}{\partial \lambda_i} Q_j(y,\{\lambda\})\right.+$$ $$\hspace{6cm}\left.\frac{1}{2-\exp [G(y,\{\lambda\})]} \frac{\partial Q_j(y,\{\lambda\})}{\partial \lambda_i} \right\}.$$ With the use of the above equation the Newton-Raphson method can proceed for all points of the hadronic space which are not close to the critical surface. Problems, however, are encountered when the quantities (46) have to be evaluated [*near*]{} and even more [*on*]{} the critiacal surface. When $T\rightarrow T_{cr}$ the function to be integrated in (46) contains a non-integrable singularity of the form $(2-\exp[G])^{-3}$. This singularity cannot be integrated even if we use the variable $z=2-\exp [G(y,\{\lambda\})]$, as we did in Refs \[7,8\]. On the other hand the quantities $R_j$ can be expressed as functions of a new set of variables. This new set can be formed if we replace the temperature T in favour of the function $\varphi$, or equivalently the $z$ variable. Then the $R_j$ can be given from $$R_j(z,\{\lambda\})=\int_1^{\tilde{z}}\frac{d\tilde{z}}{\tilde{z}-2}\cdot\left[ \frac{\textstyle Q_j (y,\{\lambda\})} y^5 \cdot\frac{\textstyle \partial \varphi (y,\{\lambda\})} {\textstyle \partial y} \right]_ {\textstyle \tilde{z}=2-\exp[G(y,\{\lambda\})]}\;\;.$$ To be able to proceed with the Newton-Raphson method, in this case, we calculate instead of (46), the derivatives of $R_j$ when $z$ is constant, i.e. derivatives of the form $\left. \frac{\partial R_j(z,\{\lambda\})}{\partial \lambda_i}\right|_ {\textstyle z}$. These derivatives should not present any singularity for any value of $z$ (even for $z=0$ when we are on the critical surface) because, as can be seen from (47), the $R_j$ can be evaluated for any values of $z$ and $\{\lambda\}$. In order to proceed with the evaluation of the derivatives of (47) we can assume that we are standing on a surface of constant $\varphi$ or equivalently constant $z$. We then let the variation to the fugacity $\lambda_i$ be $d\lambda_i$ without ever leaving the above mentioned surface. Then the variation in $z$ is $$dz=\frac{dz}{d\lambda_i}d\lambda_i=0$$ Since using the BE we have $$\frac{dG}{d\varphi}=\frac{1}{2-e^G}\;,$$ we arrive at $$\frac{dz}{d\lambda_i}=\frac{dz}{dG}\frac{dG}{d\varphi} \left.\frac{\partial\varphi}{\partial\lambda_i}\right|_z= \frac{-e^G}{2-e^G} \left.\frac{\partial\varphi}{\partial\lambda_i}\right|_z\;.$$ From the last two equations we conclude that $$\left.\frac{\partial\varphi(z,\{\lambda\})}{\partial\lambda_i}\right|_z =0\;.$$ Let us comment that on the critical surface $2-e^G=0$, so again equation (51) is to hold if equations (48) and (50) are to be fulfilled. If we then express the $z$ variable as function of the temperature $y$ and the fugacities, $z=z(y,\{\lambda\})$, then equation (51) leads to $$\frac{\partial\varphi(y,\{\lambda\})}{\partial y} \cdot \left.\frac{\partial y}{\partial \lambda_i}\right|_z + \frac{\partial\varphi(y,\{\lambda\})}{\partial\lambda_i}=0 \Rightarrow \left.\frac{\partial y}{\partial \lambda_i}\right|_z = -\frac{\partial\varphi / \partial\lambda_i}{\partial\varphi / \partial y}$$ The last relation shows us how temperature is varied with the fugacity $\lambda_i$ on a surface of constant $z$. Using the definition $$V_j(y,\{\lambda\})\equiv \frac{\textstyle Q_j (y,\{\lambda\})} {y^5 \cdot\frac{\textstyle \partial \varphi (y,\{\lambda\})} {\textstyle \partial y}}\;,$$ the derivatives we seek can be expressed as $$\left. \frac{\partial R_j(z,\{\lambda\})}{\partial \lambda_i}\right|_ {\textstyle z}=\int_1^z\frac{d \tilde{z}}{\tilde{z}-2}\cdot\left[ \frac{\textstyle V_j [y(\tilde{z},\{\lambda\}),\{\lambda\}]} {\textstyle \partial \lambda_i}\right]\;.$$ But $$\frac{\textstyle V_j [y(\tilde{z},\{\lambda\}),\{\lambda\}]} {\textstyle \partial \lambda_i}= \frac{\textstyle V_j [y,\{\lambda\}]}{\textstyle y} \left.\frac{\textstyle y}{\textstyle \lambda_i}\right|_{\textstyle z}+ \frac{\textstyle V_j [y,\{\lambda\}]}{\textstyle \partial \lambda_i}= \hspace{5cm}$$ $$=\left[\left(y^5\frac{\textstyle \partial \varphi}{\textstyle \partial y} \right)^{-1}\frac{\textstyle \partial Q_j}{\textstyle \partial y}- \frac{\textstyle 5}{\textstyle y^6} \left(\frac{\textstyle \partial \varphi}{\textstyle \partial y}\right)^{-1} Q_j- \frac{\textstyle Q_j}{\textstyle y^5} \left(\frac{\textstyle \partial \varphi}{\textstyle \partial y}\right)^{-2} \frac{\textstyle \partial^2 \varphi}{\textstyle \partial y^2}\right]\cdot \left(-\frac{\textstyle \partial \varphi/\partial\lambda_i} {\textstyle \partial \varphi/y}\right)+ \hspace{1cm}$$ $$\hspace{5cm} +\left(y^5\frac{\textstyle \partial \varphi}{\textstyle \partial y} \right)^{-1}\frac{\textstyle \partial Q_j}{\textstyle \partial \lambda_i}- \frac{\textstyle Q_j}{\textstyle y^5} \left(\frac{\textstyle \partial \varphi}{\textstyle \partial y}\right)^{-2} \frac{\textstyle \partial^2 \varphi}{\textstyle \partial y\partial \lambda_i}=$$ $$=y^{-5}\left(\frac {\partial \varphi}{\partial y}\right)^{-2}\left[ \frac{\partial \varphi}{\partial y} \frac{\partial Q_j}{\partial \lambda_i}- \frac{\partial \varphi}{\partial \lambda_i} \frac{\partial Q_j}{\partial y}+ Q_j\left(-\frac{\partial^2 \varphi}{\partial y \partial \lambda_i}+ \frac{\partial^2 \varphi}{\partial y^2} \frac{\partial\varphi / \partial\lambda_i}{\partial\varphi / \partial y} +\frac{5}{y}\frac{\partial \varphi}{\partial \lambda_i}\right)\right].$$ From the last two equations we conclude that $$\left.\frac{\textstyle \partial R_j(z,\{\lambda\})} {\textstyle \partial \lambda_i}\right|_{\textstyle z}= \int_1^z\frac{d\tilde{z}}{tilde{z}-2}\left\{y^{-5} \left(\frac {\partial \varphi}{\partial y}\right)^{-2}\left[ \frac{\partial \varphi}{\partial y} \frac{\partial Q_j}{\partial \lambda_i}- \frac{\partial \varphi}{\partial \lambda_i} \frac{\partial Q_j}{\partial y}+ \right.\right.$$ $$\left.\left. Q_j\left(-\frac{\partial^2 \varphi}{\partial y \partial \lambda_i}+ \frac{\partial^2 \varphi}{\partial y^2} \frac{\partial\varphi / \partial\lambda_i}{\partial\varphi / \partial y} +\frac{5}{y}\frac{\partial \varphi}{\partial \lambda_i}\right)\right] \right\}_{\textstyle \tilde{z}=2-\exp [G(y,\{\lambda\})]}.$$ As it is known the function to be $z$-intergrated has not so good behaviour near $z=1$. So it is better to break the above integral in two parts: $$\left.\frac{\textstyle \partial R_j(z,\{\lambda\})} {\textstyle \partial \lambda_i}\right|_{\textstyle z}= \hspace{11cm}$$ $$\int_0^{T_1}\frac{dy}{2-e^G}y^{-5} \left(\frac {\partial \varphi}{\partial y}\right)^{-1}\left[ \frac{\partial \varphi}{\partial y} \frac{\partial Q_j}{\partial \lambda_i}- \frac{\partial \varphi}{\partial \lambda_i} \frac{\partial Q_j}{\partial y}+ Q_j\left(\frac{\partial^2 \varphi}{\partial y \partial \lambda_i}- \frac{\partial^2 \varphi}{\partial y^2} \frac{\partial\varphi / \partial\lambda_i}{\partial\varphi / \partial y} +\frac{5}{y}\frac{\partial \varphi}{\partial \lambda_i}\right)\right]+$$ $$\int_{z_1}^z\frac{d\tilde{z}}{\tilde{z}-2}\left\{y^{-5} \left(\frac {\partial \varphi}{\partial y}\right)^{-2}\left[ \frac{\partial \varphi}{\partial y} \frac{\partial Q_j}{\partial \lambda_i}- \frac{\partial \varphi}{\partial \lambda_i} \frac{\partial Q_j}{\partial y}+ \right.\right.\hspace{7cm}$$ $$\hspace{4cm}\left.\left. Q_j\left(\frac{\partial^2 \varphi}{\partial y \partial \lambda_i}- \frac{\partial^2 \varphi}{\partial y^2} \frac{\partial\varphi / \partial\lambda_i}{\partial\varphi / \partial y} +\frac{5}{y}\frac{\partial \varphi}{\partial \lambda_i}\right)\right] \right\}_{\textstyle \tilde{z}=2-\exp [G(y,\{\lambda\})]}.$$ In the above relation $z_1=2-\exp [G(T_1,\{\lambda\})$ and a good choice is $z_1=0.5$. If $z>0.5$ we are not close to the critical surface and the second integral does not have to be calculated. So in general we can set $z_1=max\{z,0.5\}$. The derivatives of $R_j$ with respect to $z$ can be calculated easily. They simply read $$\left. \frac{\partial R_j(z,\{\lambda\})}{\partial z}\right|_ {\textstyle \lambda_i}= \frac{1}{z-2}\cdot y^{-5} \cdot \left\{ \frac{\textstyle \partial \varphi [y(z,\{\lambda\}),\{\lambda\}]} {\textstyle \partial y}\right\}^{-1} {\textstyle Q_j [y(z,\{\lambda\}),\{\lambda\}]}\;.$$ With the above relations the minimization of the $\chi^2$ function can proceed with the use of the Newton-Raphson method. Relation (58) can also be used to find out whether the absolute minimum of $\chi^2$ is outside or inside the critical surface. Suppose we locate the minimum value of $\chi^2=(\chi^2)_1$ [*on*]{} the critical surface and this value corresponds to the point $(z,\{\lambda\})=(0,\{\lambda^0\})$. Then the absolute minimum of $\chi^2$ is located [*inside*]{} the hadronic phase if for this point we have $$\left. \frac{\partial \chi^2 (z,\{\lambda^0\})}{\partial z}\right|_ {\textstyle z=0}<0\;.$$ If the above relation is not fulfilled the absolute minimum of $\chi^2$ lies on the [*outside*]{}. An alternative method to verify the same thing consists of locating the minimum value of $\chi^2$ on a surface near the critical one inside the hadronic phase. Let this value be $(\chi^2)_2$. The absolute minimum of $\chi^2$ is located [*inside*]{} the hadronic phase if $(\chi^2)_2<(\chi^2)_1$ and outside otherwise. For the two fits we have performed for $S+Ag$ we had to process 192 points. All these points have given us the same results with the use of the two methods. [99]{} J. Cleymans and H. Satz, Z. Phys. [**C57**]{}, 135 (1993) J. Sollfrank, M. Gaździcki, U. Heinz and J. Rafelski, Z. Phys. [**C61**]{}, 659 (1994) A. D. Panagiotou, G. Mavromanolakis and J. Tzoulis, Phys. Rev. [**C53**]{} 1353 (1996) F. Becattini and U. Heinz, Z. Phys. [**C76**]{}, 269 (1997) F. Becattini, M. Gaździcki and J. Sollfrank, Eur. Phys. J. [**C5**]{}, 143 (1998) A. S. Kapoyannis, C. N. Ktorides and A. D. Panagiotou, J. Phys. [**G23**]{}, 1921 (1997) A. S. Kapoyannis, C. N. Ktorides and A. D. Panagiotou, Phys. Rev. [**D58**]{}, 034009 (1998) A. S. Kapoyannis, C. N. Ktorides and A. D. Panagiotou, Phys. Rev. [**C58**]{}, 2879 (1998) R. Hagedorn, Nuovo Cimento Suppl., [**III**]{}, 147 (1965) R. Hagedorn, J. Ranft, Nuovo Cimento Suppl., [**VI**]{}, 169 (1968); R. Hagedorn, Nuovo Cimento Suppl., [**VI**]{}, 311 (1968) R. Hagedorn, Nuovo Cimento, [**LVI$\;$A**]{}, 1027 (1968) R. Hagedorn, 1985 “Springer Lecture Notes in Physics”, [ **221**]{} ed. K. Kajantie (Berlin, Heidelberg, New York) p.53; R. Hagedorn, 1995 “Hot Hadronic Matter” NATO-ASI-Series [**B346**]{} eds. J. Letessier et. al., p.13; R. Hagedorn, I. Montvay and J. Rafelski, [*“Hadronic Matter at Extreme Energy Density”*]{}, eds. N. Cabbibo and L. Sertorio, Plenum Press, New York, 49 (1980) R. Hagedorn and J. Rafelski, Phys. Lett. [**97B**]{}, 136 (1980) F. Becattini, J. Phys. [**G23**]{}, 1933 (1997) J. Sollfrank, J. Phys. [**G23**]{}, 1903 (1997) J. Baechler [*et al.*]{}, NA35 Coll., Eur. Phys. J. [**C2**]{}, 643 (1998) J. Baechler [*et al.*]{}, NA35 Coll., Phys. Rev. Lett. [**72**]{}, 1419 (1994) T. Alber [*et al.*]{}, NA35 Coll., Z. Phys. [**C64**]{}, 195 (1994) T. Alber [*et al.*]{}, NA35 Coll., Phys. Lett. [**B366**]{}, 56 (1996) D. Rohrich [*et al.*]{}, NA35 Coll., Nucl. Phys. [**A566**]{}, 35c (1994) D. Bailin and A. Love, Phys. Rep. [**107**]{}, 325 (1984); T. Schafer, Phys. Rev. [**D57**]{}, 3950 (1998); M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. [**B422**]{}, 247 (1998); T. Schafer and F. Wilczek, hep-ph/9810509; J. Berges and K. Rajagopal, Nucl. Phys. [**B538**]{}, 215 (1999) The fitted parameters and the $\chi^2/dof$ values for different fits through SSBM in the experimentally measured full phase space multiplicities in the $S+Ag$ interaction. In the first fit all the multiplicities are included while in the following fits we exclude each time one mutiplicity. $T_0$ is set to 190 MeV. Experimentally measured full phase space multiplicities in the $S+Ag$ interaction and their theoretically fitted values by the SSBM, with the inclusion of the $h^-$ multiplicity and without it (cases A: $T_0=182.94$ MeV and B: $T_0=183.5$ MeV). Calculation of the correction factor $f_i=\frac{(N_{IHG-BF})_i-(N_{IHG-BO})_i}{(N_{IHG-BO})_i}$ for the $i$th particle species measured in $4\pi$ phase space in the $S+Ag$ interaction. For the calculation of $f_i$ the IHG formalism has been used, while the thermodynamical variables have been extracted from the SSBM fit with $h^-$ and without $h^-$ (cases A and B). Results of the analysis by SSBM of the experimental data from the $S+Ag$ interaction ($4\pi$ phase space), with the inclusion of the $h^-$ multiplicity and without it (cases A and B). Particle ratios from the experimentally measured full phase space multiplicities for the $S+Ag$ interaction used in the analysis with $h^-$. $\;$ [**$\bf S+Ag$ (NA35) Full phase space**]{} Excluded $T$ (MeV) $\lambda_u$ $\lambda_d$ $\lambda_s$ $\gamma_s$ $VT^3/4\pi^3$ $\chi^2/dof$ ---------------------- ----------- ------------- ------------- ------------- ------------ --------------- -------------- none 170.943 1.540 1.582 1.088 0.662 2.883 10.35/3 ${K_s}^0$ 162.411 1.551 1.588 1.132 0.749 3.869 4.72/2 $\Lambda$ 170.363 1.523 1.563 1.093 0.616 3.063 9.41/2 $\overline{\Lambda}$ 158.361 1.625 1.665 1.174 0.636 4.239 7.58/2 $\overline{p}$ 175.180 1.482 1.520 1.067 0.596 2.807 4.55/2 $p-\overline{p}$ 170.626 1.538 1.579 1.090 0.665 2.904 10.33/2 $B-\overline{B}$ 172.040 1.554 1.597 1.083 0.656 2.786 8.52/2 $h^-$ 180.779 1.642 1.702 1.011 0.839 1.261 1.64/2 Table 1. [**$\bf S+Ag$ (NA35) Full phase space**]{} ---------------------- --------------- ------------------ --------------- --------------- Particles Experimental Calculated Calculated Calculated Data with $h^-$ without $h^-$ without $h^-$ (Case A) (Case B) ${K_s}^0$ $15.5\pm 1.5$ 17.613 15.181 15.155 $\Lambda$ $15.2\pm 1.2$ 14.490 15.424 15.429 $\overline{\Lambda}$ $2.6\pm 0.3$ 2.3998 2.5502 2.5538 $\overline{p}$ $2.0\pm 0.8$ 3.4614 2.3612 2.3547 $p-\overline{p}$ $43\pm 3$ 42.600 40.931 40.937 $B-\overline{B}$ $90\pm 10$ 101.39 99.307 99.325 $h^-$ $186\pm 11$ 170.68$^{\rm a}$ 128.85$^b$ 128.57$^c$ ---------------------- --------------- ------------------ --------------- --------------- $^{\rm a}$ A correction factor 1.0236 has been included for the effect of Bose statistics. $^b$ A correction factor 1.0190 has been included for the effect of Bose statistics. This multiplicity is not included in the fit. $^c$ A correction factor 1.0188 has been included for the effect of Bose statistics. This multiplicity is not included in the fit. Table 2. [**$\bf S+Ag$ (NA35) Full phase space**]{} ---------------------- --------------------- ----------------------- ----------------------- Particles ($i$) $f_i$($\%$) for the $f_i$($\%$) for the $f_i$($\%$) for the fit with $h^-$ fit without $h^-$ (A) fit without $h^-$ (B) ${K_s}^0$ $0.462$ $0.641$ $0.645$ $\Lambda$ $0.053$ $0.232$ $0.237$ $\overline{\Lambda}$ $-0.142$ $-0.410$ $-0.418$ $\overline{p}$ $0.146$ $0.105$ $0.105$ $p-\overline{p}$ $-0.388$ $-0.431$ $-0.432$ $B-\overline{B}$ $-0.434$ $-0.417$ $-0.417$ $h^-$ $2.357$ $1.897$ $1.877$ ---------------------- --------------------- ----------------------- ----------------------- Table 3. [**$\bf S+Ag$ (NA35) Full phase space**]{} --------------- -------------------- ---------------------- ---------------------- Fitted Fitted with $h^-$ Fitted without $h^-$ Fitted without $h^-$ Parameters (Case A) (Case B) $T$ (MeV) $170.6\pm 5.9$ 176.3 $176.8\pm2.1$ $\lambda_u$ $1.544\pm 0.046$ 1.640 $1.641\pm0.074$ $\lambda_d$ $1.586\pm 0.050$ 1.700 $1.701\pm0.084$ $\lambda_s$ $1.084\pm 0.036$ 1.012 $1.011\pm0.047$ $\gamma_s$ $0.670\pm 0.073$ 0.836 $0.84\pm0.12$ $VT^3/4\pi^3$ $2.74\pm 0.71$ 1.23 $1.22\pm0.60$ $\chi^2/dof$ $9.37\;/\;3$ $1.654\;/\;2^{\;e}$ $1.652\;/\;2$ $\mu_u$ (MeV) $74.1\pm5.7$ 87.3 $87.6\pm8.0$ $\mu_d$ (MeV) $78.7\pm6.0$ 93.5 $93.9\pm8.8$ $\mu_s$ (MeV) $13.8\pm5.7$ 2.2 $2.0\pm8.1$ $P_{INSIDE}$ $100\%\;(128/128)$ $48.44\%\;(31/64)$ $-$ --------------- -------------------- ---------------------- ---------------------- [$^e$ It is the minimum of $\chi^2$ within the Hadron Gas with $T_0=183$ MeV (for $\gamma_s=1$), not the absolute minimum.]{} Table 4. [**$\bf S+Ag$ (NA35) Full phase space**]{} -------------------------- -------------------- Particle ratios for Experimental the fit with $h^-$ Values $K^0_s/h^-$ $0.0833\pm 0.0095$ $\Lambda/h^-$ $0.0817\pm 0.0081$ $\overline{\Lambda}/h^-$ $0.0140\pm 0.0018$ $\overline{p}/h^-$ $0.0108\pm 0.0043$ $p-\overline{p}/h^-$ $0.231\pm 0.021$ $B-\overline{B}/h^-$ $0.484\pm 0.061$ -------------------------- -------------------- Table 5. Projections on the $(\mu_u,T)$-plane of intersections of constant $\mu_d$ and constant $\mu_s$ of the critical surface $\varphi(T,\mu_u,\mu_d,\mu_s,\gamma_s)=\ln 4-1$ for $T_0=183$ MeV at $\gamma_s=1$. Projections on the $(\mu_d,\mu_u)$-plane of intersections of constant $T$ and constant $\mu_s$ of the critical surface $\varphi(T,\mu_u,\mu_d,\mu_s,\gamma_s)=\ln 4-1$ for $T_0=183$ MeV at $\gamma_s=1$. Projections on the $(T,\mu_s)$-plane of intersections of constant $\mu_u$ and constant $\mu_d$ of the critical surface $\varphi(T,\mu_u,\mu_d,\mu_s,\gamma_s)=\ln 4-1$ for $T_0=183$ MeV at $\gamma_s=1$. Projections on the $(T,\mu_s)$-plane of intersections, at fixed $\lambda_u$ $(\mu_u/T=0.4)$, of the $<S>=0$ and $<B>=2\beta<Q>$ surfaces for the SSBM and the IHG for different values of $\beta$. For the SSBM case $T_0$ is set at 183 MeV. Projections on the $(\mu_u,\mu_d)$-plane of intersections, at fixed $\lambda_u$ $(\mu_u/T=0.4)$, of the $<S>=0$ and $<B>=2\beta<Q>$ surfaces for the SSBM and the IHG, for different values of $\beta$. For the SSBM case $T_0$ is set at 183 MeV. Projections on the $(T,\mu_u)$-plane of the intersection of the $<S>=0$ and $<B>=2\beta<Q>$ surfaces for the SSBM with the critical surface for different values of $\beta$. $T_0$ is set at 183 MeV. Projections on the $(\mu_u,\mu_d)$-plane of the intersection of the $<S>=0$ and $<B>=2\beta<Q>$ surfaces for the SSBM with the critical surface for different values of $\beta$. $T_0$ is set at 183 MeV. Projections on the $(\mu_u,\mu_s)$-plane of the intersection of the $<S>=0$ and $<B>=2\beta<Q>$ surfaces for the SSBM with the critical surface for different values of $\beta$. $T_0$ is set at 183 MeV. Experimental particle ratios in the ($\mu_u,T$)-plane for the $S+Ag$ interaction measured in $4\pi$ phase space with $\gamma_s$ set to 0.67. The point and the cross correspond to the $\chi^2$ fit with the $h^-$. The thick solid line represent the limits of the hadronic phase (HG) as set by the SSBM. Experimental particle multiplicities in the ($\mu_u,T$)-plane for $S+Ag$ interaction measured in $4\pi$ phase space with $\gamma_s$ set to 0.84 and $VT^3/4\pi^3$ set to 1.23. The point represented by the solid circle corresponds to the location of the least value within the hadron gas of $\chi^2$, without the $h^-$. The lines which correspond to $K^0_s=17$, $h^-=175$ and $h^-=197$ lie outside the hadronic domain, as set by the S-SBM. The same diagram as Fig. 10, but with an enlargement of a smaller area to show the common overlapping region (shaded area) within the hadronic phase which is compatible with all the measured multiplicities, except $h^-$. The lines wich corespond to $\Lambda=16.4$, $\bar{\Lambda}=2.9$, $\bar{p}=1.2$, $\bar{p}=2.8$ and $p-\bar{p}=46$ are outside the region of the diagram and enclose the shaded region. Comparison between the experimentally measured multiplicities in $4\pi$ phase space and the theoretically calculated values in the fit with $h^-$ and without $h^-$ (cases A and B) for the $S+Ag$ interaction. The difference is measured in units of the relevant experimental error. $(\mu_B,T)$-phase diagramme with points obtained from fits to $p+\bar{p}$ \[8\], $S+S$ \[8\] and $S+Ag$ data and corresponding critical curves given by SSBM. [^1]: Only the repulsive form of interaction has been introduced in some cases through a hard sphere model. [^2]: The latter was not included in the original SBM as the main preocupation at the time referred to nuclear matter. [^3]: In our thermodynamical context the electric charge will enter as charge number density $q$. [^4]: For purposes of comparison we have also drawn corresponding projections for the IHG model. [^5]: To ensure that the $\chi^2$ estimate is carried out without leaving the domain of analyticity of the SSBM the numerical values in table 1 correspond to $T_0=190$ MeV for $\gamma_s=1$.

--- author: - 'C. Burigana,[^1]' - 'C.S. Carvalho,' - 'T. Trombetti,' - 'A. Notari,' - | \ M. Quartin, - 'G. De Gasperis,' - 'A. Buzzelli,' - 'N. Vittorio,' - | \ G. De Zotti, - 'P. de Bernardis,' - 'J. Chluba,' - 'M. Bilicki,' - | \ L. Danese, - 'J. Delabrouille,' - 'L. Toffolatti,' - 'A. Lapi,' - | \ M. Negrello, - 'P. Mazzotta,' - 'D. Scott,' - 'D. Contreras,' - | \ A. Achúcarro, - 'P. Ade,' - 'R. Allison,' - 'M. Ashdown,' - | \ M. Ballardini, - 'A.J. Banday,' - 'R. Banerji,' - 'J. Bartlett,' - | \ N. Bartolo, - 'S. Basak,' - 'M. Bersanelli,' - 'A. Bonaldi,' - | \ M. Bonato, - 'J. Borrill,' - 'F. Bouchet,' - 'F. Boulanger,' - | \ T. Brinckmann, - 'M. Bucher,' - 'P. Cabella,' - 'Z.-Y. Cai,' - | \ M. Calvo, - 'G. Castellano,' - 'A. Challinor,' - 'S. Clesse,' - | \ I. Colantoni, - 'A. Coppolecchia,' - 'M. Crook,' - | \ G. D’Alessandro, - 'J.-M. Diego,' - 'A. Di Marco,' - | \ E. Di Valentino, - 'J. Errard,' - 'S. Feeney,' - | \ R. Fernández-Cobos, - 'S. Ferraro,' - 'F. Finelli,' - 'F. Forastieri,' - 'S. Galli,' - 'R. G[é]{}nova-Santos,' - 'M. Gerbino,' - | \ J. González-Nuevo, - 'S. Grandis,' - 'J. Greenslade,' - | \ S. Hagstotz, - 'S. Hanany,' - 'W. Handley,' - | \ C. Hernández-Monteagudo, - 'C. Hervias-Caimapo,' - 'M. Hills,' - | \ E. Hivon, - 'K. Kiiveri,' - 'T. Kisner,' - 'T. Kitching,' - 'M. Kunz,' - | \ H. Kurki-Suonio, - 'L. Lamagna,' - 'A. Lasenby,' - | \ M. Lattanzi, - 'J. Lesgourgues,' - 'M. Liguori,' - | \ V. Lindholm, - 'M. Lopez-Caniego,' - 'G. Luzzi,' - | \ B. Maffei, - 'N. Mandolesi,' - 'E. Martinez-Gonzalez,' - | \ C.J.A.P. Martins, - 'S. Masi,' - 'D. McCarthy,' - | \ A. Melchiorri, - 'J.-B. Melin,' - 'D. Molinari,' - | \ A. Monfardini, - 'P. Natoli,' - 'A. Paiella,' - 'D. Paoletti,' - | \ G. Patanchon, - 'M. Piat,' - 'G. Pisano,' - 'L. Polastri,' - | \ G. Polenta, - 'A. Pollo,' - 'V. Poulin,' - 'M. Remazeilles,' - | \ M. Roman, - 'J.-A. Rubiño-Martín,' - 'L. Salvati,' - | \ A. Tartari, - 'M. Tomasi,' - 'D. Tramonte,' - 'N. Trappe,' - | \ C. Tucker, - 'J. Väliviita,' - 'R. Van de Weijgaert,' - | \ B. van Tent, - 'V. Vennin,' - 'P. Vielva,' - 'K. Young,' - | \ M. Zannoni, - for the CORE Collaboration bibliography: - 'pecmotreferences.bib' title: 'Exploring cosmic origins with CORE: effects of observer peculiar motion' --- Introduction {#sect:Intro} ============ The peculiar motion of an observer with respect to the cosmic microwave background (CMB) rest frame gives rise to boosting effects (the largest of which is the CMB dipole, i.e., the multipole $\ell=1$ anisotropy in the Solar System barycentre frame), which can be explored by future CMB missions. In this paper, we focus on peculiar velocity effects and their relevance to the Cosmic Origins Explorer (CORE) experiment. CORE is a satellite proposal dedicated to microwave polarization and submitted to the European Space Agency (ESA) in October 2016 in response to a call for future medium-sized space mission proposals for the M5 launch opportunity of ESA’s Cosmic Vision programme. This work is part of the [*Exploring Cosmic Origins*]{} (ECO) collection of articles, aimed at describing different scientific objectives achievable with the data expected from a mission like CORE. We refer the reader to the CORE proposal [@CORE2016] and to other dedicated ECO papers for more details, in particular the mission requirements and design paper [@delabrouille_etal_ECO] and the instrument paper [@debernardis_etal_ECO], which provide a comprehensive discussion of the key parameters of CORE adopted in this work. We also refer the reader to the paper on extragalactic sources [@2016arXiv160907263D] for an investigation of their contribution to the cosmic infrared background (CIB), which is one of the key topics addressed in the present paper, as well as the papers on $B$-mode component separation [@baccigalupi_etal_ECO] for a stronger focus on polarization, and mitigation of systematic effects [@ashdown_etal_ECO] for further discussion of potential residuals included in some analyses presented in this work. Throughout this paper we use the CORE specifications summarised in Table \[tab:CORE-bands\]. The analysis of cosmic dipoles is of fundamental relevance in cosmology, being related to the isotropy and homogeneity of the Universe at the largest scales. In principle, the observed dipole is a combination of various contributions, including observer motion with respect to the CMB rest frame, the intrinsic primordial (Sachs-Wolfe) dipole and the Integrated Sachs-Wolfe dipole as well as dipoles from astrophysical (extragalactic and Galactic) sources. The interpretation that the CMB dipole is mostly (if not fully) of kinematic origin has strong support from independent studies of the galaxy and cluster distribution, in particular via the measurements of the so-called *clustering dipole*. According to the linear theory of cosmological perturbations, the peculiar velocity of an observer (as imprinted in the CMB dipole) should be related to the observer’s peculiar gravitational acceleration via $\vec{v}_{\rm lin} = \beta_{\rm rd} \vec{g}_{\rm lin}$, where $\beta_{\rm rd} \simeq \Omega_{\rm m}^{0.55} / b_{\rm g}$ is also know as the redshift-space distortion parameter ($b_{\rm g}$ and $\Omega_{\rm m}$ being, respectively, the bias of the particular galaxy sample and the matter density parameter at the present time). The peculiar velocity and acceleration of, for instance, the Local Group treated as one system, i.e., as measured from its barycentre, should thus be aligned and have a specific relation between amplitudes. The former fact has been confirmed from analyses of many surveys over the last three decades, such as IRAS [@1987MNRAS.228P...5H; @1992ApJ...397..395S], 2MASS [@2003ApJ...598L...1M; @2006MNRAS.368.1515E], or galaxy cluster samples [@1996ApJ...460..569B; @1998ApJ...500....1P]. As far as the amplitudes are concerned, the comparison has been used to place constraints on the $\beta_{\rm rd}$ parameter [@1996ApJ...460..569B; @2000MNRAS.314..375R; @2006MNRAS.368.1515E; @2006MNRAS.373.1112B; @2011ApJ...741...31B], totally independent of those from redshift-space distortions observed in spectroscopic surveys. In this context, confirming the kinematic origin of the CMB dipole, through a comparison accounting for our Galaxy’s motion in the Local Group and the Sun’s motion in the Galaxy (see e.g., Refs. [@1999AJ....118..337C; @2012MNRAS.427..274S]), would provide support for the standard cosmological model, while finding any significant deviations from this assumption could open up the possibility for other interpretations (see e.g., Refs. [@2013PhRvD..88h3529W; @2016JCAP...06..035B; @2016JCAP...12..022G; @Roldan:2016ayx]). Cosmic dipole investigations of more general type have been carried out in several frequency domains [@2012MNRAS.427.1994G], where the main signal comes from various types of astrophysical sources differently weighted in different shells in redshift. An example are dipole studies in the radio domain, pioneered by Ref. [@1984MNRAS.206..377E] and recently revisited by Ref.  performing a re-analysis of the NRAO VLA Sky Survey (NVSS) and the Westerbork Northern Sky Survey, as well as by Refs. [@2015APh....61....1T; @2015MNRAS.447.2658T] using NVSS data alone. Prospects to accurately measure the cosmic radio dipole with the Square Kilometre Array have been studied by Ref. [@2015aska.confE..32S]. Perspectives on future surveys jointly covering microwave/millimeter and far-infrared wavelengths aimed at comparing CMB and CIB dipoles have been presented in Ref. [@2011ApJ...734...61F]. The next decades will see a continuous improvement of cosmological surveys in all bands. For the CMB, space observations represent the best, if not unique, way to precisely measure this large-scale signal. It is then important to consider the expectations from (and the potential issues for) future CMB surveys beyond the already impressive results produced by [*Planck*]{}. In addition to the dipole due to the combination of observer velocity and Sachs-Wolfe and intrinsic (see ref. [@2017arXiv170400718M] for a recent study) effects, a moving observer will see velocity imprints on the CMB due to Doppler and aberration effects [@Challinor:2002zh; @Burles:2006xf], which manifest themselves in correlations between the power at subsequent multipoles of both temperature and polarization anisotropies. Precise measurements of such correlations [@Kosowsky:2010jm; @Amendola:2010ty] provide important consistency checks of fundamental principles in cosmology, as well as an alternative and general way to probe observer peculiar velocities . This type of analysis can in principle be extended to thermal Sunyaev-Zeldovich (tSZ) and CIB signals. We will discuss how these investigations could be improved when applied to data expected from a next generation of CMB missions, exploiting experimental specifications in the range of those foreseen for LiteBIRD [@2016SPIE.9904E..0XI] and CORE. Since the results from COBE [@1992ApJ...397..420B], no substantial improvements have been achieved in the observations of the CMB spectrum at $\nu \gsim 30$GHz.[^2] Absolute spectral measurements rely on ultra-precise absolute calibration. FIRAS [@1996ApJ...473..576F] achieved an absolute calibration precision of 0.57mK, with a typical inter-frequency calibration accuracy of $0.1$mK in one decade of frequencies around 300GHz. The amplitude and shape of the CIB spectrum, measured by FIRAS [@Fixsen:1998kq], is still not well known. Anisotropy missions, like CORE, are not designed to have an independent absolute calibration, but nevertheless can investigate the CMB and CIB spectra by looking at the frequency spectral behaviour of the dipole amplitude [@DaneseDeZotti1981; @Balashev2015; @2016JCAP...03..047D; @Planck_inprep]. Unavoidable spectral distortions are predicted as the result of energy injections in the radiation field occurring at different cosmic times, related to the origin of cosmic structures and to their evolution, or to the different evolution of the temperatures of matter and radiation (for a recent overview of spectral distortions within standard $\Lambda$CDM, see Ref. [@2016MNRAS.460..227C]). For quantitative forecasts we will focus on well-defined types of signal, namely Bose-Einstein (BE) and Comptonization distortions ; however, one should also be open to the possible presence of unconventional heating sources, responsible in principle for imprints larger than (and spectral shapes different from) those mentioned above, and having parameters that could be constrained through analysis of the CMB spectrum. Deciphering such signals will be a challenge, but holds the potential for important new discoveries and for constraining unexplored processes that cannot be probed by other means. At the same time, a better determination of the CIB intensity greatly contributes to our understanding of the dust-obscured star-formation phase of galaxy evolution. The rest of this paper is organised as follows. In Sect. \[sec:dipideal\] we quantify the accuracy of a mission like CORE for recovering the dipole direction and amplitude separately at a given frequency, focussing on a representative set of CORE channels. Accurate relative calibration and foreground mitigation are crucial for analysing CMB anisotropy maps. In Sect. \[sec:res\_mod\] we describe a parametric approach to modelling the pollution of theoretical maps with potential residuals. The analysis in Sect. \[sec:dipideal\] is then extended in Sect. \[sec:dipsyst\] to include a certain level of residuals. The study throughout these sections is carried out in pixel domain. In Sect. \[sect:aberration\] we describe the imprints at $\ell>1$ due to Doppler and aberration effects, which can be measured in harmonic space. Precise forecasts based on CORE specifications are presented and compared with those expected from LiteBIRD. The intrinsic signature of a boost in Sunyaev-Zeldovich and CIB maps from CORE is also discussed in this section. In Sect. \[sect:DiffCMB\] we study CMB spectral distortions and the CIB spectrum through the analysis of the frequency dependence of the dipole distortion; we introduce a method to extend predictions to higher multipoles, coupling higher-order effects and geometrical aspects. The theoretical signals are compared with sensitivity at different frequencies, in terms of angular power spectrum, for a mission like CORE. In Sect. \[sec:dip\_spec\_sim\] we exploit the available frequency coverage through simulations to forecast CORE’s sensitivity to the spectral distortion parameters and the CIB spectrum amplitude, considering the ideal case of perfect relative calibration and foreground subtraction; however, we also parametrically quantify the impact of potential residuals, in order to define the requirements for substantially improve the results beyond those from FIRAS. In Sect. \[sec:end\] we summarise and discuss the main results. The basic concepts and formalisms are introduced in the corresponding sections, while additional information and technical details are provided in several dedicated appendices for sake of completeness. The CMB dipole: forecasts for CORE in the ideal case {#sec:dipideal} ==================================================== A relative velocity, $\beta\equiv v/c$, between an observer and the CMB rest frame induces a dipole (i.e., $\ell=1$ anisotropy) in the temperature of the CMB sky through the Doppler effect. Such a dipole is likely dominated by the velocity of the Solar System, $\vec{\beta_{\rm S}}$, with respect to the CMB (Solar dipole), with a seasonal modulation due to the velocity of the Earth/satellite, $\vec{\beta_{\rm o}}$, with respect to the Sun (orbital dipole). In this work we neglect the orbital dipole (which may indeed be used for calibration), thus hereafter we will denote with $\vec{\beta}$ the relative velocity of the Solar dipole. In this section we forecast the ability to recover the dipole parameters (amplitude and direction) by performing a Markov chain Monte Carlo (MCMC) analysis in the ideal case (i.e., without calibration errors or sky residuals). Results including systematics are given in Sect. \[sec:dipsyst\]. We test the amplitude of the parameter errors against the chosen sampling resolution and we probe the impact of both instrumental noise and masking of the sky. We consider the “[*Planck*]{} common mask 76” (in temperature), which is publicly available from the Planck Legacy Archive (PLA)[^3] [@PLArefESA], and keeps 76% of the sky, avoiding the Galactic plane and regions at higher Galactic latitudes contaminated by Galactic or extragalactic sources. We exploit here an extension of this mask that excludes all the pixels at $|b| \le 30^\circ$.[^4] Additionally, we explore the dipole reconstruction ability for different frequency channels, specifically 60, 100, 145, and 220GHz. We finally investigate the impact of spectral distortions (see Sects. \[sect:DiffCMB\] and \[sec:dip\_spec\_sim\]), treating the specific case of a BE spectrum (with chemical potential $\mu_{0}=1.4\times10^{-5}$, which is several times smaller than FIRAS upper limits). ![Map of the CMB dipole used in the simulations, corresponding to an amplitude $A=3.3645$mK and a dipole direction defined by the Galactic coordinates $b_{0} =48.24^\circ$ and $l_{0}=$ $264.00^\circ$. The map is in Galactic coordinates and at a resolution of $\simeq 3.4$arcmin, corresponding to [HEALPix]{} $N_{\rm side}=1024$.[]{data-label="fig:dipolemap"}](dip.pdf){width="45.00000%"} ![Instrumental noise map and [*Planck*]{} Galactic mask (extended to cut out $\pm30^\circ$ of the Galactic plane) employed in the simulations. The noise map corresponds to 7.5$\mu$K.arcmin, as expected for the 60-GHz band. The Map is in Galactic coordinates and at resolution of $\simeq 3.4$arcmin, corresponding to [HEALPix]{} $N_{\rm side}=1024$.[]{data-label="fig:instnoisemap"}](noise.pdf "fig:"){width="45.00000%"} ![Instrumental noise map and [*Planck*]{} Galactic mask (extended to cut out $\pm30^\circ$ of the Galactic plane) employed in the simulations. The noise map corresponds to 7.5$\mu$K.arcmin, as expected for the 60-GHz band. The Map is in Galactic coordinates and at resolution of $\simeq 3.4$arcmin, corresponding to [HEALPix]{} $N_{\rm side}=1024$.[]{data-label="fig:instnoisemap"}](mask.pdf "fig:"){width="45.00000%"} We write the dipole in the form: $$d(\hat{n}) = A\, \hat{n} \cdot \hat{n}_{0} + T_{0},$$ where $\hat{n}$ and $\hat{n}_{0}$ are the unit vectors defined respectively by the Galactic longitudes and latitudes $(l,b)$ and $(l_{0},b_{0})$. In Fig. \[fig:dipolemap\] we show the dipole map we have used in our simulations, generated assuming the best-fit values of the measurements of the dipole amplitude, $A=(3.3645\pm0.002)$mK, and direction, $l_{0}=264.00^\circ \pm 0.03^\circ$ and $b_{0}=48.24^\circ \pm 0.02^\circ$, found in the [*Planck*]{} (combined result from the High Frequency Instrument, HFI, and Low Frequency Instrument, LFI) 2015 release . Assuming the dipole to be due to velocity effects only, its amplitude corresponds to $\beta \equiv |\vec{\beta}| \equiv v/c = A / T_{0} = 1.2345 \times 10^{-3}$, with $T_{0}=2.72548 \pm 0.00057\,$K being the present-day temperature of the CMB [@2009ApJ...707..916F]. In Fig. \[fig:instnoisemap\] we show the instrumental noise map and the [*Planck*]{} Galactic mask employed in the simulations. The noise map corresponds to 7.5 $\mu$K.arcmin, as expected for the 60-GHz band. We calculate the likelihoods for the parameters $A$, $l_{0}$, $b_{0}$ and $T_{0}$ using the publicly available [COSMOMC]{} generic sampler package [@2002PhRvD..66j3511L; @Metropolis53; @Hastings70]. While the monopole $T_{0}$ is not an observable of interest in this context, we include it as a free parameter, to verify any degeneracy with the other parameters and for internal consistency checks. To probe the dependence of the parameter error estimates on the sampling resolution, we investigate the dipole reconstruction at [HEALPix]{} [@2005ApJ...622..759G] $N_{\rm side}=128$, 256, 512, and 1024, eventually including the noise and the Galactic mask. The reference frequency channel for this analysis is the 60-GHz band. The corresponding likelihoods are collected in Appendix \[sec:Likelihoods\] (see Fig. \[fig:conflevel128+1024\]) for the same representative values of $N_{\rm side}$ (see also Table \[table:conflevel128+1024\] for the corresponding 68% confidence levels). In Fig. \[fig:diperrors\] we plot the 1$\sigma$ uncertainties on the parameter estimates as functions of the [HEALPix]{} $N_{\rm side}$ value. We find that the pixelization error due to the finite resolution is dominant over the instrumental noise at any $N_{\rm side}$. This means that we are essentially limited by the sampling resolution. As expected, the impact of noise is negligible, although the effect of reducing the effective sky fraction is relevant. In fact, the presence of the Galactic mask results in larger errors (for all parameters) and introduces a small correlation between the parameters $A$ and $b_{0}$, as clearly shown in these plots. The likelihood results for some of the different frequencies under analysis are collected in Figs. \[fig:conflevel100220\] of Appendix \[sec:Likelihoods\] (see also Table \[table:conflevel60220\] for the 68% confidence levels at the four considered frequencies). Here we keep the resolution fixed at [HEALPix]{} $N_{\rm side}=1024$ and consider both noise level and choice of Galactic mask. We find that the dipole parameter estimates do not significantly change among the frequency channels, which is clearly due to the sub-dominant effect of the noise. As a last test of the ideal case, we compare the dipole parameter reconstruction between the cases of a pure blackbody (BB) spectrum and a BE-distorted spectrum. The comparison of the likelihoods is presented in Fig. \[fig:conflevelBE\] of Appendix \[sec:Likelihoods\] (see also Table \[table:conflevelBE\] for the corresponding 68% confidence levels). This analysis shows that the parameter errors are not affected by the spectral distortion and that the direction of the dipole is successfully recovered. The difference found in the amplitude value is consistent with the theoretical difference of about 76nK. Parametric model for potential foreground and calibration residuals in total intensity {#sec:res_mod} ====================================================================================== In the previous section we showed that in the ideal case of pure noise, i.e., assuming perfect foreground subtraction and calibration (and the absence of systematic effects) in the sky region being analysed, pixel-sampling limitation dominates over noise limitation. Clearly, specific component-separation and calibration methods (and implementations) introduce specific types of residuals. Rather than trying to accurately characterise them (particularly in the view of great efforts carried out in the last decade for specific experiments and the progress that is expected over the coming years), we implemented a simple toy model to parametrically estimate the potential impact of imperfect foreground subtraction and calibration in total intensity (i.e., in temperature). This includes using some of the [*Planck*]{} results and products made publicly available through the PLA. The PLA provides maps in total intensity (or temperature) at high resolution ($N_{\rm side} = 2048$) of global foregrounds at each [*Planck*]{} frequency (here we use those maps based on the [COMMANDER]{} method).[^5] It provides also suitable estimates of the zodiacal light emission (ZLE) maps (in temperature) from [*Planck*]{}-HFI. Our aim is to produce templates of potential foreground residuals that are simply scalable in amplitude according to a tunable parameter. In order to estimate such emission at CORE frequencies, without relying on particular sky models, we simply interpolate linearly (in logarithmic scale, i.e., in ${\rm log} (\nu)$–${\rm log} (T))$ pixel by pixel the foreground maps and the ZLE maps, and linearly extrapolate the ZLE maps at $\nu < 100$GHz. We then create a template of signal sky amplitude at each CORE frequency, adding the absolute values in each pixel of these foreground and ZLE maps[^6] and of the CMB anisotropy map available at the same resolution in the PLA (we specifically use that based on [COMMANDER]{}). Since for this analysis we are not interested in separating CMB and astrophysical emission at $\ell \ge 3$, we then generate templates from these maps, extracting the $a_{lm}$ modes for $\ell \le 2$ only. These templates are then degraded to the desired resolution. Finally, we generate maps of Gaussian random fields at each CORE frequency, with rms amplitude given by these templates, $T_{\rm amp,for}$, multiplied by a tunable parameter, $E_{\rm for}$, which globally characterizes the potential amplitude of foreground residuals after component separation. Clearly, the choice of reasonable values of $E_{\rm for}$ depends on the resolution being considered (or on the adopted pixel size), with the same value of $E_{\rm for}$ but at smaller pixel size implying less contamination at a given angular scale. [*Planck*]{} maps reveal, at least in temperature, a greater complexity in the sky than obtained by previous experiments. The large number of frequencies of CORE is in fact designed to accurately model foreground emission components with a precision much better than [*Planck*]{}’s. Also, at least in total intensity, ancillary information will come in the future from a number of other surveys, ranging from radio to infrared frequencies. The target for CORE in the separation of diffuse polarised foreground emission corresponds to $E_{\rm for} \simeq 0.01$, i.e., to $\simeq 1$% precision at the map level for angular scales larger than about $1^\circ$ (i.e., up to multipoles $\ell \lsim 200$), where the main information on primordial $B$-modes is contained, while at larger multipoles the main limitation comes from lensing subtraction and characterization and secondarily through control of extragalactic source contributions. We note also that comparing CMB anisotropy maps available from the PLA at $N_{\rm side} = 2048$ derived with four different component-separation methods and degraded to various resolutions, shows that the rms of the six difference maps does not scale strongly with the adopted pixel size, at least if we exclude regions close to the Galactic plane. For example, outside the [*Planck*]{} common mask 76, if we pass from $N_{\rm side} = 2048$ to $N_{\rm side} = 256$ or $64$, i.e., increasing the pixel linear size by a factor of 8 or 32 (with the exception of the comparison of [SEVEM]{} versus [SMICA]{}), the rms values of the cross-comparisons range from about 8–9$\mu$K to about 3–5$\mu$K, i.e., a decreases by a factor of only about 2.5. This suggests that, at least for temperature analyses, the angular scale adopted to set $E_{\rm for}$ is not so critical. Data calibration represents one of the most delicate aspects of CMB experiments. The quality of CMB anisotropy maps does not rely on absolute calibration of the signal (as it would, for example, in experiments dedicated to absolute measurements of the CMB temperature, i.e., in the direct determination of the CMB spectrum). However, the achievement of very high accuracy in the relative calibration of the maps (sometimes referred to as absolute calibration of the anisotropy maps), as well as the inter-frequency calibration of the maps taken in different bands, is crucial for enabling the scientific goals of CMB projects. Although this calibration step could in principle benefit from the availability of precise instrumental reference calibrators (implemented for example in FIRAS [@1999ApJ...512..511M] and foreseen in PIXIE [@2011JCAP...07..025K], or – but with much less accurate requirements – in [*Planck*]{}-LFI [@2009JInst...4T2006V]), this is not necessary for anisotropy experiments, as shown for example by WMAP and [*Planck*]{}-HFI. This represents a huge simplification in the design of anisotropy experiments with respect to absolute temperature ones. [*Planck*]{} demonstrated the possibility to achieve relatively calibration of anisotropy data at a level of accuracy of about 0.1% up to about 300GHz, while recent analyses of planet flux density measurements and modelling [@2016arXiv161207151P] indicate the possibility to achieve a calibration accuracy of $\simeq 1$% even above 300GHz, with only moderate improvements over what is currently realised. The goal of CORE is to achieve a calibration accuracy level around 0.01%, while the requirement of $0.1$% is clearly feasible on the basis of current experiments, with some possible relaxation at high frequencies. Methods for improving calibration are fundamental in astrophysical and cosmological surveys, and clearly critical in CMB experiments. In principle, improvements in various directions can be pursued: from a better characterization of all instrument components to cross-correlation between different CMB surveys; from the implementation of external precise artificial calibration sources to the search for a better characterization (and increasing number) of astronomical calibration sources; and, in general, with the improvement of data analysis methods. To parametrically model potential residuals due to imperfect calibration we follow an approach similar to that described above for foreground contamination. We note that calibration uncertainty implies an error proportional to the global effective (anisotropy in our case) signal. We therefore produce templates as described above, but do so by adding the foreground, ZLE, and CMB anisotropy maps, keeping their signs and maintaining all the $a_{lm}$ modes contained in the maps. The absolute values of these templates are then multiplied by a tunable parameter, $E_{\rm cal}$ (possibly dependent on frequency), which globally characterizes the amplitude of potential residuals arising from imperfect calibration. These are then used to define the pixel-by-pixel rms amplitudes, which are adopted to construct maps, $T_{\rm res,cal}$, of Gaussian random fields at each CORE frequency. In fact, we might also expect calibration errors to affect the level of foreground residuals. Hence, as a final step, we include in the model a certain coupling between the two types of residuals. At each frequency, we multiply the above simulated maps of foreground residuals by $(1+T_{\rm res,cal}/T_{\rm amp,for})$. The CMB dipole: forecasts for CORE including potential residuals {#sec:dipsyst} ================================================================ We now extend the analysis presented in Sect. \[sec:dipideal\] by including two sources of systematic effects, namely calibration errors and sky foreground residuals. We consider two pairs of calibration uncertainty and sky residuals (parameterised by $E_{\rm for} = 0.04$ and $E_{\rm cal} = 0.004$, and by $E_{\rm for} = 0.64$ and $E_{\rm cal} = 0.064$) at $N_{\rm side} = 1024$ in order to explore different resolutions through pixel degradation. Rescaled to $N_{\rm side} = 64$, the two cases correspond to a set-up respectively better and worse by a factor of 4 with respect to the case $E_{\rm cal} = 10^{-3}$ and $E_{\rm for} = 10^{-2}$. ![Sky residual and calibration error maps (in Galactic coordinates) in the 60-GHz band employed in the simulations. Their amplitudes correspond to the pessimistic case, $E_{\rm for} = 0.64$ and $E_{\rm cal} = 0.064$, for maps at resolution [HEALPix]{} $N_{\rm side}=1024$.[]{data-label="fig:dipsystematics"}](sky_res_60_uk.pdf "fig:"){width="45.00000%"} ![Sky residual and calibration error maps (in Galactic coordinates) in the 60-GHz band employed in the simulations. Their amplitudes correspond to the pessimistic case, $E_{\rm for} = 0.64$ and $E_{\rm cal} = 0.064$, for maps at resolution [HEALPix]{} $N_{\rm side}=1024$.[]{data-label="fig:dipsystematics"}](cal_res_60_uk.pdf "fig:"){width="45.00000%"} ![1$\sigma$ errors as function of [HEALPix]{} $N_{\rm side}$ values for the parameters $A$, $b_0$, $l_0$, and $T_{0}$: dipole-only (solid black line); dipole+noise (green dot-dashed line); dipole+noise+mask (red dotted line); and dipole+noise+mask+systematics (blue dashed line). The chosen frequency channel is 60GHz and the noise map corresponds to 7.5$\mu$K.arcmin. The adopted mask is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The systematics correspond to the pessimistic expectation of calibration errors and sky (foreground, etc.) residuals. Notice that the pixelization error, due to the finite map resolution, is dominant over the noise for any $N_{\rm side}$. While the impact of noise and systematics is negligible, we find that the effect of reducing the effective sky fraction is important.[]{data-label="fig:diperrors"}](dip_nside_A.pdf "fig:") ![1$\sigma$ errors as function of [HEALPix]{} $N_{\rm side}$ values for the parameters $A$, $b_0$, $l_0$, and $T_{0}$: dipole-only (solid black line); dipole+noise (green dot-dashed line); dipole+noise+mask (red dotted line); and dipole+noise+mask+systematics (blue dashed line). The chosen frequency channel is 60GHz and the noise map corresponds to 7.5$\mu$K.arcmin. The adopted mask is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The systematics correspond to the pessimistic expectation of calibration errors and sky (foreground, etc.) residuals. Notice that the pixelization error, due to the finite map resolution, is dominant over the noise for any $N_{\rm side}$. While the impact of noise and systematics is negligible, we find that the effect of reducing the effective sky fraction is important.[]{data-label="fig:diperrors"}](dip_nside_b.pdf "fig:") ![1$\sigma$ errors as function of [HEALPix]{} $N_{\rm side}$ values for the parameters $A$, $b_0$, $l_0$, and $T_{0}$: dipole-only (solid black line); dipole+noise (green dot-dashed line); dipole+noise+mask (red dotted line); and dipole+noise+mask+systematics (blue dashed line). The chosen frequency channel is 60GHz and the noise map corresponds to 7.5$\mu$K.arcmin. The adopted mask is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The systematics correspond to the pessimistic expectation of calibration errors and sky (foreground, etc.) residuals. Notice that the pixelization error, due to the finite map resolution, is dominant over the noise for any $N_{\rm side}$. While the impact of noise and systematics is negligible, we find that the effect of reducing the effective sky fraction is important.[]{data-label="fig:diperrors"}](dip_nside_l.pdf "fig:") ![1$\sigma$ errors as function of [HEALPix]{} $N_{\rm side}$ values for the parameters $A$, $b_0$, $l_0$, and $T_{0}$: dipole-only (solid black line); dipole+noise (green dot-dashed line); dipole+noise+mask (red dotted line); and dipole+noise+mask+systematics (blue dashed line). The chosen frequency channel is 60GHz and the noise map corresponds to 7.5$\mu$K.arcmin. The adopted mask is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The systematics correspond to the pessimistic expectation of calibration errors and sky (foreground, etc.) residuals. Notice that the pixelization error, due to the finite map resolution, is dominant over the noise for any $N_{\rm side}$. While the impact of noise and systematics is negligible, we find that the effect of reducing the effective sky fraction is important.[]{data-label="fig:diperrors"}](dip_nside_T0.pdf "fig:") In Fig. \[fig:dipsystematics\] we display the maps used in the simulations (for the 60-GHz band). The amplitudes correspond to the worse expected case; the most optimistic case is not shown, since the amplitude is just rescaled by a factor 16. The corresponding likelihood plots and 68% confidence levels are collected in Appendix \[sec:Likelihoods\]. We find that the impact of systematic effects on the parameter errors is negligible. In fact, as shown in Fig. \[fig:diperrors\], calibration errors and sky residuals do not noticeably worsen the 1$\sigma$ uncertainty at any sampling resolution. Furthermore, the frequency analysis confirms that the impact of systematic effects is not relevant in any of the bands under consideration (from 60 to 220GHz). While the effect of the systematics studied here on the *precision* of the parameter reconstruction is negligible, we find instead that they may have a moderate impact on the *accuracy*, introducing a bias in the central values of the estimates. Nonetheless, the bias is usually buried within the 1$\sigma$ error, with the marginal exception of the estimate of $l_0$ for the 220-GHz band (in the case of pessimistic systematics). In conclusion, our results show that the dipole recovery (in both amplitude $A$ and direction angles $b_0$ and $l_0$) is completely dominated by the sky sampling resolution. We find that: the noise impact is negligible; the reduction of the sky fraction due to the presence of the Galactic mask impacts on the parameter error amplitude by increasing the 1$\sigma$ errors on $A$, $b_0$ and $l_0$ by a factor of about 1.5, 1.6, and 1.9, respectively; and the effect of systematics slightly worsens the accuracy of the MCMC chain without affecting the error estimate. The main point of our analysis is that, in order to achieve an increasing precision in the dipole reconstruction, high resolution measurements are required, in particular when a sky mask has to be applied. This is especially relevant for dipole spectral distortion analyses, based on the high-precision, multi-frequency observations that are necessary to study the tiny signals expected. Measuring Doppler and aberration effects in different maps {#sect:aberration} ========================================================== Boosting effects on the CMB fields ---------------------------------- As discussed in the previous sections, a relative velocity between an observer and the CMB rest frame induces a dipole in the CMB temperature through the Doppler effect. The CMB dipole, however, is completely degenerate with an [*intrinsic*]{} dipole, which could be produced by the Sachs-Wolfe effect at the last-scattering surface due to a large-scale dipolar Newtonian potential [@Roldan:2016ayx]. For $\Lambda$CDM such a dipole should be of order of the Sachs-Wolfe plateau amplitude (i.e., $10^{-5}$) [@2008PhRvD..78h3012E; @2008PhRvD..78l3529Z], nevertheless the dipole could be larger in the case of more exotic models. In addition to the dipole, a moving observer will also see velocity imprints at $\ell>1$ in the CMB due to Doppler and aberration effects [@Challinor:2002zh; @Burles:2006xf]. Such effects can be measured as correlations among different $\ell$s, as has been proposed in Refs. [@Kosowsky:2010jm; @Amendola:2010ty; @Notari:2011sb] and subsequently demonstrated in Ref. . The aberration effect changes the arrival direction of photons from $\hat{n}^{\prime}$ to $\hat{n}$, which, at linear order in $\beta$, is completely degenerate with a lensing dipole. The Doppler effect modulates the CMB (an effect that is partly degenerate with an intrinsic CMB dipole[^7]) changing the specific intensity $I'$ in the CMB rest frame to the intensity $I$ in the observer’s frame[^8] by a multiplicative, direction-dependent factor as [@1986rpa..book.....R; @Challinor:2002zh] $$\begin{aligned} I'(\nu',\,\hat{n}^{\prime})=I(\nu,\,\hat{n}) \left(\frac{\nu'}{\nu} \right)^3, \label{eq:i_boost}\end{aligned}$$ where $$\begin{aligned} \nu \,=\, \nu^{\prime}\,\gamma\,\big(1+ \vec{\beta}\cdot\hat{n}^{\prime}\big)\,,\qquad \hat{n} \,=\, \frac{\hat{n}^{\prime} + \left[\gamma\,\beta + (\gamma - 1) \big(\hat{n}^{\prime}\cdot\hat{\beta} \big)\right]\hat{\beta}}{\gamma (1+ \vec{\beta} \cdot\hat{n}^{\prime}) } \,, \label{eq:nu_boost}\end{aligned}$$ with $\gamma \equiv (1-\vec{\beta}^2)^{-1/2}$. The temperature and polarization fields $X(\hat{n})$ in the CMB rest frame (where $X$ stands for $T,\,E$ or $B$) are similarly transformed as $$\begin{aligned} X^\prime ( \hat{n}^{\prime} ) \,=\, X(\hat{n}) \gamma \big( 1 - \vec{\beta}\cdot\hat{n} \big) \,. \label{eq:Xfields}\end{aligned}$$ Decomposing Eq.  into spherical harmonics leads to an effect in the multipole $\ell$ of order $\beta^\ell.$ Although this effect is dominant in the dipole, it also introduces a small, non-negligible correction to the quadrupole, with a different frequency dependence, due to the conversion of intensity to temperature . In addition, both aberration and Doppler effects couple multipoles $\ell$ to $\ell\pm n$ [@Chluba:2011zh; @Notari:2011sb]. This coupling is largest in the correlation between $\ell$ and $\ell\pm 1$ [@Challinor:2002zh; @Kosowsky:2010jm; @Amendola:2010ty], which was measured by [*Planck*]{} at 2.8 and 4.0$\sigma$ significance for the aberration and Doppler effects, respectively . These ${\cal O}(\beta)$ couplings are present on all scales and the measurability of aberration is mostly limited by cosmic variance, which constrains our ability to assume fully uncorrelated modes for $\ell \neq \ell^\prime$. Hence, in order to improve their measurement, it is important to have as many modes as possible, which drives us to cosmic-variance-limited measurements of temperature and polarization up to very high $\ell_{\rm max}$ and coverage of a large fraction of the sky $f_{\rm sky}$. CORE probes a larger $\ell_{\rm max}$ and covers a larger effective $f_{\rm sky}$ than [*Planck*]{} (as the extra frequency channels and the better sensitivity allow for an improved capability in doing component separation), hence it should achieve a detection of almost $13\,\sigma$ even with a 1.2-m telescope, as shown below. As discussed in Ref. [@Challinor:2002zh], upon a boost of a CMB map $X$, the [$a_{\ell m}$]{} coefficients of the spherical harmonic decomposition transform as $$\label{eq:aberrated-alm} a^{X}_{\ell m} \;=\; \sum_{\ell'=0}^\infty {}_s K_{\ell' \ell m} \, a'^{X}_{\ell' m}\,,$$ where $s$ indicates the spin of the quantity $X$. For scalars (such as the temperature), $s=0$, while for spin-2 quantities (such as the polarization), $s=2.$ The kernels ${}_s K_{\ell' \, \ell\, m}$ in general cannot be computed analytically and their numerical computation is not trivial, since this involves highly oscillatory integrals [@Chluba:2011zh]. However, efficient methods using an operator approach in harmonic space have been developed [@2014PhRvD..89l3504D], although for our estimates more approximate methods will suffice. It was shown in Ref. [@Notari:2011sb; @2014PhRvD..89l3504D] that the kernels can be well approximated by Bessel functions as follows: $$\label{eq:non-linear-fit-general} \begin{aligned} K_{(\ell-n) \ell m}^X &\;\simeq\; J_n\!\left(\!-2\, \beta \left[\prod_{k=0}^{n-1} \big[(\ell-k) \;{}_sG_{(\ell-k) m} \big]\right]^{1/n} \right); \\ K_{(\ell+n) \ell m}^X &\;\simeq\; J_n\!\left(\,2\, \beta \left[\prod_{k=1}^n \big[(\ell+k) \;{}_sG_{(\ell+k) m} \big]\right]^{1/n} \right). \end{aligned}$$ Here $$\label{eq:Glm} {}_{s}G_{\ell m} \equiv \sqrt{\frac{\ell^2-m^2}{4\ell^2-1} \left[1-\frac{s^2}{\ell^2}\right]}\,,$$ and $n \ge1$ (where $n$ is the difference in multipole between a pair of coupled multipoles, namely $\ell$ and $\ell \pm n$ ). It is also assumed that $\beta \ll 1$, although the formula above can be generalised to large $\beta$ [@Notari:2011sb; @2014PhRvD..89l3504D]. These kernels couple different multipoles so that, by Taylor expanding, we find $\big<a_{\ell m}~a_{(\ell+n)m}^{\ast}\big> = {\cal O}(\beta \ell)^n$. For $\ell \ll 1/\beta,$ the most important couplings are between neighbouring multipoles, $\ell$ and $\ell \pm 1$ (e.g. [@Challinor:2002zh]). One may wonder about the importance of the couplings between non-neighbouring multipoles, i.e., $\ell$ and $ \ell \pm n$, for $\ell \gtrsim 1/\beta$. However, quite surprisingly, for $\ell \gg 1/\beta$ we find that: (1) in the $(\ell, \ell \pm 1)$ correlations, terms that are higher order in $\beta \ell$ are negligible [@Chluba:2011zh; @Notari:2011sb]; and (2) most of the correlation seems to remain in the $(\ell, \ell \pm 1)$ coupling. For these reasons, from here onwards, we will ignore terms that are higher order in $\beta$ and couplings between non-neighbouring multipoles (i.e., $n>1).$ In order to measure deviations from isotropy due to the proper motion of the observer, we therefore compute the off-diagonal correlations $\big<a_{\ell m}^{X}~a_{(\ell+1)m}^{X\ast}\big>.$ Assuming that in the rest frame the Universe is statistically isotropic and that parity is conserved, then in the boosted frame, for $\ell^{\prime}=\ell+1,$ we find that (see Refs. [@Challinor:2002zh; @Kosowsky:2010jm; @Amendola:2010ty]) $$ a_{\ell \, m}^{X} \, \simeq \, c_{\ell m}^{-} a^{' \, X}_{(\ell-1) m} + c_{\ell m}^{+} a^{' \, X}_{(\ell+1) m} \, ,$$ where $$c_{\ell m}^{+} = \beta(\ell+2 -d) {}_sG_{(\ell+1) m}\,,\qquad c_{\ell m}^{-} = -\beta(\ell-1+ d ){}_sG_{\ell m}\,,\label{eq:alm-coef}$$ and $d$ parametrizes the Doppler effect of dipolar modulation. It then follows that $$\label{eq:almcorr} \left<a_{\ell m}^{X}~a_{(\ell+1)m}^{Y\ast}\right> = \beta \left[ (\ell+2 -d) \,{}_{s_X}\! G_{(\ell+1)m} C_{\ell+1}^{XY} - (\ell+d)\,{}_{s_Y}\! G_{(\ell+1)m} C_{\ell}^{XY} \right] + O(\beta^2)\,.$$ For $\ell \gtrsim 20$, we have ${}_{2}G_{\ell m} \simeq {}_{0}G_{\ell m}$. As will be shown, large scales are not important for measuring the boost, and thus it is not important to keep the indication of the spin. Thus from here onwards, we will drop $s$. The above equation reduces to $$\left<a_{\ell m}^{X}~a_{(\ell+1)m}^{Y\ast}\right> = \beta G_{(\ell+1)m} \left[ (\ell+2 -d) C_{\ell+1}^{XY} - (\ell+d) C_{\ell}^{XY} \right] + O(\beta^2)\,, \label{eq:aberration_covariance}$$ where the angular power spectra $C^{XY}_\ell$ are measured in the CMB rest frame. For the CMB temperature and polarization, $d=1$, as observed from Eqs. –. In this case, no mixing of $E$- and $B$-polarization modes occurs, not even in higher orders in $\beta$ [@Notari:2011sb; @2014PhRvD..89l3504D]. However, for $d \neq 1$, the coupling is non-zero already at first order in $\beta$ [@Challinor:2002zh; @2014PhRvD..89l3504D]. Maps estimated from spectra that are not blackbody have different Doppler coefficients,[^9] as we discuss in the next subsection. Note that in practice one never measures temperature and polarization anisotropies directly, instead one measures anisotropies in *intensity* and then converts this to temperature and polarization. This distinction (though perhaps seeming trivial) is relevant for the Doppler effect, which induces a dipolar modulation of the CMB anisotropies, appearing with frequency-dependent factors . In particular such factors were shown to be proportional to a Compton $y$-type spectrum (exactly like the quadrupole correction  and therefore degenerate with the tSZ effect); they are measurable in the [*Planck*]{} maps at about 12$\,\sigma$ and in the CORE maps even at 25–60$\sigma$ [@2016PhRvD..94d3006N], depending on the template that is used for contamination due to the tSZ effect. Such S/N ratios are much larger than those that can be obtained in temperature and polarization and so, at first sight, they may appear to represent a better way to measure the boosting effects. However, the peculiar frequency dependence is strictly a consequence of the intensity-to-temperature (or intensity-to-polarization) conversion and thus agnostic to the source of the dipole  (i.e., whether it is from our peculiar velocity or is an intrinsic CMB dipole). For this reason we focus on the frequency-*independent* part of the dipolar modulation signal in Eq.  (with $d=1$), which is unlikely to be caused by an intrinsically large CMB dipole (see Ref. [@Roldan:2016ayx] for details), in our forecast. Going beyond the CMB maps ------------------------- Since CORE will also measure the thermal Sunyaev-Zeldovich effect, the CIB, and the weak lensing signal over a wide multipole range, it is interesting to examine if these maps could also be used to measure the aberration and Doppler couplings. The intensity of a tSZ Compton-$y$ map is given by $$\label{eq:TSZ-spectrum} I'_{tSZ}(\nu') \,=\, y \cdot g\left(\frac{h\nu'}{k_{\rm B}T_{0}}\right) \,K(\nu') \, ,$$ where $g(x')=x' \coth(x'/2)-4$, $K(\nu')$ is the conversion factor that derives from setting $T=T_{0}+\delta T$ in the Planck distribution and expanding to first order in $\delta T$, and $x' \equiv h\nu'/k_{\rm B} T_{0}$ ($T_{0}$ being the present temperature of the CMB). Explicitly $K(\nu')$ is given by $$ K(\nu') \,=\, \frac{2\,h \nu'^3}{c^2} \frac{x' \exp (x')}{(\exp (x') -1)^2} \,.$$ A boosted observer will see an intensity as defined in Eq. . Such intensity, expanded at first order in $\beta$, will contain Doppler couplings with a non-trivial frequency dependence, similarly to what happens in the case of CMB fluctuations, where frequency-dependent boost factors are generated, as discussed in the previous subsection. For simplicity we only analyse the couplings that retain the same frequency dependence of the original tSZ signal, which come from aberration,[^10] and so we here set $d=0$ in Eq. . For the intensity of the CIB map (see Sect. \[sect:DipCIB\] for further details), we assume the template obtained by Ref. [@Fixsen:1998kq], $$\label{eq:CIB-spectrum} I'_{\rm CIB} \,\propto\, \nu'^{0.64} \frac{\nu'^3}{\exp\!\left[\frac{h \nu'}{k_{\rm B}\, 18.5{\rm K}}\right]-1} \, . $$ At low frequencies, the intensity scales as $$\label{eq:CIB-spectrum-RJ} I'_{\rm CIB} = A_{\rm CIB} \, \nu'^{2.64} \, , $$ where $A_{\rm CIB}$ is a constant related to the amplitude. In the boosted frame and to lowest order in $\beta,$ we find that $$I_{\rm CIB}(\nu) = \left(\frac{\nu}{\nu'} \right)^3 A'_{\rm CIB} \, \nu'^{2.64} \simeq A'_{\rm CIB} \big[ \gamma (1- \vec{\beta} \cdot {\hat{n}}) \big]^{-0.36} \nu^{2.64} \, . $$ Therefore, the boosted amplitude is $A_{\rm CIB} \equiv A'_{\rm CIB} / \big[\gamma (1- \vec{\beta}\cdot\hat{n})\big]^{0.36}, $ which implies $d=0.36$. Note that in this case, since we work in a low-frequency approximation (relative to the peak of the CIB at around 3000GHz), we do not have any frequency-dependent boost factors. The CMB weak lensing maps can also be used to measure the boost. However, since the estimation of the weak lensing potential involves 4-point correlation functions of the CMB fields, the boost effect is more complex to estimate; hence we leave this analysis for a future study. Estimates of the Doppler and aberration effect ---------------------------------------------- For full-sky experiments, it has been shown in Ref. [@Challinor:2002zh] that, under a boost, the corrections to the power spectra are ${\cal O}(\beta^2),$ whereas for experiments with partial sky coverage there can be an ${\cal O}(\beta)$ correction [@Pereira:2010dn; @Jeong:2013sxy; @Louis:2016ahn]. Nevertheless, even for the partial-sky case, this correction to $C_{\ell}^{XY}$ would only propagate at ${\cal O}(\beta^2)$ in the correlations above. In what follows, we will neglect the effect of the sky coverage in the boost corrections. Also, since we will be restricting ourselves to ${\cal O}(\beta)$ effects, from here onwards we will drop ${\cal O}(\beta^2)$ from the equations. For the CMB fields, as it was shown in Refs. [@Amendola:2010ty; @Notari:2011sb], that the fractional uncertainty in the estimator of $ \left<a_{\ell m}^{X}~a_{(\ell+1)m}^{Y\ast}\right>$ is given by $$\label{eq:delta-beta} \left. \frac{\delta \beta}{\beta}\right|_{XY} \simeq \left[\sum_{\ell}\sum_{m=-\ell}^\ell \frac{\left< a_{\ell m}^{X}~a_{(\ell+1)m}^{Y\ast}\right>^{2}} {{\mathfrak C}^{XX}_{\ell} {\mathfrak C}^{YY}_{\ell+1}} \right]^{-1/2}$$ (see also Ref. ). Here, ${\mathfrak C}^{XX}_{\ell} \equiv (C^{XX}_{\ell} + N^{XX}_{\ell,\text{total}}) / \sqrt{f_{\rm sky}},$ where $f_{\rm sky}$ is the fraction of the sky covered by the experiment and $\,N^{XX}_{\ell,\text{total}}\,$ is the effective noise level on the map $X.$ Thus ${\mathfrak C}^{XX}_\ell$ represents the sum of instrumental noise and cosmic variance. The effective noise is obtained by taking the inverse of the sum over the different channels $i$ of the inverse of the individual $N_{\ell,i}^2$ [@Notari:2011sb], N\_[,]{}= \^[-1/2]{}. The noise in each channel is given by a constant times a Gaussian beam characterised by the beam width $\theta_{\text{FWHM}}$: \[eq:Nell-formula\] N\_[,i]{}\^[X]{} = (\^[X]{})\^2 , where $\sigma^{X}$ is the noise in $\mu$K.arcmin for the map $X.$ ![Achievable precision in estimating the velocity through aberration and Doppler effects in an ideal experiment (with $f_{\rm sky} = 1$ and limited by cosmic variance only) for different maps. *Left:* as a function of $\ell_{\rm max}$. *Right:* as a function of $\ell_{\rm min}$ (with fixed $\ell_{\rm max}=5000$). *Bottom:* for individual bins with $\Delta \ell = 200$. We see that: (i) the first hundred $\ell$s are not important for achieving a high S/N; and (ii) the non-CMB diffuse maps exhibit low precision and are not very useful for measuring $\beta$. Note that for simplicity we have assumed no primordial $B$-modes (our constraints are very weakly sensitive to this choice). \[fig:SN-Ideal\]](Ideal-CMB+tSZ+CIB-v5.pdf "fig:"){width="52.10000%"} ![Achievable precision in estimating the velocity through aberration and Doppler effects in an ideal experiment (with $f_{\rm sky} = 1$ and limited by cosmic variance only) for different maps. *Left:* as a function of $\ell_{\rm max}$. *Right:* as a function of $\ell_{\rm min}$ (with fixed $\ell_{\rm max}=5000$). *Bottom:* for individual bins with $\Delta \ell = 200$. We see that: (i) the first hundred $\ell$s are not important for achieving a high S/N; and (ii) the non-CMB diffuse maps exhibit low precision and are not very useful for measuring $\beta$. Note that for simplicity we have assumed no primordial $B$-modes (our constraints are very weakly sensitive to this choice). \[fig:SN-Ideal\]](Ideal-CMB+tSZ+CIB-lmin-v5.pdf "fig:"){width="52.10000%"} ![Achievable precision in estimating the velocity through aberration and Doppler effects in an ideal experiment (with $f_{\rm sky} = 1$ and limited by cosmic variance only) for different maps. *Left:* as a function of $\ell_{\rm max}$. *Right:* as a function of $\ell_{\rm min}$ (with fixed $\ell_{\rm max}=5000$). *Bottom:* for individual bins with $\Delta \ell = 200$. We see that: (i) the first hundred $\ell$s are not important for achieving a high S/N; and (ii) the non-CMB diffuse maps exhibit low precision and are not very useful for measuring $\beta$. Note that for simplicity we have assumed no primordial $B$-modes (our constraints are very weakly sensitive to this choice). \[fig:SN-Ideal\]](Ideal-CMB+tSZ+CIB-lbin-200-v5.pdf "fig:"){width="52.10000%"} For the tSZ signal, we assume as a fiducial spectrum the one obtained in ref. [@Aghanim:2015eva] (slightly extrapolated to higher $\ell$s). For the forecast noise spectrum we use the estimates obtained in ref. [@2017arXiv170310456M] using the NILC component separation technique (see figure 14 therein), where it was shown that residual foreground contamination is a large fraction of the total noise. For the CIB signal, we use the spectra obtained in Ref. [@Cai2013]; for the noise, we rely on the simulations carried out in Ref. [@2016arXiv160907263D]. We also make the conservative assumption that the different channels of the CIB are 100% correlated. Since different channels pick up different redshifts, effectively the correlation is not going to be total and some extra signal can be obtained from multiple channels; however, since this makes the analysis much more complex (due to the need to have all the covariance matrices) and since the CIB turns out not to be promising for measuring aberration (see Fig. \[fig:SN-Ideal\]), we neglect these corrections. We computed Eq.  for the different maps of different experiments. We compared the detection potentials of CORE (see Table \[tab:CORE-bands\]) with those expected from both [*Planck*]{} and LiteBIRD [@2016SPIE.9904E..0XI]. For the [*Planck*]{} specifications, we use the values of the 2015 release, while the LiteBIRD specifications used in this analysis are listed in Table \[tab:LiteBIRD-bands\]. In Fig. \[fig:SN-Ideal\] we show the precision that could be reached by an ideal experiment with $f_{\rm sky} = 1$ and limited by cosmic variance only. We show the results for: the range $\ell \in [2, \ell_{\rm max}]$; the range $\ell \in [\ell_{\rm min}, 5000]$; and for individual $\ell$ bins of width $\Delta\ell = 200.$ The signal-to-noise ratios in the tSZ and CIB maps are considerably lower than in the CMB maps, which is due to the fact that the spectra are smoother, as explained later. For instance, for $\ell_{\rm max}=4000,$ in the $TT$ and $EE$ maps separately we have ${\rm S/N} > 16,$ whereas in tSZ and in CIB we have ${\rm S/N} \simeq 1.$ In Fig. \[fig:SN-Core\] and Table \[tab:ston-boost\] we summarise our forecasts for CORE and compare them with both [*Planck*]{} and LiteBIRD forecasts. These results differ from the ideal case due to the inclusion of instrumental noise, foreground contamination (in the case of tSZ) and $f_{\rm sky} \neq 1.$ In the last panel we also show the total precision by combining all temperature and polarization channels assuming a negligible correlation among them (which was shown in Ref. [@Amendola:2010ty] to be a good approximation). Note also that the $TE$ and $ET$ correlation functions were shown to be independent in Ref. [@Amendola:2010ty] and both carry the same S/N. So we usually present the combined S/N for $TE+ET$, which is $\sqrt{2}$ times their individual S/N values. ![Similar to the left panel of Fig. \[fig:SN-Ideal\] but for realistic experiments (described in detail in Table \[tab:CORE-bands\]) and assuming $f_{\rm sky} = 0.8.$ In the bottom right panel we compare the total precision after combining all temperature and polarization maps, including also the case of an ideal experiment (no instrumental noise and $f_{\rm sky} = 1$). \[fig:SN-Core\]](COrE+1p2m-CMB+tsZ+CIB-v4.pdf "fig:"){width="51.00000%"} ![Similar to the left panel of Fig. \[fig:SN-Ideal\] but for realistic experiments (described in detail in Table \[tab:CORE-bands\]) and assuming $f_{\rm sky} = 0.8.$ In the bottom right panel we compare the total precision after combining all temperature and polarization maps, including also the case of an ideal experiment (no instrumental noise and $f_{\rm sky} = 1$). \[fig:SN-Core\]](LiteBIRD-all-channels-v2.pdf "fig:"){width="51.00000%"}\ ![Similar to the left panel of Fig. \[fig:SN-Ideal\] but for realistic experiments (described in detail in Table \[tab:CORE-bands\]) and assuming $f_{\rm sky} = 0.8.$ In the bottom right panel we compare the total precision after combining all temperature and polarization maps, including also the case of an ideal experiment (no instrumental noise and $f_{\rm sky} = 1$). \[fig:SN-Core\]](Planck-all-channels-v2.pdf "fig:"){width="51.00000%"} ![Similar to the left panel of Fig. \[fig:SN-Ideal\] but for realistic experiments (described in detail in Table \[tab:CORE-bands\]) and assuming $f_{\rm sky} = 0.8.$ In the bottom right panel we compare the total precision after combining all temperature and polarization maps, including also the case of an ideal experiment (no instrumental noise and $f_{\rm sky} = 1$). \[fig:SN-Core\]](Ideal-CMBTot-comparison.pdf "fig:"){width="51.00000%"} --------------------------------- --------- ---------------------- ------------------- ------ --------- ------ ------- Channel $\theta_{\rm{FWHM}}$ $\sigma^{T}$ S/N S/N S/N S/N \[GHz\] \[arcmin\] \[$\mu$K.arcmin\] $TT$ $TE+ET$ $EE$ Total [*Planck*]{} (all) $\simeq 5.5$ $\simeq 13$ 3.8 1.7 1.0 4.3 LiteBIRD (all) $\simeq 19$ $\simeq 1.7$ 2.0 1.8 1.8 3.3 60 17.87 7.5 2.1 1.9 1.8 3.4 70 15.39 7.1 2.5 2.4 2.2 4.1 80 13.52 6.8 2.8 2.8 2.6 4.8 90 12.08 5.1 3.5 3.4 3.3 5.9 100 10.92 5 3.9 3.7 3.7 6.5 115 9.56 5 4.3 4.2 4.2 7.3 130 8.51 3.9 5.1 4.9 5. 8.6 145 7.68 3.6 5.7 5.3 5.5 9.5 160 7.01 3.7 6.1 5.6 5.8 10.1 175 6.45 3.6 6.5 5.8 6.1 10.7 195 5.84 3.5 7.1 6.1 6.5 11.4 220 5.23 3.8 7.5 6.3 6.7 11.9 255 4.57 5.6 7.5 5.9 6.2 11.4 295 3.99 7.4 7.5 5.7 5.8 11. 340 3.49 11.1 7. 5.1 4.9 9.9 390 3.06 22 5.8 3.8 3.1 7.6 450 2.65 45.9 4.5 2.3 1.4 5.3 520 2.29 116.6 2.9 1. 0.3 3.1 600 1.98 358.3 1.4 0.3 0. 1.4 (all) $\simeq 4.5$ $\simeq 1.4$ 8.2 6.6 7.3 12.8 Ideal ($\ell_{\rm max}$ = 2000) (all) 0 0 5.3 7.1 8.7 12.7 Ideal ($\ell_{\rm max}$ = 3000) (all) 0 0 10 9.8 14 21 Ideal ($\ell_{\rm max}$ = 4000) (all) 0 0 16 11.4 19 29 Ideal ($\ell_{\rm max}$ = 5000) (all) 0 0 22 12.6 26 38 --------------------------------- --------- ---------------------- ------------------- ------ --------- ------ ------- : =0.4cm[**Aberration and Doppler effects with CORE.**]{} We assume $f_{\rm sky} = 0.8$ for all experiments (and $f_{\rm sky} = 1$ in the ideal cases) in order to make comparisons simpler. For CORE we assume the 1.2-m telescope configuration, but with extended mission time to match the 1.5-m noise in $\mu$K.arcmin. For CORE and LiteBIRD we assume $\sigma^{P}$ = $\sqrt{2}\sigma^{T}$, while for [*Planck*]{} we use the 2015 values. The combined channel estimates are effective values that best approximate Eq.  in the $\ell$ range of interest. Note that CORE will have ${\rm S/N} \ge 5$ in 14 different frequency bands. Also, by combining all frequencies, CORE will have similar S/N in $TT$, $TE+ET$ and $EE$. \[tab:ston-boost\] As a side note, since the estimators for $\left<a_{\ell m}^{X}~a_{(\ell+1)m}^{Y\ast}\right>$ involve a sum over all $\ell$s and $m$s and since $m$ enters through $G_{\ell m}$ only, it is useful to use the following approximations, which are valid to very good accuracy for $\ell \gtrsim 20$ [@Notari:2011sb; @2016PhRvD..94d3006N]: $$\label{eq:Glm-simple} \sum_m G_{\ell, m} = 0.39 (2\ell +1)\,; \qquad \sum_m \big[G_{\ell, m}\big]^2 = 0.408^2 (2\ell +1)\,.$$ Although we did not use these approximations in our results, they yield up to 1%-level accuracy and by allowing the sum over $m$s to be removed, they significantly simplify the calculation of the estimators. The achievable precision in $\beta$ through this method depends strongly on the shape of the power spectrum – strongly varying spectra give much lower uncertainties compared to smooth spectra. For instance, for the tSZ and CIB maps, many modes are in the cosmic-variance-limited regime, thus one might think that they would yield a good measurement of $\beta$. However, since their $C_\ell$s are smooth functions of $\ell,$ they do not carry much information on the boost. To understand this and gain some insight, we rewrite Eq.  by approximating $C_{\ell+1}$ as $C_{\ell} + {\textrm{d}}C_{\ell}/{\textrm{d}}\ell$ and adding the approximation that ${\textrm{d}}C_{\ell}/{\textrm{d}}\ell \ll C_{\ell}$ (note, however, that $ \ell {\textrm{d}}C_{\ell}/{\textrm{d}}\ell$ could be comparable to $C_{\ell}$ at small scales). We thus find that $$\label{eq:almcorr5} \sum_m \left<a_{\ell m}^{X}~a_{(\ell+1)m}^{Y\ast}\right> = 0.39 (2\ell +1) \beta \left[ (2-2d) C_{\ell}^{XY} -(\ell+d) \frac{{\textrm{d}}C_{\ell}^{XY}}{{\textrm{d}}\ell} \right]\,.$$ Assuming the cosmic-variance dominated regime (i.e., ${\mathfrak C}^{XX}_{\ell} \simeq C^{XX}_{\ell}$) for $\ell \gtrsim 20$ and putting $X=Y$, we find that $$\label{eq:delta-beta-simp3} \left. \frac{\delta \beta}{\beta}\right|_{XX} \,\simeq \, \frac{1}{0.408\beta}\left[\sum_{\ell} (2\ell+1) \left[(2-2d) - \ell \left(1 - \frac{C_{\ell+1}^{XX}}{C_{\ell}^{XX}} \right) \right]^2 \right]^{-\frac{1}{2}}.$$ For the $TE$ case, the formula is less useful. For the CMB temperature and polarization ($d=1$), only the derivative term survives: $$\label{eq:delta-beta-simp2} \left. \frac{\delta \beta}{\beta}\right|_{XX=TT,EE,BB} \,\simeq \, \frac{1}{0.408\beta}\left[\sum_{\ell} (2\ell+1) \left[\frac{{\textrm{d}}\ln C_{\ell}^{XX}}{{\textrm{d}}\ln \ell}\right]^2 \right]^{-\frac{1}{2}}.$$ Note that for the CIB the precision is smaller than for the CMB temperature and polarization, not only because the spectra are smoother, but also because there is a partial cancellation between the two terms in the summand of Eq. . In this analysis we relied only on the diffuse background components of the measured maps. Aberration and Doppler effects can in principle also be detected using point sources, since the boosting effects will change both their number counts, angular distribution, and redshift. For the upcoming CMB experiments, however, the number density of point sources is probably insufficient for a significant signal, since one needs more than about $10^6$ objects to have a detection at greater than $1\,\sigma$ [@Yoon:2015lta]. Differential approach to CMB spectral distortions and the CIB {#sect:DiffCMB} ============================================================= Using the complete description of the Compton-Getting effect [@1970PSS1825F] we compute full-sky maps of the expected effect at desired frequency. We start discussing the frequency dependence of the dipole spectrum [@DaneseDeZotti1981; @Balashev2015] and then extend the analysis beyond the dipole. The CMB dipole {#sect:DipCMB} -------------- The dipole amplitude is directly proportional to the first derivative of the photon occupation number, $\eta(\nu)$, which is related to the thermodynamic temperature, $T_{\rm therm}(\nu)$, i.e., to the temperature of the blackbody having the same $\eta(\nu)$ at the frequency $\nu$, by $$T_{\rm therm}={h\nu\over k_{\rm B}\ln(1+1/\eta(\nu))}. \label{eq:t_therm}$$ The difference in $T_{\rm therm}$ measured in the direction of motion and in the perpendicular direction is given by [@DaneseDeZotti1981]: $$\label{eq:DeltaTtherm} \Delta T_{\rm therm}={h\nu\over k}\left\{{1\over \ln\left[1+1/\eta(\nu)\right]}-{1\over \ln\left[1+1/\eta(\nu(1+\beta))\right]} \right\} \, , $$ which, to first order, can be approximated by: $$\label{eq:DeltaTtherm_firstord} \Delta T_{\rm therm} \simeq -{x \beta T_0\over (1+\eta)\ln^2(1+1/\eta)}{d\ln\eta\over d\ln x},$$ where $x\equiv h\nu/kT_0$ is the dimensionless frequency. In Fig. \[fig:Dip\_C\_BE\] we show the dipole spectrum derived for two well-defined deviations from the Planck distribution, namely the BE and Comptonization distortions induced by unavoidable energy injections in the radiation field occurring at different cosmic times, early and late, respectively. We briefly discuss below their basic properties and the signal levels expected from different processes. A BE-like distorted spectrum is produced by two distinct processes. Firstly there is the dissipation of primordial perturbations at small scales [@1994ApJ...430L...5H; @chlubasunyaev2012], which generates a positive chemical potential. Secondly we have Bose condensation of CMB photons by colder electrons, as a consequence of the faster decrease of the matter temperature relative to the radiation temperature in an expanding Universe, which generates a negative chemical potential [@2012MNRAS.419.1294C; @sunyaevkhatri2013]. The photon occupation number of the BE spectrum is given by $$\label{eq:etaBE} \eta_{\rm BE}= {1\over e^{x_{\rm e}+\mu} -1},$$ where $\mu$ is the chemical potential that quantifies the fractional energy, $\Delta \epsilon/ \varepsilon_{\rm i}$, exchanged in the plasma during the interaction,[^11] $x_{\rm e} = x / \phi (z)$, $\phi (z) = T_{\rm e}(z)/T_{\rm CMB}(z)$, with $T_{\rm e}(z)$ being the electron temperature. For a BE spectrum, $\phi = \phi_{\rm BE}(\mu)$. The dimensionless frequency $x$ is redshift invariant, since in an expanding Universe both $T_{\rm CMB}$ and the physical frequency $\nu$ scale as $(1+z)$. For small distortions, $\mu \simeq 1.4 \Delta \epsilon/ \varepsilon_{\rm i}$ and $\phi_{\rm BE} \simeq (1-1.11\mu)^{-1/4}$. The current FIRAS 95% CL upper limit is $|\mu_0|<9 \times 10^{-5}$ [@1996ApJ...473..576F], where $\mu_0$ is the value of $\mu$ at the redshift $z_1$ corresponding to the end of the kinetic equilibrium era. At earlier times $\mu$ can be significantly higher, and the ultimate limits on $\Delta \epsilon/ \varepsilon_{\rm i}$ before the thermalization redshift (when any distortion can be erased) comes from cosmological nucleosynthesis. These two kinds of distortions are characterised by a $|\mu_0|$ value in the range, respectively, $\sim 10^{-9}$–$10^{-7}$ (and in particular $\simeq 2.52 \times 10^{-8}$ for a primordial scalar perturbation spectral index $n_{\rm s}=0.96$, without running), and $\simeq 3 \times 10^{-9}$. Since very small scales that are not explored by current CMB anisotropy data are relevant in this context, a broad set of primordial spectral indices needs to be explored. A wider range of chemical potentials is found by [@chlubaal12], allowing also for variations in the amplitude of primordial perturbations at very small scales, as motivated by some inflation models. -2.cm -0.7cm ![Spectrum of dipole (in equivalent thermodynamic, or CMB, temperature) expressed as the difference between that produced by a distorted spectrum and that corresponding to the blackbody at the current temperature $T_0$. Thick solid lines (or thin three dots-dashes) correspond to positive (or negative) values. [*Left*]{}: the case of BE distortions for $\mu_0= -2.8\times10^{-9}$ (representative of adiabatic cooling; green dots, note the opposite signs with respect to the cases with positive $\mu_0$), $\mu_0= 1.4\times10^{-5}$, $1.4\times10^{-6}$ (representative of improvements with respect to FIRAS upper limits), $\mu_0= 1.12\times10^{-7}$, $2.8\times10^{-8}$, and $1.4\times10^{-9}$ (representative of primordial adiabatic perturbation dissipation). [*Right*]{}: the case of Comptonization distortions for $u=2\times10^{-6}$ (upper curves) and $u=10^{-7}$ (lower curves), representative of imprints by astrophysical or minimal reionization models, respectively. []{data-label="fig:Dip_C_BE"}](DipAmpBE-BB.pdf "fig:"){width="62.00000%"} -1.8cm ![Spectrum of dipole (in equivalent thermodynamic, or CMB, temperature) expressed as the difference between that produced by a distorted spectrum and that corresponding to the blackbody at the current temperature $T_0$. Thick solid lines (or thin three dots-dashes) correspond to positive (or negative) values. [*Left*]{}: the case of BE distortions for $\mu_0= -2.8\times10^{-9}$ (representative of adiabatic cooling; green dots, note the opposite signs with respect to the cases with positive $\mu_0$), $\mu_0= 1.4\times10^{-5}$, $1.4\times10^{-6}$ (representative of improvements with respect to FIRAS upper limits), $\mu_0= 1.12\times10^{-7}$, $2.8\times10^{-8}$, and $1.4\times10^{-9}$ (representative of primordial adiabatic perturbation dissipation). [*Right*]{}: the case of Comptonization distortions for $u=2\times10^{-6}$ (upper curves) and $u=10^{-7}$ (lower curves), representative of imprints by astrophysical or minimal reionization models, respectively. []{data-label="fig:Dip_C_BE"}](DipAmpC-BB.pdf "fig:"){width="62.00000%"} -6.5cm Cosmological reionization associated with the early stages of structure and star formation is an additional source of photon and energy production. This mechanism induces electron heating that is responsible for Comptonization distortions [@1972JETP...35..643Z]. The characteristic parameter for describing this effect is $$\label{eq:uComp} u(t)=\int_{t_{\rm i}}^{t} [(\phi-\phi_{\rm i})/\phi] (k_{\rm B}T_{\rm e}/m_{\rm e}c^2) n_{\rm e} \sigma_{\rm T} c dt \, .$$ In the case of small energy injections and integrating over the relevant epochs then $u \simeq (1/4) \Delta\varepsilon/\varepsilon_{\rm i}$. In Eq. , $\phi_{\rm i} = \phi(z_{\rm i}) = (1+ \Delta \epsilon/ \varepsilon_{\rm i})^{-1/4} \simeq 1-u$ is the ratio between the equilibrium matter temperature and the radiation temperature evaluated at the beginning of the heating process (i.e., at $z_{\rm i}$). The distorted spectrum is then $$\label{eq:etaC} \eta_{\rm C} \simeq \eta_{\rm i} + u {x / \phi_{\rm i} {\rm exp}(x/\phi_{\rm i}) \over [{\rm exp}(x/\phi_{\rm i}) - 1]^{2}} \left ( {x/\phi_{\rm i} \over {\rm tanh}(x/2\phi_{\rm i}) - 4} \right ),$$ where $\eta_{\rm i}$ is the initial photon occupation number (before the energy injection).[^12] Typically, reionization induces Comptonization distortions with [*minimal*]{} values $u \simeq 10^{-7}$ [@buriganaetal08]. In addition to this, the variety of energy injections expected in astrophysical reionization models, including: energy produced by nuclear reactions in stars and/or by nuclear activity that mechanically heats the intergalactic medium (IGM); super-winds from supernova explosions and active galactic nuclei; IGM heating by quasar radiative energy; and shocks associated with structure formation. Together these induce much larger values of $u$ ($\simeq \hbox{several}\times 10^{-6}$) [@2000PhRvD..61l3001R; @2015PhRvL.115z1301H], i.e., not much below the current FIRAS 95% CL upper limit of $|u|<1.5\times 10^{-5}$ [@1996ApJ...473..576F]. Free-free distortions associated with reionization [@2014MNRAS.437.2507T] are instead more relevant at the lowest frequencies (below $10\,$GHz), and thus we do not consider them in this paper. We could also consider the possible presence of unconventional heating sources. Decaying and annihilating particles during the pre-recombination epoch may affect the CMB spectrum, with the exact distorted shape depending on the process timescale and, in some cases, being different from the one produced by energy release. This is especially interesting for decaying particles with lifetimes $t_{X} \simeq $few$\times 10^{8}$–$10^{11}$sec [@1993PhRvL..70.2661H; @daneseburigana94; @2013MNRAS.436.2232C]. Superconducting cosmic strings would also produce copious electromagnetic radiation, creating CMB spectral distortion shapes [@ostrikerthompson87] that would be distinguishable with high accuracy measurements. Evaporating primordial black holes provide another possible source of energy injection, with the shape of the resulting distortion depending on the black hole mass function [@carretal2010]. CMB spectral distortion measurements could also be used to constrain the spin of non-evaporating black holes [@paniloeb2013]. The CMB spectrum could additionally set constraints on the power spectrum of small-scale magnetic fields [@jedamziketal2000], the decay of vacuum energy density [@BartlettSilk1990], axions [@ejllidolgov2014], and other new physical processes. The CIB dipole {#sect:DipCIB} -------------- Multi-frequency measurements of the dipole spectrum will allow us to constrain the CIB intensity spectrum [@DaneseDeZotti1981; @Balashev2015]. The spectral shape of the CIB is hard to determine directly because it requires absolute intensity measurements, which are also compromised by Galactic and other foregrounds. Although the dipole amplitude is about $10^{-3}$ of the monopole, its spatial form is already known and hence this indirect route may provide the most robust measurements of the CIB in the future. Fig. \[fig:dipole\] shows the CIB dipole spectrum computed according to Eq. , using the analytic representation of the CIB spectrum (observed at present time) given in Ref. [@Fixsen:1998kq]: $$\label{eq:eta_CIB} \eta_{\rm CIB}={c^2\over 2 h \nu^3} I_{\rm CIB}(\nu) = I_0 \left({k_{\rm B}T_{\rm CIB} \over h \nu_0}\right)^{k_F} {x^{k_{\rm F}}_{\rm CIB}\over \exp(x_{\rm CIB})-1}\, , $$ where $T_{\rm CIB}=(18.5\pm1.2)\,$K, $x_{\rm CIB}=h\nu/k_{\rm B}T_{\rm CIB}= 7.78(\nu/\nu_0)$, $\nu_0\simeq 3\times 10^{12}\,$Hz and $k_{\rm F}= 0.64 \pm 0.12$. Here $I_0$ sets the CIB spectrum amplitude, its best-fit value being $1.3 \times 10^{-5}$ [@Fixsen:1998kq]. On the other hand, the uncertainty of the CIB amplitude is currently quite high, with $I_0$ only known to a 1$\sigma$ accuracy of about 30%. The CIB dipole amplitude, in terms of thermodynamic temperature, increases rapidly with frequency, reaching $257\,\mu$K (or $652\,\hbox{Jy}\,\hbox{sr}^{-1}$) at 600GHz and $420\,\mu$K (or $1306\,\hbox{Jy}\,\hbox{sr}^{-1}$) at 800GHz. The measurement of the CIB dipole amplitude will be dependent on systematic effects from the foreground Galaxy subtraction, which has a similar spectrum to the CIB [@2011ApJ...734...61F]. Although the calibration of the dipole signal at different frequencies is not trivial (since the orbital part of the dipole will be used for calibration), the [*Planck*]{} experience is that with sufficient care the limitation is removal of the Galactic signals, not calibration uncertainty. Hence the CIB dipole should be clearly detectable by in its highest frequency bands. Such a detection will provide important constraints on the CIB intensity; its amplitude uncertainty constitutes a major current limitation in our understanding of the dust-obscured star-formation phase of galaxy evolution. -1.5cm ![Expected behaviour of the dipole spectrum. The upper lines show the spectrum of the (pure) CIB dipole, while the lower lines show the spectrum coming from the dipole pattern computed from the CIB distribution function added to the blackbody (at temperature $T_0$) distribution function, minus the dipole pattern computed by the blackbody distribution function. Thick solid lines (or thin three dots-dashes) correspond to positive (or negative) values. The analytic representation of the CIB spectrum by [@Fixsen:1998kq] is adopted here, considering the best-fit amplitude and the range of $\pm 1\,\sigma$.[]{data-label="fig:dipole"}](DipAmpCIB.pdf "fig:"){width="0.6\columnwidth"} -6.cm ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BB_060GHz_ell1.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BB_060GHz_ell2.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BB_060GHz_ell3.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_060GHz_ell1.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_145GHz_ell1.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_600GHz_ell1.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_060GHz_ell2.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_145GHz_ell2.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_600GHz_ell2.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_060GHz_ell3.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_145GHz_ell3.png "fig:"){width="\linewidth"} ![[*Top row*]{}: maps of the dipole, quadrupole and octupole computed assuming a CMB blackbody spectrum at the current temperature $T_0$, for reference. In all other cases we show the maps of the dipole ([*second row*]{}), quadrupole ([*third row*]{}), and octupole ([*bottom row*]{}) at three different frequencies (namely 60, 145, and 600GHz, from left to right), in terms of the difference between the pattern computed for a BE distortion with $\mu_0 = 1.5 \times 10^{-5}$ and that computed for a blackbody at the present-day temperature $T_0$.[]{data-label="fig:map_BE"}](BE_600GHz_ell3.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_060GHz_ell1.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_145GHz_ell1.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_600GHz_ell1.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_060GHz_ell2.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_145GHz_ell2.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_600GHz_ell2.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_060GHz_ell3.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_145GHz_ell3.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of a Comptonization distortion with $ u = 2 \times 10^{-6}$.[]{data-label="fig:map_C"}](C_600GHz_ell3.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_060GHz_ell1.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_145GHz_ell1.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_600GHz_ell1.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_060GHz_ell2.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_145GHz_ell2.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_600GHz_ell2.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_060GHz_ell3.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_145GHz_ell3.png "fig:"){width="\linewidth"} ![The same as in Fig. \[fig:map\_BE\], but for the case of the CIB with amplitude set at the best-fit value found by FIRAS. More precisely, we display the temperature pattern of the CIB distribution function added to the blackbody one, minus the temperature pattern coming from the blackbody.[]{data-label="fig:map_CIB"}](CIB_600GHz_ell3.png "fig:"){width="\linewidth"} Beyond the dipole {#sect:LowEll} ----------------- A generalization of the considerations of the previous section allows us to evaluate the effect of peculiar velocity on the whole sky. To achieve this, we generate maps and, using the Lorentz-invariance of the distribution function, we can include all orders of the effect, coupling them with the geometrical properties induced at low multipoles. To compute the maps at each multipole[^13] $\ell \ge 1$, we first derive the maps at all angular scales, both for the distorted spectra and for the blackbody at the current temperature $T_0$. From the dipole direction found in the [*Planck*]{} (HFI+LFI combined) 2015 release and defining the motion vector of the observer, we produce the maps in a pixelization scheme at a given observational frequency $\nu$ by computing the photon distribution function, $\eta^{\rm BBdist}$, for each considered type of spectrum at a frequency given by the observational frequency $\nu$ but multiplied by the product $(1 - \hat{n} \cdot \vec{\beta})/(1 - \vec{\beta}^2)^{1/2}$ to account for all the possible sky directions with respect to the observer peculiar velocity. Here the notation ‘BBdist’ stands for BB, CIB, BE, or Comptonization (C). Hence, the map of the observed signal in terms of thermodynamic temperature is given by generalising Eq. : $$T_{\rm therm}^{\rm BB/dist} (\nu, {\hat{n}}, \vec{\beta}) = \frac{xT_{0}} {{\rm{log}}(1 / (\eta(\nu, {\hat{n}}, \vec{\beta}))^{\rm BB/dist} + 1) } \, , \label{eq:eta_boost}$$ where $\eta(\nu, {\hat{n}}, \vec{\beta}) = \eta(\nu')$ with $\nu' = \nu (1 - {\hat{n}} \cdot \vec{\beta})/(1 - \vec{\beta}^2)^{1/2}$. We adopt the [HEALPix]{} pixelization scheme to discretise the sky at the desired resolution. We decompose the maps into spherical harmonics and then regenerate them considering the $a_{lm}$ only up to a desired multipole $\ell_{\rm{max}}$. We start setting $\ell_{\rm{max}}= 5$ and then iterate the process with a decreasing $\ell_{\rm{max}}$. We produce maps containing the power at a single multipole by taking the difference of the map at $\ell_{\rm{max}}$ from the map at $\ell_{\rm{max}-1}$. We then compute the difference of maps having specific spectral distortions from the purely blackbody maps. As seen in Figs. \[fig:map\_BE\]–\[fig:map\_CIB\], the expected signal is important for the dipole, can be considerable for the quadrupole and, depending on the distortion parameters, still not negligible for the octupole (although this will depend on the amplitude relative to experimental noise levels, as we discuss below). For higher-order multipoles, the signal is essentially negligible. Note that the maps present a clear and obvious symmetry with respect to the axis of the observer’s peculiar velocity.[^14] This is simply due to the angular dependence in Eq. . For coordinates in which the positive $z$-axis is aligned with the dipole, the only angular dependence comes from $\hat{n}\cdot\vec{\beta} \equiv \beta \cos{\theta_{\rm d}}$. In terms of the spherical harmonic expansion, this implies that higher-order multipoles will appear as polynomial functions of $\cos{\theta_{\rm d}}$, with different frequency-dependent factors depending on the specific type of spectral distortion being considered. In the above considerations we assumed that each multipole pattern can be isolated from that of the other multipoles. In reality, a certain leakage is expected (particularly between adjacent multipoles), especially as a result of masking for foregrounds. The sources of astrophysical emission are highly complex, and their geometrical properties mix with their frequency behaviour. Furthermore, in real data analysis, there is an interplay between the determination of the calibration and zero levels of the maps, and this issue is even more critical when data in different frequency domains are used to improve the component-separation process. The analysis of these aspects is outside the scope of the present paper, but deserves further investigation. ![Angular power spectrum of the dipole map, derived from the difference between distorted spectra and the current blackbody spectrum versus CORE sensitivity. The CORE white noise power spectrum (independent of multipole, shown as the black upper solid curve and with diamonds for different frequency channels) and its rms uncertainty (for $\ell = 1$, using dots, and for $\ell = 2$, using dashes) are plotted in black. The cross (asterisk) displays aggregated CORE noise from all channels (up to 220GHz). We shown also for comparison the LiteBIRD white noise power spectrum (red solid curve and diamonds for different frequency channels). [*Left*]{}: BE distortions for $\mu_0= -2.8\times10^{-9}$ (representative of adiabatic cooling), $\mu_0= 1.4\times10^{-5}$, $1.4\times10^{-6}$ (representative of improvements with respect to FIRAS upper limits), $\mu_0= 1.12\times10^{-7}$, $2.8\times10^{-8}$, and $1.4\times10^{-9}$ (representative of primordial adiabatic perturbation dissipation). For $\mu_0= 1.4\times10^{-5}$ we show also the angular power spectrum of the quadrupole map. [*Right*]{}: Comptonization distortions for $u=2\times10^{-6}$ (upper solid curve for the dipole map and bottom dashed curve for the quadrupole map) and $u=10^{-7}$ (lower solid curve for the dipole map), representative of imprints by astrophysical and minimal reionization models, respectively. []{data-label="fig:APS_C_BE_sens"}](cl_sens_use_newOK_ell_1_2_BE.pdf "fig:"){width="49.00000%"} ![Angular power spectrum of the dipole map, derived from the difference between distorted spectra and the current blackbody spectrum versus CORE sensitivity. The CORE white noise power spectrum (independent of multipole, shown as the black upper solid curve and with diamonds for different frequency channels) and its rms uncertainty (for $\ell = 1$, using dots, and for $\ell = 2$, using dashes) are plotted in black. The cross (asterisk) displays aggregated CORE noise from all channels (up to 220GHz). We shown also for comparison the LiteBIRD white noise power spectrum (red solid curve and diamonds for different frequency channels). [*Left*]{}: BE distortions for $\mu_0= -2.8\times10^{-9}$ (representative of adiabatic cooling), $\mu_0= 1.4\times10^{-5}$, $1.4\times10^{-6}$ (representative of improvements with respect to FIRAS upper limits), $\mu_0= 1.12\times10^{-7}$, $2.8\times10^{-8}$, and $1.4\times10^{-9}$ (representative of primordial adiabatic perturbation dissipation). For $\mu_0= 1.4\times10^{-5}$ we show also the angular power spectrum of the quadrupole map. [*Right*]{}: Comptonization distortions for $u=2\times10^{-6}$ (upper solid curve for the dipole map and bottom dashed curve for the quadrupole map) and $u=10^{-7}$ (lower solid curve for the dipole map), representative of imprints by astrophysical and minimal reionization models, respectively. []{data-label="fig:APS_C_BE_sens"}](cl_sens_use_newOK_ell_1_2_C.pdf "fig:"){width="49.60000%"} ![The same as in Fig. \[fig:APS\_C\_BE\_sens\], but for the CIB (assuming the model from [@Fixsen:1998kq]). Also shown is the quadrupole signal (dashes). The different values of $I_0$ in Eq.  are the best-fit value and deviations by $\pm1\,\sigma$.[]{data-label="fig:APS_CIB_sens"}](cl_sens_use_newOK_ell_1_2_C-BE.pdf){width="101.00000%"} ![The same as in Fig. \[fig:APS\_C\_BE\_sens\], but for the CIB (assuming the model from [@Fixsen:1998kq]). Also shown is the quadrupole signal (dashes). The different values of $I_0$ in Eq.  are the best-fit value and deviations by $\pm1\,\sigma$.[]{data-label="fig:APS_CIB_sens"}](cl_sens_use_newOK_ell_1_2_CIB.pdf){width="92.00000%"} Detectability {#sect:DiffCMB_SN} ------------- Here we discuss the detectability of the dipolar and quadrupolar signals introduced in Sects. \[sect:DipCMB\]–\[sect:LowEll\]. To this end we compare the dipole signal with the noise dipole as a function of frequency. Note that since the prediction includes the specific angular dependence of the dipole, there is no cosmic-variance related component in the noise. The noise for each frequency is determined by Table \[tab:CORE-bands\], assuming full-sky coverage for simplicity.[^15] In Fig. \[fig:APS\_C\_BE\_sens\] we show the dipole signal for BE and Comptonization distortions (left and right, respectively), defined as the temperature dipole coming from Eq.  subtracted from the CMB dipole (shown as coloured lines). In black we show the CORE noise as a function of frequency. For BE distortions, the signal is clearly above the CORE noise up to about 200GHz for $\mu_{0} \gsim 10^{-6}$ and comparable or slightly above the aggregated noise below about 100GHz for $\mu_{0} \gsim 10^{-7}$, while for Comptonization distortions, the signal is clearly above the noise up to around 500GHz for $u \gsim 2 \times 10^{-6}$ and comparable to or above the noise between approximately 100GHz and 300GHz for $u \gsim 10^{-7}$. The analogous analysis for the quadrupole (shown for simplicity only for the largest values of $\mu_0$ and $u$) shows that, for CORE sensitivity, noise dominates at any frequency for CMB spectral distortion parameters compatible with FIRAS limits, thus experiments beyond CORE are needed to use the quadrupole pattern to infer constraints on CMB spectral distortions. In Fig. \[fig:APS\_CvsBE\_sens\] we show the dipole signal of the difference between Comptonization and BE distortion maps. In Fig. \[fig:APS\_CIB\_sens\] we show the size of the dipole signal (the quadrupole is shown as dashed curves) for the CIB (where we have removed the CMB dipole) compared to noise. The signal is always above the noise except at about 100GHz. Due to the large uncertainty in the amplitude of the CIB spectra ($I_0$), we show also deviations of $\pm 1\,\sigma$ from the best-fit value of $1.3\times10^{-5}$ (as well as their difference from the best fit). The signal is orders of magnitude above the noise at high frequencies and moreover, the quadrupole is above the noise at frequencies greater than about 400GHz, although it is always much smaller than the dipole (since it is suppressed by an extra factor of $\beta$). Comparing Fig. \[fig:APS\_CIB\_sens\] with Fig. \[fig:APS\_C\_BE\_sens\], it is evident that the dipole power expected from the CIB is above those predicted for CMB spectral distortions at $\nu \gsim 200$GHz for the classes of processes and parameter values discussed here. Since the dependence of the quoted power on the CMB spectral distortion parameter is quadratic, the above statement does not hold for larger CMB distortions, even just below the FIRAS limits. Although they are not predicted by standard scenarios, they may be generated by unconventional dissipation processes, such those discussed at the end of Sect. \[sect:DipCMB\], according to their characteristic parameters. We computed for comparison the power spectrum sensitivity of LiteBIRD (see Table \[tab:LiteBIRD-bands\]): it is similar to that of CORE around 300GHz and significantly worse at $\nu \lsim 150$GHz, a range suitable in particular for BE distortions. As discussed in Sects. \[sec:dipideal\] and \[sec:dipsyst\], resolution is important to achieve the sky sampling necessary for ultra-accurate dipole analysis, thus adopting a resolution changing from a range of $\simeq 2$–18 arcmin to a range of $\simeq 0.5^{\circ}$–$1.5^{\circ}$ is certainly critical. Furthermore, the number of frequency channels is relevant, in particular (see next section) when one compares between pairs of frequencies, the number of which scales approximately as the square of the number of frequency channels. In addition, a large number of frequency channels and especially the joint analysis of frequencies around 300GHz and above 400GHz (not foreseen in LiteBIRD) is crucial for separating the various types of signals, and, in particular, to accurately control the contamination by Galactic dust emission. The analysis carried out here will be extended to include all frequency information in the following section. This will also include a discussion of the impact of residual foregrounds. Simulation results for CMB spectral distortions and CIB intensity {#sec:dip_spec_sim} ================================================================= In order to quantify the ideal CORE sensitivity to measure spectral distortion parameters and the CIB amplitude, we carried out some detailed simulations. The idea here is to simulate the sky signal assuming a certain model and to quantify the accuracy level at which (in the presence of noise and of potential residuals) we can recover the key input parameters. We consider twelve reference cases, physically or observationally motivated, based on considerations and works quoted in Sect. \[sect:DipCIB\] (cases 2–4) and Sect. \[sect:DipCMB\] (cases 8–12), namely: 0.5cm \(1) a (reference) blackbody spectrum defined by $T_0$;\ (2) a CIB spectrum at the FIRAS best-fit amplitude;\ (3) a CIB spectrum at the FIRAS best-fit amplitude plus $1\,\sigma$ error;\ (4) a CIB spectrum at the FIRAS best-fit amplitude minus $1\,\sigma$ error;\ (5) a BE spectrum with $\mu_0 = 1.12 \times 10^{-7}$, representative of a distortion induced by damping of primordial adiabatic perturbations in the case of relatively high power at small scales;\ (6) a BE spectrum with $\mu_0 = 1.4 \times 10^{-5}$, a value 6.4 times smaller than the FIRAS $95\,\%$ upper limits;\ (7) a BE spectrum with $\mu_0 = 1.4 \times 10^{-6}$, a value 64 times smaller than the FIRAS $95\,\%$ upper limits;\ (8) a BE spectrum with $\mu_0 = 1.4 \times 10^{-9}$, representative of the typical minimal distortion induced by the damping of primordial adiabatic perturbations;\ (9) a BE spectrum with $\mu_0 = 2.8 \times 10^{-8}$, representative of the typical distortion induced by damping of primordial adiabatic perturbations;\ (10) a BE spectrum with $\mu_0 = -2.8 \times 10^{-9}$, representative of the typical distortion induced by BE condensation (adiabatic cooling);\ (11) a Comptonised spectrum with $u = 10^{-7}$, representative of minimal reionization models;\ (12) a Comptonised spectrum with $u = 2 \times 10^{-6}$, representative of typical astrophysical reionization models.\ For each model listed we generate both an ideal sky (the “prediction”) and a sky with noise realizations (“simulated data”) according to the sensitivity of CORE (see Table \[tab:CORE-bands\]), at each of its 19 frequency channels. For a suitable number of cases we repeated the analysis working with maps simply containing only the dipole term and verified that the major contribution to the significance comes from the dipole, i.e., the quadrupole (the only other possibly relevant term) contributes almost negligibly,[^16] in agreement with Sect. \[sect:DiffCMB\_SN\]. For the sake of simplicity our noise realizations assume Gaussian white noise. Our simulation set consists of 10 realizations for each of the 19 CORE frequencies (giving 190 independent noise realizations). These are generated at $N_{\rm side} = 64$ (roughly $1^\circ$ resolution). We will also consider the inclusion of certain systematics in the following subsections. We then compare each theoretical prediction with all maps of our simulated data. We calculate $\Delta \chi^2$ for each combination, summarised in a $12\times12$ matrix, quantifying the significance level at which each model can be potentially detected or ruled out. We report our results in terms of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$, which directly gives the significance in terms of $\sigma$ levels, since we only consider a single parameter at a time.[^17] We perform the $\sqrt{\Delta \chi^2}$ analysis for three different approaches: 1. using each of the 19 frequency channels, assuming they are independent; 2. using the 171 ($19\times18/2$) combinations coming from the differences of the maps from pairs of frequency bands; 3. combining cases (a) and (b) together. When differences of maps from pairs of frequency bands are included in the analysis, in the corresponding contributions to the $\chi^2$ the variance comes from the sum of the variances at the two considered frequencies. Approach (a) essentially compares the amplitude of dipole of a distorted spectrum with that of the blackbody, being so sensitive to the overall difference between the two cases, while approach (b) compares the dipole signal at different frequencies for each type of spectrum, being so sensitive to its slope. Ideal case: perfect calibration and foreground subtraction {#sec:MC_ideal} ---------------------------------------------------------- [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $3650$ $4740$ $2540$ $1.63$ $175.$ $17.9$ $0.0998$ $0.532$ $-0.137$ $7.95$ $154.$ \[\] $(2)$ $ 3650$ $0$ $1100$ $1110$ $3650$ $3640$ $3650$ $3650$ $3650$ $3650$ $3650$ $3620$ \[\] $(3)$ $ 4740$ $1100$ $0$ $2210$ $4740$ $4740$ $4740$ $4740$ $4740$ $4740$ $4740$ $4710$ \[\] $(4)$ $ 2540$ $1110$ $2200$ $0$ $2540$ $2540$ $2540$ $2540$ $2540$ $2540$ $2540$ $2510$ \[\] $(5)$ $0.403$ $3650$ $4740$ $2540$ $0$ $174.$ $16.5$ $0.384$ $-0.111$ $0.445$ $6.83$ $153.$ \[\] $(6)$ $174.$ $3640$ $4740$ $2540$ $173.$ $0$ $157.$ $174.$ $174.$ $174.$ $168.$ $108.$ \[\] $(7)$ $17.0$ $3650$ $4740$ $2540$ $15.6$ $158.$ $0$ $17.0$ $16.6$ $17.0$ $11.8$ $140.$ \[\] $(8)$ $-0.0975$ $3650$ $4740$ $2540$ $1.62$ $175.$ $17.9$ $0$ $0.514$ $-0.165$ $7.93$ $154.$ \[\] $(9)$ $-0.346$ $3650$ $4740$ $2540$ $1.28$ $175.$ $17.6$ $-0.342$ $0$ $-0.352$ $7.66$ $153.$ \[\] $(10)$ $0.142$ $3650$ $4740$ $2540$ $1.67$ $175.$ $17.9$ $0.176$ $0.567$ $0$ $7.98$ $154.$ \[\] $(11)$ $ 7.25$ $3650$ $4740$ $2540$ $6.22$ $169.$ $12.8$ $7.23$ $6.98$ $7.27$ $0$ $146.$ \[\] $(12)$ $ 153.$ $3620$ $4710$ $2510$ $152.$ $108.$ $140.$ $153.$ $153.$ $153.$ $145.$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Average values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ from a Monte Carlo simulation at $N_{\rm side} = 64$, full-sky coverage, adopting perfect foreground subtraction and calibration, and considering each of the 19 frequency channels.[]{data-label="MC_corr_avg_ideal"} [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $14000$ $18200$ $9740$ $1.96$ $271.$ $27.8$ $0.124$ $0.666$ $-0.170$ $-0.229$ $43.7$ \[\] $(2)$ $14000$ $0$ $4180$ $4240$ $14000$ $14100$ $14000$ $14000$ $14000$ $14000$ $14000$ $14000$ \[\] $(3)$ $18200$ $4190$ $0$ $8430$ $18200$ $18200$ $18200$ $18200$ $18200$ $18200$ $18200$ $18200$ \[\] $(4)$ $9740$ $4240$ $8420$ $0$ $9740$ $9810$ $9750$ $9740$ $9740$ $9740$ $9740$ $9730$ \[\] $(5)$ $0.0596$ $14000$ $18200$ $9740$ $0$ $269.$ $25.6$ $0.0456$ $-0.249$ $0.0910$ $1.44$ $45.7$ \[\] $(6)$ $ 269.$ $14000$ $18200$ $9810$ $267.$ $0$ $242.$ $269.$ $269.$ $269.$ $271.$ $312.$ \[\] $(7)$ $ 26.0$ $14000$ $18200$ $9750$ $23.8$ $244.$ $0$ $26.0$ $25.4$ $26.1$ $28.1$ $69.6$ \[\] $(8)$ $-0.122$ $14000$ $18200$ $9740$ $1.93$ $271.$ $27.8$ $0$ $0.644$ $-0.205$ $-0.209$ $43.7$ \[\] $(9)$ $-0.386$ $14000$ $18200$ $9740$ $1.51$ $271.$ $27.2$ $-0.384$ $0$ $-0.391$ $0.228$ $44.2$ \[\] $(10)$ $0.178$ $14000$ $18200$ $9740$ $2.00$ $271.$ $27.9$ $0.220$ $0.708$ $0$ $-0.267$ $43.6$ \[\] $(11)$ $ 2.66$ $14000$ $18200$ $9740$ $4.69$ $273.$ $29.9$ $2.68$ $3.11$ $2.62$ $0$ $41.4$ \[\] $(12)$ $ 46.4$ $14000$ $18200$ $9730$ $48.4$ $314.$ $72.0$ $46.4$ $46.9$ $46.4$ $44.1$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_avg\_ideal\], but considering all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_cross_avg_ideal"} [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $14400$ $18800$ $10100$ $2.67$ $323.$ $33.1$ $0.157$ $0.844$ $-0.214$ $7.83$ $160.$ \[\] $(2)$ $14500$ $0$ $4320$ $4380$ $14500$ $14500$ $14500$ $14500$ $14500$ $14500$ $14500$ $14400$ \[\] $(3)$ $18800$ $4330$ $0$ $8710$ $18800$ $18800$ $18800$ $18800$ $18800$ $18800$ $18800$ $18800$ \[\] $(4)$ $10100$ $4380$ $8710$ $0$ $10100$ $10100$ $10100$ $10100$ $10100$ $10100$ $10100$ $10100$ \[\] $(5)$ $0.369$ $14400$ $18800$ $10100$ $0$ $320.$ $30.5$ $0.346$ $-0.105$ $0.413$ $7.24$ $159.$ \[\] $(6)$ $ 321.$ $14500$ $18800$ $10100$ $318.$ $0$ $288.$ $321.$ $320.$ $321.$ $319.$ $330.$ \[\] $(7)$ $ 31.1$ $14500$ $18800$ $10100$ $28.5$ $290.$ $0$ $31.0$ $30.4$ $31.1$ $30.5$ $157.$ \[\] $(8)$ $-0.153$ $14400$ $18800$ $10100$ $2.64$ $323.$ $33.1$ $0$ $0.816$ $-0.259$ $7.82$ $160.$ \[\] $(9)$ $-0.537$ $14400$ $18800$ $10100$ $2.07$ $322.$ $32.4$ $-0.532$ $0$ $-0.547$ $7.61$ $160.$ \[\] $(10)$ $0.224$ $14400$ $18800$ $10100$ $2.73$ $323.$ $33.1$ $0.277$ $0.900$ $0$ $7.85$ $160.$ \[\] $(11)$ $ 7.92$ $14400$ $18800$ $10100$ $8.03$ $321.$ $32.5$ $7.92$ $7.87$ $7.93$ $0$ $152.$ \[\] $(12)$ $ 160.$ $14400$ $18800$ $10000$ $160.$ $332.$ $157.$ $160.$ $160.$ $160.$ $152.$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_avg\_ideal\], but considering each of the 19 frequency channels independently and all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_corr_cross_avg_ideal"} Tables \[MC\_corr\_avg\_ideal\] and \[MC\_corr\_rms\_ideal\] (and Tables \[MC\_cross\_avg\_ideal\] and \[MC\_cross\_rms\_ideal\], Tables \[MC\_corr\_cross\_avg\_ideal\] and \[MC\_corr\_cross\_rms\_ideal\], respectively) report the results of approach a (approach b, and c, respectively) in terms of average and rms of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ (see Appendix \[app\_rms\_ideal\]). We find that, in general, the analysis of the difference of pairs of frequency channels (approach b) tends to substantially increase the significance of the recovery of the CIB amplitude, which is due to the very steep frequency shape of the CIB dipole spectrum. For the opposite reason, the same does not occur in general for CMB distortion parameters, and, in particular, approach (b) can make the recovery of the Comptonization distortion more difficult. These results are in agreement with the shapes displayed in Figs. \[fig:APS\_C\_BE\_sens\], \[fig:APS\_CvsBE\_sens\], and \[fig:APS\_CIB\_sens\]. It is important to note that, in general, the rms values found in approach (b) are larger than those found in approach (a), seemingly relatively more stable. We interpret this as an effect of larger susceptibility of approach (b) to realization combinations. On the other hand, for the estimation of the CIB amplitude this rms amplification does not spoil the improvement in significance. We find that combining the two approaches, as in (c), typically results in an overall advantage, with an improvement in significance larger than the possible increasing of the quoted rms. We anticipate that these results will still be valid when including potential residuals, as discussed below. We remark that in the present analysis both pure theoretical maps and maps polluted with noise are pixelised in the same way. So, the sampling problem discussed in Sects. \[sec:dipideal\] and \[sec:dipsyst\] is automatically by-passed. This is not a limitation for the present analysis, given the high resolution achieved by CORE, and because it is clear that we could in principle perform our simulations at the desired resolution. Working at roughly $1^\circ$ resolution makes our analysis feasible without supercomputing facilities, with no significant loss of information. Nonetheless, we also report some results carried out at higher resolution. In particular, in Appendix \[ideal\_highres\] we present results of the analysis repeated at $N_{\rm side} = 512$ (i.e., at about 7 arcmin resolution), for a single realization. The results are fully compatible, within the statistical variance, with those derived working at $N_{\rm side} = 64$. The matrices reported in each of these tables perhaps require a little more explanation. Firstly, we should point out that the diagonals are zero by construction. We found that the reduced $\chi^2$ ($\chi^2_{\rm r} = \chi^2 / (n_{\rm d}-1)$, where $n_{\rm d}$ is the global number of data being treated and we are considering the estimate of a single parameter, namely CMB distortion or CIB amplitude), is always extremely close to unity, which is an obvious validation cross-check. Note that, in principle, when potential residuals are included, one should specify the variance pixel-by-pixel in the estimation of $\chi^2$.[^18] This requires a precise local characterization of residuals. While this can easily be included by construction in our analyses, we explicitly avoid implementing this in the $\chi^2$ analysis, but instead perform our forecasts assuming knowledge of only the average level of the residuals in the sky region being considered. Secondly, we note that the matrices are not perfectly symmetric, due to the cross-terms in the squares (from noise and signal) entering into the $\chi^2$. Thirdly, the off-diagonal terms are sometimes negative, but with absolute values compatible with the quoted rms. These second and third effects are clearly statistical in nature. The results found in this section (see also Appendix \[ideal\_highres\]) identify the ideal sensitivity target for CMB spectral distortion parameters and CIB amplitude that are achievable from the dipole frequency behaviour. Elements[^19] (2:4, 2:4) of the matrix quantify the sensitivity to the CIB amplitude. Comparison with FIRAS in terms of the $\sigma$ level of significance can be extracted directly from the tables; the ideal improvement ranges from a factor of about 1000 to 4000. The ideal improvement found for CMB spectral distortion parameters is also impressive. Elements (1, 5:10) and (5:10, 1) and elements (1, 11:12) and (11:12, 1) refer to comparisons between the blackbody and BE and Comptonization distortions, respectively. The comparison with FIRAS is simply quoted by the element of the matrix of the table multiplied by the ratio between the FIRAS $1\,\sigma$ upper limit on $\mu_0$ or $u$ and the distortion parameter value considered in the table. The sensitivity on $u$ is clearly enough to disentangle between minimal models of reionization and a variety of astrophysical models that predict larger amounts of energy injection by various types of source. The ideal improvement with respect to FIRAS limits is about 500–600. The level of (negative) BE distortions is much lower, and the same holds also for BE distortions predicted for the damping of primordial adiabatic perturbations. Only weak, tentative constraints on models with high power at small scales could be set with this approach, for a mission with the sensitivity of CORE. Nonetheless, the ideal improvement with respect to FIRAS limits on BE distortions lies in the range 600–1000. The other elements of the matrix refer to the comparison of distorted spectra; note in particular the elements (6:7, 11:12) and (11:12, 6:7) that show how Comptonization distortions can be distinguished from BE distortions, for the two larger values considered for $\mu_0$, as suggested by Fig. \[fig:APS\_CvsBE\_sens\]. Including potential foreground and calibration residuals -------------------------------------------------------- We expect that potential residuals from imperfect foreground subtraction and calibration may affect the results presented in the previous section, depending on their level. To assess this, we have carried out a wide set of simulations in order to quantify the accuracy in recovering the CMB distortion parameters and CIB amplitude under different working assumptions. We first perform simulations adopting $E_{\rm for} = 10^{-2}$ and $E_{\rm cal} = 10^{-4}$ (defined by the parametric model introduced in Sect. \[sec:res\_mod\]) at $N_{\rm side} = 64$, and then add many tests exploring combinations of possible improvements in foreground characterization (assuming $E_{\rm for} = 10^{-3}$ or $E_{\rm for} = 10^{-2}$, but at larger $N_{\rm side}$), as well as different levels of calibration accuracy (including possible worsening at higher frequencies). [width=1]{} --------------------------- ------------ ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $13.7$ $17.7$ $9.61$ $-0.00676$ $10.4$ $0.833$ $-0.00264$ $-0.00996$ $ 0.00383$ $0.166$ $4.83$ \[\] $(2)$ $12.8$ $0$ $4.37$ $3.48$ $12.8$ $16.6$ $12.8$ $12.8$ $12.8$ $12.8$ $12.8$ $13.1$ \[\] $(3)$ $16.8$ $3.44$ $0$ $7.58$ $16.8$ $19.9$ $16.9$ $16.8$ $16.8$ $16.8$ $16.8$ $16.9$ \[\] $(4)$ $8.75$ $4.39$ $8.45$ $0$ $8.76$ $13.7$ $8.83$ $8.75$ $8.75$ $8.75$ $8.72$ $9.47$ \[\] $(5)$ $0.0391$ $13.7$ $17.7$ $9.61$ $0$ $10.3$ $0.686$ $0.0385$ $0.0307$ $0.0400$ $0.133$ $4.76$ \[\] $(6)$ $10.5$ $17.3$ $20.6$ $14.3$ $10.4$ $0$ $9.41$ $10.5$ $10.4$ $10.5$ $10.2$ $6.51$ \[\] $(7)$ $0.821$ $13.7$ $17.7$ $9.67$ $0.739$ $9.39$ $0$ $0.821$ $0.799$ $0.825$ $0.611$ $3.93$ \[\] $(8)$ $0.00270$ $13.7$ $17.7$ $9.61$ $-0.00701$ $10.4$ $0.830$ $0$ $-0.00967$ $ 0.00473$ $0.166$ $4.83$ \[\] $(9)$ $0.0139$ $13.7$ $17.7$ $9.61$ $ -0.0100$ $10.4$ $0.788$ $0.0135$ $0$ $0.0146$ $0.163$ $4.82$ \[\] $(10)$ $-0.00370$ $13.7$ $17.7$ $9.61$ $-0.00658$ $10.4$ $0.836$ $-0.00451$ $ -0.0102$ $0$ $0.166$ $4.84$ \[\] $(11)$ $0.0337$ $13.7$ $17.6$ $9.57$ $ -0.0402$ $10.2$ $0.505$ $0.0330$ $0.0210$ $0.0346$ $0$ $4.60$ \[\] $(12)$ $4.56$ $13.9$ $17.7$ $10.1$ $4.48$ $6.28$ $3.57$ $4.56$ $4.54$ $4.56$ $4.32$ $0$ \[\] --------------------------- ------------ ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Average values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ from a Monte Carlo simulation at $N_{\rm side} = 64$, with full-sky coverage, adopting $E_{\rm for} = 10^{-2}$ and $E_{\rm cal} = 10^{-4}$, and considering each of the 19 frequency channels.[]{data-label="MC_corr_avg"} [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $ 0$ $49.9$ $64.7$ $35.0$ $-0.224$ $6.67$ $-0.311$ $-0.0267$ $-0.117$ $0.0379$ $ 0.211$ $1.15$ \[\] $(2)$ $47.7$ $0$ $15.6$ $13.4$ $47.7$ $49.9$ $47.8$ $47.7$ $47.7$ $47.7$ $47.6$ $47.4$ \[\] $(3)$ $62.4$ $13.3$ $0$ $28.4$ $62.4$ $64.5$ $62.6$ $62.4$ $62.4$ $62.4$ $62.4$ $62.2$ \[\] $(4)$ $32.7$ $15.8$ $30.6$ $0$ $32.7$ $35.2$ $32.9$ $32.7$ $32.7$ $32.7$ $32.7$ $32.5$ \[\] $(5)$ $0.250$ $49.9$ $64.7$ $35.0$ $0$ $6.61$ $-0.324$ $ 0.248$ $ 0.214$ $ 0.253$ $ 0.314$ $1.21$ \[\] $(6)$ $8.51$ $52.3$ $66.9$ $37.7$ $8.44$ $0$ $7.71$ $8.51$ $8.48$ $8.51$ $8.54$ $9.31$ \[\] $(7)$ $1.15$ $50.1$ $64.9$ $35.2$ $1.08$ $5.84$ $0$ $1.15$ $1.13$ $1.15$ $1.20$ $2.06$ \[\] $(8)$ $ 0.0268$ $49.9$ $64.7$ $35.0$ $-0.223$ $6.67$ $-0.310$ $0$ $-0.114$ $0.0465$ $ 0.212$ $1.15$ \[\] $(9)$ $0.121$ $49.9$ $64.7$ $35.0$ $-0.196$ $6.66$ $-0.314$ $ 0.118$ $0$ $ 0.127$ $ 0.239$ $1.17$ \[\] $(10)$ $-0.0378$ $49.9$ $64.7$ $35.0$ $-0.227$ $6.67$ $-0.310$ $-0.0463$ $-0.122$ $0$ $ 0.208$ $1.15$ \[\] $(11)$ $-0.206$ $49.9$ $64.7$ $35.0$ $-0.289$ $6.71$ $-0.322$ $-0.207$ $-0.230$ $-0.203$ $0$ $1.11$ \[\] $(12)$ $-0.710$ $49.6$ $64.4$ $34.7$ $-0.706$ $7.40$ $-0.391$ $-0.709$ $-0.708$ $-0.711$ $-0.703$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_avg\], but considering all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_cross_avg"} [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $51.8$ $67.0$ $36.3$ $-0.191$ $12.4$ $ 0.142$ $-0.0230$ $-0.101$ $0.0327$ $ 0.309$ $5.10$ \[\] $(2)$ $49.4$ $0$ $16.2$ $13.9$ $49.4$ $52.6$ $49.5$ $49.4$ $49.4$ $49.4$ $49.3$ $49.2$ \[\] $(3)$ $64.6$ $13.7$ $0$ $29.4$ $64.7$ $67.5$ $64.8$ $64.7$ $64.7$ $64.6$ $64.6$ $64.4$ \[\] $(4)$ $33.9$ $16.4$ $31.8$ $0$ $33.9$ $37.7$ $34.1$ $33.9$ $33.9$ $33.9$ $33.9$ $33.9$ \[\] $(5)$ $0.222$ $51.8$ $67.1$ $36.3$ $0$ $12.3$ $0.0689$ $ 0.220$ $ 0.189$ $ 0.225$ $ 0.402$ $5.05$ \[\] $(6)$ $13.5$ $55.1$ $70.0$ $40.2$ $13.4$ $0$ $12.2$ $13.5$ $13.5$ $13.5$ $13.4$ $11.4$ \[\] $(7)$ $1.48$ $52.0$ $67.3$ $36.5$ $1.38$ $11.2$ $0$ $1.48$ $1.46$ $1.49$ $1.43$ $4.57$ \[\] $(8)$ $0.0231$ $51.8$ $67.0$ $36.3$ $-0.190$ $12.4$ $ 0.142$ $0$ $-0.0987$ $0.0401$ $ 0.309$ $5.10$ \[\] $(9)$ $0.105$ $51.8$ $67.0$ $36.3$ $-0.169$ $12.4$ $ 0.123$ $ 0.102$ $0$ $ 0.110$ $ 0.338$ $5.09$ \[\] $(10)$ $-0.0326$ $51.8$ $67.0$ $36.3$ $-0.193$ $12.4$ $ 0.145$ $-0.0399$ $-0.106$ $0$ $ 0.308$ $5.10$ \[\] $(11)$ $-0.143$ $51.7$ $67.0$ $36.3$ $-0.324$ $12.3$ $-0.0738$ $-0.147$ $-0.200$ $-0.136$ $0$ $4.86$ \[\] $(12)$ $4.35$ $51.5$ $66.8$ $36.2$ $4.26$ $9.77$ $3.06$ $4.35$ $4.33$ $4.35$ $4.10$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_avg\], but considering each of the 19 frequency channels independently and all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_corr_cross_avg"} ### Monte Carlo results at about $1^\circ$ resolution To understand the typical implications of different assumptions, we first perform a series of Monte Carlo simulations, identical to that described in Sect. \[sec:MC\_ideal\], but including potential foreground and calibration residuals, modelled according to Sect. \[sec:res\_mod\], assuming $E_{\rm for} = 10^{-2}$ and $E_{\rm cal} = 10^{-4}$. The main results (the average values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$) are presented in Tables \[MC\_corr\_avg\], \[MC\_cross\_avg\], and \[MC\_corr\_cross\_avg\], while the corresponding rms values are reported in Appendix \[app\_rms\_res\] (see Tables \[MC\_corr\_rms\], \[MC\_cross\_rms\], and \[MC\_corr\_cross\_rms\]). With these levels of potential residuals, the improvement with respect to FIRAS in the recovery of the CIB amplitude ranges from a factor of approximately 4 (with an rms of about 1 in the estimate of this improvement factor) for approach (a) to a factor of about 15 or 20 (with an rms of about 3) for approaches (b) and (c), respectively. The improvement found for the recovery of CMB spectral distortion parameters is also very promising. The sensitivity to $u$ improves with respect to FIRAS by a factor of $20$ (except for the less stable approach (b)), which is suitable for detecting reionization imprints (of the sort predicted in astrophysical reionization models) at about $5\,\sigma$, while the improvement on BE distortions is about a factor of 40 (approach (c)). Note that these results are derived considering the full sky, and thus we could expect to obtain improvements by applying masks to avoid regions with significant potential contamination, as discussed in the next section. ### Application of masks We performed some additional tests assuming $E_{\rm for} = 10^{-2}$ and $E_{\rm cal} = 10^{-4}$, but applying appropriate masks to the sky. Clearly, in this way we reduce the available statistical information (as we verified through tests carried out under ideal conditions of perfect foreground subtraction and calibration), but in realistic cases we may expect to improve the quality of results by reducing the impact of potential residuals. [width=1]{} --------------------------- ------------ ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $73.5$ $95.1$ $51.8$ $0.139$ $14.8$ $1.50$ $0.00844$ $ 0.0473$ $-0.0113$ $ -0.385$ $5.29$ \[\] $(2)$ $69.1$ $0$ $23.7$ $19.4$ $69.1$ $72.7$ $69.3$ $69.1$ $69.1$ $69.1$ $69.0$ $68.7$ \[\] $(3)$ $90.6$ $19.2$ $0$ $41.0$ $90.6$ $93.9$ $90.8$ $90.6$ $90.6$ $90.6$ $90.6$ $90.2$ \[\] $(4)$ $47.3$ $23.8$ $45.4$ $0$ $47.4$ $51.6$ $47.6$ $47.3$ $47.3$ $47.3$ $47.3$ $47.0$ \[\] $(5)$ $0.0920$ $73.5$ $95.1$ $51.8$ $0$ $14.7$ $1.38$ $ 0.0905$ $ 0.0611$ $ 0.0950$ $ -0.422$ $5.22$ \[\] $(6)$ $14.7$ $76.9$ $98.2$ $55.7$ $14.6$ $0$ $13.3$ $14.7$ $14.7$ $14.7$ $14.6$ $12.2$ \[\] $(7)$ $1.45$ $73.7$ $95.3$ $52.0$ $1.33$ $13.3$ $0$ $1.45$ $1.42$ $1.46$ $1.22$ $4.53$ \[\] $(8)$ $-0.00818$ $73.5$ $95.1$ $51.8$ $0.138$ $14.8$ $1.50$ $0$ $ 0.0457$ $-0.0137$ $ -0.386$ $5.29$ \[\] $(9)$ $-0.0223$ $73.5$ $95.1$ $51.8$ $0.109$ $14.8$ $1.47$ $-0.0227$ $0$ $-0.0213$ $ -0.398$ $5.27$ \[\] $(10)$ $0.0121$ $73.5$ $95.1$ $51.8$ $0.142$ $14.8$ $1.50$ $ 0.0150$ $ 0.0506$ $0$ $ -0.384$ $5.29$ \[\] $(11)$ $0.558$ $73.5$ $95.0$ $51.7$ $0.541$ $14.6$ $1.44$ $0.558$ $0.551$ $0.559$ $0$ $5.00$ \[\] $(12)$ $6.10$ $73.2$ $94.7$ $51.6$ $6.04$ $12.6$ $5.46$ $6.10$ $6.08$ $6.10$ $5.81$ $0$ \[\] --------------------------- ------------ ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ for a single realization at $N_{\rm side} = 64$, using the [*Planck*]{} mask-76 extended to exclude regions at $|b| \le 30^\circ$. We adopt $E_{\rm for} = 10^{-2}$ and $E_{\rm cal} = 10^{-4}$, and consider each of the 19 frequency channels and all 171 independent combinations of pairs of different frequencies.[]{data-label="A_corr_cross_mask76ExtGal"} We use the “[*Planck*]{} common mask 76” (in temperature) and the extension of this mask that excludes all pixels at $|b| \le 30^\circ$.[^20] Having already addressed the rms uncertainty in the $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ estimates, in this test (as well as in the following ones) we will consider a single realization only, in order to avoid repeating a huge number of unnecessary simulations. For the sake of simplicity, we omit reporting the results found in the less stable approach (b). In the case of the extended mask and including also the cross-comparisons between different frequency channels (approach (c)), we found a significant improvement (see Table \[A\_corr\_cross\_mask76ExtGal\]) with respect to the results based on the full sky; the significance of the CIB amplitude recovery improves by about 50% and that on the BE distortion improves by about 20%. This indicates the relevance of optimising the selection of the sky region for which the analysis is applied, and of comparing results obtained with different masks. ### Varying assumptions on potential foreground and calibration residuals {#var_cal_res} We now consider the implications of different levels of potential residuals, evaluating both better and worse cases with respect to the reference case analysed before. Given the results obtained in the previous section we will focus on the case of the extended mask. We present here the main outcomes of this analysis, while the tables with the corresponding numerical results are reported in Appendix \[app\_var\_cal\_res\] for sake of completeness. - [**Improving foreground subtraction**]{} We now evaluate the improvement in component separation of total intensity maps by considering the case of $E_{\rm for} = 10^{-3}$. The results, summarised in Table \[Optimistic\_corr\_cross\_mask76ExtGal\] (for approach (c)), can be compared with those of Table \[A\_corr\_cross\_mask76ExtGal\]. We find an improvement by a factor of approximately 10 in the recovery of the CIB amplitude, in line with that assumed in foreground removal, and by a factor of 5 (or 6) in the recovery of $\mu_0$ (or $u$), implying that calibration uncertainty is relatively more important for estimating CMB distortion parameters than for estimating the CIB amplitude. In fact, the CIB is better constrained at higher frequencies, where foregrounds are more relevant. - [**The case of poorer calibration**]{} We discuss here the degradation in sensitivity entailed by keeping $E_{\rm for} = 10^{-2}$, but replacing CORE’s calibration-accuracy goal of $E_{\rm cal} = 10^{-4}$ with $E_{\rm cal} = 10^{-3}$ at $\nu \le 295$GHz and $E_{\rm cal} = 10^{-2}$ at $\nu \ge 340$GHz. The results, summarised in Table \[Pessimistic\_corr\_cross\_mask76ExtGal\] (for approach (c)), can be compared with those of Table \[A\_corr\_cross\_mask76ExtGal\]. In spite of the assumed degradation in calibration accuracy at high frequencies (particularly relevant for the CIB), we find that the recovery of the CIB amplitude is very weakly affected, while the significance of the $\mu_0$ (or $u$) determination degrades by factor of approximately 2 (or 25–30%). This result strengthens the conclusion of the previous subsection that calibration uncertainty is relatively more important for estimating CMB distortion parameters than for the CIB amplitude. For the set of assumptions adopted here, we find an improvement with respect to FIRAS by factor of around 20 in the recovery of the CIB amplitude, 15 on the constraints on the Comptonization parameter $u$ (or for its detection, at a level of about 3–4$\,\sigma$ for astrophysical reionization models), and about 24 for the constraints on chemical potential $\mu_0$. We finally consider a worst case scenario with $E_{\rm for} = E_{\rm cal} = 10^{-2}$. Even in this situation, we find an improvement with respect to FIRAS by a factor of 17 in the recovery of the CIB amplitude, and a factor of a few for CMB spectral distortion parameters, specifically around 4 for BE distortions and a marginal detection of astrophysical reionization models for Comptonization distortions. - [**Poorer calibration together with improved foreground subtraction**]{} We now consider a combination of the two cases above, i.e., a further improvement in component separation of total intensity maps represented by $E_{\rm for} = 10^{-3}$ and a calibration accuracy parameterised by $E_{\rm cal} = 10^{-3}$ at $\nu \le 295$GHz and $E_{\rm cal} = 10^{-2}$ at $\nu \ge 340$GHz. The results obtained in approach (c) are summarised in Table \[Intermediate\_corr\_cross\_mask76ExtGal\]. We find that the significance of CIB amplitude recovery is intermediate between the results found in the previous cases, while the degradation due to the poorer calibration is approximately compensated by the improvement due to better foreground subtraction in the case of Comptonization distortions, but only partially compensated in the case of BE distortions. Overall, our analysis indicates that the relevance of calibration accuracy increases from CIB amplitude to Comptonization-distortion and to BE-distortion recovery, while the relevance of the quality of foreground subtraction increases from BE distortions to Comptonization distortions and to CIB amplitude recovery. This conclusion reflects the increase of the foreground level and of the relative amplitude of the imprints left by the three types of signals at increasing frequencies (for CORE). - [**Varying the reference angular scale**]{} We finally consider assumptions of errors in foreground subtraction and in calibration in the range discussed above, but at smaller angular scales, specifically at $N_{\rm side} = 256$. The corresponding pixel linear size ($\simeq 13.7$ arcmin) is similar to the FWHM resolution of CORE channels at $\nu \lsim 80$GHz that are necessary for the mitigation of low-frequency foreground emission. With this adopted set-up and considering the most advantageous approach (i.e., approach (c)), assuming a foreground mitigation parameterised by $E_{\rm for} = 10^{-3}$, we find (see Table \[standard\_corr\_cross\_mask76ExtGal\_ns256\]) an improvement with respect to FIRAS by a factor of 80–90 for the recovery of CIB amplitude, around 80 on the constraints for the Comptonization parameter $u$ (implying a precise measure of the energy injections associated to astrophysical reionization models), and about 150 on the constraints on chemical potential $\mu_0$. Adopting $E_{\rm for} = 10^{-2}$, we find instead an improvement by a factor of 75 for the recovery of CIB amplitude, 50 for the constraints on the Comptonization parameter $u$, and 80 for the constraints on the chemical potential $\mu_0$. We further consider the same set-up but at $N_{\rm side} = 128$, i.e., with a pixel side 2 times larger. As expected, we find results intermediate between those derived at $N_{\rm side} = 64$ and 256. [width=1]{} $E_{\rm cal}$ (%) $E_{\rm for}$ (%) CIB amplitude Bose-Einstein Comptonization ---------------------------- --------------------------------------------- ------------------- -------------------------- --------------------------- --------------------------- Ideal case, all sky - - $\simeq 4.4 \times 10^3$ $\simeq 10^3$ $\simeq 6.0 \times 10^2$ All sky $10^{-4}$ $10^{-2}$ $\simeq 15$ $\simeq 42$ $\simeq 18$ P76 $10^{-4}$ $10^{-2}$ $\simeq 19$ $\simeq 42$ $\simeq 18$ P76ext $10^{-2}$ $10^{-2}$ $\simeq 17$ $\sim 4$ $\sim 2$ P76ext $10^{-4}$ $10^{-2}$ $\simeq 22$ $\simeq 47$ $\simeq 21$ P76ext $10^{-4}$ $10^{-3}$ $\simeq 2.1 \times 10^2$ $\simeq 2.4 \times 10^2$ $\simeq 1.1 \times 10^2$ P76ext $10^{-3}_{(\le 295)}$–$10^{-2}_{(\ge 340)}$ $10^{-2}$ $\simeq 19$ $\simeq 26$ $\simeq 11$ P76ext $10^{-3}_{(\le 295)}$–$10^{-2}_{(\ge 340)}$ $10^{-3}$ $\simeq 48$ $\simeq 35$ $\simeq 15$ P76ext, $N_{\rm side}=128$ $10^{-3}_{(\le 295)}$–$10^{-2}_{(\ge 340)}$ $10^{-2}$ $\simeq 38$ $\simeq 51$ $\simeq 23$ P76ext, $N_{\rm side}=128$ $10^{-3}_{(\le 295)}$–$10^{-2}_{(\ge 340)}$ $10^{-3}$ $\simeq 43$ $\simeq 87$ $\simeq 39$ P76ext, $N_{\rm side}=256$ $10^{-3}_{(\le 295)}$–$10^{-2}_{(\ge 340)}$ $10^{-2}$ $\simeq 76$ $\simeq 98$ $\simeq 44$ P76ext, $N_{\rm side}=256$ $10^{-3}_{(\le 295)}$–$10^{-2}_{(\ge 340)}$ $10^{-3}$ $\simeq 85$ $\simeq 1.6 \times 10^2$ $\simeq 73$ : Predicted improvement in the recovery of the distortion parameters discussed in the text with respect to FIRAS for different calibration and foreground residual assumptions. This table summarizes the results derived with approach (c). “P06” stands for the [*Planck*]{} common mask, while “P06ext” is the extended P06 mask. When not explicitly stated, all values refer to $E_{\rm cal}$ and $E_{\rm for}$ at $N_{\rm side}=64$.[]{data-label="tab:improv_FIRAS"} Summary of simulation results {#sec:sum_simdist} ----------------------------- We have presented above a large set of simulations for different choices of the parameters characterising foreground and calibration residual levels. The main results are summarised in Table \[tab:improv\_FIRAS\] in terms of improvements with respect to FIRAS, in order to parametrically quantify the accuracy required to achieve significant improvements. For other values of $E_{\rm for}$ and $E_{\rm cal}$, we find an almost linear dependence on them for the improvement factor in parameter recovery. Discussion and conclusions {#sec:end} ========================== We have carried out a detailed investigation of three distinct scientific implications coming from exploitation of the observer’s peculiar velocity effects in CORE maps. The determination of the CMB dipole amplitude and direction is an important observable in modern cosmology. It provides information on our velocity with respect to the CMB reference frame, which is expected to dominate the effect. Related investigations in other wavebands, which exploit signals from different types of astrophysical sources, probe different shells in redshift, and together provide an important test of fundamental principles in cosmology. In particular, the alignment of the CMB dipole with those independently measured from galaxy and cluster catalogues is regarded as indirect proof of the kinematic origin of the CMB dipole. The specific relation between the amplitudes of the CMB and large-scale structure dipoles, predicted by the linear perturbation theory, has been used to obtain estimates of the redshift-space distortion parameter independent of (but consistent with) those coming from redshift surveys. It is thus important to look for possible departures from a purely kinematic character for the CMB dipole. In this context, surveys from space are clearly appealing, since they represent the best (and perhaps only) way to precisely measure this large-scale signal. We performed detailed simulations in the context of a mission like CORE, to understand the expectations, and potential issues arising from future CMB surveys beyond the already excellent results produced by [*Planck*]{}. The sampling of the sky turns out to be the main limiting factor for the precise measurement of the dipole direction and (obviously together with calibration) also a crucial limiting factor for the precise measurement of dipole amplitude. We found that the recovered uncertainty scales linearly with the map pixel linear size (i.e., inversely with $N_{\rm side}$). Although maps can be oversampled through a proper scanning strategy and by setting the sampling time of the data acquisition well below that corresponding to the beam resolution, it is clear that the experimental resolution plays a crucial role in this respect. Among CMB space missions proposed for the future, CORE has the best angular resolution. The dipole direction determination can be averaged over the various frequency channels, improving accuracy and providing cross-checks for systematics. With the assumption of a pure blackbody, the same holds for the amplitude. However, when searching for dipole spectral signatures, increasing the accuracy at each frequency (which results from a better sky sampling) turns out to be even more important. An observer moving with respect to the CMB rest frame will also see boosting imprints on the CMB at $\ell>1,$ due to Doppler and aberration effects, which can be measured in harmonic space as correlations between $\ell$ and $\ell+1$ modes (assuming that the CMB is statistically isotropic in its rest frame). Such a signal can be measured independently in temperature and polarization, which constitutes a new consistency check, with a signal-to-noise ratio of about 8 for $TT$, 7 for $TE+ET$ and 7 for $EE$. Overall, CORE can achieve a signal-to-noise ratio of almost 13, which improves on the capabilities of [*Planck*]{} (about ${\rm S/N}\simeq 4$, only in $TT$) and is essentially that of an ideal cosmic-variance-limited experiment up to $\ell \simeq 2000$. We stress the importance of performing high-sensitivity measurements at close to arcminute resolution in order to be sensitive to the correlations at high multipoles that yield most of the signal. Since CORE will also provide good measurements of the tSZ effect and the CIB, which are also assumed to be statistically isotropic in the CMB rest frame, we additionally investigated boosting effects in these maps. However, we found that the aberration effect on tSZ maps and the boosting effects on the CIB are smaller than in the CMB maps, and that the predicted signal-to-noise is less than 1 in both cases. Beyond FIRAS, great hopes are expected for PIXIE, which has been proposed to NASA to observe CMB polarization and the CMB spectrum with degree resolution and is designed to have a precision about $10^3$ times better than FIRAS, mainly relying on the achievement of extreme quality in its absolute calibration, and a corresponding similar improvement on CMB spectral distortion parameters [@2011JCAP...07..025K]. Note that even if PIXIE fails to fully achieve these ambitious goals, an improvement in calibration precision of even one or two orders of magnitude with respect to FIRAS calibration, in addition to being intrinsically interesting for strengthening the limits on CMB distortion parameters, will imply an analogous improvement for the calibration of other CMB projects. In general, combining results from experiments like PIXIE and CORE will offer a chance to have maps with substantialily improved calibration, sensitivity, and resolution. CMB anisotropy missions will not perform absolute measurements of the CMB spectrum, but can observe the frequency spectral behaviour of the CMB and CIB dipoles. We exploit the sensitivity of an experiment like CORE for the recovery of the parameters $u$ and $\mu_0$ of Comptonization and BE spectral distortions, as well as for the amplitude of the CIB spectrum. Assuming perfect relative calibration and absence of foreground contamination, the CORE sensitivity and frequency coverage, combined with its resolution (to cope with sampling uncertainty), could allow us to achieve an improvement with respect to FIRAS by a factor of around 1–$4\times 10^3$, 500–600, and 600–1000 in the recovery of the CIB spectrum amplitude, $u$, and $\mu_0$, respectively; the best results are obtained from the joint information contained in each of the frequency channels independently [*and*]{} in all the independent combinations of pairs of different frequencies. Combining pairs of different frequencies turns out to be particularly advantageous for the CIB dipole spectrum, since it exhibits a steeper frequency behaviour. As expected, foregrounds are critical in both absolute and differential methods. Relative calibration accuracy is an important limiting factor in CMB anisotropy experiments in general and even more so for analyses based on the dipole. In current data analysis pipelines the dipole itself is in fact typically used for calibration, which raises the issue of a circular argument. However, for all-sky mapping experiments (like [*WMAP*]{} and [*Planck*]{}), the [*orbital*]{} dipole from the Earth and satellite motion is ultimately used for calibration, rather than the CMB dipole itself. Precise calibration is always challenging, and it is unclear what the limiting step will be for any new experiment. Nevertheless, in principle it will be possible to measure the spectrum of the dipole with an anisotropy experiment. In general, improving and extending calibration methods is crucial for these analyses. Various approaches can be integrated into the data reduction design, ranging from a better instrumental characterization to cross-correlation between different CMB surveys and substantial refinements in astronomical calibration sources. We have carried out a large set of simulations, summarised in Table \[tab:improv\_FIRAS\], to parametrically quantify the accuracy required to achieve significant improvements with respect to FIRAS. We find that the importance of the impact of calibration errors decreases from BE distortions to Comptonization distortions and to the CIB amplitude, while the opposite holds for the impact of foreground contamination (in agreement with the increase with frequency of foreground level and of the relative amplitude of the imprints left by the three types of signal). Applying suitable masks also yields an improvement in parameter estimation. In the case of 1% accuracy (at a reference scale of about $1^\circ$) in both foreground removal and relative calibration (i.e., $E_{\rm for} = E_{\rm cal} = 10^{-2}$), CORE will be able to improve the recovery of the CIB spectrum amplitude of a factor of about 17, to achieve a marginal detection of the energy release associated with astrophysical reionization models, and to improve by a factor of approximately 4 the limits on early energy dissipations. On the other hand, an improvement of a factor of 20 for CIB amplitude, of 10 for $u$, and of around 25 for the chemical potential $\mu_0$, is found by improving the relative calibration error to $\simeq 0.1$%. Any further improvement in foreground mitigation and calibration will enable still more precise results to be achieved. Partial support by ASI/INAF Agreement 2014-024-R.1 for the [*Planck*]{} LFI Activity of Phase E2 and by ASI through the contract I-022-11-0 LSPE is acknowledged. C.H.-M. acknowledges financial support of the Spanish Ministry of Economy and Competitiveness via I+D project AYA-2015-66211-C2-2-P. J.G.N. acknowledges financial support from the Spanish MINECO for a [*Ramon y Cajal*]{} fellowship (RYC-2013-13256) and the I+D 2015 project AYA2015-65887-P (MINECO/FEDER). C.J.M. is supported by an FCT Research Professorship, contract reference IF/00064/2012, funded by FCT/MCTES (Portugal) and POPH/FSE (EC). M.Q. is supported by the Brazilian research agencies CNPq and FAPERJ. We acknowledge the use of the ESA Planck Legacy Archive. Some of the results in this paper have been derived using the [HEALPix]{} package. The use of the computational cluster at INAF-IASF Bologna is acknowledged. It is a pleasure to thank Arpine Kozmanyan for useful discussions on the [CosmoMC]{} sampler. Appendix – Likelihoods of CMB dipole parameters {#sec:Likelihoods} =============================================== For the sake of completeness, we report here the likelihoods computed for CMB dipole parameters and the confidence levels in their estimation. We limit the presentation here to the lowest and highest resolutions among those exploited in this analysis. ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$ at $N_{\rm side}=128$ (red) and at $N_{\rm side}=1024$ (blue): top left, dipole only; top right, dipole+noise; bottom left, dipole+noise+mask; and bottom right, dipole+noise+mask+systematics. The reference frequency channel is 60GHz and the noise is 7.5$\mu$K.arcmin. The mask used here is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The level of systematics correspond to the pessimistic expectation of calibration errors and sky residuals.[]{data-label="fig:conflevel128+1024"}](60_128+1024_CROP.pdf "fig:") ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$ at $N_{\rm side}=128$ (red) and at $N_{\rm side}=1024$ (blue): top left, dipole only; top right, dipole+noise; bottom left, dipole+noise+mask; and bottom right, dipole+noise+mask+systematics. The reference frequency channel is 60GHz and the noise is 7.5$\mu$K.arcmin. The mask used here is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The level of systematics correspond to the pessimistic expectation of calibration errors and sky residuals.[]{data-label="fig:conflevel128+1024"}](60_128+1024_noi_CROP.pdf "fig:") ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$ at $N_{\rm side}=128$ (red) and at $N_{\rm side}=1024$ (blue): top left, dipole only; top right, dipole+noise; bottom left, dipole+noise+mask; and bottom right, dipole+noise+mask+systematics. The reference frequency channel is 60GHz and the noise is 7.5$\mu$K.arcmin. The mask used here is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The level of systematics correspond to the pessimistic expectation of calibration errors and sky residuals.[]{data-label="fig:conflevel128+1024"}](60_128+1024_noi_maskext_CROP.pdf "fig:") ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$ at $N_{\rm side}=128$ (red) and at $N_{\rm side}=1024$ (blue): top left, dipole only; top right, dipole+noise; bottom left, dipole+noise+mask; and bottom right, dipole+noise+mask+systematics. The reference frequency channel is 60GHz and the noise is 7.5$\mu$K.arcmin. The mask used here is the [*Planck*]{} Galactic mask extended to cut out $\pm30^\circ$ of the Galactic plane. The level of systematics correspond to the pessimistic expectation of calibration errors and sky residuals.[]{data-label="fig:conflevel128+1024"}](60_128+1024_noi_maskext_badcal_badsky_CROP.pdf "fig:") [width=1]{} $N_{\rm side}=128$ $A({\rm mK})$ $b_0(^\circ)$ $l_0(^\circ)$ $T_{0}({\rm mK})$ --------------------- ----------------------------------------------- --------------------------------- ----------------------------------------------- ------------------------------------- dipole $\ensuremath{3.3644\pm0.0028}$ $48.242\pm0.047$ $\ensuremath{263.999\pm0.070}$ $2725.4793\pm0.0016$ dip+noi $\ensuremath{3.3644\pm0.0028}$ $48.240\pm0.047$ $\ensuremath{263.998\pm0.071}$ $2725.4793\pm0.0016$ dip+noi+mask $\ensuremath{3.3644\pm0.0041}$ $48.240\pm0.075$ $\ensuremath{264.00\pm0.13}$ $2725.4797\pm0.0024$ dip+noi+mask+sys $\ensuremath{3.3645\pm0.0041}$ $48.235\pm0.074$ $\ensuremath{264.00\pm0.13}$ $2725.4797\pm0.0023$ $N_{\rm side}=1024$ $A({\rm mK})$ $b_0(^\circ)$ $l_0(^\circ)$ $T_{0}({\rm mK})$ dipole $\ensuremath{3.36447\pm0.00036}$ $48.2399\pm0.0060$ $\ensuremath{264.0002\pm0.0088}$ $2725.47930\pm0.00020$ dip+noi $\ensuremath{\ensuremath{3.36450\pm0.00035}}$ $\ensuremath{48.2398\pm0.0059}$ $\ensuremath{\ensuremath{264.0005\pm0.0087}}$ $2725.47931\pm0.00020$ dip+noi+mask $\ensuremath{3.36454\pm0.00051}$ $\ensuremath{48.2387\pm0.0091}$ $264.000\pm0.017$ $2725.47966\pm0.00029$ dip+noi+mask+sys $\ensuremath{3.36451\pm0.00052}$ $\ensuremath{48.2352\pm0.0092}$ $\ensuremath{264.000\pm0.016}$ $\ensuremath{2725.47965\pm0.00029}$ : 68% confidence levels of the parameters $A$, $b_0$, $l_0$, and $T_{0}$ from Fig. \[fig:conflevel128+1024\].[]{data-label="table:conflevel128+1024"} ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$. Input maps are at $N_{\rm side}=1024$, with noise, mask and residuals (calibration errors and sky residuals). [*Top*]{}: 100GHz with optimistic (left) and pessimistic (right) systematics. [*Bottom*]{}: 220GHz with optimistic (left) and pessimistic (right) systematics.[]{data-label="fig:conflevel100220"}](100_1024_noi_maskext_goodcal_goodsky.pdf "fig:") ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$. Input maps are at $N_{\rm side}=1024$, with noise, mask and residuals (calibration errors and sky residuals). [*Top*]{}: 100GHz with optimistic (left) and pessimistic (right) systematics. [*Bottom*]{}: 220GHz with optimistic (left) and pessimistic (right) systematics.[]{data-label="fig:conflevel100220"}](100_1024_noi_maskext_badcal_badsky.pdf "fig:") ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$. Input maps are at $N_{\rm side}=1024$, with noise, mask and residuals (calibration errors and sky residuals). [*Top*]{}: 100GHz with optimistic (left) and pessimistic (right) systematics. [*Bottom*]{}: 220GHz with optimistic (left) and pessimistic (right) systematics.[]{data-label="fig:conflevel100220"}](220_1024_noi_maskext_goodcal_goodsky.pdf "fig:") ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$. Input maps are at $N_{\rm side}=1024$, with noise, mask and residuals (calibration errors and sky residuals). [*Top*]{}: 100GHz with optimistic (left) and pessimistic (right) systematics. [*Bottom*]{}: 220GHz with optimistic (left) and pessimistic (right) systematics.[]{data-label="fig:conflevel100220"}](220_1024_noi_maskext_badcal_badsky.pdf "fig:") [width=1]{} $N_{\rm side}=1024$ $A({\rm mK})$ $b_0(^\circ)$ $l_0(^\circ)$ $T_{0}({\rm mK})$ --------------------- ---------------------------------- --------------------------------- -------------------------------- ------------------------------------- 60GHz, good sys $\ensuremath{3.36454\pm0.00052}$ $\ensuremath{48.2387\pm0.0093}$ $\ensuremath{263.999\pm0.016}$ $\ensuremath{2725.47965\pm0.00029}$ 60GHz, bad sys $\ensuremath{3.36451\pm0.00052}$ $\ensuremath{48.2352\pm0.0092}$ $\ensuremath{264.000\pm0.016}$ $\ensuremath{2725.47965\pm0.00029}$ 100GHz, good sys $\ensuremath{3.36453\pm0.00053}$ $\ensuremath{48.2393\pm0.0093}$ $\ensuremath{264.000\pm0.016}$ $\ensuremath{2725.47965\pm0.00029}$ 100GHz, bad sys $\ensuremath{3.36457\pm0.00051}$ $\ensuremath{48.2406\pm0.0093}$ $\ensuremath{263.998\pm0.016}$ $\ensuremath{2725.47961\pm0.00029}$ 145GHz, good sys $\ensuremath{3.36452\pm0.00052}$ $\ensuremath{48.2391\pm0.0093}$ $\ensuremath{263.999\pm0.017}$ $\ensuremath{2725.47967\pm0.00029}$ 145GHz, bad sys $\ensuremath{3.36434\pm0.00051}$ $\ensuremath{48.2391\pm0.0091}$ $\ensuremath{263.996\pm0.017}$ $\ensuremath{2725.47965\pm0.00029}$ 220GHz, good sys $\ensuremath{3.36451\pm0.00051}$ $\ensuremath{48.2387\pm0.0092}$ $\ensuremath{263.998\pm0.016}$ $\ensuremath{2725.47966\pm0.00029}$ 220GHz, bad sys $\ensuremath{3.36434\pm0.00052}$ $\ensuremath{48.2364\pm0.0094}$ $\ensuremath{263.977\pm0.016}$ $\ensuremath{2725.47972\pm0.00029}$ : 68% confidence level of the parameters $A$, $b_0$, $l_0$, and $T_{0}$ at 60, 100, 145 and 220 GHz (see Fig. \[fig:conflevel100220\] for the likelihoods at 100 and 220 GHz).[]{data-label="table:conflevel60220"} ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$. Input maps are dipole-only, at 60GHz and at $N_{\rm side}=1024$. On the left: blackbody. On the right: Bose-Einstein (chemical potential $\mu_{0}=1.4\times10^{-5}$).[]{data-label="fig:conflevelBE"}](60_1024 "fig:") ![Marginalised likelihoods, and 68% and 95% contours of the parameters $A$, $b_0$, $l_0$, and $T_{0}$. Input maps are dipole-only, at 60GHz and at $N_{\rm side}=1024$. On the left: blackbody. On the right: Bose-Einstein (chemical potential $\mu_{0}=1.4\times10^{-5}$).[]{data-label="fig:conflevelBE"}](BE.pdf "fig:") [width=1]{} $N_{\rm side}=1024$ $A({\rm mK})$ $b_0(^\circ)$ $l_0(^\circ)$ $T_{0}({\rm mK})$ --------------------- ---------------------------------- --------------------------------- ---------------------------------- ------------------------ blackbody $\ensuremath{3.36447\pm0.00036}$ $48.2399\pm0.0060$ $\ensuremath{264.0002\pm0.0088}$ $2725.47930\pm0.00020$ Bose-Einstein $\ensuremath{3.36440\pm0.00034}$ $\ensuremath{48.2402\pm0.0059}$ $264.0002\pm0.0090$ $2725.45379\pm0.00020$ : 68% confidence level of the parameters $A$, $b_0$, $l_0$, and $T_{0}$ of Fig. \[fig:conflevelBE\].[]{data-label="table:conflevelBE"} Appendix – Rms values from Monte Carlo simulations: ideal case {#app_rms_ideal} ============================================================== We present here the tables with the estimates of the rms of the $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ quoted from a Monte Carlo simulation at $N_{\rm side} = 64$, using all-sky maps and adopting perfect foreground subtraction and calibration. [width=1]{} ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**Rms of $\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $4.83$ $6.99$ $3.16$ $0.872$ $0.568$ $0.621$ $0.129$ $0.562$ $0.183$ $0.976$ $1.06$ \[\] $(2)$ $4.22$ $0$ $4.22$ $0.00$ $4.22$ $5.27$ $5.16$ $4.22$ $4.22$ $4.22$ $5.16$ $5.16$ \[\] $(3)$ $6.99$ $3.16$ $0$ $5.16$ $6.99$ $4.22$ $6.99$ $6.99$ $6.99$ $6.99$ $6.99$ $6.75$ \[\] $(4)$ $0.00$ $3.16$ $5.16$ $0$ $0.00$ $4.22$ $3.16$ $0.00$ $0.00$ $0.00$ $3.16$ $0.00$ \[\] $(5)$ $1.06$ $4.83$ $6.99$ $3.16$ $0$ $0.675$ $0.621$ $1.06$ $0.976$ $1.07$ $1.05$ $1.06$ \[\] $(6)$ $0.667$ $6.75$ $6.32$ $4.22$ $0.738$ $0$ $0.823$ $0.667$ $0.738$ $0.568$ $0.667$ $1.08$ \[\] $(7)$ $0.636$ $4.83$ $6.75$ $3.16$ $0.636$ $0.675$ $0$ $0.636$ $0.632$ $0.635$ $0.657$ $1.14$ \[\] $(8)$ $0.129$ $4.83$ $6.99$ $3.16$ $0.871$ $0.568$ $0.610$ $0$ $0.549$ $0.224$ $0.976$ $1.06$ \[\] $(9)$ $0.578$ $4.83$ $6.99$ $3.16$ $0.813$ $0.675$ $0.619$ $0.564$ $0$ $0.606$ $0.993$ $1.06$ \[\] $(10)$ $0.182$ $4.83$ $6.99$ $3.16$ $0.873$ $0.667$ $0.613$ $0.223$ $0.587$ $0$ $0.974$ $1.06$ \[\] $(11)$ $1.04$ $5.16$ $6.75$ $3.16$ $1.14$ $0.568$ $0.620$ $1.05$ $1.07$ $1.04$ $0$ $0.994$ \[\] $(12)$ $0.994$ $5.16$ $6.32$ $3.16$ $0.919$ $1.05$ $1.08$ $0.994$ $1.07$ $0.994$ $0.966$ $0$ \[\] ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Rms values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ from a Monte Carlo simulation at $N_{\rm side} = 64$, full sky, adopting perfect foreground subtraction and calibration, and considering each of the 19 frequency channels.[]{data-label="MC_corr_rms_ideal"} [width=1]{} ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**Rms of $\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $ 0.00$ $31.6$ $42.2$ $14.8$ $2.93$ $2.67$ $2.52$ $0.342$ $1.52$ $0.485$ $2.89$ $2.18$ \[\] $(2)$ $ 31.6$ $0$ $6.75$ $6.67$ $0.00$ $51.6$ $0.00$ $31.6$ $0.00$ $31.6$ $31.6$ $31.6$ \[\] $(3)$ $ 42.2$ $5.16$ $0$ $8.43$ $42.2$ $42.2$ $31.6$ $42.2$ $42.2$ $42.2$ $42.2$ $51.6$ \[\] $(4)$ $ 10.8$ $8.23$ $11.4$ $0$ $10.8$ $12.9$ $12.9$ $10.8$ $10.8$ $10.8$ $11.7$ $12.9$ \[\] $(5)$ $ 3.11$ $31.6$ $42.2$ $14.8$ $0$ $2.59$ $2.53$ $3.09$ $2.70$ $3.15$ $4.15$ $2.20$ \[\] $(6)$ $ 2.30$ $51.6$ $42.2$ $14.5$ $2.49$ $0$ $2.33$ $2.33$ $2.67$ $2.30$ $2.50$ $2.46$ \[\] $(7)$ $ 2.59$ $31.6$ $31.6$ $14.5$ $2.64$ $2.56$ $0$ $2.62$ $2.60$ $2.60$ $2.58$ $2.31$ \[\] $(8)$ $0.343$ $31.6$ $42.2$ $14.8$ $2.91$ $2.56$ $2.51$ $0$ $1.48$ $0.594$ $2.91$ $2.19$ \[\] $(9)$ $ 1.56$ $31.6$ $42.2$ $14.8$ $2.58$ $2.59$ $2.53$ $1.52$ $0$ $1.63$ $3.29$ $2.20$ \[\] $(10)$ $0.484$ $31.6$ $42.2$ $14.8$ $2.96$ $2.67$ $2.53$ $0.593$ $1.60$ $0$ $2.85$ $2.18$ \[\] $(11)$ $ 2.39$ $31.6$ $42.2$ $14.8$ $2.93$ $2.81$ $2.48$ $2.40$ $2.60$ $2.36$ $0$ $2.20$ \[\] $(12)$ $ 2.10$ $31.6$ $52.7$ $14.0$ $2.13$ $2.59$ $2.30$ $2.09$ $2.12$ $2.12$ $2.10$ $0$ \[\] ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_rms\_ideal\], but considering all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_cross_rms_ideal"} [width=1]{} ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**Rms of $\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $52.7$ $31.6$ $31.6$ $2.69$ $2.17$ $2.14$ $0.340$ $1.50$ $0.482$ $1.40$ $1.57$ \[\] $(2)$ $51.6$ $0$ $8.43$ $6.75$ $51.6$ $0.00$ $51.6$ $51.6$ $51.6$ $51.6$ $51.6$ $31.6$ \[\] $(3)$ $31.6$ $5.16$ $0$ $11.0$ $31.6$ $52.7$ $31.6$ $31.6$ $31.6$ $31.6$ $31.6$ $51.6$ \[\] $(4)$ $31.6$ $6.32$ $12.6$ $0$ $31.6$ $0.00$ $0.00$ $31.6$ $31.6$ $31.6$ $31.6$ $51.6$ \[\] $(5)$ $3.10$ $52.7$ $31.6$ $31.6$ $0$ $2.28$ $2.14$ $3.08$ $2.68$ $3.14$ $2.15$ $1.40$ \[\] $(6)$ $2.20$ $0.00$ $48.3$ $31.6$ $2.21$ $0$ $2.17$ $2.06$ $2.21$ $2.20$ $2.21$ $2.59$ \[\] $(7)$ $2.23$ $51.6$ $31.6$ $31.6$ $2.23$ $2.27$ $0$ $2.22$ $2.22$ $2.22$ $2.57$ $1.65$ \[\] $(8)$ $0.340$ $52.7$ $31.6$ $31.6$ $2.68$ $2.17$ $2.12$ $0$ $1.46$ $0.591$ $1.40$ $1.57$ \[\] $(9)$ $1.53$ $52.7$ $31.6$ $31.6$ $2.40$ $2.28$ $2.16$ $1.49$ $0$ $1.61$ $1.58$ $1.57$ \[\] $(10)$ $0.481$ $52.7$ $31.6$ $31.6$ $2.72$ $2.31$ $2.14$ $0.589$ $1.57$ $0$ $1.38$ $1.57$ \[\] $(11)$ $1.48$ $51.6$ $31.6$ $31.6$ $2.15$ $2.18$ $2.48$ $1.49$ $1.65$ $1.46$ $0$ $1.57$ \[\] $(12)$ $1.23$ $31.6$ $52.7$ $51.6$ $1.58$ $2.72$ $1.87$ $1.23$ $1.23$ $1.23$ $1.23$ $0$ \[\] ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_rms\_ideal\], but considering each of the 19 frequency channels and all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_corr_cross_rms_ideal"} Appendix – Ideal case at high resolution {#ideal_highres} ======================================== We repeat here the same analysis carried out in the previous section, but now working at $N_{\rm side} = 512$, i.e., at about 7 arcmin resolution, and considering a single realization. The results are reported in Table \[MC\_corr\_cross\_avg\_ideal\_512\] relative to approach (c). [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $14400$ $18800$ $10100$ $-1.39$ $320.$ $30.5$ $-0.326$ $-1.31$ $0.467$ $8.65$ $161.$ \[\] $(2)$ $14500$ $0$ $4380$ $4380$ $14500$ $14500$ $14500$ $14500$ $14500$ $14500$ $14500$ $14400$ \[\] $(3)$ $18800$ $4330$ $0$ $8710$ $18800$ $18800$ $18800$ $18800$ $18800$ $18800$ $18800$ $18800$ \[\] $(4)$ $10100$ $4380$ $8710$ $0$ $10100$ $10100$ $10100$ $10100$ $10100$ $10100$ $10100$ $10000$ \[\] $(5)$ $3.90$ $14400$ $18800$ $10100$ $0$ $317.$ $27.9$ $3.86$ $3.19$ $3.97$ $9.06$ $160.$ \[\] $(6)$ $323.$ $14500$ $10100$ $10100$ $321.$ $0$ $291.$ $323.$ $323.$ $323.$ $322.$ $332.$ \[\] $(7)$ $33.8$ $18800$ $18800$ $10100$ $31.2$ $288.$ $0$ $33.8$ $33.1$ $33.8$ $33.4$ $158.$ \[\] $(8)$ $0.329$ $14400$ $18800$ $10100$ $-1.41$ $320.$ $30.4$ $0$ $-1.29$ $0.575$ $8.66$ $161.$ \[\] $(9)$ $1.60$ $14400$ $18800$ $10100$ $-1.64$ $319.$ $29.8$ $1.55$ $0$ $1.69$ $8.69$ $160.$ \[\] $(10)$ $-0.458$ $14400$ $18800$ $10100$ $-1.35$ $320.$ $30.5$ $-0.559$ $-1.36$ $0$ $8.65$ $161.$ \[\] $(11)$ $7.27$ $14400$ $18800$ $10100$ $6.56$ $318.$ $29.7$ $7.26$ $7.01$ $7.30$ $0$ $153.$ \[\] $(12)$ $159.$ $14500$ $18800$ $10000$ $159.$ $328.$ $156.$ $159.$ $159.$ $159.$ $151.$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ for a single realization at $N_{\rm side} = 512$, for the full sky, adopting perfect foreground subtraction and calibration, and considering each of the 19 frequency channels and all 171 independent combinations of pairs of different frequencies.[]{data-label="MC_corr_cross_avg_ideal_512"} Appendix – Rms values from Monte Carlo simulations: including potential residuals {#app_rms_res} ================================================================================= We report here tables with the estimates of the rms of the $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ quoted from a Monte Carlo simulation at $N_{\rm side} = 64$, with all-sky data, and including potential foreground and calibration residuals. [width=1]{} ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**Rms of $\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $0.798$ $0.804$ $0.802$ $0.274$ $0.471$ $0.673$ $0.0308$ $0.137$ $0.0435$ $0.416$ $0.469$ \[\] $(2)$ $0.838$ $ 0$ $0.792$ $0.973$ $0.846$ $0.637$ $0.819$ $0.838$ $0.838$ $0.838$ $0.838$ $0.701$ \[\] $(3)$ $0.829$ $0.981$ $ 0$ $0.871$ $0.829$ $0.682$ $0.806$ $0.829$ $0.829$ $0.829$ $0.823$ $0.753$ \[\] $(4)$ $0.859$ $0.788$ $0.799$ $ 0$ $0.857$ $0.578$ $0.837$ $0.859$ $0.860$ $0.859$ $0.841$ $0.667$ \[\] $(5)$ $0.276$ $0.798$ $0.804$ $0.800$ $ 0$ $0.480$ $0.746$ $0.274$ $0.239$ $0.280$ $0.359$ $0.472$ \[\] $(6)$ $0.490$ $0.605$ $0.657$ $0.529$ $0.495$ $ 0$ $0.486$ $0.490$ $0.480$ $0.481$ $0.484$ $0.509$ \[\] $(7)$ $0.788$ $0.795$ $0.808$ $0.768$ $0.775$ $0.477$ $ 0$ $0.788$ $0.784$ $0.789$ $0.775$ $0.474$ \[\] $(8)$ $ 0.0308$ $0.798$ $0.804$ $0.802$ $0.272$ $0.471$ $0.673$ $ 0$ $0.134$ $0.0533$ $0.415$ $0.470$ \[\] $(9)$ $0.138$ $0.798$ $0.804$ $0.802$ $0.237$ $0.478$ $0.702$ $0.134$ $ 0$ $0.145$ $0.399$ $0.469$ \[\] $(10)$ $ 0.0435$ $0.798$ $0.804$ $0.802$ $0.277$ $0.486$ $0.670$ $0.0533$ $0.144$ $ 0$ $0.418$ $0.469$ \[\] $(11)$ $0.455$ $0.798$ $0.798$ $0.784$ $0.383$ $0.482$ $0.768$ $0.454$ $0.437$ $0.457$ $ 0$ $0.469$ \[\] $(12)$ $0.539$ $0.709$ $0.733$ $0.624$ $0.542$ $0.525$ $0.591$ $0.539$ $0.539$ $0.538$ $0.543$ $0$ \[\] ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Rms values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ from a Monte Carlo simulation at $N_{\rm side} = 64$, full sky, adopting $E_{\rm for} = 10^{-2}$ and $E_{\rm cal} = 10^{-4}$, and considering each of the 19 frequency channels.[]{data-label="MC_corr_rms"} [width=1]{} ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**Rms of $\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $2.69$ $2.69$ $2.68$ $0.332$ $1.77$ $1.34$ $0.0362$ $0.164$ $0.0512$ $0.359$ $1.54$ \[\] $(2)$ $2.67$ $0$ $2.77$ $2.88$ $2.67$ $2.69$ $2.67$ $2.67$ $2.67$ $2.67$ $2.67$ $2.66$ \[\] $(3)$ $2.65$ $2.91$ $0$ $2.70$ $2.64$ $2.68$ $2.67$ $2.65$ $2.65$ $2.65$ $2.65$ $2.65$ \[\] $(4)$ $2.69$ $2.76$ $2.70$ $0$ $2.69$ $2.72$ $2.71$ $2.69$ $2.69$ $2.69$ $2.69$ $2.68$ \[\] $(5)$ $0.320$ $2.69$ $2.69$ $2.68$ $0$ $1.78$ $1.28$ $0.318$ $0.278$ $0.324$ $0.483$ $1.56$ \[\] $(6)$ $1.39$ $2.69$ $2.71$ $2.64$ $1.38$ $0$ $1.38$ $1.39$ $1.37$ $1.39$ $1.38$ $1.40$ \[\] $(7)$ $1.08$ $2.69$ $2.69$ $2.68$ $1.04$ $1.85$ $0$ $1.08$ $1.08$ $1.09$ $1.12$ $1.59$ \[\] $(8)$ $ 0.0362$ $2.69$ $2.69$ $2.68$ $0.329$ $1.77$ $1.34$ $0$ $0.160$ $0.0627$ $0.361$ $1.54$ \[\] $(9)$ $0.161$ $2.69$ $2.69$ $2.68$ $0.286$ $1.78$ $1.33$ $0.157$ $0$ $0.169$ $0.395$ $1.55$ \[\] $(10)$ $ 0.0512$ $2.69$ $2.69$ $2.68$ $0.336$ $1.77$ $1.35$ $0.0629$ $0.172$ $0$ $0.355$ $1.54$ \[\] $(11)$ $0.360$ $2.69$ $2.72$ $2.67$ $0.491$ $1.77$ $1.38$ $0.362$ $0.397$ $0.356$ $0$ $1.51$ \[\] $(12)$ $1.64$ $2.69$ $2.69$ $2.66$ $1.68$ $1.79$ $2.08$ $1.64$ $1.65$ $1.64$ $1.60$ $0$ \[\] ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_rms\], but considering all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_cross_rms"} [width=1]{} ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**Rms of $\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $ 0$ $2.79$ $2.82$ $2.79$ $0.411$ $1.10$ $1.48$ $0.0458$ $0.205$ $0.0647$ $0.469$ $0.638$ \[\] $(2)$ $ 2.79$ $0$ $2.87$ $3.03$ $2.79$ $2.74$ $2.78$ $2.79$ $2.79$ $2.79$ $2.79$ $2.75$ \[\] $(3)$ $2.77$ $3.04$ $0$ $2.84$ $2.76$ $2.75$ $2.79$ $2.76$ $2.76$ $2.77$ $2.77$ $2.75$ \[\] $(4)$ $2.82$ $2.86$ $2.81$ $0$ $2.82$ $2.73$ $2.82$ $2.82$ $2.82$ $2.82$ $2.80$ $2.74$ \[\] $(5)$ $0.406$ $2.79$ $2.79$ $2.78$ $0$ $1.10$ $1.43$ $0.404$ $0.352$ $0.411$ $0.534$ $0.649$ \[\] $(6)$ $1.02$ $2.71$ $2.76$ $2.62$ $1.02$ $0$ $1.01$ $1.02$ $1.02$ $1.02$ $1.02$ $1.34$ \[\] $(7)$ $1.18$ $2.79$ $2.79$ $2.78$ $1.14$ $1.10$ $0$ $1.18$ $1.17$ $1.18$ $1.21$ $0.853$ \[\] $(8)$ $0.0457$ $2.79$ $2.82$ $2.79$ $0.409$ $1.10$ $1.48$ $0$ $0.199$ $0.0794$ $0.470$ $0.638$ \[\] $(9)$ $0.204$ $2.79$ $2.82$ $2.79$ $0.356$ $1.08$ $1.47$ $0.199$ $0$ $0.214$ $0.475$ $0.642$ \[\] $(10)$ $ 0.0647$ $2.79$ $2.82$ $2.79$ $0.417$ $1.10$ $1.48$ $0.0794$ $0.215$ $0$ $0.467$ $0.637$ \[\] $(11)$ $ 0.503$ $2.81$ $2.82$ $2.78$ $0.547$ $1.11$ $1.54$ $0.503$ $0.513$ $0.502$ $0$ $0.636$ \[\] $(12)$ $ 0.842$ $2.76$ $2.78$ $2.74$ $0.876$ $1.55$ $1.87$ $0.843$ $0.849$ $0.840$ $0.859$ $0$ \[\] ------------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : The same as in Table \[MC\_corr\_rms\], but considering each of the 19 frequency channels and all 171 independent combinations of pairs of different frequency channels.[]{data-label="MC_corr_cross_rms"} Appendix – Results for different assumptions on potential foreground and calibration residuals {#app_var_cal_res} ============================================================================================== We report here some of the results discussed in Sect. \[var\_cal\_res\]. [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0.00$ $700.$ $911.$ $487.$ $0.859$ $75.7$ $7.85$ $0.0689$ $0.341$ $-0.0956$ $2.35$ $30.7$ \[\] $(2)$ $699.$ $0.00$ $211.$ $213.$ $699.$ $715.$ $700.$ $699.$ $699.$ $699.$ $699.$ $696.$ \[\] $(3)$ $910.$ $211.$ $0.00$ $424.$ $910.$ $926.$ $911.$ $910.$ $910.$ $910.$ $910.$ $907.$ \[\] $(4)$ $486.$ $213.$ $424.$ $0.00$ $486.$ $504.$ $487.$ $486.$ $486.$ $486.$ $486.$ $483.$ \[\] $(5)$ $-0.103$ $700.$ $911.$ $487.$ $0.00$ $75.1$ $7.24$ $-0.122$ $-0.276$ $-0.0379$ $2.19$ $30.5$ \[\] $(6)$ $75.1$ $716.$ $926.$ $505.$ $74.5$ $0.00$ $67.6$ $75.1$ $75.0$ $75.1$ $74.6$ $69.5$ \[\] $(7)$ $7.23$ $701.$ $912.$ $488.$ $6.62$ $68.2$ $0.00$ $7.22$ $7.07$ $7.24$ $7.00$ $28.7$ \[\] $(8)$ $-0.0680$ $700.$ $911.$ $487.$ $0.851$ $75.7$ $7.84$ $0.00$ $0.331$ $-0.116$ $2.35$ $30.7$ \[\] $(9)$ $-0.266$ $700.$ $911.$ $487.$ $0.697$ $75.6$ $7.70$ $-0.262$ $0.00$ $-0.275$ $2.30$ $30.7$ \[\] $(10)$ $0.0979$ $700.$ $911.$ $487.$ $0.875$ $75.7$ $7.86$ $0.121$ $0.361$ $0.00$ $2.35$ $30.8$ \[\] $(11)$ $-1.06$ $700.$ $911.$ $487.$ $-1.04$ $75.1$ $7.19$ $-1.06$ $-1.09$ $-1.05$ $0.00$ $29.3$ \[\] $(12)$ $28.5$ $697.$ $907.$ $484.$ $28.3$ $69.3$ $26.4$ $28.5$ $28.5$ $28.5$ $27.0$ $0.00$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ for a single realization at $N_{\rm side} = 64$, using [*Planck*]{} mask-76 extended to exclude regions at $|b| \le 30^\circ$. We adopt $E_{\rm for} = 10^{-3}$ and $E_{\rm cal} = 10^{-4}$, and consider each of the 19 frequency channels and all 171 independent combinations of pairs of different frequencies.[]{data-label="Optimistic_corr_cross_mask76ExtGal"} [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $65.2$ $84.3$ $46.0$ $-0.193$ $7.80$ $ 0.376$ $-0.0227$ $-0.101$ $0.0322$ $ 0.563$ $3.96$ \[\] $(2)$ $61.2$ $0$ $20.9$ $17.1$ $61.2$ $63.0$ $61.3$ $61.2$ $61.2$ $61.2$ $61.2$ $60.9$ \[\] $(3)$ $80.3$ $17.0$ $0$ $36.3$ $80.3$ $82.0$ $80.4$ $80.3$ $80.3$ $80.3$ $80.2$ $79.9$ \[\] $(4)$ $41.9$ $21.1$ $40.2$ $0$ $41.9$ $44.0$ $42.1$ $41.9$ $41.9$ $41.9$ $41.9$ $41.7$ \[\] $(5)$ $0.214$ $65.2$ $84.3$ $46.0$ $0$ $7.73$ $ 0.286$ $ 0.212$ $ 0.183$ $ 0.217$ $ 0.595$ $3.94$ \[\] $(6)$ $8.44$ $67.0$ $86.0$ $48.0$ $8.37$ $0$ $7.62$ $8.44$ $8.42$ $8.44$ $8.39$ $8.11$ \[\] $(7)$ $1.09$ $65.3$ $84.4$ $46.1$ $1.02$ $6.98$ $0$ $1.08$ $1.07$ $1.09$ $1.18$ $3.84$ \[\] $(8)$ $0.0228$ $65.2$ $84.3$ $46.0$ $-0.192$ $7.80$ $ 0.375$ $0$ $-0.0980$ $0.0395$ $ 0.563$ $3.96$ \[\] $(9)$ $0.103$ $65.2$ $84.3$ $46.0$ $-0.169$ $7.78$ $ 0.355$ $ 0.100$ $0$ $ 0.108$ $ 0.570$ $3.95$ \[\] $(10)$ $-0.0322$ $65.2$ $84.3$ $46.0$ $-0.195$ $7.80$ $ 0.378$ $-0.0394$ $-0.105$ $0$ $ 0.562$ $3.96$ \[\] $(11)$ $-0.517$ $65.2$ $84.2$ $45.9$ $-0.560$ $7.71$ $-0.481$ $-0.518$ $-0.529$ $-0.516$ $0$ $3.80$ \[\] $(12)$ $2.00$ $64.8$ $83.9$ $45.6$ $1.94$ $6.61$ $1.42$ $1.99$ $1.98$ $2.00$ $1.82$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ for a single realization at $N_{\rm side} = 64$, using [*Planck*]{} mask-76 extended to exclude regions at $|b| \le 30^\circ$. We adopt $E_{\rm for} = 10^{-2}$ and $E_{\rm cal} = 10^{-3}$ at $\nu \le 295$GHz and $E_{\rm cal} = 10^{-2}$ at $\nu \ge 340$GHz, and consider each of the 19 frequency channels and all 171 independent combinations of pairs of different frequencies.[]{data-label="Pessimistic_corr_cross_mask76ExtGal"} [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $161.$ $209.$ $112.$ $-0.601$ $8.97$ $-1.83$ $-0.0680$ $-0.303$ $0.0962$ $0.963$ $6.15$ \[\] $(2)$ $157.$ $0$ $49.8$ $46.4$ $157.$ $160.$ $157.$ $157.$ $157.$ $157.$ $157.$ $156.$ \[\] $(3)$ $205.$ $45.9$ $0$ $94.4$ $205.$ $208.$ $205.$ $205.$ $205.$ $205.$ $205.$ $204.$ \[\] $(4)$ $109.$ $50.2$ $98.2$ $0$ $109.$ $112.$ $109.$ $109.$ $109.$ $109.$ $109.$ $108.$ \[\] $(5)$ $ 0.615$ $161.$ $209.$ $112.$ $0$ $8.88$ $-1.78$ $0.611$ $0.531$ $0.622$ $1.14$ $6.17$ \[\] $(6)$ $13.1$ $164.$ $212.$ $116.$ $13.1$ $0$ $12.0$ $13.1$ $13.1$ $13.2$ $13.1$ $13.8$ \[\] $(7)$ $2.43$ $161.$ $209.$ $113.$ $2.31$ $7.81$ $0$ $2.43$ $2.40$ $2.43$ $2.59$ $6.46$ \[\] $(8)$ $0.0680$ $161.$ $209.$ $112.$ $-0.598$ $8.97$ $-1.83$ $0$ $-0.296$ $0.118$ $0.966$ $6.15$ \[\] $(9)$ $ 0.305$ $161.$ $209.$ $112.$ $-0.522$ $8.95$ $-1.82$ $0.297$ $0$ $0.320$ $1.01$ $6.15$ \[\] $(10)$ $-0.0961$ $161.$ $209.$ $112.$ $-0.609$ $8.97$ $-1.83$ $-0.118$ $-0.318$ $0$ $0.959$ $6.15$ \[\] $(11)$ $-0.909$ $161.$ $209.$ $112.$ $-1.09$ $8.87$ $-2.07$ $-0.912$ $-0.959$ $-0.904$ $0$ $5.91$ \[\] $(12)$ $1.65$ $160.$ $208.$ $111.$ $1.48$ $8.00$ $-1.60$ $1.64$ $1.61$ $1.65$ $1.27$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ for a single realization at $N_{\rm side} = 64$, using [*Planck*]{} mask-76 extended to exclude regions at $|b| \le 30^\circ$. We adopt $E_{\rm for} = 10^{-3}$ and $E_{\rm cal} = 10^{-3}$ at $\nu \le 295$GHz and $E_{\rm cal} = 10^{-2}$ at $\nu \ge 340$GHz, and consider each of the 19 frequency channels and all 171 independent combinations of pairs of different frequencies.[]{data-label="Intermediate_corr_cross_mask76ExtGal"} [width=1]{} --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- [**$\sigma$ level**]{} \[\] [**significance**]{} blackbody (units $10^{-5}$) $8 \times 10^{-8}$ $10^{-5}$ $10^{-6}$ $10^{-9}$ $2 \times 10^{-8}$ $-2 \times 10^{-9}$ $4 \times 10^{-7}$ $8 \times 10^{-6}$ $I_0^{\rm bf}$ $+ 1\,\sigma$ $ - 1\,\sigma$ \[\] $1.3$ $1.7$ $0.9$ $1.12 \times 10^{-7}$ $1.4 \times 10^{-5}$ $1.4 \times 10^{-6}$ $1.4 \times 10^{-9}$ $2.8 \times 10^{-8}$ $-2.8 \times 10^{-9}$ $10^{-7}$ $2 \times 10^{-6}$ Case (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) $(1)$ $0$ $286.$ $371.$ $200.$ $-1.12$ $48.6$ $2.74$ $-0.133$ $-0.589$ $0.189$ $2.12$ $21.3$ \[\] $(2)$ $278.$ $0$ $88.9$ $81.7$ $278.$ $289.$ $278.$ $278.$ $278.$ $278.$ $278.$ $276.$ \[\] $(3)$ $363.$ $81.0$ $0$ $167.$ $363.$ $373.$ $364.$ $363.$ $363.$ $363.$ $363.$ $361.$ \[\] $(4)$ $192.$ $89.7$ $175.$ $0$ $192.$ $205.$ $193.$ $192.$ $192.$ $192.$ $192.$ $191.$ \[\] $(5)$ $1.26$ $286.$ $371.$ $200.$ $0$ $48.2$ $2.25$ $1.25$ $1.08$ $1.28$ $2.38$ $21.2$ \[\] $(6)$ $52.1$ $297.$ $381.$ $213.$ $51.7$ $0$ $47.1$ $52.1$ $52.0$ $52.1$ $51.6$ $46.2$ \[\] $(7)$ $6.57$ $286.$ $371.$ $201.$ $6.15$ $43.5$ $0$ $6.57$ $6.47$ $6.58$ $6.52$ $19.9$ \[\] $(8)$ $ 0.134$ $286.$ $371.$ $200.$ $-1.12$ $48.5$ $2.74$ $0$ $-0.574$ $0.232$ $2.12$ $21.3$ \[\] $(9)$ $ 0.606$ $286.$ $371.$ $200.$ $-0.989$ $48.5$ $2.62$ $0.590$ $0$ $0.636$ $2.18$ $21.3$ \[\] $(10)$ $-0.189$ $286.$ $371.$ $200.$ $-1.14$ $48.6$ $2.75$ $-0.231$ $-0.617$ $0$ $2.11$ $21.3$ \[\] $(11)$ $ -1.60$ $286.$ $371.$ $200.$ $-2.06$ $48.0$ $-0.489$ $-1.61$ $-1.74$ $-1.59$ $0$ $20.3$ \[\] $(12)$ $17.7$ $284.$ $369.$ $199.$ $17.4$ $40.4$ $14.8$ $17.7$ $17.6$ $17.7$ $16.7$ $0$ \[\] --------------------------- ----------- ---------------- ------------------- ---------------- ----------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------- -------------------- -------------------- : Values of $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$ for a single realization at $N_{\rm side} = 256$, using [*Planck*]{} mask-76 extended to exclude regions at $|b| \le 30^\circ$. We adopt $E_{\rm for} = 10^{-3}$ and $E_{\rm cal} = 10^{-3}$ at $\nu \le 295$GHz and $E_{\rm cal} = 10^{-2}$ at $\nu \ge 340$GHz, and consider each of the 19 frequency channels and all 171 independent combinations of pairs of different frequencies.[]{data-label="standard_corr_cross_mask76ExtGal_ns256"} [^1]: $^*$ Corresponding author. E-mail: burigana@iasfbo.inaf.it [^2]: For recent observations at long wavelengths, see the results from the ARCADE-2 balloon [@2011ApJ...730..138S; @2011ApJ...734....6S] and from the TRIS experiment [@2008ApJ...688...24G]. [^3]: http://pla.esac.esa.int/pla/ [^4]: When we degrade the [*Planck*]{} common mask 76 to lower resolutions we apply a threshold of 0.5 for accepting or excluding pixels, so that the exact sky coverage not excluded by each mask (76–78%) slightly increases at decreasing $N_{\rm side}$. In the case of the extended masks, typical sky coverage values are 47–48%. [^5]: Adopting this choice or one of the other foreground-separation methods is not relevant for the present purpose. [^6]: Since we are not interested here in the separation of the diffuse Galactic emission and ZLE, this assumption is in principle slightly conservative. In practice, separation methods will at least distinguish between these diffuse components, which are typically treated with different approaches, e.g., analysing multi-frequency maps in the case of Galactic emission, and different surveys (or more generally, data taken at different times) for the ZLE. [^7]: It has been shown in [@Roldan:2016ayx] that, in the Gaussian case, an intrinsic large scale dipolar potential exactly mimics on large scales a Doppler modulation. [^8]: In this section we will use primes for the CMB frame and non-primes for the observer frame, following Ref. . [^9]: Note that the kernel defined as in Eq.  for $d\neq 1$ can be obtained from $_{s}K_{\ell' \ell m}$ using recursions [@2014PhRvD..89l3504D]. [^10]: Also, sub-leading contributions, namely the kinetic Sunyaev-Zeldovich effect [@1980MNRAS.190..413S] and changes in the tSZ signal induced by the observer motion relative to the CMB rest frame , as well as relativistic corrections , are specific to each particular cluster. Their inclusion could be considered in more detailed predictions in future, but represent higher-order corrections for the present study. [^11]: Here, the subscript ${\rm i}$ denotes the initial time of the dissipation process. [^12]: Here and in Eq.  we neglect the effect of photon emission/absorption processes, which is instead remarkable at low frequencies (see and ). [^13]: For the sake of generality and for the purpose of cross-checking, we also include the monopole term, which can be easily subtracted afterwards. [^14]: For real experiments, these patterns are weakly modulated (and their perfect symmetry broken) by the second-order (‘orbital dipole’) effect coming from the Earth’s motion around the Sun and (for spacecraft moving around the Earth-Sun L2 point), by the further contribution from motion in the Lissajous orbit. [^15]: Sampling variance, as specified by the adopted masks, will be taken into account in the next section. [^16]: In some cases we found it affects only the last digit (reported in the tables) of the estimated $\sqrt{|\Delta \chi^2|} \; {\rm sign}(\Delta \chi^2)$. [^17]: The adopted number of realizations allows to provide an estimate the rms of the quoted significance values suitable to check (particularly for some results based, for simplicity, on a single realization) they are not in the tail of distribution, to quantitatively compare pros and cons of the three adopted approaches, and to spot in the results the effects of coupling between signal and noise/residuals realizations. With much more realizations it is obviously possible to refine these estimates, but it is not relevant in this work that deals with wide ranges of residual parameters. [^18]: This holds also in the case that the instrument sensitivity varies across the sky because of non-uniform sky coverage from the adopted scanning strategy (an aspect that is not so crucial in the case of the relatively uniform sky coverage expected for CORE [@CORE2016; @2017MNRAS.466..425W]) is included in the analysis. Note also that, in principle, pixel-to-pixel correlations introduced by noise correlations and potential residual morphologies should be included in the $\chi^2$. This aspect, although important in the analysis of real data, is outside the scope of the present paper. Nonetheless, it does not affect the main results of our forecasts. [^19]: We adopt the convention (row index range, column index range). [^20]: We also considered a mask that excludes also all pixels within $30^\circ$ of the Ecliptic plane, to avoid zodiacal-light contamination, but the resulting map has considerably less statistical power due to the low overall sky fraction.

--- abstract: 'We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative coordinate system is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies.' address: - 'Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA' - 'Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA' author: - Taeyoung Lee - Melvin Leok - 'N. Harris McClamroch' bibliography: - 'lgvifbp.bib' title: Lie Group Variational Integrators for the Full Body Problem --- , , and Variational integrators, Lie group method, full body problem Nomenclature {#nomenclature .unnumbered} ============ ------------------- -------------------------------------------------------------------------------------------------------------- ---- $\gamma_i$ Linear momentum of the $i$th body in the inertial frame p. $\Gamma$ Relative linear momentum p. $J_i$ Standard moment of inertia matrix of the $i$th body p. $J_{d_i}$ Nonstandard moment of inertia matrix of the $i$th body p. $J_R$ Standard moment of inertia matrix of the first body with respect to the second body fixed frame p. $J_{d_R}$ Nonstandard moment of inertia matrix of the first body with respect to the second body fixed frame p. $m_i$ Mass of the $i$th body p. $m$ Reduced mass for two bodies of mass $m_1$ and $m_2$, $m=\frac{m_1m_2}{m_1+m_2}$ p. $M_i$ Gravity gradient moment on the $i$th body p. $\Omega_i$ Angular velocity of the $i$th body in its body fixed frame p. $\Omega$ Angular velocity of the first body expressed in the second body fixed frame p. $\mathbf{\Omega}$ $\mathbf{\Omega}=(\Omega_1,\Omega_2,\cdots,\Omega_n)$ p. $\Pi_i$ Angular momentum of the $i$th body in its body fixed frame p. $\Pi$ Angular momentum of the first body expressed in the second body fixed frame p. $R_i$ Rotation matrix from the $i$th body fixed frame to the inertial frame p. $R$ Relative attitude of the first body with respect to the second body p. $\mathbf{R}$ $\mathbf{R}=(R_1,R_2,\cdots,R_n)$ p. $v_i$ Velocity of the mass center of the $i$th body in the inertial frame p. $V$ Relative velocity of the first body with respect to the second body in the second body fixed frame p. $V_2$ Velocity of the second body in the second body fixed frame p. $x_i$ Position of the mass center of the $i$th body in the inertial frame p. $X$ Relative position of the first body with respect to the second body expressed in the second body fixed frame p. $X_2$ Position of the second body in the second body fixed frame p. $\mathbf{x}$ $\mathbf{x}=(x_1,x_2,\cdots,x_n)$ p. ------------------- -------------------------------------------------------------------------------------------------------------- ---- Introduction ============ Overview -------- The full body problem studies the dynamics of rigid bodies interacting under their mutual potential, and the mutual potential of distributed rigid bodies depends on both the position and the attitude of the bodies. Therefore, the translational and the rotational dynamics are coupled in the full body problem. The full body problem arises in numerous engineering and scientific fields. For example, in astrodynamics, the trajectory of a large spacecraft around the Earth is affected by the attitude of the spacecraft, and the dynamics of a binary asteroid pair is characterized by the non-spherical mass distributions of the bodies. In chemistry, the full rigid body model is used to study molecular dynamics. The importance of the full body problem is summarized in [@pro:Koon03], along with a preliminary discussion from the point of view of geometric mechanics. The full two body problem was studied by Maciejewski [@jo:macie], and he presented equations of motion in inertial and relative coordinates and discussed the existence of relative equilibria in the system. However, he does not derive the equations of motion, nor does he discuss the reconstruction equations that allow the recovery of the inertial dynamics from the relative dynamics. Scheeres derived a stability condition for the full two body problem [@jo:Scheeres02], and he studied the planar stability of an ellipsoid-sphere model [@pro:Scheeres03]. Recently, interest in the full body problem has increased, as it is estimated that up to 16% of near-earth asteroids are binaries [@jo:margot]. Spacecraft motion about binary asteroids have been discussed using the restricted three body model [@pro:Scheeres03b], [@pro:gabern], and the four body model [@jo:Scheeres05]. Conservation laws are important for studying the dynamics of the full body problem, because they describe qualitative characteristics of the system dynamics. The representation used for the attitude of the bodies should be globally defined since the complicated dynamics of such systems would require frequent coordinate changes when using representations that are only defined locally. General numerical integration methods, such as the Runge-Kutta scheme, do not preserve first integrals nor the exact geometry of the full body dynamics [@bk:hairer]. A more careful analysis of computational methods used to study the full body problem is crucial. Variational integrators and Lie group methods provide a systematic method of constructing structure-preserving numerical integrators [@bk:hairer]. The idea of the variational approach is to discretize Hamilton’s principle rather than the continuous equations of motion [@jo:marsden]. The numerical integrator obtained from the discrete Hamilton’s principle exhibits excellent energy properties [@jo:hairer], conserves first integrals associated with symmetries by a discrete version of Noether’s theorem, and preserves the symplectic structure. Many interesting differential equations, including full body dynamics, evolve on a Lie group. Lie group methods consist of numerical integrators that preserve the geometry of the configuration space by automatically remaining on the Lie group [@ic:iserles]. Moser and Vesselov [@jo:moser], Wendlandt and Marsden [@jo:wendlandt] developed numerical integrators for a free rigid body by imposing an orthogonal constraint on the attitude variables and by using unit quaternions, respectively. The idea of using the Lie group structure and the exponential map to numerically compute rigid body dynamics arises in the work of Simo, Tarnow, and Wong [@jo:simo], and in the work by Krysl [@Kr2004]. A Lie group approach is explicitly adopted by Lee, Leok, and McClamroch in the context of a variational integrator for rigid body attitude dynamics with a potential dependent on the attitude, namely the 3D pendulum dynamics, in [@pro:Lee05]. The motion of full rigid bodies depends essentially on the mutual gravitational potential, which in turn depends only on the relative positions and the relative attitudes of the bodies. Marsden et al. introduce discrete Euler-Poincaré and Lie-Poisson equations in [@jo:marsden99] and [@jo:marsden00]. They reduce the discrete dynamics on a Lie group to the dynamics on the corresponding Lie algebra. Sanyal, Shen and McClamroch develop variational integrators for mechanical systems with configuration dependent inertia and they perform discrete Routh reduction in [@jo:sanyal]. A more intrinsic development of discrete Routh reduction is given in [@leok] and [@jo:Leok]. Contributions ------------- The purpose of this paper is to provide a complete set of equations of motion for the full body problem. The equations of motion are categorized by three independent properties: continuous / discrete time, inertial / relative coordinates, and Lagrangian / Hamiltonian forms. Therefore, a total of eight types of equations of motion for the full body problem are given in this paper. The relationships between these equations of motion are shown in [Fig. \[fig:8eom\]]{}, and are further summarized in [Fig. \[fig:cubic\]]{}. Continuous equations of motion for the full body problem are given in [@jo:macie] without any formal derivation of the equations. We show, in this paper, that the equations of motion for the full body problem can be derived from Hamilton’s principle. The proper form for the variations of Lie group elements in the configuration space lead to a systematic derivation of the equations of motion. This paper develops discrete variational equations of motion for the full body model following a similar variational approach but carried out within a discrete time framework. Since numerical integrators are derived from the discrete Hamilton’s principle, they exhibit symplectic and momentum preserving properties, and good energy behavior. They are also constructed to conserve the Lie group geometry on the configuration space. Numerical simulation results for the interaction of two rigid dumbbell models are given. This paper is organized as follows. The basic idea of the variational integrator is given in section 2. The continuous equations of motion and variational integrators are developed in section 3 and 4. Numerical simulation results are given in section 5. An appendix contains several technical details that are utilized in the main development. Background ========== Hamilton’s principle and variational integrators ------------------------------------------------ The procedure to derive the Euler-Lagrange equations and Hamilton’s equations from Hamilton’s principle is shown in [Fig. \[fig:el\]]{}. When deriving the equations of motion, we first choose generalized coordinates $q$, and a corresponding configuration space $Q$. We then construct a Lagrangian from the kinetic and the potential energy. An action integral $\mathfrak{G}=\int_{t_0}^{t_f} L(q,\dot q) dt$ is defined as the path integral of the Lagrangian along a time-parameterized trajectory. After taking the variation of the action integral, and requiring it to be stationary, we obtain the Euler-Lagrange equations. If we use the Legendre transformation defined as $$\begin{aligned} p\cdot\delta\dot{q}&=\mathbb{F}L(q,\dot q) \cdot\delta\dot{q},\nonumber\\& = \frac{d}{d\epsilon} \bigg|_{\epsilon=0} L(q,\dot q+\epsilon \delta\dot{q}),\label{eqn:FL}\end{aligned}$$ where $\delta\dot{q}\in T_q Q$, then we obtain Hamilton’s equations in terms of momenta variables $p=\mathbb{F}L(q,\dot{q})\in T^*Q$. These equations are equivalent to the Euler-Lagrange equations [@bk:marsden]. There are two issues that arise in applying this procedure to the full body problem. The first is that the configuration space for each rigid body is the semi-direct product, ${\ensuremath{\mathrm{SE(3)}}}=\Rset^3 \,\textcircled{s}\,{\ensuremath{\mathrm{SO(3)}}}$, where ${\ensuremath{\mathrm{SO(3)}}}$ is the Lie group of orthogonal matrices with determinant 1. Therefore, variations should be carefully chosen such that they respect the geometry of the configuration space. For example, a varied rotation matrix $R^\epsilon\in{\ensuremath{\mathrm{SO(3)}}}$ can be expressed as $$\begin{aligned} R^\epsilon = R e^{\epsilon\eta},\end{aligned}$$ where $\epsilon\in\Rset$, and $\eta\in{\ensuremath{\mathfrak{so}(3)}}$ is a variation represented by a skew-symmetric matrix. The variation of the rotation matrix $\delta R$ is the part of $R^\epsilon$ that is linear in $\epsilon$: $$\begin{aligned} R^\epsilon = R + \epsilon \delta R + \mathcal{O}(\epsilon^2).\end{aligned}$$ More precisely, $\delta R$ is given by $$\begin{aligned} \delta R = \frac{d}{d\epsilon}\bigg|_{\epsilon=0}R^\epsilon=R\eta.\label{eqn:deltaR0}\end{aligned}$$ The second issue is that reduced variables can be used to obtain equations of motion expressed in relative coordinates. The variations of the reduced variables are constrained as they must arise from the variations of the unreduced variables while respecting the geometry of the configuration space. The proper variations of Lie group elements and reduced quantities are computed while deriving the continuous equations of motion. Generally, numerical integrators are obtained by approximating the continuous Euler-Lagrange equation using a finite difference rule such as $\dot{q}_k=(q_{k+1}-q_k)/h$, where $q_k$ denotes the value of $q(t)$ at $t=hk$ for an integration step size $h\in\Rset$ and an integer $k$. A variational integrator is derived by following a procedure shown in the right column of [Fig. \[fig:el\]]{}. When deriving a variational integrator, the velocity phase space $(q,\dot q)\in TQ$ of the continuous Lagrangian is replaced by $(q_k,q_{k+1})\in Q\times Q$, and the discrete Lagrangian $L_{d}$ is chosen such that it approximates a segment of the action integral $$\begin{aligned} L_d(q_k,q_{k+1})\approx\int_{0}^{h} L{\ensuremath{\left( q_{k,k+1}(t),\dot{q}_{k,k+1}(t) \right)}} dt,\end{aligned}$$ where $q_{k,k+1}(t)$ is the solution of the Euler-Lagrange equation satisfying boundary conditions $q_{k,k+1}(0)=q_k$ and $q_{k,k+1}(h)=q_{k+1}$. Then, the discrete action sum $\mathfrak{G}_d=\sum L_d(q_k, q_{k+1})$ approximates the action integral $\mathfrak{G}$. Taking the variations of the action sum, we obtain the discrete Euler Lagrange equation $$\begin{aligned} \mathbf{D}_{q_k}L_d(q_{k-1},q_k)+\mathbf{D}_{q_k} L_d(q_k,q_{k+1})=0,\end{aligned}$$ where $\mathbf{D}_{q_k}L_d$ denotes the partial derivative of $L_d$ with respect to $q_k$. This yields a discrete Lagrangian map $F_{L_d}:(q_{k-1},q_k)\mapsto(q_k,q_{k+1})$. Using a discrete analogue of the Legendre transformation, referred to as a discrete fiber derivative $\mathbb{F}L_d:Q\times Q\rightarrow T^* Q$, variational integrators can be expressed in Hamiltonian form as $$\begin{aligned} p_k & =-\mathbf{D}_{q_k} L_d (q_k,q_{k+1}),\label{eqn:dLtk}\\ p_{k+1} & = \mathbf{D}_{q_{k+1}} L_d (q_k,q_{k+1}).\label{eqn:dLtkp}\end{aligned}$$ This yields a discrete Hamiltonian map $\tilde{F}_{L_d}:(q_{k},p_k)\mapsto(q_{k+1},p_{k+1})$. A complete development of variational integrators can be found in [@jo:marsden]. Notation -------- Variables in the inertial and the body fixed coordinates are denoted by lower-case and capital letters, respectively. A subscript $i$ is used for variables related to the $i$th body. The relative variables have no subscript and the $k$th discrete variables have the second level subscript $k$. The letters $x,v,\Omega$ and $R$ are used to denote the position, velocity, angular velocity and rotation matrix, respectively. The trace of $A\in\Rset^{n\times n}$ is denoted by $$\begin{aligned} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ A \right]}}}}=\sum_{i=1}^{n}\, [A]_{ii},\end{aligned}$$ where $[A]_{ii}$ is the $i,i$th element of $A$. It can be shown that $$\begin{gathered} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ AB \right]}}}}={\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ BA \right]}}}}={\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ B^TA^T \right]}}}}={\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ A^TB^T \right]}}}},\label{eqn:trAB}\\ {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ A^TB \right]}}}}=\sum_{p,q=1}^3 [A]_{pq} [B]_{pq},\label{eqn:trSum}\\ {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ PQ \right]}}}}=0,\label{eqn:trskew}\end{gathered}$$ for matrices $A,B\in\Rset^{n\times n}$, a skew-symmetric matrix $P=-P^T\in\Rset^{n\times n}$, and a symmetric matrix $Q=Q^T\in\Rset^{n\times n}$. The map $S:\Rset^3\mapsto\Rset^{3\times 3}$ is defined by the condition that $S(x)y=x\times y$ for $x,y\in\Rset^3$. For $x=(x_1,x_2,x_3)\in\Rset^3$, $S(x)$ is given by $$\begin{aligned} S(x)=\begin{bmatrix}0&-x_3&x_2\\x_3&0&-x_1\\-x_2&x_1&0\end{bmatrix}.\end{aligned}$$ It can be shown that $$\begin{gathered} S(x)^T=-S(x),\label{eqn:Sskew}\\ S(x\times y)=S(x)S(y)-S(y)S(x)=yx^T-xy^T,\label{eqn:Scross}\\ S(Rx)=RS(x)R^T,\label{eqn:SR}\\ S(x)^TS(x)={\ensuremath{\left( x^T x \right)}} I_{3\times 3} - x x^T={\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ xx^T \right]}}}}I_{3\times 3}-x x^T,\label{eqn:STS}\end{gathered}$$ for $x,y\in\Rset^3$ and $R\in{\ensuremath{\mathrm{SO(3)}}}$. Using homogeneous coordinates, we can represent an element of ${\ensuremath{\mathrm{SE(3)}}}$ as follows: $$\begin{aligned} {\ensuremath{\begin{bmatrix}R&x\\0&1\end{bmatrix}}}\in{\ensuremath{\mathrm{SE(3)}}}\end{aligned}$$ for $x\in\Rset^3$ and $R\in{\ensuremath{\mathrm{SO(3)}}}$. Then, an action on ${\ensuremath{\mathrm{SE(3)}}}$ is given by the usual matrix multiplication in $\Rset^{4\times 4}$. For example $$\begin{aligned} {\ensuremath{\begin{bmatrix}R_1&x_1\\0&1\end{bmatrix}}}{\ensuremath{\begin{bmatrix}R_2&x_2\\0&1\end{bmatrix}}}={\ensuremath{\begin{bmatrix}R_1R_2&R_1x_2+x_1\\0&1\end{bmatrix}}}.\end{aligned}$$ for $x_1,x_2\in\Rset^3$ and $R_1,R_2\in{\ensuremath{\mathrm{SO(3)}}}$. The action of an element of ${\ensuremath{\mathrm{SE(3)}}}$ on $\Rset^3$ can be expressed using a matrix-vector product, once we identify $\Rset^3$ with $\Rset^3\times\{1\}\subset\Rset^4$. In particular, we see from $${\ensuremath{\begin{bmatrix}R&x_1\\0&1\end{bmatrix}}} \begin{bmatrix} x_2 \\ 1\end{bmatrix} = \begin{bmatrix} Rx_2 + x_1\\ 1\end{bmatrix}$$ that $x_2\mapsto Rx_2+x_1$. Continuous time full body models ================================ In this section, the continuous time equations of motion for the full body problem are derived in inertial and relative coordinates, and are expressed in both Lagrangian and in Hamiltonian form. We define $O-e_1e_2e_3$ as an inertial frame, and $O_{\mathcal{B}_i}-E_{i_1}E_{i_2}E_{i_3}$ as a body fixed frame for the $i$th body, $\mathcal{B}_i$. The origin of the $i$th body fixed frame is located at the center of mass of body $\mathcal{B}_i$. The configuration space of the $i$th rigid body is ${\ensuremath{\mathrm{SE(3)}}}=\Rset^3\,\textcircled{s}\,{\ensuremath{\mathrm{SO(3)}}}$. We denote the position of the center of mass of $\mathcal{B}_i$ in the inertial frame by $x_i\in\Rset^3$,\[no:xi\] and we denote the attitude of $\mathcal{B}_i$ by $R_i\in{\ensuremath{\mathrm{SO(3)}}}$,\[no:Ri\] which is a rotation matrix from the $i$th body fixed frame to the inertial frame. Inertial coordinates -------------------- *Lagrangian:* To derive the equations of motion, we first construct a Lagrangian for the full body problem. Given $(x_i,R_i)\in{\ensuremath{\mathrm{SE(3)}}}$, the inertial position of a mass element of $\mathcal{B}_i$ is given by $x_i+R_i\rho_i$, where $\rho_i\in\Rset^3$ denotes the position of the mass element in the body fixed frame. Then, the kinetic energy of $\mathcal{B}_i$ can be written as $$\begin{aligned} T_i & = \frac{1}{2}\int_{\mathcal{B}_i} \|\dot{x}_i+\dot{R}_i\rho_i\|^2\, dm_i.\end{aligned}$$ Using the fact that $\int_{\mathcal{B}_i} \rho_i dm_i=0$ and the kinematic equation $\dot{R}_i=R_iS(\Omega_i)$, the kinetic energy $T_i$ can be rewritten as $$\begin{aligned} T_i(\dot{x}_i,\Omega_i) & = \frac{1}{2}\int_{\mathcal{B}_i} {\ensuremath{\left\| \dot{x}_i \right\|}}^2 +{\ensuremath{\left\| S(\Omega_i)\rho_i \right\|}}^2 dm_i,\nonumber\\ & = \frac{1}{2}m_i {\ensuremath{\left\| \dot{x}_i \right\|}}^2 + \frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega_i)J_{d_i}S(\Omega_i)^T \right]}}}},\label{eqn:Ti}\end{aligned}$$ where $m_i\in\Rset$ is the total mass of $\mathcal{B}_i$\[no:mi\], $\Omega_i\in\Rset^3$ is the angular velocity of $\mathcal{B}_i$ in the body fixed frame,\[no:Omegai\] and $J_{d_i}=\int_{\mathcal{B}_i}\rho_i\rho_i^Tdm_i\in\Rset^{3\times 3}$\[no:Jdi\] is a nonstandard moment of inertia matrix. Using [(\[eqn:STS\])]{}, it can be shown that the standard moment of inertia matrix $J_{i}=\int_{\mathcal{B}_i}S(\rho_i)^TS(\rho_i)dm_i\in\Rset^{3\times 3}$\[no:Ji\] is related to $J_{d_i}$ by $$\begin{aligned} J_i={\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ J_{d_i} \right]}}}}I_{3\times 3} - J_{d_i},\end{aligned}$$ and that the following condition holds for any $\Omega_i\in\Rset^3$ $$\begin{aligned} S(J_i\Omega_i)=S(\Omega_i)J_{d_i}+J_{d_i}S(\Omega_i).\label{eqn:JdJ}\end{aligned}$$ We first derive equations using the nonstandard moment of inertia matrix $J_{d_i}$, and then express them in terms of the standard moment of inertia $J_i$ by using [(\[eqn:JdJ\])]{}. The gravitational potential energy $U:{\ensuremath{\mathrm{SE(3)}}}^n\mapsto\Rset$ is given by $$\begin{aligned} U(x_1,x_2,\cdots,x_n,R_1,R_2,\cdots,R_n) =-\frac{1}{2}\sum_{\substack{i,j=1\\i\neq j}}^{n}\int_{\mathcal{B}_i}\int_{\mathcal{B}_j}\frac{G dm_i dm_j}{{\ensuremath{\left\| x_i+R_i\rho_i - x_j - R_j\rho_j \right\|}}},\label{eqn:U}\end{aligned}$$ where $G$ is the universal gravitational constant. Then, the Lagrangian for $n$ full bodies can be written as $$\begin{aligned} L(\mathbf{x},\mathbf{\dot x},\mathbf{R},\mathbf{\Omega})=\sum_{i=1}^{n} \frac{1}{2}m_i {\ensuremath{\left\| \dot{x}_i \right\|}}^2 + \frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega_i)J_{d_i}S(\Omega_i)^T \right]}}}} - U(\mathbf{x},\mathbf{R}),\label{eqn:L}\end{aligned}$$ where bold type letters denote ordered $n$-tuples of variables. For example, $\mathbf{x}\in(\Rset^3)^n$, $\mathbf{R}\in{\ensuremath{\mathrm{SO(3)}}}^n$, and $\mathbf{\Omega}\in(\Rset^3)^n$ are defined as $\mathbf{x}=(x_1,x_2,\cdots,x_n)$,\[no:bfx\] $\mathbf{R}=(R_1,R_2,\cdots,R_n)$,\[no:bfR\] and $\mathbf{\Omega}=(\Omega_1,\Omega_2,\cdots,\Omega_n)$, respectively.\[no:bfOmega\] *Variations of variables:* Since the configuration space is ${\ensuremath{\mathrm{SE(3)}}}^n$, the variations should be carefully chosen such that they respect the geometry of the configuration space. The variations of $x_i:[t_0,t_f]\mapsto\Rset^3$ and $\dot{x}_i:[t_0,t_f]\mapsto\Rset^3$ are trivial, namely $$\begin{aligned} x_i^\epsilon = x_i + \epsilon\delta x_i + \mathcal{O}(\epsilon^2),\\ \dot{x}_i^\epsilon = \dot{x}_i + \epsilon\delta \dot{x}_i + \mathcal{O}(\epsilon^2),\end{aligned}$$ where $\delta x_i:[t_0,t_f]\mapsto\Rset^3$, $\delta \dot{x}_i:[t_0,t_f]\mapsto\Rset^3$ vanish at the initial time $t_0$ and at the final time $t_f$. The variation of $R_i:[t_0,t_f]\mapsto{\ensuremath{\mathrm{SO(3)}}}$, as given in [(\[eqn:deltaR0\])]{}, is $$\begin{aligned} \delta R_i = R_i \eta_i\label{eqn:delRi},\end{aligned}$$ where $\eta_i:[t_0,t_f]\mapsto{\ensuremath{\mathfrak{so}(3)}}$ is a variation with values represented by a skew-symmetric matrix ($\eta_i^T=-\eta_i$) vanishing at $t_0$ and $t_f$. The variation of $\Omega_i$ can be computed from the kinematic equation $\dot{R}_i=R_iS(\Omega_i)$ and [(\[eqn:delRi\])]{} to be $$\begin{aligned} S(\delta \Omega_i) & = \frac{d}{d\epsilon}\bigg|_{\epsilon=0} R_i^{\epsilon T}\dot{R}_i^\epsilon=\delta R_i^T \dot{R}_i + R_i^T \delta \dot{R}_i,\nonumber\\ & = -\eta_iS(\Omega_i) + S(\Omega_i)\eta_i +\dot{\eta}_i .\label{eqn:delOmegai}\end{aligned}$$ ### Equations of motion: Lagrangian form If we take variations of the Lagrangian using [(\[eqn:delRi\])]{} and [(\[eqn:delOmegai\])]{}, we obtain the equations of motion from Hamilton’s principle. We first take the variation of the kinetic energy of $\mathcal{B}_i$. $$\begin{aligned} \delta T_i & = \frac{d}{d\epsilon}\bigg|_{\epsilon=0}T_i(\dot{x}_i+\epsilon\delta\dot{x}_i,\Omega_i+\epsilon\delta\Omega_i),\nonumber\\ & = m_i \dot{x}_i^T \delta\dot{x}_i + \frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\delta\Omega_i)J_{d_i}S(\Omega_i)^T+S(\Omega_i)J_{d_i}S(\delta\Omega_i)^T \right]}}}}.\end{aligned}$$ Substituting [(\[eqn:delOmegai\])]{} into the above equation and using [(\[eqn:trAB\])]{}, we obtain $$\begin{aligned} \delta T_i & = m_i \dot{x}_i^T \delta\dot{x}_i + \frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ -\dot\eta_i {\ensuremath{\left\{ J_{d_i}S(\Omega_i)+S(\Omega_i)J_{d_i} \right\}}} \right]}}}}\nonumber\\ & \quad + \frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_i{\ensuremath{\left\{ S(\Omega_i){\ensuremath{\left\{ J_{d_i}S(\Omega_i)+S(\Omega_i)J_{d_i} \right\}}}-{\ensuremath{\left\{ J_{d_i}S(\Omega_i)+S(\Omega_i)J_{d_i} \right\}}}S(\Omega_i) \right\}}} \right]}}}}.\end{aligned}$$ Using [(\[eqn:Scross\])]{} and [(\[eqn:JdJ\])]{}, $\delta T_i$ is given by $$\begin{aligned} \delta T_i & = m_i \dot{x}_i^T \delta\dot{x}_i +\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ -\dot{\eta}_iS(J_i\Omega_i)+\eta_iS(\Omega_i\times J_i\Omega_i) \right]}}}}.\label{eqn:delTi}\end{aligned}$$ The variation of the potential energy is given by $$\begin{aligned} \delta U & = \frac{d}{d\epsilon}\bigg|_{\epsilon=0} U(\mathbf{x}+\epsilon\delta\mathbf{x},\mathbf{R}+\epsilon\delta\mathbf{R}),$$ where $\delta\mathbf{x}=(\delta x_1,\delta x_2,\cdots,\delta x_n)$, $\delta \mathbf{R}=(\delta R_1,\delta R_2,\cdots,\delta R_n)$. Then, $\delta U$ can be written as $$\begin{aligned} \delta U & =\sum_{i=1}^n{\ensuremath{\left( \sum_{p=1}^3{\ensuremath{\frac{\partial U}{\partial [x_i]_p}}}[\delta x_i]_p + \sum_{p,q=1}^{3}{\ensuremath{\frac{\partial U}{\partial [R_i]_{pq}}}}[R_i\eta_i]_{pq} \right)}},\nonumber\\ & =\sum_{i=1}^n{\ensuremath{\left( {\ensuremath{\frac{\partial U}{\partial x_i}}}^T\delta x_i-{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_iR^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}} \right]}}}} \right)}},\label{eqn:delU}\end{aligned}$$ where $[A]_{pq}$ denotes the $p,q$th element of a matrix $A$, and ${\ensuremath{\frac{\partial U}{\partial x_i}}},\,{\ensuremath{\frac{\partial U}{\partial R_i}}}$ are given by $[{\ensuremath{\frac{\partial U}{\partial x_i}}}]_p={\ensuremath{\frac{\partial U}{\partial [x_i]_p}}}$, $[{\ensuremath{\frac{\partial U}{\partial R_i}}}]_{pq}={\ensuremath{\frac{\partial U}{\partial [R_i]_{pq}}}}$, respectively. The variation of the Lagrangian has the form $$\begin{aligned} \delta L = \sum_{i=1}^{n} \delta T_i - \delta U,\label{eqn:delL}\end{aligned}$$ which can be written more explicitly by using [(\[eqn:delTi\])]{} and [(\[eqn:delU\])]{}. The action integral is defined to be $$\begin{aligned} \mathfrak{G} = \int_{t_0}^{t_f} L(\mathbf{x},\mathbf{\dot x},\mathbf{R},\mathbf{\Omega})\,dt.\label{eqn:caction}\end{aligned}$$ The variation of the action integral can be written as $$\begin{aligned} \delta \mathfrak{G} = \sum_{i=1}^{n} \int_{t_0}^{t_f} & m_i \dot{x}_i^T \delta\dot{x}_i-{\ensuremath{\frac{\partial U}{\partial x_i}}}^T\delta x_i+\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ -\dot{\eta}_iS(J_i\Omega_i)+\eta_i{\ensuremath{\left\{ S(\Omega_i\times J_i\Omega_i)+2R^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}} \right\}}} \right]}}}}\,dt.\end{aligned}$$ Using integration by parts, $$\begin{aligned} \delta \mathfrak{G} & = \sum_{i=1}^{n}\; m_i\dot{x}_i^T \delta x_i \bigg|_{t_0}^{t_f}-\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_iS(J_i\Omega_i) \right]}}}}\bigg|_{t_0}^{t_f}+\int_{t_0}^{t_f} -m_i\ddot{x}_i^T \delta x_i +\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_iS(J_i\dot\Omega_i) \right]}}}}\,dt\\& \quad + \sum_{i=1}^{n} \int_{t_0}^{t_f}-{\ensuremath{\frac{\partial U}{\partial x_i}}}^T\delta x_i+\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_i{\ensuremath{\left\{ S(\Omega_i\times J_i\Omega_i)+2R^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}} \right\}}} \right]}}}}\,dt.\end{aligned}$$ Using the fact that $\delta x_i$ and $\eta_i$ vanish at $t_0$ and $t_f$, the first two terms of the above equation vanish. Then, $\delta\mathfrak{G}$ is given by $$\begin{aligned} \delta \mathfrak{G} & = \sum_{i=1}^{n} \int_{t_0}^{t_f} - \delta x_i^T {\ensuremath{\left\{ m_i\ddot{x}_i+{\ensuremath{\frac{\partial U}{\partial x_i}}} \right\}}}+\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_i{\ensuremath{\left\{ S(J_i\dot{\Omega}_i+\Omega_i\times J_i\Omega_i)+2R^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}} \right\}}} \right]}}}} \,dt.\end{aligned}$$ From Hamilton’s principle, $\delta\mathfrak{G}$ should be zero for all possible variations $\delta x_i:[t_0,t_f]\mapsto\Rset^3$ and $\eta_i:[t_0,t_f]\mapsto{\ensuremath{\mathfrak{so}(3)}}$. Therefore, the expression in the first brace should be zero. Furthermore, since $\eta_i$ is skew symmetric, we have by [(\[eqn:trskew\])]{}, that the expression in the second brace should be symmetric. Then, we obtain the continuous equations of motion as $$\begin{gathered} m_i \ddot{x}_i = -{\ensuremath{\frac{\partial U}{\partial x_i}}},\nonumber\\ S(J_i\dot{\Omega}_i+\Omega_i\times J_i\Omega_i)+2R^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}}=S(J_i\dot{\Omega}_i+\Omega_i\times J_i\Omega_i)^T+2{\ensuremath{\frac{\partial U}{\partial R_i}}}^TR.\label{eqn:SS}\end{gathered}$$ Using [(\[eqn:Sskew\])]{}, we can simplify [(\[eqn:SS\])]{} to be $$\begin{gathered} S(J_i\dot{\Omega}_i+\Omega_i\times J_i\Omega_i)={\ensuremath{\frac{\partial U}{\partial R_i}}}^TR-R^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}}.\end{gathered}$$ Note that the right hand side expression in the above equation is also skew symmetric. Then, the moment due to the gravitational potential on the $i$th body, $M_i\in\Rset^3$ is given by $S(M_i)={\ensuremath{\frac{\partial U}{\partial R_i}}}^TR_i-R^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}}$. This moment $M_i$ can be expressed more explicitly as the following computation shows. $$\begin{aligned} S(M_i) & = {\ensuremath{\frac{\partial U}{\partial R_i}}}^TR_i-R^T_i{\ensuremath{\frac{\partial U}{\partial R_i}}},\\ & = \begin{bmatrix} & & \\u^T_{ri_1}&u^T_{ri_2}&u^T_{ri_3}\\& & \end{bmatrix} \begin{bmatrix}&r_{i_1}&\\&r_{i_2}&\\&r_{i_3}\end{bmatrix} -\begin{bmatrix} & & \\r_{i_1}^T&r_{i_2}^T&r_{i_3}^T\\& & \end{bmatrix} \begin{bmatrix}&u_{ri_1}&\\&u_{ri_2}&\\&u_{ri_3}\end{bmatrix},\\ & = {\ensuremath{\left( u^T_{ri_1}r_{i_1}-r_{i_1}^Tu_{ri_1} \right)}}+{\ensuremath{\left( u^T_{ri_2}r_{i_2}-r_{i_2}^Tu_{ri_2} \right)}}+{\ensuremath{\left( u^T_{ri_3}r_{i_3}-r_{i_3}^Tu_{ri_3} \right)}},\end{aligned}$$ where $r_{i_p},u_{ri_p}\in\Rset^{1\times 3}$ are the $p$th row vectors of $R_i$ and ${\ensuremath{\frac{\partial U}{\partial R_i}}}$, respectively. Substituting $x=r_{i_p}^T$, $y=u_{ri_p}^T$ into [(\[eqn:Scross\])]{}, we obtain $$\begin{aligned} S(M_i) & = S(r_{i_1}\times u_{ri_1})+S(r_{i_2}\times u_{ri_2})+S(r_{i_3}\times u_{ri_3}),\nonumber\\ &=S(r_{i_1}\times u_{ri_1}+r_{i_2}\times u_{ri_2}+r_{i_3}\times u_{ri_3}),\label{eqn:SMi}\end{aligned}$$ Then, the gravitational moment $M_i$ is given by $$\begin{aligned} M_i=r_{i_1}\times u_{ri_1}+r_{i_2}\times u_{ri_2}+r_{i_3}\times u_{ri_3}.\label{eqn:Mi}\end{aligned}$$\[no:Mi\] In summary, *the continuous equations of motion for the full body problem, in Lagrangian form,* can be written for $i\in(1,2,\cdots,n)$ as $$\begin{gathered} \dot{v}_i=-\frac{1}{m_i}{\ensuremath{\frac{\partial U}{\partial x_i}}},\label{eqn:vidot}\\ J_i\dot{\Omega}_i+\Omega_i\times J_i\Omega_i = M_i,\\ \dot{x}_i=v_i,\\ \dot{R}_i=R_iS(\Omega_i),\label{eqn:Ridot}\end{gathered}$$ where the translational velocity $v_i\in\Rset^3$ is defined as $v_i=\dot{x}_i$.\[no:vi\] ### Equations of motion: Hamiltonian form We denote the linear and angular momentum of the $i$th body $\mathcal{B}_i$ by $\gamma_i\in\Rset^3$\[no:gammai\], and $\Pi_i\in\Rset^3$\[no:Pii\], respectively. They are related to the linear and angular velocities by $\gamma_i = m_i v_i$, and $\Pi_i = J_i \Omega_i$. Then, the equations of motion can be rewritten in terms of the momenta variables. *The continuous equations of motion for the full body problem, in Hamiltonian form,* can be written for $i\in(1,2,\cdots,n)$ as $$\begin{gathered} \dot{\gamma}_i=-{\ensuremath{\frac{\partial U}{\partial x_i}}},\label{eqn:gamidot}\\ \dot{\Pi}_i+\Omega_i\times\Pi_i = M_i,\\ \dot{x}_i=\frac{\gamma_i}{m_i},\\ \dot{R}_i=R_iS(\Omega_i).\label{eqn:Ridoth}\end{gathered}$$ Relative coordinates -------------------- The motion of the full rigid bodies depends only on the relative positions and the relative attitudes of the bodies. This is a consequence of the property that the gravitational potential can be expressed using only these relative variables. Physically, this is related to the fact that the total linear momentum and the total angular momentum about the mass center of the bodies are conserved. Mathematically, the Lagrangian is invariant under a left action of an element of ${\ensuremath{\mathrm{SE(3)}}}$. So, it is natural to express the equations of motion in one of the body fixed frames. In this section, the equations of motion for the full two body problem are derived in relative coordinates. This result can be readily generalized to the $n$ body problem. *Reduction of variables:* In [@jo:macie], the reduction is carried out in stages, by first reducing position variables in $\Rset^3$, and then reducing attitude variables in ${\ensuremath{\mathrm{SO(3)}}}$. This is equivalent to directly reducing the position and the attitude variables in ${\ensuremath{\mathrm{SE(3)}}}$ in a single step, which is a result that can be explained by the general theory of Lagrangian reduction by stages [@jo:cemara]. The reduced position and the reduced attitude variables are the relative position and the relative attitude of the first body with respect to the second body. In other words, the variables are reduced by applying the inverse of $(R_2,x_2)\in{\ensuremath{\mathrm{SE(3)}}}$ given by $(R_2^T,-R_2^Tx_2)\in{\ensuremath{\mathrm{SE(3)}}}$, to the following homogeneous transformations: $$\begin{aligned} {\ensuremath{\begin{bmatrix}R_2^T&-R_2^Tx_2\\0&1\end{bmatrix}}}{\ensuremath{\left( {\ensuremath{\begin{bmatrix}R_1&x_1\\0&1\end{bmatrix}}},{\ensuremath{\begin{bmatrix}R_2&x_2\\0&1\end{bmatrix}}} \right)}}& = {\ensuremath{\left( {\ensuremath{\begin{bmatrix}R_2^TR_1&R_2^T(x_1-x_2)\\0&1\end{bmatrix}}},{\ensuremath{\begin{bmatrix}R_2^TR_2&R_2^T(x_2-x_2)\\0&1\end{bmatrix}}} \right)}},\nonumber\\ & = {\ensuremath{\left( {\ensuremath{\begin{bmatrix}R_2^TR_1&R_2^T(x_1-x_2)\\0&1\end{bmatrix}}},{\ensuremath{\begin{bmatrix}I_{3\times 3}&0\\0&1\end{bmatrix}}} \right)}}.\label{eqn:HT}\end{aligned}$$ This motivates the definition of the reduced variables as $$\begin{aligned} X & = R_2^T(x_1-x_2),\label{eqn:X}\\ R & = R_2^TR_1,\label{eqn:R}\end{aligned}$$ where $X\in\Rset^3$ is the relative position of the first body with respect to the second body expressed in the second body fixed frame, and $R\in{\ensuremath{\mathrm{SO(3)}}}$ is the relative attitude of the first body with respect to the second body. The corresponding linear and angular velocities are also defined as $$\begin{aligned} V & = R_2^T(\dot{x}_1-\dot{x}_2),\\ \Omega & = R\Omega_1,\end{aligned}$$ where $V\in\Rset^3$ represents the relative velocity of the first body with respect to the second body in the second body fixed frame\[no:V\], and $\Omega\in\Rset^3$ is the angular velocity of the first body expressed in the second body fixed frame\[no:Omega\]. Here, the capital letters denote variables expressed in the second body fixed frame. For convenience, we denote the inertial position and the inertial velocity of the second body, expressed in the second body fixed frame by $X_2,V_2\in\Rset^3$: $$\begin{aligned} X_2 & = R_2^T x_2,\label{eqn:X2}\\ V_2 & = R_2^T \dot{x}_2.\label{eqn:V2}\end{aligned}$$ *Reduced Lagrangian:* The equations of motion in relative coordinates are derived in the same way used to derive the equations in the inertial frame. We first construct a reduced Lagrangian. The reduced Lagrangian $l$ is obtained by expressing the original Lagrangian [(\[eqn:L\])]{} in terms of the reduced variables. The kinetic energy is given by $$\begin{aligned} T_1 + T_2 & =\frac{1}{2}m_1{\ensuremath{\left\| \dot{x}_1 \right\|}}^2 +\frac{1}{2}m_2{\ensuremath{\left\| \dot{x}_2 \right\|}}^2+\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega_1)J_{d_1}S(\Omega_1)^T \right]}}}} +\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega_2)J_{d_2}S(\Omega_2)^T \right]}}}},\\ & =\frac{1}{2}m_1{\ensuremath{\left\| V+V_2 \right\|}}^2 +\frac{1}{2}m_2{\ensuremath{\left\| V_2 \right\|}}^2+\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega)J_{d_R}S(\Omega)^T \right]}}}} +\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega_2)J_{d_2}S(\Omega_2)^T \right]}}}},\end{aligned}$$ where [(\[eqn:SR\])]{} is used, and $J_{d_{R}}\in\Rset^{3\times 3}$ is defined as $J_{d_{R}}=RJ_{d_{1}}R^T$, which is an expression of the nonstandard moment of inertia matrix of the first body with respect to the second body fixed frame.\[no:JdR\] Note that $J_{d_{R}}$ is not a constant matrix. Using [(\[eqn:SR\])]{}, it can be shown that $J_{d_{R}}$ also satisfies a property similar to [(\[eqn:JdJ\])]{}, namely $$\begin{aligned} S(J_{R}\Omega)=S(\Omega)J_{d_{R}}+J_{d_{R}}S(\Omega),\label{eqn:JdRJR}\end{aligned}$$ where $J_R=RJ_1R^T\in\Rset^{3\times 3}$ is the standard moment of inertia matrix of the first body with respect to the second body fixed frame.\[no:JR\] Using [(\[eqn:U\])]{}, the gravitational potential $U$ of the full two bodies is given by $$\begin{aligned} U(x_1,x_2,R_1,R_2) =- \int_{\mathcal{B}_1}\int_{\mathcal{B}_2}\frac{G dm_1 dm_2}{{\ensuremath{\left\| x_1+R_1\rho_1 - x_2 - R_2\rho_2 \right\|}}},\end{aligned}$$ and it is invariant under an action of an element of ${\ensuremath{\mathrm{SE(3)}}}$. Therefore, the gravitational potential can be written as a function of the relative variables only. By applying the inverse of $(R_2,x_2)\in{\ensuremath{\mathrm{SE(3)}}}$ as given in [(\[eqn:HT\])]{}, we obtain $$\begin{aligned} U(x_1,x_2,R_1,R_2) & = U(R_2^T(x_1-x_2),0,R_2^TR_1,I_{3\times 3}),\\ & = - \int_{\mathcal{B}_1}\int_{\mathcal{B}_2}\frac{G dm_1 dm_2}{{\ensuremath{\left\| R_2^T(x_1-x_2)+R_2^T R_1\rho_1 - I_{3\times 3} \rho_2 \right\|}}},\\ & = - \int_{\mathcal{B}_1}\int_{\mathcal{B}_2}\frac{G dm_1 dm_2}{{\ensuremath{\left\| X+R\rho_1 - \rho_2 \right\|}}}, \\ & \triangleq U(X,R).\end{aligned}$$ Here we abuse notation slightly by using the same letter $U$ to denote the gravitational potential as a function of the relative variables. Then, the reduced Lagrangian $l$ is given by $$\begin{aligned} l (R,X,\Omega,V,\Omega_2,V_2)& = \frac{1}{2} m_1 \|V+V_2\|^2 + \frac{1}{2} m_2 \|V_2\|^2 \nonumber \\ & \quad+ \frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega)J_{d_{R}}S(\Omega)^T \right]}}}} + \frac{1}{2} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega_2)J_{d_2} S(\Omega_2)^T \right]}}}}- U(X,R).\label{eqn:l}\end{aligned}$$ *Variations of reduced variables:* The variations of the reduced variables must be restricted to those that can arise from the variations of the original variables. For example, the variation of the relative attitude $R$ is given by $$\begin{aligned} \delta R & = \frac{d}{d\epsilon}\bigg|_{\epsilon=0} R_2^{\epsilon T}R_1^\epsilon,\\ & = \delta R_2^TR_1 + R_2^T \delta R_1.\end{aligned}$$ Substituting [(\[eqn:delRi\])]{} into the above equation, $$\begin{aligned} \delta R & = -\eta_2R_2^TR_1 + R_2^T R_1\eta_1,\\ & = -\eta_2R + \eta R,\end{aligned}$$ where a reduced variation $\eta\in{\ensuremath{\mathfrak{so}(3)}}$ is defined as $\eta=R\eta_1R^T$. The variations of other reduced variables can be obtained in a similar way. The detailed derivations are given in \[apprv\], and we summarize the results as follows: $$\begin{aligned} \delta R & = \eta R - \eta_2 R,\label{eqn:delR}\\ \delta X & = \chi - \eta_2 X,\label{eqn:delX}\\ S(\delta\Omega) & = \dot\eta -S(\Omega)\eta+\eta S(\Omega)+S(\Omega)\eta_2-\eta_2 S(\Omega)+S(\Omega_2)\eta-\eta S(\Omega_2),\label{eqn:delOmega}\\ \delta V & = \dot \chi + S(\Omega_2) \chi -\eta_2 V,\label{eqn:delV}\\ S(\delta\Omega_2) & = \dot\eta_2 +S(\Omega_2)\eta_2-\eta_2 S(\Omega_2),\label{eqn:delOmega2}\\ \delta V_2 & = \dot \chi_2 + S(\Omega_2) \chi_2 -\eta_2 V_2,\label{eqn:delV2}\end{aligned}$$ where $\chi,\chi_2 :[t_0,t_f]\mapsto\Rset^3$ and $\eta,\eta_2 :[t_0,t_f]\mapsto{\ensuremath{\mathfrak{so}(3)}}$ are variations that vanish at the end points. These Lie group variations are the key elements required to obtain the equations of motion in relative coordinates. ### Equations of motion: Lagrangian form The reduced equations of motion can be computed from the reduced Lagrangian using the reduced Hamilton’s principle. By taking the variation of the reduced Lagrangian [(\[eqn:l\])]{} using the constrained variations given by [(\[eqn:delR\])]{} through [(\[eqn:delV2\])]{}, we can obtain the equations of motion in the relative coordinates. Following a similar process to the derivation of $\delta T_i, \delta U$ as in [(\[eqn:delTi\])]{} and [(\[eqn:delU\])]{}, the variation of the reduced Lagrangian $\delta l$ can be obtained as $$\begin{aligned} \delta l & = \dot\chi^T {\ensuremath{\left[ m_1(V+V_2) \right]}} - \chi^T{\ensuremath{\left[ m_1 \Omega_2 \times (V+V_2) \right]}}\nonumber\\ & \quad + \dot\chi_2^T {\ensuremath{\left[ m_1(V+V_2)+m_2V_2 \right]}}-\chi_2^T{\ensuremath{\left[ m_1 \Omega_2 \times (V+V_2)+m_2 \Omega_2 \times V_2 \right]}}\nonumber\\ & \quad +\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ -\dot\eta S(J_R \Omega)+\eta S(\Omega_2\times J_R \Omega) \right]}}}}+\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ -\dot\eta_2S(J_2\Omega_2)+\eta_2 S(\Omega_2\times J_2\Omega_2) \right]}}}}\nonumber\\ &\quad - \chi^T {\ensuremath{\frac{\partial U}{\partial X}}} + {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_2X{\ensuremath{\frac{\partial U}{\partial X}}}^T \right]}}}} + {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_2R{\ensuremath{\frac{\partial U}{\partial R}}}^T-\eta R {\ensuremath{\frac{\partial U}{\partial R}}}^T \right]}}}},\label{eqn:dell}\end{aligned}$$ where we used the identities [(\[eqn:Scross\])]{}, [(\[eqn:JdJ\])]{} and [(\[eqn:JdRJR\])]{}, and the constrained variations [(\[eqn:delR\])]{} through [(\[eqn:delV2\])]{}. The action integral in terms of the reduced Lagrangian is $$\begin{aligned} \mathfrak{G} = \int_{t_0}^{t_f} l (R,X,\Omega,V,\Omega_2,V_2) \,dt.\label{eqn:craction}\end{aligned}$$ Using integration by parts together with the fact that $\chi,\chi_2,\eta$ and $\eta_2$ vanish at $t_0$ and $t_f$, the variation of the action integral can be expressed from [(\[eqn:dell\])]{} as $$\begin{aligned} \delta\mathfrak{G} & = - \int_{t_0}^{t_f} \chi^T {\ensuremath{\left\{ m_1(\dot V+\dot V_2)+m_1 \Omega_2 \times (V+V_2)+{\ensuremath{\frac{\partial U}{\partial X}}} \right\}}}\,dt\\ & \quad -\int_{t_0}^{t_f} \chi_2^T {\ensuremath{\left\{ m_1(\dot V+\dot V_2)+m_2\dot V_2+m_1 \Omega_2 \times (V+V_2)+m_2 \Omega_2 \times V_2 \right\}}}\,dt\\ & \quad +\frac{1}{2}\int_{t_0}^{t_f}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta {\ensuremath{\left\{ S(\dot{(J_R\Omega)}+\Omega_2\times J_R\Omega)-2R {\ensuremath{\frac{\partial U}{\partial R}}}^T \right\}}} \right]}}}}\,dt\\ & \quad +\frac{1}{2}\int_{t_0}^{t_f}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_2{\ensuremath{\left\{ S(J_2\dot{\Omega}_2+\Omega_2\times J_2\Omega_2) +2X{\ensuremath{\frac{\partial U}{\partial X}}}^T+2R{\ensuremath{\frac{\partial U}{\partial R}}}^T \right\}}} \right]}}}}\,dt.\end{aligned}$$ From the reduced Hamilton’s principle, $\delta\mathfrak{G}=0$ for all possible variations $\chi,\chi_2:[t_0,t_f]\mapsto\Rset^3$ and $\eta,\eta_2:[t_0,t_f]\mapsto{\ensuremath{\mathfrak{so}(3)}}$ that vanish at $t_0$ and $t_f$. Therefore, in the above equation, the expressions in the first two braces should be zero and the expressions in the last two braces should be symmetric since $\eta,\eta_2$ are skew symmetric. Then, we obtain the following equations of motion, $$\begin{gathered} m_1(\dot V+\dot V_2)+m_1 \Omega_2 \times (V+V_2)=-{\ensuremath{\frac{\partial U}{\partial X}}},\label{eqn:Vdot0}\\ m_2\dot V_2+m_2\Omega_2 \times V_2={\ensuremath{\frac{\partial U}{\partial X}}},\label{eqn:V2dot0}\\ S(\dot{(J_R\Omega)}+\Omega_2\times J_R\Omega)=-S(M),\nonumber\\ S(J_2\dot{\Omega}_2+\Omega_2\times J_2\Omega_2)={\ensuremath{\frac{\partial U}{\partial X}}}X^T-X{\ensuremath{\frac{\partial U}{\partial X}}}^T+S(M),\label{eqn:Pi2dot0}\end{gathered}$$ where $M\in\Rset^3$ is defined by the relation $S(M)={\ensuremath{\frac{\partial U}{\partial R}}}R^T-R {\ensuremath{\frac{\partial U}{\partial R}}}^T$. By a procedure analogous to the derivation of [(\[eqn:SMi\])]{}, $M$ can be written as $$\begin{aligned} M = r_1 \times u_{r_1} + r_2 \times u_{r_2} + r_3 \times u_{r_3},\label{eqn:M}\end{aligned}$$ where $r_p,u_{r_p}\in\Rset^{3}$ are the $p$th column vectors of $R$ and ${\ensuremath{\frac{\partial U}{\partial R}}}$, respectively. Equation [(\[eqn:Vdot0\])]{} can be simplified using [(\[eqn:V2dot0\])]{} as $$\begin{aligned} \dot{V}+\Omega_2\times V = -\frac{m_1+m_2}{m_1m_2}{\ensuremath{\frac{\partial U}{\partial X}}}.\end{aligned}$$ For reconstruction of the motion of the second body, it is natural to express the motion of the second body in the inertial frame. Since $\dot{V}_2=\dot{R}_2^T\dot{x}_2+R_2^T\ddot{x}_2=-S(\Omega_2)V+R_2^T\dot{v}_2$, [(\[eqn:V2dot0\])]{} can be written as $$\begin{aligned} m_2 R_2^T\dot{v}_2 = {\ensuremath{\frac{\partial U}{\partial X}}}.\end{aligned}$$ Equation [(\[eqn:Pi2dot0\])]{} can be simplified using the property ${\ensuremath{\frac{\partial U}{\partial X}}}X^T-X{\ensuremath{\frac{\partial U}{\partial X}}}^T=S(X\times{\ensuremath{\frac{\partial U}{\partial X}}})$ from [(\[eqn:Scross\])]{}. The kinematics equations for $\dot{R}$ and $\dot{X}$ can be derived in a similar way. In summary, *the continuous equations of relative motion for the full two body problem, in Lagrangian form,* can be written as $$\begin{gathered} \dot{V}+\Omega_2\times V = -\frac{1}{m}{\ensuremath{\frac{\partial U}{\partial X}}},\label{eqn:Vdot}\\ \dot{(J_R\Omega)}+\Omega_2\times J_R\Omega=-M,\\ J_2\dot{\Omega}_2+\Omega_2\times J_2\Omega_2=X \times {\ensuremath{\frac{\partial U}{\partial X}}}+M,\label{eqn:Pi2dot}\\ \dot{X}+\Omega_2 \times X=V,\\ \dot{R}=S(\Omega)R-S(\Omega_2)R,\label{eqn:Rdot}\end{gathered}$$ where $m=\frac{m_1m_2}{m_1+m_2}$.\[no:m\] The following equations can be used for reconstruction of the motion of the second body in the inertial frame: $$\begin{gathered} \dot{v}_2 = \frac{1}{m_2}R_2 {\ensuremath{\frac{\partial U}{\partial X}}},\label{eqn:v2dot}\\ \dot{x}_2 = v_2,\\ \dot{R_2}=R_2 S(\Omega_2).\label{eqn:R2dot}\end{gathered}$$ These equations are equivalent to those given in [@jo:macie]. However, [(\[eqn:v2dot\])]{} is not given in [@jo:macie]. Equations [(\[eqn:Vdot\])]{} though [(\[eqn:R2dot\])]{} give a complete set of equations for the reduced dynamics and reconstruction. Furthermore, they are derived systematically in the context of geometric mechanics using proper variational formulas given in [(\[eqn:delR\])]{} through [(\[eqn:delV2\])]{}. This result can be readily generalized for $n$ bodies. ### Equations of motion: Hamiltonian form Define the linear momenta $\Gamma,\gamma_2\in\Rset^3$,\[no:Gamma\] and the angular momenta $\Pi,\Pi_2\in\Rset^3$\[no:Pi\] as $$\begin{aligned} \Gamma & = m V,\\ \gamma_2 & = m v_2,\\ \Pi& =J_R\Omega=RJ_1\Omega_1,\\ \Pi_2&=J_2\Omega_2.\end{aligned}$$ Then, the equations of motion can be rewritten in terms of these momenta variables. *The continuous equations of relative motion for the full two body problem, in Hamiltonian form,* can be written as $$\begin{gathered} \dot{\Gamma}+\Omega_2\times \Gamma = -{\ensuremath{\frac{\partial U}{\partial X}}},\label{eqn:Gamdot}\\ \dot{\Pi}+\Omega_2\times \Pi=-M,\\ \dot{\Pi}_2+\Omega_2\times \Pi_2=X \times {\ensuremath{\frac{\partial U}{\partial X}}}+M,\\ \dot{X}+\Omega_2 \times X=\frac{\Gamma}{m},\\ \dot{R}=S(\Omega)R-S(\Omega_2)R,\end{gathered}$$ where $m=\frac{m_1m_2}{m_1+m_2}$. The following equations can be used to reconstruct the motion of the second body in the inertial frame: $$\begin{gathered} \dot{\gamma}_2 = R_2 {\ensuremath{\frac{\partial U}{\partial X}}},\\ \dot{x}_2 = \frac{\gamma_2}{m_2},\\ \dot{R_2}=R_2 S(\Omega_2)\label{eqn:R2doth}.\end{gathered}$$ Variational integrators ======================= A variational integrator discretizes Hamilton’s principle rather than the continuous equations of motion. Taking variations of the discretization of the action integral leads to the discrete Euler-Lagrange or discrete Hamilton’s equations. The discrete Euler-Lagrange equations can be interpreted as a discrete Lagrangian map that updates the variables in the configuration space, which are the positions and the attitudes of the bodies. A discrete Legendre transformation relates the configuration variables with the linear and angular momenta variables, and yields a discrete Hamiltonian map, which is equivalent to the discrete Lagrangian map. In this section, we derive both a Lagrangian and Hamiltonian form of variational integrators for the full body problem in inertial and relative coordinates. The second level subscript $k$ denotes the value of variables at $t=kh+t_0$ for an integration step size $h\in\Rset$ and an integer $k$. The integer $N$ satisfies $t_f=kN+t_0$, so $N$ is the number of time-steps of length $h$ to go from the initial time $t_0$ to the final time $t_f$. Inertial coordinates -------------------- *Discrete Lagrangian:* In continuous time, the structure of the kinematics equations [(\[eqn:Ridot\])]{}, [(\[eqn:Rdot\])]{} and [(\[eqn:R2dot\])]{} ensure that $R_i$, $R$ and $R_2$ evolve on ${\ensuremath{\mathrm{SO(3)}}}$ automatically. Here, we introduce a new variable $F_{i_{k}}\in{\ensuremath{\mathrm{SO(3)}}}$ defined such that $R_{i_{k+1}}=R_{i_{k}}F_{i_{k}}$, i.e. $$\begin{aligned} F_{i_{k}}=R_{i_{k}}^TR_{i_{k+1}}.\label{eqn:Fik}\end{aligned}$$ Thus, $F_{i_{k}}$ represents the relative attitude between two integration steps, and by requiring that $F_{i_{k}}\in{\ensuremath{\mathrm{SO(3)}}}$, we guarantee that $R_{i_{k}}$ evolves in ${\ensuremath{\mathrm{SO(3)}}}$. Using the kinematic equation $\dot{R}_i=R_iS(\Omega_i)$, the skew-symmetric matrix $S(\Omega_{_{k}})$ can be approximated as $$\begin{aligned} S(\Omega_{i_{k}}) & = R_{i_{k}}^T \dot{R}_{i_{k}}\approx R_{i_{k}}^T \frac{R_{i_{k+1}}-R_{i_{k}}}{h}= \frac{1}{h}(F_{i_{k}}-I_{3\times 3}).\label{eqn:SOmegaik}\end{aligned}$$ The velocity, $\dot{x}_{i_{k}}$ can be approximated simply by $(x_{i_{k+1}}-x_{i_{k}})/h$. Using these approximations of the angular and linear velocity, the kinetic energy of the $i$th body given in [(\[eqn:Ti\])]{} can be approximated as $$\begin{aligned} T_i(\dot{x}_i,\Omega_i)& \approxT_i{\ensuremath{\left( \frac{1}{h}(x_{i_{k+1}}-x_{i_{k}}),\frac{1}{h}(F_{i_{k}}-I_{3\times 3}) \right)}},\\ & =\frac{1}{2h^2}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{i_{k}} \right\|}}^2+\frac{1}{2h^2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (F_{i_{k}}-I_{3\times 3})J_{d_i}(F_{i_{k}}-I_{3\times 3})^T \right]}}}},\\ & =\frac{1}{2h^2}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{i_{k}} \right\|}}^2+\frac{1}{h^2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I_{3\times 3}-F_{i_{k}})J_{d_i} \right]}}}},\end{aligned}$$ where [(\[eqn:trAB\])]{} is used. A discrete Lagrangian $L_d(\mathbf{x}_{_{k}},\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k}},\mathbf{F}_{_{k}})$ is constructed such that it approximates a segment of the action integral [(\[eqn:caction\])]{}, $$\begin{aligned} L_d & = \frac{h}{2}L{\ensuremath{\left( \mathbf{x}_{_{k}},\frac{1}{h}(\mathbf x_{_{k+1}}-\mathbf x_{_{k}}),\mathbf{R}_{_{k}},\frac{1}{h}(\mathbf F_{_{k}}-\mathbf I) \right)}}\nonumber\\& \quad + \frac{h}{2}L{\ensuremath{\left( \mathbf{x}_{_{k+1}},\frac{1}{h}(\mathbf x_{_{k+1}}-\mathbf x_{_{k}}),\mathbf{R}_{_{k+1}},\frac{1}{h}(\mathbf F_{_{k}}-\mathbf I) \right)}}\nonumber,\\ & = \sum_{i=1}^{n} \frac{1}{2h}m_i{\ensuremath{\left\| x_{i_{k+1}}-x_{i_{k}} \right\|}}^2+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I_{3\times 3}-F_{i_{k}})J_{d_i} \right]}}}}\nonumber\\& \quad -\frac{h}{2}U(\mathbf{x}_{_{k}},\mathbf{R}_{_{k}})-\frac{h}{2}U(\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k+1}}),\label{eqn:Ld}\end{aligned}$$ where $\mathbf{x}_{_k}\in(\Rset^3)^n$, $\mathbf{R}_{_k}\in{\ensuremath{\mathrm{SO(3)}}}^n$, and $\mathbf{F}_{_k}\in(\Rset^3)^n$, and $\mathbf{I}\in(\Rset^{3\times 3})^n$ are defined as $\mathbf{x}_{_k}=(x_{1_k},x_{2_k},\cdots,x_{n_k})$, $\mathbf{R}_{_k}=(R_{1_k},R_{2_k},\cdots,R_{n_k})$, $\mathbf{F}_{_k}=(F_{1_k},F_{2_k},\cdots,F_{n_k})$, and $\mathbf{I}=(I_{3\times 3},I_{3\times 3},\cdots,I_{3\times 3})$, respectively. This discrete Lagrangian is self-adjoint [@bk:hairer], and self-adjoint numerical integration methods have even order, so we are guaranteed that the resulting integration method is at least second-order accurate. *Variations of discrete variables:* The variations of the discrete variables are chosen to respect the geometry of the configuration space ${\ensuremath{\mathrm{SE(3)}}}$. The variation of $x_{i_{k}}$ is given by $$\begin{aligned} x_{i_{k}}^\epsilon = x_{i_{k}} + \epsilon\delta x_{i_{k}} +\mathcal{O}(\epsilon^2),\end{aligned}$$ where $\delta x_{i_{k}}\in\Rset^3$ and vanishes at $k=0$ and $k=N$. The variation of $R_{i_{k}}$ is given by $$\begin{aligned} \delta R_{i_{k}}=R_{i_{k}}\eta_{i_{k}},\label{eqn:delRik}\end{aligned}$$ where $\eta_{i_{k}}\in{\ensuremath{\mathfrak{so}(3)}}$ is a variation represented by a skew-symmetric matrix and vanishes at $k=0$ and $k=N$. The variation of $F_{i_{k}}$ can be computed from the definition $F_{i_{k}}=R_{i_{k}}^TR_{i_{k+1}}$ to give $$\begin{aligned} \delta F_{i_{k}} & = \delta R_{i_{k}}^TR_{i_{k+1}} + R_{i_{k}}^T\delta R_{i_{k+1}},\nonumber\\ & = -\eta_{i_{k}} R_{i_{k}}^TR_{i_{k+1}} + R_{i_{k}}^TR_{i_{k+1}} \eta_{i_{k+1}},\nonumber\\ & = -\eta_{i_{k}}F_{i_{k}}+F_{i_{k}}\eta_{i_{k+1}}.\label{eqn:delFik}\end{aligned}$$ ### Discrete equations of motion: Lagrangian form To obtain the discrete equations of motion in Lagrangian form, we compute the variation of the discrete Lagrangian from [(\[eqn:delU\])]{}, [(\[eqn:delRik\])]{} and [(\[eqn:delFik\])]{}, to give $$\begin{aligned} \delta L_d = \sum_{i=1}^{n}\, & \frac{1}{h} m_i (x_{i_{k+1}}-x_{i_{k}})^T(\delta x_{i_{k+1}}-\delta x_{i_{k}})+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ {\ensuremath{\left( \eta_{i_{k}}F_{i_{k}}-F_{i_{k}}\eta_{i_{k+1}} \right)}}J_{d_i} \right]}}}}\nonumber\\& -\frac{h}{2}{\ensuremath{\left( {\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}}^T\delta x_{i_{k}}+{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial x_{i_{k+1}}}}}^T\delta x_{i_{k+1}} \right)}}+\frac{h}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{i_{k}}R_{i_{k}}^T{\ensuremath{\frac{\partial U_{_{k}}}{\partial R_{i_{k}}}}}+\eta_{i_{k+1}}R_{i_{k+1}}^T{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial R_{i_{k+1}}}}} \right]}}}},\label{eqn:delLd}\end{aligned}$$ where $U_{_{k}}=U(\mathbf{x}_{_{k}},\mathbf{R}_{_{k}})$ denotes the value of the potential at $t=kh+t_0$. Define the action sum as $$\begin{aligned} \mathfrak{G}_d = \sum_{k=0}^{N-1}L_d(\mathbf{x}_{_{k}},\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k}},\mathbf{F}_{_{k}}).\label{eqn:daction}\end{aligned}$$ The discrete action sum $\mathfrak{G}_d$ approximates the action integral [(\[eqn:caction\])]{}, because the discrete Lagrangian approximates a segment of the action integral. Substituting [(\[eqn:delLd\])]{} into [(\[eqn:daction\])]{}, the variation of the action sum is given by $$\begin{aligned} \delta\mathfrak{G}_d = \sum_{k=0}^{N-1}\sum_{i=1}^{n}\; &\delta x_{i_{k+1}}^T {\ensuremath{\left\{ \frac{1}{h} m_i (x_{i_{k+1}}-x_{i_{k}})-\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial x_{i_{k+1}}}}} \right\}}}\\ & +\delta x_{i_{k}}^T {\ensuremath{\left\{ -\frac{1}{h} m_i (x_{i_{k+1}}-x_{i_{k}})-\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}} \right\}}}\\ & +{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{i_{k+1}}{\ensuremath{\left\{ -\frac{1}{h}J_{d_i}F_{i_k}+\frac{h}{2}R_{i_{k+1}}^T{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial R_{i_{k+1}}}}} \right\}}} \right]}}}}\\ & +{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{i_{k}}{\ensuremath{\left\{ \frac{1}{h}F_{i_k}J_{d_i} +\frac{h}{2}R_{i_{k}}^T{\ensuremath{\frac{\partial U_{_{k}}}{\partial R_{i_{k}}}}} \right\}}} \right]}}}}.\end{aligned}$$ Using the fact that $\delta x_{i_{k}}$ and $\eta_{i_{k}}$ vanish at $k=0$ and $k=N$, we can reindex the summation, which is the discrete analogue of integration by parts, to yield $$\begin{aligned} \delta \mathfrak{G}_d = \sum_{k=1}^{N-1}\sum_{i=1}^n\,&-\delta x_{i_{k}} {\ensuremath{\left\{ \frac{1}{h}m_i{\ensuremath{\left( x_{i_{k+1}}-2x_{i_{k}}+x_{i_{k-1}} \right)}}+h{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}} \right\}}}\\ &+{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{i_{k}}{\ensuremath{\left\{ \frac{1}{h}{\ensuremath{\left( F_{i_{k}}J_{d_i}-J_{d_i}F_{i_{k-1}} \right)}}+h R_{i_{k}}^T{\ensuremath{\frac{\partial U_{_{k}}}{\partial R_{i_{k}}}}} \right\}}} \right]}}}}.\end{aligned}$$ Hamilton’s principle states that $\delta\mathfrak{G}_d$ should be zero for all possible variations $\delta x_{i_{k}}\in\Rset^3$ and $\eta_{i_{k}}\in{\ensuremath{\mathfrak{so}(3)}}$ that vanish at the endpoints. Therefore, the expression in the first brace should be zero, and since $\eta_{i_{k}}$ is skew-symmetric, the expression in the second brace should be symmetric. Thus, we obtain *the discrete equations of motion for the full body problem, in Lagrangian form,* for $i\in(1,2,\cdots,n)$ as $$\begin{gathered} \frac{1}{h}{\ensuremath{\left( x_{i_{k+1}}-2x_{i_{k}}+x_{i_{k-1}} \right)}}=-h{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}},\label{eqn:updatexik}\\ \frac{1}{h}{\ensuremath{\left( F_{i_{k+1}}J_{d_i}-J_{d_i}F_{i_{k+1}}^T-J_{d_i}F_{i_{k}}+F_{i_{k}}^TJ_{d_i} \right)}}=h S(M_{i_{k+1}}),\label{eqn:findFik}\\ R_{i_{k+1}}=R_{i_{k}}F_{i_{k}},\label{eqn:updateRik}\end{gathered}$$ where $M_{i_{k}}\in\Rset^3$ is defined in [(\[eqn:Mi\])]{} as $$\begin{aligned} M_{i_k}=r_{i_1}\times u_{ri_1}+r_{i_2}\times u_{ri_2}+r_{i_3}\times u_{ri_3},\label{eqn:Mik}\end{aligned}$$ where $r_{i_p},u_{ri_p}\in\Rset^{1\times 3}$ are $p$th row vectors of $R_{i_k}$ and ${\ensuremath{\frac{\partial U_{_k}}{\partial R_{i_k}}}}$, respectively. Given the initial conditions $(x_{i_{0}},R_{i_{0}},x_{i_{1}},R_{i_{1}})$, we can obtain $x_{i_{2}}$ from [(\[eqn:updatexik\])]{}. Then, $F_{i_{0}}$ is computed from [(\[eqn:updateRik\])]{}, and $F_{i_{1}}$ can be obtained by solving the implicit equation [(\[eqn:findFik\])]{}. Finally, $R_{i_{2}}$ is found from [(\[eqn:updateRik\])]{}. This yields an update map $(x_{i_{0}},R_{i_{0}},x_{i_{1}},R_{i_{1}})\mapsto(x_{i_{1}},R_{i_{1}},x_{i_{2}},R_{i_{2}})$, and this process can be repeated. ### Discrete equations of motion: Hamiltonian form As discussed above, equations [(\[eqn:updatexik\])]{} through [(\[eqn:updateRik\])]{} defines a discrete Lagrangian map that updates $x_{i_{k}}$ and $R_{i_{k}}$. The discrete Legendre transformation given in [(\[eqn:dLtk\])]{} and [(\[eqn:dLtkp\])]{} relates the configuration variables $x_{i_{k}}$, $R_{i_{k}}$ and the corresponding momenta. This induces a discrete Hamiltonian map that is equivalent to the discrete Lagrangian map. The discrete Hamiltonian map is particularly convenient if the initial conditions are given in terms of the positions and momenta at the initial time, $(x_{i_{0}},v_{i_{0}},R_{i_{0}},\Omega_{i_{0}})$. Before deriving the variational integrator in Hamiltonian form, consider the momenta conjugate to $x_i$ and $R_i$, namely $P_{v_i}\in\Rset^3$ and $P_{\Omega_i}\in\Rset^{3\times 3}$. From the definition [(\[eqn:FL\])]{}, $\mathbb{F}_{v_i}L$ is obtained by taking the derivative of $L$, given in [(\[eqn:L\])]{}, with $\dot{x}_i$ while holding other variables fixed. $$\begin{aligned} \delta\dot{x}_i^TP_{v_i}&=\mathbb{F}_{v_i}L(\mathbf{x},\mathbf{\dot x},\mathbf{R},\mathbf{\Omega}),\\ &=\frac{d}{d\epsilon}\bigg|_{\epsilon=0}L(\mathbf{x},\mathbf{\dot x}+\epsilon\delta\mathbf{\dot{x}}_i,\mathbf{R},\mathbf{\Omega}),\\ & = \frac{d}{d\epsilon}\bigg|_{\epsilon=0}T_i(\dot{x}_i+\epsilon\delta\dot{x}_i,\Omega_i),\\ & = \delta\dot{x}_i^T {\ensuremath{\left( m_i \dot{x}_i \right)}},\end{aligned}$$ where $\delta\mathbf{\dot{x}}_i\in(\Rset^3)^n$ denotes $(0,0,\cdots,\delta\dot{x}_i,\cdots,0)$, and $T_i$ is given in [(\[eqn:Ti\])]{}. Then, we obtain $$\begin{aligned} P_{v_i} & = m_i v_i = \gamma_i, \label{eqn:Pvi}\end{aligned}$$ which is equal to the linear momentum of $\mathcal{B}_i$. Similarly, $$\begin{aligned} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\delta\Omega_i)^TP_{\Omega_i} \right]}}}}& =\mathbb{F}_{\Omega_i}L(\mathbf{x},\mathbf{\dot x},\mathbf{R},\mathbf{\Omega}),\\ & = \frac{d}{d\epsilon}\bigg|_{\epsilon=0}T_i(\dot{x}_i,\Omega_i+\epsilon\delta\Omega_i),\\ & =\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\delta\Omega_i)J_{d_i}S(\Omega_i)^T+S(\Omega_i)J_{d_i}S(\delta\Omega_i)^T \right]}}}},\\ & = \frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\delta\Omega_i)^TS(J_i\Omega_i) \right]}}}},\end{aligned}$$ where [(\[eqn:trAB\])]{} and [(\[eqn:JdJ\])]{} are used. Now, we obtain $$\begin{aligned} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\delta\Omega_i)^T{\ensuremath{\left\{ P_{\Omega_i}-\frac{1}{2}S(J_i\Omega_i) \right\}}} \right]}}}}=0.\end{aligned}$$ Since $S(\Omega_i)$ is skew-symmetric, the expression in the braces should be symmetric. This implies that $$\begin{aligned} P_{\Omega_i}-P_{\Omega_i}^T& =S(J_i\Omega_i) = S(\Pi_i).\label{eqn:POmegai}\end{aligned}$$ Equations [(\[eqn:Pvi\])]{} and [(\[eqn:POmegai\])]{} give expressions for the momenta conjugate to $x_i$ and $R_i$. Consider the discrete Legendre transformations given in [(\[eqn:dLtk\])]{} and [(\[eqn:dLtkp\])]{}. Then, $$\begin{aligned} \delta x_{i_{k}}^T\mathbf{D}_{x_{i,k}}L_d(\mathbf{x}_{_{k}},\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k}},\mathbf{F}_{_{k}})&=\frac{d}{d\epsilon}\bigg|_{\epsilon=0}L_d(\mathbf{x}_{_{k}}+\epsilon\delta\mathbf{x}_{i_k},\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k}},\mathbf{F}_{_{k}}),\nonumber\\ & = -\delta x_{i_{k}}^T {\ensuremath{\left[ \frac{1}{h} m_i (x_{i_{k+1}}-x_{i_{k}}) +\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}} \right]}},\label{eqn:DxikLd0}\end{aligned}$$ where $\delta\mathbf{x_{i_k}}\in(\Rset^3)^n$ denotes $(0,0,\cdots,\delta x_{_{i_k}},\cdots,0)$. Therefore, we have $$\begin{aligned} \mathbf{D}_{x_{i,k}}L_d(\mathbf{x}_{_{k}},\mathbf{x}_{_{k+1}},\mathbf{R}_{_{k}},\mathbf{F}_{_{k}})&=-\frac{1}{h} m_i (x_{i_{k+1}}-x_{i_{k}}) -\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}}.\label{eqn:DxikLd}\end{aligned}$$ From the discrete Legendre transformation given in [(\[eqn:dLtk\])]{}, $P_{v_{i,k}}=-\mathbf{D}_{x_{i,k}}L_d$. Using [(\[eqn:Pvi\])]{} and [(\[eqn:DxikLd\])]{}, we obtain $$\begin{aligned} \gamma_{i_k}=\frac{1}{h} m_i (x_{i_{k+1}}-x_{i_{k}}) +\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}}.\label{eqn:mvk}\end{aligned}$$ Using the discrete Legendre transformation given in [(\[eqn:dLtkp\])]{}, $P_{v_{i,k+1}}=\mathbf{D}_{x_{i,k+1}}L_d$, we can derive the following equation similarly: $$\begin{aligned} \gamma_{i_{k+1}}=\frac{1}{h} m_i (x_{i_{k+1}}-x_{i_{k}}) -\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial x_{i_{k+1}}}}}.\label{eqn:mvkp}\end{aligned}$$ Equations [(\[eqn:mvk\])]{} and [(\[eqn:mvkp\])]{} define the variational integrator in Hamiltonian form for the translational motion. Now, consider the rotational motion. We have $$\begin{aligned} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{i_{k}}{\mathbf{D}_{R_{i,k}}L_d}^T \right]}}}}&={\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{i_{k}}{\ensuremath{\left\{ \frac{1}{h}F_{i_{k}}J_{d_i}+\frac{h}{2}R_{i_{k}}^T{\ensuremath{\frac{\partial U_{_{k}}}{\partial R_{i_{k}}}}} \right\}}} \right]}}}},\label{eqn:DRikLd0}\end{aligned}$$ where the right side is obtained by taking the variation of $L_d$ with respect to $R_{i_{k}}$, while holding other variables fixed. Since $\eta_{i_{k}}$ is skew-symmetric, $$\begin{aligned} -\mathbf{D}_{R_{i,k}}L_d+{\mathbf{D}_{R_{i,k}}L_d}^T =\frac{1}{h}{\ensuremath{\left( F_{i_{k}}J_{d_i}-J_{d_i}F_{i_{k}}^T \right)}} -\frac{h}{2}S(M_{i_k}),\label{eqn:DRikLd}\end{aligned}$$ where $M_{i_i}\in\Rset^3$ is defined in [(\[eqn:Mik\])]{}. From the discrete Legendre transformation given in [(\[eqn:dLtk\])]{}, $P_{\Omega_{i,k}}=-\mathbf{D}_{R_{i,k}}L_d$, we obtain the following equation by using [(\[eqn:POmegai\])]{} and [(\[eqn:DRikLd\])]{}, $$\begin{aligned} S(\Pi_{i_k})=\frac{1}{h}{\ensuremath{\left( F_{i_{k}}J_{d_i}-J_{d_i}F_{i_{k}}^T \right)}} -\frac{h}{2}S(M_{i_k}).\label{eqn:SJOmegaik}\end{aligned}$$ Using the discrete Legendre transformation given in [(\[eqn:dLtkp\])]{}, $P_{\Omega_{i,k+1}}=\mathbf{D}_{R_{i,k+1}}L_d$, we can obtain the following equation: $$\begin{aligned} S(\Pi_{i_{k+1}})=\frac{1}{h}F_{i_{k}}^T{\ensuremath{\left( F_{i_{k}}J_{d_i}-J_{d_i}F_{i_{k}}^T \right)}}F_{i_{k}} +\frac{h}{2}S(M_{i_{k+1}}).\label{eqn:SJOmegaikp}\end{aligned}$$ By using [(\[eqn:SR\])]{} and substituting [(\[eqn:SJOmegaik\])]{}, we can reduce [(\[eqn:SJOmegaikp\])]{} to the following equation in vector form. $$\begin{aligned} \Pi_{i_{k+1}}=F_{i_{k}}^T\Pi_{i_k}+\frac{h}{2}F_{i_{k}}^TM_{i_k}+\frac{h}{2}M_{i_{k+1}}. \label{eqn:JOmegaikp}\end{aligned}$$ Equations [(\[eqn:SJOmegaik\])]{} and [(\[eqn:JOmegaikp\])]{} define the variational integrator in Hamiltonian form for the rotational motion. In summary, using [(\[eqn:mvk\])]{}, [(\[eqn:mvkp\])]{}, [(\[eqn:SJOmegaik\])]{} and [(\[eqn:JOmegaikp\])]{}, *the discrete equations of motion for the full body problem, in Hamiltonian form,* can be written for $i\in(1,2,\cdots,n)$ as $$\begin{gathered} x_{i_{k+1}} = x_{i_{k}} + \frac{h}{m_i} \gamma_{i_k}-\frac{h^2}{2m_i}{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}},\label{eqn:updatexikH}\\ \gamma_{i_{k+1}}=\gamma_{i_k}-\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k}}}{\partial x_{i_{k}}}}} -\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial x_{i_{k+1}}}}},\label{eqn:updatevik}\\ h S(\Pi_{i_k}+\frac{h}{2}M_{i_k})=F_{i_{k}}J_{d_i}-J_{d_i}F_{i_{k}}^T,\label{eqn:findFikH}\\ \Pi_{i_{k+1}}=F_{i_{k}}^T\Pi_{i_k}+\frac{h}{2}F_{i_{k}}^TM_{i_k}+\frac{h}{2}M_{i_{k+1}}. \label{eqn:updatePiik},\\ R_{i_{k+1}}=R_{i_{k}}F_{i_{k}}.\label{eqn:updateRikH}\end{gathered}$$ Given $(x_{i_0},\gamma_{i_0},R_{i_0},\Pi_{i_0})$, we can find $x_{i_1}$from [(\[eqn:updatexikH\])]{}. Solving the implicit equation [(\[eqn:findFikH\])]{} yields $F_{i_0}$, and $R_{i_1}$ is computed from [(\[eqn:updateRikH\])]{}. Then, [(\[eqn:updatevik\])]{} and [(\[eqn:updatePiik\])]{} gives $\gamma_{i_1}$, and $\Pi_{i_1}$. This defines the discrete Hamiltonian map, $(x_{i_0},\gamma_{i_0},R_{i_0},\Pi_{i_0})\mapsto(x_{i_1},\gamma_{i_1},R_{i_1},\Pi_{i_1})$, and this process can be repeated. Relative coordinates -------------------- In this section, we derive the variational integrator for the full two body problem in relative coordinates by following the procedure given before. This result can be readily generalized to $n$ bodies. *Reduction of discrete variables:* The discrete reduced variables are defined in the same way as the continuous reduced variables, which are given in [(\[eqn:X\])]{} through [(\[eqn:V2\])]{}. We introduce $F_{_k}\in{\ensuremath{\mathrm{SO(3)}}}$ such that $R_{_{k+1}}=R_{2_{k+1}}^TR_{1_{k+1}}=F_{2_k}^T F_{_k} R_{_k}$. i.e. $$\begin{aligned} F_{_k} = R_{_k} F_{1_k} R_{_k}^T.\label{eqn:Fk}\end{aligned}$$ *Discrete reduced Lagrangian:* The discrete reduced Lagrangian is obtained by expressing the original discrete Lagrangian given in [(\[eqn:Ld\])]{} in terms of the discrete reduced variables. From the definition of the discrete reduced variables given in [(\[eqn:X\])]{} and [(\[eqn:X2\])]{}, we have $$\begin{aligned} x_{1_{k+1}}-x_{1_{k}}&=R_{2_{k+1}}(X_{_{k+1}}+X_{2_{k+1}})-R_{2_{k}}(X_{_{k}}+X_{2_{k}}),\nonumber\\ &=R_{2_{k}}{\ensuremath{\left\{ F_{2_{k}}(X_{_{k+1}}+X_{2_{k+1}})-(X_{_{k}}+X_{2_{k}}) \right\}}},\label{eqn:delx1k}\\ x_{2_{k+1}}-x_{2_{k}}&=R_{2_{k}}{\ensuremath{\left\{ F_{2_{k}}X_{2_{k+1}}-X_{2_k} \right\}}}.\label{eqn:delx2k}\end{aligned}$$ From [(\[eqn:SOmegaik\])]{}, $S(\Omega_{1_k})$ and $S(\Omega_{2_k})$ are expressed as $$\begin{aligned} S(\Omega_{1_k})&=\frac{1}{h}{\ensuremath{\left( F_{1_{k}}-I_{3\times 3} \right)}},\nonumber\\ & =\frac{1}{h}R_{_k}^T {\ensuremath{\left( F_{_k} - I_{3\times 3} \right)}} R_{_k},\label{eqn:SOmega1k}\\ S(\Omega_{2_{k}}) & = \frac{1}{h}{\ensuremath{\left( F_{2_{k}}-I_{3\times 3} \right)}}.\label{eqn:SOmega2k}\end{aligned}$$ Substituting [(\[eqn:delx1k\])]{} through [(\[eqn:SOmega2k\])]{} into [(\[eqn:Ld\])]{}, we obtain the discrete reduced Lagrangian. $$\begin{aligned} l_{d_k}& = l_d(X_{_{k}},X_{_{k+1}},X_{2_{k}},X_{2_{k+1}},R_{_{k}},F_{_{k}},F_{2_{k}})\nonumber\\ & = \frac{1}{2h}m_1{\ensuremath{\left\| F_{2_{k}}(X_{_{k+1}}+X_{2_{k+1}})-(X_{_k}+X_{2_{k}}) \right\|}}^2+\frac{1}{2h}m_2{\ensuremath{\left\| F_{2_{k}}X_{2_{k+1}}-X_{2_k} \right\|}}^2\nonumber\\ & \quad + \frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I_{3\times 3}-F_{_{k}})J_{{dR}_k} \right]}}}} + \frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I_{3\times 3}-F_{2_{k}})J_{d_2} \right]}}}}-\frac{h}{2}U(X_{_{k}},R_{_{k}})-\frac{h}{2}U(X_{_{k+1}},R_{_{k+1}}),\label{eqn:ld}\end{aligned}$$ where $J_{{dR}_k}\in\Rset^{3\times 3}$ is defined to be $J_{{dR}_k}=R_{_{k}}J_{d_1}R_{_{k}}^T$, which gives the nonstandard moment of inertia matrix of the first body with respect to the second body fixed frame at $t=kh+t_0$. *Variations of discrete reduced variables:* The variations of the discrete reduced variables can be derived from those of the original variables. The variations of $R_{_{k}}$, $X_{_{k}}$, and $F_{2_{k}}$ are the same as given in [(\[eqn:delR\])]{}, [(\[eqn:delX\])]{}, and [(\[eqn:delFik\])]{}, respectively. The variation of $F_{_{k}}$ is computed in \[appdrv\]. In summary, the variations of discrete reduced variables are given by $$\begin{aligned} \delta R_{_k} & = \eta_{_k} R_{_k} - \eta_{2_k} R_{_k},\label{eqn:delRk}\\ \delta X_{_k} & = \chi_{_k} -\eta_{2_k}X_{_k},\label{eqn:delXk}\\ \delta F_{_k} & =-\eta_{2_{k}}F_{_{k}} +F_{2_{k}}\eta_{_{k+1}}F_{2_{k}}^TF_{_{k}}+F_{_{k}}{\ensuremath{\left( -\eta_{_{k}} +\eta_{2_{k}} \right)}},\label{eqn:delFk}\\ \delta X_{2_k} & = \chi_{2_k} -\eta_{2_k}X_{2_k},\label{eqn:delX2k}\\ \delta F_{2_k} & =-\eta_{2_k}F_{2_k}+F_{2_k}\eta_{2_{k+1}}.\label{eqn:delF2k}\end{aligned}$$ These Lie group variations are the main elements required to derive the variational integrator equations. ### Discrete equations of motion: Lagrangian form As before, we can obtain the discrete equations of motion in Lagrangian form by computing the variation of the discrete reduced Lagrangian which, by using [(\[eqn:delRk\])]{} through [(\[eqn:delF2k\])]{}, is given by $$\begin{aligned} \delta l_{d_k} & = \frac{1}{h}\chi_{_{k+1}}^T{\ensuremath{\left[ m_1(X_{_{k+1}}+X_{2_{k+1}})-m_1F_{2_{k}}^T(X_{_k}+X_{2_{k}}) \right]}}\nonumber\\& \quad +\frac{1}{h}\chi_{_{k}}^T{\ensuremath{\left[ m_1(X_{_{k}}+X_{2_{k}})-m_1F_{2_{k}}(X_{_{k+1}} +X_{2_{k+1}}) \right]}}\nonumber\\& \quad + \frac{1}{h}\chi_{2_{k+1}}^T{\ensuremath{\left[ m_1(X_{_{k+1}}+X_{2_{k+1}})-m_1F_{2_{k}}^T(X_{_k}+X_{2_{k}})+m_2X_{2_{k+1}}-m_2F_{2_{k}}^TX_{2_{k}} \right]}}\nonumber\\& \quad + \frac{1}{h} \chi_{2_{k}}^T{\ensuremath{\left[ m_1(X_{_{k}}+X_{2_{k}})-m_1F_{2_{k}}(X_{_{k+1}}+X_{2_{k+1}})+m_2X_{2_{k}}-m_2F_{2_{k}}X_{2_{k+1}} \right]}}\nonumber\\& \quad -\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{_{k+1}}F_{2_{k}}^TF_{_{k}} J_{{dR}_k} F_{2_{k}} \right]}}}}+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{_{k}}F_{_{k}}J_{{dR}_k} \right]}}}} -\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{2_{k+1}}J_{d_2}F_{2_{k}} \right]}}}}+\frac{1}{h}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{2_{k}}F_{2_{k}}J_{d_2} \right]}}}}\nonumber\\ & \quad - \frac{h}{2}\chi_{_{k}}^T{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}}+\frac{h}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{2_{k}}X_{_{k}}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}}^T \right]}}}}- \frac{h}{2}\chi_{_{k+1}}^T{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial X_{_{k+1}}}}}+\frac{h}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{2_{k+1}}X_{_{k+1}}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial X_{_{k+1}}}}}^T \right]}}}}\nonumber\\& \quad +\frac{h}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{2_{k}}R_{_{k}}{\ensuremath{\frac{\partial U_{_k}}{\partial R_{_{k}}}}}^T-\eta_{_{k}}R_{_{k}}{\ensuremath{\frac{\partial U_{_k}}{\partial R_{_{k}}}}}^T \right]}}}}+\frac{h}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{2_{k+1}}R_{_{k+1}}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial R_{_{k+1}}}}}^T-\eta_{_{k+1}}R_{_{k+1}}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial R_{_{k+1}}}}}^T \right]}}}}\label{eqn:delldk}.\end{aligned}$$ The action sum expressed in terms of the discrete reduced Lagrangian has the form $$\begin{aligned} \mathfrak{G}_d = \sum_{k=0}^{N-1}l_d(X_{_{k}},X_{_{k+1}},X_{2_{k}},X_{2_{k+1}},R_{_{k}},F_{_{k}},F_{2_{k}}).\label{eqn:draction}\end{aligned}$$ The discrete action sum $\mathfrak{G}_d$ approximates the action integral [(\[eqn:craction\])]{}, because the discrete Lagrangian approximates a piece of the integral. Using the fact that the variations $\chi_{_{k}},\chi_{2_{k}},\eta_{_{k}},\eta_{2_{k}}$ vanish at $k=0$ and $k=N$, the variation of the discrete action sum can be expressed as $$\begin{aligned} \delta \mathfrak{G}_d = & \sum_{k=1}^{N-1} \frac{1}{h}\chi_{_{k}}^T\bigg\{-m_1F_{2_{k-1}}^T(X_{_{k-1}}+X_{2_{k-1}})+2m_1(X_{_{k}}+X_{2_{k}})\\ & \hspace{1.8cm} -m_1F_{2_{k}}(X_{_{k+1}} +X_{2_{k+1}})-h^2{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}}\bigg\}\\& + \sum_{k=1}^{N-1} \frac{1}{h}\chi_{2_{k}}^T\bigg\{-m_1F_{2_{k-1}}^T(X_{_{k-1}}+X_{2_{k-1}})+2m_1(X_{_{k}}+X_{2_{k}})-m_1F_{2_{k}}(X_{_{k+1}} +X_{2_{k+1}})\\& \hspace{2.2cm} -m_2F_{2_{k-1}}^TX_{2_{k-1}}+2m_2X_{2_{k}}-m_2F_{2_{k}}X_{2_{k+1}}\bigg\}\\& + \sum_{k=1}^{N-1} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{_{k}}{\ensuremath{\left\{ \frac{1}{h}{\ensuremath{\left( -F_{2_{k-1}}^TF_{_{k-1}} R_{_{k-1}}J_{d_1}R_{_{k-1}}^T F_{2_{k-1}}+F_{_{k}}R_{_{k}}J_{d_1}R_{_{k}}^T \right)}}-hR_{_{k}}{\ensuremath{\frac{\partial U_{_k}}{\partial R_{_{k}}}}}^T \right\}}} \right]}}}}\\ & + \sum_{k=1}^{N-1} {\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ \eta_{2_{k}}{\ensuremath{\left\{ \frac{1}{h}{\ensuremath{\left( -J_{d_2}F_{2_{k-1}}+F_{2_{k}}J_{d_2} \right)}}+hX_{_{k}}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}}^T+hR_{_{k}}{\ensuremath{\frac{\partial U_{_k}}{\partial R_{_{k}}}}}^T \right\}}} \right]}}}}.\end{aligned}$$ From Hamilton’s principle, $\delta \mathfrak{G}_d$ should be zero for all possible variations $\chi_{_{k}},\chi_{2_{k}}\in\Rset^3$ and $\eta_{_{k}},\eta_{2_{k}}\in{\ensuremath{\mathfrak{so}(3)}}$ which vanish at the endpoints. Therefore, in the above equation, the expressions in the first two braces should be zero, and the expressions in the last two braces should be symmetric since $\eta_{_{k}},\eta_{2_{k}}$ are skew-symmetric. After some simplification, we obtain *the discrete equations of relative motion for the full two body problem, in Lagrangian form,* as $$\begin{gathered} F_{2_{k}}X_{_{k+1}}-2X_{_{k}}+F_{2_{k-1}}^TX_{_{k-1}}=-\frac{h^2}{m}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}},\label{eqn:updateXl}\\ F_{_{k+1}}J_{dR_{k+1}}-J_{dR_{k+1}}F_{_{k+1}}^T=F_{2_{k}}^T {\ensuremath{\left( F_{_{k}}J_{dR_{k}}-J_{dR_{k}}F_{_{k}}^T \right)}}F_{2_{k}}-h^2S(M_{k+1}),\label{eqn:findFl}\\F_{2_{k+1}}J_{d_2}-J_{d_2}F_{2_{k+1}}^T =F_{2_{k}}^T{\ensuremath{\left( F_{2_{k}}J_{d_2}-J_{d_2}F_{2_{k}}^T \right)}}F_{2_{k}} +h^2X_{_{k+1}}\times{\ensuremath{\frac{\partial U}{\partial X_{_{k+1}}}}}+h^2S(M_{k+1}) ,\label{eqn:findF2l}\\ R_{_{k+1}}=F_{2_{k}}^TF_{_{k}}R_{_{k}},\label{eqn:updateRl}\\ R_{2_{k+1}}=R_{2_{k}}F_{2_{k}}.\label{eqn:updateR2l}\end{gathered}$$ It is natural to express equations of motion for the second body in the inertial frame. $$\begin{gathered} x_{2_{k+1}}-2x_{2_{k}}+x_{2_{k-1}}=\frac{h^2}{m_2}R_{_k}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}}.\label{eqn:updatex2l}\end{gathered}$$ Given $(X_{_{0}},R_{_0},R_{2_{0}},X_{_{1}},R_{_1},R_{2_{1}})$, we can determine $F_{_0}$ and $F_{2_0}$ from [(\[eqn:updateRl\])]{} and [(\[eqn:updateR2l\])]{}. Solving the implicit equations [(\[eqn:findFl\])]{} and [(\[eqn:findF2l\])]{} gives $F_{_1}$ and $F_{2_1}$. Then $X_{_2}$, $R_{_2}$ and $R_{2_2}$ are found from [(\[eqn:updateXl\])]{}, [(\[eqn:updateRl\])]{} and [(\[eqn:updateR2l\])]{}, respectively. This yields the discrete Lagrangian map $(X_{_{0}},R_{_0},R_{2_{0}},X_{_{1}},R_{_1},R_{2_{1}})\mapsto (X_{_{1}},R_{_1},R_{2_{1}},X_{_{2}},R_{_2},R_{2_{2}})$ and this process can be repeated. We can separately reconstruct $x_{2_k}$ using [(\[eqn:updatex2l\])]{}. ### Discrete equations of motion: Hamiltonian form Using the discrete Legendre transformation, we can obtain the Hamiltonian map, in terms of reduced variables, that is equivalent to the Lagrangian map given in [(\[eqn:updateXl\])]{} through [(\[eqn:updatex2l\])]{}. We will only sketch the procedure as it is analogous to the approach of the previous section. First, we find expressions for the conjugate momenta variables corresponding to [(\[eqn:Pvi\])]{} and [(\[eqn:POmegai\])]{}. We compute the discrete Legendre transformation by taking the variation of the discrete reduced Lagrangian as in [(\[eqn:DxikLd0\])]{} and [(\[eqn:DRikLd0\])]{}. Then, we obtain the discrete equations of motion in Hamiltonian form using [(\[eqn:dLtk\])]{} and [(\[eqn:dLtkp\])]{}. *The discrete equations of relative motion for the full two body problem, in Hamiltonian form,* can be written as $$\begin{gathered} X_{_{k+1}}=F_{2_{k}}^T{\ensuremath{\left( X_{_{k}}+h\frac{\Gamma_{_{k}}}{m}-\frac{h^2}{2m}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}} \right)}}, \label{eqn:updateX}\\ \Gamma_{_{k+1}}=F_{2_{k}}^T{\ensuremath{\left( \Gamma_{_{k}}-\frac{h}{2}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}} \right)}}-\frac{h}{2}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial X_{_{k+1}}}}}, \label{eqn:updateV}\\ \Pi_{_{k+1}}=F_{2_{k}}^T {\ensuremath{\left( \Pi_{_{k}}-\frac{h}{2}M_{_{k}} \right)}}-\frac{h}{2}M_{_{k+1}}, \label{eqn:updatePi}\\ \Pi_{2_{k+1}}=F_{2_{k}}^T{\ensuremath{\left( \Pi_{_{k}}+\frac{h}{2}X_{_{k}}\times{\ensuremath{\frac{\partial U}{\partial X_{_{k}}}}}+\frac{h}{2}M_{_{k}} \right)}}+\frac{h}{2}X_{_{k+1}}\times{\ensuremath{\frac{\partial U}{\partial X_{_{k+1}}}}}+\frac{h}{2}M_{_{k+1}}, \label{eqn:updatePi2}\\ R_{_{k+1}}=F_{2_{k}}^TF_{_{k}}R_{_{k}},\label{eqn:updateR}\\h S{\ensuremath{\left( \Pi_k-\frac{h}{2}M_{_{k}} \right)}} = F_{_{k}}J_{dR_k}-J_{dR_k}F_{_{k}}^T,\label{eqn:findF}\\ h S{\ensuremath{\left( \Pi_{2_{k}}+\frac{h}{2}X_{_{k}}\times{\ensuremath{\frac{\partial U}{\partial X_{_{k}}}}}+\frac{h}{2}M_{_{k}} \right)}}=F_{2_{k}}J_{d_2}-J_{d_2}F_{2_{k}}^T.\label{eqn:findF2}$$ It is natural to express equations of motion for the second body in the inertial frame for reconstruction: $$\begin{gathered} x_{2_{k+1}}=x_{2_{k}}+h\frac{\gamma_{2_{k}}}{m_2}+\frac{h^2}{2m_2}R_{_k}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}}, \label{eqn:updateX2}\\ \gamma_{2_{k+1}}=\gamma_{2_{k}}+\frac{h}{2}R_{_k}{\ensuremath{\frac{\partial U_{_k}}{\partial X_{_{k}}}}}+\frac{h}{2}R_{_{k+1}}{\ensuremath{\frac{\partial U_{_{k+1}}}{\partial X_{_{k+1}}}}}, \label{eqn:updateV2}\\ R_{2_{k+1}}=R_{2_{k}}F_{2_{k}}.\label{eqn:updateR2}\end{gathered}$$ Given $(R_{_0},X_{_0},\Pi_{_0},\Gamma_{_0},\Pi_{2_0})$, we can determine $F_{_0}$ and $F_{2_0}$ by solving the implicit equations [(\[eqn:findF\])]{} and [(\[eqn:findF2\])]{}. Then, $X_{_1}$ and $R_{_1}$ are found from [(\[eqn:updateX\])]{} and [(\[eqn:updateR\])]{}, respectively. After that, we can compute $\Gamma_{_1}$, $\Pi_{_1}$, and $\Pi_{2_1}$ from [(\[eqn:updateV\])]{}, [(\[eqn:updatePi\])]{} and [(\[eqn:updatePi2\])]{}. This yields a discrete Hamiltonian map $(R_{_0},X_{_0},\Pi_{_0},\Gamma_{_0},\Pi_{2_0})\mapsto (R_{_1},X_{_1},\Pi_{_1},\Gamma_{_1},\Pi_{2_1})$, and this process can be repeated. $x_{2_k}$, $\gamma_{2_k}$ and $R_{2_k}$ can be updated separately using [(\[eqn:updateX2\])]{}, [(\[eqn:updateV2\])]{} and [(\[eqn:updateR2\])]{}, respectively, for reconstruction. Numerical considerations ------------------------ *Properties of the variational integrators:* Variational integrators exhibit a discrete analogue of Noether’s theorem [@jo:marsden], and symmetries of the discrete Lagrangian result in conservation of the corresponding momentum maps. Our choice of discrete Lagrangian is such that it inherits the symmetries of the continuous Lagrangian. Therefore, all the conserved momenta in the continuous dynamics are preserved by the discrete dynamics. The proposed variational integrators are expressed in terms of Lie group computations [@ic:iserles]. During each integration step, $F_{i_k} \in {\ensuremath{\mathrm{SO(3)}}}$ is obtained by solving an implicit equation, and $R_{i_k}$ is updated by multiplication with $F_{i_k}$. Since ${\ensuremath{\mathrm{SO(3)}}}$ is closed under matrix multiplication, the attitude matrix $R_{i_{k+1}}$ remains in [$\mathrm{SO(3)}$]{}. We make this more explicit in section \[comp\] by expressing $F_{i_k}$ as the exponential function of an element of the Lie algebra $\mathfrak{so}(3)$. An adjoint integration method is the inverse map of the original method with reversed time-step. An integration method is called self-adjoint or symmetric if it is identical with its adjoint; a self-adjoint method always has even order. Our discrete Lagrangian is chosen to be self-adjoint, and therefore the corresponding variational integrators are second-order accurate. *Higher-order methods:* While the numerical methods we present in this paper are second-order, it is possible to apply the symmetric composition methods, introduced in [@jo:Yoshida], to construct higher-order versions of the Lie group variational integrators introduced here. Given a basic numerical method represented by the flow map $\Phi_h$, the composition method is obtained by applying the basic method using different step sizes, $$\Psi_h = \Phi_{\lambda_s h}\circ\ldots\circ\Phi_{\lambda_1 h},$$ where $\lambda_1,\lambda_2,\cdots,\lambda_s\in\Rset$. In particular, the Yoshida symmetric composition method for composing a symmetric method of order 2 into a symmetric method of order 4 is obtained when $s=3$, and $$\lambda_1=\lambda_3=\frac{1}{2-2^{1/3}},\qquad \lambda_2=-\frac{2^{1/3}}{2-2^{1/3}}.$$ Alternatively, by adopting the formalism of higher-order Lie group variational integrators introduced in [@jo:Leok2004] in conjunction with the Rodrigues formula, one can directly construct higher-order generalizations of the Lie group methods presented here. *Reduction of orthogonality loss due to roundoff error:* In the Lie group variational integrators, the numerical solution is made to automatically remain on the rotation group by requiring that the numerical solution is updated by matrix multiplication with the exponential of a skew symmetric matrix. Since the exponential of the skew symmetric matrix is orthogonal to machine precision, the numerical solution will only deviate from orthogonality due to the accumulation of roundoff error in the matrix multiplication, and this orthogonality loss grows linearly with the number of timesteps taken. One possible method of addressing this issue is to use the Baker-Campbell-Hausdorff (BCH) formula to track the updates purely at the level of skew symmetric matrices (the Lie algebra). This allows us to find a matrix $C(t)$, such that, $$\exp(tA)\exp(tB)=\exp C(t).$$ This matrix $C(t)$ satisfies the following differential equation, $$\dot C=A+B +\frac{1}{2}[A-B,C]+\sum_{k \geq 2}\frac{B_k}{k!}\operatorname{ad}_C^k(A+B),$$ with initial value $C(0)=0$, and where $B_k$ denotes the Bernoulli numbers, and $\operatorname{ad}_C A = [C,A]=CA-AC$. The problem with this approach is that the matrix $C(t)$ is not readily computable for arbitrary $A$ and $B$, and in practice, the series is truncated, and the differential equation is solved numerically. An error is introduced in truncating the series, and numerical errors are introduced in numerically integrating the differential equations. Consequently, while the BCH formula could be used solely at the reconstruction stage to ensure that the numerical attitude always remains in the rotation group to machine precision, the truncation error would destroy the symplecticity and momentum preserving properties of the numerical scheme. However, by combining the BCH formula with the Rodrigues formula in constructing the discrete variational principle, it might be possible to construct a Lie group variational integrator that tracks the reconstructed trajectory on the rotation group at the level of a curve in the Lie algebra, while retaining its structure-preservation properties. Computational approach {#comp} ---------------------- The structure of the discrete equations of motion given in [(\[eqn:findFik\])]{}, [(\[eqn:findFikH\])]{}, [(\[eqn:findFl\])]{}, [(\[eqn:findF2l\])]{}, [(\[eqn:findF\])]{}, and [(\[eqn:findF2\])]{} suggests a specific computational approach. For a given $g\in\Rset^3$, we have to solve the following Lyapunov-like equation to find $F\in{\ensuremath{\mathrm{SO(3)}}}$ at each integration step. $$\begin{aligned} FJ_d - J_d F^T = S(g).\label{eqn:findf}\end{aligned}$$ We now introduce an iterative approach to solve [(\[eqn:findf\])]{} numerically. An element of a Lie group can be expressed as the exponential of an element of its Lie algebra, so $F\in{\ensuremath{\mathrm{SO(3)}}}$ can be expressed as an exponential of $S(f)\in{\ensuremath{\mathfrak{so}(3)}}$ for some vector $f\in\Rset^3$. The exponential can be written in closed form, using Rodrigues’ formula, $$\begin{aligned} F &= e^{S(f)},\nonumber\\& = I_{3\times3} + \frac{\sin{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}} S(f) + \frac{1-\cos{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}^2}S(f)^2.\label{eqn:rodc}\end{aligned}$$ Substituting [(\[eqn:rodc\])]{} into [(\[eqn:findf\])]{}, we obtain $$\begin{aligned} S(g) & = \frac{\sin{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}} S(Jf) + \frac{1-\cos{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}^2} S(f\times Jf),\end{aligned}$$ where [(\[eqn:Scross\])]{} and [(\[eqn:JdJ\])]{} are used. Thus, [(\[eqn:findf\])]{} is converted into the equivalent vector equation $g=G(f)$, where $G : \Rset^3 \mapsto \Rset^3$ is $$\begin{aligned} G(f) & = \frac{\sin{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}}\, J f + \frac{1-\cos{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}^2}\, f \times J f.\end{aligned}$$ We use the Newton method to solve $g=G(f)$, which gives the iteration $$\begin{aligned} f_{i+1} = f_i + \nabla G(f_i)^{-1} {\ensuremath{\left( g- G(f_i) \right)}}.\label{eqn:newton}\end{aligned}$$ We iterate until ${\ensuremath{\left\| g- G(f_i) \right\|}} < \epsilon$ for a small tolerance $\epsilon > 0$. The Jacobian $\nabla G(f)$ in [(\[eqn:newton\])]{} can be expressed as $$\begin{aligned} \nabla G(f) & = \frac{\cos{\ensuremath{\left\| f \right\|}}{\ensuremath{\left\| f \right\|}}-\sin{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}^3}Jff^T + \frac{\sin{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}} J\\ & \quad + \frac{\sin{\ensuremath{\left\| f \right\|}}{\ensuremath{\left\| f \right\|}}-2(1-\cos{\ensuremath{\left\| f \right\|}})}{{\ensuremath{\left\| f \right\|}}^4}{\ensuremath{\left( f \times Jf \right)}}f^T\\ & \quad +\frac{1-\cos{\ensuremath{\left\| f \right\|}}}{{\ensuremath{\left\| f \right\|}}^2} {\ensuremath{\left\{ -S(Jf)+S(f)J \right\}}}.\end{aligned}$$ Numerical simulations show that 3 or 4 iterations are sufficient to achieve a tolerance of $\epsilon=10^{-15}$. Numerical simulations ===================== The variational integrator in Hamiltonian form given in [(\[eqn:updateX\])]{} through [(\[eqn:updateR2\])]{} is used to simulate the dynamics of two simple dumbbell bodies acting under their mutual gravity. Full body problem defined by two dumbbell bodies ------------------------------------------------ Each dumbbell model consists of two equal rigid spheres and a massless rod as shown in [Fig. \[fig:dumbbell\]]{}. The gravitational potential of the two dumbbell models is given by $$\begin{aligned} U(X,R) = -\sum_{p,q=1}^{2}\frac{Gm_1m_2/4}{{\ensuremath{\left\| X+\rho_{2_p}+R\rho_{1_q} \right\|}}},\end{aligned}$$ where $G$ is the universal gravitational constant, $m_i\in\Rset$ is the total mass of the $i$th dumbbell, and $\rho_{i_p}\in\Rset^3$ is a vector from the origin of the body fixed frame to the $p$th sphere of the $i$th dumbbell in the $i$th body fixed frame. The vectors $\rho_{i_1}=[l_i/2,0,0]^T$, $\rho_{i_2}=-\rho_{i_1}$, where $l_i$ is the length between the two spheres. $$\begin{xy} \xyWARMprocessEPS{dumbbells}{eps}\xyMarkedImport{}\xyMarkedMathPoints{1-6} \end{xy}$$ *Normalization:* Mass, length and time dimensions are normalized as follows, $$\begin{aligned} \bar{m}_i & = \frac{m_i}{m},\\ \bar{X}_i & = \frac{X_i}{l},\\ \bar{t} & = \sqrt{\frac{G(m_1+m_2)}{l^3}}\,t,\end{aligned}$$ where $m=\frac{m_1m_2}{m_1+m_2}$, and $l$ is chosen as the initial horizontal distance between the center of mass of the two dumbbells. The time is normalized so that the orbital period is of order unity. Over-bars denote normalized variables. We can expresses the equations of motion in terms of the normalized variables. For example, [(\[eqn:Vdot\])]{} can be written as $$\begin{gathered} \bar{V}'+\bar{\Omega}_2\times \bar{V} = -{\ensuremath{\frac{\partial \bar{U}}{\partial \bar{X}}}},\end{gathered}$$ where $'$ denotes a derivative with respect to $\bar{t}$. The normalized gravitational potential and its partial derivatives are given by $$\begin{aligned} \bar{U}&=-\frac{1}{4}\sum_{p,q=1}^{2}\frac{1}{{\ensuremath{\left\| \bar X+\bar \rho_{2_p}+R\bar\rho_{1_q} \right\|}}},\\ {\ensuremath{\frac{\partial \bar{U}}{\partial \bar X}}}&=\frac{1}{4}\sum_{p,q=1}^{2}\frac{\bar X+\bar \rho_{2_p}+R\bar\rho_{1_q}}{{\ensuremath{\left\| \bar X+\bar \rho_{2_p}+R\bar\rho_{1_q} \right\|}}^3},\\ {\ensuremath{\frac{\partial \bar{U}}{\partial R}}}&=\frac{1}{4}\sum_{p,q=1}^{2}\frac{(\bar X+\bar \rho_{2_p})\bar\rho_{1_q}^T}{{\ensuremath{\left\| \bar X+\bar \rho_{2_p}+R\bar\rho_{1_q} \right\|}}^3}.\\\end{aligned}$$ *Conserved quantities:* The total energy $E$ is conserved: $$\begin{aligned} E & =\frac{1}{2}m_1{\ensuremath{\left\| V+V_2 \right\|}}^2 +\frac{1}{2}m_2{\ensuremath{\left\| V_2 \right\|}}^2\\& \quad +\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega)J_{d_R}S(\Omega)^T \right]}}}} +\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ S(\Omega_2)J_{d_2}S(\Omega_2)^T \right]}}}}+U(X,R).\end{aligned}$$ The total linear momentum $\gamma_T\in\Rset^3$, and the total angular momentum about the mass center of the system $\pi_T\in\Rset^3$, in the inertial frame, are also conserved: $$\begin{aligned} \gamma_T & = R_2{\ensuremath{\left\{ m_1{\ensuremath{\left( V+V_2 \right)}}+m_2 V_2 \right\}}},\\ \pi_T & = R_2{\ensuremath{\left\{ m X\times V + J_R \Omega + J_2 \Omega_2 \right\}}}.\end{aligned}$$ Simulation results ------------------ The properties of the two dumbbell bodies are chosen to be $$\begin{aligned} {3} \bar{m}_1&=1.5,&\quad \bar l_1&=0.25,&\quad \bar J_1&=\mathrm{diag}\begin{bmatrix}0.0004&0.0238&0.0238\end{bmatrix},\\ \bar{m}_2&=3,&\quad \bar l_2&=0.5,&\quad \bar J_2&=\mathrm{diag}\begin{bmatrix}0.0030&0.1905&0.1905\end{bmatrix}.\end{aligned}$$ The mass and length of the second dumbbell are twice that of the first dumbbell. The initial conditions are chosen such that the total linear momentum in the inertial frame is zero and the total energy is positive. $$\begin{aligned} {4} \bar X_{_0} & = \begin{bmatrix}1&0&0.3\end{bmatrix},&\quad\bar V_{_0} & = \begin{bmatrix}0&1&0\end{bmatrix},&\quad\bar \Omega_{1_0} & = \begin{bmatrix}0&0&9\end{bmatrix},&\quadR_{_0} & = I_{3\times 3},\\ \bar x_{2_0} & = \begin{bmatrix}-0.33&0&-0.1\end{bmatrix},&\bar v_{2_0} & = \begin{bmatrix}0&-0.33&0\end{bmatrix},&\bar \Omega_{2_0} & =\begin{bmatrix}0&0&0\end{bmatrix},&R_{2_0} & = I_{3\times 3}.\end{aligned}$$ Simulation results obtained using the Lie group variational integrator are given in [Fig. \[fig:flyby3d\]]{} and [Fig. \[fig:flybyER\]]{}. [Fig. \[fig:flyby3d\]]{} shows the trajectory of the two dumbbells in the inertial frame. [Fig. \[fig:flybyE\]]{} shows the evolution of the normalized energy, where the upper figure gives the history of the translational kinetic energy and the rotational kinetic energy, and the lower figure shows the interchange between the total kinetic energy and the gravitational potential energy. [Fig. \[fig:flybyR\]]{} shows the evolution of the theoretically conserved quantities, where the upper figure is the history of the total energy, and the lower figure is the error in the rotation matrix. ![Trajectory in the inertial frame[]{data-label="fig:flyby3d"}](flyby3d){width="90.00000%"} Initially, the first dumbbell rotates around the vertical $e_3$ axis, and the second dumbbell does not rotate. Since the angular velocity of the first dumbbell is relatively large, the rotational kinetic energy initially exceeds the translational kinetic energy. As the two dumbbells orbit around each other, the second dumbbell starts to rotate, the rotational kinetic energy increases, and the translational kinetic energy decreases slightly for about 6 normalized units of time. At 9 units of time, the distance between the two dumbbells reaches its minimal separation, and the potential energy is transformed into kinetic energy, especially translational kinetic energy. After that, two dumbbells continue to move apart, and the translational energy and the rotational energy equalize. (A simple animation of this motion can be found at <http://www.umich.edu/~tylee>.) This shows some of the interesting dynamics that the full body problem can exhibit. The non-trivial interchange between rotational kinetic energy, translational kinetic energy, and potential energy may yield complicated motions that cannot be observed in the classical two body problem. The Lie group variational integrator preserves the total energy and the geometry of the configuration space. The maximum deviation of the total energy is $2.6966\times 10^{-7}$, and the maximum value of the rotation matrix error ${\ensuremath{\left\| I-R^TR \right\|}}$ is $2.8657\times 10^{-13}$. As a comparison, [Fig. \[fig:flybyERrk\]]{} shows simulation results obtained by numerically integrating the continuous equations of motion [(\[eqn:Gamdot\])]{}-[(\[eqn:R2doth\])]{} using a standard Runge-Kutta method. The rotational and the translational kinetic energy responses are similar to those given in [Fig. \[fig:flybyER\]]{} prior to the close encounter. However, it fails to simulate the rapid interchange of the energy near the minimal separation of the two dumbbells. The deviation of the total energy is relatively large, with a maximum deviation of $1.1246\times 10^{-2}$. Also, the energy transfer is quite different from that given in [Fig. \[fig:flybyE\]]{}. The Runge-Kutta method does not preserve the geometry of the configuration space, as the discrete trajectory rapidly drifts off the rotation group to give a maximum rotation matrix error of $2.2435\times 10^{-2}$. As the gravity and momentum between the two dumbbells depend on the relative attitude, the errors in the rotation matrix limits the applicability of standard techniques to long time simulations. Conclusions =========== Eight different forms of the equations of motion for the full body problem are derived. The continuous equations of motion and variational integrators are derived both in inertial coordinates and in relative coordinates, and each set of equations of motion is expressed in both Lagrangian and Hamiltonian form. The relationships between these equations of motion are summarized in [Fig. \[fig:cubic\]]{}. This commutative cube was originally given in [@jo:Leok]. In the figure, dashed arrows represent discretization from the continuous systems on the left face of the cube to the discrete systems on the right face. Vertical arrows represent reduction from the full (inertial) equations on the top face to the reduced (relative) equations on the bottom face. Front and back faces represent Lagrangian and Hamiltonian forms, respectively. The corresponding equation numbers are also indicated in parentheses. It is shown that the equations of motion for the full body problem can be derived systematically, using proper Lie group variations, from Hamilton’s principle. The proposed variational integrators preserve the momenta and symplectic form of the continuous dynamics, exhibit good energy properties, and they also conserve the geometry of the configuration space since they are based on Lie group computations. The main contribution of this paper is the combination of variational integrators and Lie group computations, developed for the full body problem. Hence, the resulting numerical integrators conserve the first integrals as well as the geometry of the configuration space of the full body dynamics. Appendix ======== Variations of reduced variables {#apprv} ------------------------------- The variations of the reduced variables given in [(\[eqn:delX\])]{} through [(\[eqn:delV2\])]{} are derived in this section. The variations of the reduced variables can be obtained from the definitions of the reduced variables, and the variations of the original variables. The variation of $X=R_2^T(x_1-x_2)$ is given by $$\begin{aligned} \delta X & = \delta R_2^T (x_1-x_2) + R_2 (\delta x_1-\delta x_2).\end{aligned}$$ Substituting [(\[eqn:delRi\])]{} into the above equation, we obtain $$\begin{aligned} \delta X & = - \eta_2 R_2^T (x_1-x_2) + R_2 (\delta x_1-\delta x_2),\\ & = -\eta_2 X + \chi,\end{aligned}$$ where the reduced variation $\chi:[t_0,t_f]\mapsto\Rset^3$ is defined to be $\chi=R_2(\delta x_1-\delta x_2)$. From the definition of $\Omega=R\Omega_1$ and [(\[eqn:SR\])]{}, $S(\delta\Omega)$ is given by $$\begin{aligned} S(\delta\Omega)&=\frac{d}{d\epsilon}\bigg|_{\epsilon=0}S(R^\epsilon \Omega_1^\epsilon)=\frac{d}{d\epsilon}\bigg|_{\epsilon=0}R^\epsilon S(\Omega_1^\epsilon)R^{\epsilon T},\\ & = \delta R S(\Omega_1) R^T + R S(\delta \Omega_1) R^T+ R S(\Omega_1) \delta R^T.\end{aligned}$$ Substituting [(\[eqn:delR\])]{} and [(\[eqn:delOmegai\])]{} into the above equation, we obtain $$\begin{aligned} S(\delta\Omega)&={\ensuremath{\left\{ \eta-\eta_2 \right\}}}RS(\Omega_1)R^T+ R {\ensuremath{\left\{ \dot\eta_1 + S(\Omega_1)\eta_1-\eta_1 S(\Omega_1) \right\}}} R^T\\& \quad + R S(\Omega_1) R^T {\ensuremath{\left\{ -\eta+\eta_2 \right\}}},\\ &={\ensuremath{\left\{ \eta-\eta_2 \right\}}}S(R\Omega_1)+ R\dot\eta_1R^T + S(R\Omega_1)R\eta_1R^T-R\eta_1R^T S(R\Omega_1)\\& \quad + S(R\Omega_1) {\ensuremath{\left\{ -\eta+\eta_2 \right\}}}.\end{aligned}$$ Since $\eta=R\eta_1R^T$ and $\Omega=R\Omega_1$, the above equation reduces to $$\begin{aligned} S(\delta\Omega) = -\eta_2 S(\Omega) + R\dot{\eta}_1R^T +S(\Omega)\eta_2.\label{eqn:delOmega0}\end{aligned}$$ From the definition of $R=R_2^TR_1$, $\dot{R}$ is given by $$\begin{aligned} \dot{R} &= \dot{R}_2^T R_1 + R_2^T \dot{R}_1,\nonumber\\ & = -S(\Omega_2) R + S(\Omega)R.\label{eqn:Rdot0}\end{aligned}$$ Then, $\dot{\eta}$ can be written as $$\begin{aligned} \dot\eta &= R\dot{\eta}_1R^T + \dot{R}\eta_1R^T + R\eta_1\dot{R}^T,\nonumber\\ & = R\dot{\eta}_1R^T + {\ensuremath{\left\{ S(\Omega)-S(\Omega_2) \right\}}}\eta - \eta {\ensuremath{\left\{ S(\Omega)-S(\Omega_2) \right\}}}.\label{eqn:etadot}\end{aligned}$$ Substituting [(\[eqn:etadot\])]{} into [(\[eqn:delOmega0\])]{}, we obtain $S(\delta \Omega)$ in terms of $\eta,\eta_2$ as $$\begin{aligned} S(\delta\Omega) & = \dot\eta -S(\Omega)\eta+\eta S(\Omega)+S(\Omega)\eta_2-\eta_2 S(\Omega)+S(\Omega_2)\eta-\eta S(\Omega_2),\end{aligned}$$ which is equivalent to [(\[eqn:delOmega\])]{}. The variation of $V=R_2^T (\dot{x}_1-\dot{x}_2)$ is given by $$\begin{aligned} \delta V & = \delta R_2^T (\dot{x}_1-\dot{x}_2)+ R_2^T(\delta\dot{x}_1-\delta\dot{x}_2),\nonumber\\ & = -\eta_2 V + R_2^T(\delta\dot{x}_1-\delta\dot{x}_2).\label{eqn:delV0}\end{aligned}$$ From the definition of $\chi=R_2^T(\delta x_1-\delta x_2)$, $\dot{\chi}$ is given by $$\begin{aligned} \dot{\chi} & = \dot{R}_2^T(\delta x_1-\delta x_2)+R_2^T(\delta x_1-\delta x_2),\nonumber\\ & = -S(\Omega_2)\chi+R_2^T(\delta x_1-\delta x_2).\label{eqn:chidot}\end{aligned}$$ Substituting [(\[eqn:chidot\])]{} into [(\[eqn:delV0\])]{}, we obtain $$\begin{aligned} \delta V & =-\eta_2 V +\dot\chi + S(\Omega_2)\chi,\end{aligned}$$ which is equivalent to [(\[eqn:delV\])]{}. The variation $\delta V_2$ can be derived in the same way, and $S(\delta\Omega_2)$ is given in [(\[eqn:delOmegai\])]{}. Variations of discrete reduced variables {#appdrv} ---------------------------------------- The variation of the reduced variables $\delta F_{_{k}}$ given in [(\[eqn:delFk\])]{} is derived in this section. From [(\[eqn:delFik\])]{} and [(\[eqn:Fk\])]{}, the variation $\delta F_{1_{k}}$ is written as $$\begin{aligned} \delta F_{1_{k}} & = -\eta_{1_{k}} F_{1_{k}} + F_{1_{k}} \eta_{1_{k+1}},\\ & = -R_{_{k}}^T \eta_{_{k}}F_{_{k}}R_{_{k}}+R_{_{k}}^T F_{_{k}} R_{_{k}} R_{_{k+1}}^T \eta_{_{k+1}} R_{_{k+1}},\end{aligned}$$ where $\eta_{_{k}}\in{\ensuremath{\mathfrak{so}(3)}}$ is defined as $\eta_{_{k}}=R_{_{k}}\eta_{1_{k}}R_{_{k}}^T$. Since $F_{_{k}} R_{_{k}} R_{_{k+1}}^T=F_{_{k}} R_{_{k}} (R_{_{k}}^TF_{_{k}}^TF_{2_{k}})=F_{2_{k}}$, we have $$\begin{aligned} \delta F_{1_{k}} & = R_{_{k}}^T {\ensuremath{\left( -\eta_{_{k}}F_{_{k}}+F_{2_{k}} \eta_{_{k+1}}F_{2_{k}}^TF_{_{k}} \right)}} R_{_{k}}.\end{aligned}$$ Then, the variation $\delta F_{_{k}}$ is given by $$\begin{aligned} \delta F_{_{k}} & = \delta R_{_{k}} F_{1_{k}} R_{_{k}}^T+R_{_{k}} \delta F_{1_{k}} R_{_{k}}^T+R_{_{k}} F_{1_{k}} \delta R_{_{k}}^T,\nonumber\\& = -\eta_{2_{k}}F_{_{k}} +F_{2_{k}}\eta_{_{k+1}}F_{2_{k}}^TF_{_{k}}+F_{_{k}}{\ensuremath{\left( -\eta_{_{k}} +\eta_{2_{k}} \right)}},\end{aligned}$$ which is equivalent to [(\[eqn:delFk\])]{}.

--- abstract: 'Coherent transport promises to be the basis for an emerging new technology. Notwithstanding, a mechanistic understanding of the fundamental principles behind optimal scattering media is still missing. Here, complex network analysis is applied for the characterization of geometries that result in optimal coherent transport. The approach is tailored towards the elucidation of the subtle relationship between transport and geometry. Investigating systems with a different number of elementary units allows us to identify classes of structures which are common to all system sizes and which possess distinct robustness features. In particular, we find that small groups of two or three sites closely packed together that do not carry excitation at any time are fundamental to realize efficient and robust excitation transport. Features identified in small systems recur also in larger systems, what suggests that such strategy can efficiently be used to construct close-to-optimal transport properties irrespective of the system size.' author: - Stefano Mostarda - Federico Levi - 'Diego Prada-Gracia' - Florian Mintert - Francesco Rao bibliography: - 'mostarda2012bisse.bib' title: 'Optimal, robust geometries for coherent excitation transport' --- Introduction ============ Energy and charge transport are of fundamental importance for technological innovation as well as biological processes such as photosynthesis [@Scholes2011; @Renger2001]. If the dynamics is coherent, transport can be enhanced due to constructive interference. This, however, relies on well defined phase relations which get modified easily if the scattering medium is subject to external or internal sources of noise, even for small perturbations. Consequently, interference is destructive in most “real world" cases so that the efficiency is reduced to the point where transport might be completely suppressed [@Lattices1956; @Kramer1993; @Chin2010; @Rebentrost2009c]. The relationship between the detailed spatial configuration of the medium and its functional dynamical properties is subtle [@Baumann1986; @Renger2009]: two structures with similar geometries can possess strongly different transport properties and, vice versa, two structures with comparable transport properties may not share any evident common geometrical feature [@Scholak2011]. Clearly, a mechanistic understanding of the relationship between structure and transport efficiency would be necessary to use quantum coherence as a physical mechanism to develop new technological applications as well as understand photosynthesis at a fundamental level [@Li2012; @Leegwater1996; @Chachisvilis1997]. A recent application of complex network analysis on a set of randomly arranged excitable sites provided a systematic framework to characterize the structural properties of efficient transport [@Mostarda2013]. With much of a surprise, results provided strong evidence for the positive role of a structural motif formed by pair sites that are tightly packed together; although never significantly excited, they assure high transport efficiency and robustness against random displacements of the sites. This partition into excitation carriers and inactive pairs defines a dynamical separation that is reflected in the Hamiltonian, which is approximately composed of two weakly coupled blocks. While not necessarily emerging from the same geometrical features, such a dynamical arrangement has been located in some natural light harvesting complexes such as FMO [@Brixner2005; @Adolphs2006a]. It is then interesting to understand whether such a active/inactive modular arrangement is a truly general principle or if it is rather a peculiarity of systems of [*e.g.*]{} specific size. In this contribution, we therefore consider a paradigmatic system with variable number of randomly disposed excitable sites. Structures with outstanding transport properties are scrutinized, their common geometrical features determined through complex network analyses and their dynamical characteristics studied via inverse participation ratio and eigenvalue distributions. Comparison of the results obtained for different system sizes confirms the presence of specific structural classes for efficient transport that can be differentiated by their robustness properties. This outcome reinforces the idea that tightly packed sites which are not actively involved in the excitation transfer play a fundamental role in the transport, as they make the whole system efficient and robust under perturbations. Methods ======= Tight binding model ------------------- We analyze the transport properties of discrete systems, comprised by a set of $N$ excitable sites that are modeled as two-level systems. The interactions are described by the tight-binding Hamiltonian $$H=\sum_{i\neq j}^N \frac{Jr^{3}_{0}}{|\vec r_i-\vec r_j|^3}\sigma_i^{-}\sigma_j^{+}\ , \label{eq:ham}$$ where $J$ is the dimensionless coupling constant and $\sigma_i^{-/+}$ describe the annihilation/creation of an excitation at site $i$. The interaction rate decays cubically with the inter-site distance in accordance with dipole-dipole interaction. Within this model, a *structure* is defined by the positions of the $N$ sites. The initially excited site (input) and the site where the excitation is sought to arrive (output) are located at the diagonally opposite corners of a cube of side $r_0$, while the remaining $N-2$ sites are placed randomly within this cube. The system is initialized with an excitation on the input site; transport efficiency is defined as the maximal probability to find the excitation at the output site within a short time interval after initialization $$\epsilon=\mathrm{max}_{t\in[0,\mathcal{T}]}|\langle in|\mathrm{e}^{iHt}|out\rangle|^2\ . \label{eq:emaxtau}$$ The states $| in \rangle$/ $| out \rangle$ denote the situation where the input/ output site is excited and all other sites are in their ground state. In order to target exclusively fast transport that necessarily results from constructive interference, we choose $\mathcal{T} = \frac{1}{10} \frac{2\pi \hbar}{J} \frac{r_{in-out}^3}{r_{0}^3}$, [*i.e.*]{} a time-scale ten times shorter than the interval associated with direct interaction between input and output sites [@Scholak2011; @Mostarda2013]. For longer times the excitation would oscillate back and forth between input and output because the dynamics is purely coherent. With a sufficiently short time window, however, only a single oscillation is taken into consideration. Inverse Participation Ratio (IPR) --------------------------------- Under a coherent dynamics induced by a Hamiltonian of the form given in equation (\[eq:ham\]) the excitation will get delocalized over the sites of the system. This delocalization can be quantified in terms of the inverse participation ratio (IPR) defined as $$\mathrm{IPR}(t)=\frac{1}{\sum_{i=1}^{N} q_{i}^2(t)}\ , \label{eq:ipr}$$ where $q_i$ is the probability for site $i$ to be excited. A value for the IPR which is larger than $K-1$ implies that the excitation is delocalised over at least $K$ sites. The maximum value of the IPR is $N$, which is obtained in the case of even delocalization over the whole $N$ constituents. On the other hand, if the excitation is completely localized (e.g. at $t=0$ in our case), the IPR adopts its minimal value of 1. Efficiency Network ------------------ To unravel the structure-dynamics relationship, we apply a set of tools based on complex networks. Originally, these tools had been developed for the characterization of molecular systems [@Rao2004; @Gfeller2007]. However, since these methods are designed to analyze large ensembles of configurations, they prove very useful for our present purposes as they allow a systematic classification of structures which lead to exceptional transport. We generate a complex network where structures with $\epsilon>0.9$ represent the nodes and a link is placed between them if two structures are geometrically similar independently on the specific dynamics of the excitation. The parameter used to estimate structural similarity depends on the relative distances of the excitable sites of two structures under comparison. The sites are indistinguishable, thus all different permutations of the site labels need to be performed [^1]. In addition, a rotational symmetry around the in-out axis and an additional mirror symmetry has to be taken into consideration. The measure $S$ of similarity between configurations A and B is thus defined as $$S^2=\min\sum^n_{i=1} \frac{d_i ^2}{n}\ ,$$ where $d_i$ is the difference of the coordinates of the $i-$th site in the two configurations, and the minimization is performed over all permutations, rotations around the in-out axis and the mirror symmetry. A link is placed in the network only if $S$ lies below a certain threshold value $S^{*}$ which is going to be discussed in detail in the next sections. ![The structural superposition algorithm (here on the first cluster with $N=6$, 10731 structures) makes geometrical features emerge from the noise.[]{data-label="fig:superimposition"}](superimposition_1.pdf){width="47.50000%"} Network Clusterization ---------------------- Densely connected regions of the network indicate the presence of groups of structures with common geometrical motifs [@Mostarda2013]. We identify these regions using a network clusterization algorithm based on a self-consistency criterion in terms of network random walks, the Markov clusterization algorithm (MCL). The network is in this way split into different clusters comprised of structures with similar sites arrangements [@Gfeller2007; @Van2008]. The method consists of four steps: (a) start with the transition matrix $A$ of the network, where each column is normalized to 1; (b) compute $A^{2}$; (c) take the [*p*]{}-th power ($p>1$) of every element of $A^{2}$, normalize each column; (d) go back to step (b). After some iterations of the MCL, $A$ converges to $A_{MCL}$, where only one entry for each column is non-zero. Clusters are defined by the connected regions of the percolation network. In the limit of $p=1$, only one cluster is detected. On the other hand, the parameter [*p*]{} is related to the granularity of the clustering process. Large values of [*p*]{} generate several small clusters. Structural superposition ------------------------ A structural representation of the clusters is obtained in the following way: for each cluster, the most connected structure is taken as reference and all the others are superimposed. For each structure, we represent the one obtained with the combination of labeling, rotation and mirror state which minimizes the similarity parameter $S$ (see [*Network Creation*]{} section). In order to reduce noise, the coordinates of the sites are averaged with the ones from two other structures of the cluster taken at random. Structural rendering is done with VMD [@humphrey1996vmd]. An example of the effects of such algorithm is shown in Fig. \[fig:superimposition\]. We follow this procedure to depict all the clusters in Supp. Fig. \[fig:supp:VMDrepr1\],\[fig:supp:VMDrepr2\],\[fig:supp:VMDrepr3\]. ![Distribution of the similarity parameter $S$ for systems with different number of sites. (a) The average similarity between two given structures decreases ($S$ increases) with the number of sites. (b) The chosen cutoffs (in grey) show a dependence on N that is similar to that of the average similarity $S$ (in black).[]{data-label="fig:Sdistrib"}](Sdistribuwithcutoffs_2.png){width="47.50000%"} Consistency parameter $C$ -------------------------- In order to monitor whether the clusterization procedure is consistent while varying the granularity parameter $p$, we introduce here a “consistency parameter” $C$. If the clusterization is accurate, an increase in the granularity breaks big clusters into smaller clusters, without mixing them. In fact, every cluster obtained for a given value of $p$ should be fully (or almost fully) included in only one single cluster generated with a smaller value $p -\Delta p$. If this is the case, the clusterization is consistent and the value for $C$ will be maximal. On the other hand, the worst case is a completely random clusterization: the structures of each cluster for a given $p$ are equally distributed between the $n$ clusters generated with $p-\Delta p$. This would correspond to the minimal value of $C$. For the computation of $C$, we first perform the clusterization analysis from $p=1.1$ to $p=2.2$ in steps of $\Delta p = 0.1$ (from very low to very high values of $p$). For every step $p$, we calculate for each cluster $\mathcal{I}$ the largest portion $C_{\mathcal{I}}$ of its population $\mathcal{P}_{\mathcal{I}}$ included in a single cluster obtained at $p - \Delta p$. $C$ is then calculated as the average of $C_{\mathcal{I}}$ weighted over the relative populations $\mathcal{P}_{\mathcal{I}} / \sum_{\mathcal{I}} \mathcal{P}_{\mathcal{I}}$. In this way, the maximum value of $C$ is always 1, which corresponds to a perfectly consistent clusterization. The minimum of $C$ at a given $p$ is $1/n$, where $n$ is the number of clusters generated at $p -\Delta p$. To make the value of $C$ independent of $n$, we normalize it such that $1/n$ corresponds to 0 and rescale the $[1,1/n]$ segment linearly to $[1,0]$. Practical advices for the parameters choice =========================================== Network creation and clusterization depend on two parameters: the similarity cutoff $S^{*}$ which sets the accepted degree of similarity between different structures and the granularity parameter $p$ which determines the degree of coarse-graining in the clusterization. It is important to note that finding the correct value of these parameters for structural comparison is an open and unsolved problem in the broader field of complex systems. Apparently, there is no single [*right*]{} choice, as those parameters probe the system at different resolutions. Best practice suggests a scanning in parameter space in order to asses the robustness of the observations on a particular data set. In this section, we discuss cut-off choices in some detail. In Fig. \[fig:Sdistrib\]-a the distributions of $S$ are shown for the most efficient structures ($\epsilon>0.9$) obtained for $N=4-8$. Interestingly, two behaviors are present. The case $N=4$ is compatible with an almost homogeneous ensemble, where any two structures are very similar to each other ($S<0.15$ for the 94% of the links). On the other hand, in the systems with $N=6,7,8$ the number of pairs of compatible structures is instead very small, i.e. the ensemble is deeply heterogeneous ($S>0.15$ for the 92%, 98% and 99% of the links for $N=6,7,8$, respectively). The case $N=5$ shows an intermediate behavior. ![Parameters choice in the clusterization procedure for $N=6$. (a) Relative cluster populations for $p=1.2,1.4,1.6$ are shown in black, dark and light grey, respectively. Significant clusters separate from the noise which results in an exponential tail (fitted dashed lines). (b) Consistency parameter $C$ as a function of $p$.[]{data-label="fig:loyaltyparam"}](consistencyparam_3.pdf){width="47.50000%"} If only one system is considered, the cut-off needs just to be self consistent, i.e. the results should not vary too much with $S^{*}$. The problem arises when one wishes to compare different networks, [*i.e.*]{} different distance distributions. A fixed value for $S^{*}$ for the different cases would create networks with very different connectivities, which makes the comparison very hard. In order to set the thresholds in a compatible way, $S^{*}$ is taken as the minimal value of $S$ for which the networks are fully connected (99.9% of nodes have been considered). The resulting values are $0.02$, $0.06$, $0.11$, $0.16$ and $0.18 \ r_0$ for $N=4-8$, respectively. These values increase in a similar manner as the average value of the distance $S$ (see Fig. \[fig:Sdistrib\]-b). $S^{*}$ lies just above the tails of the pairwise distance distributions. Consequently, only the most similar structures are linked together. Lower values of the cut-off would generate a disconnected network, while values too close to the maximum of the distributions would put links between structures that are not very similar. For the clusterization process the goal is to separate the bulk of the signal from the statistical noise. To this aim, one can look at the population of the clusters obtained, ranked by decreasing size. Typically, the signal is formed by a small number of populated clusters, while the noise is composed of a large number of small clusters which follow an exponential tail. Fig. \[fig:loyaltyparam\]-a depicts as an example the results for the $N=6$ case obtained with different $p$. At $p=1.2$ (black curve) the algorithm detects two big clusters with 74.6% and 25.1% of the population plus two satellites due to noise with 0.3% of cumulative population. With $p=1.4$ (dark grey curve) the second cluster at $p=1.2$ splits into eight clusters with smaller relative populations ranging from 1.0% to 7.8%, while the noise is composed by the remaining 20 clusters (nicely fitted by an exponential function in Fig. \[fig:loyaltyparam\]-a). At $p=1.6$ (light grey curve) only four significant clusters are detected. Their populations are 41.5%, 14.9%, 8.5% and 7.1% of the total population (cumulatively the 72.0%), while the remaining 80 clusters have a cumulative relative population of 28.0% and constitute noise. With even higher values of $p$ the network breaks more and more into small noisy clusters. These three scenarios show how changing the granularity parameter $p$ leads to different signal to noise ratio. This behaviour is not necessarily monotonic: incrementing $p$ at first increases the number of significative clusters up to a maximum after which the noise grows and becomes dominant. However, similarly to the choice of $S^{*}$, our priority is to compare different networks. Therefore, the choice of $p$ which maximizes the signal to noise ratio for each network might not be the best for this purpose. We thus employ the *consistency parameter* $C$, which we calculate for a wide range of $p$ (see Methods for details). This quantity monitors whether the clusterization procedure is accurate and provides a way to consistently compare different networks. {width="99.00000%"} Let us illustrate the behaviour of $C$ for the $N=6$ case in detail as an example (see Fig. \[fig:loyaltyparam\]-b): at $p=1$ only one cluster is present, so $C=1$ by definition. At $p=1.1$ we have 2 clusters, but they are obviously both fully contained in the cluster of the former step, thus $C=1$. The first non trivial value of $C$ is at $p=1.2$ where the 4 clusters are quite well identifiable with the 2 clusters at $p=1.1$. There is a slight drop of $C$, but the value $C=0.99$ is sufficiently close to unity to warrant consistency. This regime is valid up to $p=1.4$, while at $p=1.5$ the value of $C$ drops to $0.95$ (the values of $C$ are normalized); this implies that the clusterization loses some consistency. The biggest drop of $C$ occurs at $p=1.7$, where $C=0.87$ and the consistency of the clusterization process is lost. For each choice of $N$, $C$ has a different behavior (this can be seen in Fig. \[figsupp:fidelityparam\] in Supp.Mat.). This means that we cannot choose a unique value $p$ to use in all the clusterization processes, but we need to investigate case by case the dependence of $C$ on $p$. We then select the highest value of $p$ for which $C=1$ for a given system size $N$: this systematic choice of $p$ allows a first qualitative understanding of the geometrical characterization of the system. The values correspond to $p=1.3,1.3,1.1,1.1$ for $N=4-7$, respectively. For the case $N=8$, the choice of $p=1.4$ obtained following the mentioned criterion leads to a single cluster. This is probably due to the high value of the $S^{*}$ chosen for this system, which creates a more densely connected network, hard to break into clusters ([*i.e.*]{} a higher value of $p$ would be needed). We therefore increase in this case the value slightly to $p=1.5$. Results ======= Structural characterization of the clusters {#sect:STRCHAR} ------------------------------------------- We analyze quantum transport for a large sample ($10^8$) of randomly generated structures with different number of sites ($N=4-8$, see Methods for details). The case $N=3$ has not been studied, since it never leads to efficiencies higher than 37% [@scholakphd]. Our analysis focuses on structures with $\epsilon>0.9$. Within this reduced set, the number of efficient structures is 3530, 7368, 14280, 5896, 6688 for $N=4$ to $N=8$, respectively (in Fig. \[figsupp:effdistrib\] in Supp. Mat. the probability of generating efficient structures is shown for $N=4-8$). Most of the sets of efficient structures are highly *heterogeneous*, which means that two structures with similar efficiency do not necessarily share any evident common pattern. This structural heterogeneity prevents a straightforward identifications of the geometrical features that are compatible with efficient transport. To uncover these features, we apply the protocol based on complex networks described in Methods. All the clusters we identify are shown in Supp. Fig. \[fig:supp:VMDrepr1\],\[fig:supp:VMDrepr2\],\[fig:supp:VMDrepr3\]. A sketch of the most relevant ones is depicted in Fig. \[fig:N4cl12\]. According to the network analysis, systems with $N=4$ and $N=5$ are quite homogeneous, and few geometries are compatible with efficient transport. The configurations obtained are shown in Fig. \[fig:N4cl12\]. In both cases the clusterization algorithm gives two clusters, where the intermediate sites of the less populated cluster are more strongly aligned than the others. For $N=4$ we label the two clusters $\tau_a$ and $\tau_b$ (first row of Fig. \[fig:N4cl12\]); they represent 63.1% and 36.9% of the total population, respectively. In both cases the four sites are equidistant from each other. However, in the latter case, the two intermediate sites are arranged along the input-output axis while in the former case they are slightly offset. For $N=5$, the situation is similar, but there is an extra intermediate site. Two clusters, named $\pi_a$ and $\pi_b$ (second row of Fig. \[fig:N4cl12\]), represent 83.5% and 15.8% of the total population (the remaining 0.7% is noise). In this case there is a slight deviation from an equidistant distribution of the inter-site distances. In the cluster $\pi_b$ the 3 intermediate sites are aligned along an axis which is rotated with respect to the in-out one; in $\pi_a$ these three sites form a triangle. {width="99.00000%"} Interestingly, the structures found for $N=4$ and $N=5$ constitute the building blocks for the higher dimensional cases (see Supp. Fig. \[fig:supp:VMDrepr1\],\[fig:supp:VMDrepr2\],\[fig:supp:VMDrepr3\]). In fact, systems with $N>5$ present a higher degree of heterogeneity and a prototypical modular structure. The first module is comprised by four/five sites approximately lined up along the in-out axis; this defines a structural backbone. In all cases this module is compatible with either $\tau_a$/$\tau_b$ or $\pi_a$/$\pi_b$. The second module is essentially formed by the remaining two to four sites, organized in tightly packed *pairs* or *triplets*. Backbone sites are approximately equally spaced between input and output with typical inter-site distances of around $0.50-0.60 \ r_0$, depending on the specific case. Pair/triplet sites instead are always very close to each other with a inter-site distance of around $0.25 \ r_0$, depending on the particular organization. It is worth noting that the backbone arrangement of all mentioned systems is symmetric under input output inversion [@Mostarda2013; @Walschaers2013]. The organization of $\tau_b$ constitutes the backbone of the most populated cluster of $N=6$, $\tau_6$ (75.1%, third row of Fig. \[fig:N4cl12\]). In this cluster, we identify a pair whose position is less well defined than the position of the backbone sites. A backbone of four sites that resembles $\tau_b$ is present also in the second cluster for $N=7$, $\tau_7$ (46.9%, fourth row of Fig. \[fig:N4cl12\], on the right), where the remaining three sites lie close together at comparable reciprocal distances, [*i.e.*]{} they form a *triplet*. This triplet is located more heterogeneously than the intermediate sites, in a region comparable to the one occupied by the pair module in $\tau_6$. Lastly, the most populated cluster for $N=8$, $\tau_8$ (58.4%), is formed by a backbone of four sites as in $\tau_6$ and $\tau_7$, with the remaining four sites organized into two pairs. At an increase of the granularity parameter to the value $p=1.7$, this cluster breaks into two smaller clusters that differ only in the location of the pairs: the more populated cluster $\tau_8^a$ (47.1%) has the two pairs on the same side of the backbone, where they form a triangle with the two intermediate sites of the backbone (Fig. \[fig:N4cl12\], fifth row right); the two pairs of the smaller cluster $\tau_8^b$ (10.3%) are instead diametrically opposed with respect to the backbone axis. Backbones composed of five sites emerge for $N=7$. The most populated cluster $\pi_7$ (53,1%, fourth row of Fig. \[fig:N4cl12\], on the left), is in fact formed by five sites organized in a backbone geometry similar to $\pi_a$ with an additional pair similar to the case of $\tau_6$. With this choice for the granularity parameters, the remaining clusters are less well defined. The second cluster $v_6$ for $N=6$ (24.8%), is composed of heterogeneous structures which are hard to reconduct to a single structural motif. In this case, increasing $p$ to $1.4$ separates this cluster into $7$ smaller more homogeneous clusters, with sites either disposed on a line or in a sparse manner. A more detailed discussion of the case $N=6$ can be found in [@Mostarda2013]. Also the second cluster $v_8$ for $N=8$ (36.9% of the population) is poorly identifiable. Cluster $v_8$ is in fact composed by a well defined backbone-like module with five sites and the remaining three sites in a sparse configuration. With the chosen granularity value $p=1.5$, the latter module is not compatible with a triplet. Subgroups at higher $p$, but with a very small population, present a triplet in a similar manner as in $\tau_7$. Altogether these results provide evidence for the presence of a modular arrangement in the geometries of efficient structures. Inverse participation ratio --------------------------- So far, we have constructed and clusterized a complex network of efficient structures on purely geometrical grounds. Now we move to investigate the dynamics of these structures, to better understand whether the common geometrical features identified correspond to dynamical similarities. To quantitatively characterize the dynamical behavior of the identified structures, the inverse participation ratio (IPR, see Methods) is calculated at every instant of time for each structure. In Fig. \[fig:inverseparticip\] the distributions of the maxima of the IPR within $t\in(0,\mathcal{T})$ from $N=4$ to $N=8$ are shown, divided into clusters. Remarkably, the maxima of the IPR spontaneously group into two well defined distributions. The cases $N=4$ and $N=5$ are basically homogeneous, with negligible differences between the two clusters in both systems. The corresponding values lie around $3.3$ and $4.2$, respectively, which means that the excitation is shared between approximately four or five sites. The two values are prototypical for the IPR distributions of clusters with bigger number of sites. In fact, structures in cluster $\tau_6$ have IPR maxima values similar to those in $\tau_a$ and $\tau_b$, while values for structures in $v_6$ have values close to those in $\pi_a$ and $\pi_b$. The distributions for triplet cluster in $N=7$ and the double pair clusters in $N=8$ (red curves) correspond to those for $\tau_a$ and $\tau_b$, while the pair cluster in $N=7$ and the sparse cluster in $N=8$ (blue curves) have the same IPR distribution as $\pi_a$ and $\pi_b$. Strikingly, the distribution of IPR supports that indeed the backbones of the $\tau_X$ and $\pi_X$ clusters (where $X$ stands for any $N$) for $N>5$ correspond to $\tau_a$/$\tau_b$ and $\pi_a$/$\pi_b$ respectively, not only from a geometric point of view, but also dynamically. This is evidenced by the excellent overlap of the distributions for different $N$ (bottom right panel of Fig. \[fig:inverseparticip\]). Inactive sites enhanced transport --------------------------------- The distributions of the IPR maxima reveal that for $N>5$ only a subset of the sites is substantially excited at the same time. In fact, the sites arranged in a backbone and those forming a pair or triplet possess a different dynamical role; while the former carry the excitation actively, sites closely packed together are never significantly populated by the excitation. Such a behavior emerges systematically for all system sizes, such that the pairs identified before for the case $N=6$ [@Mostarda2013] are just one example. In the following sections, we will thus refer to the backbone and to the pairs/triplets as to the [*active*]{} and [*inactive*]{} modules of the clusters. ![Efficiency loss upon removal of inactive modules as a function of the original efficiency $\epsilon$ for the clusters with $N=7$ and $N=8$. Error bars are calculated according to the standard deviation. All cases but $v_8$ are compatible with the pair effect found for $\tau_6$.[]{data-label="fig:efflossinactivemod"}](efficiencylossvertical_6.pdf){width="44.00000%"} Removal of the inactive module results in a systematic efficiency loss. This is shown in Fig. \[fig:efflossinactivemod\], where all clusters, apart from $v_8$, behave similarly: the contribution of the inactive modules is particularly important for the most efficient realizations due to the sensitivity of perfect constructive interference. The loss upon pair removal typically ranges from $0.05$ to $0.15$, depending on the initial value of the efficiency. As we suggested from geometrical considerations, $v_8$ is a very noisy cluster and cannot be considered completely composed of a 5-sites backbone plus a triplet. In fact, removal of these three sites causes an efficiency drop up to 60%, which indicates that the triplet in $v_8$ cannot be considered an inactive module. While the triplets in the first cluster with $N=7$ play a role similar to the pair in $\tau_6$, it is not obvious whether the presence of two pairs in $\tau_{8}^a$ and $\tau_{8}^b$ is necessary or if only one of them is enough to obtain the same effect. In fact, the two pairs show a small degree of collectiveness, which means that one is dominant and the other one has a close-to-negligible effect (Fig. \[fig:efflossinactivemodonebyone\](c-d) in Supp.Mat.). This is confirmed by the fact that the efficiency loss upon removal of the two pairs at the same time is only slightly larger than the sum of efficiency losses upon removal either pair(Fig. \[fig:efflossinactivemodonebyone\](a-b) in Supp. Mat.). Inactive modules induce eigenvalue shift ---------------------------------------- The mechanism behind the influence of the inactive modules on the exciton dynamics can be understood from the distribution of the energy eigenvalues with and without the inactive sites as displayed in Fig. \[fig:lambdashift\]. Because only a single excitation is present in the system at any time, there are $N$ energy eigenvalues to study. Given the weak interaction between the backbone and $k$ pairs or a triplet there are $N-2k$ or $N-3$ eigenstates whose amplitudes are highly localized on the backbone. The amplitudes of the remaining $2k$ or $3$ eigenstates are instead localized on the inactive sites. The interaction between the backbone and the inactive sites results in a shift of the eigenfrequencies of the former $N-2k$ or $N-3$ eigenstates (denoted by $\mathit{\lambda}_i$ in Fig. \[fig:lambdashift\]), such that their differences are close to integer multiples of a fundamental frequency. With this shift, the excitation is transferred to the output site essentially perfectly after one period of this fundamental frequency. This is true for all the clusters, ranging from the pair and triplet clusters with $N=7$ ($\pi_7$ in Fig. \[fig:lambdashift\]) to the clusters with two pairs in the $N=8$ case ($\tau_{8}^a$ and $\tau_{8}^b$ in Fig. \[fig:lambdashift\]), what strongly suggests that this mechanism [@Mostarda2013] works independently of the system size. ![Shift of the first backbone eigenvalue upon removal of inactive modules. In systems with both $N=7$ (top row) and $N=8$ (bottom row) the removal of the inactive sites causes a shift in only the first eigenvalue. In the former case, the mechanism is similar for both the pair and the triplet.[]{data-label="fig:lambdashift"}](eigshift_7.pdf){width="44.00000%"} Robustness ---------- The analysis presented so far shows the emergence of two classes of geometrical and dynamical behavior, characterized by an arrangement into active and inactive modules. In the following, we explore this separation with respect to the robustness properties of the various clusters. {width="99.00000%"} Transport robustness is probed by random displacements of the individual sites of a structure. With displacements confined to a cube of side $0.05 \ r_0$ centered around the original position of the site, $\mathit{\Delta} \epsilon_{\text{rand}}$ is calculated for each structure as the difference between the original efficiency and the average efficiency obtained from 1000 site-randomizations. In this scheme, structures are kept rigid which corresponds to the assumption that the dynamics occurs on a much faster time scale than low-frequency fluctuations of the entire system (e.g. in the context of biological systems this would be equivalent to large-scale protein breathing). The distributions of $\mathit{\Delta} \epsilon_{\text{rand}}$ for $N=4-8$ are shown in Fig. \[fig:stabilitytot\]. For $N=4$, both $\tau_a$ and $\tau_b$ are very robust under random displacement, the former is slightly more stable than the latter with $\mathit{\Delta} \epsilon_{\text{rand}} = 0.041$ as compared to $\mathit{\Delta} \epsilon_{\text{rand}} = 0.044$. Also in the case of $N=5$, $\pi_a$ and $\pi_b$ present overall a quite similar pattern of robustness: $\mathit{\Delta} \epsilon_{\text{rand}}$ are $0.094$ and $0.107$ for the first and second cluster, respectively. It is however in the clusters obtained for system of size from $N=6$ to $N=8$ that we detect the largest separation in response to random displacements. In all these cases, the loss in efficiency for structures in $\tau_X$ clusters is roughly half of the efficiency of the losses in $\pi_X$ or $v_X$ clusters. Exact values can be found in the caption of Fig. \[fig:stabilitytot\], where can be also visually noticed that the two curves separate well from each other in all cases. Overall, the efficiency loss upon random displacement, which represents the robustness of our randomly generated structures, spontaneously group into two distributions, independently on the number of total sites. This is clearly shown by the overlap of all the curves into a single plot (bottom right panel of Fig. \[fig:stabilitytot\]). Two behaviors are present, depending on the number of sites that build the backbone. Clusters whose backbone is composed by four sites (red data in Fig. \[fig:stabilitytot\]) show good robustness under random displacement of the sites ($\mathit{\Delta} \epsilon_{\text{rand}}$ peaked around $0.06$), while the efficiency loss of backbones with a larger number of sites (typically $5$) peaks around $0.10$ in all cases (blue data). Poorly defined clusters $v_6$ and $v_8$ share the same response to noise as the $\pi_X$ clusters. Overall, this result suggests that the backbone size is already a good indicator on the robustness of a given efficient structure. In agreement with the IPR analysis, all the robustness distributions overlap very well (compare the bottom right panels of Fig. \[fig:inverseparticip\] for IPR and Fig. \[fig:stabilitytot\] for robustness). This provides strong evidence for a clear correlation between robustness, backbone size and inverse participation ratio. Conclusions =========== As shown here, the application of advanced statistical techniques from complex network analysis permit to find a geometrical characterization of efficient structures. The analysis of efficient transport in systems with a variable number of excitable sites from $N=4$ to $N=8$ highlights the emergence of clear structural signatures related to high efficiency, independently of system size. For growing $N$, a modular arrangement appears. The first is a backbone-like module, typically formed by four or five sites that actively carry the excitation. The remaining sites are arranged in one or more inactive modules composed by tightly packed sites whose function is to tune the eigenvalues of the backbone to realize constructive interference and enhance transport. This mechanism is statistically dominant: only the 2% of the structures with $N=7$ or $N=8$ does not possess any inactive module. Remarkably, common geometrical and dynamical features evidence the recursiveness of these modules. Efficient structures for smaller systems ($N=4,5$) are identified as building blocks for larger structures ($N\geq6$). The addition of inactive modules to these prototypical backbones seem to represent an effective general strategy for the construction of structures in which high efficient transport is achieved by means of constructive interference. The analysis presented so far has been performed for a purely coherent case, i.e. without any source of noise. This choice is consistent, because results would not change qualitatively in the presence of incoherent effects. It has been in fact emphazised before that as long as the interest lies in the characterization of [*fast*]{} transport, which is what motivates the current definition of efficiency equation (\[eq:emaxtau\]), environmental noise would decrease the efficiency of every structure with no specific distinction [@Mostarda2013], irrespective of the environment considered. The modularity identified here holds great promise for an explicit exploitation as design principle: the construction of large optimized system seems feasible if it can be decomposed into smaller, individually optimized units, whereas a simultaneous optimization over all degrees of freedom easily turns impractical. Existing aspects of such modularity in actual LHC’s underline the feasibility to obtain such optimal structures through evolutionary optimization. It should thus be expected that the features classified here pave a practical roadmap towards the design of systems that achieve highly efficient transport in a potentially robust fashion. [**SUPPLEMENTARY FIGURES**]{} {width="80mm"} {width="80mm"} {width="100mm"} {width="130mm"} {width="130mm"} {width="130mm"} [^1]: There are $(N-2)!$ permutations since input- and output-site are distinguished from the other sites.

--- abstract: 'We present a sample of 131 quasars from the Sloan Digital Sky Survey at redshifts $0.8<z<1.6$ with double peaks in either of the high-ionization narrow emission lines [\[\]$\lambda3426$]{} or [\[\]$\lambda3869$]{}. These sources were selected with the intention of identifying high-redshift analogs of the $z<0.8$ active galactic nuclei (AGN) with double-peaked [\[\]$\lambda5007$]{} lines, which might represent AGN outflows or dual AGN. Lines of high-ionization potential are believed to originate in the inner, highly photoionized portion of the narrow line region (NLR), and we exploit this assumption to investigate the possible kinematic origins of the double-peaked lines. For comparison, we measure the [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} double peaks in low-redshift ($z<0.8$) [\[\]]{}-selected sources. We find that [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} show a correlation between line-splitting and line-width similar to that of [\[\]$\lambda5007$]{} in other studies; and the velocity-splittings are correlated with the quasar Eddington ratio. These results suggest an outflow origin for at least a subset of the double-peaks, allowing us to study the high-ionization gas kinematics around quasars. However, we find that a non-neligible fraction of our sample show no evidence for an ionization stratification. For these sources, the outflow scenario is less compelling, leaving the dual AGN scenario as a viable possibility. Finally, we find that our sample shows an anti-correlation between the velocity-offset ratio and luminosity ratio of the components, which is a potential dynamical argument for the presence of dual AGN. Therefore, this study serves as a first attempt at extending the selection of candidate dual AGN to higher redshifts.' author: - 'R. Scott Barrows, Claud H. Sandberg Lacy, Julia Kennefick, Julia M. Comerford, Daniel Kennefick, and Joel C. Berrier' title: 'Identification of Outflows and Candidate Dual Active Galactic Nuclei in SDSS Quasars at $z=0.8-1.6$' --- Introduction {#intro} ============ Mergers of gas-rich galaxies are likely to play a key role in the growth of supermassive black holes (SMBHs) through accretion, particularly in the triggering of quasar phases [@Sanders:1988; @Treister:2010]. Quasars may also represent an important phase in the evolution of galaxies due to radiative feedback [@Silk:Rees:1998; @Kauffman:2000]. This sequence of events is a potential scenario linking the evolution of galaxies and the growth of SMBHs [@Di_Matteo:2005; @Hopkins05]. Interestingly, both the merger and quasar phases can manifest themselves as active galactic nuclei (AGN) with double-peaked narrow emission lines in their spectra. Specifically, during a galaxy merger, when the SMBHs are separated by $\sim$1 kpc and are actively accreting as a dual AGN [@Comerford2009a] they will each produce Doppler-shifted emission from their narrow line regions (NLRs), resulting in a double-peaked profile in the integrated spectrum. In the feedback scenario, radiation from a single quasar can drive bi-conical outflows of the NLR [@Arribas:1996; @Veilleux:2001; @Crenshaw:2010b], producing similar emission line profiles [@Zheng90; @Fischer:2011]. AGN whose spectra exhibit double-peaked narrow emission lines (i.e. they are best fit by two components) represent a sub-class of the AGN population. Extended, offset and double-peaked line profiles have been observed in the narrow emission lines of AGN since the earliest studies of AGN NLRs [@Heckman1981]. These complex narrow line profiles are observed in both Type 1 and Type 2 AGN [@Crenshaw:2010a], and they are generally most pronounced in forbidden transition lines of high ionization potentials (I.P.s). In particular, such observations have often focused on the 5007Å transition line of [\[\]]{} ($I.P.=35.15$ eV) since it is a relatively intense emission line produced by the ionizing continuum of AGN and is accessible in optical spectra (see @Veilleux:1991 and @Whittle:1992 for examples). However, there is even variation among the high-ionization lines, with those of the highest ionization potentials, such as [\[\]]{} ($I.P.=41.07$ eV), and [\[\]]{} ($I.P.=97.16$ eV), displaying the largest velocity offsets [@De_Robertis:1984; @Sturm:2002; @Spoon:Holt:2009]. In the dual AGN scenario, the two emission line peaks are produced by the orbital motion of two AGN within a single merger remnant galaxy. This interpretation is intriguing since SMBHs reside in the bulges of galaxies, and dual SMBHs (kpc-scale separations) are a stage of galaxy mergers before the SMBHs coalesce. The existence of dual AGN has been confirmed observationally in several serendipitous cases, most notably in ultra-luminous infrared galaxies as pairs of X-ray point sources [@Komossa2003; @Guainazzi2005; @Hudson2006; @Bianchi2008; @Piconcelli2010; @Koss:2011; @Mazzarella:2012]. Systematic searches for dual AGN in large spectroscopic databases, such as the Sloan Digital Sky Survey [SDSS; @Abazajian09] and DEEP2 [@Newman:2012] have involved identifying AGN with double emission line systems [@Comerford2009a; @Wang2009; @Liu2010a; @Smith:2010; @Ge:2012]. Promising results are being obtained through follow-up observations in the form of high-resolution optical imaging [@Comerford2009b], near-infrared (NIR) adaptive optics imaging [@Liu2010b; @Fu:2011a; @Rosario:2011; @Shen:2011b; @Barrows:2012], spatially resolved spectroscopy [@McGurk:2011; @Fu:2012], hard X-ray observations [@Comerford:2011; @Civano:2012; @Liu:2012], radio observations [@Fu:2011], and diagnostics deduced statistically from longslit spectroscopy [@Comerford:2012]. A sample of dual AGN will be useful in studying the connection between galaxy interactions/mergers and AGN activity [@Green2010; @Liu:2011; @Liu:2012a], and for refining the galaxy merger rate [@Conselice:2003; @Berrier:2006; @Lotz:2011; @Berrier:2012]. In the outflow scenario, offset or double-peaked narrow emission lines are often attributed to radially flowing NLR gas driven by energy from the AGN. Furthermore, this effect is also thought to produce a stratification of the NLR since the incident ionizing flux and electron density should diminish with increasing distance from the nuclear source. This will result in the production of different lines in varying proportions as a function of radius from the AGN [@Veilleux:1991]. Observationally, luminous quasars are known to have extended NLRs driven by energy from the AGN coupled to the interstellar medium gas [@Bennert:2002], and powerful radio galaxies often show complex, spatially extended NLRs with multiple components aligned along the radio axis. Models for this alignment include reflection of the AGN emission [@Tadhunter:1988] or material entrained in a radio jet [@Holt2003; @Holt2008; @Komossa:2008b]. In AGN with sufficiently high Eddington ratios, radiation pressure acting on gas and dust [@Everett2007b] or a hot wind that entrains the NLR clouds [@Everett2007a] are possible mechanisms capable of driving the line splitting. Whichever mechanism is the dominant driver of outflows in an individual source, they may represent cases of AGN feedback, which might be important in quenching star formation in galaxies following a merger and in establishing the observed correlations between SMBH masses and host galaxy properties [@Ferrarese:Merritt:2000; @Marconi:Hunt:2003]. Whether AGN with double-peaked emission lines represent dual AGN or powerful AGN outflows, they have proven useful for investigating several aspects of AGN evolution, including AGN triggering and feedback. Systematic searches for double-peaked AGN have primarily utilized the [\[\]$\lambda5007$]{} emission line and have therefore been limited to below redshifts of $z\approx0.80$ since [\[\]$\lambda5007$]{} is not accessible in optical spectra at higher redshifts. However, there is evidence that at high redshifts galaxy mergers were more prevalent, and AGN outflows might have played an important role in the evolution of galaxies at $z\ge1$, including massive radio galaxies [@Nesvadba:2008]. Furthermore, there is significant controversy over the role that galaxy mergers play in AGN activity and SMBH growth at high redshifts [@Cisternas:2011; @Treister:2012; @Kocevski:2012]. Therefore, a sample of high redshift double-peaked narrow line AGN will be important for investigating these questions. So far only one such candidate dual AGN has been identified above $z\sim0.8$ as a serendipitous discovery at $z=1.175$ through double-peaked UV and optical emission lines, particularly evident in [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} [@Barrows:2012]. Motivated by this discovery, we have conducted a systematic search for additional AGN at high redshift ($z>0.8$) with double-peaked [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} emission lines. These emission lines have relatively high ionization potentials and trace the gas photoionized by the AGN continuum. Furthermore, they are likely to originate in the inner (more highly ionized) portion of the NLR, allowing for the identification of AGN driven outflows at $0.8<z<1.6$. This study will also serve as a first attempt at extending the identification of candidate dual AGN to higher redshifts. In Section \[analysis\] we describe our parent sample and how we selected our final sample and low-redshift comparison sample. In Section \[redshifts\] we describe the systemic redshifts we will use throughout the paper. In Section \[sec:offsets\] we describe the general properties of our sample and our estimates of selection completeness. In Section \[kinematics\] we investigate several correlations among the emission line properties that aid in illuminating the origin of the double-peaked line profiles. In Section \[radio\] we describe the radio properties of our sample and compare them to our parent sample. In Section \[interpretation\] we discuss the most likely physical mechanisms driving the line-splitting, particularly focusing on the scenarios of AGN outflows and dual AGN. In Section \[conclusions\] we summarize our main conclusions. Throughout the paper we adopt the cosmological parameters $\Omega_{\Lambda}=0.728$, $\Omega_{b}=0.0455$, $\Omega_{m}h^{2}=0.1347$, and $H_{0}=70.4$ km s$^{?1}$ Mpc$^{?1}$. This corresponds to the maximum likelihood cosmology from the combined WMAP+BAO+H0 results from the WMAP 7 data release of @Komatsu:2011. Generating the Sample {#analysis} ===================== Parent Sample {#sample} ------------- Our parent sample consists of archival spectra drawn from the quasar catalog of the SDSS Data Release 7 (DR7) which is described in detail in @Schneider:2010. The typical resolution of the SDSS spectra is $\lambda/\Delta \lambda \sim$2000. In short, inclusion in the catalog requires luminosities brighter than $M_{i}=-22.0$, at least one emission line with FWHM $>1000$ km s$^{-1}$ or complex absorption features, apparent magnitudes fainter than $i\approx 15.0$, and have highly reliable redshifts (see Section \[redshifts\] for a discussion of the redshifts). We restricted the lower redshift limit of the sample to $z\ge0.80$ to only include sources *not* in the parent samples of [\[\]]{}-selected double-peaked emitters from the SDSS [@Wang2009; @Liu2010a; @Smith:2010; @Ge:2012] since our intention is to select sources which are not identifiable by their methods. Additionally, we required that at least [\[\]$\lambda3426$]{} be accessible in the SDSS wavelength range ($3800-9200$ Å), which imposes an upper limit of $z\sim1.7$. We did not make any selection cuts based on the signal-to-noise ratios (S/N). This resulted in a parent sample of 39,876 sources.  \ Initial Selection {#sec:initial} ----------------- Our initial selection involved visually identifying quasars from the parent sample (Section \[sample\]) with detectable double emission line peaks in [\[\]$\lambda3426$]{} and/or [\[\]$\lambda3869$]{}. These two lines were used since they are accessible in the SDSS optical wavelength range at $z>0.80$, are relatively strong narrow lines in quasar spectra, and are not severely blended with any other strong lines. We did not require that two explicit peaks be detectable in both of those emission lines for two reasons: 1) the ionizing continuum may be such that [\[\]$\lambda3426$]{} is too weak to be detected whereas [\[\]$\lambda3869$]{} is detectable; and 2) a difference in line ratios and/or velocity-splittings between the two lines may result in one pair being more blended than the other. Therefore, since the purpose of this analysis is to investigate the origin of the double-peaked emission lines, we did not want to exclude those sources which show variations among the line properties. While [\[\]$\lambda3727$]{} is another strong emission line accessible in most of our parent sample, we did not select sources based on this line since [\[\]$\lambda3727$]{} is a doublet ($\lambda3726,3729$ Å) with $\sim$200 km s$^{-1}$ separation between the transition wavelengths and is difficult to discern from true peaks, particularly with the limited spectral resolution of the SDSS (see @Smith:2010 for a similar discussion). In sources where the double peak detections are ambiguous in each line, corresponding peaks in both [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} are needed for confirmation. Out of the parent sample, our visual selection process resulted in 181 preliminary sources.\ $ \begin{array}{c} \hspace*{-0.1in} \includegraphics[width=7in, height=1.8in]{Figure1a.eps} \\ \hspace*{-0.1in} \includegraphics[width=7in, height=1.8in]{Figure1b.eps} \\ \hspace*{-0.1in} \includegraphics[width=7in, height=1.8in]{Figure1c.eps} \\ \hspace*{-0.1in} \includegraphics[width=7in, height=1.8in]{Figure1d.eps} \end{array} $ $ \begin{array}{cc} \hspace*{-0.2in} \includegraphics[width=1.8in, height=2.7in]{Figure2a.eps} & \hspace*{-0.1in} \includegraphics[width=1.8in, height=2.7in]{Figure2b.eps} \end{array} $ Modeling the Spectra and Selection of the Final Sample {#modeling} ------------------------------------------------------ To generate the final sample to be used in our subsequent analyses, we modeled each spectrum yielded by the initial selection stage (Section \[sec:initial\]) in order to determine if a two-Gaussian model significantly improves the line profile fit over a single-Gaussian model. This modeling proceeded in two stages: 1) continuum modeling, and 2) emission line modeling. ### Continuum Modeling {#subsec:continuum} The continuum was modeled by masking all detectable emission lines and simultaneously fitting a power-law function ($F_{\lambda}\sim \lambda^{-\alpha_{\lambda}}$) for the underlying AGN contiuum radiation plus a pseudo-continuum of broadened   emission lines from the empirical templates of @Tsuzuki:2006 ($\lambda_{\rm{obs}} < 3500$ Å) and @veron_cetty:2004 ($\lambda_{\rm{obs}} > 3500$ Å) which were developed from the spectrum of the narrow-line Seyfert 1 galaxy I Zw 1. Due to the use of two separate  templates, the powerlaw function was allowed to have a break at 3500 Å, and the  normalization allowed to vary independently below and above the break wavelength. The  pseudo-continuum is composed of many blended  transitions which are generally believed to originate in or near the classical broad line region (BLR), and as such the redshift and broadening of the   emission should be comparable to that of the broad emission lines. Therefore, in our modeling the  emission was fixed at the redshift of the quasar being modeled (see Section \[redshifts\] for a detailed discussion of the redshifts); we broadened the  template by convolving with a Gaussian of $FWHM_{\rm{conv}}$ where $FWHM_{\rm{FeII}}^{2}=FWHM_{\rm{conv}}^{2}+FWHM_{\rm{I~Zw~1}}^{2}$ and $FWHM_{\rm{I~Zw~1}}=900$ km s$^{-1}$. Use of this template necessarily limits the minimum $FWHM_{FeII}$ best-fit to 900 km s$^{-1}$, though this was not a problem since these sources were in the quasar catalog based on the presence of broad lines with $FWHM>1000$ km s$^{-1}$. The lower rest-wavelength end of the continuum+ fitting window was 2750 Å and the upper rest-wavelength end was 3950 Å (just redward of [\[\]$\lambda3869$]{}) if accessible, or otherwise the red end of the spectral coverage. This fitting window allowed proper detection of the   emission since it has a relatively large equivalent width near [$\lambda2800$]{}. We allowed $FWHM_{\rm{FeII}}$ to vary in steps of 100 km s$^{-1}$, and in general, the solutions are in the range $\sim$2000-9000 km s$^{-1}$, though in most cases the quality of the fits were not strongly dependent on the broadening. ![[]{data-label="chi2"}](Figure3.eps "fig:"){width="3.5in" height="3.15in"}\ ### Narrow Emission Line Modeling We fit each emission line ([\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}) with both a single and a double Gaussian model, and required that sources in our final sample have fits to at least one of these two lines significantly improved by the double Gaussian model ($\Delta \chi^{2} \ge \Delta n_{DOF}$). In seven individual line models, an additional broad component of line width $FWHM\approx700-1500$ km s$^{-1}$ was required for a satisfactory fit. Three of the [\[\]$\lambda3426$]{} fits required such a component, where in two of those cases the broad component is consistent with the blue narrow peak (J074242.18+374402.0: $FWHM=1470$ km s$^{-1}$; and J105035.57+190544.2: $FWHM=1220$ km s$^{-1}$), and in the third case it is consistent with the red narrow peak (J150243.93+281739.9: $FWHM=1320$ km s$^{-1}$). Four of the [\[\]$\lambda3869$]{} fits required a broad component, where in three of those cases it is consistent with the blue narrow peak (J105035.57+190544.2: $FWHM=1520$ km s$^{-1}$; J105634.56+121023.5: $FWHM=820$ km s$^{-1}$; and J145659.27+503805.4: $FWHM=700$ km s$^{-1}$) and in one those cases it is located between the blue and red peaks (J085205.91+183922.2: $FWHM=1020$ km s$^{-1}$). We note that in only one of these sources (J105035.57+190544.2) is a broad component seen in both [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, and it is consistent with the blue peak. Properly accounting for the contribution of flux from  emission is often important in determining the profiles of the emission lines of interest in this wavelength range. In particular, there is a local peak in the broadened  emission just blueward of [\[\]$\lambda3426$]{}, between 3390 Å and 3410 Å, which has the potential to complicate the isolation of the blue [\[\]$\lambda3426$]{} component. Therefore, careful attention was paid to the modeling in this region, and any sources for which the model was ambiguous were not admitted into the final sample. However, there is no significant  emission near [\[\]$\lambda3869$]{}, and only a small local peak redward of [\[\]$\lambda3727$]{} but with which it is not blended. From our spectral modeling, there are 38 sources in our sample for which we have measured robust double-peaks in *both* [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, 42 sources for which we have *only* robust [\[\]$\lambda3426$]{} measurements, and 51 sources for which we have *only* robust [\[\]$\lambda3869$]{} measurements. Thus, there are a total of 131 double-peaked sources in our sample, making the fraction of double-peaked AGN detected in this manner $\sim$0.3% of the parent sample. See Figure \[examples\] for examples of the spectra in our final sample showing the best-fit models over the rest-wavelength range containing [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}. Additionally, Figure \[examples\_close\] compares the individual [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} line profiles in velocity-space. The best-fit parameters of the emission line modeling are listed in Table \[line\_props1\] ([\[\]$\lambda3426$]{}) and Table \[line\_props2\] ([\[\]$\lambda3869$]{}). All errors correspond to $1\sigma$ uncertainties and have been propagated throughout any further calculations. The reduced $\chi^{2}$ values for each double Gaussian model are listed in Tables \[line\_props1\] and \[line\_props2\] and shown as a distribution in Figure \[chi2\]. The emission line modeling was performed with SPECFIT [@Kriss94]. Our best-fitting  FWHMs are listed in Table \[tab:general\], along with other relevant quasar properties (redshifts, [$\lambda2800$]{} FWHMs, and Eddington ratios) which will be described in subsequent sections. Throughout the rest of the paper, the velocity offsets of the individual blue and red components will refer to the offsets from the systemic velocity. For [\[\]$\lambda3426$]{}, these blue and red velocity offsets will be defined as $\Delta V_{\rm{[NeV],blue}}$ and $\Delta V_{\rm{[NeV],red}}$, respectively. For [\[\]$\lambda3869$]{}, the blue and red velocity offsets will be defined as $\Delta V_{\rm{[NeIII],blue}}$ and $\Delta V_{\rm{[NeIII],red}}$, respectively. Blueshifts will correspond to positive velocities. The velocity-splittings will then be the velocity difference between the lines, i.e. $\Delta V_{\rm{NeV}}\equiv \Delta V_{\rm{[NeV],blue}}-\Delta V_{\rm{[NeV],red}}$ and $\Delta V_{\rm{NeIII}}\equiv \Delta V_{\rm{[NeIII],blue}}-\Delta V_{\rm{[NeIII],red}}$. The redshifts corresponding to the systemic velocity of each source are described in Section \[redshifts\]. Comparison Sample ----------------- To generate a $z<0.8$ comparison sample to be used in our analyses, we examined the 89 [\[\]]{}-selected Type 1 sources in @Smith:2010 and measured the line properties of double-peaked [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} in cases for which two peaks are detectable in those lines. We chose this sample because, as opposed to other double-peaked [\[\]$\lambda5007$]{} samples, it includes Type 1 AGN which were selected from the SDSS quasar catalog. Therefore, it provides a $z<0.8$ sample which can be used for direct comparison. For the measurements on the [\[\]]{}-selected sample, we selected those with robust [\[\]]{}/[\[\]]{} double-peaks in the same way as for our high-redshift sample, and likewise modeled the spectra as described above. This resulted in 18 sources in the [\[\]]{}-selected comparison sample for which we have measured robust double-peaks in *both* [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, 4 sources for which we have *only* have robust [\[\]$\lambda3426$]{} measurements, and 20 sources for which we have *only* have robust [\[\]$\lambda3869$]{} measurements. Thus, there are a total of 42 double-peaked sources in our comparison sample. Redshifts ========= Some sections of our analysis require a knowledge of the individual quasar redshifts. For the sources in our sample, we have redshifts available from several different measurement techniques. In this section we describe those redshift estimates which are useful for our scientific interests. SDSS Redshifts -------------- Redshifts for spectroscopically-detected sources in the SDSS are determined by the ‘spectro1d’ code described [lccccccc]{} J003159.87$+$063518.8 & $\phantom{00} 5.4\pm 4\phantom{00}$ & $ \phantom{0} 334\pm124 \phantom{0}$ & $ \phantom{0} 457\pm268 \phantom{0}$ & $\phantom{00} 6.9\pm 4\phantom{00}$ & $ \phantom{0}- 128\pm 6\phantom{000}$ & $ \ 465\pm203 \phantom{0}$ & 0.58 (10)\ J014933.86$+$143142.6 & $\phantom{00} 4.7\pm 2\phantom{00}$ & $ \phantom{0} 959\pm 55 \phantom{00}$ & $ \phantom{0} 260\pm 97\phantom{00}$ & $ \phantom{0} 18.7\pm 4\phantom{00}$ & $ \phantom{0-} 319\pm 57 \phantom{0}$ & $ \ 731\pm209 \phantom{0}$ & 1.25 (42)\ J015325.74$+$145233.4 & $ \phantom{0} 28.3\pm 8\phantom{00}$ & $ \phantom{0} 504\pm 49 \phantom{00}$ & $ \phantom{0} 403\pm119 \phantom{0}$ & $ \phantom{0} 33.9\pm 11 \phantom{0}$ & $ \phantom{0}- 176\pm 68 \phantom{00}$ & $ \ 585\pm249 \phantom{0}$ & 0.70 (32)\ J021648.36$-$092534.3 & $\phantom{00} 7.0\pm 2\phantom{00}$ & $ \phantom{0} 930\pm 35 \phantom{00}$ & $ \phantom{0} 245\pm 50\phantom{00}$ & $ \phantom{0} 34.9\pm 5\phantom{00}$ & $ \phantom{0-} 184\pm 41 \phantom{0}$ & $ \ 802\pm131 \phantom{0}$ & 0.65 (43)\ J021703.10$-$091031.1 & $ \phantom{0} 16.4\pm 4\phantom{00}$ & $ \phantom{0} 871\pm 35 \phantom{00}$ & $ \phantom{0} 364\pm 86\phantom{00}$ & $ \phantom{0} 28.0\pm 5\phantom{00}$ & $ \phantom{0-} 155\pm 51 \phantom{0}$ & $ \ 671\pm156 \phantom{0}$ & 0.82 (30)\ J034222.54$-$055727.9 & $ \phantom{0} 23.8\pm 6\phantom{00}$ & $ \phantom{0} 746\pm 73 \phantom{00}$ & $ \phantom{0} 641\pm206 \phantom{0}$ & $ \phantom{0} 24.4\pm 6\phantom{00}$ & $ \phantom{0}- 136\pm 83 \phantom{00}$ & $ \ 682\pm176 \phantom{0}$ & 0.90 (85)\ J035230.55$-$071102.3 & $ \phantom{}149.1\pm100 \phantom{}$ & $ \phantom{0} 690\pm270 \phantom{0}$ & $ \phantom{0} 933\pm267 \phantom{0}$ & $ \phantom{}244.1\pm108 \phantom{}$ & $ \phantom{00}- 15\pm110 \phantom{0}$ & $ \ 788\pm185 \phantom{0}$ & 0.72 (79)\ J073408.62$+$411901.1 & $ \phantom{0} 10.8\pm 4\phantom{00}$ & $ \phantom{0} 589\pm 81 \phantom{00}$ & $ \phantom{0} 364\pm214 \phantom{0}$ & $ \phantom{0} 20.1\pm 5\phantom{00}$ & $ \phantom{00}- 11\pm 68 \phantom{00}$ & $ \ 469\pm151 \phantom{0}$ & 1.95 (35)\ J074242.18$+$374402.0\* & $\phantom{00} 9.9\pm 5\phantom{00}$ & $ \phantom{0} 198\pm 34 \phantom{00}$ & $ \phantom{0} 236\pm125 \phantom{0}$ & $ \phantom{0} 10.3\pm 4\phantom{00}$ & $ \phantom{0}- 116\pm 30 \phantom{00}$ & $ \ 209\pm 74\phantom{00}$ & 0.81 (33)\ J074641.70$+$352645.6 & $ \phantom{0} 38.9\pm 22 \phantom{0}$ & $ \phantom{} 1070\pm327 \phantom{0}$ & $ \phantom{}1306\pm476 \phantom{0}$ & $ \phantom{0} 22.8\pm 21 \phantom{0}$ & $ \phantom{0-} 124\pm254 \phantom{}$ & $ \ 961\pm373 \phantom{0}$ & 0.84 (58) \[line\_props1\] [lccccccc]{} J000531.41$+$001455.9 & $ \phantom{0} 28.0\pm 7\phantom{00}$ & $ \phantom{0} 146\pm 53 \phantom{00}$ & $ \phantom{0} 372\pm125 \phantom{0}$ & $ \phantom{0} 11.2\pm 5\phantom{00}$ & $ \phantom{0}- 179\pm 32 \phantom{00}$ & $ \ 125\pm 49\phantom{00}$ & 2.15 (30)\ J003159.87$+$063518.8 & $\phantom{00} 6.2\pm 11 \phantom{0}$ & $ \phantom{0} 446\pm139 \phantom{0}$ & $ \phantom{0} 223\pm278 \phantom{0}$ & $ \phantom{0} 18.1\pm 12 \phantom{0}$ & $ \phantom{00-} 61\pm232 \phantom{}$ & $ \ 560\pm274 \phantom{0}$ & 2.07 (31)\ J004305.92$-$004637.6 & $\phantom{00} 7.8\pm 6\phantom{00}$ & $ \phantom{0} 395\pm 74 \phantom{00}$ & $ \phantom{0} 236\pm276 \phantom{0}$ & $\phantom{00} 9.5\pm 6\phantom{00}$ & $ \phantom{0-} 111\pm 55 \phantom{0}$ & $ \ 209\pm151 \phantom{0}$ & 1.20 (32)\ J004312.70$+$005605.0 & $\phantom{00} 9.3\pm 2\phantom{00}$ & $ \phantom{0} 215\pm 31 \phantom{00}$ & $ \phantom{0} 215\pm 65\phantom{00}$ & $ \phantom{0} 16.6\pm 2\phantom{00}$ & $ \phantom{0}- 140\pm 13 \phantom{00}$ & $ \ 212\pm 24\phantom{00}$ & 1.32 (33)\ J014822.62$+$132142.7 & $ \phantom{0} 37.8\pm 15 \phantom{0}$ & $ \phantom{0} 883\pm 78 \phantom{00}$ & $ \phantom{}1048\pm438 \phantom{0}$ & $ \phantom{0} 27.6\pm 6\phantom{00}$ & $ \phantom{0}- 376\pm102 \phantom{0}$ & $ \ 779\pm149 \phantom{0}$ & 0.83 (47)\ J015734.24$+$003405.5 & $ \phantom{0} 17.1\pm 5\phantom{00}$ & $ \phantom{0} 806\pm 50 \phantom{00}$ & $ \phantom{0} 260\pm 93\phantom{00}$ & $ \phantom{0} 70.6\pm 12 \phantom{0}$ & $ \phantom{00}- 69\pm 42 \phantom{00}$ & $ \ 672\pm131 \phantom{0}$ & 1.05 (37)\ J021648.36$-$092534.3 & $\phantom{00} 8.9\pm 2\phantom{00}$ & $ \phantom{0} 538\pm 27 \phantom{00}$ & $ \phantom{0} 185\pm 50\phantom{00}$ & $ \phantom{0} 35.8\pm 5\phantom{00}$ & $\phantom{000}- 4\pm 36 \phantom{00}$ & $ \ 546\pm 95\phantom{00}$ & 1.06 (24)\ J023234.33$-$091053.0 & $ \phantom{0} 51.4\pm 17 \phantom{0}$ & $ \phantom{} 1462\pm157 \phantom{0}$ & $ \phantom{}1565\pm506 \phantom{0}$ & $ \phantom{0} 34.4\pm 10 \phantom{0}$ & $ \phantom{00-} 31\pm151 \phantom{}$ & $ 1121\pm238 \phantom{0}$ & 0.73 (66)\ J034222.54$-$055727.9 & $ \phantom{0} 28.8\pm 9\phantom{00}$ & $ \phantom{0} 480\pm 0\phantom{000}$ & $ \phantom{0} 925\pm302 \phantom{0}$ & $ \phantom{0} 58.4\pm 7\phantom{00}$ & $ \phantom{0}- 323\pm 37 \phantom{00}$ & $ \ 656\pm 81\phantom{00}$ & 1.09 (78)\ J035230.55$-$071102.3 & $ \phantom{}120.0\pm 42 \phantom{0}$ & $ \phantom{0} 382\pm 68 \phantom{00}$ & $ \phantom{0} 433\pm135 \phantom{0}$ & $ \phantom{}201.2\pm 40 \phantom{0}$ & $ \phantom{00}- 70\pm 33 \phantom{00}$ & $ \ 379\pm 57\phantom{00}$ & 0.95 (32) \[line\_props2\] [lccccc]{} J000531.41$+$001455.9 & $0.9918\pm 0.0009$ & $0.9931\pm 0.0018$ & 9000 & $ 5424.76\pm 1192.59$ & 0.109\ J003159.87$+$063518.8 & $1.0921\pm 0.0010$ & $1.0935\pm 0.0016$ & 1000 & $ 2475.19\pm 1745.19$ & 0.085\ J004305.92$-$004637.6 & $0.8482\pm 0.0015$ & $0.8488\pm 0.0015$ & 8400 & $ 2828.74\pm 247.77\phantom{0}$ & 0.094\ J004312.70$+$005605.0 & $0.9036\pm 0.0010$ & $0.9047\pm 0.0014$ & 8500 & $ 6223.22\pm 283.39\phantom{0}$ & 0.105\ J014822.62$+$132142.7 & $0.8767\pm 0.0018$ & $0.8759\pm 0.0019$ & 8000 & $ 6063.22\pm 134.05\phantom{0}$ & 0.146\ J014933.86$+$143142.6 & $0.9024\pm 0.0013$ & $0.9027\pm 0.0013$ & 4500 & $ 2969.71\pm 244.58\phantom{0}$ & 0.172\ J015325.74$+$145233.4 & $1.1755\pm 0.0020$ & $1.1762\pm 0.0021$ & 5100 & $ 5219.56\pm 553.29\phantom{0}$ & 0.202\ J015734.24$+$003405.5 & $1.0834\pm 0.0013$ & $1.0825\pm 0.0020$ & 3300 & $ 6846.37\pm 1636.32$ & 0.109\ J021648.36$-$092534.3 & $0.8834\pm 0.0013$ & $0.8834\pm 0.0014$ & 2700 & $ 4288.12\pm 632.18\phantom{0}$ & 0.119\ J021703.10$-$091031.1 & $0.8752\pm 0.0013$ & $0.8757\pm 0.0015$ & 2700 & $ 2598.31\pm 432.35\phantom{0}$ & 0.289 \[tab:general\] in @Stoughton:2002. In short, two separate redshifts are determined, an emission line redshift ($z_{EL}$) and a cross-correlation redshift ($z_{XC}$), and the final value adopted by the code is the redshift solution with the highest confidence level (though each individual value is stored and available through the archive). Additionally, a small fraction of the quasar redshifts were re-determined after visual inspection as described in @Schneider:2010. $z_{EL}$ is measured by the identification of common galaxy and quasar emission lines with known rest-wavelength values, and therefore for our sample the highest confidence $z_{EL}$ values are almost exclusively determined by sets of the strong quasar emission lines [$\lambda1549$]{}, [\]$\lambda1909$]{}, [$\lambda2800$]{}, [\[\]$\lambda3727$]{}, H$\delta$, H$\gamma$, and H$\beta$. Likewise, the template which provided the highest confidence $z_{XC}$ values for our sample is the composite quasar template from @Berk:2001. For 107 of our sources, $z_{XC}$ and $z_{EL}$ are consistent with each other within their errors. For the remaining 24 redshifts, 19 are from $z_{XC}$, 4 are from $z_{EL}$, and 1 was determined by hand. We note that for those sources with disagreeing values of $z_{XC}$ and $z_{EL}$, the poorer of the two values is clearly incorrect and not usable. Throughout the rest of this paper, we refer to the final redshifts produced by the ‘spectro1d’ code as $z_{SDSS}$, and these values are listed in Table \[tab:general\]. ![[]{data-label="fig:z_offsets"}](Figure4.eps){width="3.5in" height="3.15in"} Redshifts from [$\lambda2800$]{} {#subsec:z_MGII} -------------------------------- In principle, individual emission lines can provide independent redshift estimates corresponding to the physical regions where those lines originated, e.g. the broad emission lines provide redshifts for the BLR. For our purposes, it is useful to know the redshift of the central SMBH (under the assumption of a single, active SMBH in the host galaxy) which can potentially be traced by the BLR if the gas is virialized. Fortuitously, given the redshift range of our sample, for all of our sources we have spectral access to the broad [$\lambda2800$]{} emission line, which has been shown to be virialized in the potential of the central SMBH [@McLure:Jarvis:2002]. Therefore, we have used the [$\lambda2800$]{} emission line models from the catalog of SDSS DR7 quasar properties [@Shen:2011a] to obtain estimates of the BLR redshifts ($z_{MgII}$). ![ []{data-label="fig:redshifts"}](Figure5.eps){width="3.5in" height="3.15in"} However, we would like to note several potential technical difficulties involved in measuring the [$\lambda2800$]{} line centroid which limit its application to our work. These difficulties include blending with  emission, UV absorption features, and the often ambiguous presence of a narrow emission line component (which is actually the doublet $\lambda\lambda2797,2802$). We also note that the [$\lambda2800$]{} line centroids may be complicated if the low-ionization broad emission lines have the characteristic double-peaked profile of disk emitters [@EH03; @strateva03]. Furthermore, since we are considering the scenario of two AGN within the same host galaxy, we must consider the possibility that the broad [$\lambda2800$]{} line is a blending of two BLRs from two Type 1 AGN. Though the components of a dual AGN would not be close enough for the line-splitting to exceed the FWHM (several thousand km s$^{-1}$) and produce explicit double-peaked profiles [@Shen:2010a], two broad [$\lambda2800$]{} components separated by several hundred km s$^{-1}$ will still result in a relatively broadened, and possibly asymmetric, [$\lambda2800$]{} profile, thereby complicating the centroid measurement. We have visually inspected the [$\lambda2800$]{} line profile for each source in our sample in order to characterize their structure. There are several sources which show evidence for asymmetric structure, though it is generally difficult to discern if any of the profiles contain two broad components as would be the case if there are dual BLRs. Though many of those redshifts have rather large errors, a subset of them have robust line centroids with percent errors less than $1\%$ (73 sources) and percent errors less than $0.1\%$ (52 sources). The [$\lambda2800$]{} redshifts are listed in Table \[tab:general\]. Comparison of $z_{SDSS}$ and $z_{MgII}$ --------------------------------------- Figure \[fig:z\_offsets\] shows the distribution of $(z_{SDSS}-z_{MgII})/(1+z_{SDSS})$ where the [$\lambda2800$]{} redshifts appear to be systematically larger than the cross correlation redshifts in both our sample and the parent sample. This same effect is clearly apparent in the SDSS redshift analysis of @Hewett:2010, which suggests that we are seeing the same systematic trend in our analysis. While @Hewett:2010 provide redshifts which correct for this systematic offset in a statistical sense, these corrections may not be appropriate for some individual quasars. Therefore, we do not use these improved redshifts since our sample only contains 131 sources. Based upon our comparison between $z_{SDSS}$ and $z_{MgII}$, for the rest of our analysis we have chosen to use the $z_{SDSS}$ values and their associated errors. These SDSS redshifts will serve as the systemic velocities for our sources. All quoted line-splittings between the two line peaks ($\Delta V_{\rm{[NeV]}}$ and $\Delta V_{\rm{[NeIII]}}$) and velocity offsets for the blue ($\Delta V_{\rm{[NeV],blue}}$ and $\Delta V_{\rm{[NeIII],blue}}$) and red ($\Delta V_{\rm{[NeV],red}}$ and $\Delta V_{\rm{[NeIII],red}}$) components are relative to these SDSS redshifts. The range of redshifts in our sample is $z=0.80$ (low-limit cutoff discussed in Section \[sample\]) to $z=1.53$ (highest redshift source in our sample). In Section \[sec:dynamics\], for which our analysis is heavily dependent upon the choice of redshift, we will also use the robust $z_{MgII}$ measurements mentioned in Section \[subsec:z\_MGII\] for comparison. General Properties of the Sample {#sec:offsets} ================================ In this section we describe several of the general properties of our sample. We compare the distributions of these properties (redshifts, S/N, and velocity offsets) to the parent sample or our low-redshift comparison sample in order to show our selection biases. Additionally, we discuss the results of our completeness estimates with respect to several of these and other properties. Distributions ------------- The SDSS redshifts we have adopted for our sample (Section \[redshifts\]) are shown and compared to the parent sample in Figure \[fig:redshifts\]. The redshift distribution shows a peak near the low redshift cutoff of the sample, with a gradual decline out to higher redshifts, in contrast to the increasing population of the parent SDSS quasar sample. This strong dependence on redshift may reflect a dependence on equivalent width or a luminosity bias, which we discuss further below. Figure \[SN\] shows the distribution of the continuum S/N per pixel, where the mean values are 18.5 and 14.5 in the wavelength regions adjacent to [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, respectively. Compared to the parent sample, our sample shows a deficit at the low S/N regime, and a stronger tail out to higher S/N values. These distributions show that our selection is biased toward spectra of good quality which reflect the fact that detection of double-peaked structure in [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} is sensitive to the S/N and generally difficult at S/N$<$5. Furthermore, the larger mean value of the continuum S/N near [\[\]$\lambda3426$]{} is reflective of the fact that [\[\]$\lambda3426$]{} is generally a weaker line than [\[\]$\lambda3869$]{} in AGN spectra [@Osterbrock:2006] and therefore a stronger signal is required for the detection of double-peaked structure. The dependence of our selection completeness on S/N will be addressed below. ![ []{data-label="SN"}](Figure6.eps "fig:"){width="3.5in" height="3.15in"}\ Since this study is motivated by the identification of high-redshift analogs of AGN with double-peaked [\[\]$\lambda5007$]{} lines, we compare the distribution of line properties of our sample to those of the [\[\]]{}-selected comparison sample. The SDSS spectral resolution, S/N, and the intensity of the emission lines determine whether or not we are able to reliably detect double-peaks at a given separation in velocity-space. [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} are fainter than [\[\]$\lambda5007$]{} in AGN spectra by a factor of $\sim4$ [@Ferland:Osterbrock:1986], which means that selection based on [\[\]]{}/[\[\]]{} will be biased toward larger line-splittings relative to [\[\]$\lambda5007$]{}. For example, double-peaked samples selected through [\[\]$\lambda5007$]{} have sources with line splittings down to $\sim$200 km s$^{-1}$, and the mean of the [\[\]$\lambda5007$]{} line-splittings in the Type 1 sample of @Smith:2010 is $\sim$420 km s$^{-1}$. On the other hand, the mean of the [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} line-splittings of our sample is $\sim$700 and $\sim$700 km s$^{-1}$, respectively. {width="7.in" height="3.5in"}\ Figure \[offsets\] shows the distribution of velocity offsets from the systemic redshift for [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} in both our sample and the [\[\]]{}-selected sample. While the measurements on the comparison sample were performed in the same way as for our sample (Section \[analysis\]), these sources were selected from a sample of previously determined double-peaked AGN. Therefore, we were able to more confidently identify two peaks in [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} at relatively smaller [$\Delta V$]{}s for the [\[\]]{}-selected sample, as shown in Figure \[offsets\]. A Kolmogorov-Smirnov (KS) test indicates that the blue components of our sample come from a different distribution than those of the [\[\]$\lambda5007$]{}-selected sample ($D_{\rm{[NeV]}}=0.56$, $D_{\rm{[NeIII]}}=0.43$, and probabilities that they come from the same distribution $P_{\rm{[NeV]}}=3\times 10^{-5}$, $P_{\rm{[NeIII]}}=7\times 10^{-5}$). A KS test also reveals a (less significant) difference in the red components ($D_{\rm{[NeV]}}=0.43$, $D_{\rm{[NeIII]}}=0.33$, and probabilities of $P_{\rm{[NeV]}}=2.3\times 10^{-3}$, $P_{\rm{[NeIII]}}=5.6\times 10^{-3}$). This indicates that we are finding similar double-peaked narrow line sources to those selected through [\[\]$\lambda5007$]{} but that we are preferentially missing a portion of those with relatively small [$\Delta V$]{}’s. Selection Completeness {#completeness} ---------------------- To estimate what fraction of sources with double-peaked [\[\]$\lambda3426$]{} and/or [\[\]$\lambda3869$]{} we are missing due to the potential selection biases discussed above, we have estimated our selection completeness through simulations. Specifically, we generated artificial spectra designed to mimic SDSS spectra in the [\[\]]{}/[\[\]]{} wavelength ranges (a similar test was performed by @Liu2010a for their [\[\]]{}-selected sample). Using the SDSS spectral dispersion, we modeled the continuum as a power-law and the emission lines as Gaussians with randomly assigned values for the parameters of equivalent width (EW), FWHM, continuum S/N per pixel, and line offsets from the systemic velocity. The ranges of allowed values for those parameters were made uniform and at least slightly larger than the distributions of our final sample in order to ensure that we sampled the relevant parameter space for the analysis. While the EW distribution of our sample only extended to $\sim$35 Å for the blue and red components, we allowed our simulations to have EWs of up to 50 Å for the individual lines (100 Å total EW). The simulated FWHMs ranged from 100 to 1800 km s$^{-1}$, $\sim$200 km s$^{-1}$ larger than the largest FWHM of our sample. The simulated continuum S/N values ranged from 2 to 33, just larger than the range of our sample (see Figure \[SN\]). Finally, the simulated line offsets ranged from 0 to 2000 km s$^{-1}$, $\sim$500 km s$^{-1}$ larger than the largest line offsets of our sample. We generated a total of 10,000 spectra, with 2,000 of them having double-peaked emission lines, and the other 8,000 having single-peaked emission lines. Out of these simulated spectra, randomly distributed, we selected a final sample of double-peaked emission lines in the same manner as described in Section \[analysis\]. We see a positive trend between completeness and EW, and find that we are highly incomplete over the EW range of our sample. For example, we estimate a completeness of just $\sim$1.2$\%$ for a total EW of 10 Å and $\sim$4.5$\%$ for a total EW of 30 Å. We see a negative trend between completeness and FWHM which can be interpreted as the result of peak blending at large line widths. We see a moderate positive trend between completeness and S/N, with a completeness of $\sim$42$\%$ at S/N=5 and $\sim$50$\%$ at S/N=30. Finally, we see a strong dependence on velocity-splitting, with a completeness of $\sim$55% at $\Delta V=900$ km s$^{-1}$ and $\sim$34% at $\Delta V=500$ km s$^{-1}$. In contrast, @Liu2010a find a completeness of $\sim$75% at $\Delta V=900$ km s$^{-1}$ and $\sim$50% at $\Delta V=500$ km s$^{-1}$ for their [\[\]$\lambda5007$]{}  sample. Over comparable parameter ranges, we are less complete than @Liu2010a for all of the parameters ($\sim$50-80% of their completeness), with the exception of FWHM, for which we have an equivalent or slightly greater completeness at the large FWHM end of their distribution ($\sim$900 km s$^{-1}$). This is likely due to the fact that we allowed for a larger range of line offsets resulting in our ability to identify broader double peaks. The smaller completeness in our selection compared to those of @Liu2010a is likely due to the relatively smaller equivalent widths of [\[\]]{}/[\[\]]{} allowed in our simulations which is reflective of the weaker intensities compared to [\[\]$\lambda5007$]{}. Of the parameters investigated, we find that our completeness is most strongly dependent on and most drastically different from @Liu2010a in the velocity-offsets (Figure \[offsets\]). This result is again consistent with the notion that we are preferentially missing a portion of those with relatively small [$\Delta V$]{}s. Since false double-peaks are generated by noise, false positives are most likely to occur in low S/N spectra in which random noise peaks are difficult to discern from the emission line signal. However, our conservative selection criteria effectively required that the S/N be sufficiently large, as can be seen in Figure \[SN\], with $S/N_{\rm{[NeV]}}<5$ for only 4 sources (3%) and $S/N_{\rm{[NeIII]}}<5$ for only 11 sources (8%). This is reflected in the fact that we did not select any false positives in our completeness analysis. $ \begin{array}{c c} \hspace*{-0.2in} \includegraphics[width=3.5in,height=5in]{Figure8.eps} \end{array} $ Correlations among Sample Properties {#kinematics} ==================================== In the following section, we examine several trends related to the velocity offsets of the individual blue and red components from the systemic redshift and line-splittings between the blue and red components. We are interested in uncovering the prevailing mechanism(s) producing the double-peaked emission lines in our sample. In particular, we investigate the following four relationships: line-splitting vs line width, line-splitting vs quasar Eddington ratio, [\[\]$\lambda3426$]{}  velocity offsets vs [\[\]$\lambda3869$]{}  velocity offsets, and blue/red velocity-offset ratio vs blue/red luminosity ratio. In the following analyses we combine the [\[\]]{}/[\[\]]{}  measurements of our sample with those of the [\[\]]{}-selected sample in order to increase both the sample size and the dynamic range of velocity offsets and line-splittings. A Correlation Between Line-Splittings and Line Widths {#sec:relv_width} ----------------------------------------------------- In AGN with emission lines offset from the systemic velocity, there is often an observed positive correlation between the line peak offset (blueshifts being positive offsets) and line width. For example, in samples of so-called ‘blue outliers’ (AGN with [\[\]$\lambda5007$]{} lines blueshifted by $>100$ km s$^{-1}$), there is an observed positive correlation between the blueshift and line width of [\[\]$\lambda5007$]{} [@Komossa:2008b]. A similar trend was also seen by @Liu2010a for their sample of Type 2 double-peaked [\[\]$\lambda5007$]{} sources, with the offsets being the line-splitting between the two peaks. In a similar fashion, we have plotted the line splittings against the line widths (FWHM) for the double-peaked [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} lines in both our sample and in the [\[\]]{}-selected sample (Figure \[relv\_width\]). For both [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, statistically significant positive correlations between the line-splitting and FWHM are evident. The correlations are strong and the line-splittings are generally larger than the line widths, and both of these results were also found by @Liu2010a with respect to [\[\]$\lambda5007$]{} emission lines. $ \begin{array}{cc} \hspace*{-0.1in}\includegraphics[width=7in,height=5.6in]{Figure9.eps} \end{array} $ The possible physical interpretations of these trends often includes an ionization stratification of the NLR in which the higher ionization lines originate closer to the ionizing radiation source [@Zamanov:2002; @Komossa:2008b]. In such a scenario, if there is a radially decelerating outflowing component of the NLR, then the higher ionization lines are accelerated to higher velocities. The emission lines produced in the inner portions of the NLR will also be more broadened as their motion would be dominated by the bulge gravitational potential [@Nelson:Whittle:1996]. Since the ionizing potentials of [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} ($I.P.=41.07$ and $97.16$ eV, respectively) are larger than that of [\[\]$\lambda5007$]{} ($I.P.=35.15$ eV), we might expect the correlation between line-splitting and FWHM to be stronger among these emission lines. This is also expected based on the higher critical electron densities for collisional de-excitation ($n_{\rm{crit}}$) of [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} ($n_{\rm{crit}}=1.3\times 10^{7}$ cm$^{-3}$ and $n_{crit}=9.5\times 10^{6}$ cm$^{-3}$, respectively) compared to [\[\]$\lambda5007$]{} ($n_{\rm{crit}}=6.8\times 10^{5}$ cm$^{-3}$) since electron densities will increase toward the nuclear region. Indeed, we find that the strengths of these correlations are generally comparable to or stronger than those for [\[\]$\lambda5007$]{} in the sample of @Liu2010a and in the narrow line Seyfert 1 (NLS1) sample from @Komossa:2008b. These trends in our sample might be more reflective of the even stronger trends among the subset of ‘blue-outliers’ from @Komossa:2008b. $ \begin{array}{c c} \hspace*{-0.2in} \includegraphics[width=3.5in, height=2.4in]{Figure10a.eps} & \includegraphics[width=3.5in, height=2.4in]{Figure10b.eps} \\ \hspace*{-0.2in} \includegraphics[width=3.5in, height=2.4in]{Figure10c.eps} & \includegraphics[width=3.5in, height=2.4in]{Figure10d.eps} \end{array} $ A Correlation Between Velocity-Splitting and Eddington Ratio {#ledd} ------------------------------------------------------------ Motivated by the outflow interpretation often used for samples of double-peaked emission line AGN, we investigate the relationship between the velocity-splittings and the quasar Eddington ratios ($f_{\rm{Edd}}=L_{\rm{bol}}/L_{\rm{Edd}}$). We calculated $L_{\rm{Edd}}$ using the standard derivation of the Eddington luminosity, $L_{\rm{Edd}}=4\pi cGM_{\rm{BH}}\mu_{e}/\sigma_{T}$, where $G$ is the gravitational constant, $\mu_{e}$ is the mass per unit electron, and $\sigma_{T}$ is the Thomson scattering cross-section [@Krolik:1999]. $M_{\rm{BH}}$ was estimated for each Type 1 AGN using the SMBH mass-scaling relationships from broad emission line widths and monochromatic luminosities: $FWHM_{\rm{H\beta}}$ and $L$[5100 Å]{} ($z<0.8$) or $FWHM_{\rm{MgII}}$ and $L$[3000 Å]{} ($z>0.8$) from McClure and Dunlop (2004). The line widths are mostly from the SDSS DR7 catalog of quasar properties @Shen:2011a, though we visually inspected and re-fit several of them for which we were able to improve on the models. For each AGN with an estimate of $M_{\rm{BH}}$, we estimated $L_{\rm{bol}}$ from the monochromatic luminosities $L$[5100 Å]{} ($z<0.8$) or $L$[3000 Å]{} ($z>0.8$) and the bolometric corrections of Richards et al. (2006). Figure \[fig:ledd\] shows $f_{\rm{Edd}}$ plotted against three quantities: velocity-splittings, blue velocity-offsets and red velocity-offsets. For both [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, a Spearman rank test reveals statistically significant correlations between $f_{\rm{Edd}}$ and each of these quantities. When comparing [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, it is apparent that the correlations are stronger for [\[\]$\lambda3426$]{} than for [\[\]$\lambda3869$]{} in all correlations tested. Additionally, for both [\[\]$\lambda3426$]{}  and [\[\]$\lambda3869$]{} the correlations are strongest between $f_{\rm{Edd}}$ and the blue velocity-offsets. We note that the low velocity-offsets are dominated by the [\[\]]{}-selected sources, which simply reflects the selection bias discussed in Section \[sec:offsets\]. If the velocity-splittings represent a NLR outflow in a significant number of these sources, then this result suggests that the radiation generated by the quasar accretion rate may play a crucial role in driving the line-splittings. Additionally, it appears that the blue component is more strongly dominated by outflowing material than the red component, which has implications for obscuration. $ \begin{array}{cc} \hspace*{-0.1in}\includegraphics[width=7in,height=4.2in]{Figure11.eps} \end{array} $ Ionization Stratification {#variation} ------------------------- The difference in I.P. between [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} is $\sim$56 eV and can be used to provide insight on the gas dynamics in different regions of the NLR. Since the line offsets have been shown to correlate with the quasar Eddington ratios (Section \[ledd\]), suggesting an outflowing component, we would like to search for evidence of an ionization stratification. Figures \[fig:strat\]A and \[fig:strat\]B show $\Delta V_{\rm{[NeV], blue}}$ versus $\Delta V_{\rm{[NeIII], blue}}$ and $\Delta V_{\rm{[NeV],red}}$ versus $\Delta V_{\rm{[NeIII],red}}$ for all sources in our sample and the [\[\]]{}-selected sample for which there are detectable double-peaks in both lines. Though the true nature of the gas kinematics in NLRs is likely to be complex, under the simplest pictures of NLR ionization stratifications it is generally not expected that lines of lower ionization potential will show blue velocity-offsets larger than those of greater ionization potentials. Indeed, from Figure \[fig:strat\]A it can be seen that all of the points are consistent with having equivalent blue velocity-offsets in the two lines, or otherwise with a larger blue offset in [\[\]$\lambda3426$]{}. This is also apparent in Figure \[fig:strat\]C, with a mean $\Delta V_{\rm{[NeV],blue}}-\Delta V_{\rm{[NeIII],blue}}$ value of 195 km s$^{-1}$. Accounting for the $1 \sigma$ uncertainties, from our sample 15 sources (40%) are consistent with equivalent blue velocity-offsets for both [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, and 23 sources (60%) have significantly larger [\[\]$\lambda3426$]{} blue velocity offsets, consistent with the presence of an ionization stratification. Comparable fractions are seen in the [\[\]]{}-selected sample. The sources in the small velocity-offset portion of the plot are dominated by the [\[\]]{}-selected sample, which again reflects the selection bias discussed in Section \[sec:offsets\]. In contrast, the difference in the red offsets has a narrower distribution and a mean value more consistent with $\Delta V_{\rm{[NeV],red}}=\Delta V_{\rm{[NeIII],red}}$ (Figures \[fig:strat\]B and \[fig:strat\]D). This suggests that the red systems are generally less stratified and have a different origin. Also plotted are several additional double-peaked sources for which follow-up observations have been obtained, including extended and unresolved NLRs from @Fu:2012, dual AGN from @Fu:2012 and @Comerford:2011, and a candidate dual AGN from [@Barrows:2012]. Note that the extended/unresolved NLRs and the dual AGN are [\[\]]{}-selected, and the candidate dual AGN was selected based on [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}. With the exception of the dual AGN SDSS J150243.1+111557, these additional sources are all consistent with no apparent ionization stratification in either the blue or red system by the $1\sigma$ criteria we have used above. Dynamical Relation {#sec:dynamics} ------------------ In this section we examine a dynamical argument, originally proposed by @Wang2009, for the presence of dual AGN in our sample. Under the simplest picture of Keplerian orbital motion, a binary system of masses should show an inverse correlation between the ratio of their velocities, $V_{1}/V_{2}$, and the ratio of their masses, $M_{1}/M_{2}$, i.e. $V_{1}/V_{2}=M_{2}/M_{1}$. As shown in @Wang2009 for their sample of double-peaked [\[\]$\lambda5007$]{} AGN, the ratio of the line offsets is equivalent to the velocity ratio, and the mass ratio can be approximated by the luminosity ratio multiplied by a factor representing the ratio of the accretion rates, $\epsilon_{1,2}$. This yields the relation $L_{b}/L_{r}=\epsilon_{b,r}\Delta V_{r}/\Delta V_{b}$ between the blue and red components. For our sample, we have measured the requisite observational properties to perform this same analysis, but with the emission lines [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}. Figure \[fig:flux\_shift\] shows $\Delta V_{b}/\Delta V_{r}$ plotted against $L_{b}/L_{r}$ for both [\[\]$\lambda3426$]{}  and [\[\]$\lambda3869$]{}. The plots include both our sample and the low-redshift comparison sample. Also over-plotted are the same confirmed/strong dual AGN candidates and extended/unresolved NLR AGN shown in Figure \[fig:strat\] and discussed in Section \[variation\]. From a Spearman rank correlation test, it is clear that there is a strong negative correlation between $\Delta V_{b}/\Delta V_{r}$ and $L_{b}/L_{r}$, which is in the same sense as expected based on the theoretical relation of the binary, Keplerian orbit. Specifically, a strong correlation is seen individually in our sample, the comparison sample, and in the combined sample. For both [\[\]$\lambda3426$]{}  and [\[\]$\lambda3869$]{}, the comparison sample shows slightly stronger correlations than for our sample. As argued in @Wang2009, in a major merger one might expect that the SMBHs will be in similar environments and therefore, on average, should have similar Eddington ratios ($\epsilon_{b,r}=1$). Therefore, we over-plotted the theoretical relation for $\epsilon_{b,r}=1$, which has the following form in base-10 logarithm space: $\log_{10}[L_{b}/L_{r}]=\log_{10}[\epsilon_{b,r}]-\log_{10}[V_{b}/V_{r}]$. For both [\[\]$\lambda3426$]{}  and [\[\]$\lambda3869$]{}  the colored/starred sources are generally consistent with the theoretical and best-fit lines, and they span a large enough dynamic range to display the same general trend as our sample. The best-fit linear relations (weighted by the X and Y errors) are $\log_{10}[L_{b}/L_{r}]=(0.16\pm0.02)-(0.75\pm0.09)\times \log_{10}[V_{b}/V_{r}]$ ([\[\]$\lambda3426$]{}) and $\log_{10}[L_{b}/L_{r}]=(0.05\pm0.02)-(0.51\pm0.04)\times \log_{10}[V_{b}/V_{r}]$ ([\[\]$\lambda3869$]{}). These relations are similar to the theoretical relation for $\epsilon_{b,r}=1$, though they are technically not consistent when accounting for the $1\sigma$ errors, with both fits yielding shallower slopes. The shallower slopes appear to be caused by some of the sources at the low $L_{b}/L_{r}$ end which extend out to larger $V_{b}/V_{r}$ ratios than the rest of the sample. In this portion of the plot, our sources are systematically shifted above the theoretical relation. Finally, we have over-plotted the theoretical relation expected for a biconical outflow, $L_{b}/L_{r}=(\Delta V_{b}/\Delta V_{r})^{3}$ (developed in @Wang2009). This relation is in the nearly the opposite sense as the for the binary relation and is far from agreeing with the best-fit relations. In Section \[sec:test\_dyn\] we will discuss how this correlation and the offset from the theoretical relation may be consistent with the presence of both dual AGN and outflows in our sample. While the version of this analysis presented in @Wang2009 utilized host galaxy redshifts (obtained through fitting template galaxy spectra), we can not obtain that information for our sample since the quasar continuum outshines the galaxy starlight. Instead, the velocity offsets used in our analysis here are relative to the SDSS redshifts ($z_{SDSS}$) discussed in Section \[redshifts\]. However, we have attempted to investigate the dependence of our results on the choice of redshift. Recalling our discussion in Section \[redshifts\], we also have redshifts based upon [$\lambda2800$]{}  which (in the single SMBH scenario) should trace the central, active SMBH redshift. Therefore, we examined $V_{b}/V_{r}$ versus $L_{b}/L_{r}$ for the subset of our sample with robust (percent error $<0.1\%$) $z_{MgII}$ values. We find that this does not introduce a significant change in the best-fit coefficients for [\[\]$\lambda3426$]{}  or [\[\]$\lambda3869$]{}. Radio Loudness {#radio} ============== Since $91\%$ (119/131) of our sources are in the FIRST footprint ($5\sigma$ flux limit of $\sim$750 mJy), to determine their radio-loudness we have adopted the commonly used definition of radio to optical luminosity ratio $\mathcal{R}=L$[5GHz]{}$/L$[2500 Å]{}, with $\mathcal{R}=10$ being the cutoff value for the radio-loud classification [@Ho:2002]. From our parent sample, the fraction of radio-loud SDSS quasars in the FIRST footprint is 10%, whereas that fraction is 23% in our final double-peaked [\[\]]{}/[\[\]]{}  sample. For comparison, @Smith:2010 obtained 9% (parent sample at $z<0.8$) and 27% (double-peaked [\[\]$\lambda5007$]{}  sample), respectively. This preferential selection of radio sources over the parent quasar population in both studies suggests that the origin of the double-peaks might be related to the presence of radio jets in some sources, as in the case of SDSS J151709.20+335324.7 [@Rosario:2010]. Interpretation ============== In this section, we synthesize the results of the previous sections to highlight the most likely physical scenarios that produce the line splitting and line offsets in our sample of quasars with high-ionization, double-peaked narrow emission lines. In particular, we examine the possibilities of AGN outflows and dual AGN. Examining the Outflow Hypothesis {#outflows} -------------------------------- The correlation between line-splitting and line width evident in the blue and red systems of both [\[\]$\lambda3426$]{}  and [\[\]$\lambda3869$]{} (Figure \[relv\_width\]) indicates that the mechanisms producing both the line-splittings and the line broadening are related, an observation which is consistent with the two emission line components originating near the same AGN. In general, emission line velocity-offsets from the host galaxy or quasar redshift are often interpreted as evidence for outflowing photoionized gas; powerful AGN, including quasars, are known to be capable of driving high-velocity and/or large-scale outflows [@Fischer:2011]. We will discuss how properties of our sample are consistent with some of the mechanisms known to produce outflows, and how they result in stratified NLRs. Furthermore, we will discuss our interpretation within the context of a proposed outflow and stratification geometry shown in Figure \[fig:cartoon\]. $ \begin{array}{cc} \hspace*{-0.in}\includegraphics[width=3.5in,height=2.625in]{Figure12.eps} \end{array} $ ### Mechanisms Producing AGN Outflows Two of the commonly proposed mechanisms for driving outflows in powerful AGN are radiation pressure from the accretion disk and radio jets. Figure \[fig:ledd\] shows that the velocity-splittings are correlated with the quasar Eddington ratio, $f_{\rm{Edd}}$, for [\[\]$\lambda3426$]{}  and for [\[\]$\lambda3869$]{}. This result is consistent with the notion that outflows can be driven by radiation pressure from an accretion disk, and that more actively accreting SMBHs will drive stronger outflows. This scenario is shown in Figure \[fig:cartoon\] with the radiation emanating from the accretion disk in the commonly assumed bi-conical shape [@Antonucci:1993]. Our result is also consistent with the results of @Komossa:2008b who find that their sample of NLS1s with offset [\[\]$\lambda5007$]{} lines have relatively large Eddington ratios which might be reflective of the radiation driving the outflow. Finally, that the correlation with $f_{\rm{Edd}}$ is stronger for [\[\]$\lambda3426$]{}  than for [\[\]$\lambda3869$]{}  fits with a picture in which the NLR gas closest to the central engine is more strongly accelerated by the radiation pressure, as indicated in Figure \[fig:cartoon\] by the larger velocity vectors on material closest to the SMBH. However, it is possible that not all of the outflows are driven by radiation pressure. In particular, radio jets are known to drive outflows in AGN by entrainment of NLR material, and our sample (and the [\[\]]{}-selected sample) have radio loud fractions of $\sim$25% (Section \[radio\]). Therefore, the high fraction of radio loud quasars in our sample relative to the parent quasar sample is a strong indication that the double-peaked sample we have compiled includes quasars with outflowing components. ### Evidence for Stratified NLRs If outflows are a common mechanism for driving line-splitting in NLRs, then this should naturally result in a stratified NLR since lines of greater I.P. and $n_{\rm{crit}}$ will be preferentially produced nearer to the AGN where they are accelerated to higher velocities relative to lines of lower I.P and $n_{\rm{crit}}$. The relative origins and velocities of emission lines in the stratification scenario are illustrated in Figure \[fig:cartoon\], where [\[\]$\lambda3426$]{} originates closer to the SMBH than [\[\]$\lambda3869$]{}. We examined this scenario by looking at the relationship between the velocity-offsets of [\[\]$\lambda3426$]{}  and [\[\]$\lambda3869$]{}  in Section \[variation\]. In that analysis, Figure \[fig:strat\]A shows that 60% of the sources have a significantly larger blue velocity offset in [\[\]$\lambda3426$]{} compared to [\[\]$\lambda3869$]{}, suggestive of a stratified NLR. We are only seeing the projected velocity-offsets, so this percentage might represent a lower limit on the number of sources with stratified NLRs when accounting for random orientations of the outflow axes. We note that in Figure \[fig:strat\] the largest stratifications are seen at the largest velocities. This could also be a result of projection effects, since orientations which reduce the radial velocity components will also reduce the observed stratification. However, this trend may also be due, in part, to the physical effect of stronger outflows (larger velocities) producing larger stratifications. For most of the sources in our sample we can only place upper limits on the fluxes and velocity offsets of blue [\[\]$\lambda3727$]{}  components. This is partly due to the blending of the $\lambda3726,3729$Å  doublet (Section \[sec:initial\]). However, if the double emission line components are driven by outflows and the NLR is stratified, one might expect that [\[\]$\lambda3727$]{}, with a relatively small ionization potential of $I.P.=13.614$ eV and critical density of $n_{\rm{crit}}=3.4\times 10^{3}$ cm$^{-3}$, will be strongest in a portion of the NLR relatively further from the central AGN. As a result, it will not be accelerated to high velocities, resulting in a small line-splitting. With the outflow axis oriented at some angle intermediate to edge-on or face-on, the redshifted NLR emission will consequently be more attenuated than the blueshifted emission, as illustrated by the ‘obscuring ISM’ labeled in Figure \[fig:cartoon\]. The portions of the NLR with the largest line of sight velocity components will be most obscured. Conversely, the portions with the smallest line of sight velocity components will be the least obscured. The result is that, in the presence of such attenuation, the most redshifted portion of the emission lines will obscured, moving the observed position of the red emission peak closer to the systemic, i.e. non-Doppler shifted, redshift. In this case the red component resembles the ‘classical’ NLR. This is consistent with observations in which offset narrow emission lines in AGN are usually blueward of the systemic velocity, indicating that we are able to view the outflowing component moving toward the observer, while the component moving away from the observer is obscured by a larger column of dust. For example, from Figure \[offsets\] it appears that the mean magnitude of the red component velocity-offsets from the systemic redshift are generally smaller than those of the blue components, consistent with the notion that the red components are less dominated by outflows. Note that in Figure \[fig:cartoon\], with sufficient attenuation even the NLR emission which is least Doppler-shifted will be obscured, resulting an apparent blueshifting of the red component, as is occasionally seen in some of our sources and in other studies [@Spoon:Holt:2009]. For example, the blue [\[\]$\lambda3426$]{} component may be emitted from a portion of the NLR on the observer’s side which is closest to the central source and moving at the greatest velocity (e.g. white dots in Figure \[fig:cartoon\]), while the red [\[\]$\lambda3426$]{} component is from a portion which is further from the central source (e.g. grey dots in Figure \[fig:cartoon\]). We did find in Section \[ledd\] evidence for a mild positive correlation between the red line offsets and the quasar Eddington ratio. This suggests that, while the red system tends to represent the ‘classical’ NLR, it is still effected by the radiation pressure since it must originate close enough to the central source where the ionizing flux is sufficient. Additionally, Figures \[fig:strat\]B and \[fig:strat\]C show that there is some evidence for stratification of the red systems (though much less significant than for the blue systems). Implications for Dual AGN at High-Redshift ------------------------------------------ It is possible that some of the sources in our sample may host two SMBHs following a galaxy merger. In this case, the double-peaks may be from two distinct NLRs that each accompanies its own active SMBH, or perhaps two NLR peaks are influenced by the orbital motion of two SMBHs [@Blecha:2012]. So far there are only a handful of known plausible merger remnants hosting two AGN at redshifts comparable to our sample: $z\sim0.709$ [@Gerke2007], $z\sim0.78$ [@Comerford2009a], and $z\sim1.175$ [@Barrows:2012]. Since galaxy mergers were more frequent at higher redshifts, we would like to investigate the dual AGN scenario for the sources in our sample. ### Sources with No Apparent Ionization Stratification The two [\[\]]{}-selected confirmed dual AGN for which we measured double peaks in [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} are consistent with no apparent ionization stratification (Figure \[fig:strat\]). Additionally, there is only one candidate dual AGN identified through the double-peaked profile of [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, and it is also consistent with no apparent ionization stratification [@Barrows:2012]. In these sources, the evidence for outflowing NLR material is less compelling, and the line-splitting may instead be produced by orbital motion of two AGN about each other. Likewise, the $40\%$ percent of our sources plotted in Figure \[fig:strat\] with no apparent ionization stratification may include cases where the double-peaks are the result of two SMBHs following a galaxy merger. Additionally, we see explicit double [\[\]$\lambda3727$]{} peaks for a subset (11) of our sources which perhaps suggests that the outflow scenario is less likely in these sources since [\[\]$\lambda3727$]{} should originate at a greater distance from the central AGN, as mentioned in Section \[outflows\]. We note, however, that the lack of an apparent ionization stratification does not preclude the possibility of an outflow, or general gas kinematic origin of the double-peaked emission lines. For example, the two extended NLR AGN from @Fu:2012 have no measurable ionization stratification but the double emission components are known to be produced by the NLR around a single AGN based on integral-field spectroscopy and high-resolution imaging. Conversely, a stratified NLR (or two stratified NLRs) does not preclude the presence of two AGN. For example, as discussed in Section \[variation\], our measurement of [\[\]]{}/[\[\]$\lambda3869$]{}  in the confirmed dual AGN SDSS J150243.1+111557 shows some evidence for a stratification.  \ ### Dual AGN with Large Velocities As is evident from Figure \[offsets\] and discussed in Section \[sec:offsets\], the velocity-splittings in our sample are generally larger than those from [\[\]]{}-selected samples, which might tend to select against likely dual AGN candidates at kpc-scale separations since they would not be bound to the merging galaxy system with such large velocities. Most strong dual AGN candidates have [$\Delta V$]{}s less than $500$ km s$^{-1}$: $\Delta V=150$ km s$^{-1}$ [@Comerford2009b; @Civano2010], $\Delta V=500$ km s$^{-1}$ [@XK09], $\Delta V=350$ km s$^{-1}$ [@Comerford:2011], and $\Delta V=420$ km s$^{-1}$ [@McGurk:2011; @Fu:2012]. However, there are several candidates with velocities $>500$ km s$^{-1}$: $\Delta V=630$ km s$^{-1}$ [@Gerke2007] and $\Delta V=700$ km s$^{-1}$ [@Barrows:2012]. Additionally, the dual AGN hypothesis could be allowed for larger [$\Delta V$]{}s if the AGN pairs are at small separations. For example, @Blecha:2012 find in their simulations that large [$\Delta V$]{}s ($>500$ km s$^{-1}$) are often associated with dual AGN at sub-kpc separations during pericentric passages. For comparison, $23\%$ of our sample have $\Delta V<500$ km s$^{-1}$, $45\%$ have $500<\Delta V<800$ km s$^{-1}$, and $32\%$ have $\Delta V>800$ km s$^{-1}$, with a maximum of $\Delta V=1665$ km s$^{-1}$. Therefore, though a fraction of our sample exhibit [$\Delta V$]{}s higher than expected for dual AGN, $2/3$ fall in the range expected for either kpc or sub-kpc separation AGN pairs. We note that recent numerical simulations suggest dual activation of the SMBHs following a galaxy merger is most likely to occur at separations smaller than 1-10 kpc [@Van_Wassenhove:2012]. Therefore, under this picture of dual activation, the correlation between Eddington ratio and line-splitting seen in Figure \[fig:ledd\] would naturally emerge for a sample of dual AGN. As discussed in Section \[redshifts\], for dual AGN with sufficiently large orbital velocities the broad emission line profiles may be significantly broader than expected if both components are Type 1 AGN. To test for such additional broadening, we have compared the [$\lambda2800$]{} FWHMs with those of , but find no evidence for systematically broadened [$\lambda2800$]{} compared to . Furthermore, a K-S test does not indicate a significant difference between the [$\lambda2800$]{} FWHM distribution of our sample and that of the parent sample. However, we note that the mean $FWHM_{\rm{MgII}}$ value for our sample (4740 km s$^{-1}$) is slightly larger than that of the parent sample (4580 km s$^{-1}$) which is perhaps suggestive of additional [$\lambda2800$]{} broadening. ### Testing the Dynamical Argument {#sec:test_dyn} In Section \[sec:dynamics\] we tested a dynamical argument for the presence of dual AGN in our sample. The results are generally consistent with the theoretical expectation for a binary, Keplerian orbit (Figure \[fig:flux\_shift\]). However, to understand the extent to which we can interpret this result, we must also strongly consider the role of alternative physical scenarios in producing such a correlation. First, we note that for both [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}, based on the Spearman rank test, the comparison sample shows slightly stronger correlations than for our sample. If this is an indication that the comparison sample shows stronger evidence for dual AGN, it would be consistent with the notion that larger velocity-splittings are less likely to be associated with dual AGN since our sample has larger velocity-splittings than the comparison sample. Second, it is worth noting that the coefficients for the best-fit linear relations of [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} are in disagreement. This is consistent with the correlation being produced by outflows (or at least some of the sources experiencing outflows) since the red component of [\[\]$\lambda3869$]{} would originate at a greater distance from the observer (compared to the red [\[\]$\lambda3426$]{} component) and therefore be even more obscured relative to the blue component (Figure \[fig:cartoon\]). In this case, the $L_{b}/L_{r}$ ratio should be even larger for [\[\]$\lambda3869$]{}, making the slope shallower as observed. However, it is possible that if our sample contains some combination of outflows and dual AGN then the outflows are responsible for deviations from the theoretical binary relation. For example, as discussed in Section \[sec:dynamics\] many of the sources at the low $L_{b}/L_{r}$, high $V_{b}/V_{r}$ portion of Figure \[fig:flux\_shift\] are offset above the theoretical relation. These sources may be more likely to represent outflows since they are trending in the same direction as the outflow relation. Additionally, they have large $V_{b}/V_{r}$ ratios because the red component is near the systemic redshift, consistent with attenuation of the redshifted outflow component. Lastly, the $L_{b}/L_{r}$ ratios are smaller, indicating that the red component is stronger, and the blue component is a lower luminosity, extended wing as is often seen in outflows and is seen a few of our sources. If these deviant sources are most likely to be outflows, then the remainder would be more consistent with the theoretical relation. Additionally, the remainder would have a $L_{b}/L_{r}$ distribution consistent with $\epsilon_{b,r}=1$, similar to the result of @Wang2009. We note that an additional source of scatter in the correlation could be due to stochastic accretion, such that the luminosity ratio does not accurately reflect the true mass ratio. This effect could be particularly significant when the SMBHs are at larger separations when gas is less efficiently funneled to the nuclear regions. Interestingly, at $z=0.8-1.6$, the 3” fiber diameter of the SDSS spectrograph corresponds to $\sim$22-25 kpc, so that our sample may contain such early-stage mergers. ### Estimating the Fraction of Dual AGN in Our Sample While the completeness of our selection process as discussed in Section \[sec:offsets\] suggests that there is a significant number of double-peaked emitters that we have missed, especially at small [$\Delta V$]{}s, we can not correct for the true number since we do not know the shape of the underlying distribution. However, at the least we can use our [\[\]]{}-selected comparison sample to estimate what fraction of [\[\]$\lambda5007$]{} double-peaked emitters that we missed based on selection through [\[\]$\lambda3426$]{} or [\[\]$\lambda3869$]{}. Of the 57 Type 1 AGN from [@Smith:2010], we could reliably measure double [\[\]$\lambda3869$]{} peaks for $67\%$ (we note that this fraction is consistent with the comparison between our completeness estimates and those of @Liu2010a in Section \[completeness\]). Based on this fraction, we arrive at a corrected number of 195 double-peaked [\[\]$\lambda5007$]{} sources and a double-peaked [\[\]$\lambda5007$]{} AGN fraction of $\sim0.5\%$ at $0.8<z<1.6$. This fraction is likely to be a lower estimate because the double-peaked [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} lines of the [\[\]]{}-selected sample were not selected in exactly the same way (i.e. we had a prior knowledge of the double-peaked separation). Therefore, the double-peaked [\[\]$\lambda5007$]{} AGN fraction at $0.8<z<1.6$ is likely to be larger, potentially making it consistent with the fraction of $\sim1\%$ found by @Liu2010a at $z<0.8$. To actually estimate the expected fraction of dual AGN in our sample we need the true fractions of double-peaked AGN and of dual AGN out of all AGN at $0.8<z<1.6$, neither of which are known. However, we may make several reasonable assumptions that provide a rough estimate. First, in order to determine the true fraction of double-peaked AGN out of all AGN at $0.8<z<1.6$, we need to correct for both our selection incompleteness and random projection effects. This fraction was estimated by @Shen:2011b in which they determined that the fraction of detectable double-peaked [\[\]$\lambda5007$]{} AGN is only $20\%$ of the actual number of AGN with double [\[\]$\lambda5007$]{} components. Correcting our estimated double-peaked [\[\]$\lambda5007$]{} fraction of $0.5\%$ yields a ‘true’ double [\[\]$\lambda5007$]{} fraction of $\sim$2.5% at $0.8<z<1.6$. The influence of inclination and phase angle for selection of dual AGN through double-peaked emission line profiles was also investigated by @Wang:2012 in which they found that, at a phase angle of $\phi\approx 50^{\circ}$, we miss at least 50% of all AGN with double emission components. We note that this 50% correction is a lower limit since it does not account for instrumental resolution, and that applying additional corrections based on our completeness estimates would likely make the correction estimated in this manner similar to that of @Shen:2011b. Finally, if we take the dual AGN fraction at $z=1.2$ to be $\sim0.05\%$ [@Yu:2011], then we estimate the fraction of dual AGN out of double-peaked AGN at $0.8<z<1.6$ to be $2\%$. This fraction is several times smaller than the results of @Fu:2012 (4.5-12%) and @Shen:2011b ($\sim10\%$) which were obtained from follow-up observations of double-peaked [\[\]$\lambda5007$]{} AGN. However, the difference can be attributed, at least in part, to the small expected number of dual AGN at $z=1.2$ estimated by @Yu:2011 which is due to the redshift evolution of galaxy morphology in their analysis which yields more late-type galaxies with smaller initial SMBH masses at higher redshift. A direct test of this through follow-up observations is therefore crucial in understanding the frequency of galaxy mergers at redshifts $z>0.8$ and their role in AGN triggering. For example, NIR spectroscopy will be capable of accessing the redshifted [\[\]$\lambda5007$]{} emission line for sources in our sample, allowing for a direct comparison with the $z<0.8$ samples of double-peaked AGN. This was done with the $z=1.175$ dual AGN candidate CXOXBJ142607.6+353351 in @Barrows:2012 which was initially selected through double-peaked [\[\]]{}/[\[\]]{} but for which follow-up NIR spectroscopy provided access to [\[\]$\lambda5007$]{}. The additional spatial information of [\[\]$\lambda5007$]{} provided by 2D longslit spectroscopy would enable one to determine if any of these sources are strong dual AGN candidates. Follow-up high-resolution imaging, such as radio observations, would be capable of resolving the two AGN cores, if present. Conclusions =========== We have compiled a sample of 131 quasars at $z=0.8-1.6$ which show double emission line components in either of the high-ionization narrow lines [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{}. The purpose of this search was to identify high-redshift analogs of the double-peaked [\[\]$\lambda5007$]{} sources found in several previous studies. Those double-peaked [\[\]$\lambda5007$]{} sources are believed to represent complex gas kinematics, large-scale outflows, or in a few cases dual AGN. Given the increased frequency of galaxy mergers at higher redshifts and their importance in models of galaxy evolution, we have investigated these phenomena at higher redshifts using our sample, with the following conclusions: - There is a clear bias towards selecting double-peaks with large velocity-splittings. This bias was made apparent by our comparison of the velocity-offsets of the blue and red components in our sample to those of [\[\]]{}-selected samples, and it is corroborated by the results of our completeness simulations. This selection bias is not surprising, and it is imposed by the relatively weaker intensities of [\[\]$\lambda3426$]{} and [\[\]$\lambda3869$]{} compared to [\[\]$\lambda5007$]{}. - We have found two results suggesting that both the blue and red systems are influenced by kinematics in the NLR. First, the line-widths of both the blue and red components are strongly correlated with the line-splittings, suggesting a common origin. Second, we find that the individual offsets for both the ‘blue’ and ‘red’ systems are positively correlated with the quasar Eddington ratio, suggesting that the SMBH accretion rate and therefore the radiation pressure is responsible for driving the line-offsets in the blueward direction for both line components. - We find evidence suggesting that the observed kinematics are strongest in the blue systems. This is suggested because the blue systems’ have larger velocity shifts from the quasar redshift, those velocity offsets show the strongest correlations with the Eddington ratio, and the blue systems show the highest degree of ionization stratification. This further suggests that the red outflowing components are generally more obscured. - We have found that a significant fraction ($\sim23\%$) are radio loud, compared to the $10\%$ radio loud fraction of the parent sample. Taken together, the previous conclusions paint a picture in which the blue systems originate in a portion of the NLR much closer to the AGN where they are accelerated by the accretion disk radiation pressure or radio jets to high velocities. This explains the large blueshifts, the stronger correlation with Eddington ratio and the pronounced ionization stratification. The red system originates further from the AGN where it is not accelerated to the high velocities of the blue system but is nevertheless close enough so it’s bulk velocity offset is also influenced by the AGN radiation pressure. This explains the smaller velocity offsets, the much weaker correlation with Eddington ratio, and the relatively less pronounced ionization stratification compared to the blue systems. A generalized schematic of this scenario in Figure \[fig:cartoon\]. The enhanced radio loud fraction relative to the parent sample also suggests that radio jets may be another mechanism which is capable of accelerating the NLR clouds to produce the line offsets. This sample can be used to study outflows from luminous AGN at relatively high redshifts when AGN feedback may have been an important factor in the growth of massive galaxies. There are several interesting results which leave open the possibility for dual AGN in the sample. In particular, several of our correlations can be thought of as consistent with the dual AGN scenario, and even suggest that the sample is likely to include dual AGN:\ - The correlation between velocity-splitting and Eddington ratio, while consistent with the picture of radiatively-driven outflows, could plausibly be consistent with orbiting SMBH pairs in which enhanced accretion is more likely to occur at smaller separations where the SMBH orbital velocities will be largest. - We have found that a subset of our sample ($40\%$) are consistent with no measurable ionization stratification between [\[\]$\lambda3426$]{}  and [\[\]$\lambda3869$]{}, similar to other dual AGN and strong candidate dual AGN. - We have found that our sample shows a correlation between the velocity-offset ratio and the luminosity ratio of the blue and red components. This correlation is broadly consistent with the theoretical expectation for a binary, Keplerian orbit, though our sample seems to be systematically offset above the relation. We have shown how this deviation could be produced in a sample which includes a combination of AGN outflows and dual AGN.\ - We have estimated the fraction of dual AGN out of double-peaked AGN that we expect at $0.8<z<1.6$, finding a fraction (2%) which is smaller than that estimated at lower redshifts. However, we caution that a significant - and perhaps the primary - reason for this lower fraction is the small estimated number of high redshift dual AGN that we adopt. Follow-up NIR observations to access the [\[\]$\lambda5007$]{}  line in our sources would allow for a direct comparison with the [\[\]$\lambda5007$]{}  velocity and spatial profiles of the $z<0.80$ samples, allowing for a more robust assessment of the origin of the double-peaks in the high-ionization narrow emission lines. Therefore, this sample represents an initial step toward extending the study of double-peaked emission line AGN to higher redshifts. We would like to thank an anonymous referee for very helpful comments that improved the quality of the paper. We would also like to acknowledge Daniel Stern for assistance on the analysis of our data and Laura Blecha for highly useful suggestions regarding the interpretation of our results. Finally, we acknowledge constructive discussion from members of the Arkansas Galaxy Evolution Survey (AGES) and the Arkansas Center for Space and Planetary Sciences, including Douglas Shields, Benjamin Davis, Adam Hughes and Jennifer Hanley. This research has made use of NASA’s Astrophysics Data System and the Sloan Digital Sky Survey. Support for this work was provided in part by the Arkansas NASA EPSCoR program (grant number NNX08AW03A).  \ [93]{} natexlab\#1[\#1]{} , K. N., [et al.]{} 2009, ApJ, 182, 543 , R. 1993, , 31, 473 , S., [Mediavilla]{}, E., & [Garcia-Lorenzo]{}, B. 1996, , 463, 509 , R. S., [et al.]{} 2012, , 744, 7 , N., [Falcke]{}, H., [Schulz]{}, H., [Wilson]{}, A. S., & [Wills]{}, B. J. 2002, , 574, L105 , J. C., [Bullock]{}, J. S., [Barton]{}, E. J., [Guenther]{}, H. D., [Zentner]{}, A. R., & [Wechsler]{}, R. H. 2006, , 652, 56 , J. C., & [Cooke]{}, J. 2012, ArXiv e-prints , S., [Chiaberge]{}, M., [Piconcelli]{}, E., [Guainazzi]{}, M., & [Matt]{}, G. 2008, , 386, 105 , L., [Loeb]{}, A., & [Narayan]{}, R. 2012, ArXiv e-prints , M., [et al.]{} 2011, , 726, 57 , F., [et al.]{} 2010, , 717, 209 —. 2012, , 752, 49 , J. M., [Gerke]{}, B. F., [Stern]{}, D., [Cooper]{}, M. C., [Weiner]{}, B. J., [Newman]{}, J. A., [Madsen]{}, K., & [Barrows]{}, R. S. 2012, , 753, 42 , J. M., [Griffith]{}, R. L., [Gerke]{}, B. F., [Cooper]{}, M. C., [Newman]{}, J. A., [Davis]{}, M., & [Stern]{}, D. 2009, , 702, L82 , J. M., [Pooley]{}, D., [Gerke]{}, B. F., & [Madejski]{}, G. M. 2011, , 737, L19+ , J. M., [et al.]{} 2009, , 698, 956 , C. J., [Bershady]{}, M. A., [Dickinson]{}, M., & [Papovich]{}, C. 2003, , 126, 1183 , D. M., [Kraemer]{}, S. B., [Schmitt]{}, H. R., [Jaff[é]{}]{}, Y. L., [Deo]{}, R. P., [Collins]{}, N. R., & [Fischer]{}, T. C. 2010, , 139, 871 , D. M., [Schmitt]{}, H. R., [Kraemer]{}, S. B., [Mushotzky]{}, R. F., & [Dunn]{}, J. P. 2010, , 708, 419 , M. M., & [Osterbrock]{}, D. E. 1984, , 286, 171 , T., [Springel]{}, V., & [Hernquist]{}, L. 2005, , 433, 604 , M., & [Halpern]{}, J. P. 2003, ApJ, 599, 886 , J. E. 2007, , 311, 269 , J. E., & [Murray]{}, N. 2007, , 656, 93 , G. J., & [Osterbrock]{}, D. E. 1986, , 300, 658 , L., & [Merritt]{}, D. 2000, , 539, L9 , T. C., [Crenshaw]{}, D. M., [Kraemer]{}, S. B., [Schmitt]{}, H. R., [Mushotsky]{}, R. F., & [Dunn]{}, J. P. 2011, , 727, 71 , H., [Myers]{}, A. D., [Djorgovski]{}, S. G., & [Yan]{}, L. 2011, , 733, 103 , H., [Yan]{}, L., [Myers]{}, A. D., [Stockton]{}, A., [Djorgovski]{}, S. G., [Aldering]{}, G., & [Rich]{}, J. A. 2012, , 745, 67 , H., [et al.]{} 2011, , 740, L44 , J.-Q., [Hu]{}, C., [Wang]{}, J.-M., [Bai]{}, J.-M., & [Zhang]{}, S. 2012, , 201, 31 , B. F., [et al.]{} 2007, , 660, L23 , P. J., [Myers]{}, A. D., [Barkhouse]{}, W. A., [Mulchaey]{}, J. S., [Bennert]{}, V. N., [Cox]{}, T. J., & [Aldcroft]{}, T. L. 2010, , 710, 1578 , M., [Piconcelli]{}, E., [Jim[é]{}nez-Bail[ó]{}n]{}, E., & [Matt]{}, G. 2005, , 429, L9 , T. M., [Miley]{}, G. K., [van Breugel]{}, W. J. M., & [Butcher]{}, H. R. 1981, , 247, 403 , P. C., & [Wild]{}, V. 2010, , 405, 2302 , L. C. 2002, , 564, 120 , J., [Tadhunter]{}, C. N., & [Morganti]{}, R. 2003, , 342, 227 —. 2008, , 387, 639 , P. F., [Hernquist]{}, L., [Cox]{}, T. J., [Di Matteo]{}, T., [Martini]{}, P., [Robertson]{}, B., & [Springel]{}, V. 2005, ApJ, 630, 705 , D. S., [Reiprich]{}, T. H., [Clarke]{}, T. E., & [Sarazin]{}, C. L. 2006, , 453, 433 , G., & [Haehnelt]{}, M. 2000, , 311, 576 , D. D., [et al.]{} 2012, , 744, 148 , E., [et al.]{} 2011, , 192, 18 , S., [Burwitz]{}, V., [Hasinger]{}, G., [Predehl]{}, P., [Kaastra]{}, J. S., & [Ikebe]{}, Y. 2003, , 582, L15 , S., [Xu]{}, D., [Zhou]{}, H., [Storchi-Bergmann]{}, T., & [Binette]{}, L. 2008, , 680, 926 , M., [et al.]{} 2011, , 735, L42+ , G. 1994, Astronomical Data Analysis Software and Systems, 3, 437 , J. H. 1999, [Active galactic nuclei : from the central black hole to the galactic environment]{}, ed. [Krolik, J. H.]{} , X., [Civano]{}, F., [Shen]{}, Y., [Green]{}, P. J., [Greene]{}, J. E., & [Strauss]{}, M. A. 2012, ArXiv e-prints , X., [Greene]{}, J. E., [Shen]{}, Y., & [Strauss]{}, M. A. 2010, , 715, L30 , X., [Shen]{}, Y., & [Strauss]{}, M. A. 2012, , 745, 94 , X., [Shen]{}, Y., [Strauss]{}, M. A., & [Greene]{}, J. E. 2010, , 708, 427 , X., [Shen]{}, Y., [Strauss]{}, M. A., & [Hao]{}, L. 2011, , 737, 101 , J. M., [Jonsson]{}, P., [Cox]{}, T. J., [Croton]{}, D., [Primack]{}, J. R., [Somerville]{}, R. S., & [Stewart]{}, K. 2011, , 742, 103 , A., & [Hunt]{}, L. K. 2003, , 589, L21 , J. M., [et al.]{} 2012, ArXiv e-prints , R. C., [Max]{}, C. E., [Rosario]{}, D. J., [Shields]{}, G. A., [Smith]{}, K. L., & [Wright]{}, S. A. 2011, , 738, L2 , R. J., & [Jarvis]{}, M. J. 2002, , 337, 109 , C. H., & [Whittle]{}, M. 1996, , 465, 96 , N. P. H., [Lehnert]{}, M. D., [De Breuck]{}, C., [Gilbert]{}, A. M., & [van Breugel]{}, W. 2008, , 491, 407 , J. A., [et al.]{} 2012, ArXiv e-prints , D. E., & [Ferland]{}, G. J. 2006, [Astrophysics of gaseous nebulae and active galactic nuclei]{}, ed. [Osterbrock, D. E. & Ferland, G. J.]{} , E., [et al.]{} 2010, ArXiv e-prints , D. J., [McGurk]{}, R. C., [Max]{}, C. E., [Shields]{}, G. A., [Smith]{}, K. L., & [Ammons]{}, S. M. 2011, , 739, 44 , D. J., [Shields]{}, G. A., [Taylor]{}, G. B., [Salviander]{}, S., & [Smith]{}, K. L. 2010, , 716, 131 , D. B., [Soifer]{}, B. T., [Elias]{}, J. H., [Madore]{}, B. F., [Matthews]{}, K., [Neugebauer]{}, G., & [Scoville]{}, N. Z. 1988, , 325, 74 , D. P., [et al.]{} 2010, , 139, 2360 , Y., [Liu]{}, X., [Greene]{}, J. E., & [Strauss]{}, M. A. 2011, , 735, 48 , Y., & [Loeb]{}, A. 2010, , 725, 249 , Y., [et al.]{} 2011, , 194, 45 , J., & [Rees]{}, M. J. 1998, , 331, L1 , K. L., [Shields]{}, G. A., [Bonning]{}, E. W., [McMullen]{}, C. C., [Rosario]{}, D. J., & [Salviander]{}, S. 2010, , 716, 866 , H. W. W., & [Holt]{}, J. 2009, , 702, L42 , C., [et al.]{} 2002, , 123, 485 , I. V., [et al.]{} 2003, AJ, 126, 1720 , E., [Lutz]{}, D., [Verma]{}, A., [Netzer]{}, H., [Sternberg]{}, A., [Moorwood]{}, A. F. M., [Oliva]{}, E., & [Genzel]{}, R. 2002, , 393, 821 , C. N., [Fosbury]{}, R. A. E., [di Serego Alighieri]{}, S., [Bland]{}, J., [Danziger]{}, I. J., [Goss]{}, W. M., [McAdam]{}, W. B., & [Snijders]{}, M. A. J. 1988, , 235, 403 , E., [Natarajan]{}, P., [Sanders]{}, D. B., [Urry]{}, C. M., [Schawinski]{}, K., & [Kartaltepe]{}, J. 2010, Science, 328, 600 , E., [Schawinski]{}, K., [Urry]{}, C. M., & [Simmons]{}, B. D. 2012, , 758, L39 , Y., [Kawara]{}, K., [Yoshii]{}, Y., [Oyabu]{}, S., [Tanab[é]{}]{}, T., & [Matsuoka]{}, Y. 2006, , 650, 57 , S., [Volonteri]{}, M., [Mayer]{}, L., [Dotti]{}, M., [Bellovary]{}, J., & [Callegari]{}, S. 2012, , 748, L7 , D. E., [et al.]{} 2001, , 122, 549 , S. 1991, , 369, 331 , S., [Shopbell]{}, P. L., & [Miller]{}, S. T. 2001, , 121, 198 , M.-P., [Joly]{}, M., & [V[é]{}ron]{}, P. 2004, , 417, 515 , J., [Chen]{}, Y., [Hu]{}, C., [Mao]{}, W., [Zhang]{}, S., & [Bian]{}, W. 2009, , 705, L76 , X.-W., & [Zhou]{}, H.-Y. 2012, , 757, 124 , M. 1992, , 387, 109 , D., & [Komossa]{}, S. 2009, ApJ, 705, L20 , Q., [Lu]{}, Y., [Mohayaee]{}, R., & [Colin]{}, J. 2011, , 738, 92 , R., [Marziani]{}, P., [Sulentic]{}, J. W., [Calvani]{}, M., [Dultzin-Hacyan]{}, D., & [Bachev]{}, R. 2002, , 576, L9 , W., [Sulentic]{}, J. W., & [Binette]{}, L. 1990, ApJ, 365, 115

--- author: - 'A.A. Baykalov' title: | [MSC 20G40, 20D60, 20D06]{} Intersections of conjugate solvable subgroups in classical groups of Lie type\ --- Introduction ============ Assume that a finite group $G$ acts on a set $\Omega.$ An element $x \in \Omega$ is called a $G$[**-regular point**]{} if $|xG|=|G|$, i.e. if the stabilizer of $x$ is trivial. Define the action of the group $G$ on $\Omega^k$ by $$g: (i_1, \ldots,i_k)\mapsto(i_1g,\ldots,i_kg).$$ If $G$ acts faithfully and transitively on $\Omega$ then the minimal number $k$ such that the set $\Omega^k$ contains a $G$-regular point is called the [**base size**]{} of $G$ and is denoted by $b(G).$ For a positive integer $m$ denote the number of $G$-regular orbits on $\Omega^m$ by $Reg(G,m)$ (this number equals 0 if $m < b(G)$). If $H$ is a subgroup of $G$ and $G$ acts by the right multiplication on the set $\Omega$ of right cosets of $H$ then $G/H_G$ acts faithfully and transitively on the set $\Omega.$ (Here $H_G=\cap_{g \in G} H^g.$) In this case, we denote $b(G/H_G)$ and $Reg(G/H_G,m)$ by $b_H(G)$ and $Reg_H(G,m)$ respectively. In this work we consider Problem 17.41 from ”Kourovka notebook”[@kt]: Let $H$ be a solvable subgroup of finite group $G$ and $G$ does not contain nontrivial normal solvable subgroups. Are there always exist five subgroups conjugated with $H$ such that their intersection is trivial? Let $A$ and $B$ be subgroups of $G$ such that $B \trianglelefteq A.$ Then $N_G(A/B) := N_G(A) \cap N_G(B)$ is the [**normalizer**]{} of $A/B$ in $G$. If $x\in N_G(A/B)$, then $x$ induces an automorphism of $A/B$ by $Ba \mapsto Bx^{-1}ax.$ Thus there exists a homomorphism $N_G(A/B) \to Aut(A/B).$ The image of $N_G(A/B)$ under this homomorphism is denoted by $Aut_G(A/B)$ and is called a [**group of $G$-induced automorphisms of $A/B$**]{}. The problem is reduced to the case when $G$ is almost simple in [@vd]: \[sved\] Let $G$ be a group and let $$\{e\}=G_0<G_1< G_2< \ldots < G_n=G \mbox{ ---}$$ be a composition series of $G$ that is a refinement of a chief series. Assume that for some $k$ the following condition holds: If $G_i/G_{i-1}$is nonabelian, then for every solvable subgroup $T$ of $Aut_G(G_i/G_{i-1})$ we have $$b_T(Aut_G(G_i/G_{i-1})) \le k \mbox{ and } Reg_T(Aut_G(G_i/G_{i-1}),k)\ge5.$$ Then, for every maximal solvable subgroup $H$ of $G$ we have $b_H(G)\le k.$ In [@bay] the author prove the inequality $Reg_H(G,5)\ge 5$ is proved for almost simple groups with socle $G_0$ isomorphic to alternating group $A_n,$ $n\ge 5.$ Fix a prime number $p$ and $q=p^f$, where $f$ is a positive integer. Denote a finite field with $q$ elements by $F_q$ and its multiplicative group by $F^{\times}_q.$ Denote by $GL_n^{\epsilon}(q)$ groups $GL_n(q)$ and $GU_n(q)\le GL_n(q^2)$ for $\epsilon$ equal to $+$ and $-$ respectively. Groups $SL_n^{\epsilon}(q), PGL_n^{\epsilon}(q)$ and $PSL_n^{\epsilon}(q)$ are defined in the same way. For the classical almost simple groups of Lie type T. Burness has proved the following [*[@burness Theorem 1.1]*]{}\[bernclass\] Let $G$ be a finite almost simple classical group in a faithful primitive non- standard action. Then either $b(G) \le 4$, or $G = U_6 (2) \cdot 2$, $H = U_4 (3) \cdot 2^2$ and $b(G) = 5$. The definition of non-standard action is given in [@burness Definition 1.1]. If the simple socle of an almost simple group $G$ is isomorphic to $PSL_n^{\epsilon}(q) $, then an action is non-standard if the point stabilizer in $PSL_n^{\epsilon}(q) $ is not parabolic, i.e. does not lie in Aschbacher’s class $\mathscr{C}_1$. If a solvable subgroup of $G$ lies in a maximal subgroup $M$ and the action of $G$ on $G/M$ is non-standard, then using Theorem \[bernclass\] we obtain that $Reg_H(G,5)\ge5.$ Therefore, it is necessary to study the case when solvable subgroup lies in a maximal subgroup $M$ such that the action of $G$ on $G/M$ is standard. In the present paper we consider the case when a solvable subgroup $H$ lies in a maximal subgroup of $GL_n^{\epsilon}(q)$ from the Aschbacher’s class $\mathscr{C}_1$ and either $H$ is completely reducible or $H$ contains a Sylow $p$-subgroup, where $p$ is the characteristic of the field $F_q$. In the first case we consider groups ${\mathrm{Sin}}_n^{\epsilon}(q)$ which are cyclic subgroups of $GL_n^{\epsilon}(q)$ of order $q^n-(\epsilon 1)^n$. In case $\epsilon =+$ the subgroup ${\mathrm{Sin}}_n(q)={\mathrm{Sin}}_n^{+}(q)$ is a well known Singer cycle, so we will call the ${\mathrm{Sin}}_n^{\epsilon}(q)$ Singer cycle in general case. The subgroup ${\mathrm{Sin}}_n^{\epsilon}(q)$ is a maximal torus of $GL_n^{\epsilon}(q)$ (see [@seminar G] for the definition) and $N_{GL_n^{\epsilon}(q)}({\mathrm{Sin}}_n^{\epsilon}(q))/{\mathrm{Sin}}_n^{\epsilon}(q)$ is a cyclic group of order $n$ [@singnorm Lemma 2.7]. Also the well known fact is that ${\mathrm{Sin}}_n^{+}(q)\cup\{0\}$ is isomorphic to a finite field $F_{q^n}$, so the set ${\mathrm{Sin}}_n^{+}(q)\cup{0}$ is closed under addition and multiplication. If $n$ is odd then ${\mathrm{Sin}}_n^{-}(q)$ is a subgroup of ${\mathrm{Sin}}_n^{+}(q^2)$ and if $n$ is even then ${\mathrm{Sin}}_n^{-}(q)$ is a subdirect product of ${\mathrm{Sin}}_{n/2}^{+}(q^2) \times {\mathrm{Sin}}_{n/2}^{+}(q^2)$ and lies in an algebra which is isomorphic to $F_{q^n} \times F_{q^n}$. Moreover, if the Hermitian form corresponding to the unitary group has the form $\left( \begin{smallmatrix} 0 & E\\ E & 0\\ \end{smallmatrix} \right)$ then ${\mathrm{Sin}}_n^{-}(q)$ is conjugate to the group consisting of matrices of the form $\left( \begin{smallmatrix} A & 0\\ 0 & \overline{A^{-1}}^T\\ \end{smallmatrix} \right)$, where $A \in {\mathrm{Sin}}_{n/2}^{+}(q^2)$, $A=(a_{ij})$, $\overline{A}= (\overline{a_{ij}})$ and $\overline{\phantom{A}}$ is the field automorphism of $F_{q^2}$ of order 2. The normalizer $N_{GL_n^{\epsilon}(q)}({\mathrm{Sin}}_n^{\epsilon}(q))$ is equal to ${\mathrm{Sin}}_n^{\epsilon}(q) \rtimes \langle \varphi \rangle$, where $\varphi :g \mapsto g^q$ if $\epsilon=+$, $\varphi :g \mapsto g^{-q}$ if $\epsilon=-$ and $n$ is even; $\varphi :g \mapsto g^{q^2}$ if $\epsilon=-$ and $n$ is odd. The main result in this case is the following theorem: \[TH1\] Let $p$ be prime and $q=p^t$. Let $G$ be isomorphic to $GL_n^{\epsilon}(q)$ and $H$ be a subgroup of $G$ such that $H$ is block diagonal with blocks isomorphic to $Sin_i^{\epsilon}(q) \rtimes \langle \varphi_{n_i} \rangle$, where $ i=1, \ldots, k $ and $ \sum_{i=1}^k n_i=n.$ Then $b_H(G)\le 4$. In the second case either $H$ is contained in Borel subgroup and by [@vd2 Lemma 8] we have $b_H(G)\le4$, or the following case is realized: \[TH2\] Let $G = GL_n(q) \rtimes \langle \tau \rangle $, where $q=2$ or $q=3$, $n$ is even, $\tau$ is an automorphism which acts by $\tau: A \mapsto (A^{-1})^{T}$ for $A \in GL_n(q)$. Let $H$ be the normalizer in $G$ of $P \le GL_n(q)$, where $P$ is the stabilizer of the chain of subspaces: $$\langle v_n, v_{n-1}\rangle < \langle v_n, v_{n-1}, v_{n-2}, v_{n-3} \rangle < \ldots <\langle v_n, v_{n-1}, \ldots , v_2, v_1 \rangle .$$ Then $Reg_H(G,5)\ge5.$ In the forthcoming papers we plan to use considered cases as a base to prove the statement in general case. Preliminary results =================== If a group $G$ acts on a set $\Omega$ then define $C_{\Omega}(x)$ to be the set of points in $\Omega$ fixed by an element $x \in G$. If $G$ and $\Omega$ are finite then we define the [**fixed point ratio**]{} of $x$ ( denote it by ${{\mathrm{fpr}}}(x)$), to be the proportion of points in $\Omega$ fixed by $x$, i.e. ${{\mathrm{fpr}}}(x) = |C_{\Omega}(x)|/|\Omega|.$ The following result is known and we give its proof here for completeness. If $G$ acts on a set $\Omega$ transitively and $H$ is the stabilizer of a point then $${{\mathrm{fpr}}}(x)= \frac{|x^{G} \cap H|}{|x^G|}$$ for all $x \in G.$ Let $\{1=g_1, g_2, \ldots , g_k\}$ be a right transversal of $G$ by $H$. The action is transitive, so $\{H, H^{g_2}, \ldots , H^{g_k}\}$ contains stabilizers of all points. Then $$|C_{\Omega}(x)|=|\{i \mid x \in H^{g_i}\}|= \frac{|\{g \mid x^{g^{-1}} \in H\}|}{|H|}= \frac{|x^G \cap H||C_{G}(x)|}{|H|}.$$ On the other hand, $|\Omega|=|G:H|=\frac{|x^G||C_{G}(x)|}{|H|}.$ Thus $${{\mathrm{fpr}}}(x)=\frac{|C_{\Omega}(x)|}{|\Omega|}=\frac{|x^G \cap H|}{|x^G|},$$ and the lemma follows. The papers [@fpr; @fpr2; @fpr3; @fpr4] are devoted to the study of the fixed point ratio in classical groups, and we will use the notation from these papers. Recall some elementary observations from [@fpr]. Let a group $G$ acts on the set $\Omega$ of right cosets of a subgroup $H$ of $G.$ Let $Q(G, c)$ be the probability that a randomly chosen $c$-tuple of points in $\Omega$ is not a base for $G$, so $G$ admits a base of size $c$ if and only if $Q(G, c) < 1$. Of course, a $c$-tuple in $\Omega$ fails to be a base if and only if it is fixed by an element $x \in G\backslash H_G$(here $H_G=\cap_{g\in G}H^g$) of prime order, and we note that the probability that a random $c$-tuple is fixed by $x$ is at most ${{\mathrm{fpr}}}(x)^c$. Let $\mathscr{P}$ be the set of elements of prime order in $G\backslash H_G$, and let $x_1, \ldots,x_k$ be a set of representatives for the $G$-classes of elements in $\mathscr{P}$. Since $G$ is transitive, fixed point ratios are constant on conjugacy classes and it follows that $$\label{ver} Q(G,c) \le \sum_{x \in \mathscr{P}}{{\mathrm{fpr}}}(x)^c = \sum_{i=1}^{k}|x_i^G|\cdot|{{\mathrm{fpr}}}(x_i)^c|=:\widehat{Q}(G,c).$$ Suppose that there exists $\xi$ such that for any prime order element $x \in \mathscr{P}$ we have ${{\mathrm{fpr}}}(x)\le|x^G|^{-\xi}.$ Then we obtain $\widehat{Q}(G,c)\le \sum_{i=1}^{k}|x_i^{G}|^{1-c\xi}.$ Let $\mathscr{C}$ be the set of conjugacy classes of prime order elements in $G\backslash H_G$. For $t \in \mathbb{R}$ define $\eta_G(t)$ by $$\eta_G(t)= \sum_{C \in \mathscr{C}}|C|^{-t}$$ and define $T_G \in (0, 1)$ such that $\eta_G(T_G) = 1.$ If $G$ acts on $\Omega$ transitively , ${{\mathrm{fpr}}}(x)\le|x^G|^{-\xi}$ for all $x \in \mathscr{P}$ and $T_G < c\xi - 1$ then $b(G) \le c$. We follow the proof of [@burness Proposition 2.1]. Let $x_1,...,x_k$ be representatives for the $G$-classes of prime order elements in $G$, and let $Q(G, c)$ be the probability that a randomly chosen $c$-tuple of points in $\Omega$ is not a base for $G$. Evidently, $G$ admits a base of size $c$ if and only if $Q(G, c) < 1.$ By we obtain that $$Q(G,c) \le \sum_{i=1}^{k}|x_i^G|\cdot|{{\mathrm{fpr}}}(x_i)^c| = \eta_G(c\xi - 1)$$ and the result follows since $\eta_G(t) < 1$ for all $t>T_G.$ It is proven in [@burness Proposition 2.2] that if $G$ is an almost simple group of Lie type and $n \ge 6$ then $T_G<1/3$. It is easy to see that $T_G<1/3$ holds for $G=GL^{\pm}_n(q)$ too. Indeed, if $\overline{G}=PGL^{\pm}_n(q)$ and an element $x \in {G}$ is a prime order element then $|C_G(x)|\le |C_{\overline{G}}(\overline{x})||Z(G)|$, so $|x^G| \ge |\overline{x}^{\overline{G}}|$ and $\eta_{\overline{G}}(t)\ge \eta_G(t)$ for $t>0$. Since $\eta_G(t)$ is a monotonically decreasing function we have $T_{\overline{G}} \ge T_G.$ Thus if $$\label{0} {{\mathrm{fpr}}}(x)<|x^G|^{-\frac{4}{3c}}$$ then $\xi \ge \frac{4}{3c}$ and $c\xi -1 \ge 1/3 >T_G$ and there is a base of size $c$. First consider the case when $G$ is isomorphic to $GL_n(q)$ and the point stabilizer $H$ is conjugate to $ Sin_n(q) \rtimes \langle \varphi \rangle \simeq F_{q^n}^{\times} \rtimes \mathbb{Z}_n $, where $Sin_n(q)$ is a Singer cycle of $GL_n(q)$ and $\varphi$ is an element from $GL_n(q)$ inducing a field automorphism of $Sin_n(q)\simeq F_{q^n}^{\times}$ such that $\varphi:g \mapsto g^q$ for $g \in Sin_n(q).$ Since $Sin_n(q)\rtimes \langle \varphi \rangle$ is isomorphic to $F_{q^n}^{\times} \rtimes \mathbb{Z}_n$, it is convenient to express an element of $H$ in the form $(\lambda, j),$ where $\lambda \in F_{q^n}^{\times}$ and $j \in \mathbb{Z}_n$. Now we discuss how an element $h=(\lambda,j)\in H$ acts on $V=F_{q}^{n}.$ Let $\theta $ be a generating element of $F_{q^n}^{\times}$ then a vector space $F_q^n$ can be identified with elements of a field $F_{q^n}$. Furthermore, a nonzero element of $F_{q^n}$ can be written in the form $\theta^s$ for suitable $s \in \mathbb{Z}_{q^n-1}$. Using this identification it is possible to define action of $(\lambda, j)$ on element $v \in F_{q^n}$ by $v \cdot (\lambda, j) = (v\lambda )^{q^j}$. It is clear that if $y =(\lambda , j) \in F_{q^n}^{\times} \rtimes \mathbb{Z}_n$, then $y^r=(\lambda^{q^{(n-j)(r-1)} + q^{(n-j)(r-2)} + \ldots +q^{(n-j)}+1},rj)$, so if $|y|=r$ and $r$ is prime then $rj \equiv 0 \pmod{n}.$ Thus either $y \in F_{q^n}^{\times} $ or $r$ divides $n$. Note that for $n$ even ${\mathrm{Sin}}_n^-(q)$ is isomorphic to $F_{q^n}^{\times} \rtimes \mathbb{Z}_n$ with the action $v \cdot (\lambda, j) = (v\lambda )^{(-q)^j}$ for $v \in F_{q^n}$. So, we obtain $y^r=(\lambda^{(-q)^{(n-j)(r-1)} + (-q)^{(n-j)(r-2)} + \ldots +(-q)^{(n-j)}+1},rj)$. \[form\] Denote by $\theta$ a generating element of $F_{q^n}^{\times}$. Let $G=GL_n^{\epsilon}(q),$ $H={\mathrm{Sin}}_n^{\epsilon}(q) \simeq F_{q^n}^{\times} \rtimes \mathbb{Z}_n $. If $y =(\lambda , j) \in H$, $|y|=r$ is a prime number and $r$ divide $n$ then $y$ can be written as $(\theta^{j_1((\epsilon q)^{n/r}-1)} ,rj)$ for some $j_1.$ We start the proof with case $\epsilon=+$. Choose $t$ so that the identity $\lambda = \theta^t$ holds. Since $|y|=r$, it follows that $y^r = (1, 0)$ in the above notation. Then $$\label{pol} \lambda^{q^{r(n-j)(r-1)/r}+q^{r(n-j)(r-2)/r}+ \ldots + q^{r(n-j)/r}+1}=1$$ Since $r$ divides $n$, there is some $j_2 <n$ such that $j=nj_2/r$ and therefore $$r(n-j)=r(n -nj_2/r)=rn-nj_2=n(r-j_2).$$ Denote $(r-j_2)$ by $m.$ Thus $$q^{r(n-j)(r-1)/r}+q^{r(n-j)(r-2)/r}+ \ldots + q^{r(n-j)/r}+1=q^{nm(r-1)/r}+\ldots+q^{nm/r}+1$$ and because of we obtain $$1=\theta^{t\cdot(q^{nm(r-1)/r}+\ldots+q^{nm/r}+1)},$$ and thus $$t\cdot(q^{nm(r-1)/r}+\ldots+q^{nm/r}+1)\equiv 0 \pmod{q^n-1}.$$ Now it is sufficient to prove that $(q^{nm(r-1)/r}+\ldots+q^{nm/r}+1,q^{n/r}-1)=1$ to obtain that $t=j_1(q^{n/r}-1)$, since $q^{n/r}-1$ is a divisor of $q^n-1.$ So, consider the greatest common divisor $$(q^{nm(r-1)/r}+\ldots+q^{nm/r}+1,q^{n/r}-1)$$ and add the second number to the first, we have $$(q^{nm(r-1)/r}+\ldots+q^{nm/r}+1 +q^{n/r}-1,q^{n/r}-1)=(q^{n/r}(q^{\frac{nm(r-1)-n}{r}}+\ldots+q^{\frac{nm-n}{r}}+1),q^{n/r}-1)$$ $$=((q^{\frac{nm(r-1)-n}{r}}+\ldots+q^{\frac{nm-n}{r}}+1),q^{n/r}-1),$$ since $q^{n/r}$ and $q^{n/r}-1$ are co-prime. After $m$ iterations we have $$((q^{\frac{nm(r-2)}{r}}+\ldots+2),q^{n/r}-1),$$ after $mr$ iterations we obtain $$(r,q^{n/r}-1).$$ So the greatest common divisor can be equal to either $1$ or $r$. Notice that $q^{nm(r-1)/r}+\ldots+q^{nm/r}+1\equiv 1 \pmod{r}$, therefore we obtain the statement of the lemma. The statement of the lemma is true for $\epsilon =-$ since if $n$ is odd then $H$ is a subgroup of ${\mathrm{Sin}}_n^{+}(q^2) \rtimes \langle \varphi \rangle$ (here $\varphi : g \mapsto g^{q^2}$), and if $n$ is even then it is sufficient to replace $q$ by $(-q)$ in the proof above. \[nep\] Let $G=GL_n^{\epsilon}(q),$ $H={\mathrm{Sin}}_n^{\epsilon}(q) \simeq F_{q^n}^{\times} \rtimes \mathbb{Z}_n $. An element of $H$ of prime order $r$ can have a subspace of fixed points of dimension either $0$ or $n/r.$ We start the proof with case $\epsilon=+$. Let $y \in H$ be an element of prime order $r$, where $r$ divide $n$ and $v\cdot y=v$ where $v = \theta^i \in F_{q^n}.$ As we explain above, $y$ can be written as $(\lambda, j)$, $\lambda \in F_{q^n}^{\times}, j \in \mathbb{Z}_n.$ By Lemma \[form\] we can write $y = (\theta^{j_1(q^{n/r}-1)},nj_2/r).$ So, equality $v \cdot y =v$ is equivalent to $$\begin{gathered} \label{sr} (i+ j_1(q^{n/r}-1))q^{nj_2/r}))\equiv i \pmod{q^n-1}\\ i(q^{nj_2/r}-1)\equiv -j_1(q^{n/r}-1) \pmod{q^n-1}.\notag\end{gathered}$$ Indeed, it is easy to check that $$(q^n -1)= (q^{n/r}-1)(q^{\frac{n(r-1)}{r}}+q^{\frac{n(r-2)}{r}} + \ldots + q^{\frac{n}{r}}+1)$$ $$(q^{nj_2/r} -1)= (q^{n/r}-1)(q^{\frac{n(j_2-1)}{r}}+q^{\frac{n(j_2-2)}{r}} + \ldots + q^{\frac{n}{r}}+1)$$ So, the congruence is equivalent to $$i\frac{(q^{nj_2/r}-1)}{(q^{n/r}-1)}\equiv -j_1 \pmod{\frac{q^n-1}{(q^{n/r}-1)}}$$ where $0\le i< q^n-1$. Now show that $\left(\frac{(q^{nj_2/r}-1)}{(q^{n/r}-1)},\frac{q^n-1}{(q^{n/r}-1)}\right)=1.$ If $j_2>r$ then $$\begin{gathered} \left(\frac{q^n-1}{(q^{n/r}-1)},\frac{(q^{nj_2/r}-1)}{(q^{n/r}-1)}\right)=((q^{\frac{n(r-1)}{r}}+q^{\frac{n(r-2)}{r}} + \ldots + q^{\frac{n}{r}}+1),(q^{\frac{n(j_2-1)}{r}}+q^{\frac{n(j_2-2)}{r}} + \ldots + q^{\frac{n}{r}}+1))=\\ \backslash \mbox{ and we can take the first argument from the second}\backslash \\ =((q^{\frac{n(r-1)}{r}} + \ldots + q^{\frac{n}{r}}+1),(q^{\frac{n(j_2-1)}{r}}+ \ldots + q^{\frac{nr}{r}}))=\\ =((q^{\frac{n(r-1)}{r}} + \ldots + q^{\frac{n}{r}}+1),q^n(q^{\frac{n(j_2-r-1)}{r}}+ \ldots + 1)) = ((q^{\frac{n(r-1)}{r}} + \ldots + q^{\frac{n}{r}}+1),(q^{\frac{n(j_2-r-1)}{r}}+ \ldots + 1))= \ldots =\\ =((q^{\frac{n(r-1)}{r}} + \ldots + q^{\frac{n}{r}}+1),(q^{\frac{n(s-1)}{r}}+ \ldots + 1)) \end{gathered}$$ where $j_2=ar+s,$ $a,s \in \mathbb{N}, s<r$. Thus exponents of the first term in $((q^{\frac{n(r-1)}{r}} + \ldots + q^{\frac{n}{r}}+1)$ and the first term in $(q^{\frac{n(s-1)}{r}}+ \ldots + 1))$ are changing according to the Euclid’s algorithm for $r$ and $j_2$, so $$\left(\frac{(q^{nj_2/r-1})}{(q^{n/r}-1)},\frac{q^n-1}{(q^{n/r}-1)}\right)=\left((q^{\frac{n(d-1)}{r}} + \ldots +1),(q^{\frac{n(t-1)}{r}}+ \ldots + 1)\right)$$ for suitable $t$ and $d=(j_2,r)$. If $d \ne 1$ then it is obvious that the initial congruence have solutions if and only if $j_1(q^{n/r}-1) \equiv 0 \pmod{q^n-1}$, but in this case $y=1$. If $d=1$ then $$((1,(q^{\frac{n(t-1)}{r}}+ \ldots + 1))=1,$$ so $\frac{(q^{nj_2/r-1})}{(q^{n/r}-1)}$ modulo $\frac{q^n-1}{(q^{n/r}-1)}$ is invertible in $\mathbb{Z}_{\frac{q^n-1}{(q^{n/r}-1)}}$ and we have the unique solution modulo $\frac{q^n-1}{(q^{n/r}-1)}$. Therefore there are $\frac{q^n-1}{(q^{n/r}-1)}$ solutions for $0\le i< q^n-1$. Thus an element of $H$ of prime order $r$ can have a subspace of fixed points of dimension either $0$ or $n/r.$ The statement of the lemma is true for $\epsilon =-$ since if $n$ is odd then $H$ is a subgroup of ${\mathrm{Sin}}_n^{+}(q^2) \rtimes \langle \varphi \rangle$ (here $\varphi : g \mapsto g^{q^2}$), and if $n$ is even then it is sufficient to replace $q$ by $(-q)$ in the proof above. \[prop1\] If $y$ is an element of ${\mathrm{Sin}}_n^{\epsilon}(q)$ of prime order $r$ not dividing $n$ then $|y^G \cap H|\le n.$ Let $y$ be an element of $Sin_n^{\epsilon}(q)$ of prime order $r$ not dividing $n$ and $y^g$ lies in $y^G \cap H$ for some $g \in G.$ Then $y^g$ can be represented in the form $(\lambda_1, i)$ for suitable $\lambda_1 \in F_{q^n}^{\times}$ and $i \in \mathbb{Z}_n$, so $(y^g)^r=(\lambda_2, ri)$ and $ri \equiv 0 \pmod{n}$. Thus $i\equiv 0 \pmod{n}$ and $y^g \in Sin_n^{\epsilon}(q)$ since $(r,n)=1$. Now $Sin_n^{\epsilon}(q)$ is a maximal torus of $GL_n^{\epsilon}$, so by [@carter Corollary 3.7.2] if $y, y^g \in Sin_n(q)$ then there is $g_1 \in H$ such that $y^g=y^{g_1}.$ Let $g_1 =(\mu,s )$ then $y^{(\mu, s)}= y^{(1,s)}$. Thus $|y^G \cap H|\le |\varphi|=n.$ For an odd prime integer $r$ such that $(r,q)=1$ denote the multiplicative order of $q$ in $\mathbb{Z}_r$ by $e(r,q)$, i.e. $e(r,q)>0$ is the minimal number such that $q^{e(r,q)} \equiv 1 \pmod{r}$. (In particular $e(r,q)<r$.) \[ir\] If $x \in G$ has prime order $r$ and has no invariant proper subspaces then $e(r,q)=n$. First observe that the minimal polynomial $\mu(t)$ of $x$ is irreducible over $F_q$. Assume the contrary that $\mu(t)=f(t)g(t)$ where $\deg(g(t))\ge \deg(f(t))>0$. Then $$0< Im(g(x))\le Ker(f(x))<Ker(\mu(x)),$$ so $Ker(f(x))$ is a nontrivial $x$-invariant proper subspace that contrary with irreducibility of $x$. Since $e=e(r,q)$ is the multiplicative order of $q$ in $\mathbb{Z}_r$ we obtain that $x^{q^e}-x=0.$ For the spectrum of $x$ this means that $Spec(x) \subseteq F_{q^e}.$ Then for irreducible $\mu(t)$ it is true that $$\deg(\mu(t))\le |F_{q^e}:F|=e.$$ Since $\mu(t)$ is the minimal polynomial, for a nonzero vector $v$ we have $L(v, vx, vx^2, \ldots , vx^{\deg(\mu(t)) -1 })$ is an $x$-invariant subspace. Thus $n=e$ and $\{v, vx, vx^2, \ldots , vx^{\deg(\mu(t)) -1 }\}$ is a basis. Structure of element of prime order $r \ne p$ in general follows from Lemma \[ir\]. Indeed, it follows by the Maschke’s theorem that $x$ is completely reducible. Thus $x$ consist of several cells of size $e$ and a cell which is an identity matrix. $$\begin{gathered} \label{kletk} x = \begin{pmatrix} * & \ldots & * & & & & & & & \\ \vdots & e\times e & \vdots & & & & & & & \\ * & \ldots & * & & & & & & & \\ & & & \ddots & & & & & & \\ & & & & * & \ldots & * & & & \\ & & & & \vdots & e\times e & \vdots & & & \\ & & & & * & \ldots & * & & & \\ & & & & & & & 1 & & \\ & & & & & & & & \ddots & \\ & & & & & & & & & 1 \end{pmatrix}\end{gathered}$$ Note that $x$ is always diagonalizable for $r =2 \ne p$, so we have that $e=1$ in this case. Proof of Theorem \[TH1\] ======================== We use some calculations in GAP [@gap]. Programs and logs can be found by the link: [ goo.gl/gI6nna]{} We begin the proof of Theorem \[TH1\] by assuming that $G=GL_n(q)$. By the structure of element of prime order $r \ne p$ we have $$C_G(x) \le GL_k(q^e) \times GL_t(q),$$ where $n=ke+t.$ Thus $$\begin{gathered} |x^G|=\frac{|G|}{|C_G(x)|}=\frac{q^{\frac{n(n-1)}{2}}(q^n-1)(q^{n-1}-1)\ldots(q-1)}{ q^{\frac{ek(k-1)}{2}+\frac{t(t-1)}{2}}(q^{ek}-1)(q^{e(k-1)}-1) \ldots (q^e-1)(q^t-1)(q^{t-1}-1)\ldots(q-1)}=\\= q^{n(n-1)/2 -ek(k-1)/2-t(t-1)/2}\cdot \frac{\prod_{i\ne me, m\le k}^n(q^{i}-1)}{\prod_{i=1}^t(q^i-1)}\end{gathered}$$ $$\prod_{i\ne me, m\le k}^n(q^{i}-1) \ge \prod_{i\ne me, m\le k}^{n-1}q^{i}=q^{n(n-1)/2 - (e(k+1)-1)k/2}$$ $$\prod_{i=1}^t(q^i-1)\le \prod_{i=1}^t q^i=q^{t(t+1)/2}$$ and we obtain that $$\label{ozenk} \log_q(|x^G|)\ge n(n-1) - {ek^2} + k -t^2.$$ \[b6\] Let $G=GL_n(q)$, $H= Sin_n(q) \rtimes \langle \varphi \rangle$ and $n\ge6$. Then $b_H(G)\le 2.$ It is enough to verify inequality for an element $x \in H\backslash H_G$ of prime order and $c=2$ to obtain the statement. Let $n$ be equal to $ek+t$, where $t$ is exactly the dimension of $C_V(x)$, so by Lemma \[nep\] we have $t\le n/r\le n/2$, where $r=|x|$ . [**Case 1. ($|x|\ne p$)**]{} Assume first that $e\ge2$ and $k\le n/2$. The inequality takes the form $$\label{q2} |x^G\cap H|^3 <|x^G|$$ in our case. If $x \in Sin_n(q)$ and $(|x|,n)=1$ then by Proposition \[prop1\] it is enough to verify the inequality $$n^3 <|x^G|.$$ Indeed, $$n^3 <q^{\frac{n^2}{2} -n}\le (q^{n(n-1)-nn/2 }) \le (q^{n(n-1)-{ek^2}+k -t^2}) \le |x^G|$$ since $|x^G \cap H| \le n$ and $n=e\cdot k$, $e>1$. Obviously, this inequality holds for $q\ge 2$ and $n \ge 6.$ Now let $x \in H$ and $|x|$ divide $n$. Then $$|x^G|\ge q^{n(n-1)-{ek^2}+k -t^2}\ge q^{n(n-1) -{ek\cdot n/2} +k -(n/2)^2} \ge q^{n^2/4-n }.$$ Consider the left-hand side of inequality $$|x^G\cap H|^3\le |H|^3 = (q^n-1)^3n^3.$$ Thus holds if $$\label{eq1} (q^n-1)^3n^3 < q^{n^2/4-n }.$$ Direct calculation shows that inequality holds for $q\ge 2$ and $n>20.$ Now consider the cases $n=6, \ldots, 20$. It is routine to check that inequality $(q^n-1)^3n^3< (q^{n(n-1)-{ek^2}+k -t^2})$ holds for all possible decomposition $n=ke+t$(such that $e=e(r,q)$, $r$ divide $n$ and $t$ is equal to $0$ or $n/r$) in cases $n=7, \ldots, 20$. Consider more carefully the case $n=6.$ First, we find a stronger estimate for $|x^G \cap H|. $ Recall that we can represent $x$ as $(\lambda, j) \in F_{q^n}^{\times} \leftthreetimes \mathbb{Z}_n.$ If $r=2$ then $e=1$ and we consider this case beneath. Now let $r=3$ then $(\lambda,j)^3=(\lambda^{q^{2(n-j)} +q^{n-j}+1},3j)=(1,0)$, i.e. $j=4,2,0.$ If $j=4$ then $\lambda^{q^4+q^2+1}=1$. If $j=2$ then $\lambda^{q^8+q^4+1}=1$. Note that $q^8 +q^4 +1= (q^6-1)q^2 + (q^4+q^2+1)$. Therefore in both cases we have $\lambda^{q^4+q^2+1}=1$. Because of $(q^6-1)=(q^4+q^2+1)(q^2-1)$ we obtain that $\lambda = \theta^{m(q^2-1)}$, where $m=1, \ldots, (q^4 +q^2+1).$ If $j=0$ then $x$ is an element of ${\mathrm{Sin}}_6(q)$, and there is only two elements of order 3 in ${\mathrm{Sin}}_6(q)$. In total we get $2(q^4+q^2+2)$ elements of order 3, thus $|x^G \cap H|\le 2(q^4+q^2+2)$. Now we verify inequality for this case. For $r=3$ there are three cases $$1) e=2, k=2, t=2;$$ $$2) e=3, k=2, t=0;$$ $$3) e=2, k=3, t=0.$$ For the first two cases inequality $|x^G \cap H|^3< (q^{n(n-1)-{ek^2}+k -t^2})$ holds for all $q\ge 2$. For the rest, the inequality holds for $q>3$, for $q=2$ the statement of the lemma can be checked by GAP. Now let $e=1$, i.e. $x$ is conjugate with diagonal matrix. Denote the dimension of a maximal eigenspace of $x$ by $d$. Let $\alpha$ be the corresponding eigenvalue. Recall that $H={\mathrm{Sin}}_n(q) \rtimes \langle \varphi \langle,$ so we write $x$ as $x=a\varphi^j$, where $a \in {\mathrm{Sin}}_n(q)$, $j<n$. Consider the element $\alpha^{-1}x=\alpha^{-1}(a\varphi^j)=(\alpha^{-1}a)\varphi^j$, which lies in $ H$, because the set ${\mathrm{Sin}}_n(q) \cup \{0\}$ is an algebra over $F_q$ and closed under the multiplication by scalar. Obviously, the order of the element $\alpha^{-1}x$ is also equal to $r$, and dimension of subspace of fixed points also have to be $n/r$ by Lemma \[nep\]. Thus $d=n/r.$ Let $\overline{\phantom{G}}:GL_n(q) \mapsto PGL_n(q)$ be the canonical homomorphism. Consider $x$ as an element of the algebraic group $\hat{G}=PGL_n(\overline{F}_q)$ over the algebraic closure $\overline{F}_q$ of the field $F_q$. Let $s=s(x)$ be the co-dimension of maximal eigenspase of $x$. In our case we have $s=n-d\ge n/2.$ Let $n$ be not prime, i.e. $r\le n/2$. Then by [@fpr2 Proposition 3.38] we obtain that $$|x^G|\ge |\overline{x}^{\overline{G}}|> \frac{1}{2r}q^{ns} > \frac{1}{n}q^{n^2/2}.$$ The inequality $$|x^G \cap H|^3 \le |H|^3=(q^n-1)^3n^3<\frac{1}{n}q^{n^2/2}$$ holds for $n \ge 8 $ and $q\ge2$ if $r\ne p.$ Now let $r=p$ and $s=n-1$. Then by [@fpr2 Proposition 3.38] we obtain that $$|x^G|\ge |\overline{x}^{\overline{G}}|> \frac{1}{2r}q^{ns} > \frac{1}{2n}q^{n^2-n}.$$ The inequality $$(q^n-1)^3n^3<\frac{1}{2n}q^{n^2-n}$$ holds for $n \ge 7$ and $q \ge 2.$ Let $n=6$. Each element $x$ of order $2$ can be written as $(\lambda, j) \in F_{q^n}^{\times} \rtimes \mathbb{Z}_n$ then $(\lambda, j)^2=(\lambda^{q^{(n-j)}+1}, 2j)=1$. Then $j=3$ and $\lambda^{q^3+1}=1.$ Thus $\lambda= \theta^{m\cdot(q^3-1)}$ where $m=1, \ldots, q^3+1.$ Therefore there are $q^3$ involutions which are not scalar matrix in $H.$ It is easy to see that inequality $|x^G \cap H|^3\le (q^3-1)^3 <\frac{1}{4}q^{6^2/2} \le |x^G|$ holds for $q\ge 3.$ Let $r=3$ then $s=4$ and by [@fpr2 Proposition 3.38] we obtain $(q^n-1)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}| \le |x^G|$ for $q>3$, the case $q=2$ can be checked by GAP. [**Case 2. ($|x|= p$)**]{} Let $\overline{\phantom{G}}:GL_n(q) \mapsto PGL_n(q)$ be the canonical homomorphism. Consider $x$ as an element of the algebraic group $\hat{G}=PGL_n(\overline{F}_q)$ over the algebraic closure $\overline{F}_q$ of the field $F_q$. Let $s=s(x)$ be the codimention of maximal eigenspase of $x$. Thus $s=n-t$ because $x$ is unipotent. Therefore $s\ge n/2$ by Lemma \[nep\]. So, by [@fpr2 Proposition 3.38] $$|x^G|\ge |\overline{x}^{\overline{G}}| > \frac{1}{2p}q^{ns}\ge\frac{1}{2p}q^{n^2/2}\ge \frac{1}{2}q^{n^2/2-1},$$ and $(q^n-1)^3n^3<|x^G|$ holds for $q\ge2$ and $n\ge 9.$ For $n=8$ this inequality holds for $q\ge3$, the case $q=2$ can be checked by GAP. If $n=7$ then $(q^n-1)^3n^3 <\frac{1}{2}q^{n^2/2-1}$ holds for $q\ge 13$, so since $r=p=7$ we obtain $s=6$ and by [@fpr2 Proposition 3.38] we have $(q^n-1)^3n^3<\frac{1}{14}q^{7\cdot6}<|\overline{x}^{\overline{G}}| \le |x^G|$. This inequality holds for $n\ge7.$ If $n=6$ there are two possibilities: $r=3$ and $r=2.$ Let $r=3$ then $s=4$ and by [@fpr2 Proposition 3.38] we obtain $(q^n-1)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}| \le |x^G|$ for $q>3$, case $q=3$ can be checked by GAP. If $r=p=2$ then $s=3$ and $|x^G|>\frac{1}{4}q^{6\cdot3}.$ Consider $\langle \varphi^3 \rangle =S\in Syl_2(H).$ Well known that the number of Sylow subgoup $n_2(H)$ is equal to $|H:N_H(S)|.$ Let $a \in Sin_6(q)$ and $a^{-1}\varphi^3a=\varphi^u$ then $\varphi^{-3}a^{-1}\varphi^3a=\varphi^{-3}\varphi^u$ and $(a^{-1})^{q^3}a=\varphi^{-3}\varphi^u=1$ because $(a^{-1})^{q^3}a \in Sin_6(q).$ Thus $a = \theta^{m(q^3+1)}$ where $m= 1 \ldots (q^3-1)$. Therefore $|N_H(S)|=6\cdot(q^3-1)$, so $n_2(H)=(q^3+1)$. Obviously $|x^G \cap H|\le n_2(H)$ and we need to verify that $(q^3+1)^3<|x^G|$. This inequality $(q^3+1)^3<\frac{1}{4}q^{6\cdot3}$ holds for all $q\ge 2.$ Consider now the case when $G=GU_n(q)\le GL_n(q^2).$ We want to obtain the estimate on $|x^G|$ similar to previous case. Let $x$ be a prime order element of $G$. Then $C_G(x)\le GU_k(q^e) \times GU_t(q)$. $$\begin{gathered} |x^G|=\frac{G}{C_G(x)}=\\=\frac{q^{\frac{n(n-1)}{2}}(q^n-(-1)^n)(q^{n-(-1)^{n-1}}-1)\ldots(q+1)}{ q^{\frac{ek(k-1)}{2}+\frac{t(t-1)}{2}}(q^{ek}-(-1)^{k})(q^{e(k-1}-(-1)^{k-1}) \ldots (q^e+1)(q^t-(-1)^t)(q^{t-1}-(-1)^{t-1})\ldots(q+1)}=\\= q^{n(n-1)/2 -ek(k-1)/2-t(t-1)/2}\cdot \frac{\prod_{i\ne me, m\le k}^n(q^{i}-(-1)^i)}{\prod_{i=1}^t(q^i-(-1)^t)}= q^{n(n-1)/2 -ek(k-1)/2-t(t-1)/2}\cdot \frac{\Pi_1}{\Pi_2}\end{gathered}$$ $$\prod_{i=1}^n(q^i-(-1)^i)\ge q\cdot q \cdot q^3\cdot q^3 \ldots = \begin{cases} (\prod_{i=1, i \text{is odd}}^{n-2}q^{2i})\cdot q^n =q^{(n^2+1)/2}, & n \text{ is odd;} \\ \prod_{i=1, i \text{is odd}}^{n}q^{2i}=q^{n^2/2}, & n \text{ is even.} \end{cases} \Biggr\} \ge q^{n^2/2}$$ Now to get the estimate for $\Pi_1$ we have to remove from the product $q\cdot q \cdot q^3\cdot q^3 \ldots$ the multipliers which are corresponding to $i=me,$ $m \le k.$ If $e$ is even, we have $$\prod_{i= me, m\le k}q^{i-1}=q^{(e(k+1)-2)k/2}.$$ If $e$ is odd, we have $$\prod_{i= me, m\le k \text{is even}}q^{i-1} \cdot \prod_{i= me, m\le k \text{is odd}}q^{i}= \begin{cases} q^{(e(k+2)-2)k/4 + (ek)k/4}=q^{(e(k+1)-1)k/2} & k \text{ is even;} \\ q^{(e(k+1)-2)(k-1)/4 + e(k+1)(k+1)/4}=q^{e(k+1)k/2-(k-1)/2} & k \text{ odd.} \end{cases}$$ Therefore $\Pi_1 \ge q^{n^2/2 - (e(k+1)-1)k/2}$. For $\Pi_2$ we have $$\Pi_2 \le \prod_{i \text{is even}}q^i \cdot \prod_{i \text{is odd}}q^{i+1}= \begin{cases} q^{(t+2)t/4+(t+2)t/4}=q^{(t+2)t/2} & t \text{ is even;} \\ q^{(t+1)(t-1)/4 +(t+3)(t+1)/4}=q^{(t+1)^2/2} & t \text{ odd.} \end{cases}$$ So, after all, we obtain $log_q(|x^G|)\ge n(n-1)/2 -ek(k-1)/2 -t(t-1)/2 +n^2/2 -(e(k+1)-1)k/2-(t+1)^2=n(n-1) -ek^2+k-t^2 +(n-k -t -1)/2$. Let $G=GL_n^-(q)$, $H= {\mathrm{Sin}}_n^-(q) \rtimes \langle \varphi \rangle$ and $n\ge6$ then $b_H(G)\le 2.$ It is enough to verify for an element $x \in H\backslash H_G$ of prime order and $c=2$ to obtain the statement. Let $n$ be equal to $ek+t$, where $t$ is exactly the dimension of $C_V(x)$, so $t\le n/r\le n/2$ by Lemma \[nep\]. [**Case 1. ($|x|\ne p$)**]{} Assume first that $e\ge2$ and $k\le n/2$. The inequality takes the form $$\label{q21} |x^G\cap H|^3 <|x^G|$$ in our case. Denote $|x|$ by $r.$ If $x \in Sin_n(q)$ and $(|x|,n)=1$ then by Proposition \[prop1\] it is enough to verify the inequalities $$n^3 <q^{\frac{n^2}{2} -n}\le (q^{n(n-1)-nn/2 }) \le (q^{n(n-1)-{ek^2}+k -t^2+(n-k -t -1)/2}) \le |x^G|$$ since $|x^G \cap H| \le n$ and $n=e\cdot k$, $e>1$. Obviously, this inequality holds for $q\ge 2$ and $n \ge 6.$ Now let $x \in H$ and $|x|$ divide $n$. Then $$|x^G|\ge q^{n(n-1)-{ek^2}+k -t^2+(n-k -t -1)/2}\ge q^{n(n-1) -{ek\cdot n/2} +k -(n/2)^2+(n-k -t -1)/2} \ge q^{n^2/4-n }.$$ Consider the left-hand side of inequality $$|x^G\cap H|^3\le |H|^3 = (q^n-(-1)^n)^3n^3.$$ Thus the statement of the lemma is true if $$\label{eq11} (q^n-(-1)^n)^3n^3 < q^{n^2/4-n }.$$ Direct calculation shows that inequality holds for $q\ge 2$ and $n>20.$ Now consider cases $n=6, \ldots, 20$. It is routine to check that inequality $(q^n-(-1)^n)^3n^3< (q^{n(n-1)-{ek^2}+k -t^2+(n-k -t -1)/2})$ holds for all possible decomposition $n=ke+t$(such that $e=e(r,q^2)$, $r$ divide $n$ and $t$ is equal to $0$ or $n/r$) in cases $n=7, \ldots, 20$. Consider more carefully the case $n=6.$ First, we find stronger estimate for $|x^G \cap H|. $ We represent $x$ as $(\lambda, j) \in F_{q^n}^{\times} \leftthreetimes \mathbb{Z}_n.$ If $r=2$ then $e=1$ and we consider this case beneath. Now let $r=3$ then $1=(\lambda,j)^3=(\lambda^{(-q)^{2(n-j)} +(-q)^{n-j}+1},3j)$, i.e. $j=4,2,0.$ If $j=4$ then $\lambda^{q^4+q^2+1}=1$. If $j=2$ then $\lambda^{q^8+q^4+1}=1$. Note that $q^8 +q^4 +1= (q^6-1)q^2 + (q^4+q^2+1)$. Therefore in both cases we have $\lambda^{q^4+q^2+1}=1$. Because of $(q^6-1)=(q^4+q^2+1)(q^2-1)$ we obtain that $\lambda = \theta^{m(q^2-1)}$, where $m=1, \ldots, (q^4 +q^2+1).$ If $j=0$ then $x$ is an element of ${\mathrm{Sin}}_6(q)$, and there is only two elements of order 3 in ${\mathrm{Sin}}_6(q)$. In total we get $2(q^4+q^2+2)$ elements of order 3, thus $|x^G \cap H|\le 2(q^4+q^2+2)$. Now we verify inequality for this case for all possible decompositions $n=ek+t$ with $e>1$. For $r=3$ there are two cases $e=2, k=2, t=2$, and $e=2, k=3, t=0.$ The inequality $|x^G \cap H|^3< (q^{n(n-1)-{ek^2}+k -t^2})$ holds for $q>3$, for $q=2$ the statement of the lemma can be checked by GAP. Now let $e=1$, i.e. $x$ is conjugate with diagonal matrix. Denote the dimension of a maximal eigenspase of $x$ by $d$. Let $\alpha$ be the corresponding eigenvalue. If $n$ is odd then $x \in Sin_n^{+}(q^2) \rtimes \langle \varphi \rangle.$ We write $x$ as $x=a\varphi^j$, where $a \in {\mathrm{Sin}}_n^{+}(q^2)$, $j<n$. Consider the element $\alpha^{-1}x=\alpha^{-1}(a\varphi^j)=(\alpha^{-1}a)\varphi^j$, which lies in $ Sin_n^{+}(q^2) \rtimes \langle \varphi \rangle$, because the set ${\mathrm{Sin}}_n^{+}(q^2) \cup \{0\}$ is an algebra over $F_{q^2}$ and closed under the multiplication by scalar. Note, that if $n$ is even then ${\mathrm{Sin}}_n^{-}(q) \cup \{0\}$ is also closed under the multiplication by the stracture of ${\mathrm{Sin}}_n^{-}(q)$ described in the introduction, so in this case $\alpha^{-1}x$ lies in $Sin_n^{-}(q) \rtimes \langle \varphi \rangle$. Obviously, the order of the element $\alpha^{-1}x$ is also equal to $r$, and so the dimension of subspace of fixed points also have to be $n/r$ by Lemma \[nep\]. Thus $d=n/r.$ Let $\overline{\phantom{G}}:GU_n(q) \mapsto PGU_n(q)$ be the canonical homomorphism. Consider $x$ as an element of the algebraic group $\hat{G}=PGU_n(\overline{F}_q)$ over the algebraic closure $\overline{F}_q$ of the field $F_q$. Let $s=s(x)$ be the co-dimension of maximal eigenspase of $x$. In our case we have $s=n-d\ge n/2.$ Let $n$ be not prime, in particular $r\le n/2$. Then by [@fpr2 Proposition 3.38] we obtain that $$|x^G|\ge |\overline{x}^{\overline{G}}|> \frac{1}{2r}q^{ns} > \frac{1}{n}q^{n^2/2}.$$ The inequality $$(q^n-1)^3n^3<\frac{1}{n}q^{n^2/2}$$ holds for $n \ge 8 $ and $q\ge2$ if $r\ne p.$ Now let $n$ be a prime and $s=n-1$. Then by [@fpr2 Proposition 3.38] we obtain that $$|x^G|\ge |\overline{x}^{\overline{G}}|> \frac{1}{2r}q^{ns} > \frac{1}{2n}q^{n^2-n}.$$ The inequality $$(q^n-(-1)^n)^3n^3<\frac{1}{2n}q^{n^2-n}$$ holds for $n \ge 7$ and $q \ge 2.$ Let $n=6$ and $(\lambda, j) \in F_{q^n}^{\times} \rtimes \mathbb{Z}_n$ be chosen so that $(\lambda, j)^2=(\lambda^{(-q)^{(n-j)}+1}, 2j)=1$. Then $j=3$ and $$\lambda^{(-q)^3+1}=\lambda^{(-q)^3+1 +q^6-1}=\lambda^{q^3(q^3-1)}=1,$$ so, since $(q^3, q^6-1)=1,$ we obtain that $\lambda = \theta^{m(q^3-1)}$, where $m=1, \ldots, (q^3 +1).$ Thus $|x^G \cap H|\le (q^3 +1)$. It is easy to see that inequality $|x^G \cap H|^3\le (q^3+1)^3 <\frac{1}{4}q^{6^2/2} \le |x^G|$ holds for $q\ge 3.$ Let $r=3$ then $s=4$ and by [@fpr2 Proposition 3.38] we obtain $(q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}| \le |x^G|$ for $q>3$, the case $q=2$ can be checked by GAP. [**Case 2. ($|x|= p$)**]{} Let $\overline{\phantom{G}}:GU_n(q) \mapsto PGU_n(q)$ be the canonical homomorphism. Consider $x$ as an element of the algebraic group $\hat{G}=PGU_n(\overline{F}_q)$ over the algebraic closure $\overline{F}_q$ of the field $F_q$. Let $s=s(x)$ be the codimention of maximal eigenspase of $x$. Thus $s=n-t$ because $x$ is unipotent. Therefore $s\ge n/2$ by Lemma \[nep\]. So, by [@fpr2 Proposition 3.38] $$|x^G|\ge |\overline{x}^{\overline{G}}| > \frac{1}{2p}q^{ns}\ge\frac{1}{2p}q^{n^2/2}\ge \frac{1}{2}q^{n^2/2-1},$$ and $(q^n-(-1)^n)^3n^3<|x^G|$ holds for $q\ge2$ and $n\ge 9.$ For $n=8$ this inequality holds for $q\ge3$, the case $q=2$ can be checked by GAP. If $n=7$ then $(q^n-(-1)^n)^3n^3 <\frac{1}{2}q^{n^2/2-1}$ holds for $q\ge 13$, so since $r=p=7$ we obtain $s=6$ and by [@fpr2 Proposition 3.38] we have $(q^n-(-1)^n)^3n^3<\frac{1}{14}q^{7\cdot6}<|\overline{x}^{\overline{G}}| \le |x^G|$. This inequality holds for $q\ge7.$ If $n=6$ there are two possibilities: $r=3$ and $r=2.$ Let $r=3$ then $s=4$ and by [@fpr2 Proposition 3.38] we obtain $(q^n-(-1)^n)^3n^3<\frac{1}{6}q^{6\cdot4}<|\overline{x}^{\overline{G}}| \le |x^G|$ for $q>3$, case $q=3$ can be checked by GAP. If $r=p=2$ then $s=3$ and $|x^G|>\frac{1}{4}q^{6\cdot3}.$ Represent $x$ as $(\lambda, j) \in F_{q^n}^{\times} \rtimes \mathbb{Z}_n$ such that $(\lambda, j)^2=(\lambda^{q^{(n-j)}+1}, 2j)=1$. Then $j=3$ and $\lambda^{q^3+1}=1,$ so $\lambda = \theta^{m(q^3-1)}$, where $m=1, \ldots, (q^3 +1).$ Thus $|x^G \cap H|\le (q^3 +1)$ and we need to verify that $(q^3 +1)^3<|x^G|$. This inequality $(q^3 +1)^3<\frac{1}{4}q^{6\cdot3}$ holds for all $q\ge 2.$ \[m6\] Let $G=GL_n^{\epsilon}(q)$, $H= Sin_n^{\epsilon}(q) \rtimes \langle \varphi_{n} \rangle$ and $n\le5$ then $b_H(G)\le 3.$ The statement follows from [@burness Proposition 4.1]. \[m7\] Let $S$ be a Singer cycle ${\mathrm{Sin}}_n(q)={\mathrm{Sin}}_n^+(q)$. If $g \in S$ is reducible then $g$ is a scalar matrix. It is well known that $S$ is irreducible and primitive linear group. Since S is abelian, $\langle g \rangle $ is normal in $S$. By Clifford’s theorem we obtain that $\langle g \rangle $ is completely reducible and $F_q^n=V = V_1 \oplus \ldots \oplus V_k$, where $V_i$ is an irreducible $g$-module of dimension $n/k$ for $i=1, \ldots , k.$ So, $S \le GL(V_1) \wr Sym_k$ therefore $S$ is not irreducible if $\dim V_i >1.$ Thus, up to conjugation, $g$ is diagonal. Let $g =\left( \begin{smallmatrix} \alpha_1 & & \\ & \ddots & \\ & & \alpha_n\\ \end{smallmatrix} \right) $. We know that $S$ contains all scalar matrices and $S \cup \{0\}$ is closed under addition, so $g- \alpha_1E \in S \cup \{0\}$, but this matrix is degenerative, so $g- \alpha_1E=0$ and $g$ is scalar. Denote ${\mathrm{Sin}}_{n_i}^{\epsilon}(q) \rtimes \langle \varphi_i \rangle$ by $R_i$. By Lemma \[b6\] and Lemma \[m6\] we know that for $R_i$ there exist $x_i, y_i$ such that $R_i \cap R_i^{x_i} \cap R_i^{y_i}\le Z(GL_{n_i}^{\epsilon}(q))$. Let $x=x_1 \cdot \ldots \cdot x_k$ and $y=y_1 \cdot \ldots \cdot y_k.$ Consider the intersection $H \cap H^x \cap H^y$, and denote it by $K$. Let $g$ be an element of $K$ then $g$ is a diagonal matrix with scalar matrix in each cage: $$\begin{pmatrix} \alpha_1 & 0 & \dots & 0 \\ 0 & \ddots & & \vdots \\ \vdots & & \ddots & 0 & & & {\text{ \Huge 0}}& &\\ 0 & \dots & 0 &\alpha_1 \\ & & & & \ddots\\ & & & & & \alpha_k & 0 & \dots & 0 \\ & & {\text{ \Huge 0}} & & & 0 & \ddots & & \vdots \\ & & & & & \vdots & & \ddots & 0\\ & & & & & 0 & \dots & 0 &\alpha_k \\ \end{pmatrix}$$ Moreover, if there are $i_0, \ldots, i_l$ such that $n_{i_j}=1 ; 0 \le j \le l$ then up to conjugation in $G$ we can assume that $\{i_0, \ldots, i_l\}=\{k-l, \ldots k \}$. Then the group $L={\mathrm{Sin}}_{n_{k-l}}^{\epsilon}(q) \rtimes \langle \varphi_{k-l}\rangle \times \ldots \times {\mathrm{Sin}}_{n_k}^{\epsilon}(q) \rtimes \langle \varphi_k \rangle = {\mathrm{Sin}}_{n_{k-l}}^{\epsilon}(q) \times \ldots \times {\mathrm{Sin}}_{n_k}^{\epsilon}(q) $ is abelian. Thus by [@zen Theorem 1] there is $u \in GL_{l+1}^{\epsilon}(q)$ such that $L\cap L^u \le Z(GL_{l+1}^{\epsilon}(q))$. So, we can define $x$ and $y$ as $x_1 \cdot \ldots \cdot x_{k-l+1} \cdot u$ and $y_1 \cdot \ldots \cdot y_{k-l+1}$ respectively. Then we obtain than in $g$ $\alpha_{k-l}= \ldots = \alpha_k$.Therefore without loss of generality we can assume that there is only one Singer cycle of degree $1$ and $n_k=1.$ It is sufficient to find $z \in G$ such that for every $g$ lying in $K$ and in $H^z$ we have $\alpha_i = \alpha _{i+1}$ for $i =1 \ldots k-1.$ We start the proof with the case $G=GL_n(q)$. Let $z$ be the permutation matrix $(1,2, \ldots , n)$. If $g \in H^z$ then $g^{z^{-1}} \in K^{z^{-1}} \cap H $, and the matrix $g'=\left( \begin{smallmatrix} \alpha_1 & 0 & 0 & 0\\ 0 & \ddots & 0 & 0\\ 0& 0& \alpha_1 &0\\ 0& 0& 0& \alpha_2\\ \end{smallmatrix} \right)$ lies in $Sin_{n_1}^{\epsilon}(q)\rtimes<\varphi_1>.$ Thus $g'$ normalizes $Sin_{n_1}^{\epsilon}(q).$ Let $T$ be a generating element of $Sin_{n_1}^{\epsilon}(q)$ then $T^{g'}=T^m$ for an integer $m$. If $T = \left( \begin{smallmatrix} A & B\\ C & D\\ \end{smallmatrix} \right)$ where $A$ is $(n_1-1) \times (n_1-1)$ matrix, $B^{T}$ and $C$ are $n_1-1$ rows, and $D$ is an integer then $T^{g'}= \left( \begin{smallmatrix} A & \alpha_1^{-1}\alpha_2 B\\ \alpha_2^{-1}\alpha_1 C & D\\ \end{smallmatrix} \right)$. Thus $T-T^{g'}$ is a degenerate matrix for $n_1>2$. We have $T-T^{g'} \in Sin_{n_1}(q) \cup \{0\}$ because $Sin_{n_1}(q) \cup \{0\}$ is closed under addition, thus $T-T^{g'}=0$ and $T=T^{g'}$. Therefore we have $\alpha_1 = \alpha_2$, so $g'$ is a scalar matrix. The same arguments work for all $n_i > 2; i =1 \ldots k,$ so $\alpha_i$ equals $\alpha_{i+1}$ if $n_i>2.$ For $n_1=2$. Denote $ \alpha_1^{-1}\alpha_2$ and $\alpha_2^{-1}\alpha_1$ by $\beta_1$ and $\beta_2$ respectively. We have that $T-\beta_1 T^{g'}$ is equal to $T^l$ for some integer because ${\mathrm{Sin}}_{n_1}(q)$ is also an algebra under $GF(q)$. Thus $T^l =\left( \begin{smallmatrix} (1-\beta_1)A & (1-\beta_1^{2}) B\\ \text{ \large 0}& (1-\beta_1)D\\ \end{smallmatrix} \right)$ for suitable $l$ and $T^l$ is a scalar matrix by Lemma \[m7\]. It means that $1-\beta_1^{2}=0$ because if $B=0$ then $T$ is reducible and it can not generate ${\mathrm{Sin}}_{n_1}(q).$ So, $\beta_1=\beta_2$. If $p=2$ then $0=(1-\beta_1^{2})=(1-\beta_1)^{2}$, so $\beta_1=1$ and $\alpha_1=\alpha_2$. Let $p>2$, we have $\beta_1=\beta_2$ and $\beta_1^2=1$, so $\alpha_1^2=\alpha_2^2.$ Note that in this case we have two possibilities:\ If $g' \in Sin_n(q)$ then by Lemma \[m7\] $g$ is scalar, so $\alpha_1 =\alpha_2$.\ If $g'=s\cdot \varphi_1$, $s \in Sin_n(q)$ then for an element $g \in Sin_n(q)$ we obtain $$\label{phi} g^{g'}=g^{\varphi_1}=g^q.$$ Also, $g'^2 \in {\mathrm{Sin}}_n(q)$ but $g'^2$ is diagonal, so it is scalar by Lemma \[m7\]. Thus without loss of generality we can assume that if $\alpha_1 \ne \alpha_2$ then $g'^2=1$, so $\alpha_1^2=\alpha_2^2=1$ and $\alpha_1=1$; $\alpha_2=-1$. Therefore $T^{g'}=\left( \begin{smallmatrix} A & -B\\ -C & D\\ \end{smallmatrix} \right)$. Thus it is sufficient to prove that there is ${\mathrm{Sin}}_2(q)$ such that $T^{g'}=\left( \begin{smallmatrix} A & -B\\ -C & D\\ \end{smallmatrix} \right) \ne T^q.$ Let $a\ne 1$ be an element of $F_q$ such that the equation $x^2=a$ does not have solutions, and let $M= \left( \begin{smallmatrix} 0 & 1\\ a & 0\\ \end{smallmatrix} \right)$. It is easy to see that $C_G(M)$ is a Singer cycle. Consider the group $\langle M \rangle \rtimes g'$ which is a subgroup of normalizer of Singer cycle. Conjugate this subgroup with the matrix $ \left( \begin{smallmatrix} 1 & 1\\ 0 & 1\\ \end{smallmatrix} \right)$. We also have a subgroup of a Singer cycle: $$\langle \left( \begin{smallmatrix} a & 1-a\\ a & -a\\ \end{smallmatrix} \right) \rangle \ltimes \langle \left( \begin{smallmatrix} -1 & 1-2\\ 0 & -1\\ \end{smallmatrix} \right) \rangle.$$ It is easy to see that $ \left( \begin{smallmatrix} a & 1-a\\ a & -a\\ \end{smallmatrix} \right)^q$ is not equal to $ \left( \begin{smallmatrix} a & 1-a\\ a & -a\\ \end{smallmatrix}\right)^{g'}$. Indeed, $$\left( \begin{smallmatrix} a & 1-a\\ a & -a\\ \end{smallmatrix} \right)^q= a^{(q-1)/2} \left( \begin{smallmatrix} a & 1-a\\ a & -a\\ \end{smallmatrix}\right) \ne \left( \begin{smallmatrix} a & a-1\\ -a & -a\\ \end{smallmatrix}\right)= \left( \begin{smallmatrix} a & 1-a\\ a & -a\\ \end{smallmatrix}\right)^{g'}.$$ So, for suitable Singer cycle the matrix $\left( \begin{smallmatrix} 1 & 0\\ 0& -1\\ \end{smallmatrix}\right)$ does not normalize it. Thus $g'$ is a scalar matrix in this case and $\alpha_1 =\alpha_2$. The same arguments show that for all $n_i = 2; i =1 \ldots k$ $\alpha_i$ equals $\alpha_{i+1}$. This yields that $\alpha_i=\alpha_j$ for all $i,j =1 \ldots k,$ and we obtain that $g$ is a scalar matrix. Now consider the case $G=GL_n^{-}(q)=GU_n(q).$ The proof is similar to the proof of the previous case. For the convenience of proof that $\alpha_i=\alpha_{i+1}$ when $n_i=2$ we will consider $G$ as the group of unitary transformations with respect to the hermitian form with the matrix $$\begin{pmatrix} I_1 & 0 & \dots & 0 \\ 0 & I_2 & & \vdots \\ \vdots & & \ddots & 0 \\ 0 & \dots & 0 &I_k \end{pmatrix},$$ where $I_i$ is a $n_i \times n_i$-matrix, and $I_i$ is identity matrix if $n_i \ne 2$ and $I_i= \left( \begin{smallmatrix} 0 & 1\\ 1 & 0 \end{smallmatrix} \right)$ if $n_i=2.$ Let $p$ be odd. Consider the matrix $D =\frac{1}{2} \left( \begin{smallmatrix} 1 +\alpha & 1 -\alpha\\ 1 -\alpha & 1 +\alpha \end{smallmatrix} \right),$ where $\alpha \in F_{q^2}$ is that $\alpha \overline{\alpha}=-1$ (such $\alpha$ always exists in characteristic not equal to 2; if $\theta$ is a generating element of $F_{q^2}$ then $\alpha=\theta^{\frac{q-1}{2}}$). It is routine to check that $D \overline{D}^{t}=\left( \begin{smallmatrix} 0 & 1\\ 1 & 0 \end{smallmatrix} \right) $. So, if denote by ${\bf v}=\{v_1, v_2\}$ and $ {\bf w}=\{w_1,w_2\}$ the basises of $GU_2(q)$ with respect to hermitian forms with matrices $\left( \begin{smallmatrix} 1 & 0\\ 0 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & 1\\ 1 & 0 \end{smallmatrix} \right)$ respectively then $D$ is the transition matrix from the basis ${\bf v}$ to the basis ${\bf w}$. Denote by $S$ the matrix $$\begin{pmatrix} S_1 & 0 & \dots & 0 \\ 0 & S_2 & & \vdots \\ \vdots & & \ddots & 0 \\ 0 & \dots & 0 &S_k \end{pmatrix},$$ where $S_i$ is $n_i \times n_i$-matrix and it is equal to the identity matrix if $n_i\ne 2$ and equal to $D$ if $n_i=2$. Recall that $z$ stands for the permutation matrix $(1,2, \ldots , n)$. This matrix is unitary with respect to a hermitian form with identity matrix. It follows, as easy to see, that $z'= Sz(S^{-1})$ lies in $G.$ Consider now $g \in K \cap H^{z'}$. If $n_1 >2$ then matrix $g'=\left( \begin{smallmatrix} \alpha_1 & 0 & 0 & 0\\ 0 & \ddots & 0 & 0\\ 0& 0& \alpha_1 &0\\ 0& 0& 0& \alpha_2\\ \end{smallmatrix} \right) \in Sin_{n_1}^{\epsilon}(q)\rtimes<\varphi_1>.$ Thus $g'$ normalizes $Sin_{n_1}^{\epsilon}(q).$ Let $T$ be a generating element of $Sin_{n_1}^{\epsilon}(q)$ then as was mentioned above $T-T^{g'}$ is a degenerate matrix. If ${\epsilon=-}$ and $n_1$ is odd than $T-T^{g'} \in Sin_{n_1}(q^2) \cup \{0\}$, thus $T-T^{g'}=0$ and $T=T^{g'}$. If ${\epsilon=-}$ and $n_1$ is even than $T-T^{g'} \in Sin_{n_1/2}(q^2) \times Sin_{n_1/2}(q^2) \cup \{0\}$, and by the structure of $Sin_n^-(q)$ described in the introduction if $T-T^{g'}$ is a degenerate matrix then $T-T^{g'}=0$ and $T=T^{g'}$. Then we have $\alpha_1 = \alpha_2$, so $g'$ is a scalar matrix. The same arguments work for all $n_i > 2; i =1 \ldots k,$ so $\alpha_i$ equals $\alpha_{i+1}$ if $n_i>2.$ Let now $n_1=2$. Then $g' = S \left( \begin{smallmatrix} \alpha_1 & 0\\ 0 & \alpha_2 \end{smallmatrix} \right)S^{-1} \in Sin_{n_1}^{\epsilon}(q)\rtimes<\varphi_1>$. Calculations show that $$g'=1/4 \begin{pmatrix} (1+ \alpha)(1-\overline{\alpha})\alpha_1 + (1+\overline{\alpha})(1-\alpha)\alpha_2 & (1+ \overline{\alpha})(1+\alpha)\alpha_1 + (1-\alpha)(1-\overline{\alpha})\alpha_2 \\ (1- \alpha)(1-\overline{\alpha})\alpha_1 + (1+\alpha)(1+\overline{\alpha})\alpha_2 & (1- \alpha)(1+\overline{\alpha})\alpha_1 + (1+\alpha)(1-\overline{\alpha})\alpha_2 \end{pmatrix}.$$ Let $\theta$ be a generating element of $F_{q^2}$, consider the matrix $ U=\left( \begin{smallmatrix} \theta & 0\\ 0& \overline{\theta}^{-1}\\ \end{smallmatrix}\right)$. Without loss of generality we can assume that $${\mathrm{Sin}}_2^-(q) \rtimes \langle \varphi_1 \rangle = \langle U \rangle \rtimes \langle \left( \begin{smallmatrix} 0& 1\\ 1& 0\\ \end{smallmatrix}\right)\rangle.$$ If $g' \in {\mathrm{Sin}}_2^-(q)$ then the element $(1+ \overline{\alpha})(1+\alpha)\alpha_1 + (1-\alpha)(1-\overline{\alpha})\alpha_2=(\overline{\alpha} + \alpha)\alpha_1 - ( \overline{\alpha} + \alpha)\alpha_2 $ must be equal to $0$. The element $( \overline{\alpha} + \alpha)$ can not be equal to $0$ because then $\overline{\alpha} \alpha = -\alpha^2$ and $\alpha^2=1$, but $\alpha^2 = (\theta^{\frac{q-1}{2}})=\theta^{{q-1}} \ne 1.$ So, $\alpha_1=\alpha_2$. If $g' \in {\mathrm{Sin}}_2^-(q) \rtimes \langle \varphi_1 \rangle \backslash {\mathrm{Sin}}_2^-(q) $ then the diagonal elements of matrix $g'$ are equal to $0$. $$\label{a1a2} \begin{cases} (2+ \alpha -\overline{\alpha})\alpha_1 + (2- \alpha +\overline{\alpha})\alpha_2=0\\ (2- \alpha +\overline{\alpha})\alpha_1 + (2+ \alpha -\overline{\alpha})\alpha_2=0 \end{cases}$$ Take a sum of that two equations and get $4\alpha_1+4\alpha_2=0.$ So $\alpha_1=-\alpha_2$. Substituting this solution in the first equation of the system we obtain that $\overline{\alpha} +\alpha=0$ but, as we already know, this is not true. Therefore the system does not have solutions and $g'$ is not an element from ${\mathrm{Sin}}_2^-(q) \rtimes \langle \varphi_1 \rangle \backslash {\mathrm{Sin}}_2^-(q) $. The same arguments show that for all $n_i = 2; i =1 \ldots k$ $\alpha_i$ equals $\alpha_{i+1}$. This yields that $\alpha_i=\alpha_j$ for all $i,j =1 \ldots k,$ and we obtain that $g$ is a scalar matrix. Now, let $p=2$. The proof is the similar as proof when p is add with different matrix $D$. In this case $$D=\left( \begin{matrix} \frac{1}{\beta+1}& \frac{\beta}{\beta+1}\\ \frac{\beta}{\beta+1} & \frac{1}{\beta+1} \end{matrix} \right),$$ where $\beta$ is an element of $F_{q^2}$ such that $\beta \overline{\beta}=1$ (such element always exist, for example if $\beta = \theta^{q-1}$ then $\beta \overline{\beta}=\theta^{q-1}\theta^{q^2-q}=\theta^{q^2-1} =1$ ). The proof that if $n_i >2$ then $\alpha_i=\alpha_{i+1}$ is just the same as in case of odd $p$. Assume $n_1=2$ then $g' = S \left( \begin{smallmatrix} \alpha_1 & 0\\ 0 & \alpha_2 \end{smallmatrix} \right)S^{-1} \in Sin_{n_1}^{-}(q)\rtimes \langle \varphi_1 \rangle$. Calculations show that $$g'= \begin{pmatrix} \frac{ \overline{\beta}\alpha_1 + \beta \alpha_2}{\overline{\beta}+\beta } & \frac{\alpha_1 + \alpha_2}{\overline{\beta}+\beta } \\ \frac{\alpha_1 + \alpha_2}{\overline{\beta}+\beta } & \frac{\beta \alpha_1 + \overline{\beta}\alpha_2 }{\overline{\beta}+\beta } \end{pmatrix}.$$ We still assume that $${\mathrm{Sin}}_2^-(q) \rtimes \langle \varphi_1 \rangle = \langle U \rangle \rtimes \langle \left( \begin{smallmatrix} 0& 1\\ 1& 0\\ \end{smallmatrix}\right)\rangle.$$ So, if $g' \in {\mathrm{Sin}}_2^-(q)$ then obviously $\alpha_1=\alpha_2$. Let $g'$ be an element of the set ${\mathrm{Sin}}_2^-(q) \rtimes \langle \varphi_1 \rangle \backslash {\mathrm{Sin}}_2^-(q) $ then $$\label{a1a22} \begin{cases} \overline{\beta}\alpha_1 + \beta \alpha_2=0\\ \beta \alpha_1 + \overline{\beta}\alpha_2 =0 \end{cases}$$ From the first equation we obtain that $\alpha_2=\alpha_1 \beta^{q-1}$ and using the first equation we obtain that $ \beta^{q-1}=1$. That is a contradiction. The same arguments show that for all $n_i = 2; i =1 \ldots k$ $\alpha_i$ equals $\alpha_{i+1}$. This yields that $\alpha_i=\alpha_j$ for all $i,j =1 \ldots k,$ and we obtain that $g$ is a scalar matrix. Proof of Theorem \[TH2\] ======================== Let $G = GL_n(q) \rtimes \langle \tau \rangle $ where $q=2$ or $q=3$, $n$ is even, $\tau$ is an automorphism which acts by $\tau: A \mapsto (A^{-1})^{T}$ for $A \in GL_n(q)$. Let $H$ be the normalizer in $G$ of subgroup $P \le GL_n(q)$, where $P$ is the stabilizer of the chain of subspaces: $$\langle v_n, v_{n-1}\rangle < \langle v_n, v_{n-1}, v_{n-2}, v_{n-3} \rangle < \ldots <\langle v_n, v_{n-1}, \ldots , v_2, v_1 \rangle .$$ $$P \thicksim \begin{pmatrix} * &*&* &* &* &* \\ * &*& *&*&* &*\\ & & * & * & * &*\\ & & * & * & * &*\\ & {\text{ \Huge 0}} & & \ddots & \vdots &\vdots\\ &&&& *&*\\ &&&&*&*\\ \end{pmatrix}$$ with $GL_2(q)$-boxes on the diagonal in the basis $\{v_1, v_2, \ldots, v_n\}$. Since $P=N_{GL_n(q)}(P)$, we obtain that $H \cap GL_n(q)=P.$ Consider the intersection $$K=H\cap H^x \cap (H \cap H^x)^y,$$ where $x$ and $y$ are permutation matrices corresponding to permutations $(1,n)(2,n-1) \ldots (n/2, n/2+1)$ and $(1,2, \ldots,n)$ respectively. If $g \in P\cap P^x \cap (P \cap P^x)^y$ it is clear that $g$ is diagonal. Indeed, the group $ P\cap HP^x$ stabilize subspaces $$\langle v_n, v_{n-1}\rangle , \langle v_{n-2}, v_{n-3} \rangle , \ldots ,\langle v_2, v_1 \rangle .$$ Meanwhile, the group $ (P \cap P^x)^y$ stabilize subspaces $$\langle v_1, v_{n}\rangle , \langle v_{n-1}, v_{n-2} \rangle , \ldots ,\langle v_3, v_2 \rangle .$$ So, $GL_n(q) \cap H\cap H^x \cap (H \cap H^x)^y$ stabilize $\langle v_i \rangle$ for $i \in \{1,2, \ldots , n\}$. Obviously, $\tau$ is not in $H$ because it does not normalize $P$. Let $g\tau$ be an element of $K$. Then $g\tau$ have to normalize $P$. If $ P^{g\tau}= P$ then $P^g=P^{\tau} =P^{T}$, so $g$ maps upper triangular matrix to lower triangular matrix.We know that $x$ also maps upper triangular matrix to lower triangular matrix thus $P^{gx}=P$ and $gx$ lies in $N_{GL_n(q)}(P)$ furthermore $gx \in P$ because $P$ is selfnomalizing in $GL_n(q)$. It means that $g \in Px$ and $$\label{1} g \thicksim \begin{pmatrix} * &*&* &* &* &* \\ * &*& *&*&* &*\\ *& *& * & * & &\\ *& *& * & * & &\\ \vdots& \vdots &\Ddots & & {\text{ \Huge 0}} &\\ *&*&&& &\\ *&*&&&&\\ \end{pmatrix}$$ Also $g\tau$ must normalize $P \cap P^x$ because $g\tau \in H\cap H^x.$ It is easy to see that $\tau$ normalizes $P \cap P^x$ therefore $g \in N_{GL_n(q)}(P\cap P^x)=GL_2(q) \wr S_{n/2}.$ From these and we obtain $$g \thicksim \begin{pmatrix} && & &* &* \\ &{\text{ \huge 0}}& &&* &*\\ & & * & * & &\\ & & * & * & &\\ & &\Ddots & & {\text{ \huge 0}} &\\ *&*&&& &\\ *&*&&&&\\ \end{pmatrix}$$ In the same way $g\tau$ normalizes the intersection $$P \cap P^x \cap P^y \thicksim \begin{pmatrix} * &0& & & &&&& \\ * &*& && &&&&\\ & & * & * & &&&&\\ & & 0 & * & &&&&\\ & & & &* &*&&&\\ & & & & 0 &*&&&\\ & & & & & & \ddots&&\\ &&&& & & &* &*&\\ &&& & & & & 0 &*&\\ \end{pmatrix}$$ and $\tau$ move zero to the opposite corner of matrix in each box so $g$ does. Thus $$\label{2} g \thicksim \begin{pmatrix} &&&&& & &0 &*&\\ &&& & & & & * &*&\\ && & & &* &*&&&\\ & & & & & * &0&&&\\ & & & * & * & &&&&\\ & & & * & 0 & &&&&\\ & &\Ddots & & & & &&\\ *&*&&&&&&& \\ *&0&&&&&&&\\ \end{pmatrix}$$ Similarly $g\tau$ normalizes $P^x \cap P^{xy} \cap P$ and we get $$\label{3} g \thicksim \begin{pmatrix} &&&&& & &* &*&\\ &&& & & & & * &0&\\ && & & &0 &*&&&\\ & & & & & * &*&&&\\ & & & 0 & * & &&&&\\ & & & * & * & &&&&\\ & &\Ddots & & & & &&\\ 0&*&&&&&&& \\ *&*&&&&&&&\\ \end{pmatrix}$$ From and we obtain that $g$ is secondary diagonal matrix. The same arguments show that if $h\tau \in H\cap H^x \cap H^{y^{-1}} \cap H^{xy^{-1}}$ and $h \in GL_n(q)$ then $h$ is secondary diagonal. Now we have $$(g\tau)^{y^{-1}}=g^{y^{-1}}\tau \in K^{y^{-1}}=H\cap H^x \cap H^{y^{-1}} \cap H^{xy^{-1}}$$ and $q^{y^{-1}}$ have to be secondary diagonal but it is easy to see that if $g =secdiag(\alpha_1, \ldots, \alpha_n)$ then $$g^{y^{-1}}= \begin{pmatrix} 0 &0& \ldots &0 &\alpha_2 &0&0&& \\ 0 &\ldots & 0 & \alpha_3& 0&0&0&&\\ & & \Ddots & & &&&&\\ & \Ddots & & &&&&&\\ \alpha_{n-1}&0 & \ldots & \ldots &\ldots &\ldots&0&&\\ 0&\ldots & \ldots & \ldots & \ldots &0& \alpha_n&&\\ 0 &\ldots &\ldots &\ldots & 0 &\alpha_1&0&&\\ \end{pmatrix}$$ which is not secondary diagonal matrix thus $K$ is a subgroup of group of diagonal matrices. Thus in case when $q=2$ we have $b_H(G)\le 4.$ Let $q=3$. There is $z$ such that $H^z$ stabilises $\langle u_1= v_1 + v_2 +v_5 + \ldots + v_n, u_2= v_1 + v_3 + v_4 + \ldots + v_n \rangle$. In other words, $v_n^{z}=u_1$, $v_{n-1}^z=u_2.$ Here $\{v_1,v_2, \ldots, v_n\}$ is the initial basis. Consider $g$ from $H \cap H^x \cap H^y \cap H^{xy} \cap H^z$. The element $g$ is a diagonal matrix in the initial basis because it lies in $K$, so $g =diag\{\alpha_1, \ldots, \alpha_n\}$. Then $$u_1^g = \alpha_1 v_1 + \alpha_2 v_2 +\alpha_5 v_5 + \ldots + \alpha_n v_n.$$ Since $u_1^g$ does not involve $v_3$, we obtain that $u_1^g= \alpha u_1 +0 \cdot u_2$ i.e. $u_1^g$ is equal to $\alpha u_1$ for appropriate $\alpha.$ Hence $ \alpha_1 = \alpha_2 =\alpha_5 = \ldots = \alpha_n$. Also, $$u_2^g = \alpha_1 v_1 + 2 \alpha_3 v_3 +\alpha_4 v_4 + \ldots + \alpha_n v_n.$$ Since $u_2^g$ does not involve $v_2$, we obtain that $u_2^g= 0 \cdot u_1 +\beta u_2$ i.e. $u_2^g$ is equal to $\beta u_2$ for appropriate $\beta.$ Hence $ \alpha_1 = \alpha_3 =\alpha_4 = \ldots = \alpha_n$. Suchwise we have $H \cap H^x \cap H^y \cap H^{xy} \cap H^z \le Z(G).$ Consider $z_1 \in G$ such that $v_n^{z_1}=v_3 + v_1+v_5 + \ldots + v_n$, $v_{n-1}^{z_1}=v_1+v_2 + v_4+v_5 + \ldots + v_n$. It is easy to see that $H \cap H^x \cap H^y \cap H^{xy} \cap H^{z_1} \le Z(G).$ The points $(H,Hx,Hy,Hxy,Hz)$ and $(H,Hx,Hy,Hxy,Hz_1)$ lies in the same orbit of $\Omega^5$, where $\Omega$ is the set of all right cosets of $H$, if and only if there is an element $t \in G$ such that $t \in H \cap H^x \cap H^y \cap H^{xy} \cap z^{-1}Hz_1.$ So, in this case $t$ muct be diagonal matrix. Let $t = diag\{\tau_1, \tau_2, \ldots , \tau_n\}$. Since $t \in z^{-1}Hz_1$ we have $t=z^{-1}hz_1$ for some $h \in H$. Consider $u_1^t= u_1^{z^{-1}hz_1}=v_n^{hz_1}=(\delta v_n +\gamma v_{n-1})^{z_1}=\delta v_1 + \delta v_3 + \gamma v_1 + \gamma v_2 + \gamma v_4 + (\delta +\gamma)v_5 + \ldots + (\delta +\gamma)v_n. $ Since $t$ is diagonal we obtain that $\delta v_3=0$ and $\gamma v_4=0$ then $\delta=\gamma=0$ and $t=0$ which contradicts the assumption, so points $(H,Hx,Hy,Hxy,Hz)$ and $(H,Hx,Hy,Hxy,Hz_1)$ lies in distinct orbits. The same argument shows that points $$\begin{gathered} (H,Hx,Hy,Hxy,Hz); \\ (H,Hx,Hy,Hxy,Hz_1); \\ (H,Hx,Hy,Hxy,Hz_2);\\ (H,Hx,Hy,Hxy,Hz_3); \\ (H,Hx,Hy,Hxy,Hz_4); \end{gathered}$$ lies in the distinct five orbits. Where $$\begin{gathered} v_n^{z_2}=v_1 +v_4;\\ v_{n-1}^{z_2}=v_1+v_2+v_3;\\ v_n^{z_3}=v_2 +v_3;\\ v_{n-1}^{z_3}=v_1+v_2+v_4;\\ v_n^{z_4}=v_2 +v_4;\\ v_{n-1}^{z_4}=v_1+v_2+v_3.\\ \end{gathered}$$ Thus $Reg_H(G,5)\ge 5.$ [1]{} A.A. Baykalov, [*Intersection of conjugate solvable subgroups in symmetric groups*]{}, to appear. See also http://arxiv.org/abs/1701.04231 A. Borel, R. Carter, C. W. Curtis, N. Iwahori T. A. Springer, R. Steinberg [*Seminar on Algebraic Groups and Related Finite Groups* ]{} Timothy C. Burness [*Fixed point ratios in actions of finite classical groups, I* ]{} Journal of Algebra 309 (2007) 69–79 Timothy C. Burness [*Fixed point ratios in actions of finite classical groups, II* ]{} Journal of Algebra 309 (2007) 80–138 Timothy C. Burness [*Fixed point ratios in actions of finite classical groups, III* ]{} Journal of Algebra 314 (2007) 693–748 Timothy C. Burness [*Fixed point ratios in actions of finite classical groups, IV* ]{} Journal of Algebra 309 (2007) 749–788 Timothy C. Burness [*On base sizes for actions of finite classical groups* ]{} J. London Math. Soc. (2) 75 (2007) 545–562 R.W. Carter [*Finite groups of Lie type: Conjugacy Classes and Complex Characters* ]{} Victor D. Mazurov and Evgeny I. Khukhro [ *Unsolved problems in group theory. The Kourovka notebook*]{} No 18. URL: http://arxiv.org/abs/1401.0300 A. Previtali, M. C. Tamburini and E.P. Vdovin [*The Carter subgroups of some classical groups*]{} Bull. London Math. Soc. 36 (2004) 145–155 E. P. Vdovin, [*On the base size of a transitive group with solvable point stabilizer*]{} [Journal of Algebra and Application]{}, v. **11** (2012), N 1, 1250015 (14 pages) , [*On intersections of solvable Hall subgroups in finite simple exceptional groups of Lie type*]{} (Russian) Proc. Steklov Inst. Math. (2014) 285(Suppl 1): 183–190 , [*Intersections of abelian subgroups in finite groups*]{} (Russian) Mat. Zametki 56 (1994), no. 2, 150–152; translation in Math. Notes 56 (1994), no. 1–2, 869–871 (1995) , Version 4.7.8 of 09-Jun-2015 (free software, GPL) http://www.gap-system.org

--- author: - 'Simon Brendle and Fernando C. Marques' title: 'Blow-up phenomena for the Yamabe equation II' --- Introduction ============ Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$. The Yamabe problem is concerned with finding metrics of constant scalar curvature in the conformal class of $g$. This problem leads to a semi-linear elliptic PDE for the conformal factor. More precisely, a conformal metric of the form $u^{\frac{4}{n-2}} \, g$ has constant scalar curvature $c$ if and only if $$\label{yamabe.pde} \frac{4(n-1)}{n-2} \, \Delta_g u - R_g \, u + c \, u^{\frac{n+2}{n-2}} = 0,$$ where $\Delta_g$ is the Laplace operator with respect to $g$ and $R_g$ denotes the scalar curvature of $g$. Every solution of (\[yamabe.pde\]) is a critical point of the functional $$\label{yamabe.functional} E_g(u) = \frac{\int_M \big ( \frac{4(n-1)}{n-2} \, |du|_g^2 + R_g \, u^2 \big ) \, dvol_g}{\big ( \int_M u^{\frac{2n}{n-2}} \, dvol_g \big )^{\frac{n-2}{n}}}.$$ In this paper, we address the question whether the set of all solutions to the Yamabe PDE is compact in the $C^2$-topology. It has been conjectured that this should be true unless $(M,g)$ is conformally equivalent to the round sphere (see [@Schoen1],[@Schoen2],[@Schoen3]). The case of the round sphere $S^n$ is special in that (\[yamabe.pde\]) is invariant under the action of the conformal group on $S^n$, which is non-compact. It follows from a theorem of Obata [@Obata] that every solution of the Yamabe PDE on $S^n$ is minimizing, and the space of all solutions to the Yamabe PDE on $S^n$ can be identified with the unit ball $B^{n+1}$. Note that the round sphere is the only compact manifold for which the set of minimizing solutions is non-compact. The Compactness Conjecture has been verified in low dimensions and in the locally conformally flat case. If $(M,g)$ is locally conformally flat, compactness follows from work of R. Schoen [@Schoen1],[@Schoen2]. Moreover, Schoen proposed a strategy, based on the Pohozaev identity, for proving the conjecture in the non-locally conformally flat case. In [@Li-Zhu], Y.Y. Li and M. Zhu followed this strategy to prove compactness in dimension $3$. O. Druet [@Druet] proved the conjecture in dimensions $4$ and $5$. The case $n \geq 6$ is more subtle, and requires a careful analysis of the local properties of the background metric $g$ near a blow-up point. The Compactness Conjecture is closely related to the Weyl Vanishing Conjecture, which asserts that the Weyl tensor should vanish to an order greater than $\frac{n-6}{2}$ at a blow-up point (see [@Schoen3]). The Weyl Vanishing Conjecture has been verified in dimensions $6$ and $7$ by F. Marques [@Marques] and, independently, by Y.Y. Li and L. Zhang [@Li-Zhang1]. Using these results and the positive mass theorem, these authors were able to prove compactness for $n \leq 7$. Moreover, Li and Zhang showed that compactness holds in all dimensions provided that $|W_g(p)| + |\nabla W_g(p)| > 0$ for all $p \in M$. In dimensions $10$ and $11$, it is sufficient to assume that $|W_g(p)| + |\nabla W_g(p)| + |\nabla^2 W_g(p)| > 0$ for all $p \in M$ (see [@Li-Zhang2]). Very recently, M. Khuri, F. Marques and R. Schoen [@Khuri-Marques-Schoen] proved the Weyl Vanishing Conjecture up to dimension $24$. This result, combined with the positive mass theorem, implies the Compactness Conjecture for those dimensions. After proving sharp pointwise estimates, they reduce these questions to showing a certain quadratic form is positive definite. It turns out the quadratic form has negative eigenvalues if $n \geq 25$. In a recent paper [@Brendle1], it was shown that the Compactness Conjecture fails for $n \geq 52$. More precisely, given any integer $n \geq 52$, there exists a smooth Riemannian metric $g$ on $S^n$ such that set of constant scalar curvature metrics in the conformal class of $g$ is non-compact. Moreover, the blowing-up sequences obtained in [@Brendle1] form exactly one bubble. The construction relies on a gluing procedure based on some local model metric. These local models are directions in which the quadratic form of [@Khuri-Marques-Schoen] is negative definite. We refer to [@Brendle2] for a survey of this and related results. In the present paper, we extend these counterexamples to the dimensions $25 \leq n \leq 51$. Our main theorem is: \[main.theorem\] Assume that $25 \leq n \leq 51$. Then there exists a Riemannian metric $g$ on $S^n$ (of class $C^\infty$) and a sequence of positive functions $v_\nu \in C^\infty(S^n)$ ($\nu \in \mathbb{N}$) with the following properties: - $g$ is not conformally flat - $v_\nu$ is a solution of the Yamabe PDE (\[yamabe.pde\]) for all $\nu \in \mathbb{N}$ - $E_g(v_\nu) < Y(S^n)$ for all $\nu \in \mathbb{N}$, and $E_g(v_\nu) \to Y(S^n)$ as $\nu \to \infty$ - $\sup_{S^n} v_\nu \to \infty$ as $\nu \to \infty$ (Here, $Y(S^n)$ denotes the Yamabe energy of the round metric on $S^n$.) We note that O. Druet and E. Hebey [@Druet-Hebey1] have constructed blow-up examples for perturbations of (\[yamabe.pde\]) (see also [@Druet-Hebey2]). In Section 2, we describe how the problem can be reduced to finding critical points of a certain function $\mathcal{F}_g(\xi,\varepsilon)$, where $\xi$ is a vector in $\mathbb{R}^n$ and $\varepsilon$ is a positive real number. This idea has been used by many authors (see, e.g., [@Ambrosetti], [@Ambrosetti-Malchiodi], [@Berti-Malchiodi], [@Brendle1]). In Section 3, we show that the function $\mathcal{F}_g(\xi,\varepsilon)$ can be approximated by an auxiliary function $F(\xi,\varepsilon)$. In Section 4, we prove that the function $F(\xi,\varepsilon)$ has a critical point, which is a strict local minimum. Finally, in Section 5, we use a perturbation argument to construct critical points of the function $\mathcal{F}_g(\xi,\varepsilon)$. From this the non-compactness result follows. The authors would like to thank Professor Richard Schoen for constant support and encouragement. The first author was supported by the Alfred P. Sloan foundation and by the National Science Foundation under grant DMS-0605223. The second author was supported by CNPq-Brazil, FAPERJ and the Stanford Mathematics Department.\ Lyapunov-Schmidt reduction ========================== In this section, we collect some basic results established in [@Brendle1]. Let $$\mathcal{E} = \bigg \{ w \in L^{\frac{2n}{n-2}}(\mathbb{R}^n) \cap W_{loc}^{1,2}(\mathbb{R}^n): \int_{\mathbb{R}^n} |dw|^2 < \infty \bigg \}.$$ By Sobolev’s inequality, there exists a constant $K$, depending only on $n$, such that $$\bigg ( \int_{\mathbb{R}^n} |w|^{\frac{2n}{n-2}} \bigg )^{\frac{n-2}{n}} \leq K \, \int_{\mathbb{R}^n} |dw|^2$$ for all $w \in \mathcal{E}$. We define a norm on $\mathcal{E}$ by $\|w\|_{\mathcal{E}}^2 = \int_{\mathbb{R}^n} |dw|^2$. It is easy to see that $\mathcal{E}$, equipped with this norm, is complete.\ Given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$, we define a function $u_{(\xi,\varepsilon)}: \mathbb{R}^n \to \mathbb{R}$ by $$u_{(\xi,\varepsilon)}(x) = \Big ( \frac{\varepsilon}{\varepsilon^2 + |x - \xi|^2} \Big )^{\frac{n-2}{2}}.$$ The function $u_{(\xi,\varepsilon)}$ satisfies the elliptic PDE $$\Delta u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} = 0.$$ It is well known that $$\int_{\mathbb{R}^n} u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}} = \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}$$ for all $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$. We next define $$\varphi_{(\xi,\varepsilon,0)}(x) = \Big ( \frac{\varepsilon}{\varepsilon^2 + |x - \xi|^2} \Big )^{\frac{n+2}{2}} \, \frac{\varepsilon^2 - |x - \xi|^2}{\varepsilon^2 + |x - \xi|^2}$$ and $$\varphi_{(\xi,\varepsilon,k)}(x) = \Big ( \frac{\varepsilon}{\varepsilon^2 + |x - \xi|^2} \Big )^{\frac{n+2}{2}} \, \frac{2\varepsilon \, (x_k - \xi_k)}{\varepsilon^2 + |x - \xi|^2}$$ for $k = 1,\hdots,n$. Finally, we define a closed subspace $\mathcal{E}_{(\xi,\varepsilon)} \subset \mathcal{E}$ by $$\mathcal{E}_{(\xi,\varepsilon)} = \bigg \{ w \in \mathcal{E}: \int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \, w = 0 \quad \text{for $k = 0,1,\hdots,n$} \bigg \}.$$ Clearly, $u_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$. \[linearized.operator\] Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) = \exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on $\mathbb{R}^n$ satisfying $h(x) = 0$ for $|x| \geq 1$. There exists a positive constant $\alpha_0 \leq 1$, depending only on $n$, with the following significance: if $|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha_0$ for all $x \in \mathbb{R}^n$, then, given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$ and any function $f \in L^{\frac{2n}{n+2}}(\mathbb{R}^n)$, there exists a unique function $w=G_{(\xi,\varepsilon)}(f) \in \mathcal{E}_{(\xi,\varepsilon)}$ such that $$\int_{\mathbb{R}^n} \Big ( \langle dw,d\psi \rangle_g + \frac{n-2}{4(n-1)} \, R_g \, w \, \psi - n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w \, \psi \Big ) = \int_{\mathbb{R}^n} f \, \psi$$ for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Moreover, we have $\|w\|_{\mathcal{E}} \leq C \, \|f\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}$, where $C$ is a constant that depends only on $n$. \[fixed.point.argument\] Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) = \exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on $\mathbb{R}^n$ satisfying $h(x) = 0$ for $|x| \geq 1$. Moreover, let $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$. There exists a positive constant $\alpha_1 \leq \alpha_0$, depending only on $n$, with the following significance: if $|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha_1$ for all $x \in \mathbb{R}^n$, then there exists a function $v_{(\xi,\varepsilon)} \in \mathcal{E}$ such that $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$ and $$\int_{\mathbb{R}^n} \Big ( \langle dv_{(\xi,\varepsilon)},d\psi \rangle_g + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \, \psi - n(n-2) \, |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)} \, \psi \Big ) = 0$$ for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Moreover, we have the estimate $$\begin{aligned} &\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{\mathcal{E}} \\ &\leq C \, \Big \| \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)},\end{aligned}$$ where $C$ is a constant that depends only on $n$. We next define a function $\mathcal{F}_g: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ by $$\begin{aligned} \mathcal{F}_g(\xi,\varepsilon) &= \int_{\mathbb{R}^n} \Big ( |dv_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)}^2 - (n-2)^2 \, |v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} \Big ) \\ &- 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}.\end{aligned}$$ If we choose $\alpha_1$ small enough, then we obtain the following result:\ \[reduction.to.a.finite.dimensional.problem\] The function $\mathcal{F}_g$ is continuously differentiable. Moreover, if $(\bar{\xi},\bar{\varepsilon})$ is a critical point of the function $\mathcal{F}_g$, then the function $v_{(\bar{\xi},\bar{\varepsilon})}$ is a non-negative weak solution of the equation $$\Delta_g v_{(\bar{\xi},\bar{\varepsilon})} - \frac{n-2}{4(n-1)} \, R_g \, v_{(\bar{\xi},\bar{\varepsilon})} + n(n-2) \, v_{(\bar{\xi},\bar{\varepsilon})}^{\frac{n+2}{n-2}} = 0.$$ An estimate for the energy of a “bubble" ======================================== Throughout this paper, we fix a real number $\tau$ and a multi-linear form $W: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$. The number $\tau$ depends only on the dimension $n$. The exact choice of $\tau$ will be postponed until Section 4. We assume that $W_{ijkl}$ satisfies all the algebraic properties of the Weyl tensor. Moreover, we assume that some components of $W$ are non-zero, so that $$\sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 > 0.$$ For abbreviation, we put $$H_{ik}(x) = \sum_{p,q=1}^n W_{ipkq} \, x_p \, x_q$$ and $$\overline{H}_{ik}(x) = f(|x|^2) \, H_{ik}(x),$$ where $f(s) = \tau + 5s - s^2 + \frac{1}{20} \, s^3$. It is easy to see that $H_{ik}(x)$ is trace-free, $\sum_{i=1}^n x_i \, H_{ik}(x) = 0$, and $\sum_{i=1}^n \partial_i H_{ik}(x) = 0$ for all $x \in \mathbb{R}^n$.\ We consider a Riemannian metric of the form $g(x) = \exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on $\mathbb{R}^n$ satisfying $h(x) = 0$ for $|x| \geq 1$, $$|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha_1$$ for all $x \in \mathbb{R}^n$, and $$h_{ik}(x) = \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x)$$ for $|x| \leq \rho$. We assume that the parameters $\lambda$, $\mu$, and $\rho$ are chosen such that $\mu \leq 1$ and $\lambda \leq \rho \leq 1$. Note that $\sum_{i=1}^n x_i \, h_{ik}(x) = 0$ and $\sum_{i=1}^n \partial_i h_{ik}(x) = 0$ for $|x| \leq \rho$. Given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$, there exists a unique function $v_{(\xi,\varepsilon)}$ such that $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$ and $$\int_{\mathbb{R}^n} \Big ( \langle dv_{(\xi,\varepsilon)},d\psi \rangle_g + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \, \psi - n(n-2) \, |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)} \, \psi \Big ) = 0$$ for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$ (see Proposition \[fixed.point.argument\]). For abbreviation, let $$\Omega = \bigg \{ (\xi,\varepsilon) \in \mathbb{R}^n \times \mathbb{R}: |\xi| < 1, \, \frac{1}{2} < \varepsilon < 2 \bigg \}.$$ The following result is proved in the Appendix A of [@Brendle1]. A similar formula is derived in [@Ambrosetti-Malchiodi]. \[Taylor.expansion.of.scalar.curvature\] Consider a Riemannian metric on $\mathbb{R}^n$ of the form $g(x) = \exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on $\mathbb{R}^n$ satisfying $|h(x)| \leq 1$ for all $x \in \mathbb{R}^n$. Let $R_g$ be the scalar curvature of $g$. There exists a constant $C$, depending only on $n$, such that $$\begin{aligned} &\Big | R_g - \partial_i \partial_k h_{ik} + \partial_i(h_{il} \, \partial_k h_{kl}) - \frac{1}{2} \, \partial_i h_{il} \, \partial_k h_{kl} + \frac{1}{4} \, \partial_l h_{ik} \, \partial_l h_{ik} \Big | \\ &\leq C \, |h|^2 \, |\partial^2 h| + C \, |h| \, |\partial h|^2.\end{aligned}$$ \[estimate.for.error.term\] Assume that $(\xi,\varepsilon) \in \lambda \, \Omega$. Then we have $$\begin{aligned} &\Big \| \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\ &\leq C \, \lambda^8 \, \mu + C \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}\end{aligned}$$ and $$\begin{aligned} &\Big \| \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \\ &\hspace{10mm} + \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\ &\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.\end{aligned}$$ **Proof.** Note that $\sum_{i=1}^n \partial_i h_{ik}(x) = 0$ for $|x| \leq \rho$. Hence, it follows from Proposition \[Taylor.expansion.of.scalar.curvature\] that $$|R_g(x)| \leq C \, |h(x)|^2 \, |\partial^2 h(x)| + C \, |\partial h(x)|^2 \leq C \, \mu^2 \, (\lambda + |x|)^{14}$$ for $|x| \leq \rho$. This implies $$\begin{aligned} &\Big | \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big | \\ &= \Big | \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik}) \, \partial_k u_{(\xi,\varepsilon)} \big ] - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} \Big | \\ &\leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda + |x|)^{8-n}\end{aligned}$$ and $$\begin{aligned} &\Big | \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} + \sum_{i,k=1}^n h_{ik} \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big | \\ &= \Big | \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik} + h_{ik}) \, \partial_k u_{(\xi,\varepsilon)} \big ] - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} \Big | \\ &\leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda + |x|)^{16-n} \\ &\leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda + |x|)^{\frac{8(n+2)}{n-2}-n}\end{aligned}$$ for $|x| \leq \rho$. From this the assertion follows.\ \[estimate.for.v.1\] The function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}$ satisfies the estimate $$\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \lambda^8 \, \mu + C \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}$$ whenever $(\xi,\varepsilon) \in \lambda \, \Omega$. **Proof.** It follows from Proposition \[fixed.point.argument\] that $$\begin{aligned} &\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\ &\leq C \, \Big \| \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)},\end{aligned}$$ where $C$ is a constant that depends only on $n$. Hence, the assertion follows from Proposition \[estimate.for.error.term\].\ We next establish a more precise estimate for the function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}$. Applying Proposition \[linearized.operator\] with $h = 0$, we conclude that there exists a unique function $w_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$ such that $$\begin{aligned} &\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle - n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \, \psi \Big ) \\ &= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \, \psi\end{aligned}$$ for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. \[properties.of.w\] The function $w_{(\xi,\varepsilon)}$ is smooth. Moreover, if $(\xi,\varepsilon) \in \lambda \, \Omega$, then the function $w_{(\xi,\varepsilon)}$ satisfies the estimates $$\begin{aligned} &|w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda + |x|)^{10-n} \\ &|\partial w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda + |x|)^{9-n} \\ &|\partial^2 w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda + |x|)^{8-n}\end{aligned}$$ for all $x \in \mathbb{R}^n$. **Proof.** There exist real numbers $b_k(\xi,\varepsilon)$ such that $$\begin{aligned} &\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle - n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \, \psi \Big ) \\ &= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \, \psi \\ &+ \sum_{k=0}^n b_k(\xi,\varepsilon) \, \int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \, \psi\end{aligned}$$ for all test functions $\psi \in \mathcal{E}$. Hence, standard elliptic regularity theory implies that $w_{(\xi,\varepsilon)}$ is smooth. It remains to prove quantitative estimates for $w_{(\xi,\varepsilon)}$. To that end, we consider a pair $(\xi,\varepsilon) \in \lambda \, \Omega$. One readily verifies that $$\Big \| \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \, \lambda^8 \, \mu.$$ As a consequence, the function $w_{(\xi,\varepsilon)}$ satisfies $\|w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \lambda^8 \, \mu$. Moreover, we have $\sum_{k=0}^n |b_k(\xi,\varepsilon)| \leq C \, \lambda^8 \, \mu$. This implies $$\begin{aligned} &\big | \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big | \\ &= \bigg | \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} - \sum_{k=0}^n b_k(\xi,\varepsilon) \, \int_{\mathbb{R}^n} \varphi_{(\xi,\varepsilon,k)} \bigg | \\ &\leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda+|x|)^{8-n} \end{aligned}$$ for all $x \in \mathbb{R}^n$. We claim that $$\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^8 \, \mu.$$ To show this, we fix a point $x_0 \in \mathbb{R}^n$. Let $r = \frac{1}{2} \, (\lambda + |x_0|)$. Then $$u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \leq C \, r^{-2}$$ and $$\big | \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big | \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, r^{8-n}$$ for all $x \in B_r(x_0)$. Hence, it follows from standard interior estimates that $$\begin{aligned} r^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x_0)| &\leq C \, \|w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(B_r(x_0))} \\ &+ C \, r^{\frac{n+2}{2}} \, \big \| \Delta w_{(\xi,\varepsilon)} + n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \big \|_{L^\infty(B_r(x_0))} \\ &\leq C \, \lambda^8 \, \mu + C \, \lambda^{\frac{n-2}{2}} \, \mu \, r^{-\frac{n-18}{2}} \\ &\leq C \, \lambda^8 \, \mu. \end{aligned}$$ Therefore, we have $$\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^8 \, \mu,$$ as claimed. Since $\sup_{x \in \mathbb{R}^n} |x|^{\frac{n-2}{2}} \, |w_{(\xi,\varepsilon)}(x)| < \infty$, we can express the function $w_{(\xi,\varepsilon)}$ in the form $$\label{convolution.formula} w_{(\xi,\varepsilon)}(x) = -\frac{1}{(n-2) \, |S^{n-1}|} \int_{\mathbb{R}^n} |x - y|^{2-n} \, \Delta w_{(\xi,\varepsilon)}(y) \, dy$$ for all $x \in \mathbb{R}^n$. We are now able to use a bootstrap argument to prove the desired estimate for $w_{(\xi,\varepsilon)}$. It follows from (\[convolution.formula\]) that $$\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^\beta \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\beta+2} \, |\Delta w_{(\xi,\varepsilon)}(x)|$$ for all $0 < \beta < n-2$. Since $$\begin{aligned} |\Delta w_{(\xi,\varepsilon)}(x)| &\leq n(n+2) \, u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \, |w_{(\xi,\varepsilon)}(x)| \\ &+ C \, \lambda^{\frac{n-2}{2}} \, \mu \, (\lambda+|x|)^{8-n} \end{aligned}$$ for all $x \in \mathbb{R}^n$, we conclude that $$\begin{aligned} \sup_{x \in \mathbb{R}^n} (\lambda + |x|)^\beta \, |w_{(\xi,\varepsilon)}(x)| &\leq C \, \lambda^2 \, \sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{\beta-2} \, |w_{(\xi,\varepsilon)}(x)| \\ &+ C \, \lambda^{\beta-\frac{n-18}{2}} \, \mu\end{aligned}$$ for all $0 < \beta \leq n-10$. Iterating this inequality, we obtain $$\sup_{x \in \mathbb{R}^n} (\lambda + |x|)^{n-10} \, |w_{(\xi,\varepsilon)}(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu.$$ The estimates for the first and second derivatives of $w_{(\xi,\varepsilon)}$ follow now from standard interior estimates.\ \[estimate.for.v.2\] The function $v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}$ satisfies the estimate $$\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}$$ whenever $(\xi,\varepsilon) \in \lambda \, \Omega$. **Proof.** Consider the functions $$B_1 = \sum_{i,k=1}^n \partial_i \big [ (g^{ik} - \delta_{ik}) \, \partial_k w_{(\xi,\varepsilon)} \big ] - \frac{n-2}{4(n-1)} \, R_g \, w_{(\xi,\varepsilon)}$$ and $$B_2 = \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)}.$$ By definition of $w_{(\xi,\varepsilon)}$, we have $$\begin{aligned} &\int_{\mathbb{R}^n} \Big ( \langle dw_{(\xi,\varepsilon)},d\psi \rangle_g + \frac{n-2}{4(n-1)} \, R_g \, w_{(\xi,\varepsilon)} \, \psi - n(n+2) \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, w_{(\xi,\varepsilon)} \, \psi \Big ) \\ &= -\int_{\mathbb{R}^n} (B_1 + B_2) \, \psi\end{aligned}$$ for all functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Since $w_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$, it follows that $$w_{(\xi,\varepsilon)} = -G_{(\xi,\varepsilon)}(B_1 + B_2).$$ Moreover, we have $$v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} = G_{(\xi,\varepsilon)} \big ( B_3 + n(n-2) \, B_4 \big ),$$ where $$B_3 = \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}}$$ and $$B_4 = |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} - \frac{n+2}{n-2} \, u_{(\xi,\varepsilon)}^{\frac{4}{n-2}} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}).$$ Thus, we conclude that $$v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)} = G_{(\xi,\varepsilon)} \big ( B_1 + B_2 + B_3 + n(n-2) \, B_4 \big ),$$ where $G_{(\xi,\varepsilon)}$ denotes the solution operator constructed in Proposition \[linearized.operator\]. As a consequence, we obtain $$\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \big \| B_1 + B_2 + B_3 + n(n-2) \, B_4 \big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)}.$$ It follows from Proposition \[properties.of.w\] that $$|B_1(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda + |x|)^{16-n} \leq C \, \lambda^{\frac{n-2}{2}} \, \mu^2 \, (\lambda+|x|)^{\frac{8(n+2)}{n-2}-n}$$ for $|x| \leq \rho$ and $$|B_1(x)| \leq C \, \lambda^{\frac{n-2}{2}} \, \mu \, |x|^{8-n}$$ for $|x| \geq \rho$. This implies $$\|B_1\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \rho^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.$$ Moreover, observe that $$\|B_2 + B_3\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^2 + C \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}$$ by Proposition \[estimate.for.error.term\]. Finally, Corollary \[estimate.for.v.1\] implies that $$\begin{aligned} \|B_4\|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} &\leq C \, \|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}^{\frac{n+2}{n-2}} \\ &\leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n+2}{2}}.\end{aligned}$$ Putting these facts together, we obtain $$\|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \lambda^{\frac{8(n+2)}{n-2}} \, \mu^{\frac{n+2}{n-2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}.$$ This completes the proof.\ \[term.1\] We have $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} \Big ( |dv_{(\xi,\varepsilon)}|_g^2 - |du_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \, (v_{(\xi,\varepsilon)}^2 - u_{(\xi,\varepsilon)}^2) \Big ) \\ &\hspace{10mm} + \int_{\mathbb{R}^n} n(n-2) \, (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \, v_{(\xi,\varepsilon)} \\ &\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \, (|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \\ &\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \, w_{(\xi,\varepsilon)} \bigg | \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}\end{aligned}$$ whenever $(\xi,\varepsilon) \in \lambda \, \Omega$. **Proof.** By definition of $v_{(\xi,\varepsilon)}$, we have $$\begin{aligned} &\int_{\mathbb{R}^n} \Big ( |dv_{(\xi,\varepsilon)}|_g^2 - \langle du_{(\xi,\varepsilon)},dv_{(\xi,\varepsilon)} \rangle_g + \frac{n-2}{4(n-1)} \, R_g \, v_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) \Big ) \\ &\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \, |v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} \, v_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) = 0.\end{aligned}$$ Using Proposition \[estimate.for.error.term\] and Corollary \[estimate.for.v.1\], we obtain $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} \Big ( \langle du_{(\xi,\varepsilon)},dv_{(\xi,\varepsilon)} \rangle_g - |du_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) \Big ) \\ &\hspace{10mm} - \int_{\mathbb{R}^n} n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) \\ &\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}) \bigg | \\ &\leq \Big \| \Delta_g u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)} + n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{n+2}{n-2}} \\ &\hspace{10mm} + \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \Big \|_{L^{\frac{2n}{n+2}}(\mathbb{R}^n)} \\ &\hspace{5mm} \cdot \|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.\end{aligned}$$ Moreover, we have $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)} \, (v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}) \bigg | \\ &\leq C \, \lambda^8 \, \mu \, \|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)} - w_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}}\end{aligned}$$ by Corollary \[estimate.for.v.2\]. Putting these facts together, the assertion follows.\ \[term.2\] We have $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \, v_{(\xi,\varepsilon)} - \frac{2}{n} \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \bigg | \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^n\end{aligned}$$ whenever $(\xi,\varepsilon) \in \lambda \, \Omega$. **Proof.** We have the pointwise estimate $$\begin{aligned} &\Big | (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \, v_{(\xi,\varepsilon)} - \frac{2}{n} \, (|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \Big | \\ &\leq C \, |v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}},\end{aligned}$$ where $C$ is a constant that depends only on $n$. This implies $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{4}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{4}{n-2}}) \, u_{(\xi,\varepsilon)} \, v_{(\xi,\varepsilon)} - \frac{2}{n} \int_{\mathbb{R}^n} (|v_{(\xi,\varepsilon)}|^{\frac{2n}{n-2}} - u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}}) \bigg | \\ &\leq C \, \|v_{(\xi,\varepsilon)} - u_{(\xi,\varepsilon)}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)}^{\frac{2n}{n-2}} \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^n \end{aligned}$$ by Corollary \[estimate.for.v.1\].\ \[term.3\] We have $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} \Big ( |du_{(\xi,\varepsilon)}|_g^2 + \frac{n-2}{4(n-1)} \, R_g \, u_{(\xi,\varepsilon)}^2 - n(n-2) \, u_{(\xi,\varepsilon)}^{\frac{2n}{n-2}} \Big ) \\ &\hspace{10mm} - \int_{B_\rho(0)} \frac{1}{2} \, \sum_{i,k,l=1}^n h_{il} \, h_{kl} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \\ &\hspace{10mm} + \int_{B_\rho(0)} \frac{n-2}{16(n-1)} \, \sum_{i,k,l=1}^n (\partial_l h_{ik})^2 \, u_{(\xi,\varepsilon)}^2 \bigg | \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}\end{aligned}$$ whenever $(\xi,\varepsilon) \in \lambda \, \Omega$. **Proof.** Note that $$\begin{aligned} &\Big | g^{ik}(x) - \delta_{ik} + h_{ik}(x) - \frac{1}{2} \, \sum_{l=1}^n h_{il}(x) \, h_{kl}(x) \Big | \\ &\leq C \, |h(x)|^3 \leq C \, \mu^3 \, (\lambda + |x|)^{24} \leq C \, \mu^3 \, (\lambda+|x|)^{\frac{16n}{n-2}}\end{aligned}$$ for $|x| \leq \rho$. This implies $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} \big ( |du_{(\xi,\varepsilon)}|_g^2 - |du_{(\xi,\varepsilon)}|^2 \big ) + \int_{\mathbb{R}^n} \sum_{i,k=1}^n h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \\ &\hspace{10mm} - \int_{B_\rho(0)} \frac{1}{2} \sum_{i,k,l=1}^n h_{il} \, h_{kl} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \bigg | \\ &\leq C \, \lambda^{n-2} \, \mu^3 \, \int_{B_\rho(0)} (\lambda+|x|)^{\frac{16n}{n-2}+2-2n} + C \, \lambda^{n-2} \, \int_{\mathbb{R}^n \setminus B_\rho(0)} (\lambda + |x|)^{2-2n} \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.\end{aligned}$$ By Proposition \[Taylor.expansion.of.scalar.curvature\], the scalar curvature of $g$ satisfies the estimate $$\begin{aligned} &\Big | R_g(x) + \frac{1}{4} \sum_{i,k,l=1}^n (\partial_l h_{ik}(x))^2 \Big | \\ &\leq C \, |h(x)|^2 \, |\partial^2 h(x)| + C \, |h(x)| \, |\partial h(x)|^2 \\ &\leq C \, \mu^3 \, (\lambda + |x|)^{22} \leq C \, \mu^3 \, (\lambda + |x|)^{\frac{16n}{n-2}-2}\end{aligned}$$ for $|x| \leq \rho$. This implies $$\begin{aligned} &\bigg | \int_{\mathbb{R}^n} R_g \, u_{(\xi,\varepsilon)}^2 + \int_{B_\rho(0)} \frac{1}{4} \sum_{i,k,l=1}^n (\partial_l h_{ik})^2 \, u_{(\xi,\varepsilon)}^2 \bigg | \\ &\leq C \, \lambda^{n-2} \, \mu^3 \, \int_{B_\rho(0)} (\lambda+|x|)^{\frac{16n}{n-2}+2-2n} + C \, \lambda^{n-2} \, \int_{\mathbb{R}^n \setminus B_\rho(0)} (\lambda + |x|)^{4-2n} \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^3 + C \, \rho^2 \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.\end{aligned}$$ At this point, we use the formula $$\begin{aligned} &\partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} - \frac{n-2}{4(n-1)} \, \partial_i \partial_k (u_{(\xi,\varepsilon)}^2) \\ &= \frac{1}{n} \, \Big ( |du_{(\xi,\varepsilon)}|^2 - \frac{n-2}{4(n-1)} \, \Delta (u_{(\xi,\varepsilon)}^2) \Big ) \, \delta_{ik}.\end{aligned}$$ Since $h_{ik}$ is trace-free, we obtain $$\sum_{i,k=1}^n h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} = \frac{n-2}{4(n-1)} \sum_{i,k=1}^n h_{ik} \, \partial_i \partial_k (u_{(\xi,\varepsilon)}^2),$$ hence $$\int_{\mathbb{R}^n} \sum_{i,k=1}^n h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} = \int_{\mathbb{R}^n} \frac{n-2}{4(n-1)} \sum_{i,k=1}^n \partial_i \partial_k h_{ik} \, u_{(\xi,\varepsilon)}^2.$$ Since $\sum_{i=1}^n \partial_i h_{ik}(x) = 0$ for $|x| \leq \rho$, it follows that $$\bigg | \int_{\mathbb{R}^n} \sum_{i,k=1}^n h_{ik} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \bigg | \leq C \int_{\mathbb{R}^n \setminus B_\rho(0)} u_{(\xi,\varepsilon)}^2 \leq C \, \rho^2 \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}.$$ Putting these facts together, the assertion follows.\ \[key.estimate\] The function $\mathcal{F}_g(\xi,\varepsilon)$ satisfies the estimate $$\begin{aligned} &\bigg | \mathcal{F}_g(\xi,\varepsilon) - \int_{B_\rho(0)} \frac{1}{2} \, \sum_{i,k,l=1}^n h_{il} \, h_{kl} \, \partial_i u_{(\xi,\varepsilon)} \, \partial_k u_{(\xi,\varepsilon)} \\ &\hspace{10mm} + \int_{B_\rho(0)} \frac{n-2}{16(n-1)} \, \sum_{i,k,l=1}^n (\partial_l h_{ik})^2 \, u_{(\xi,\varepsilon)}^2 \\ &\hspace{10mm} - \int_{\mathbb{R}^n} \sum_{i,k=1}^n \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x) \, \partial_i\partial_k u_{(\xi,\varepsilon)} \, w_{(\xi,\varepsilon)} \bigg | \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}\end{aligned}$$ whenever $(\xi,\varepsilon) \in \lambda \, \Omega$. Finding a critical point of an auxiliary function ================================================= We define a function $F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ as follows: given any pair $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$, we define $$\begin{aligned} F(\xi,\varepsilon) &= \int_{\mathbb{R}^n} \frac{1}{2} \sum_{i,k,l=1}^n \overline{H}_{il}(x) \, \overline{H}_{kl}(x) \, \partial_i u_{(\xi,\varepsilon)}(x) \, \partial_k u_{(\xi,\varepsilon)}(x) \\ &- \int_{\mathbb{R}^n} \frac{n-2}{16(n-1)} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, u_{(\xi,\varepsilon)}(x)^2 \\ &+ \int_{\mathbb{R}^n} \sum_{i,k=1}^n \overline{H}_{ik}(x) \, \partial_i \partial_k u_{(\xi,\varepsilon)}(x) \, z_{(\xi,\varepsilon)}(x),\end{aligned}$$ where $z_{(\xi,\varepsilon)} \in \mathcal{E}_{(\xi,\varepsilon)}$ satisfies the relation $$\begin{aligned} &\int_{\mathbb{R}^n} \Big ( \langle dz_{(\xi,\varepsilon)},d\psi \rangle - n(n+2) \, u_{(\xi,\varepsilon)}(x)^{\frac{4}{n-2}} \, z_{(\xi,\varepsilon)} \, \psi \Big ) \\ &= -\int_{\mathbb{R}^n} \sum_{i,k=1}^n \overline{H}_{ik} \, \partial_i \partial_k u_{(\xi,\varepsilon)} \, \psi\end{aligned}$$ for all test functions $\psi \in \mathcal{E}_{(\xi,\varepsilon)}$. Our goal in this section is to show that the function $F(\xi,\varepsilon)$ has a critical point. \[symmetry\] The function $F(\xi,\varepsilon)$ satisfies $F(\xi,\varepsilon) = F(-\xi,\varepsilon)$ for all $(\xi,\varepsilon) \in \mathbb{R}^n \times (0,\infty)$. Consequently, we have $\frac{\partial}{\partial \xi_p} F(0,\varepsilon) = 0$ and $\frac{\partial^2}{\partial \varepsilon \, \partial \xi_p} F(0,\varepsilon) = 0$ for all $\varepsilon > 0$ and $p = 1, \hdots, n$. **Proof.** This follows immediately from the relation $\overline{H}_{ik}(-x) = \overline{H}_{ik}(x)$.\ \[integral.identity.1\] We have $$\begin{aligned} &\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \, x_p \, x_q \\ &= \frac{2}{n(n+2)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+3} \\ &+ \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \, r^{n+3}\end{aligned}$$ and $$\begin{aligned} &\int_{\partial B_r(0)} \sum_{i,k=1}^n H_{ik}(x)^2 \, x_p \, x_q \\ &= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+5} \\ &+ \frac{1}{2n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \, r^{n+5}.\end{aligned}$$ **Proof.** See [@Brendle1], Proposition 16.\ \[integral.identity.2\] We have $$\begin{aligned} &\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, x_p \, x_q \\ &= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\ &\hspace{10mm} \cdot r^{n+3} \, \Big [ (n+4) \, f(r^2)^2 + 8r^2 \, f(r^2) \, f'(r^2) + 4r^4 \, f'(r^2)^2 \Big ] \\ &+ \frac{1}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\ &\hspace{10mm} \cdot r^{n+3} \, \Big [ (n+4) \, f(r^2)^2 + 4r^2 \, f(r^2) \, f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ].\end{aligned}$$ **Proof.** Using the identity $$\partial_l \overline{H}_{ik}(x) = f(|x|^2) \, \partial_l H_{ik}(x) + 2 \, f'(|x|^2) \, H_{ik}(x) \, x_l$$ and Euler’s theorem, we obtain $$\begin{aligned} &\sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \\ &= f(|x|^2)^2 \, \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \\ &+ 4 \, f(|x|^2) \, f'(|x|^2) \, \sum_{i,k,l=1}^n H_{ik}(x) \, x_l \, \partial_l H_{ik}(x) \\ &+ 4 \, |x|^2 \, f'(|x|^2)^2 \, \sum_{i,k=1}^n H_{ik}(x)^2 \\ &= f(|x|^2)^2 \, \sum_{i,k,l=1}^n (\partial_l H_{ik}(x))^2 \\ &+ \big [ 8 \, f(|x|^2) \, f'(|x|^2) + 4 \, |x|^2 \, f'(|x|^2)^2 \big ] \, \sum_{i,k=1}^n H_{ik}(x)^2.\end{aligned}$$ Hence, the assertion follows from the previous proposition.\ \[integral.identity.3\] We have $$\begin{aligned} &\int_{\partial B_r(0)} \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \\ &= \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\ &\hspace{10mm} \cdot r^{n+1} \, \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \, f'(r^2) + 2 \, r^4 \, f'(r^2)^2 \Big ].\end{aligned}$$ \[formula.for.F\] We have $$\begin{aligned} F(0,\varepsilon) &= -\frac{n-2}{16n(n-1)(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \, r^{n+1} \\ &\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \, f'(r^2) + 2 \, r^4 \, f'(r^2)^2 \Big ] \, dr.\end{aligned}$$ **Proof.** Note that $z_{(0,\varepsilon)}(x) = 0$ for all $x \in \mathbb{R}^n$. This implies $$F(0,\varepsilon) = -\int_{\mathbb{R}^n} \frac{n-2}{16(n-1)} \, \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{2-n} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2.$$ Using Corollary \[integral.identity.3\], we obtain $$\begin{aligned} &\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{2-n} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \\ &= \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \, r^{n+1} \\ &\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \, f'(r^2) + 2 \, r^4 \, f'(r^2)^2 \Big ].\end{aligned}$$ This proves the assertion.\ \[i\] The function $F(0,\varepsilon)$ can be written in the form $$\begin{aligned} F(0,\varepsilon) &= -\frac{n-2}{16n(n-1)(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \\ &\hspace{10mm} \cdot I(\varepsilon^2) \, \int_0^\infty (1 + r^2)^{2-n} \, r^{n+7} \, dr,\end{aligned}$$ where $$\begin{aligned} I(s) &= \frac{n-12}{n+6} \, \frac{n-10}{n+4} \, (n-8) \, \tau^2 \, s^2 + 10 \, \frac{n-12}{n+6} \, (n-10) \, \tau \, s^3 \\ &+ \Big ( 25 \, \frac{n-12}{n+6} \, (n+8) - 2(n-12) \, \tau \Big ) \, s^4 + \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, s^5 \\ &+ \frac{n+8}{n-14} \, \frac{3n+52}{2} \, s^6 - \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+24}{10} \, s^7 \\ &+ \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+12}{n-18} \, \frac{n+32}{400} \, s^8.\end{aligned}$$ **Proof.** It is straightforward to check that $$\begin{aligned} &(n+2) \, f(s)^2 + 4s \, f(s) \, f'(s) + 2s^2 \, f'(s)^2 \\ &= (n+2)\tau^2 + 10(n+4)\tau \, s + \Big ( 25(n+8) - 2(n+6)\tau \Big ) \, s^2 \\ &+ \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, s^3 + \frac{3n+52}{2} \, s^4 - \frac{n+24}{10} \, s^5 + \frac{n+32}{400} \, s^6.\end{aligned}$$ This implies $$\begin{aligned} &\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \, r^{n+1} \\ &\hspace{10mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \, f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ] \, dr \\ &= (n+2)\tau^2 \, \varepsilon^4 \int_0^\infty (1+r^2)^{2-n} \, r^{n+1} \, dr \\ &+ 10(n+4)\tau \, \varepsilon^6 \int_0^\infty (1+r^2)^{2-n} \, r^{n+3} \, dr \\ &+ \Big ( 25(n+8) - 2(n+6)\tau \Big ) \, \varepsilon^8 \int_0^\infty (1+r^2)^{2-n} \, r^{n+5} \, dr \\ &+ \Big ( \frac{n+8}{10} \, \tau - 10(n+12) \Big ) \, \varepsilon^{10} \int_0^\infty (1+r^2)^{2-n} \, r^{n+7} \, dr \\ &+ \frac{3n+52}{2} \, \varepsilon^{12} \int_0^\infty (1+r^2)^{2-n} \, r^{n+9} \, dr \\ &- \frac{n+24}{10} \, \varepsilon^{14} \int_0^\infty (1+r^2)^{2-n} \, r^{n+11} \, dr \\ &+ \frac{n+32}{400} \, \varepsilon^{16} \int_0^\infty (1+r^2)^{2-n} \, r^{n+13} \, dr.\end{aligned}$$ Using the identity $$\int_0^\infty (1+r^2)^{2-n} \, r^{\beta+2} \, dr = \frac{\beta+1}{2n-\beta-7} \int_0^\infty (1+r^2)^{2-n} \, r^\beta \, dr,$$ we obtain $$\begin{aligned} &\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{2-n} \, r^{n+1} \\ &\hspace{10mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \, f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ] \, dr \\ &= I(\varepsilon^2) \, \int_0^\infty (1+r^2)^{2-n} \, r^{n+7} \, dr.\end{aligned}$$ This completes the proof.\ In the next step, we compute the Hessian of $F$ at $(0,\varepsilon)$.\ \[Hessian.of.F.1\] The second order partial derivatives of the function $F(\xi,\varepsilon)$ are given by $$\begin{aligned} \frac{\partial^2}{\partial \xi_p \, \partial \xi_q} F(0,\varepsilon) &= \int_{\mathbb{R}^n} (n-2)^2 \, \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{-n} \, \sum_{l=1}^n \overline{H}_{pl}(x) \, \overline{H}_{ql}(x) \\ &- \int_{\mathbb{R}^n} \frac{(n-2)^2}{4} \, \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{-n} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, x_p \, x_q \\ &+ \int_{\mathbb{R}^n} \frac{(n-2)^2}{8(n-1)} \, \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{1-n} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, \delta_{pq}.\end{aligned}$$ **Proof.** See [@Brendle1], Proposition 21.\ \[Hessian.of.F.2\] The second order partial derivatives of the function $F(\xi,\varepsilon)$ are given by $$\begin{aligned} &\frac{\partial^2}{\partial \xi_p \, \partial \xi_q} F(0,\varepsilon) \\ &= -\frac{2(n-2)^2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\ &- \frac{(n-2)^2}{2n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\ &+ \frac{(n-2)^2}{4n(n-1)(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{1-n} \, r^{n+5} \, f'(r^2)^2 \, dr.\end{aligned}$$ **Proof.** Using the identity $$\begin{aligned} &\int_{\partial B_r(0)} \sum_{l=1}^n \overline{H}_{pl}(x) \, \overline{H}_{ql}(x) \\ &= \int_{\partial B_r(0)} \sum_{i,j,k,l,m=1}^n W_{ipkl} \, W_{jqml} \, x_i \, x_j \, x_k \, x_m \, f(|x|^2)^2 \\ &= \frac{1}{n(n+2)} \, |S^{n-1}| \\ &\hspace{10mm} \cdot \sum_{i,j,k,l,m=1}^n W_{ipkl} \, W_{jqml} \, (\delta_{ij} \, \delta_{km} + \delta_{ik} \, \delta_{jm} + \delta_{im} \, \delta_{jk}) \, r^{n+3} \, f(r^2)^2 \\ &= \frac{1}{2n(n+2)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \, r^{n+3} \, f(r^2)^2,\end{aligned}$$ we obtain $$\begin{aligned} &\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{-n} \, \sum_{i,k,l=1}^n \overline{H}_{pl}(x) \, \overline{H}_{ql}(x) \\ &= \frac{1}{2n(n+2)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3} \, f(r^2)^2 \, dr.\end{aligned}$$ Similarly, it follows from Proposition \[integral.identity.2\] that $$\begin{aligned} &\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{-n} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, x_p \, x_q \\ &= \frac{2}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,k,l=1}^n (W_{ipkl} + W_{ilkp}) \, (W_{iqkl} + W_{ilkq}) \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3} \\ &\hspace{15mm} \cdot \Big [ (n+4) \, f(r^2)^2 + 8r^2 \, f(r^2) \, f'(r^2) + 4r^4 \, f'(r^2)^2 \Big ] \, dr \\ &+ \frac{1}{n(n+2)(n+4)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3} \\ &\hspace{15mm} \cdot \Big [ (n+4) \, f(r^2)^2 + 4r^2 \, f(r^2) \, f'(r^2) + 2r^4 \, f'(r^2)^2 \Big ] \, dr.\end{aligned}$$ Moreover, we have $$\begin{aligned} &\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{1-n} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, \delta_{pq} \\ &= \frac{1}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{1-n} \, r^{n+1} \\ &\hspace{15mm} \cdot \Big [ (n+2) \, f(r^2)^2 + 4 \, r^2 \, f(r^2) \, f'(r^2) + 2 \, r^4 \, f'(r^2)^2 \Big ] \, dr\end{aligned}$$ by Corollary \[integral.identity.3\]. A straightforward calculation yields $$\begin{aligned} &(\varepsilon^2+r^2)^{1-n} \, r^{n+1} \, \big [ (n+2) \, f(r^2)^2 + 4r^2 \, f(r^2) \, f'(r^2) \big ] \\ &= 2(n-1) \, (\varepsilon^2+r^2)^{-n} \, r^{n+3} \, f(r^2)^2 + \frac{d}{dr} \big [ (\varepsilon^2+r^2)^{1-n} \, r^{n+2} \, f(r^2)^2 \big ].\end{aligned}$$ This implies $$\begin{aligned} &\int_{\mathbb{R}^n} \varepsilon^{n-2} \, (\varepsilon^2 + |x|^2)^{1-n} \, \sum_{i,k,l=1}^n (\partial_l \overline{H}_{ik}(x))^2 \, \delta_{pq} \\ &= \frac{2(n-1)}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+3} \, f(r^2)^2 \, dr \\ &+ \frac{2}{n(n+2)} \, |S^{n-1}| \, \sum_{i,j,k,l=1}^n (W_{ijkl} + W_{ilkj})^2 \, \delta_{pq} \\ &\hspace{5mm} \cdot \int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{1-n} \, r^{n+5} \, f'(r^2)^2 \, dr.\end{aligned}$$ Putting these facts together, the assertion follows.\ \[j\] We have $$\begin{aligned} &\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\ &= J(\varepsilon^2) \, \int_0^\infty (1+r^2)^{-n} \, r^{n+9} \, dr,\end{aligned}$$ where $$\begin{aligned} J(s) &= 10 \, \frac{n-10}{n+8} \, \frac{n-8}{n+6} \, \tau \, s^2 + \frac{n-10}{n+8} \, (75-4\tau) \, s^3 \\ &+ \Big ( \frac{3}{10} \, \tau - 50 \Big ) \, s^4 + \frac{23}{2} \, \frac{n+10}{n-12} \, s^5 - \frac{11}{10} \, \frac{n+10}{n-12} \, \frac{n+12}{n-14} \, s^6 \\ &+ \frac{3}{80} \, \frac{n+10}{n-12} \, \frac{n+12}{n-14} \, \frac{n+14}{n-16} \, s^7.\end{aligned}$$ **Proof.** Note that $$\begin{aligned} &2 \, f(s) \, f'(s) + s \, f'(s)^2 \\ &= 10\tau + (75-4\tau) \, s + \Big ( \frac{3}{10} \, \tau - 50 \Big ) \, s^2 + \frac{23}{2} \, s^3 - \frac{11}{10} \, s^4 + \frac{3}{80} \, s^5.\end{aligned}$$ This implies $$\begin{aligned} &\int_0^\infty \varepsilon^{n-2} \, (\varepsilon^2 + r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr \\ &= 10\tau \, \varepsilon^4 \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+5} \, dr \\ &+ (75-4\tau) \, \varepsilon^6 \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+7} \, dr \\ &+ \Big ( \frac{3}{10} \, \tau - 50 \Big ) \, \varepsilon^8 \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+9} \, dr \\ &+ \frac{23}{2} \, \varepsilon^{10} \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+11} \, dr \\ &- \frac{11}{10} \, \varepsilon^{12} \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+13} \, dr \\ &+ \frac{3}{80} \, \varepsilon^{14} \, \int_0^\infty (1 + r^2)^{-n} \, r^{n+15} \, dr.\end{aligned}$$ Hence, the assertion follows from the identity $$\int_0^\infty (1+r^2)^{-n} \, r^{\beta+2} \, dr = \frac{\beta+1}{2n-\beta-3} \int_0^\infty (1+r^2)^{-n} \, r^\beta \, dr.$$ Assume that $25 \leq n \leq 51$. Then we can choose $\tau \in \mathbb{R}$ such that $I'(1) = 0$, $I''(1) < 0$, and $J(1) < 0$. **Proof.** The condition $I'(1) = 0$ is equivalent to $$a_n \, \tau^2 + b_n \, \tau + c_n = 0,$$ where $$\begin{aligned} a_n &= 2 \, \frac{n-12}{n+6} \, \frac{n-10}{n+4} \, (n-8) \\ b_n &= 30 \, \frac{n-12}{n+6} \, (n-10) - 8(n-12) + \frac{n+8}{2} \\ c_n &= 100 \, \frac{n-12}{n+6} \, (n+8) - 50(n+12) + 3 \, \frac{n+8}{n-14} \, (3n+52) \\ &- 7 \, \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+24}{10} + \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+12}{n-18} \, \frac{n+32}{50}.\end{aligned}$$ By inspection, one verifies that $49 \, a_n - 7 \, b_n + c_n < 0$ for $25 \leq n \leq 51$. Since $a_n$ is positive, there exists a unique real number $\tau < -7$ such that $a_n \, \tau^2 + b_n \, \tau + c_n = 0$. Moreover, we have $$I''(1) - I'(1) = \alpha_n \, \tau + \beta_n$$ and $$J(1) = \gamma_n \, \tau + \delta_n,$$ where $$\begin{aligned} \alpha_n &= 30 \, \frac{n-12}{n+6} \, (n-10) - 16(n-12) + \frac{3(n+8)}{2} \\ \beta_n &= 200 \, \frac{n-12}{n+6} \, (n+8) - 150(n+12) + 12 \, \frac{n+8}{n-14} \, (3n+52) \\ &- 35 \, \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+24}{10} + 3 \, \frac{n+8}{n-14} \, \frac{n+10}{n-16} \, \frac{n+12}{n-18} \, \frac{n+32}{25} \\[2mm] \gamma_n &= 10 \, \frac{n-10}{n+8} \, \frac{n-8}{n+6} - \frac{4(n-10)}{n+8} + \frac{3}{10} \\ \delta_n &= 75 \, \frac{n-10}{n+8} - 50 + \frac{23}{2} \, \frac{n+10}{n-12} - \frac{11}{10} \, \frac{n+10}{n-12} \, \frac{n+12}{n-14} \\ &+ \frac{3}{80} \, \frac{n+10}{n-12} \, \frac{n+12}{n-14} \, \frac{n+14}{n-16}.\end{aligned}$$ By inspection, one verifies that $7\alpha_n > \beta_n > 0$ and $7\gamma_n > \delta_n > 0$ for $25 \leq n \leq 51$. This implies $I''(1) = \alpha_n \, \tau + \beta_n < -7\alpha_n + \beta_n < 0$ and $J(1) = \gamma_n \, \tau + \delta_n < -7\gamma_n + \delta_n < 0$. This completes the proof.\ \[strict.local.minimum\] Assume that $\tau$ is chosen such that $I'(1) = 0$, $I''(1) < 0$, and $J(1) < 0$. Then the function $F(\xi,\varepsilon)$ has a strict local minimum at $(0,1)$. **Proof.** Since $I'(1) = 0$, we have $\frac{\partial}{\partial \varepsilon} F(0,1) = 0$. Therefore, $(0,1)$ is a critical point of the function $F(\xi,\varepsilon)$. Since $J(1) < 0$, we have $$\int_0^\infty (1 + r^2)^{-n} \, r^{n+5} \, \Big [ 2 \, f(r^2) \, f'(r^2) + r^2 \, f'(r^2)^2 \Big ] \, dr < 0$$ by Proposition \[j\]. Hence, it follows from Proposition \[Hessian.of.F.2\] that the matrix $\frac{\partial^2}{\partial \xi_p \, \partial \xi_q} F(0,1)$ is positive definite. Using Proposition \[i\] and the inequality $I''(0) < 0$, we obtain $\frac{\partial^2}{\partial \varepsilon^2} F(0,1) > 0$. Consequently, the function $F(\xi,\varepsilon)$ has a strict local minimum at $(0,1)$.\ Proof of the main theorem ========================= \[perturbation.argument\] Assume that $25 \leq n \leq 51$. Moreover, let $g$ be a smooth metric on $\mathbb{R}^n$ of the form $g(x) = \exp(h(x))$, where $h(x)$ is a trace-free symmetric two-tensor on $\mathbb{R}^n$ such that $|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha \leq \alpha_1$ for all $x \in \mathbb{R}^n$, $h(x) = 0$ for $|x| \geq 1$, and $$h_{ik}(x) = \mu \, \lambda^6 \, f(\lambda^{-2} \, |x|^2) \, H_{ik}(x)$$ for $|x| \leq \rho$. If $\alpha$ and $\rho^{2-n} \, \mu^{-2} \, \lambda^{n-18}$ are sufficiently small, then there exists a positive function $v$ such that $$\Delta_g v - \frac{n-2}{4(n-1)} \, R_g \, v + n(n-2) \, v^{\frac{n+2}{n-2}} = 0,$$ $$\int_{\mathbb{R}^n} v^{\frac{2n}{n-2}} < \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}},$$ and $\sup_{|x| \leq \lambda} v(x) \geq c \, \lambda^{\frac{2-n}{2}}$. Here, $c$ is a positive constant that depends only on $n$. **Proof.** By Corollary \[strict.local.minimum\], the function $F(\xi,\varepsilon)$ has a strict local minimum at $(0,1)$. It follows from Proposition \[formula.for.F\] that $F(0,1) < 0$. Hence, we can find an open set $\Omega' \subset \Omega$ such that $(0,1) \in \Omega'$ and $$F(0,1) < \inf_{(\xi,\varepsilon) \in \partial \Omega'} F(\xi,\varepsilon) < 0.$$ Using Corollary \[key.estimate\], we obtain $$\begin{aligned} &|\mathcal{F}_g(\lambda\xi,\lambda\varepsilon) - \lambda^{16} \, \mu^2 \, F(\xi,\varepsilon)| \\ &\leq C \, \lambda^{\frac{16n}{n-2}} \, \mu^{\frac{2n}{n-2}} + C \, \lambda^8 \, \mu \, \Big ( \frac{\lambda}{\rho} \Big )^{\frac{n-2}{2}} + C \, \Big ( \frac{\lambda}{\rho} \Big )^{n-2}\end{aligned}$$ for all $(\xi,\varepsilon) \in \Omega$. This implies $$\begin{aligned} &|\lambda^{-16} \, \mu^{-2} \, \mathcal{F}_g(\lambda\xi,\lambda\varepsilon) - F(\xi,\varepsilon)| \\ &\leq C \, \lambda^{\frac{32}{n-2}} \, \mu^{\frac{4}{n-2}} + C \, \rho^{\frac{2-n}{2}} \, \mu^{-1} \, \lambda^{\frac{n-18}{2}} + C \, \rho^{2-n} \, \mu^{-2} \, \lambda^{n-18}\end{aligned}$$ for all $(\xi,\varepsilon) \in \Omega$. Hence, if $\rho^{2-n} \, \mu^{-2} \, \lambda^{n-18}$ is sufficiently small, then we have $$\mathcal{F}_g(0,\lambda) < \inf_{(\xi,\varepsilon) \in \partial \Omega'} \mathcal{F}_g(\lambda\xi,\lambda\varepsilon) < 0.$$ Consequently, there exists a point $(\bar{\xi},\bar{\varepsilon}) \in \Omega'$ such that $$\mathcal{F}_g(\lambda\bar{\xi},\lambda\bar{\varepsilon}) = \inf_{(\xi,\varepsilon) \in \Omega'} \mathcal{F}_g(\lambda\xi,\lambda\varepsilon) < 0.$$ By Proposition \[reduction.to.a.finite.dimensional.problem\], the function $v = v_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}$ is a non-negative weak solution of the partial differential equation $$\Delta_g v - \frac{n-2}{4(n-1)} \, R_g \, v + n(n-2) \, v^{\frac{n+2}{n-2}} = 0.$$ Using a result of N. Trudinger, we conclude that $v$ is smooth (see [@Trudinger], Theorem 3 on p. 271). Moreover, we have $$\begin{aligned} 2(n-2) \int_{\mathbb{R}^n} v^{\frac{2n}{n-2}} &= 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}} + \mathcal{F}_g(\lambda\bar{\xi},\lambda\bar{\varepsilon}) \\ &< 2(n-2) \, \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}.\end{aligned}$$ Finally, it follows from Proposition \[fixed.point.argument\] that $\|v - u_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^n)} \leq C \, \alpha$. This implies $$|B_\lambda(0)|^{\frac{n-2}{2n}} \, \sup_{|x| \leq \lambda} v(x) \geq \|v\|_{L^{\frac{2n}{n-2}}(B_\lambda(0))} \geq \|u_{(\lambda\bar{\xi},\lambda\bar{\varepsilon})}\|_{L^{\frac{2n}{n-2}}(B_\lambda(0))} - C \, \alpha.$$ Hence, if $\alpha$ is sufficiently small, then we obtain $\lambda^{\frac{n-2}{2}} \, \sup_{|x| \leq \lambda} v(x) \geq c$.\ Let $25 \leq n \leq 51$. Then there exists a smooth metric $g$ on $\mathbb{R}^n$ with the following properties: - $g_{ik}(x) = \delta_{ik}$ for $|x| \geq \frac{1}{2}$ - $g$ is not conformally flat - There exists a sequence of non-negative smooth functions $v_\nu$ ($\nu \in \mathbb{N}$) such that $$\Delta_g v_\nu - \frac{n-2}{4(n-1)} \, R_g \, v_\nu + n(n-2) \, v_\nu^{\frac{n+2}{n-2}} = 0$$ for all $\nu \in \mathbb{N}$, $$\int_{\mathbb{R}^n} v_\nu^{\frac{2n}{n-2}} < \Big ( \frac{Y(S^n)}{4n(n-1)} \Big )^{\frac{n}{2}}$$ for all $\nu \in \mathbb{N}$, and $\sup_{|x| \leq 1} v_\nu(x) \to \infty$ as $\nu \to \infty$. **Proof.** Choose a smooth cutoff function $\eta: \mathbb{R} \to \mathbb{R}$ such that $\eta(t) = 1$ for $t \leq 1$ and $\eta(t) = 0$ for $t \geq 2$. We define a trace-free symmetric two-tensor on $\mathbb{R}^n$ by $$h_{ik}(x) = \sum_{N=N_0}^\infty \eta(4N^2 \, |x - y_N|) \, 2^{-4N} \, f(2^{N} \, |x - y_N|^2) \, H_{ik}(x - y_N),$$ where $y_N = (\frac{1}{N},0,\hdots,0) \in \mathbb{R}^n$. It is straightforward to verify that $h(x)$ is $C^\infty$ smooth. Moreover, if $N_0$ is sufficiently large, then we have $h(x) = 0$ for $|x| \geq \frac{1}{2}$ and $|h(x)| + |\partial h(x)| + |\partial^2 h(x)| \leq \alpha$ for all $x \in \mathbb{R}^n$. (Here, $\alpha$ is the constant that appears in Proposition \[perturbation.argument\].) We now define a Riemannian metric $g$ by $g(x) = \exp(h(x))$. The assertion is then a consequence of Proposition \[perturbation.argument\].\ [99]{} A. Ambrosetti, *Multiplicity results for the Yamabe problem on $S^n$,* Proc. Natl. Acad. Sci. USA 99, 15252–15256 (2002) A. Ambrosetti and A. Malchiodi, *A multiplicity result for the Yamabe problem on $S^n$,* J. Funct. Anal. 168, 529–561 (1999) M. Berti and A. Malchiodi, *Non-compactness and multiplicity results for the Yamabe problem on $S^n$,* J. Funct. Anal. 180, 210–241 (2001) S. Brendle, *Blow-up phenomena for the Yamabe equation,* J. Amer. Math. Soc. 21, 951–979 (2008) S. Brendle, *On the conformal scalar curvature equation and related problems,* Surveys in Differential Geometry (to appear) O. Druet, *Compactness for Yamabe metrics in low dimensions,* Internat. Math. Res. Notices 23, 1143–1191 (2004) O. Druet and E. Hebey, *Blow-up examples for second order elliptic PDEs of critical Sobolev growth,* Trans. Amer. Math. Soc. 357, 1915–1929 (2005) O. Druet and E. Hebey, *Elliptic equations of Yamabe type,* International Mathematics Research Surveys 1, 1–113 (2005) M. Khuri, F.C. Marques and R. Schoen, *A compactness theorem for the Yamabe problem,* J. Diff. Geom. (to appear) Y.Y. Li and L. Zhang, *Compactness of solutions to the Yamabe problem II,* Calc. Var. PDE 24, 185–237 (2005) Y.Y. Li and L. Zhang, *Compactness of solutions to the Yamabe problem III,* J. Funct. Anal. 245, 438–474 (2007) Y.Y. Li and L. Zhu, *Yamabe type equations on three-dimensional Riemannian manifolds,* Commun. Contemp. Math. 1, 1–50 (1999) F.C. Marques, *A-priori estimates for the Yamabe problem in the non-locally conformally flat case,* J. Diff. Geom. 71, 315–346 (2005) M. Obata, *The conjectures on conformal transformations of Riemannian manifolds,* J. Diff. Geom. 6, 247–258 (1972) R.M. Schoen, *Variational theory for the total scalar curvature functional for Riemannian metrics and related topics,* Topics in the calculus of variations (ed. by Mariano Giaquinta), Lecture Notes in Mathematics, vol. 1365, Springer Verlag 1989, 120–154 R.M. Schoen, *On the number of constant scalar curvature metrics in a conformal class,* Differential geometry (ed. by H. Blaine Lawson, Jr., and Keti Tenenblat), Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 52, Longman Scientific & Technical 1991, 311–320 R.M. Schoen, *A report on some recent progress on nonlinear problems in geometry,* In: Surveys in differential geometry, Lehigh University, Bethlehem, PA, 1991, 201–241 N. Trudinger, *Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,* Annali Scuola Norm. Sup. Pisa 22, 265–274 (1968)

--- abstract: 'We present Rabi oscillation measurements of a Nb/AlO$_\text{x}$/Nb dc superconducting quantum interference device (SQUID) phase qubit with a 100 ${\ensuremath{\mu {\ensuremath{\mathrm{m}}}}}^2$ area junction acquired over a range of microwave drive power and frequency detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the qubit dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi oscillations were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed oscillation frequencies with a simplified model constructed from the full phase qubit Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi frequency, and leakage to the higher excited states.' author: - 'S. K. Dutta' - 'Frederick W. Strauch' - 'R. M. Lewis' - Kaushik Mitra - Hanhee Paik - 'T. A. Palomaki' - Eite Tiesinga - 'J. R. Anderson' - 'Alex J. Dragt' - 'C. J. Lobb' - 'F. C. Wellstood' title: Multilevel effects in the Rabi oscillations of a Josephson phase qubit --- Introduction {#SIntro} ============ Over the past decade, a variety of qubits based on superconducting Josephson junctions has been proposed and experimentally realized.[@Makhlin01a; @Devoret04a; @You05a; @Wendin07a; @Zagoskin07a] Among them is the phase qubit, which can take the form of a single current-biased junction,[@Ramos01a; @Yu02a] a flux-biased rf superconducting quantum interference device (SQUID),[@Simmonds04a] a low inductance dc SQUID,[@Claudon04a] a large inductance dc SQUID where one of the junctions is thought of as the qubit,[@Martinis02a] or a vortex in an annular junction.[@Wallraff03c] The behaviors of the devices are similar, as they all employ a large area junction, where the dynamics of the quantum mechanical phase difference across the junction (or the orientation of the vortex in the last case) are determined by a tilted washboard-like potential. The two lowest states in a well of this potential serve as the qubit basis. Among the possible benefits of the phase qubit are its relative insensitivity to charge and flux noise and its ability to operate over a wide range of parameters. Recent demonstrations include fast readout,[@Claudon04a] Rabi oscillations,[@Yu02a; @Martinis02a; @Claudon04a; @Strauch06a; @Lisenfeld07a] and simple capacitive coupling of two qubits as measured through spectroscopy,[@Xu05a] simultaneous state measurement,[@McDermott05a] and state tomography.[@Steffen06b] In the phase qubit, higher excited states can impact the dynamics strongly, due to the nearly harmonic level structure within the potential well. The presence of these quantized energy levels has been clearly detected by monitoring the decay of a thermal population[@Silvestrini97a] and by microwave spectroscopy with single[@Martinis85a] and multiphoton[@Wallraff03a; @Strauch06a] transitions. Multilevel systems can, for example, be exploited for state initialization[@Valenzuela06a] and readout,[@Martinis02a] quantum logic gates[@Amin03a; @Strauch03a; @Wei08a] and algorithms,[@Ahn00a] and cryptography.[@Groblacher06a] However, when controlling the state of the phase qubit with a microwave current, off-resonant excitation of higher levels leads to leakage out of the desired qubit space and therefore a loss of quantum information. Rabi oscillations serve as both a standard demonstration of quantum state manipulation and a diagnostic for decoherence and control fidelity. While these oscillations take on a simple form in a two-level system, several effects can occur in a multilevel system: the oscillations can distort, the higher states can become populated, and the basic resonance properties of the oscillations can undergo subtle shifts. Ultimately, to achieve fast and accurate control of the qubit state, all of these effects need to be carefully characterized. Progress in this area has been predominantly theoretical in nature, pointing out the conditions under which errors are introduced and methods to minimize their effects.[@Goorden03a; @Steffen03a; @Meier05a; @Amin06a; @Strauch06a; @Shevchenko07a] There has, however, been less direct experimental evidence confirming the validity of models upon which these predictions are based,[@Claudon04a; @Claudon07a; @Strauch06a; @Lucero08a] particularly with regard to leakage. In this paper, we investigate the influence of the higher states of a phase qubit by examining the behavior of Rabi oscillations at high power, where their impact is greatest. In Sec. \[SPhase\], we describe the design of our qubit and a Hamiltonian that approximates its dynamics. Also discussed are various experimental details and the readout scheme that allows measurement of very small excited state populations. Section \[SStrong\] describes Rabi oscillations taken at a range of power and detuning from resonance, along with comparisons to a simple theory, while in Sec. \[SDecoherence\] we develop a more complete density-matrix model including the effects of decoherence and noise. Finally, Sec. \[SSummary\] contains a summary of the key points of our findings. SQUID Phase Qubit {#SPhase} ================== ![\[FDevice\]The dc SQUID phase qubit. (a) The qubit junction $J1$ (with critical current $I_{01}$ and capacitance $C_1$) is isolated from the current bias leads by an auxiliary junction $J2$ (with $I_{02}$ and $C_2$) and geometrical inductances $L_1$ and $L_2$. The device is controlled with a current bias $I_b$ and a flux current $I_f$ which generates flux $\Phi_a$ through mutual inductance $M$. Transitions can be induced by a microwave current [$I_{r\!f}$]{}, which is coupled to $J1$ via [$C_{r\!f}$]{}. (b) When biased appropriately, the dynamics of the phase difference $\gamma_1$ across the qubit junction are analogous to those of a ball in a one-dimensional tilted washboard potential $U$. The metastable state [$\left| n \right>$]{} differs in energy from [$\left| m \right>$]{} by $\hbar {\ensuremath{\omega_{n m}}}$ and tunnels to the voltage state with a rate [$\Gamma_{n}$]{}. (c) The photograph shows a Nb/AlO$_\text{x}$/Nb device. Not seen is an identical SQUID coupled to this device intended for two-qubit experiments; the second SQUID was kept unbiased throughout the course of this work.](Devicev3.eps){width="3.0in"} Figure \[FDevice\](a) shows the circuit schematic for our dc SQUID phase qubit.[@Palomaki06a] The qubit junction $J1$ (with critical current $I_{01}$ and capacitance $C_1$) is shown on the left. It is isolated from the current bias source $I_b$ by geometrical inductances $L_1$ and $L_2$ and the second junction $J2$ (with $I_{02}$ and $C_2$). In order to independently control the currents in the two arms of the resulting dc SQUID, a current source $I_f$ applies a flux $\Phi_a$ to the SQUID loop through mutual inductance $M$. Good isolation of the qubit junction is obtained when $L_1 / M \gg 1$ and $L_1 / {\ensuremath{\left( L_2 + L_{J2} \right)}} \gg 1$, where $L_{J2}$ is the Josephson inductance of the isolation junction.[@Martinis02a] For arbitrary values of the bias current $I_b$ and flux current $I_f$, the dynamics of a dc SQUID can be described by 2 degrees of freedom corresponding to the phase differences across each of the junctions. We, however, operate the device by increasing $I_b$ by $\Delta I_b$ while simultaneously increasing $I_f$ by $L_1 \Delta I_b / M$. This nominally keeps the total current through the isolation junction $J2$ near zero, so that the qubit junction current is roughly $I_b$. Furthermore, one obtains a weak dynamical coupling between the junctions by choosing $L_1$ to be large and biasing the SQUID so that the two junctions are well out of resonance with each other.[@Palomaki06a] In this case, the dynamics of the phase difference $\gamma_1$ across the qubit junction are governed to a good approximation[@Mitra06a] by the Hamiltonian of a single current-biased junction,[@Fulton74a; @Leggett87a] $${\ensuremath{\mathcal{H}}}= \frac{4 E_C}{\hbar^2} p_1^2 - E_J {\ensuremath{\left( \cos \gamma_1 + \frac{I_b - {\ensuremath{I_{r\!f}}}\cos {\ensuremath{\omega_{r\!f}}}t}{I_{01}} \gamma_1 \right)}}. \label{eqa}$$ Here, $E_C = e^2 / 2 C_1$ and $E_J = I_{01} {\ensuremath{\Phi_0}}/ 2 \pi$ are the charging energy and Josephson coupling energy of the qubit junction, $p_1 = {\ensuremath{\left( {\ensuremath{\Phi_0}}/ 2 \pi \right)}}^2 C_1 \dot{\gamma_1}$ is the momentum conjugate to $\gamma_1$, and [$I_{r\!f}$]{} is the amplitude of a microwave drive current of frequency [$\omega_{r\!f}$]{}. In quantizing ${\ensuremath{\mathcal{H}}}$, $\gamma_1$ and $p_1$ become operators with ${\ensuremath{\left[ \gamma_1, p_1 \right]}} = i \hbar$. The second term on the right-hand side of [Eq. [(\[eqa\])]{}]{} defines a one-dimensional tilted washboard potential $U$, sketched in Fig. \[FDevice\](b). Each well is characterized by the classical plasma frequency $\omega_p$ and barrier height $\Delta U$, which can also be expressed as the dimensionless quantity $N_s = \Delta U / \hbar \omega_p$. A single potential well supports roughly $N_s$ metastable states [$\left| n \right>$]{}, where the ground state [$\left| 0 \right>$]{} and first excited state [$\left| 1 \right>$]{} serve as the basis for quantum computation.[@Ramos01a] Motivated by this, the Hamiltonian can be expressed in a discrete representation as $${\ensuremath{\mathcal{H}}}_N = \sum_{n=0}^{N-1} \hbar {\ensuremath{\omega_{0 n}}} {\ensuremath{\left| n \right>}} {\ensuremath{\left< n \right|}} + \sum_{n, m = 0}^{N-1} \hbar {\ensuremath{\Omega_{n m}}} {\ensuremath{\left| n \right>}} {\ensuremath{\left< m \right|}} \cos {\ensuremath{\omega_{r\!f}}}t, \label{eqc}$$ where $N$ is the number of states in the well being considered, $\hbar {\ensuremath{\omega_{n m}}}$ is the energy-level spacing between states [$\left| n \right>$]{} and [$\left| m \right>$]{} (tunable through $I_b$), and $${\ensuremath{\Omega_{n m}}} = \frac{{\ensuremath{\Phi_0}}}{2 \pi} \frac{{\ensuremath{I_{r\!f}}}}{\hbar} {\ensuremath{\left< n \left| \, \gamma_1 \, \right| m \right>}} \label{eqf}$$ is a bare Rabi frequency. The simplified Hamiltonian in [Eq. [(\[eqc\])]{}]{} neglects tunneling through the potential barrier,[@Larkin86a; @Leggett87a; @Chow88a; @Kopietz88a] a process corresponding to the SQUID spontaneously switching from the supercurrent to the voltage state; tunneling from [$\left| n \right>$]{} occurs at an escape rate [$\Gamma_{n}$]{}. As ${\ensuremath{\Gamma_{n+1}}} / {\ensuremath{\Gamma_{n}}} \sim 10^3$ for values of $I_b$ that we studied, tunneling can be exploited to perform state readout. Note also typically ${\ensuremath{\Gamma_{n}}} \ll {\ensuremath{\omega_{n, n+1}}}$ for the lowest few levels; for states near the barrier, this is not true. The total escape rate of the system to the voltage state is $$\Gamma = \frac{1}{\rho_{tot}} \sum_{n=0}^{N-1} \rho_{nn} {\ensuremath{\Gamma_{n}}}, \label{eqg}$$ where $\rho_{nn}$ is the occupation probability of state [$\left| n \right>$]{} and $\rho_{tot} = \sum \rho_{nn}$ is the probability of the system being in the supercurrent state; all of the quantities in the equation are time dependent during a bias ramp. This description of the qubit makes several simplifications. For one, we have ignored the quantum states of the isolation junction $J2$.[@Mitra06a] For typical bias conditions, we keep the current through this junction small and its first excited state is two to three times higher in energy than that of the qubit. However, the higher excited states of the qubit junction may come into resonance with states of the isolation junction. This approach further ignores the role of $J2$ in determining the bias conditions. Quantum mechanical simulations of the SQUID show that the current through the qubit junction weakly depends on its quantum state;[@Mitra06a] shifts in the qubit current are predicted to be less than 5 nA (much smaller than $I_{01} \approx 18\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$) for our situation, so we have not considered this correction below. As a result, the junction parameters in [Eq. [(\[eqa\])]{}]{} need not be equal to the actual values for the qubit junction. The simulations described in the sections that follow require values of [$\Gamma_{n}$]{}, [$\omega_{n m}$]{}, and [$\left< n \left| \, \gamma_1 \, \right| m \right>$]{}, all of which are specified by $I_b$, $I_{01}$, and $C_1$. In the absence of dissipation, analytical expressions for these quantities can be obtained by applying perturbation theory to the cubic approximation of the tilted washboard potential.[@Strauch04a] For the escape rates and energy levels, we instead use numerical solutions to the exact potential, which are more reliable for states near the top of the barrier. These solutions come from two methods that give consistent results: (i) solving the Schrödinger equation with transmission boundary conditions[@Shibata91a] and (ii) complex scaling.[@Moiseyev98a] We performed experiments on a dc SQUID phase qubit fabricated by Hypres, Inc.[@NIST] using a $30\ {\ensuremath{{\ensuremath{\mathrm{A}}} / {\ensuremath{\mathrm{cm}}}^2}}$ Nb/AlO$_\text{x}$/Nb trilayer process \[see Fig. \[FDevice\](c)\]. The qubit and isolation junctions had areas of $10 \times 10~{\ensuremath{\mu {\ensuremath{\mathrm{m}}}}}^2$ and $7 \times 7~{\ensuremath{\mu {\ensuremath{\mathrm{m}}}}}^2$, respectively, and the inductances were roughly $L_1 = 3.4\ {\ensuremath{{\ensuremath{\mathrm{nH}}}}}$, $L_2 = 30\ {\ensuremath{{\ensuremath{\mathrm{pH}}}}}$, and $M = 13.4\ {\ensuremath{{\ensuremath{\mathrm{pH}}}}}$.[@Palomaki07a] The error in these and other fit parameters reported in this paper is about one unit in the least significant digit. We have not attempted a thorough analysis of the uncertainties in all the parameters due to the complexities of the nonlinear functions involved. The microwave current [$I_{r\!f}$]{} was carried inside the refrigerator on a single length of lossy stainless coax. It was coupled to the qubit via an on-chip $1\ {\ensuremath{{\ensuremath{\mathrm{fF}}}}}$ capacitor [$C_{r\!f}$]{} \[see Fig. \[FDevice\](a)\]; while the large impedance mismatch this produced potentially improved the isolation, it did make an independent calibration of the power reaching the qubit difficult. The device was mounted in a superconducting aluminum box that was attached to the mixing chamber of a dilution refrigerator with a base temperature of 20 mK. The refrigerator, located in an rf shielded room, was surrounded by a mu-metal cylinder. In addition, the measurement and bias lines were protected from noise at room temperature by discrete $LC$ filters and copper powder filters at the mixing chamber. The escape rate of the qubit junction was measured by simultaneously increasing $I_b$ and $I_f$ as described above, while monitoring the voltage on the current bias line of the SQUID. We measured the time interval between the start of the ramps and when the system tunneled to the voltage state. Repeating this procedure many times ($\sim 10^5$) at a rate of about 250 Hz yielded a histogram of switching times, from which the escape rate could be calculated.[@Fulton74a] Decreasing the repetition rate did not yield a significant change in the escape rate, suggesting that heating due to the device switching to the voltage state had a minimal impact on the measurements. Because the large loop inductance of the SQUID resulted in about 20 possible flux states, the device was initialized to the zero trapped flux state with a flux shaking procedure before each repetition of the measurement cycle.[@Palomaki06a] We typically applied 50 oscillations of the flux current, which yielded a success rate better than 98%. ![\[FEnergyGamma\] Qubit energy-level spectroscopy and tunneling escape rates. (a) Open circles show the resonance frequency of the transition between the ground and first excited states of the qubit, measured at 20 mK. The scatter in the values is indicative of the uncertainty in the measurement. Also plotted are theoretical values of [$\omega_{0 1}$]{} (solid), [$\omega_{1 2}$]{} (dashed), and [$\omega_{2 3}$]{} (dotted) for $I_{01} = 17.930\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ and $C_1 = 4.50\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$. (b) At high bias, the measured background escape rate (open circles) agrees with the predicted ground-state escape rate [$\Gamma_{0}$]{} (solid line) for the junction parameters given above. Calculated [$\Gamma_{1}$]{}, [$\Gamma_{2}$]{}, and [$\Gamma_{3}$]{} are plotted as dashed, dotted, and dashed-dotted lines. In both plots, the bottom axes show the total bias current $I_b$, while the top axes indicate the normalized barrier height $N_s$, calculated using the extracted junction parameters.](EnergyGammav3.eps){width="3.0in"} Figure \[FEnergyGamma\](a) shows a spectrum of transitions for the qubit junction. With a microwave drive of fixed frequency applied to the device, the total escape rate was measured while ramping the biases. The open circles indicate the values of $I_b$ where the microwave drive caused the largest enhancement in the escape rate (over its background value in the absence of microwaves). As shown by the solid line, we fit these points to the theoretical values of [$\omega_{0 1}$]{}, yielding $I_{01} = 17.930\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ and $C_1 = 4.50\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$. These values are likely to differ from the actual critical current and capacitance of the qubit junction, both in principle, due to the simplifications in the Hamiltonian mentioned above, and in practice, as this fitting procedure is sensitive to inaccuracies in the simultaneous ramping of $I_b$ and $I_f$, constant offset flux that biases the SQUID (from, for example, trapped flux near the device), and drifts in the detection electronics. Nonetheless, when we repeated the measurement at elevated temperature, the location of the transitions between higher levels agreed to within 0.2% of predictions for [$\omega_{1 2}$]{} (dashed) and [$\omega_{2 3}$]{} (dotted) obtained with the effective parameters.[@Strauch06a] In addition, two-photon transitions between these levels occurred at high power at the expected frequencies. Multiphoton transitions between the ground and first excited states of a variety of junction qubits have previously been observed and characterized.[@Nakamura01a; @Wallraff03a; @Saito04a] Over the course of three months (during which the refrigerator remained near its base temperature), the junction parameters found from the spectral fits varied by roughly 1%. Thus, we repeated the spectroscopy measurement often to obtain the values of $I_{01}$ and $C_1$ needed for the simulations; these values are listed in the caption of each figure. The open circles of Fig. \[FEnergyGamma\](b) show the measured background escape rate in the absence of microwaves. The solid line indicates the theoretical value of [$\Gamma_{0}$]{}, calculated with the values of $I_{01}$ and $C_1$ extracted from the spectrum. At low bias current, the measured escape rate exceeds [$\Gamma_{0}$]{}, suggesting the presence of excited state population even though the refrigerator was at 20 mK. Based on experiments described elsewhere,[@Palomaki07a] we believe that features such as the peak near $I_b = 17.73\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ are due to population in [$\left| 2 \right>$]{} generated when noise on the leads at the resonant frequency of the isolation junction matches the [$0 \rightarrow 2$]{} transition frequency of the qubit. At high bias, the escape rates exceed the excitation rates of the noise and the total escape rate converges to $\Gamma_0$; a similar effect occurs for thermal excitations.[@Dutta04a] While noise complicates the situation, the overall behavior suggests that both [$\omega_{0 1}$]{} and [$\Gamma_{0}$]{} of the dc SQUID are described by the same one-dimensional tilted washboard potential. Also shown in Fig. \[FEnergyGamma\](b) are numerical predictions for [$\Gamma_{1}$]{} (dashed), [$\Gamma_{2}$]{} (dotted), and [$\Gamma_{3}$]{} (dashed-dotted). We observed Rabi oscillations between the ground and first excited states by turning the microwave current [$I_{r\!f}$]{} on when $I_b$ was at the value where ${\ensuremath{\omega_{0 1}}} = {\ensuremath{\omega_{r\!f}}}$ and measuring the time-resolved escape rate while [$I_{r\!f}$]{} remained at a constant value. This serves as a simple method of monitoring the evolution of the state populations.[@Yu02a] Although the biases continued to increase during the Rabi oscillations, the ramp rates were reduced (with $d I_b / dt \approx 0.01\ {\ensuremath{{\ensuremath{\mathrm{A}}} / {\ensuremath{\mathrm{s}}}}}$) before [$I_{r\!f}$]{} was turned on, so that the level spacing ${\ensuremath{\omega_{0 1}}}$ changed by a negligible amount during the escape rate measurement. The symbols in Fig. \[F031806DGtot\] show $\Gamma$ due to a 6.2 GHz microwave drive for a range of powers ${\ensuremath{P_S}}$ at the microwave generator. As expected, the oscillation frequency increases with power and decoherence causes the amplitude of the oscillations to decay with time. However, there are two unexpected features in the data. First, the escape rate [$\Gamma_{\infty}$]{} at long time (once the oscillations have decayed away) increases with [$P_S$]{}, whereas for an ideal two-level system, the excited state population saturates at high power. Furthermore, the highest measured escape rates far exceed the value of $2.2 \times 10^6\ {\ensuremath{{\ensuremath{\mathrm{1/s}}}}}$ predicted for [$\Gamma_{1}$]{} at this bias current, strongly suggesting that the states [$\left| 2 \right>$]{} and higher are becoming occupied. As the escape rates from these levels are very large, only a small population would be required to account for the observed $\Gamma$. Second, at high power, the oscillation maxima increase over the first few cycles \[see Fig. \[F031806DGtot\](a)\]; this is due to the rise time of the microwave current amplitude, which will be discussed in Sec. \[SDecoherence\]. In Sec. \[SStrong\], we will show how more complete measurements reveal that the higher excited states impact the Rabi oscillation frequencies in ways that are in quantitative agreement with predictions from the Hamiltonian of [Eq. [(\[eqc\])]{}]{}, but are difficult to see in Fig. \[F031806DGtot\]. As shown by the solid lines in the figure, most of the features of the measured escape rate are captured by a multilevel density-matrix simulation which will be described in Sec. \[SDecoherence\]. ![\[F031806DGtot\] Rabi oscillations in the escape rate $\Gamma$ were induced at $I_b = 17.746\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ by switching on a microwave current at $t = 0$ with a frequency of 6.2 GHz (resonant with the [$0 \rightarrow 1$]{} transition) and source powers [$P_S$]{} between $-12$ and $-32$ dBm, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with $I_{01} = 17.930\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$, $C_1 = 4.50\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$, $T_1 = 17\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$, and $T_\phi = 16\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$.](031806DGtotv3.eps){width="3.0in"} Detuning and Strong Field Effects {#SStrong} ================================= We first model Rabi oscillations using the rotating wave approximation, in the absence of tunneling and dissipation. This provides a way of predicting the oscillation frequency for a wide range of experimental parameters.[@Steffen03a; @Claudon04a; @Meier05a; @Amin06a; @Strauch06a; @Claudon07b] In the rotating frame corresponding to the drive frequency ${\ensuremath{\omega_{r\!f}}}\approx {\ensuremath{\omega_{0 1}}}$, the Hamiltonian of [Eq. [(\[eqc\])]{}]{} simplifies to[@Strauch06a] $${\ensuremath{\mathcal{H}}}_N^{RW} = \sum_{n=0}^{N-1} \hbar \Delta_n {\ensuremath{\left| n \right>}} {\ensuremath{\left< n \right|}} + \frac{1}{2} \sum_{\substack{n,m=0 \\ n \neq m}}^{N-1} \hbar \Omega_{nm}^\prime {\ensuremath{\left| n \right>}} {\ensuremath{\left< m \right|}}, \label{eqd}$$ where $\Delta_n = {\ensuremath{\omega_{0 n}}} - n {\ensuremath{\omega_{r\!f}}}$ and $$\Omega_{nm}^\prime = {\ensuremath{\Omega_{n m}}} \sum_{s=0}^1 J_{m-n+2s-1} {\ensuremath{\left( \frac{{\ensuremath{\Omega_{n n}}} - {\ensuremath{\Omega_{m m}}}}{{\ensuremath{\omega_{r\!f}}}} \right)}}$$ for $n < m$ and $\Omega_{mn}^\prime = \Omega_{nm}^\prime$. Here, $J_n {\ensuremath{\left( x \right)}}$ is the $n$th order Bessel function and $\Omega_{nm}$ is defined by [Eq. [(\[eqf\])]{}]{}. Differences between the $N$ eigenvalues of this Hamiltonian specify effective multilevel Rabi frequencies or modes of the system. We find it convenient to label these frequency differences by the states [$\left| n \right>$]{} with the largest weight in the two corresponding eigenfunctions. For the parameter regime of interest here, the differences can be uniquely classified by two states, which we denote by [$\Omega_{R,n m}$]{}. For example, [$\Omega_{R,0 2}$]{} denotes the Rabi oscillation frequency between eigenstates that describe a two-photon transition between [$\left| 0 \right>$]{} and [$\left| 2 \right>$]{}. While approximate analytical solutions can be obtained for a three-level junction system,[@Strauch06a] we numerically found the eigenvalues of the rotating wave Hamiltonian for systems with up to seven levels in the simulations that follow. ![\[F031806Dfosc\] Rabi oscillation frequency [$\Omega_{R,0 1}$]{} at fixed bias as a function of microwave current [$I_{r\!f}$]{}. Extracted values from data (including the plots in Fig. \[F031806DGtot\]) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using $I_{01} = 17.930\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ and $C_1 = 4.50\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$ with ${\ensuremath{\omega_{r\!f}}}/ 2 \pi = 6.2\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$.](031806Dfoscv3.eps){width="3.0in"} We fit the escape rates in Fig. \[F031806DGtot\] (and additional data for other powers not shown) to a decaying sinusoid with an offset. The extracted frequencies are shown with circles in Fig. \[F031806Dfosc\]. To compare to theory, [$\Omega_{R,0 1}$]{}, calculated using the rotating wave solution for a system with five levels, is shown with a solid line. The implied assumption that the oscillation frequencies of $\Gamma$ and $\rho_{11}$ are equal, even at high power in a multilevel system, will be addressed in Sec. \[SSummary\]. In plotting the data, we have introduced a single fitting parameter $117\ {\ensuremath{{\ensuremath{{\ensuremath{\mathrm{nA}}}}}/ \sqrt{{\ensuremath{{\ensuremath{\mathrm{mW}}}}}}}}$ that converts the power [$P_S$]{} at the microwave source to the current amplitude [$I_{r\!f}$]{} at the qubit. Good agreement is found over the full range of power. As [$I_{r\!f}$]{} increases in Fig. \[F031806Dfosc\], the oscillation frequency is smaller than the expected linear relationship for a two-level system (dashed line). This effect is a hallmark of a multilevel system and has been previously observed in a similar phase qubit.[@Claudon04a; @Claudon07b] There are two distinct phenomena that affect [$0 \rightarrow 1$]{} Rabi oscillations in such a device.[@Goorden03a; @Meier05a; @Strauch06a; @Shevchenko07a] To describe these, we must first define resonance as occurring when the Rabi frequency [$\Omega_{R,0 1}$]{} is at a minimum, as the detuning between the microwave drive frequency [$\omega_{r\!f}$]{} and level spacing [$\omega_{0 1}$]{} is varied (for a fixed drive power). In a two-level system, this happens for ${\ensuremath{\omega_{0 1}}} = {\ensuremath{\omega_{r\!f}}}$. For our phase qubit, resonance occurs when ${\ensuremath{\omega_{0 1}}} < {\ensuremath{\omega_{r\!f}}}$ due to the decreasing level spacings with increasing state [$\left| n \right>$]{} \[see Fig. \[FEnergyGamma\](a)\]. This shift in resonance is an analog of the ac Stark effect. Resonance shifts occur under strong driving in other superconducting qubits as well.[@Schuster05a; @Dutton06a] However, this shift does not explain the data shown in Fig. \[F031806Dfosc\], which were taken at a fixed bias current (and thus off resonance at high power). The higher levels also affect the frequency of the Rabi oscillations. We will refer to the minimum value of [$\Omega_{R,n m}$]{} as the on-resonance Rabi frequency [$\Omega^{min}_{R,n m}$]{}. The suppression of [$\Omega^{min}_{R,0 1}$]{} below [$\Omega_{0 1}$]{} leads to the effect seen in Fig. \[F031806Dfosc\]. Both of these effects become significant as [$\Omega_{0 1}$]{} approaches ${\ensuremath{\omega_{0 1}}} - {\ensuremath{\omega_{1 2}}}$, which is a measure of the anharmonicity of the system;[@Meier05a; @Amin06a; @Strauch06a] in the case of Fig. \[F031806Dfosc\], ${\ensuremath{\omega_{0 1}}} / 2 \pi = 6.2\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$ and ${\ensuremath{\omega_{1 2}}} / 2 \pi = 5.5\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$. Clearly, these shifts need to be considered when working at high power or at low current bias. In order to follow experimentally the shift of the resonance condition, it was necessary to measure Rabi oscillations for different detunings of the microwave drive. We chose to do this by keeping the drive frequency [$\omega_{r\!f}$]{} fixed and changing the level spacing [$\omega_{0 1}$]{} (through $I_b$), because the power transmitted by the microwave lines had a nontrivial frequency dependence. Figure \[F010206H1\](a) shows a grayscale plot of Rabi oscillations measured from such an experiment, where black represents a high escape rate. Each horizontal line is the escape rate versus time due to a microwave current of 6.5 GHz and $-11$ dBm, which was turned on at the value of the current bias $I_b$ indicated on the vertical axis. While the measurements were performed at 110 mK, this is not expected to have a significant impact on the Rabi oscillations, as the temperature was well below $\hbar {\ensuremath{\omega_{0 1}}} / k_B \approx 325\ {\ensuremath{{\ensuremath{\mathrm{mK}}}}}$.[@Lisenfeld07a] ![\[F010206H1\] (Color online) Multiphoton, multilevel Rabi oscillations plotted in the time and frequency domains. (a) The escape rate $\Gamma$ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, $-11$ dBm microwave drive was turned on and the current bias $I_b$ of the qubit; $\Gamma$ ranges from 0 (white) to $3 \times 10^8\ {\ensuremath{{\ensuremath{\mathrm{1/s}}}}}$ (black). (b) The normalized power spectral density of the time-domain data from $t = 1$ to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi frequencies obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters $I_{01} = 17.828\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ and $C_1 = 4.52\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$, and microwave current ${\ensuremath{I_{r\!f}}}= 24.4\ {\ensuremath{{\ensuremath{\mathrm{nA}}}}}$. Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d).](010206H1v3.eps){width="3.0in"} From Fig. \[F010206H1\](a), we see that the oscillation frequency depends on the current bias. This variation can be seen more readily in Fig. \[F010206H1\](b), which shows the power spectral density (calculated as the absolute square of the discrete Fourier transform) of the escape rate data in Fig. \[F010206H1\](a) between $t = 1$ and $45\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$. For this plot, each horizontal line has been normalized to its maximum value (black) in order to emphasize the location of the dominant frequency. Three distinct bands are visible. For this data set, the level spacing ${\ensuremath{\omega_{0 1}}} / 2 \pi$ is equal to the microwave frequency ${\ensuremath{\omega_{r\!f}}}/ 2 \pi = 6.5\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$ at $I_b = 17.614\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$. The band with the highest current in Fig. \[F010206H1\](b) is centered about $I_b = 17.624\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$, suggesting that [$0 \rightarrow 1$]{} Rabi oscillations are the dominant process near this bias. For slightly higher or lower $I_b$, the oscillation frequency increases as ${\ensuremath{\Omega_{R,0 1}}} \approx \sqrt{\Omega_{01}^{\prime2} + {\ensuremath{\left( {\ensuremath{\omega_{r\!f}}}- {\ensuremath{\omega_{0 1}}} \right)}}^2}$, in agreement with simple two-level Rabi theory, leading to the curved band in the grayscale plot. The other bands correspond to oscillations between the ground state and higher excited states. Due to the anharmonic level structure, ${\ensuremath{\omega_{0 2}}} / 4 \pi$ is 6.5 GHz at a smaller current bias $I_b = 17.594\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$; thus a second band appears there, corresponding to two-photon [$0 \rightarrow 2$]{} Rabi oscillations. Similarly, three-photon [$0 \rightarrow 3$]{} oscillations are visible near $17.572\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ where ${\ensuremath{\omega_{0 3}}} / 6 \pi = 6.5\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$. Finally, large escape rates occur near $17.549\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ corresponding to a four-photon [$0 \rightarrow 4$]{} transition, although no oscillations are apparent in Fig. \[F010206H1\](a). Note that the effective junction parameters $I_{01}$ and $C_1$ (given in the caption) used to predict [$\omega_{n m}$]{} and other level properties are slightly different than those for Figs. \[FEnergyGamma\]–\[F031806Dfosc\], as the data sets were taken two months apart. The rotating wave solution provides a simple way to predict the oscillation frequencies. Calculations of [$\Omega_{R,0 n}$]{} using [Eq. [(\[eqd\])]{}]{} for a seven-level system are shown as dashed curves in Fig. \[F010206H1\](b) for the four lowest multiphoton transitions ($n = 1, 2, 3, 4$). The microwave amplitude ${\ensuremath{I_{r\!f}}}= 24.4\ {\ensuremath{{\ensuremath{\mathrm{nA}}}}}$ is the only free parameter in the calculation and the rotating wave solution using this value reproduces the oscillation frequencies of the different processes well, even at large detuning. Figure \[F010206H1\](b) shows that the minimum oscillation frequency ${\ensuremath{\Omega^{min}_{R,0 1}}} / 2 \pi = 540\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$ of the first (experimental) band occurs at $I_b = 17.624\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$, for which ${\ensuremath{\omega_{0 1}}} / 2 \pi = 6.4\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$. This again indicates an ac Stark shift of this transition, which we denote by $\Delta {\ensuremath{\omega_{0 1}}} \equiv {\ensuremath{\omega_{r\!f}}}- {\ensuremath{\omega_{0 1}}} \approx 2 \pi \times 100\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$. In addition, the higher levels have suppressed the oscillation frequency below the bare Rabi frequency of ${\ensuremath{\Omega_{0 1}}} / 2 \pi = 620\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$ \[calculated with [Eq. [(\[eqf\])]{}]{}\]. ![\[F032206MNstats\] (Color online) The (a) on-resonance Rabi oscillation frequencies [$\Omega^{min}_{R,0 1}$]{} and [$\Omega^{min}_{R,0 2}$]{} and (b) resonance frequency shifts $\Delta {\ensuremath{\omega_{0 1}}} = {\ensuremath{\omega_{r\!f}}}- {\ensuremath{\omega_{0 1}}}$ and $\Delta {\ensuremath{\omega_{0 2}}} = 2 {\ensuremath{\omega_{r\!f}}}- {\ensuremath{\omega_{0 2}}}$ are plotted as a function of the microwave current, for data taken at $110\ {\ensuremath{{\ensuremath{\mathrm{mK}}}}}$ with a microwave drive of frequency ${\ensuremath{\omega_{r\!f}}}/ 2 \pi = 6.5\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$ and powers ${\ensuremath{P_S}}= -23, -20, -17, -15, -10\ {\ensuremath{{\ensuremath{\mathrm{dBm}}}}}$. Values extracted from data for the [$0 \rightarrow 1$]{} ([$0 \rightarrow 2$]{}) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with $I_{01} = 17.736\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ and $C_1 = 4.49\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$ are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system.](032206MNstatsv3.eps){width="3.0in"} We repeated this analysis of resonance for five different microwave powers, with data taken at a later date. Figure \[F032206MNstats\] shows experimental results for the on-resonance Rabi frequencies ${\ensuremath{\Omega^{min}_{R,n m}}}$ and Stark shifts $\Delta {\ensuremath{\omega_{n m}}}$ with the corresponding results from a five-level rotating wave solution for the [$0 \rightarrow 1$]{} (circles for data and solid lines for theory) and two-photon [$0 \rightarrow 2$]{} (squares and dashed lines) transitions. Here, the power calibration is $84\ {\ensuremath{{\ensuremath{{\ensuremath{\mathrm{nA}}}}}/ \sqrt{{\ensuremath{{\ensuremath{\mathrm{mW}}}}}}}}$ for ${\ensuremath{\omega_{r\!f}}}/ 2 \pi = 6.5\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$. Figure \[F032206MNstats\](a) differs from Fig. \[F031806Dfosc\], because in the former, $I_b$ was varied at each power to give the minimum oscillation frequency; by staying on resonance in this way, the effect of the higher levels on the [$0 \rightarrow 1$]{} oscillations is maximized. The resonant oscillation frequencies [$\Omega^{min}_{R,0 1}$]{} and [$\Omega^{min}_{R,0 2}$]{} in Fig. \[F032206MNstats\](a) are well described by [Eq. [(\[eqd\])]{}]{} over the full range of [$I_{r\!f}$]{}. The deviation between the [$0 \rightarrow 1$]{} oscillation frequency and the values expected in a two-level system (dotted line) increases with [$I_{r\!f}$]{}. Similar measurements taken at a higher $I_b$ show a smaller frequency suppression over a similar range of [$I_{r\!f}$]{},[@Strauch06a] as expected for a system with stronger anharmonicity. The resonance shifts $\Delta {\ensuremath{\omega_{0 1}}}$ and $\Delta {\ensuremath{\omega_{0 2}}}$ in Fig. \[F032206MNstats\](b), however, differ significantly from the model predictions. As a change of 10 MHz in ${\ensuremath{\omega_{0 1}}} / 2 \pi$ corresponds to a roughly 1 nA change in $I_b$, the discrepancy is difficult to see in Fig. \[F010206H1\](b). The indicated uncertainty in the experimental points in Fig. \[F032206MNstats\](b) is roughly 5 MHz due to errors in the calibration of ${\ensuremath{\omega_{0 1}}} {\ensuremath{\left( I_b \right)}}$. In addition the uncertainty is somewhat larger at low power, where the relatively small total escape rates result in poor counting statistics, and high power, where the weak dependence on detuning makes it difficult to identify the resonant level spacing. While the general trend of the shifts is consistent with the model, the scatter in the data is large and further work would be needed to determine if there are true deviations from the multilevel theory. Decoherence {#SDecoherence} =========== We now model the time dependence of the escape rate measured in the experiment. In order to do this, several additions have to be made to the treatment of the system given in Sec. \[SStrong\]: the effects of tunneling, other sources of damping and noise, and experimental limitations. The density-matrix formalism[@Smith78a; @Blum96a] provides a straightforward scheme for including nonunitary processes and has been previously applied to the lowest two or three levels of the phase qubit.[@Yu02a; @Kosugi05a; @Meier05a; @Amin06a; @Claudon07b; @Shevchenko07a] For a system with $N$ levels, we assume that the evolution of the qubit’s reduced density matrix [$\rho$]{} is given by the modified Liouville–von Neumann equation $${\ensuremath{\frac{\partial {\ensuremath{\rho}}}{\partial t}}} = -\frac{i}{\hbar} {\ensuremath{\left[ {\ensuremath{\mathcal{H}}}_N, {\ensuremath{\rho}}\right]}} - G {\ensuremath{\rho}}- R {\ensuremath{\rho}}- D {\ensuremath{\rho}}. \label{eqb}$$ Here, we use the discrete Hamiltonian in [Eq. [(\[eqc\])]{}]{}, so that the diagonal elements $\rho_{nn}$ give the occupancy of the states [$\left| n \right>$]{}. Tunneling is characterized by the tensor $G$, where ${\ensuremath{\left[ G {\ensuremath{\rho}}\right]}}_{nm} = {\ensuremath{\left( {\ensuremath{\Gamma_{n}}} + {\ensuremath{\Gamma_{m}}} \right)}} \rho_{nm} / 2$, which leads to a decay of all elements of [$\rho$]{}. The tensors $R$ and $D$ account for two distinct decoherence mechanisms. The Bloch-Redfield tensor $R$ models the effects of the system being in equilibrium with a thermal bath at temperature $T$.[@Burkard04a] This tensor leads to the decay of the diagonal elements of $\rho$ (dissipation) as well as the off-diagonal elements (decoherence). The coupling to the bath, assumed linear in $\gamma_1$, is parametrized by $R_1 {\ensuremath{\left( \omega \right)}}$, which is the inverse of the real part of the total admittance that shunts the qubit junction, evaluated at angular frequency $\omega$. In [Eq. [(\[eqb\])]{}]{}, $${\ensuremath{\left[ R {\ensuremath{\rho}}\right]}}_{nm} = \sum_{k,l=0}^{N-1} R_{nmkl} \rho_{kl}, \label{eqe}$$ where $$R_{nmkl} = - \gamma_{lmnk} - \gamma_{knml} + \delta_{lm} \sum_{r=0}^{N-1} \gamma_{nrrk} + \delta_{nk} \sum_{r=0}^{N-1} \gamma_{mrrl}$$ and $$\gamma_{lmnk} = \frac{1}{2 \hbar} {\ensuremath{\left( \frac{{\ensuremath{\Phi_0}}}{2 \pi} \right)}}^2 \frac{{\ensuremath{\left< l \left| \, \gamma_1 \, \right| m \right>}} {\ensuremath{\left< n \left| \, \gamma_1 \, \right| k \right>}}} {R_1 {\ensuremath{\left( {\ensuremath{\omega_{n k}}} \right)}}} {\ensuremath{\left[ {\ensuremath{\left( 1 - \delta_{nk} \right)}} {\ensuremath{\omega_{n k}}} \frac{\exp {\ensuremath{\left( -\frac{\hbar {\ensuremath{\omega_{n k}}} \mathrm{sgn} {\ensuremath{\left( n - k \right)}}} {2 k_B T} \right)}}} {\sinh {\ensuremath{\left( \hbar {\ensuremath{\omega_{n k}}} / 2 k_B T \right)}}} + \delta_{nk} \frac{2 k_B T}{\hbar} \right]}}.$$ In the following, we assume that $R_1$ is independent of frequency, in which case we can define a dissipation time $T_1 = R_1 C_1$. In this limit, [Eq. [(\[eqe\])]{}]{} gives well known interlevel transition rates.[@Larkin86a; @Chow88a; @Xu05b] For example, thermal excitation from [$\left| n \right>$]{} to [$\left| m \right>$]{} (with $m > n$) occurs at a rate $$\begin{aligned} W_{mn}^+ & = & 2 \gamma_{nmmn} \nonumber \\ & = & \frac{2}{\hbar} {\ensuremath{\left( \frac{{\ensuremath{\Phi_0}}}{2 \pi} \right)}}^2 {\ensuremath{\left( \frac{{\ensuremath{\omega_{n m}}}}{R_1} \right)}} \frac{{\ensuremath{\left| {\ensuremath{\left< n \left| \, \gamma_1 \, \right| m \right>}} \right|}}^2} {\exp {\ensuremath{\left( \hbar {\ensuremath{\omega_{n m}}} / k_B T \right)}} - 1}.\end{aligned}$$ As required by detailed balance, decay from [$\left| m \right>$]{} to [$\left| n \right>$]{} occurs at a rate $W_{nm}^- = W_{mn}^+ \exp {\ensuremath{\left( \hbar {\ensuremath{\omega_{n m}}} / k_B T \right)}}$. In addition, [Eq. [(\[eqe\])]{}]{} specifies the decoherence due to this dissipation. Given ${\ensuremath{\omega_{n m}}}$ and the matrix elements of $\gamma_1$, all of the thermal rates are specified by $T_1$, which is set to $17\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ for the simulations below. This value comes from additional measurements of the escape rate of the device in the absence of microwaves over a range of temperatures.[@Dutta04a] We find that this treatment of dissipation alone is insufficient to capture the decay of Rabi oscillations, suggesting that an additional decoherence mechanism is present. This we model by the tensor $D$ in [Eq. [(\[eqb\])]{}]{}, which has the form $$D {\ensuremath{\rho}}= \sum_n \lambda_n {\ensuremath{\left( L_n {\ensuremath{\rho}}L_n^\dag - \frac{1}{2} L_n^\dag L_n {\ensuremath{\rho}}- \frac{1}{2} {\ensuremath{\rho}}L_n^\dag L_n \right)}}, \label{eqh}$$ where $L_n$ are Lindblad operators with strengths $\lambda_n$. The best overall agreement with the measurements (see further discussion below) is found for a set of operators $L_n = {\ensuremath{\left| n \right>}} {\ensuremath{\left< n \right|}}$, where $\lambda_n = 1 / T_\phi$ and $n$ ranges from 0 to $N-1$. This leads to an exponential decay of each of the off-diagonal elements of [$\rho$]{} with a common time constant given by the dephasing time $T_\phi$ and no change in the diagonal elements. In the simulations of this section, we set $T_\phi = 16\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$, yielding a coherence time $T_2 = {\ensuremath{\left[ 1 / {\ensuremath{\left( 2 T_1 \right)}} + 1 / T_\phi \right]}}^{-1} = 10.9\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$. For $N = 2$, [Eq. [(\[eqb\])]{}]{} reduces to the optical Bloch equations,[@Smith78a; @Blum96a] for which analytical solutions exist. However, for $N > 2$ the master equation is easily numerically integrated to obtain the time dependence of [$\rho$]{}, without making the rotating wave approximation. With the state occupation probabilities $\rho_{nn}$ in hand, the total escape rate can be calculated with [Eq. [(\[eqg\])]{}]{} and compared with experiment. As mentioned earlier, one of the more striking features of the series of [$0 \rightarrow 1$]{} Rabi oscillations in Fig. \[F031806DGtot\] is that [$\Gamma_{\infty}$]{}, which we define as the steady escape rate the system approaches as the oscillations decay away, increases over the full range of measured power. The circles in Fig. \[F031806DGeq\] show experimental values of [$\Gamma_{\infty}$]{} as a function of the bare Rabi frequency [$\Omega_{0 1}$]{}, which was calculated from the microwave source power by using the fit in Fig. \[F031806Dfosc\]. We will now use the density-matrix simulations to understand these escape rates. ![\[F031806DGeq\] Long-time escape rate [$\Gamma_{\infty}$]{} as a function of the bare Rabi frequency [$\Omega_{0 1}$]{}. (a) [$\Gamma_{\infty}$]{} is plotted for a density-matrix simulation of a system with two (dashed-dotted), three (dashed), four (dotted), and five (solid) levels, $I_{01} = 17.930\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$, and $C_1 = 4.50\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$. (b) [$\Gamma_{\infty}$]{} at low power (for a five-level system) is affected by the inclusion of a small microwave current at [$\omega_{0 2}$]{} (dashed) and inhomogeneous broadening in addition to this noise current (solid). One set of experimental data, for source powers between $-68$ and $-11\ {\ensuremath{{\ensuremath{\mathrm{dBm}}}}}$ at 6.2 GHz, is plotted in both panels with open circles.](031806DGeqv3.eps){width="3.0in"} The simulations suggest that [$\Gamma_{\infty}$]{} is not sensitive to the time evolution of [$\rho$]{}, and is only weakly dependent on $T_1$ and $T_\phi$ at high power. However, [$\Gamma_{\infty}$]{} does depend on the number of states that participate in the dynamics and their individual escape rates [$\Gamma_{n}$]{}. It should be noted that the actual populations do not reach steady state, but continuously decay due to tunneling. The dashed-dotted line in Fig. \[F031806DGeq\](a) shows the calculated [$\Gamma_{\infty}$]{} for a two-level system ($N = 2$). It increases up to a value of ${\ensuremath{\Gamma_{1}}} / 2$ at roughly ${\ensuremath{\Omega_{0 1}}} = 1 / \sqrt{T_1 T_2}$; the data show only a subtle shoulder, masked by an overall steady increase, near this escape rate. Results for three (dashed), four (dotted), and five (solid) levels are also plotted in the figure, displaying increasing agreement with experiment. This agreement between theory and experiment at high power is an indication that transitions to the higher states, or leakage in the context of quantum computation, are occurring as expected. As power increases, the occupation of higher excited states increases and this effect is magnified by their larger escape rates. For example, with the five-level simulation at ${\ensuremath{\Omega_{0 1}}} / 2 \pi = 1\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$ the state [$\left| 3 \right>$]{} has a 2% occupation probability, but accounts for 60% of [$\Gamma_{\infty}$]{}. Note that microwave pulse shaping will not reduce this effect. That is, while leakage can be minimized at the end of a single qubit operation, higher states are always populated during the pulse.[@Steffen03a; @Meier05a; @Amin06a] Such pulses will have enhanced escape rates comparable to those seen here for a constant [$I_{r\!f}$]{}. These simulations do not do a good job of explaining the value of [$\Gamma_{\infty}$]{} at very low power. The reason for this can be seen in Fig. \[FEnergyGamma\](b), where the measured escape rate in the absence of microwaves exceeds the predicted value of [$\Gamma_{0}$]{}. In particular, at the bias current of $I_b = 17.746\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ where the Rabi oscillations were performed, the measured escape rate is $\sim 2 \times 10^4\ {\ensuremath{{\ensuremath{\mathrm{1/s}}}}}$. While this excess escape rate could be attributed to a thermal population at $T = 56\ {\ensuremath{{\ensuremath{\mathrm{mK}}}}}$ (whereas the refrigerator thermometer read 20 mK), raising the temperature of the simulation leads to a significant enhancement in [$\Gamma_{\infty}$]{} that extends up to moderate microwave power, an effect not seen in the data. Instead, separate measurements[@Palomaki07a] indicate that the spurious features in the measured $\Gamma$ are largely due to a population in [$\left| 2 \right>$]{}, beyond that expected from a thermal bath at 20 mK. Assuming that this was a result of noise on the bias lines, we included in the simulation an additional microwave source at a frequency of [$\omega_{0 2}$]{}; a microwave current amplitude of 0.2 nA reproduces the background escape rate measured at $I_b = 17.746\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ (corresponding to Figs. \[F031806DGtot\], \[F031806Dfosc\], and \[F031806DGeq\]) and leads to a population $\rho_{22} = 3 \times 10^{-5}$. The coherent effects of this drive are negligible, as ${\ensuremath{\Omega_{0 2}}} / 2 \pi = 0.5\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$, which is much smaller than $1/T_1$ or $1/T_\phi$. The results of a five-level simulation with the extra source included are shown with a dashed line in Fig. \[F031806DGeq\](b). Compared to the simpler simulations in Fig. \[F031806DGeq\](a), they show improved agreement at the lowest powers, with little change above ${\ensuremath{\Omega_{0 1}}} / 2 \pi = 10\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$. To this point, we have not included effects from inhomogeneous broadening, although our spectroscopic measurements suggest its presence. For $T_2 = 10.9\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$, which describes many of the experiments below, the expected full width at half maximum of a resonance peak is roughly 30 MHz. However, we commonly find peak widths of 50 MHz at bias currents where tunneling makes a negligible contribution. The remaining broadening may be due to current noise at frequencies much lower than $1 / T_1$, whose contribution to the width scales with the spectral slope $d {\ensuremath{\omega_{0 1}}} / d I_b$.[@Berkley03b] This can be modeled by taking into account the frequency content of the noise and using the stochastic Bloch equations.[@Xu05b] Instead, we mimicked the inferred spectroscopic broadening simply by running the simulation for a range of bias currents and then convolving the resulting escape rate (at a given time) and a Gaussian with standard deviation $\sigma_I = 1.5\ {\ensuremath{{\ensuremath{\mathrm{nA}}}}}$ (corresponding to a 35 MHz spread in ${\ensuremath{\omega_{0 1}}} / 2 \pi$). Calculations of [$\Gamma_{\infty}$]{} for a five-level system with noise at [$\omega_{0 2}$]{} and this inhomogeneous broadening are drawn with a solid line in Fig. \[F031806DGeq\](b). The extra broadening has negligible effect at high power (where Rabi oscillations are observed), but does bring the simulation into better agreement with data near ${\ensuremath{\Omega_{0 1}}} / 2 \pi = 10\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$. The small remaining discrepancy could be due to detuning arising from a misidentification of the mean value of $I_b$. For example, performing the simulation at 17.745 rather than $17.746\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ results in an underestimate of the measured values. Nevertheless, the overall agreement over three decades in the Rabi frequency is very good. While the calibration of the microwave current [$I_{r\!f}$]{} came from a fit to data, one conversion factor reproduces both the oscillation frequencies (Fig. \[F031806Dfosc\]) and long-time escape rates (Fig. \[F031806DGeq\]). ![\[F032106FN10\] (Color online) Response to a microwave pulse. (a) The measured escape rate $\Gamma$ (circles), due to a 6.2 GHz, $-20\ {\ensuremath{{\ensuremath{\mathrm{dBm}}}}}$ microwave pulse nominally 30 ns long, shows Rabi oscillations followed by a decay governed by multiple time constants. The solid line is the result of a five-level density-matrix simulation for a junction with $I_{01} = 17.730\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ and $C_1 = 4.46\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$, using the measured microwave pulse amplitude (with a maximum of 11.7 nA) shown in the inset. The simulation also produces (b) the contribution to the escape rate and (c) the normalized occupation for each of the five levels, as labeled.](032106FN10v3.eps){width="3.0in"} We next consider the time dependence of the escape rate for the data plotted in Fig. \[F032106FN10\]. Here, a 6.2 GHz microwave pulse nominally 30 ns long was applied on resonance with the [$0 \rightarrow 1$]{} transition of the qubit junction. The measured escape rate shows Rabi oscillations followed by a decay back to the ground state once the microwave drive has turned off. This decay appears to be governed by three time constants. Nontrivial decays have previously been reported in phase qubits[@Martinis02a] and we have found them in several of our devices. Accurately simulating this experiment requires knowledge of the time dependence of the microwave pulse, which was created by the internal gate of a Hewlett-Packard 83731B synthesized source,[@NIST] without any further filtering. We measured the pulse envelope at the source’s output using a digital sampling oscilloscope \[see inset of Fig. \[F032106FN10\](a)\]. Ignoring any distortion of the pulse before it reached the qubit junction, this was taken as the microwave amplitude ${\ensuremath{I_{r\!f}}}{\ensuremath{\left( t \right)}}$. For the microwave power used in Fig. \[F032106FN10\], the maximum value of [$I_{r\!f}$]{} was taken to be 11.7 nA (a value obtained from the power calibration in Fig. \[F031806Dfosc\]). The solid line in Fig. \[F032106FN10\](a) shows the calculated escape rate for a five-level simulation in the presence of noise at [$\omega_{0 2}$]{} and inhomogeneous broadening. For this time-domain plot and others discussed below, we have convolved the simulation and a Gaussian with a full width of 150 ps to remove very fast, small oscillations (due to highly detuned multiphoton processes) that could not be seen in the experiment due to insufficient time resolution. The main part of the Rabi oscillation is reproduced well. In particular, the second oscillation maximum has a larger escape rate than the first, due to the 7 ns it takes for [$I_{r\!f}$]{} to reach its maximum. The first part of the decay is also reproduced, which from the simulation should correspond to the emptying of state [$\left| 2 \right>$]{}. However, the data also show a slow decay constant longer than 50 ns that the simulation does not explain. The longer time is inconsistent with our thermal measurement of $T_1$ and instead may be indicative of the qubit interacting with an additional quantum system. It appears, though, that this extra degree of freedom does not significantly affect the description of the Rabi oscillations. Note that if this longer time constant were the dominant relaxation process at high power, two-level saturation would have occurred at a lower power in Fig. \[F031806DGeq\](a). The density-matrix model can also be used to predict the escape rates for the power series in Fig. \[F031806DGtot\]; calculations are plotted with solid lines in that figure. The presence of the noise signal at ${\ensuremath{\omega_{0 2}}}$ has little effect and inhomogeneous broadening decreases the escape rate at the lowest microwave power slightly. The maximum value of ${\ensuremath{I_{r\!f}}}{\ensuremath{\left( t \right)}}$ was again calculated for each power using the fit in Fig. \[F031806Dfosc\]. The $T_\phi$ value of 16 ns used in the simulations was chosen to best reproduce the decay envelopes over the full range of powers. At the highest power, there is a discrepancy in the oscillation maxima and minima, perhaps indicative of inaccurate parameters for levels [$\left| 3 \right>$]{} and [$\left| 4 \right>$]{}. The envelope of the microwave turn-on is also reflected in the shape of the escape rate. As a final test of the model, we examine multiphoton transitions in the system. Figure \[F010206H1\](c) shows a grayscale plot of the escape rate calculated with a seven-level density-matrix simulation. Nearly all of the features seen in the data of Fig. \[F010206H1\](a) are present in the simulation. As the gray scales are identical, a small quantitative disagreement is visible, particularly for the [$0 \rightarrow 3$]{} three-photon transition. Figure \[F010206H1\](d) shows the normalized power spectral density calculated from Fig. \[F010206H1\](c); it agrees well with the data in Fig. \[F010206H1\](b). Figure \[F010206H1lines\] shows line cuts of the time-domain data (circles) and simulation (solid lines) from Fig. \[F010206H1\] at bias currents of (a) 17.623, (b) 17.596, and (c) 17.571 [$\mu {\ensuremath{\mathrm{A}}}$]{}. These values of $I_b$ correspond to the resonances of the one, two, and three-photon transitions at the high power at which the data were taken. While the decay time and the long-time escape rate [$\Gamma_{\infty}$]{} are reproduced well for the three transitions, the first few nanoseconds are not captured fully. This could be due to distortion of the microwave current by the coupling capacitor [$C_{r\!f}$]{} or other line mismatches. ![\[F010206H1lines\] (Color online) Comparison of possible damping scenarios for multi-photon transitions. The measured escape rates from Fig. \[F010206H1\](a) for the (a) [$0 \rightarrow 1$]{}, (b) [$0 \rightarrow 2$]{}, and (c) [$0 \rightarrow 3$]{} transitions are plotted with circles. The solid lines were calculated with the same simulation (with a common dephasing time $T_\phi$) whose results are shown in Fig. \[F010206H1\](c). The dotted lines were calculated for $T_1 = 10\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ and no additional dephasing, while the dashed lines correspond to $T_1 = 17\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ and pure dephasing with $T_\phi = 16\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ due to low frequency noise (see text).](010206H1linesv3.eps){width="3.0in"} As with all of the simulations discussed in this section so far, the solid lines in Fig. \[F010206H1lines\] were calculated with $T_1 = 17\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ and $T_\phi = 16\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$. This gives $T_2 = 10.9\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ and a Rabi decay time roughly equal to the two-level value[@Torrey49a; @Smith78a] of $T^\prime = {\ensuremath{\left[ 1/{\ensuremath{\left( 2 T_1 \right)}} + 1/{\ensuremath{\left( 2 T_2 \right)}} \right]}}^{-1} = 13.3\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$, consistent with measurements of the decay envelope of the escape rate. If pure dephasing were not present then $T^\prime$ would be 22.7 ns, which is significantly longer than what is observed. Although unlikely, our thermal measurement of $T_1 = 17\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ could be incorrect. To examine this possibility, simulations with $T_1 = 10\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ and $T_2 = 2 T_1$ (which also give $T^\prime = 13.3\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$) are shown with dotted lines in Fig. \[F010206H1lines\]. They are nearly identical to the curves calculated with dephasing, although [$\Gamma_{\infty}$]{} is somewhat smaller. Thus this data set alone cannot rule out dissipation-limited decoherence. However, the shorter $T_1$ reduces the prediction for [$\Gamma_{\infty}$]{} by roughly 15% over the full range of measured powers in Fig. \[F031806DGeq\], suggesting that additional dephasing is instead affecting the Rabi oscillations. While dephasing is needed to faithfully reproduce features of the experimental measurements, its origin is unclear. In the simulations, we assume that each off-diagonal term of the density matrix decays with a common dephasing time of $T_\phi = 16\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ (along with decoherence due to dissipation, which is transition dependent). If this dephasing at high power were due to low frequency noise (similar to the effects of inhomogeneous broadening discussed before),[@Martinis03a; @Xu05b] the corresponding dephasing operator $D$ in [Eq. [(\[eqh\])]{}]{} could be expressed with $L = \sum {\ensuremath{\omega_{0 n}}} / {\ensuremath{\omega_{0 1}}} {\ensuremath{\left| n \right>}} {\ensuremath{\left< n \right|}}$ and $\lambda = 2 / T_\phi$ (similar to the harmonic oscillator number operator). With this choice of operator $L$ and strength $\lambda$, the [$0 \rightarrow 1$]{} dephasing is unchanged. Simulations with this sort of damping are shown with dashed lines in Fig. \[F010206H1lines\]. While the [$0 \rightarrow 1$]{} Rabi oscillations are modeled well, there is far too much decoherence for the higher order transitions. This should also be true for other implementations of low frequency noise that have dephasing rates that scale as $(d {\ensuremath{\omega_{n m}}}/d I_b)^2$. Thus having the ability to measure a wide range of transitions can reveal additional information about decoherence. Discussion and Summary {#SSummary} ====================== We have presented measurements taken on a dc SQUID operated in such a way that one of the junctions behaves much like a simple current-biased junction. We find that the simplified Hamiltonian of [Eq. [(\[eqa\])]{}]{} gives an accurate description of the qubit dynamics. Our observation of subtle features of this model provides further confidence that it can be applied to design the gates needed for quantum computation. The analysis presented here can also be extended to describe the behavior of the other types of superconducting qubits mentioned in Sec. \[SIntro\]. We performed several checks on the model, including measuring the Rabi oscillation frequency for one and two-photon transitions and comparing the data with predictions based on the rotating wave Hamiltonian of [Eq. [(\[eqd\])]{}]{}. We find good agreement, despite the model not containing any damping. Thus the resonance shifts in multilevel Rabi oscillation frequencies are mainly determined by the anharmonic level structure and not the $T_1$ and $T_2$ of the qubit. These multilevel effects were identified under strong driving where it is clear that they could lead to errors in simple two-state rotations. Future experiments could mitigate these effects by proper pulse shaping and operating at low microwave power.[@Steffen03a; @Amin06a; @Lucero08a] We also used a density-matrix simulation, which included the effects of decoherence, to calculate the tunneling escape rate. The results of the simulation agreed well with experimental Rabi oscillation data which were acquired with the device biased so that even the ground-state escape rate [$\Gamma_{0}$]{} was measurable. While this simple measurement (which ends with the qubit in the finite voltage state on every repetition) is not an ideal projective measurement to the qubit basis states [$\left| 0 \right>$]{} and [$\left| 1 \right>$]{}, it is particularly well-suited for the purposes of the current work. For one, it is extremely sensitive to excited state population due to a microwave drive or thermal transitions. The leakage information contained in Fig. \[F031806DGeq\] would be considerably more difficult to obtain from the pulsed single-shot measurements[@Cottet02a; @Claudon04a] that excel at determining the total population not in the ground state; nonetheless, these techniques have been successfully used to measure the second excited state population precisely.[@Palomaki07a; @Claudon07a; @Lucero08a] With the simple tunneling measurement, the large ratio ${\ensuremath{\Gamma_{n+1}}} / {\ensuremath{\Gamma_{n}}}$ provides a natural way of distinguishing which excited state is populated. In addition, as nothing is done to the qubit to initiate a measurement, this method is not limited by a measurement fidelity and does not suffer from the potential problems associated with changing the bias location in order to perform state readout.[@Cooper04a; @Zhang06a; @Palomaki07a; @Claudon07a] The main drawback to the escape rate measurement is that it does not directly produce individual state populations. However, the density-matrix simulations do provide each level’s contribution to the total escape rate and its occupation probability. An example for a [$0 \rightarrow 1$]{} oscillation is shown in Figs. \[F032106FN10\](b) and \[F032106FN10\](c). The plots suggest that for high power, the escape rate during a Rabi oscillation is dominated by the contributions $\rho_{22} {\ensuremath{\Gamma_{2}}}$ and $\rho_{33} {\ensuremath{\Gamma_{3}}}$ from states [$\left| 2 \right>$]{} and [$\left| 3 \right>$]{}. The figure also shows that $\rho_{22} {\ensuremath{\left( t \right)}}$ and $\rho_{33} {\ensuremath{\left( t \right)}}$ have nearly the same form as $\rho_{11} {\ensuremath{\left( t \right)}}$, albeit with a much smaller oscillation amplitude.[@Amin06a] Thus changes in $\Gamma$ are reflective of the underlying oscillation of [$\left| 1 \right>$]{}, which is why we believe the frequency analysis in Fig. \[F031806Dfosc\] is valid. The populations in Fig. \[F032106FN10\](c) have been normalized at each time $t$. While the probability of the junction leaving the supercurrent state during a microwave pulse depends strongly on [$I_{r\!f}$]{} and the bias conditions, $\rho_{tot} = 0.42$ at $t = 30\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ for this data set. Thus, while we wish to use tunneling as a probe of $\rho_{nn}$, it is clearly playing a large role in the evolution of the system. However, repeating the density-matrix simulation in the absence of tunneling (*i.e.*, $G {\ensuremath{\rho}}= 0$) results in a less than 5% change in the normalized populations of [$\left| 0 \right>$]{} and [$\left| 1 \right>$]{} during the oscillation. The subsequent decay is significantly different without the fast decays due to escape. These insights from the simulation hinge on the assumption that tunneling does not affect $T_1$ and $T_\phi$. A few remarks should be made about the many input parameters required for the simulations. For example, the seven-level density-matrix calculation shown in Fig. \[F010206H1\](c) needed seven escape rates, six energy-level spacings, and 28 matrix elements of $\gamma_1$, all of which are functions of $I_b$. We determined these from the properties of a tilted washboard potential defined by $I_{01}$ and $C_1$, which were measured independently by low power spectroscopy. To accurately reproduce the measured escape rate (particularly at low power), we also had to add several other features to the simulation, each motivated by separate measurements. In particular, we added a microwave source at [$\omega_{0 2}$]{} to mimic noise (whose magnitude was found from the microwave-free escape rate), inhomogeneous broadening due to bias noise (estimated from spectroscopic resonance widths), and a finite time resolution (consistent with the bandwidth of the detection electronics). The dissipation time $T_1 = 17\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ was estimated by an independent thermal escape rate experiment,[@Dutta04a] while the dephasing time $T_\phi = 16\ {\ensuremath{{\ensuremath{\mathrm{ns}}}}}$ (and the way it was incorporated into the simulation) was chosen to maximize the agreement with the Rabi data. For simplicity, both time constants were assumed to be independent of frequency. Removing this condition in the analysis could yield additional information about the decoherence in the system, as the oscillations are most sensitive to noise at the Rabi frequency.[@Martinis03a; @Ithier05a] Finally, the conversion between the microwave power at its generator and the current through the qubit junction was itself calibrated by the observed oscillation frequency, and the resulting factor was allowed to be a function of microwave frequency. Thus a wide range of multilevel phenomena was explained with just a few truly free parameters. In addition, we found that the output of the simulations (whether that be an oscillation frequency or escape rate) converged as the number of levels $N$ was increased. For any of the five or seven-level simulations discussed here, an additional level produced a change too small to be detected by the experiment; more levels were required at high power or low $I_b$ for the solution to converge satisfactorily. Surprisingly, the highest included levels are predicted to lie above the barrier. However, small errors in the energy levels or matrix elements for the highest levels do not produce large changes in the final escape rates. Certain features of the data were not reproduced by the simulations. Although not discussed here, the transition spectra of our Nb qubits show a series of splittings (all less than 10 MHz wide), similar to but smaller than those reported in other superconducting qubits.[@Simmonds04a; @Plourde05a; @Claudon07a; @Dong07a] The extra degrees of freedom responsible for these features were not included in the device Hamiltonian, except perhaps in some effective way through $T_1$ and $T_\phi$ of the density-matrix simulation. For weak coupling and high microwave power, the impact of individual two-level systems on Rabi oscillations is expected to be small.[@Meier05a; @Ashhab06a] However, it is unclear if a bath of quantum systems could be responsible for the anomalous decay in Fig. \[F032106FN10\]. As other groups have observed strong dissipation in Nb trilayer junctions,[@Martinis02a; @Lisenfeld07a] it is likely that the wiring insulation or other details of the fabrication are playing a critical role.[@Martinis05a] While further work is needed to identify (and potentially eliminate) the actual microscopic sources of decoherence in this system, the multilevel features in the dynamics of our qubit are largely understood. We have benefited greatly from discussions with B. K. Cooper, P. R. Johnson, H. Kwon, J. Matthews, K. D. Osborn, B. S. Palmer, A. J. Przybysz, R. C. Ramos, C. P. Vlahacos, and H. Xu. The work was funded by the National Science Foundation through the QuBIC Program, the National Security Agency, and the state of Maryland through the Center for Nanophysics and Advanced Materials, formerly the Center for Superconductivity Research. 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--- author: - | , Waseem Kamleh, Derek B. Leinweber and Benjamin J. Owen\ Centre for the Subatomic Structure of Matter,\ Department of Physics,\ University of Adelaide, SA 5005\ E-mail: bibliography: - 'database.bib' title: 'Electromagnetic Form Factors of Excited Nucleons via Parity-Expanded Variational Analysis' --- Introduction ============ In order to evaluate the form factors and transition moments of baryon excitations in lattice QCD, it is necessary to isolate these states at finite momentum. Excited baryons have been isolated on the lattice through a combination of parity projection and variational analysis techniques [@Kiratidis:2015vpa; @Mahbub:2013ala; @Mahbub:2012ri; @Mahbub:2010rm; @Edwards:2012fx; @Edwards:2011jj; @Lang:2016hnn; @Lang:2012db; @Alexandrou:2014mka]. At zero momentum, these techniques are well established and can isolate the states of interest. However, at non-zero momentum, these techniques are vulnerable to opposite parity contaminations. To resolve this issue, we developed the Parity-Expanded Variational Analysis (PEVA) technique [@Menadue:2013kfi]. By introducing a novel Dirac projector and expanding the operator basis used to construct the correlation matrix, we are able to cleanly isolate states of both parities at finite momentum. Utilising the PEVA technique, we are able to present here the world’s first lattice QCD calculations of nucleon excited state form factors free from opposite parity contaminations. Specifically, the Sachs electromagnetic form factors of a localised negative parity nucleon excitation are examined. Furthermore, we clearly demonstrate the efficacy of variational analysis techniques at providing access to ground state form factors with extremely good control over excited state effects. Conventional Variational Analysis ================================= We begin by briefly highlighting where opposite parity contaminations enter into conventional variational analysis techniques and motivate the PEVA technique. In these proceedings we use the Pauli representation for Dirac matrices. In order to discuss opposite parity contaminations, we need to be able to categorise states by their parity. However, eigenstates of non-zero momentum are not eigenstates of parity, so we must categorise boosted states by their rest-frame parity. To perform a conventional variational analysis on spin-1/2 baryons, we take a basis of $n$ conventional baryon operators ${\chi^i}$, which couple to states of both parities, $$\begin{aligned} \braket{\Omega | \chi^i | B^+} = \lambda^i_{B^+} \sqrt{\frac{m_{B^+}}{E_{B^+}}} \, {u_{B^+}(p, s)} \, , \quad \braket{\Omega | \chi^i | B^-} = \lambda^i_{B^-} \sqrt{\frac{m_{B^-}}{E_{B^-}}} \, \gamma_5 \, {u_{B^-}(p, s)}\,,\end{aligned}$$ and use this basis to form an $n\times n$ matrix of two-point correlation functions $$\begin{aligned} \mathcal{G}^{ij}(\vect{p};\, t) \definedby &\sum_{\vect{x}} \ee^{\ii\vect{p}\cdot\vect{x}}\, \braket{\Omega | \chi^{i}(x) \, \adjoint\chi^{j}(0) | \Omega} \,.\end{aligned}$$ This correlation matrix contains states of both parities, so we introduce the ‘parity projector’ ${\Gamma_{\!\pm}} = \left(\gamma_4\pm\mathbb{I}\right)/2$, and take the spinor trace, defining the projected correlation matrix $G^{ij}({\Gamma_{\!\pm}};\, \vect{p},\, t) \definedby \tr\left({\Gamma_{\!\pm}} \, \mathcal{G}^{ij}(\vect{p},\, t)\right)$. By inserting a complete set of states between the two operators, and noting the use of Euclidean time, we can rewrite this projected correlation matrix as $$\begin{aligned} G^{ij}({\Gamma_{\!\pm}};\, \vect{p},\, t) =\, & \sum_{B^{+}} \ee^{-E_{B^{+}}(\vect{p})\,t} \, \lambda^i_{B^{+}} \, \adjoint\lambda^j_{B^{+}} \frac{E_{B^{+}}\!(\vect{p}) \,\pm m_{B^{+}}}{2E_{B^{+}}\!(\vect{p})} + \sum_{B^{-}} \ee^{-E_{B^{-}}(\vect{p})\,t} \, \lambda^i_{B^{-}} \, \adjoint\lambda^j_{B^{-}} \frac{E_{B^{-}}\!(\vect{p}) \,\mp m_{B^{-}}}{2E_{B^{-}}\!(\vect{p})}\,,\end{aligned}$$ At zero momentum, $E_{B}(\vect{0}) = m_{B}$ and the projected correlation matrices will each contain terms of a single parity. However, at non-zero momentum $E_{B}(\vect{p}) \ne m_{B}$ and the projected correlation matrices contain $O\left((E - m)/2E\right)$ opposite parity contaminations. These opposite parity contaminations were investigated in Ref. [@Lee:1998cx]. Parity-Expanded Variational Analysis {#sec:PEVA} ==================================== To solve the problem of opposite parity contaminations at finite momentum, we developed the PEVA technique [@Menadue:2013kfi]. In this section, we summarise the PEVA technique, and describe how it applies to form factor calculations. The PEVA technique works by expanding the operator basis of the correlation matrix to isolate energy eigenstates of both rest-frame parities simultaneously while still retaining a signature of this parity. By considering the Dirac structure of the unprojected correlation matrix, we construct the novel momentum-dependent projector ${{\Gamma_{\!\vect{p}}}} \definedby \frac{1}{4} \left(\identity + \gamma_4\right) \left(\identity - \ii \gamma_5 \gamma_k \unitvect{p}_k\right)$. This allows us to construct a set of “parity-signature” projected operators $\left\{\chi^i_{\vect{p}} = {{\Gamma_{\!\vect{p}}}} \, \chi^i \, , \; \chi^{i'}_{\vect{p}} = {{\Gamma_{\!\vect{p}}}} \, \gamma_5 \, \chi^i\right\}$, where the primed indices denote the inclusion of $\gamma_5$, inverting the way the operators transform under parity. Unlike the conventional opertors ${\,\chi^i\,}$, the inclusion of ${{\Gamma_{\!\vect{p}}}}$ ensures that the operators $\chi^i_{\vect{p}}$ and $\chi^{i'}_{\vect{p}}$ have definite parity at zero momentum without requiring projection by $\Gamma_{\pm}$. By performing a variational analysis with this expanded basis [@Menadue:2013kfi], we construct optimised operators $\phi^{\alpha}_{\vect{p}}(x)$ that couple to each state $\alpha$. We can then use these operators to calculate the three point correlation function $$\begin{aligned} \mathcal{G}^{\mu}_{+}(\vect{p'}, \vect{p};\, t_2, t_1;\, \alpha) \definedby &\sum_{\vect{x_2},\vect{x_1}} e^{-\ii \vect{p'}\cdot\vect{x_2}} \, e^{\ii (\vect{p'} - \vect{p})\cdot\vect{x_1}} \braket{\,\phi^{\alpha}_{\vect{p'}}(x_2)\,|\,J^{\mu}(x_1)\,|\,\adjoint\phi^{\alpha}_{\vect{p}}(0)\,}\,,\end{aligned}$$ where $J^{\mu}$ is the $O(a)$-improved [@Martinelli:1990ny] conserved vector current used in [@Boinepalli:2006xd], inserted with some momentum transfer $q = p' - p$. We can take the spinor trace of this with some projector $\Gamma$ to get the projected three point correlation function $G^{\mu}_{+}(\vect{p'}, \vect{p};\, t_2, t_1;\, \Gamma;\, \alpha) \definedby \tr\left(\Gamma \, \mathcal{G}^{\mu}_{+}(\vect{p'}, \vect{p};\, t_2, t_1;\, \alpha) \right)$. There is an arbitrary sign choice in the definition of ${{\Gamma_{\!\vect{p}}}}$, so it is convenient to define ${{\Gamma_{\!\vect{p}}}}' \definedby \frac{1}{4} \left(\identity + \gamma_4\right) \left(\identity + \ii \gamma_5 \gamma_k \unitvect{p}_k\right) = {{\Gamma_{\!\vect{-p}}}}$, which is equally valid. We can then use this alternate projector in constructing an alternate sink operator $\phi'^{\alpha}_{\vect{p}}(x)$, while leaving the source operator unchanged. This gives us an alternate three point correlation function, $\mathcal{G}^{\mu}_{-}(\vect{p'}, \vect{p};\, t_2, t_1;\, \alpha)$, leading to an alternate projected three point correlation function, $G^{\mu}_{-}(\vect{p'}, \vect{p};\, t_2, t_1;\, \Gamma;\, \alpha)$. We can then construct the reduced ratio, $$\begin{aligned} \adjoint{R}_{\pm}(\vect{p'}, \vect{p};\, \alpha;\, r, s) \definedby \, & \sqrt{\left|\frac{r_{\mu} \, G^{\mu}_{\pm}(\vect{p'}, \vect{p};\, t_2, t_1;\, s_{\nu} \, {\Gamma_{\!\nu}};\, \alpha) \; r_{\rho} \, G^{\rho}_{\pm}(\vect{p}, \vect{p'};\, t_2, t_1;\, s_{\sigma} \, {\Gamma_{\!\sigma}};\, \alpha)}{G(\vect{p'};\, t_2;\, \alpha) \, G(\vect{p};\, t_2;\, \alpha)}\right|} \\ & \quad \times \mathrm{sign}\left(r_{\gamma} \, G^{\gamma}_{\pm}(\vect{p'}, \vect{p};\, t_2, t_1;\, s_{\delta}\, \Gamma_{\delta};\, \alpha)\right) \sqrt{\frac{2 E_{\alpha}(\vect{p})}{E_{\alpha}(\vect{p})+m_{\alpha}}} \, \sqrt{\frac{2 E_{\alpha}(\vect{p'})}{E_{\alpha}(\vect{p'})+m_{\alpha}}}\,,\end{aligned}$$ where $r_{\mu}$ and $s_{\mu}$ are coefficients selected to determine the form factors. By investigating the $r_{\mu}$ and $s_{\mu}$ dependence of $\adjoint{R}_{\pm}$, we find that the clearest signals are given by $$\begin{aligned} R^{T}_{\pm} &= \frac{2}{1 \pm\, \unitvect{p} \cdot \unitvect{p'}} \; \adjoint{R}_{\pm}\left(\vect{p'}, \vect{p};\, \alpha;\, (1, \vect{0}), (1, \vect{0})\right) \,,\;\mathrm{and} \\ R^{S}_{\mp} &= \frac{2}{1 \pm\, \unitvect{p} \cdot \unitvect{p'}} \; \adjoint{R}_{\mp}\left(\vect{p'}, \vect{p};\, \alpha;\, (0, \unitvect{r}), (0, \unitvect{s})\right) \,,\end{aligned}$$ where $\unitvect{s}$ is chosen such that $\vect{p} \cdot \unitvect{s} = 0 = \vect{p'} \cdot \unitvect{s}$, $\unitvect{r}$ is equal to $\unitvect{q} \times \unitvect{s}$, and the sign $\pm$ is chosen such that $1 \pm \unitvect{p} \cdot \unitvect{p'}$ is maximised. We can then find the Sachs electric and magnetic form factors $$\begin{aligned} G_E(Q^2) =\, &\left[Q^2 \left(E_{\alpha}(\vect{p'}) + E_{\alpha}(\vect{p})\right)\, \left(\left(E_{\alpha}(\vect{p}) + m_{\alpha}\right)\,\left(E_{\alpha}(\vect{p'}) + m_{\alpha}\right) \mp \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr|\right) \, R^{T}_\pm \right. \\ & \left. \quad \pm\, 2 \bigl|\vect{q}\bigr| \left(1 \mp\, \unitvect{p} \cdot \unitvect{p'}\right) \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr| \left(\left(E_{\alpha}(\vect{p}) + m_{\alpha}\right)\,\left(E_{\alpha}(\vect{p'}) + m_{\alpha}\right) \pm \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr|\right) \, R^{S}_{\mp} \right] \\ &/ \left[ 4 m_{\alpha} \left[ \left( E_{\alpha}(\vect{p}) \, E_{\alpha}(\vect{p'}) + m_{\alpha}^2 \mp\, \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr| \right) \bigl|\vect{q}\bigr|^2 + 4 \bigl|\vect{p}\bigr|^2 \bigl|\vect{p'}\bigr|^2 \left(1 \mp\, \unitvect{p} \cdot \unitvect{p'}\right) \right] \right] \,,\;\mathrm{and} \\ G_M(Q^2) =\, &\left[\pm\, 2 \left(1 \mp\, \unitvect{p} \cdot \unitvect{p'}\right) \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr| \left(\left(E_{\alpha}(\vect{p}) + m_{\alpha}\right)\,\left(E_{\alpha}(\vect{p'}) + m_{\alpha}\right) \pm \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr|\right) \, R^{T}_\pm \right. \\ & \left. \quad - \bigl|\vect{q}\bigr| \left(E_{\alpha}(\vect{p'}) + E_{\alpha}(\vect{p})\right)\, \left(\left(E_{\alpha}(\vect{p}) + m_{\alpha}\right)\,\left(E_{\alpha}(\vect{p'}) + m_{\alpha}\right) \mp \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr|\right) \, R^{S}_{\mp} \right] \\ &/ \left[ 2 \left[ \left( E_{\alpha}(\vect{p}) \, E_{\alpha}(\vect{p'}) + m_{\alpha}^2 \mp\, \bigl|\vect{p}\bigr| \bigl|\vect{p'}\bigr| \right) \bigl|\vect{q}\bigr|^2 + 4 \bigl|\vect{p}\bigr|^2 \bigl|\vect{p'}\bigr|^2 \left(1 \mp\, \unitvect{p} \cdot \unitvect{p'}\right) \right] \right]\,.\end{aligned}$$ The details of this procedure will be presented in full in Ref. [@stokes:formfactors]. Results ======= In this section, we present world-first lattice QCD calculations of the Sachs electric and magnetic form factors of the ground state nucleon and first negative-parity excitation with good control over opposite parity contaminations. We compare the results obtained by the PEVA technique to an analysis using conventional parity projection. These results are calculated on the second heaviest PACS-CS $(2+1)$-flavour full-QCD ensemble [@Aoki:2008sm], made available through the ILDG [@Beckett:2009cb]. This ensemble uses a $32^3 \times 64$ lattice, and employs an Iwasaki gauge action with $\beta = 1.90$ and non-perturbatively $O(a)$-improved Wilson quarks. We use the $m_{\pi}=\SI{570}{\mega\electronvolt}$ PACS-CS ensemble, and set the scale using the Sommer parameter with $r_0 = \SI{0.49}{\femto\meter}$, giving a lattice spacing of $a = \SI{0.1009(23)}{\femto\meter}$. With this scale, our pion mass is . We used 343 gauge field configurations, with a single source location on each configuration. $\chi^2 / \mathrm{dof}$ is calculared with the full covariance matrix, and all fits have $\chi^2 / \mathrm{dof}\ < 1.2$. For the analyses in this section, we start with a basis of eight operators, by taking two conventional spin-$\nicefrac{1}{2}$ nucleon operators ($\,\chi_1 = \epsilon^{abc} \, [{u^a}\transpose (C\gamma_5) \, d^b] \, u^c$, and $\chi_2 = \epsilon^{abc} \, [{u^a}\transpose (C) \, d^b] \, \gamma_5 \, u^c\,$), and applying 16, 35, 100, and 200 sweeps of gauge invariant Gaussian smearing when creating the propagators [@Mahbub:2013ala]. For the conventional variational analysis, we take this basis of eight operators and project with ${\Gamma_{\!\pm}}$, and for the PEVA analysis, we parity expand the basis to sixteen operators and project with ${{\Gamma_{\!\vect{p}}}}$ and ${{\Gamma_{\!\vect{p}}}}'$. To extract the form factors, we fix the source at time slice 16, and the current insertion at time slice 21. We choose time slice 21 by inspecting the two point correlation functions associated with each state and observing that excited state contaminations are strongly suppressed by time slice 21. We then extract the form factors from the ratios given in Sec. \[sec:PEVA\] for every possible sink time and look for a plateau consistent with a single-state ansatz. Beginning with the ground state, in Fig. \[fig:groundstate:GE\] we plot $G_E(Q^2)$ and in Fig. \[fig:groundstate:GM\] we plot $G_M(Q^2)$ with respect to sink time, at $Q^2 = \SI{0.144}{\giga\electronvolt^2}$. There are only slight differences between the results extracted by PEVA and the results extracted by a conventional variational analysis in this case. We believe this is because the opposite parity contaminations are small, and come from heavier states that are suppressed by Euclidean time evolution. For both the PEVA and conventional variational analysis, we see very clear and clean plateaus in the form factors, indicating very good control over excited state contaminations. This supports previous work demonstrating the utility of variational analysis in calculating baryon matrix elements [@Dragos:2016rtx; @Owen:2012ts]. By using such techniques we are able to cleanly isolate precise values for the Sachs electric and magnetic form factors of the ground state nucleon. Moving on to the first negative parity excited state, in Fig. \[fig:firstneg:GE\] we plot $G_E(Q^2)$ at $Q^2 = \SI{0.146}{\giga\electronvolt^2}$ vs the sink time. We see that the PEVA analysis (closed points) allows us to fit a plateau (shaded bands with solid lines) at a much earlier time, and with a significantly different value to the conventional analysis (open points, dashed lines). This demonstrates the effectiveness of the PEVA technique at removing opposite parity contaminations. The localised nature of this state is manifest in the large values for $G_E(Q^2)$, similar to that for the proton. We also see similar improvements in $G_M(Q^2)$, as presented in Fig. \[fig:firstneg:GM\]. Conclusion ========== We have demonstrated the effectiveness of the PEVA technique specifically and variational analysis techniques in general at controlling excited state effects. For the ground state nucleon, conventional variational analysis techniques are sufficient to provide clean plateaus that allow for the effective extraction of form factors. However, for excited states, opposite parity contaminations have a clear and significant effect. The PEVA technique allows us to remove these contaminations and calculate the form factors of excited nucleons for the first time.

--- abstract: 'Experimental data indicate small spin-orbit splittings in hadrons. For heavy-light mesons we identify a relativistic symmetry that suppresses these splittings. We suggest an experimental test in electron-positron annihilation. Furthermore, we argue that the dynamics necessary for this symmetry are possible in QCD.' author: - | P.R. Page[^1], T. Goldman[^2], and J.N. Ginocchio[^3]\ [*MS B283, Theoretical Division, Los Alamos National Laboratory*]{}\ [*Los Alamos, New Mexico 87545*]{} date: November 2000 title: '\' --- epsf =-.17in =0.15in =6ex plus 0.2pt minus 0.2pt plus 0.2pt minus 0.2pt Introduction ============ Recently, Isgur[@nathan] has re-emphasized the experimental fact that spin-orbit splittings in meson and baryon systems, which might be expected to originate from one-gluon-exchange (OGE) effects between quarks, are absent from the observed spectrum. He conjectures that this is due to a fairly precise, but accidental, cancellation between OGE and Thomas precession effects, each of which has “splittings of hundreds of MeV” [@nathan]. Taking the point of view that precise cancellations reflect symmetries rather than accidents, we have examined what dynamical requirements would lead to such a result. One of us recently observed[@gino] that a relativistic symmetry is the origin of pseudospin degeneracies first observed in nuclei more than thirty years ago[@kth; @aa]. We find that a close relative of that dynamics can account for the spin degeneracies observed in hadrons composed of one light quark (antiquark) and one heavy antiquark (quark). Below, we first elucidate the experimental evidence for small spin-orbit splittings. Then we identify the symmetry involved in terms of potentials in the Dirac Hamiltonian for heavy-light quark systems, and note the relation to the symmetry for pseudospin. We show that the former symmetry predicts that the Dirac momentum space wavefunctions will be identical for the two states in the doublet, leading to a proposed experimental test. Finally, we argue that the required relation between the potentials may be plausible from known features of QCD. Experimental and Lattice QCD Spectrum ===================================== In the limit where the heavy (anti)quark is infinitely heavy, the angular momentum of the light degrees of freedom, $j$, is separately conserved[@hqet]. The states can be labelled by $l_{j}$, where $l$ is the orbital angular momentum of the light degrees of freedom. In non–relativistic models of conventional mesons the splitting between $l_{l+\frac{1}{2}}$ and $l_{l-\frac{1}{2}}$ levels, e.g. the $p_{\frac{3}{2}}$ and $p_{\frac{1}{2}}$ or $d_{\frac{5}{2}}$ and $d_{\frac{3}{2}}$ levels, can [*only*]{} arise from spin-orbit interactions[@nathan]. The $p_{\frac{1}{2}}$ level corresponds to two degenerate broad states with different total angular momenta $J=j\pm s_Q$ (here $j=\frac{1}{2})$, where $s_Q$ is the spin of the heavy (anti)quark[@hqet]. For example, in the case of $D$-mesons, $s_Q=\frac{1}{2}$ and the two states are called $D^{\ast}_0$ and $D_1'$. There are also two degenerate narrow $p_{\frac{3}{2}}$ states $D_1$ and $D^{\ast}_2$[@hqet]. The degenerate states separate as one moves slightly away from the heavy quark limit, and their spin-averaged mass remains approximately equal to the mass before separation. For the $D$–mesons, the CLEO collaboration claims a broad $J^P=1^+$ state at $2461^{+41}_{-34}\pm 10 \pm 32$ MeV[@cleo], belonging to the $p_{\frac{1}{2}}$ level, in close vicinity to the $D^{\ast}_2$ at $2459\pm 2$ MeV[@pdg98], belonging to the $p_{\frac{3}{2}}$ level, indicating a remarkable $p_{\frac{3}{2}}$-$p_{\frac{1}{2}}$ spin-orbit degeneracy of $-2\pm 50$ MeV. It is appropriate to extract the spin-orbit splitting this way since: Firstly, the charm quark behaves like a heavy quark. Secondly, the difference between the $D_1'$ and $D^{\ast}_2$ levels is the best indicator[@l3] of the $p_{\frac{3}{2}}$-$p_{\frac{1}{2}}$ splitting in the absence of experimental data[^4] on the $D^{\ast}_0$, as opposed to the difference between the $D_1'$ and spin-averaged $p_{\frac{3}{2}}$ level at $2446\pm 2$ MeV. Spin-averaged masses are determined from experiment[@pdg98]. For the $K$-mesons, the $p_{\frac{1}{2}}$ level is at $1409\pm 5$ MeV, with $p_{\frac{3}{2}}$ nearby at $1371\pm 3$ MeV, corresponding to a $p_{\frac{3}{2}}$-$p_{\frac{1}{2}}$ splitting of $-38\pm 6$ MeV. The splitting between the higher-lying $d_{\frac{5}{2}}$ and $d_{\frac{3}{2}}$ levels is $-4\pm 14$ MeV or $41\pm 13$ MeV, depending on how the states are paired into doublets. These results indicate a near spin-orbit degeneracy if the strange quark can be treated as heavy, although it has certainly not been established that such a treatment is valid. For $B$-mesons, both L3 [@l3] and OPAL [@opal] have performed analyses, using input from theoretical models and heavy quark effective theory, to determine that the $p_{\frac{3}{2}}$-$p_{\frac{1}{2}}$ splitting is $97\pm 11$ MeV (L3) or $-109\pm 14$ MeV (OPAL). Note that these are [*not*]{} model-independent experimental results. In the same analyses the mass difference between the $B_2^{\ast}$ and $B_0^{\ast}$, an approximate indicator of the $p_{\frac{3}{2}}$-$p_{\frac{1}{2}}$ splitting, is $110\pm 11$ MeV (L3) or $-89\pm 14$ MeV (OPAL). The L3 result agrees with lattice QCD estimates of $155^{+9}_{-13} \pm 32$ MeV[@gupta] and $183\pm 34$ MeV[@ali]. However, according to other estimates[@lewis], the splitting is less than 100 MeV, and consistent with zero. Recently, $31\pm 18$ MeV was calculated [@lewis1]. One lattice QCD study found evidence for a change of sign in the splitting somewhere between the charm and bottom quark masses, albeit with large error bars [@hein]. A splitting of 40 MeV serves as a typical example of model predictions [@godfrey], although there is variation in the range -155 to 72 MeV [@models], summarized in ref. [@lewis1]. In order to more quantitatively measure the spin-orbit splitting, define $$r = \frac{(p_{\frac{3}{2}}-p_{\frac{1}{2}})}{((4 p_{\frac{3}{2}}+ 2 p_{\frac{1}{2}})/6 - s_{\frac{1}{2}})},$$ where all entries refer to masses. The experimental data on $D$, $K$ and $B$ mesons give respectively $r=0.00\pm 0.10, \; -0.06\pm 0.00$ and $0.23 \pm 0.04$ (L3) or $-0.23\pm 0.03$ (OPAL). For the Dirac equation with arbitrary vector and scalar Coulomb potentials, the only cases for which the relevant analytic solutions are known, $-0.7 {\stackrel{<}{\scriptstyle \sim}}r {\stackrel{<}{\scriptstyle \sim}}0.6$. It is hence evident that the spin-orbit splittings extracted from experimental results are indeed small. There is also evidence in light quark mesons and baryonic systems that the spin-orbit interaction is small[@nathan]. In non-relativistic models, meson and “two-body” baryon spin-orbit interactions are related and, for a specific class of baryons, the spin-orbit interaction is small for exactly the same reasons that it is small in mesons[@nathan]. A Dynamical Symmetry for the Dirac Hamiltonian ============================================== If we consider a system of a (sufficiently) heavy antiquark (quark) and light quark (antiquark), the dynamics may well be represented by the motion of the light quark (antiquark) in a fixed potential provided by the heavy antiquark (quark). Let us assume that both vector and scalar potentials are present. Then the Dirac Hamiltonian describing the motion of the light quark is $$H = {\vec{\alpha}} \cdot {\vec p} + \beta (m + V_S) + V_V + M , \label{dirac}$$ where we have set $\hbar = c =1$, ${\vec \alpha}$, $\beta $ are the usual Dirac matrices, $\vec p$ is the three-momentum, $m$ is the mass of the light quark and $M$ is the mass of the heavy quark. This one quark Dirac Hamiltonian follows from the two-body Bethe-Salpeter equation in the equal time approximation, the spectator (Gross) equation with a simple kernel, and a two quark Dirac equation, in the limit that $M$ is large[@bethe; @gross; @pik]. If the vector potential, $V_{V}(\vec r)$, is equal to the scalar potential plus a constant potential, $U$, which is independent of the spatial location of the light quark relative to the heavy one, i.e., $V_{V}(\vec r) = V_{S}(\vec r) + U$, then the Dirac Hamiltonian is invariant under a spin symmetry[@bell; @ami], $[\,H\,,\, {\hat { S}}_i\,] = 0$, where the generators of that symmetry are given by, $${\hat {S}}_i = \left ( {{\hat {s}_i} \atop 0 } { 0 \atop { {\hat{\tilde s}}_i}}\right ). \label{sgen}$$ where ${\hat s}_i = \sigma_i/2$ are the usual spin generators, $\sigma_i$ the Pauli matrices, and ${\hat {\tilde s}}_i = U_p\ {\hat {s}}_i \ U_p $ with $U_p = \, {{ \vec \sigma\cdot \vec p} \over p}$. Thus Dirac eigenstates can be labeled by the orientation of the spin, even though the system may be highly relativistic, and the eigenstates with different spin orientation will be degenerate. For spherically symmetric potentials, $V_{V}(\vec r) = V_{V}(r),\; V_{S}(\vec r) = V_{s}( r)$, the Dirac Hamiltonian has an additional invariant algebra; namely, the orbital angular momentum, $${\hat {L}}_i = \left ( {{\hat {\ell}_i} \atop 0 } { 0 \atop { {\hat{\tilde \ell}}_i}}\right ), \label{jgen}$$ where ${\hat {\tilde \ell}}_i = U_p\, {\hat \ell}_i$ $U_p$ and ${\hat\ell}_i = (\vec r \times \vec p)_i$. This means that the Dirac eigenstates can be labeled with orbital angular momentum as well as spin, and the states with the same orbital angular momentum projection will be degenerate. Thus, for example, the $n_r\ p_{1/2}$ and $n_r\ p_{3/2}$ states will be degenerate, where $n_r$ is the radial quantum number. Thus, we have identified a symmetry in the heavy-light quark system which produces spin-orbit degeneracies independent of the details of the potential. If this potential is strong, the heavy-light quark system will be very relativistic; that is, the lower component for the light quark will be comparable in magnitude to the upper component of the light quark. It is remarkable that non-relativistic behaviour of energy levels can arise for such fully relativistic systems. This symmetry is similar to the relativistic symmetry[@gino] identified as being responsible for pseudospin degeneracies observed in nuclei[@kth; @aa]. In contrast to spin symmetry, pseudospin symmetry has the pairs of states $((n_r-1)s_{1/2}, n_rd_{3/2})$,\ $((n_r-1)p_{3/2}, n_rf_{5/2})$, etc. degenerate, making the origin of this symmetry less transparent. The pseudospin generators are $${\hat {\tilde S}}_i = \left ( {{\hat {\tilde s}_i} \atop 0 } { 0 \atop { {\hat { s}}_i}}\right ). \label{gen}$$ For pseudospin symmetry, the nuclear mean scalar and vector potential must be equal in magnitude and opposite in sign, up to a constant, $V_{V} = -V_{S}+U$. Relativistic mean field representations of the nuclear potential do have this property; that is, $V_{S}~\approx~-V_{V}$[@dirk; @mad]. We will return later to the question of whether the relation $V_{V} = V_{S} + U$ arises in QCD. It has previously been observed that pseudospin symmetry improves with increasing energy of the states, for various potentials[@gino]. A similar behaviour may be expected for spin symmetry, consistent with the experimental observations that spin–orbit splittings decrease for higher mass states[@nathan; @pdg98]. The Dirac Hamiltonian (\[dirac\]) encompasses the effects of the OGE and Thomas precession spin-dependent terms customarily included in non-relativistic models[@nathan]. Experimental Test ================= In the spin symmetry limit, the radial wavefunctions of the upper components of the Dirac wavefunction of the two states in the spin doublet will be identical, behaving “non-relativistically”, whereas the lower components will have different radial wavefunctions. This follows from the form of the spin generators given in Equation (\[sgen\]). The $(1,1)$ entry of the operator matrix is simply the non-relativistic spin operator which relates the upper component of the Dirac wavefunction of one state in the doublet to the upper component of the other state in the doublet. Since this operator does not affect the radial wavefunction, the two radial wavefunctions must be the same. By contrast, the lower component wavefunction is operated on by $U_p$ which does operate on the radial wavefunction because of the momentum operator. As an example, we show in Figure 1 the upper and lower components for Dirac wavefunctions of the $p_{1/2} - p_{3/2}$ doublet. The scalar and vector potentials were determined by matching the available spectral data of the $D$-mesons, assuming a $p_{\frac{3}{2}}-p_{\frac{1}{2}}$ splitting at the lower end of the range defined by the experimental value of $-2\pm 50 $ MeV. This maximizes the wavefunction differences. In this realistic case, $V_V \approx V_S + U$, so the radial wavefunctions for the upper components are not exactly identical but are very close, whereas the radial wavefunctions for the lower components are very different. Likewise the momentum space wavefunctions for the upper components will be very similar, as seen in Figure 2, again because the spin operator does not affect the wavefunction. However, since $U_p$ depends only on the angular part of the momentum, $ \hat p = {\vec p \over p}$, it does not affect the radial momentum space wavefunction. In Figure 2 we see that the radial momentum space wavefunctions are very similar for the lower components as well. This prediction of the symmetry can be tested in the following experiment. The annihilation $e^+e^- \rightarrow D^{\ast}_0 D^{\ast}_0$, $D^{\ast}_0 D^{\ast}_2$ and $D^{\ast}_2 D^{\ast}_2$ allows for the extraction of the $D^{\ast}_0$ and $D^{\ast}_2$ electromagnetic static form factors and the $D^{\ast}_0$ to $D^{\ast}_2$ electromagnetic transition form factor. The photon interaction ensures that all radial wavefunctions of the light quark are accessed. When spin symmetry is realised, there are only two independent radial momentum space wavefunctions, which should enable the prediction of one of the three form factors in terms of the other two. This should enable the verification of the predictions of spin symmetry. On the other hand, non-relativistic models, with no lower components for the wavefunctions, have only one independent radial wavefunction, which will lead to the prediction of two of the form factors in terms of the remaining one. This might be too restrictive. The proposed experiment can be carried out at the Beijing Electron Positron Collider at an energy of approximately 1 GeV above the $\psi(4040)$ peak in the final state $DD\pi\pi$. An equivalent experiment for K-mesons would involve detection of the $KK\pi\pi$ final state, which has already been measured[@dm2]. The wavefunctions of $K$-mesons fitting the experimental spectrum show similar behaviour to the $D$-mesons, with the $p_{\frac{3}{2}}$ and $p_{\frac{1}{2}}$ wavefunctions even more similar than in Figures 1a and 2. If B-mesons do also exhibit spin symmetry, one can do equivalent experiments around 1 GeV above the $\Upsilon$(3S) peak at the SLAC, KEK or CESR B-factories. QCD Origins =========== If such a dynamical symmetry can explain the suppression of spin-orbit splitting in the hadron spectrum, the question remains as to why it might be expected to appear in QCD. To address this, we first recall the ongoing argument as to whether confinement corresponds to a vector or scalar potential[@refs]. The first natural expectation was that confinement reflected the infrared growth of the QCD coupling constant, enhancing the color-Coulomb interaction at large distances, see e.g. Ref.([@BuchTye]). An involved two- (or multi-) gluon effect has been proposed[@goldh] to account for the origin of a scalar confining potential. The existence of one or the other of these vector and scalar potentials is not necessarily exclusionary – they may both be realised. The arguments in Ref.([@Goldman]) suggest further that they are related, with the scalar exceeding the vector by an amount which may be approximately constant as one saturates into the linear confining region at large separations. We very briefly reiterate the basic argument of Ref.([@Goldman]) here. The starting point is to accept the standard approach[@BuchTye] that renormalization-group-improved single-gluon-exchange produces a linearly increasing vector potential between a quark and an antiquark. One then considers what to expect for multiple gluon exchange, starting with two gluons. Since two gluons are attracted to each other in a color singlet channel, and also have a zero mass threshold (as for massless quark-antiquark pairs), it is reasonable to conclude that a (Lorentz and color) scalar gluonic condensate develops, along with a mass gap for a glueball state. These developments are indeed observed in lattice QCD calculations. Ref.([@Goldman]) goes on to argue that renormalization-group-improved single-glueball-exchange involves the square of the QCD coupling and so, despite the massiveness of the object exchanged, also leads to a (now scalar) confining potential between quarks and antiquarks. This further implies that the ratio of the slopes of the two potentials in their common linear (confining) region is given by the square of the ratio of the QCD scale for growth of the coupling constant to the value of the mass gap of the condensate formation. This ratio may be expected to be of order one as both quantities are determined by the underlying QCD scale. If the two potentials do indeed have similar slopes in the region outside that dominated by the color Coulomb interaction, they would necessarily differ only by an approximately constant value, in that region. Thus, the origin of the dynamical symmetry may not be unreasonable, and may indeed be a natural outcome of non-perturbative QCD. On the other hand, [*identically*]{} equal vector and scalar potentials, except for a constant difference, would appear to be coincidental. An ameliorating effect is that to produce an approximation to the spin symmetry of Eq. (2) this condition need only hold in regions where the wavefunctions are substantial. The determination of QCD potentials, from models like the minimal area law, stochastic vacuum model, or dual QCD, and from lattice QCD, is hampered by the problem of rigorously defining the concept of a potential from QCD when one quark is light. It suffices to say that there is no agreement on the mixed Lorentz character of the potential even between two heavy quarks[@ebert], where the potential can be rigorously defined, although lattice QCD results are consistent with simply a vector Coulomb and scalar linear potential[@bali]. Summary ======= The observation of “accidental” spin-orbit degeneracies observed in heavy-light quark mesons can be explained by a relativistic symmetry of the Dirac Hamiltonian which occurs when the vector and scalar potentials exerted on the light quark by the heavy antiquark differ approximately by a constant, $V_V \approx V_S + U$. Conversely, if future experiments determine that spin-orbit splittings are small not only for the lowest excited states in mesons but are small throughout the meson spectrum, this experimental fact dictates that the effective QCD vector and scalar potentials between a quark and antiquark are approximately equal up to a constant, which would be a significant observation about the nature of non-perturbative QCD. Furthermore, the approximate symmetry predicts that the spatial Dirac wavefunction for the spin doublets will be approximately equal in momentum space, a feature which can be tested in electron-positron annihilation. We have argued that $V_V \approx V_S + U$ may occur in QCD, particularly for regions of space dominated by the light quark wavefunction. Work is in progress to extend this symmetry to purely light quark systems. This research is supported by the Department of Energy under contract W-7405-ENG-36. [99]{} N. Isgur, nucl-th/9908028; [*Phys. Rev.*]{} [**D62**]{} (2000) 014025. J.N. Ginocchio, [*Phys. Rev. Lett.*]{} [**78**]{} (1997) 436; [*Phys. Rep.*]{} [**315**]{} (1999) 231. K.T. Hecht and A. Adler, [*Nucl. Phys.*]{} [**A137**]{} (1969) 129. A. Arima, M. Harvey and K. Shimizu, [*Phys. Lett.*]{} [**30B**]{} (1969) 517. N. Isgur and M.B. Wise, [*Phys. Lett.*]{}[**B232**]{} (1989) 113; [*Phys. Rev. Lett.*]{} [**66**]{} (1991) 1130. S. Anderson [*et al.*]{} (CLEO Collab.), [*Nucl. Phys.*]{} [**A663-664**]{} (2000) 647. Particle Data Group (C. Caso [*et al.*]{}), [*Eur. Phys. J.*]{} [**C3**]{} (1998) 1. M. Acciari [*et al.*]{} (L3 Collab.), [*Phys. Lett.*]{} [**B465**]{} (1999) 323-334. G. Abbiendi [*et al.*]{} (OPAL Collab.), hep-ex/0010031. R. Gupta, [*Nucl. Phys. Proc. Suppl.*]{} [**83**]{} (2000) 295; A. Ali Khan [*et al.*]{}, [*Phys. Rev.*]{} [**D62**]{} (2000) 054505. A. Ali-Khan [*et al.*]{}, [*Nucl. Phys. Proc. Suppl.*]{} [**B73**]{} (1999) 345. R. Lewis and R.M. Woloshyn, [*Nucl. Phys. Proc. Suppl.*]{} [**83**]{} (2000) 316, and references therein; hep-lat/0010001. R. Lewis and R.M. Woloshyn, hep-lat/0003011.on 24 oct 00 Table XXI, J. Hein [*et al.*]{}, [*Phys. Rev.*]{} [ **D62**]{} (2000) 074503. S. Godfrey and R. Kokoski, [*Phys. Rev.*]{} [**D43**]{} (1991) 1679. H.J. Schnitzer, [*Phys. Lett.*]{} [**B76**]{} (1978) 461; [*Nucl. Phys.*]{} [**B207**]{} (1982) 131; Y. Dai, C. Huang and H. Jin, [*Phys. Lett.*]{} [**B331**]{} (1994) 174; P.C. Tiemeyer and J.A. Tjon, [*Phys. Rev.*]{} [**C49**]{} (1994) 494; S.N. Gupta and J.M. Johnson, [*Phys. Rev.*]{} [**D51**]{} (1995) 168; J.W. van Orden, J. Zeng and W. Roberts, [*Phys. Rev.*]{} [**D52**]{} (1995) 5229; N. Isgur, [*Phys. Rev.*]{} [**D57**]{} (1998) 4041; D. Ebert, V.O. Galkin and R.N. Faustov, [*Phys. Rev.*]{} [**D57**]{} (1998) 5663; erratum [*ibid.*]{} [**D59**]{} (1999) 019902; A.H. Orsland and H. Høgaasen, [*Eur. Phys. J.*]{} [**C9**]{} (1999) 503; T.A. Lähde, C.J. Nyfält and D.O. Riska, [*Nucl. Phys.*]{} [**A674**]{} (2000) 141; M. Di Pierro and E. Eichten, hep-ph/0009177. V. B. Mandelzweig and S. J. Wallace, [*Phys. Lett.*]{} [**197B**]{} (1987) 469. F. Gross, [*Phys. Rev.*]{} [**186C**]{} (1969) 1448. H. Pilkuhn, [*J. Phys.* ]{} [**B25**]{} (1992) 289. J.S. Bell and H. Ruegg, [*Nucl. Phys.*]{} [**B98**]{} (1975) 151. J.N. Ginocchio and A. Leviatan, [*Phys. Lett. B*]{} [**425**]{} (1998) 1. B.D. Serot and J.D. Walecka, [*The Relativistic Nuclear Many - Body Problem*]{} in [*Advances in Nuclear Physics*]{}, eds. J.W. Negele and E. Vogt, Vol. [**16**]{} (Plenum, New York, 1986). B.A. Nikolaus, T. Hoch and D.G. Madland, [*Phys. Rev.*]{} [**C46**]{} (1992) 1757. D. Bisello [*et al.*]{} (DM2 Collab.), LAL-90-35 (1990). See for example, Yu.S. Kalashnikova and A.V. Nefediev, [*Phys. Lett.*]{} [**B414**]{} (1997) 149 and references therein; T.J. Allen, M.G. Olsson, S. Veseli and K. Williams, [*Phys. Rev.*]{} [**D55**]{} (1997) 5408; M.G. Olsson, [*Int. J. Mod. Phys.*]{} [**A12**]{} (1997) 4099; M. Avila, hep-ph/9710241; hep-ph/9711500; Yu.A. Simonov, [*Few Body Syst.*]{} [**25**]{} (1998) 45. W. Buchmüller and S.H.H. Tye, [*Phys.  Rev.*]{}[**D24**]{} (1981) 132; A.K. Grant, J.L. Rosner and E. Rynes, [*Phys. Rev.*]{}[**D47**]{} (1993) 1981. A.S. Goldhaber and T. Goldman, [*Phys. Lett.*]{} [**B344**]{} (1995) 319. T. Goldman, in [*Proc. of the Int. Workshop on Quark Confinement and the Hadron Spectrum II*]{} (June 1996, Como, Italy), eds. N. Brambilla and G.M. Prosperi (World Scientific, Singapore, 1997), p. 349, hep-ph/9703222. See discussion in D. Ebert, R.N. Faustov and V.O. Galkin, [*Phys. Rev.*]{} [**D62**]{} (2000) 034014. G.S. Bali, K. Schilling and A. Wachter, [*Phys. Rev.*]{} [**D56**]{} (1997) 2566. S. Sarwar, contribution to “The Meeting of the Division of Particles and Fields of the American Physical Society” (DPF2000), 9-12 Aug. 2000, Columbus, OH. [**FIGURES**]{} Figure 1: (a) The square of the Dirac radial wavefunction of the upper component times $r^2$. (b) The square of the Dirac radial wavefunction of the lower component times $r^2$. $p_{\frac{3}{2}}$ is the solid line and $p_{\frac{1}{2}}$ is the dashed line. Note that the lower component is comparable to the upper component. The wavefunctions are solutions of the Dirac equation (see Eq. (\[dirac\])) with Coulomb potentials $V_S(r) = \frac{\alpha_S}{r}+U_S$ and $V_V(r) = \frac{\alpha_V}{r}+U_V$, where $\alpha_S = -1.279,\; U_S = 506$ MeV, $\alpha_V = -0.779,\; U_V = 515$ MeV, $m=330$ MeV and $M = 1480$ MeV. This corresponds to a $p_{\frac{3}{2}}-p_{\frac{1}{2}}$ splitting of -52 MeV. Figure 2: (a) The square of the Dirac momentum space wavefunction of the upper component times $q^2$. (b) The square of the Dirac momentum space wavefunction of the lower component times $q^2$. Other conventions are the same as in Figure 1. [^1]: *E-mail: prp@lanl.gov* [^2]: *E-mail: tgoldman@lanl.gov* [^3]: *E-mail: gino@t5.lanl.gov* [^4]: The FOCUS collaboration preliminarily found $D^{\ast}_0$ at a mass of $2420$ MeV [@dpf]. The error on the mass was not reported.

--- abstract: 'In recent literature it is claimed that BitCoin price behaves more likely to a volatile stock asset than a currency and that changes in its price are influenced by sentiment about the BitCoin system itself; in @MainDrivers the author analyses transaction based as well as popularity based potential drivers of the BitCoin price finding positive evidence. Here, we endorse this finding and consider a bivariate model in continuous time to describe the price dynamics of one BitCoin as well as a second factor, affecting the price itself, which represents a sentiment indicator. We prove that the suggested model is arbitrage-free under a mild condition and, based on risk-neutral evaluation, we obtain a closed formula to approximate the price of European style derivatives on the BitCoin. By applying the same approximation technique to the joint likelihood of a discrete sample of the bivariate process, we are also able to fit the model to market data. This is done by using both the *Volume* and the number of *Google searches* as possible proxies for the sentiment factor. Further, the performance of the pricing formula is assessed on a sample of market option prices obtained by the website deribit.com.' address: - 'Alessandra Cretarola, Department of Mathematics and Computer Science, University of Perugia, Via Luigi Vanvitelli, 1, I-06123 Perugia, Italy.' - 'Gianna Figà Talamanca, Department of Economics, University of Perugia, Via Alessandro Pascoli, I-06123 Perugia, Italy.' - 'Marco Patacca, Department of Economics, University of Perugia, Via Alessandro Pascoli, I-06123 Perugia, Italy.' author: - Alessandra Cretarola - 'Gianna Figà-Talamanca' - Marco Patacca bibliography: - 'biblio\_BS1.bib' date: - - 'This is an updated version (September 2017) of the paper *A confidence-based model for asset and derivatives pricing in the BitCoin market*' title: 'A sentiment-based model for the BitCoin: theory, estimation and option pricing' --- [**Keywords**]{}: BitCoin, sentiment, stochastic models, equivalent martingale measure, option pricing, likelihood. Introduction ============ The BitCoin was first introduced as an electronic payment system between peers. It is based on an open source software which generates a peer to peer network. This network includes a high number of computers connected to each other through the Internet and complex mathematical procedures are implemented both to check the truthfulness of the transaction and to generate new BitCoins. Opposite to traditional transactions, which are based on the trust in financial intermediaries, this system relies on the network, on the fixed rules and on cryptography. The open source software was created in 2009 by a computer scientist known under the pseudonym Satoshi Nakamoto, whose identity is still unknown. BitCoin has several attractive properties for consumers: it does not rely on central banks to regulate the money supply and it enables essentially anonymous transactions. BitCoins can be purchased on appropriate websites that allow to change usual currencies in BitCoins. Further, payments can be made in BitCoins for several online services and goods and its use is increasing. Special applications have been designed for smartphones and tablets for transactions in BitCoins and some ATM have appeared all over the world (see Coin ATM radar) to change traditional currencies in BitCoins. At very low expenses it is also possible to send cryptocurrency internationally. However, the downside of BitCoin is that, due to anonymous transactions, it has been labeled as an exchange for organized crime and money laundering. It is worth to mention the recent Wannacry malware which last May has infected the informatic systems of many huge companies as well as thousands of computers around the world. The hackers which have spread this malware asked a ransom of 300 to 600 USD to be payed in BitCoins in order to get each computer rid of the infection. Besides, BitCoins can be only deposited in a digital wallet which is costly and possibly subject to hacking attacks, thefts and other issues related to cyber-security. In spite of all the above critics, BitCoin has experienced a rapid growth both in value and in the number of transactions. A number of competitors, so called alt-coins, have also appeared recently without reaching the popularity of BitCoin; the most successful among these is Ethereum. Economic and financial aspects of BitCoin have been frequently addressed by financial blogs and by financial media but, until recently, researchers in Academia were primarily focused on the underlying technology and on safety and legal issues such as double spending. From an economic viewpoint, one of the main concerns about BitCoin is whether it should be considered a currency, a commodity or a stock. In @Yermack, the author performs a detailed qualitative analysis of BitCoin behavior. He remarks that a currency is usually characterized by three properties: a medium of exchange, a unit of account and a store of value. BitCoin is indeed a medium of exchange, though limited in relative volume of transactions and essentially restricted to online markets; however it lacks the other two properties. BitCoin value is rather volatile and traded for different prices in different exchanges, making it unreliable as a unit of account. The conclusion in @Yermack is that BitCoin behaves as a high volatility stock and that most transactions on BitCoins are aimed to speculative investments. In recent years several papers have also appeared in order to analyze which are the main drivers of its price evolution in time; many authors claim that the high volatility in BitCoin prices may depend on sentiment and popularity about the BitCoin market itself; of course sentiment and popularity on BitCoin are not directly observed but several variables may be considered as indicators, from the more traditional volume or number of transactions to the number of Google searches or Wikipedia requests about the topic, in the period under investigation. Main references in this area are @MainDrivers [@GoogleTrends; @SentimentAnalysis]. Alternatively, in @bukovina2016sentiment a sentiment measure related to the BitCoin system is obtained from the website Sentdex.com. This website collect data on sentiment through an algorithm, based on Natural Language Processing techniques, which is capable of identifying string of words conveying positive, neutral or negative sentiment on a topic (BitCoin in this case). The authors of the paper develop a model in discrete time and show that excessive confidence on the system may boost a Bubble on the BitCoin price. Motivated by the evidences in the above quoted papers we introduce a bivariate model in continuous time to describe both the dynamics of a BitCoin sentiment indicator and of the corresponding BitCoin price. From the theoretical viewpoint we give three contributions: the model is proven to be arbitrage-free under proper conditions and its statistical properties are investigated. Then, based on risk-neutral evaluation, a quasi-closed formula is derived for any European style derivative on the BitCoin. It is worth noticing that a market for derivatives on BitCoin has recently raised on appropriate websites such as *https://coinut.com* and *https://deribit.com* trading European Calls and Puts as well as Binary option endorsing the idea in @Yermack that BitCoins are likely to be used for speculative purposes. Further, the likelihood for a discrete sample of the model is computed and an approximated closed formula is derived so that maximum likelihood estimates can be obtained for model parameters. Precisely, we suggest a two-step maximum likelihood method, the profile likelihood described in @Davison:statmod [@Pawitan:allLik] to fit the model to market data. From the empirical viewpoint we contribute to the literature by fitting the suggested model to market data considering both the Volume and the number of Google searches as proxies for the sentiment factor. Besides the performance of the pricing formula is assesses on a sample of market option prices obtained by the website *https://deribit.com*.\ The rest of the paper is structured as follows. In Section \[sec:model\] we describe the model for the BitCoin price dynamics and show that the market is arbitrage-free under a mild condition. In Section \[sec:loglike\] we compute the joint distribution of the discretely sampled model as well as a closed form approximation. Then, we propose a statistical estimation procedure based on the corresponding approximated profile likelihood. In Section \[sec:option-pricing\] we prove a quasi-closed formula for European-style derivatives with detailed computations for Plain Vanilla and Binary option prices. Section \[sec:fit\] is devoted to test the possible proxies for the sentiment indicator, such as the number and volume of BitCoin transactions or the internet searches on Google and Wikipedia and to apply the whole estimation procedure to market data obtained from *http://blockchain.info*. In Section \[sec:assessing\] we evaluate model performance for option pricing considering options traded on *https://deribit.com* for some “test” days. Finally, in Section \[sec:remarks\] we give some concluding remarks and draw directions for interesting future investigations. The Appendices collect a brief description of the Levy approximation approach and the proofs of most of the technical results. The BitCoin market model {#sec:model} ======================== We fix a probability space $(\Omega,\F,\P)$ endowed with a filtration $\bF = \{\F_t,\ t \ge 0\}$ that satisfies the usual conditions of right-continuity and completeness. On the given probability space, we consider a main market in which heterogeneous agents buy or sell BitCoins and denote by $S = \{S_t,\ t \geq 0\}$ the price process of the cryptocurrency. We assume that the BitCoin price dynamics is described by the following equation: $$\label{eq:S} {\mathrm d}S_t = \mu_S P_{t-\tau} S_t {\mathrm d}t+\sigma_S \sqrt{P_{t-\tau}}S_t{\mathrm d}W_t,\quad S_0=s_0 \in \R_+,$$ where $\mu_{S} \in \R \setminus \{0\}$, $\sigma_{S} \in \R_+,\ \tau \in \R_+$ represent model parameters; $W = \{W_t,\ t \ge 0\}$ is a standard $\bF$-Brownian motion on $(\Omega,\F,\P)$ and $P = \{P_t,\ t \geq 0\}$ is a stochastic factor, representing the sentiment index in the BitCoin market, satisfying $$\label{eq:BSdyn} {\mathrm d}P_t =\mu_P P_t{\mathrm d}t+\sigma_P P_t{\mathrm d}Z_t, \quad P_t = \phi(t),\ t \in [-L,0].$$ Here, $\mu_P \in \R \setminus \{0\}$, $\sigma_P \in \R_+$, $L \in \R_+$, $Z = \{Z_t,\ t \geq 0\}$ is a standard $\bF$-Brownian motion on $(\Omega,\F,\P)$, which is $\P$-independent of $W$, and $\phi:[-L,0] \to [0,+\infty)$ is a continuous (deterministic) initial function. Note that, the non negative property of the function $\phi$ corresponds to require that the minimum level for sentiment is zero. It is worth noticing that in we also consider the effect of the past, since we assume that the sentiment factor $P$ affects explicitly the BitCoin price $S_t$ up to a certain preceding time $t-\tau$. Assuming that $\tau<L$ and that factor $P$ is observed in the period $[-L,0]$ makes the bivariate model jointly feasible. It is well-known that the solution of is available in closed form and that $P_t$ has a lognormal distribution for each $t > 0$, see @black1973pricing. Here, $P$ stands for an exogenous factor affecting the instantaneous variance of the BitCoin price changes modulated by $\sigma_S$. It is worth noticing that the instantaneous variance of the BitCoin price process increases with the delayed process $P$; this may appear counter-intuitive if $P$ is interpreted only as a *positive sentiment* indicator. However, in our perspective, the factor $P$ is mathematically a non-negative variable but does not necessary represent a *positive sentiment* indicator. Examples are the volume or number of transactions; these are non negative values which increase with both short (fear) and long (enthusiasm) positions in BitCoins. Similarly, the number of internet searches within a fixed time period cannot go negative but internet searches may increase both with enthusiasm and fear about the BitCoin System, or whatever other financial asset we would like to model with the dynamics we suggest here. Hence, an increase in $P$ may be actually related to an increase in the uncertainty about the the price $S$. In order to visualize the dynamics implied by the model in equations and , we plot in Figure \[CambioTau\] a possible simulated path of daily observations for the sentiment factor $P$ and the corresponding BitCoin prices $S$ within one year horizon by letting $\tau$ vary; as expected, market reaction to sentiment is delayed when $\tau$ increases. ![An example of BitCoin price dynamics given the evolution of the sentimente index (red): $\tau=1$ day (black), $\tau=10$ days (blue). Model parameters are set to $\mu_P=0.03,\sigma_P=0.35$, $\mu_S=10^{-5},\sigma_S=0.04$.[]{data-label="CambioTau"}](%D:/LavoriRicerca/InCorso/BitCoin/Nostro/FigurePaper/ IndipVariTau.jpg){width="13cm" height="6cm"} The suggested model is motivated by the outcomes in @MainDrivers [@GoogleTrends; @SentimentAnalysis; @bukovina2016sentiment] where the authors relate the BitCoin price dynamics to some sentiment factors and does not take into account special features of the BitCoin market such as the underlying technology or the order mechanism. Besides, BitCoin is treated as a financial stock as suggested in @Yermack. Possibly, the model can be applied to any other financial assets whether one believes their price to depend on suitably identified sentiment indicators. We assume that the reference filtration $\bF=\{\F_t,\ t \geq 0\}$, describing the information on the BitCoin market, is of the form $$\F_t=\F_t^W \vee \F_t^Z, \quad t \ge 0,$$ where $\F_t^W$ and $\F_t^Z$ denote the $\sigma$-algebras generated by $W_t$ and $Z_t$ respectively up to time $t \geq 0$. Note that $\F_t^Z=\F_t^P$, for each $t \geq 0$, with $\F_t^P$ being the $\sigma$-algebra generated by $P_t$ up to time $t \geq 0$. Since at any time $t$ the BitCoin price dynamics is affected by the sentiment index only up to time $t-\tau$, to describe the traders information on the digital market, we consider the filtration $\widetilde \bF=\{\widetilde \F_t,\ t \geq 0\}$, defined by $$\widetilde \F_t = \F_t^W \vee \F_{t-\tau}^P, \quad t \geq 0.$$ We also remark that all filtrations satisfy the usual conditions of completeness and right continuity (see e.g. @protter2005stochastic). Now, we introduce the [*integrated information process*]{} $X^\tau=\{X_{t}^\tau,\ t \ge 0\}$ associated to the sentiment index $P$, defined as follows: $$\label{eq:int_info} X_t^\tau := \left\{ \begin{array}{ll} \int_0^t P_{u-\tau} {\mathrm d}u =\int_{-\tau}^{0} \phi(u) {\mathrm d}u + \int_0^{t-\tau} P_u {\mathrm d}u= X_\tau^\tau + \int_0^{t-\tau} P_u {\mathrm d}u, & \quad 0 \leq \tau \leq t,\\ \int_{-\tau}^{t-\tau} \phi(u) {\mathrm d}u, & \quad 0 \leq t \leq \tau. \end{array} \right.$$ Note that, for $t \in [0,\tau]$, we have $X_t^\tau=\int_{-\tau}^{t-\tau} \phi(u) {\mathrm d}u$ which is deterministic. In addition, for a finite time horizon $T>0$, let us define the corresponding variation over the interval $[t,T]$, for $t\leq T$, as $X_{t,T}^\tau:=X_T^\tau-X_t^\tau$. Obviously, $X_{T,T}^\tau=0$; moreover, for $t<T$, $$\label{eq:int2_info} X_{t,T}^\tau := \left\{ \begin{array}{ll} \int_{t-\tau}^{T-\tau} P_u {\mathrm d}u & \quad \mbox{ if }\ 0 \leq \tau \leq t < T, \\ \int_{t-\tau}^{0} \phi(u) {\mathrm d}u + \int_0^{T-\tau} P_u {\mathrm d}u & \quad \mbox{ if } \ 0 \leq t \leq \tau < T, \\ \int_{t-\tau}^{T-\tau} \phi(u) {\mathrm d}u & \quad \mbox{ if }\ 0\leq t < T \leq \tau. \\ \end{array} \right.$$ Again, note that for $T\leq \tau$, we get $X_{t,T}^\tau=\int_{t-\tau}^{T-\tau} \phi(u) {\mathrm d}u$ which is deterministic. The following lemma establishes basic statistical properties for the integrated information process as well as for its variation in case they are not fully deterministic. \[th:means\] In the market model outlined above we have: - For $t > \tau$, $$\begin{aligned} {\mathbb E\left[X_t^\tau\right]} & = X_\tau^\tau +\frac{\phi(0)}{\mu_{P}}\left(e^{\mu_{P}(t-\tau)} -1\right);\\ \mathbb V{\rm ar}[X_t^\tau] & = \frac{2\phi^{2}(0)}{\left(\mu_{P}+\sigma_{P}^{2}\right)\left(2\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)(t-\tau)} -1\right)\\ & \qquad -\frac{2\phi^{2}(0)}{\mu_{P}\left(\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\mu_{P}(t-\tau)}-1\right)-\left(\frac{\phi(0)}{\mu_{P}}\left(e^{\mu_{P}(t-\tau)}-1\right)\right)^2.\end{aligned}$$ - For $ \tau \leq t < T $, $$\begin{aligned} {\mathbb E\left[X_{t,T}^\tau\right]} & = \frac{\phi(0)e^{\mu_{P}(t-\tau)}}{\mu_{P}}\left(e^{\mu_{P}(T-t)}-1\right);\\ \mathbb V{\rm ar}[X_{t,T}^\tau] & = \frac{2\phi^{2}(0)e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)(t-\tau)}}{\left(\mu_{P}+\sigma_{P}^{2}\right)\left(2\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)(T-t)} -1\right)\\ & \qquad -\frac{2\phi^{2}(0)e^{\mu_{P}(t-\tau)}}{\mu_{P}\left(\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\mu_{P}(T-t)} -1\right)-\left(\frac{\phi(0)e^{\mu_{P}(t-\tau)}}{\mu_{P}}\left(e^{\mu_{P}(T-t)} -1\right)\right)^2.\end{aligned}$$ - For $ t \leq \tau < T $, $$\begin{aligned} {\mathbb E\left[X_{t,T}^\tau\right]} & = \int_{t-\tau}^0 \phi\left(u\right){\mathrm d}u + \frac{\phi(0)}{\mu_{P}}\left(e^{\mu_{P}(T-\tau)}-1\right);\\ \mathbb V{\rm ar}[X_{t,T}^\tau] & = \frac{2\phi^{2}(0)}{\left(\mu_{P}+\sigma_{P}^{2}\right)\left(2\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)(T-\tau)} -1\right)\\ & \qquad -\frac{2\phi^{2}(0)}{\mu_{P}\left(\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\mu_{P}(T-\tau)} -1\right)-\left(\frac{\phi(0)}{\mu_{P}}\left(e^{\mu_{P}(T-\tau)} -1\right)\right)^2.\end{aligned}$$ In [@hull1987pricing] similar outcomes are claimed for $\tau=0$ without providing a proof; for the sake of clarity, we give a self-contained proof in Appendix \[appendix:technical\].\ The system given by equations and is well-defined in $\R_+$ as stated in the following theorem, which also provides its explicit solution. \[th:sol\] In the market model outlined above, the followings hold: - the bivariate stochastic delayed differential equation $$\label{eq:Bivdyn} \left\{ \begin{array}{ll} {\mathrm d}S_t = \mu_S P_{t-\tau} S_t {\mathrm d}t+\sigma_S \sqrt{P_{t-\tau}}S_t{\mathrm d}W_t, \quad S_0=s_0 \in \R_+, \\ {\mathrm d}P_t =\mu_P P_t{\mathrm d}t+\sigma_P P_t{\mathrm d}Z_t, \quad P_t = \phi(t),\ t \in [-L,0], \\\end{array} \right.$$ has a continuous, $\bF$-adapted, unique solution $(S,P)=\{(S_t,P_t),\ t \geq 0\}$ given by $$\begin{aligned} S_t & =s_0e^{\left(\mu_{S}-\frac{\sigma_{S}^{2}}{2}\right)\int_0^t P_{u-\tau} {\mathrm d}u+\sigma_S \int_0^t \sqrt{P_{u-\tau}}{\mathrm d}W_u},\quad t \ge 0,\label{eq:sol_S}\\ P_t & = \phi(0)e^{\left(\mu_{P}-\frac{\sigma_{P}^{2}}{2}\right)t+\sigma_P Z_t}, \quad t \ge 0. \label{eq:sol_P}\end{aligned}$$ More precisely, $S$ can be computed step by step as follows: for $k=0,1,2,\ldots$ and $t \in [k\tau,(k+1)\tau]$, $$\label{eq:sol_S1} S_t = S_{k\tau}e^{\left(\mu_{S}-\frac{\sigma_{S}^{2}}{2}\right)\int_{k\tau}^t P_{u-\tau} {\mathrm d}u+\sigma_S \int_ {k\tau}^t \sqrt{P_{u-\tau}}{\mathrm d}W_u}.$$ In particular, $P_t \ge 0$ $\P$-a.s. for all $t \geq 0$. If in addition, $\phi(0) > 0$, then $P_t > 0$ $\P$-a.s. for all $t \geq 0$. - Further, for every $t \ge 0$, the conditional distribution of $S_{t}$, given the integrated information $X_t^\tau$, is log-Normal with mean $\log \left(s_0\right) + \left(\mu_{S}-\frac{\sigma_{S}^{2}}{2}\right)X_t^\tau$ and variance $\sigma_{S}^{2}X_t^\tau$. - Finally, for every $t \in [0,\tau]$, the random variable $\log \left( S_t \right)$ has mean $\log \left(s_0\right) + \left(\mu_{S}-\frac{\sigma_{S}^{2}}{2}\right)X_t^\tau$ and variance $\sigma_{S}^{2}X_t^\tau$; for every $t > \tau$, $\log \left(S_t\right)$ has mean and variance respectively given by $$\begin{aligned} {\mathbb E\left[\log \left( S_t \right)\right]} & = \log \left(s_0\right) + \left(\mu_S-\frac{\sigma_S^2}{2}\right) {\mathbb E\left[X_t^\tau\right]};\\ \mathbb V{\rm ar}\left[{\log \left( S_t \right)}\right] & = \left(\mu_S-\frac{\sigma_S^2}{2}\right)^2\mathbb V{\rm ar}[X_t^\tau]+ \sigma_S^2 {\mathbb E\left[X_t^\tau\right]},\end{aligned}$$ where ${\mathbb E\left[X_t^\tau\right]}$ and $\mathbb V{\rm ar}[X_t^\tau]$ are both provided by Lemma \[th:means\], point (i). [**Point (i)**]{}. Clearly, $S$ and $P$, given in and respectively, are $\bF$-adapted processes with continuous trajectories. Similarly to @mao2013delay [Theorem 2.1], we provide existence and uniqueness of a strong solution to the pair of stochastic differential equations in system by using forward induction steps of length $\tau$, without the need of checking any assumptions on the coefficients, e.g. the local Lipschitz condition and the linear growth condition. First, note that the second equation in the system does not depend on $S$, and its solution is well known for all $t \ge 0$. Clearly, equation says that $P_t \ge 0$ $\P$-a.s. for all $t \ge 0$ and that $\phi(0) > 0$ implies that the solution $P$ remains strictly greater than $0$ over $[0,+\infty)$, i.e. $P_t > 0$, $\P$-a.s. for all $t \ge 0$.\ Next, by the first equation in and applying Itô’s formula to $\log \left(S_t\right)$, we get $$\label{eq:logS} {\mathrm d}\log \left(S_t\right) =\left(\mu_S-\frac{\sigma_S^2}{2}\right)P_{t-\tau} {\mathrm d}t + \sigma_S\sqrt{P_{t-\tau}} {\mathrm d}W_t,$$ or equivalently, in integral form $$\label{eq:log_int} \log\left(\frac{S_{t}}{s_{0}}\right) = \left(\mu_S-\frac{\sigma_S^2}{2}\right)\int_0^t P_{u-\tau} {\mathrm d}u + \sigma_S\int_0^t\sqrt{P_{u-\tau}} {\mathrm d}W_u,\quad t \ge 0.$$ For $t \in [0,\tau]$, can be written as $$\label{eq:log1} \log\left(\frac{S_{t}}{s_{0}}\right) = \left(\mu_S-\frac{\sigma_S^2}{2}\right)\int_0^t \phi \left(u-\tau \right) {\mathrm d}u + \sigma_S\int_0^t\sqrt{\phi \left(u-\tau \right)} {\mathrm d}W_u,$$ that is, holds for $k = 0$. Given that $S_t$ is now known for $t \in [0,\tau]$, we may restrict the first equation in on $t \in [\tau, 2\tau]$, so that it corresponds to consider for $t \in [\tau, 2\tau]$. Equivalently, in integral form, $$\label{eq:log} \log\left(\frac{S_{t}}{S_{\tau}}\right) = \left(\mu_S-\frac{\sigma_S^2}{2}\right)\int_\tau^t P_{u-\tau} {\mathrm d}u + \sigma_S\int_\tau^t\sqrt{P_{u-\tau}} {\mathrm d}W_u.$$ This shows that holds for $k = 1$. Similar computations for $k=2,3,\ldots$, give the final result. [**Point (ii)**]{}. Set $Y_t:=\int_0^t\sqrt{\phi \left( t-\tau \right)} {\mathrm d}W_u $, for $t \in [0,\tau]$ and $Y_t:= Y_{k\tau}+ \int_{k\tau}^t\sqrt{P_{u-\tau}} {\mathrm d}W_u $, for $t \in [k\tau,(k+1)\tau]$, with $k=1,2,\ldots$. Then, by applying the outcomes in Point (i) and the decomposition $$\log\left(\frac{S_{t}}{s_{0}}\right) =\log\left(\frac{S_{t}}{S_{k\tau}}\right)+\sum_{j=0}^{k-1}\log\left(\frac{S_{(j+1)\tau}}{S_{j\tau}}\right), $$ for $t\in [k\tau,(k+1)\tau]$, with $k=1,2,\ldots$, we can write $$\label{eq:LOG} \log\left(S_{t}\right)= \log(s_0)+\left(\mu_S-\frac{\sigma_S^2}{2}\right)X_t^\tau +\sigma_S Y_t,\quad t \ge 0.$$ To complete the proof, it suffices to show that, for each $t\ge 0$ the random variable $Y_t$, conditional on $X_t^\tau$, is Normally distributed with mean $0$ and variance $X_t^\tau$. This is straightforward from if $t \in [0,\tau]$. Otherwise, we first observe that since $Z_{u-\tau}$ is independent of $W_u$ for every $\tau < u \leq t$, the distribution of $Y_t$, conditional on $\{Z_{u-\tau}:\ \tau < u \leq t-\tau\}=\{P_{u-\tau}:\ \tau < u \leq t-\tau\} =\F_{t-\tau}^P$, is Normal with mean $0$ and variance $\sigma_S^2X_t^\tau$.\ Now, for each $t > \tau$, the moment-generating function of $Y_t$, conditioned on the history of the process $P$ up to time $t-\tau$, is given by $$\begin{aligned} {\mathbb E\left[e^{aY_t}\Big{|}\F_{t-\tau}^{P}\right]} & = e^{\int_0^t \frac{a^2}{2} P_{u-\tau} {\mathrm d}u} = e^{\frac{a^2}{2} \int_0^t P_{u-\tau} {\mathrm d}u}\\ & = e^{\frac{a^2}{2}\left(\sqrt{X_t^\tau}\right)^2}, \quad a \in \R,\end{aligned}$$ that only depends on its integrated information $X_t^\tau$ up to time $t$, that is, $${\mathbb E\left[e^{aY_t}\Big{|}\F_{t-\tau}^{P}\right]}={\mathbb E\left[e^{aY_t}\Big{|}X_t^\tau\right]}, \quad t > \tau.$$ [**Point (iii)**]{}. The proof is trivial for $t \in [0,\tau]$. If $t >\tau$, and Lemma \[th:means\] together with the null-expectation property of the Itô integral, give $$\begin{aligned} {\mathbb E\left[\log \left(S_t\right)\right]} & = \log \left(s_0\right) + \left(\mu_S-\frac{\sigma_S^2}{2}\right) {\mathbb E\left[X_t^\tau\right]}\\ & = \log \left(s_0\right) + \left(\mu_S-\frac{\sigma_S^2}{2}\right)\left(X_\tau^\tau +\frac{\phi(0)}{\mu_{P}}\left(e^{\mu_{P}(t-\tau)} -1\right)\right).\end{aligned}$$ Now, we compute the variance of $\log \left(S_t\right)$. Since for each $t > \tau$ the random variable $Y_t $ has mean $0$ conditional on $\F_{t-\tau}^P$, we have $$\begin{aligned} \mathbb V{\rm ar}\left[\log \left(S_t\right)\right] & = {\mathbb E\left[\log^2\left(S_t\right)\right]} - \left({\mathbb E\left[\log\left(S_t\right)\right]}\right)^2\\ & = \left(\mu_S-\frac{\sigma_S^2}{2}\right)^2{\mathbb E\left[(X_t^\tau)^2\right]} + 2\left(\mu_S-\frac{\sigma_S^2}{2}\right){\mathbb E\left[X_t^\tau {\mathbb E\left[Y_t\Big{|}\F_{t-\tau}^P\right]}\right]}\\ & \qquad \qquad + \sigma_S^2 {\mathbb E\left[X_t^\tau\right]} - \left(\mu_S-\frac{\sigma_S^2}{2}\right)^2 {\mathbb E\left[X_t^\tau\right]}^2\\ & = \left(\mu_S-\frac{\sigma_S^2}{2}\right)^2\mathbb V{\rm ar}[X_t^\tau]+ \sigma_S^2 {\mathbb E\left[X_t^\tau\right]}.\end{aligned}$$ Thus, the proof is complete. Existence of a risk-neutral probability measure ----------------------------------------------- Let us fix a finite time horizon $T>0$ and assume the existence of a riskless asset, say the money market account, whose value process $B=\{B_t,\ t \in [0,T]\}$ is given by $$B_t=e^{\int_0^t r(s){\mathrm d}s},\quad t \in [0,T],$$ where $r:[0,T] \to \R$ is a bounded, deterministic function representing the instantaneous risk-free interest rate. To exclude arbitrage opportunities, we need to check that the set of all equivalent martingale measures for the BitCoin price process $S$ is non-empty. More precisely, it contains more than a single element, since $P$ does not represent the price of any tradeable asset, and therefore the underlying market model is incomplete. \[lem:measure\] Let $\phi(t) > 0$, for each $t \in [-L,0]$, in . Then, every equivalent martingale measure $\Q$ for $S$ defined on $(\Omega,\F_T)$ has the following density $$\label{def:Q} \frac{{\mathrm d}\Q}{{\mathrm d}\P}\bigg{|}_{\F_T}=:L_T^\Q, \quad \P-\mbox{a.s.}, $$ where $L_T^\Q$ is the terminal value of the $(\bF,\P)$-martingale $L^\Q=\{L_t^\Q,\ t \in [0,T]\}$ given by $$\label{eq:L} L_t^\Q :={\mathcal E\left(-\int_0^\cdot \frac{\mu_S P_{s-\tau}-r(s)}{\sigma_S\sqrt{P_{s-\tau}}} {\mathrm d}W_s - \int_0^\cdot\gamma_s{\mathrm d}Z_s\right)}_t, \quad t \in [0,T], $$ for a suitable $\bF$-progressively measurable process $\gamma=\{\gamma_t,\ t \in [0,T]\}$. The proof is postponed to Appendix \[appendix:technical\]. Here $\E (Y)$ denotes the Doleans-Dade exponential of an $(\bF, \P)$-semimartingale $Y$. In the rest of the paper, suppose that $\phi(t) > 0$, for each $t \in [-L,0]$, in . Then, Lemma \[lem:measure\] ensures that the space of equivalent martingale measures for $S$ is described by . More precisely, it is parameterized by the process $\gamma$ which governs the change of drift of the $(\bF,\P)$-Brownian motion $Z$. Note that the sentiment factor dynamics under $\Q$ in the BitCoin market is given by $${\mathrm d}P_t = (\mu_P - \sigma_P \gamma_t)P_t{\mathrm d}t + \sigma_P P_t {\mathrm d}Z_t^\Q, \quad P_t = \phi(t),\ t \in [-L,0].$$ The process $\gamma$ can be interpreted as the risk perception associated to the future direction or future possible movements of the BitCoin market. One simple example of a candidate equivalent martingale measure is the so-called [*minimal martingale measure*]{} (see e.g. @follmer1991hedging, @follmer2010minimal), denoted by $\widehat \P$, whose density process $L = \{L_t,\ t \in [0,T]\}$, is given by $$\label{eq:Lp} L_t := e^{-\int_0^t \frac{\mu_S P_{s-\tau}-r(s)}{\sigma_S\sqrt{P_{s-\tau}}} {\mathrm d}W_s - \frac{1}{2}\int_0^t\left(\frac{\mu_S P_{s-\tau}-r(s)}{\sigma_S\sqrt{P_{s-\tau}}}\right)^2{\mathrm d}s},\quad t \in [0,T].$$ This is the probability measure which corresponds to the choice $\gamma \equiv 0$ in . Intuitively, under the minimal martingale measure, say $\widehat \P$, the drift of the Brownian motion driving the BitCoin price process $S$ is modified to make $S$ an $(\bF,\widehat \P)$-martingale, while the drift of the Brownian motion which is strongly orthogonal to $S$ is not affected by the change measure from $\P$ to $\widehat \P$. More precisely, under the change of measure from $\P$ to $\widehat \P$, we have two independent $(\bF,\widehat \P)$-Brownian motions $\widehat W=\{\widehat W_t,\ t \in [0,T]\}$ and $\widehat Z=\{\widehat Z_t,\ t \in [0,T]\}$ defined respectively by $$\begin{aligned} \widehat W_t & := W_t + \int_0^t\frac{\mu_S P_{s-\tau}-r(s)}{\sigma_S\sqrt{P_{s-\tau}}} {\mathrm d}s,\quad t \in [0,T],\\ \widehat Z_t & := Z_t, \quad t \in [0,T]. \label{def:hat_Z}\end{aligned}$$ Denote by $\widetilde S_t=\{\widetilde S_t,\ t \in [0,T]\}$ the discounted BitCoin price process defined as ${\displaystyle}\widetilde S_t:=\frac{S_t}{B_t}$, for each $t \in [0,T]$. Then, on the probability space $(\Omega,\F,\widehat \P)$, the pair $(\widetilde S,P)$ satisfies the following system of stochastic delayed differential equations: $$\label{eq:Bivdyn-MMM} \left\{ \begin{array}{ll} {\mathrm d}\widetilde S_t = \sigma_S \sqrt{P_{t-\tau}}\widetilde S_t{\mathrm d}\widehat W_t, \quad \widetilde S_0=s_0 \in \R_+, \\ {\mathrm d}P_t =\mu_P P_t{\mathrm d}t+\sigma_P P_t{\mathrm d}Z_t, \quad P_t = \phi(t),\ t \in [-L,0], \\\end{array} \right.$$ By Theorem \[th:sol\], point (i), the explicit expression of the solution to , which provides the discounted BitCoin price $\widetilde S_t$, at any time $t \in [0,T]$, is given by $$\label{def:tilde_S} \widetilde S_t=s_0e^{\sigma_S\int_0^t\sqrt{P_{u-\tau}}{\mathrm d}\widehat W_u - \frac{\sigma_S^2}{2}\int_0^t P_{u-\tau} {\mathrm d}u},\quad t \in [0,T],$$ with the representation of the sentiment factor $P$ still provided by . Under our assumptions, the dynamics of the (non-discounted) BitCoin price under the minimal martingale measure is given by $$\label{eq:Bivdyn-Q} \left\{ \begin{array}{ll} {\mathrm d}S_t = r(t) {\mathrm d}t + \sigma_S \sqrt{P_{t-\tau}} S_t{\mathrm d}\widehat W_t, \quad S_0=s_0 \in \R_+, \\ {\mathrm d}P_t =\mu_P P_t{\mathrm d}t+\sigma_P P_t{\mathrm d}Z_t, \quad P_t = \phi(t),\ t \in [-L,0], \\\end{array} \right.$$ where $r(t)$ is the risk-free interest rate at time $t$. The above dynamics was assumed in @hull1987pricing to describe price changes for a stock and its instantaneous variance (for which $\sigma_S=1$ and $\tau=0$ by definition). However, the authors assumed from the very beginning a risk-neutral framework without defining the dynamics under the physical measure and with no proof of the existence of any equivalent martingale measure. In Section \[sec:option-pricing\], we derive option pricing formulas via the risk-neutral evaluation procedure based on the minimal martingale measure above defined. As usual, pricing formulas depend on model parameters which have to be estimated on market data. A common approach, when a closed formula for option is available, is the so called calibration of parameters; their value is obtained in order to minimize a proper distance between model an market prices for options. However, this method is particularly of interest when there is a standardized and liquid market for options. Of course, this is not the case for the BitCoin so we will fit the model directly to a time series of BitCoin prices with a more classical statistical procedure based on the approximation of the probability density function of a discrete sample for model described by equations and . To this end we need to know the dynamics of the BitCoin price under the physical measure and derive statistical properties for a discrete sample of the process $P$ given in . Statistical properties of discretely observed quantities and parameter estimation {#sec:loglike} ================================================================================= In this section, we introduce basic statistical properties for a sample of discretely observed prices and suggest a possible closed form approximation for the joint probability density of the discrete sample. Let us fix a discrete observation step $\Delta$ and consider the discrete time process $\{S_i,\ i \in \bN\}$, where $S_i:=S_{i\Delta}$. Define the corresponding logarithmic returns process $\{R_i,\ i\in \bN \}$ as $$\label{eq:Ri} R_i=\log (S_{i})-\log (S_{i-1}).$$ By , we get $$\label{def:R} R_i= \left(\mu_S-\frac{\sigma_S^2}{2}\right)\int_{(i-1)\Delta}^{i\Delta} P_{u-\tau} du + \sigma_S \int_{(i-1)\Delta}^{i\Delta} \sqrt{P_{u-\tau}} {\mathrm d}W_u, \quad i \in \bN.$$ Setting $Y_t:=\int_0^t \sqrt{P_{u-\tau}} {\mathrm d}W_u$, with $t \in [0,T]$, as in the proof of Theorem \[th:sol\], we define $$\label{eq:Yi} M_i:=Y_{i\Delta}-Y_{(i-1)\Delta}=\int_{(i-1)\Delta}^{i\Delta} \sqrt{P_{u-\tau}} {\mathrm d}W_u,\,\,\, \forall i \in \bN,$$ so that, can be written as $$R_i= \left(\mu_S-\frac{\sigma_S^2}{2}\right)A_i^\tau + \sigma_S M_i, \quad i \in \bN,$$ where $A_i^\tau:=X_{(i-1)\Delta,\; i\Delta}^\tau$, with $X_{t,T}^\tau$ being the variation of the integrated information process introduced in ; since $\tau$ is fixed we omit hereafter the dependence on it and, without loss of generality we assume $\tau < \Delta$ so that $A_1=X_\tau^\tau + \int_0^{\Delta-\tau} P_u {\mathrm d}u$. Note that if $j\Delta \leq \tau<(j+1)\Delta $ the quantities $A_1,\ldots,A_j$ are deterministic and the outcomes in what follows still hold if $A_1$ is replaced by the first non deterministic value $A_{j+1}$. Let us consider a finite time horizon $T=n\Delta$; under model assumptions the conditional probability distribution of the vector $\mathbf{M}=\left(M_1,M_2,\dots,M_n\right)$, given the vector $\mathbf{A}=\left(A_1,A_2,\dots,A_n\right)$, is a multi-variate normal with covariance matrix $Diag(A_1,A_2,\dots,A_n)$. Hence, the vector of discretely observed logarithmic returns $\mathbf{R}=\left(R_1,R_2,\dots,R_n\right)$, conditionally on $\mathbf{A}$, is jointly normal with covariance matrix $\Sigma=\sigma_S^2 Diag(A_1,A_2,\dots,A_n)$. The application of Bayes’s rule allows to write the unconditional joint probability distribution of $\left(\mathbf{R},\mathbf{A}\right)$, i.e. the density function $f_{\left(\mathbf{R},\mathbf{A}\right)}:\R^n \times \R_+^n \longrightarrow \R$ as $$\label{aprdens} f_{\left(\mathbf{R},\mathbf{A}\right)}(\mathbf{r},\mathbf{a})=f_{A_1}(a_1)\prod_{i=2}^nf_{\left(A_i|A_{i-1}\right)}(a_i) \prod_{i=1}^n \frac{1}{\sqrt{2\pi \sigma_S^2 a_i}} e^{-\frac{1}{2}\frac{\left(r_i-\left(\mu_S-\frac{\sigma_S^2}{2}\right)a_i\right)^2}{\sigma_S^2 a_i}}.$$ with $\mathbf{r}=\left(r_1,r_2,\dots,r_n\right) \in \R^n$ and , $\mathbf{a}=\left(a_1,a_2,\dots,a_n\right) \in \R_+^n$. The probability distribution functions $f_{A_1}(.)$ and $f_{A_i|A_{i-1}}(.) \mbox{ for } i=2,3,\dots,n$ are not available in closed form; though, several approximations exist among which those introduced in @levy1992pricing and @MilPosner. Of course any approximation available for such densities can be applied in order to find a closed formula approximating the joint density $f_{\left(\mathbf{R},\mathbf{A}\right)}\left(\mathbf{r},\mathbf{a}\right)$; in what follows we adopt the one suggested in @levy1992pricing, see Appendix \[appendix:levy\] for further details. Note that the inverse gamma approach suggested in @MilPosner holds in the limit when $T$ tends to infinity, a condition which is not at all consistent with the applications we have in mind; further discussion on the approximating distribution to select is beyond the scope of our paper. The approximated likelihood --------------------------- One of the pillar in statistical inference is the maximum likelihood (in short ML) estimation approach where model parameters are estimated so as to maximize the probability of the the realized sample to be extracted randomly; the likelihood function shares the same mathematical expression of the probability density function but it is computed “ex-post” when a realization of involved random variables is available and assuming the underlying model parameters to be unknown. It is well known that ML estimates are consistent and asymptotically normal and they achieve efficiency, i.e. they have the lowest variance among estimators sharing the same asymptotic properties (see @Davison:statmod). By applying the approximation of @levy1992pricing, we prove the following Lemma. \[lemma:distr\] Let $\phi(t) > 0$, for each $t \in [-L,0]$, in and $\tau<\Delta$. Then, in the market model outlined in Section \[sec:model\], we have - the distribution of $A_1-X_\tau^\tau$ is approximated by a log-normal with mean $\alpha_1$ and variance $\nu_1^2$ given by $$\begin{gathered} \alpha_1 = \log \phi(0) + 2\log \frac{e^{\mu_P(\Delta-\tau)}-1}{\mu_P} -\frac{1}{2}\log\left( \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{(2\mu_P+\sigma_P^2)(\Delta-\tau)}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P(\Delta-\tau)}-1}{\mu_P} \right] \right) \\ \nu_1^2=\log \left( \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{(2\mu_P+\sigma_P^2)(\Delta-\tau)}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P(\Delta-\tau)}-1}{\mu_P} \right] \right)-2\log \left(\frac{e^{\mu_P(\Delta-\tau)}-1}{\mu_P} \right)\end{gathered}$$ - the distribution of $A_i$ given $A_{i-1}$ (shortly $A_i|A_{i-1}$), for $i=1,\dots,n$, is approximated by a log-normal with means $\alpha_i$ and variances $\nu_i^2$ given by $$\begin{gathered} \alpha_i=\log \left(A_{i-1}\right)+ \left(\mu_P-\frac{\sigma_P^2}{2}\right)\Delta, \quad \mbox{ for } i=1,\dots,n,\\ \nu_i^2=\sigma_P^2\Delta, \quad \mbox{ for } i=1,\dots,n.\end{gathered}$$ The proof is postponed to Appendix \[appendix:technical\]. Now, we are in the position to state the following theorem. \[likelihood\] Under the same assumptions of Lemma \[lemma:distr\], given the realized sample $\left(\bar{\mathbf{r}},\bar{\mathbf{a}}\right)$, the log-likelihood function $\log{\L_\mathbf{R,A}(\mu_P,\mu_S,\sigma_P,\sigma_S)}:\R^2 \times \R_+^2 \longrightarrow \R$ can be approximated by $$\label{eq:likli} \begin{split} \log{\L_\mathbf{R,A}(\mu_P,\mu_S,\sigma_P,\sigma_S)} & =\sum_{i=1}^n\left[\log\left(\frac{1}{\sqrt{2\pi \sigma_S^2 a_i}}\right)-\frac{1}{2}\frac{\left(r_i-\left(\mu_S-\frac{\sigma_S^2}{2}\right)a_i\right)^2}{\sigma_S^2 a_i}\right]\\ & +\sum_{i=1}^n\left[\log\left( \frac{1}{a_i\nu_i\sqrt{2\pi}}\right) -\frac{\left(\log (a_i)-\alpha_i \right)^2}{2\nu_i^2}\right], \end{split}$$ where upper case letters are used for random variables and lowercase for the corresponding realizations. First, recall that the likelihood function of a parameter corresponds to the probability density function where random variables are replaced by they realizations and parameters are unknown. Then, by simply applying the logarithmic function to we get $$\label{eq:lik1} \begin{split} \log{f_{(\mathbf{R,A})}(\mathbf{r},\mathbf{a})}& =\log{\left(\prod_{i=1}^n \left[ \frac{1}{\sqrt{2\pi \sigma_S^2 a_i}}e^{-\frac{1}{2}\frac{\left(r_i-\left(\mu_S-\frac{\sigma_S^2}{2}\right)a_i\right)^2}{\sigma_S^2 a_i}} \right] f_{A_1}(a_1)\prod_{i=2}^nf_{A_i|A_{i-1}}(a_i)\right)}\\ & =\sum_{i=1}^n\left[\log\left(\frac{1}{\sqrt{2\pi \sigma_S^2 a_i}}\right)-\frac{1}{2}\frac{\left(r_i-\left(\mu_S-\frac{\sigma_S^2}{2}\right)a_i\right)^2}{\sigma_S^2 a_i}\right]+\log \left(f_{A_1}(a_1)\right) \\ & \qquad \qquad +\sum_{i=2}^n\log \left(f_{A_i|A_{i-1}}(a_i)\right). \end{split}$$ Replacing the unknown densities in according to Lemma \[lemma:distr\] gives the desired result. Maximum likelihood estimates for the model can be obtained by maximizing the log-likelihood approximation in i.e. $$\label{eq:maxLik} (\widehat{\mu}_P,\widehat{\mu}_S,\widehat{\sigma}_P,\widehat{\sigma}_S)=\arg\max_{\substack{\mu_P,\mu_S \\ \sigma_P,\sigma_S}}\log{\L_\mathbf{R,A}(\mu_P,\mu_S,\sigma_P,\sigma_S)}.$$ In this case the methodology is referred to as Quasi-Maximum likelihood since the exact expression of the likelihood is not available; under suitable conditions, quasi-maximum likelihood estimates are asymptotically equivalent to the maximum likelihood estimates, see e.g. @White [@Gourieroux]. We also performed a simulation study to assess finite sample behavior of the estimates. It is worth to stress that the above estimation method does not assume the process $P$ to be observed, as far as $X_\tau^\tau$ and $A_i$, $i \geq 1$ are observed (note that $A_i$ is the cumulative of $P$ along the time interval $[(i-1)\Delta-\tau,i\Delta-\tau]$). Finite sample behavior of QML estimates {#sec:liksimulstudy} --------------------------------------- In order to check the goodness of the log-likelihood approximation introduced in Theorem \[likelihood\], we apply the proposed estimation method to simulated data and assume, for the sake of simplicity, $\tau=0$. We simulate $m$ samples of length $n$ for the processes in and assuming a constant finer observation step $\delta$; we extract corresponding samples for $\mathbf{R,A}$ at a lower frequency, with observation step $\Delta=r\delta$. In the numerical exercise we choose $n=730$, $m=1000$, $\delta=\frac{1}{365}$ (daily observations), $\Delta=7\delta$ (weekly observations); parameters values are set as $\mu_P=2$, $\sigma_P=0.5$ $\mu_S=0.05$, $\sigma_S=0.3$. We end with $1000$ samples of $104$ observations for $\left(\mathbf{R,A}\right)$; for each sample we estimate the parameters by means of the quasi-maximum likelihood as suggested in previous subsection. The results are summed up in Table \[tab:fitpar\]. \[tab:fitpar\] Variable Theor. value Fitted value Std. error t-value P($>\lvert t \rvert$) RMSE ------------ -------------- -------------- ------------ --------- ----------------------- --------- $\mu_P$ 2.0000 1.9759 0.3675 -0.0655 0.9478 11.6475 $\sigma_P$ 0.5000 0.4089 0.0302 -3.0170 0.0026 3.0358 $\mu_S$ 0.0500 0.0497 0.0112 -0.0297 0.9763 0.3544 $\sigma_S$ 0.3000 0.2978 0.0210 -0.1032 0.9178 0.6687 : Parameter fit with simulated data of QML method. We also performed a $t$-test in order to check for estimation bias. The fitted values of $\mu_P,\mu_S,\sigma_S$ are close in mean to their theoretical value and with a reasonable standard deviation; the $p$-values of the t-test confirm that estimated are not biased. Different conclusions are in order as for parameter $\sigma_P$ which estimations is by no doubt biased. In Figure \[fig:hist\_par\] we plot the histograms of the estimated as well as the fitted normal distribution and the expected mean of the asymptotic distribution. Pictures confirm the biasedness of the estimator for $\sigma_P$ but all other estimates perform well and outcomes may become better by increasing the sample length. The simulation exercise have been repeated by letting the parameters values, the number and the sample length vary obtaining analogous qualitative results. ![Histogram of parameter fit with simulated data of QML method.[]{data-label="fig:hist_par"}](hist_fitRMSElik.png) In order to disentangle the contribution of the Levy approximation [@levy1992pricing] to the estimation bias, we suggest to apply the method of moments to estimate $\mu_P$ and $\sigma_P$ considering the whole sample for $P$ generated at the finer observation step $\delta$ to compute the sample mean and sample variance of the sentiment realizations. If we then we plug the estimated values in the likelihood in order to estimate $\lbrace \mu_S, \sigma_S \rbrace$ these two estimates remain unchanged; in fact the likelihood may be maximized separately with respect to $\mu_P,\sigma_P$ and $\mu_S,\sigma_S$ since each of the two addend in the likelihood expression depends on just one of this pairs. The results of this alternative estimation method are reported in Table \[tab:fitpar\_momPi\]. \[tab:fitpar\_momPi\] Variable Theor. value Fitted value Std. error t-value P($>\lvert t \rvert$) RMSE ------------ -------------- -------------- ------------ --------- ----------------------- --------- $\mu_P$ 2.0000 2.0104 0.3638 0.0286 0.9772 11.5034 $\sigma_P$ 0.5000 0.4995 0.0135 -0.0385 0.9693 0.4282 : Parameter fit with simulated data of Moments method. ![Histogram of parameter $\lbrace \mu_P, \sigma_P \rbrace$ fit with simulated data of QML method and Moments method at finer step $\delta$.[]{data-label="fig:hist_par_QMLmomPi"}](hist_fitRMSEQMLmomPi.png) To visualize the bias of $\lbrace \sigma_P \rbrace$ we plot in Figure \[fig:hist\_par\_QMLmomPi\] the histogram of parameter $\lbrace \mu_P, \sigma_P \rbrace$ fit with simulated data using the two methods. As we can see using the two step procedure we obtain better estimates of $\lbrace \sigma_P \rbrace$ both in terms of expected value and standard deviation. It is evident from Table \[tab:fitpar\_momPi\] and Figure \[fig:hist\_par\_QMLmomPi\] the that the estimation of $\sigma_P$ is not biased in this case hence the estimation bias may essentially be attributed to the aggregation of the sentiment over time intervals and to the corresponding approximating distribution. Hence, whether the sentiment factor is observed at a finer step than the price, the above separate estimation is more reliable. Estimation of the delay parameter --------------------------------- The delay parameter $\tau$ directly affects the definition of the discrete process $A_i$. Hence, in order to proceed with its estimation we need to observe the process $P$ at a finer observation step $\delta$ with respect to the log-returns. In what follows we set $\Delta=\delta r$ and we adopt a two step estimation procedure known as Profile Likelihood in order to estimate the delay. We briefly describe the Profile Likelihood approach to estimation and its application in our specific case; interested readers are referred to @Davison:statmod [@Pawitan:allLik] for details on the profile likelihood. The basic idea of this approach is to split the parameter vector which has to be estimated, say $\theta$, in two sub-vectors, one representing the parameter of interest and the other the so called nuisance parameter i.e. $\theta=(\gamma,\lambda)$; to estimate $\gamma$ and $\lambda$ jointly we should maximize at once the likelihood i.e. $$\max_{\substack{\gamma,\lambda}}\log{\L\left(\left(\gamma,\lambda\right)\right)}.$$ When this is not feasible and provided the likelihood computed with respect to the nuisance parameter vector $\lambda$ is available and it is easy to maximize we can apply a two step procedure by maximizing, $\forall \gamma$ in its parametric space, $$\label{eq:proflik} \mathcal{L}_p(\gamma)=\max_{\substack{\lambda}}\mathcal{L}(\gamma,\lambda)=\mathcal{L}(\gamma,\widehat{\lambda}_\tau),\,\,\,$$ where $\widehat{\lambda}_\gamma$ is the maximum likelihood estimate of $\lambda$ for a fixed $\gamma$, then the best estimate for $\tau$ is $$\widehat{\gamma}=\arg \max_{\substack{\gamma}}\log{\mathcal{L}_p(\gamma)}.$$ Classical confidence intervals cannot be defined in this setting; indeed, it is possible to obtain a confidence region for $\tau$ using the likelihood ratio statistics (see @Davison:statmod), defined as $$W_p(\gamma_0)=2\left\lbrace \mathcal{L}(\widehat{\gamma},\widehat{\lambda}) - \mathcal{L}(\gamma_0,\widehat{\lambda}_{\gamma_0}) \right\rbrace,$$ where $$W_p(\gamma_0) \xrightarrow{ \;D\; } \chi_p^2$$ and $\gamma_0$ is an assigned value for $\gamma$. These results imply that the confidence region for $\gamma$ is the set $$\label{eq:confreg} \left\lbrace \gamma : \mathcal{L}_p(\gamma)\geq \mathcal{L}_p(\widehat{\gamma})-\frac{1}{2} c_p(1-2\alpha) \right\rbrace ,$$ with $c_p(\alpha)$ is the $\alpha$ quantile of the $\chi_p^2$ distribution. In our exercise we split $\theta:=(\mu_P,\mu_S,\sigma_P,\sigma_S,\tau)$ in $\theta=(\tau,\lambda)$ where $\tau$ is the parameter on which we are focusing and $\lambda=(\mu_P,\mu_S,\sigma_P,\sigma_S)$ is the nuisance parameter vector. The Profile Likelihood approach is feasible in our case since a closed approximating expression for the likelihood with respect to the nuisance parameter is indeed available. The parametric space for $\gamma:=\tau$ is the interval $[0,L]$ in this case but, for practical purposes, $\tau$ is chosen on a grid i.e. $\tau\in\lbrace \tau_0,\tau_1,\tau_2,\ldots,\tau_k\rbrace$; the maximization of the likelihood $\log{\L_\mathbf{R,A}(\mathbf{r},\mathbf{a})}$ is then performed with respect to $\lambda$ for each value $\tau_j$ in the grid, obtaining $\mathcal{L}_p(\tau_j)$ for $j=0,1,\dots,k$. An estimate for $\tau$ is then obtained as $\widehat{\tau}=\arg\max_j \mathcal{L}_p(\tau_j)$. Finally we get $\widehat{\theta}=\left(\widehat{\tau},\widehat{\lambda}_{\widehat{\tau}} \right)$. Of course the estimation error decreases with the mesh of the grid so that it sufficiently spans the parametric set for $\tau$. Risk neutral evaluation of European-type contingent claims {#sec:option-pricing} ========================================================== Let $H=\varphi(S_T)$ be an $\widetilde \F_T$-measurable random variable representing the payoff a European-type contingent claim with date of maturity $T$, which can be traded on the underlying market. Here $\varphi : \R \to \R$ is a a Borel-measurable function such that $H$ is integrable under $\widehat \P$. The function $\varphi$ is usually referred to as the [*contract function*]{}. The following result provides a risk-neutral pricing formula under the minimal martingale measure $\widehat \P$ for any $\widehat \P$-integrable European contingent claim. Since the martingale measure is fixed, the risk-neutral price agrees with the arbitrage free price for those options which can be replicated by investing on the underlying market. Recall that $X_{t,T}^\tau=X_T^\tau-X_t^\tau$, for each $t \in [0,T)$, refers to the variation of the process $X^\tau$ defined in , over the interval $[t,T]$. Then, denote by ${\mathbb E^{\bar \P}\left[\cdot\Big{|}\widetilde \F_t\right]}$ the conditional expectation with respect to $\widetilde \F_t$ under the probability measure $\widehat \P$ and so on. \[th:cont\_claim\] Let $H=\varphi(S_T)$ be the payoff a European-type contingent claim with date of maturity $T$. Then, the risk-neutral price $\Phi_t(H)$ at time $t$ of $H$ is given by $$\label{eq:gen_option} \Phi_t(H) = {\mathbb E^{\widehat \P}\left[\psi(t,S_t,X_{t,T}^\tau)\Bigg{|} S_t\right]}, \quad t \in [0,T),$$ where $\psi: [0,T) \times \R_+ \times \R_+ \longrightarrow \R$ is a Borel-measurable function such that $$\label{def:psi} \psi(t,S_t,X_{t,T}^\tau)=B_t{\mathbb E^{\widehat \P}\left[\frac{1}{B_T}G\left(t,S_t,X_{t,T}^\tau,Y_{t,T}\right)\Bigg{|}\F_t^W \vee \F_{T-\tau}^P\right]},$$ for a suitable function $G$ depending on the contract such that $G\left(t,S_t,X_{t,T}^\tau,Y_{t,T}\right)$ is $\widehat \P$-integrable. For the sake of simplicity suppose that $\tau<T$ and set $Y_{t,T}:= \int_t^T\sqrt{P_{u-\tau}}{\mathrm d}\widehat W_u$, for each $t \in [0,T)$. Then, the risk-neutral price $\Phi_t(H)$ at time $t$ of a European-type contingent claim with payoff $H=\varphi(S_T)$ is given by $$\begin{aligned} \Phi_t(H) & = B_t {\mathbb E^{\widehat \P}\left[\frac{\varphi(S_T)}{B_T}\bigg{|}\widetilde \F_t\right]}\\ & = B_t {\mathbb E^{\widehat \P}\left[{\mathbb E^{\widehat \P}\left[\frac{\varphi\left(S_te^{\int_t^T r(u){\mathrm d}u-\frac{\sigma_S^2}{2}X_{t,T}^\tau+ \sigma_S Y_{t,T}}\right)}{B_T}\left\vert\vphantom{\frac{\varphi\left(S_te^{\int_t^T r(u){\mathrm d}u-\frac{\sigma_S^2}{2}X_{t,T}^\tau+ \sigma_S Y_{t,T}}\right)}{B_T}}\right. \F_t^W \vee \F_{T-\tau}^P\right]}\left\vert\vphantom{\frac{\varphi\left(S_te^{\int_t^T r(u){\mathrm d}u-\frac{\sigma_S^2}{2}X_{t,T}^\tau+ \sigma_S Y_{t,T}}\right)}{B_T}}\right.\widetilde \F_t\right]}, \label{eq:phi_S} $$ where ${\mathbb E^{\widehat \P}\left[\cdot\Big{|}\widetilde \F_t\right]}$ denotes the conditional expectation with respect to $\widetilde \F_t$ under the minimal martingale measure $\widehat \P$. More generally, can be written as $$\label{eq:G} \Phi_t(H) = B_t {\mathbb E^{\widehat \P}\left[{\mathbb E^{\widehat \P}\left[\frac{G(t,S_t,X_{t,T}^\tau, Y_{t,T})}{B_T}\Bigg{|}\F_t^W \vee \F_{T-\tau}^P\right]}\Bigg{|}\widetilde \F_t\right]},$$ for a suitable function $G$ depending on the contract function $\varphi$. Since the $(\bF,\P)$-Brownian motion $Z$ driving the factor $P$ is not affected by the change of measure from $\P$ to $\widehat \P$ by the definition of minimal martingale measure, we have that $Z$ is also an $(\bF,\widehat \P)$-Brownian motion independent of $\widehat W$, see . Hence, we can apply the same arguments used in point (ii) of the proof of Theorem \[th:sol\], to get that, for each $t \in [0,T)$, the random variable $Y_{t,T}$ conditioned on $\F_{T-\tau}^P$ is Normally distributed with mean $0$ and variance $X_{t,T}^\tau$. Then, we can write (in law) that $Y_{t,T}= \sqrt{X_{t,T}^\tau} \epsilon$, where $\epsilon$ is a standard Normal random variable and this allows to find a function $\psi$ such that holds, which means that the conditional expectation with respect to $\F_t^W \vee \F_{T-\tau}^P$ in only depends on $S_t$ and $X_{t,T}^\tau$, for every $t \in [0,T)$. Consequently, the risk-neutral price $\Phi_t(H)$ can be written as $$\begin{aligned} \Phi_t(H) & = {\mathbb E^{\widehat \P}\left[\psi(t,S_t,X_{t,T}^\tau)\bigg{|}\widetilde \F_t\right]}= {\mathbb E^{\widehat \P}\left[\psi(t,S_t,X_{t,T}^\tau)\Bigg{|} S_t\right]}, \label{eq:pricing}\end{aligned}$$ where the last equality holds since $S$ is $\widetilde \bF$-adapted and $X_{t,T}^\tau$ is independent of $\widetilde \F_t$, for each $t \in [0,T)$, see e.g. @pascucci2011pde [Lemma A.108]. More precisely, we have $${\mathbb E^{\widehat \P}\left[\psi(t,S_t,X_{t,T}^\tau) \bigg{|}\widetilde \F_t\right]} ={\mathbb E^{\widehat \P}\left[\psi(t,S_t,X_{t,T}^\tau) \bigg{|} S_t\right]}=g(S_t),$$ where $$g(s)={\mathbb E^{\widehat \P}\left[\psi(t,s,X_{t,T}^\tau) \bigg{|} S_t=s\right]},\quad s \in \R_+.$$ \[rem:sigma\] It is worth to remark that $\psi(t,S_t,x)$, with $x \in \R_+$, represents the risk-neutral price at time $t \in [0,T)$ of the contract $H=\varphi(S_T)$ in a Black & Scholes framework, where the constant volatility parameter $\sigma^{BS}$ is defined by $$\sigma^{BS}:=\sigma_S\sqrt{\frac{x}{T-t}}.$$ This is proved explicitly in Corollary \[th:call2\] below for the special case of a *plain vanilla* European Call option. A pricing formula analogous to is conjectured in @hull1987pricing for a special example of the model suggested here ($\tau=0$ and $\sigma_S=1$). As already noticed the authors start from the very beginning under a risk neutral framework. Theorem \[th:cont\_claim\] extends their results to the more general case and give a rigorous proof. A Black & Scholes-type option pricing formula --------------------------------------------- Let us consider a European Call option with strike price $K$ and maturity $T$ and define the function $C^{BS}$ as follows $$\label{def:pricing_function} C^{BS}(t,s,x):=s\mathcal N(d_1(t,s,x)) - Ke^{-\int_0^t r(u) {\mathrm d}u}\mathcal N(d_2(t,s,x)),$$ where $$\label{def:d1} d_1(t,s,x)=\frac{\log\left(\frac{s}{K}\right) + \int_0^t r(u){\mathrm d}u + \frac{\sigma_S^2}{2}x}{\sigma_S \sqrt{x}}$$ and $d_2(t,s,x)=d_1(t,s,x)-\sigma_S \sqrt{x}$, or more explicitly $$\label{def:d2} d_2(t,s,x)=\frac{\log\left(\frac{s}{K}\right) + \int_0^t r(u){\mathrm d}u - \frac{\sigma_S^2}{2}x}{\sigma_S \sqrt{x}}.$$ Here, $\mathcal N$ stands for the standard Gaussian cumulative distribution function $$\mathcal N(y)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{y} e^{-\frac{z^2}{2}}{\mathrm d}z, \quad \forall\ y \in \R.$$ \[th:call2\] The risk-neutral price $C_t$ at time $t$ of a European Call option written on the BitCoin with price $S$ expiring in $T$ and with strike price $K$ is given by the formula $$\label{eq:call} C_t={\mathbb E^{\widehat \P}\left[C^{BS}(t,S_t,X_{t,T}^\tau)\bigg{|} S_t\right]}, \quad t \in [0,T), $$ where the function $C^{BS}:[0,T) \times \R_+ \times \R_+ \longrightarrow \R$ is given by and the functions $d_1$, $d_2$ are respectively given by -. As in the proof of Theorem \[th:cont\_claim\], let us assume that $\tau<T$. Under the minimal martingale measure $\widehat \P$, the risk-neutral price $C_t$ at time $t \in [0,T)$ of a European Call option written on the BitCoin with price $S$ expiring in $T$ and with strike price $K$, is given by $$\begin{aligned} C_t & = B_t {\mathbb E^{\widehat \P}\left[\frac{\max \left( S_T-K,0\right)}{B_T}\bigg{|}\widetilde \F_t\right]}\\ & = B_t{\mathbb E^{\widehat \P}\left[\widetilde S_T\I_{\{S_T>K\}}\Big{|}\widetilde \F_t\right]}-K e^{-\int_t^T r(u) {\mathrm d}u}{\mathbb E^{\widehat \P}\left[\I_{\{S_T>K\}}\Big{|}\widetilde \F_t\right]}\\ & = B_t J_1 - Ke^{-\int_t^T r(u) {\mathrm d}u}J_2,\end{aligned}$$ where we have set $J_1:={\mathbb E^{\widehat \P}\left[\widetilde S_T\I_{\{S_T>K\}}\Big{|}\widetilde \F_t\right]}$ and $J_2:={\mathbb E^{\widehat \P}\left[\I_{\{S_T>K\}}\Big{|}\widetilde \F_t\right]}$. Recall that $Y_{t,T}= \int_t^T\sqrt{P_{u-\tau}}{\mathrm d}\widehat W_u$, for every $t \in [0,T)$. Then, the term $J_2$ can be written as $$\begin{aligned} J_2 & = {\mathbb E^{\widehat \P}\left[{\mathbb E^{\widehat \P}\left[\I_{\{S_T>K\}}|\F_t^W \vee \F_{T-\tau}^P\right]}\Big{|}\widetilde \F_t\right]}\nonumber \\ & = {\mathbb E^{\widehat \P}\left[\widehat \P\left(S_te^{\int_t^T r(u){\mathrm d}u-\frac{\sigma_S^2}{2}X_{t,T}^\tau+\sigma_S Y_{t,T}}>K \bigg{|}\F_t^W \vee \F_{T-\tau}^P \right)\bigg{|}\widetilde \F_t\right]}\\ & = {\mathbb E^{\widehat \P}\left[\widehat \P\left(-\frac{Y_{t,T}}{ \sqrt{X_{t,T}^\tau}} < \frac{\log\left(\frac{S_t}{K}\right)+ \int_t^T r(u){\mathrm d}u - \frac{\sigma_S^2}{2}X_{t,T}^\tau}{ \sigma_S \sqrt{X_{t,T}^\tau}} \bigg{|}\F_t^W \vee \F_{T-\tau}^P\right) \bigg{|}\widetilde \F_t\right]}\\ & = {\mathbb E^{\widehat \P}\left[\mathcal N\left( d_2(t,S_t,X_{t,T}^\tau) \right) \bigg{|}\widetilde \F_t\right]}, \end{aligned}$$ as for each $t \in [0,T)$, the random variable ${\displaystyle}-\frac{Y_{t,T}}{\sqrt{X_{t,T}^\tau}}$ has a standard Gaussian law $\mathcal N(0,1)$ given $\F_t^W \vee \F_{T-\tau}^P$ under the minimal martingale measure $\widehat \P$. Concerning $J_1$, consider the auxiliary probability measure $\bar \P$ on $(\Omega,\F_T)$ defined as follows: $$\label{def:barP} \frac{{\mathrm d}\bar \P}{{\mathrm d}\widehat \P} := e^{-\frac{\sigma_S^2}{2} \int_0^T P_{u-\tau}{\mathrm d}u + \sigma_S \int_0^T \sqrt{P_{u-\tau}}{\mathrm d}\widehat W_u}, \quad \widehat \P-\mbox{a.s.}.$$ By Girsanov’s Theorem, we get that the process $\bar W=\{\bar W_t,\ t \in [0,T]\}$, given by $$\label{def:barW} \bar W_t := \widehat W_t - \sigma_S\int_0^t\sqrt{P_{u-\tau}}{\mathrm d}u, \quad t \in [0,T],$$ follows a standard $(\bF,\bar \P)$-Brownian motion. In addition, using , we obtain $$\label{def:tildeST} \widetilde S_T = \widetilde S_t e^{\sigma_S\int_t^T\sqrt{P_{u-\tau}}{\mathrm d}\bar W_u + \frac{\sigma_S^2}{2}\int_t^T P_{u-\tau} {\mathrm d}u},$$ for every $t \in [0,T]$. Since $S$ is $\widetilde \bF$-adapted, by and the Bayes formula on the change of probability measure for conditional expectation, for every $t \in [0,T)$ we get $$\begin{aligned} J_1 & = {\mathbb E^{\widehat \P}\left[\widetilde S_T \I_{\{S_T>K\}}\Big{|}\widetilde \F_t\right]}\nonumber\\ & = \widetilde S_t \frac{{\mathbb E^{\widehat \P}\left[e^{-\frac{\sigma_S^2}{2} X_{T}^\tau + \sigma_S Y_{0,T}}\I_{\{S_T>K\}}\bigg{|}\widetilde \F_t\right]}}{e^{-\frac{\sigma_S^2}{2} X_{t}^\tau + \sigma_S Y_{0,t}}}\nonumber\\ & = \widetilde S_t {\mathbb E^{\bar \P}\left[\I_{\left\{\widetilde S_T > K B_T^{-1}\right\}}\bigg{|}\widetilde \F_t\right]}\nonumber\\ & = \widetilde S_t {\mathbb E^{\bar \P}\left[ {\mathbb E^{\bar \P}\left[\I_{\left\{ \sigma_S \bar Y_{t,T} > \log\left(\frac{K}{S_t}\right)- \int_t^T r(u){\mathrm d}u - \frac{\sigma_S^2}{2}X_{t,T}^\tau \right\}}\bigg{|}\F_t^W \vee \F_{T-\tau}^P\right]} \bigg{|}\widetilde \F_t\right]}\nonumber\\ & = \widetilde S_t {\mathbb E^{\bar \P}\left[\bar \P \left( -\frac{\bar Y_{t,T}}{\sqrt{X_{t,T}^\tau}} < \frac{\log\left(\frac{S_t}{K}\right) + \int_t^T r(u){\mathrm d}u + \frac{\sigma_S^2}{2}X_{t,T}^\tau}{\sigma_S \sqrt{X_{t,T}^\tau}} \bigg{|}\F_t^W \vee \F_{T-\tau}^P\right) \bigg{|}\widetilde \F_t\right]}\nonumber\\ & = \widetilde S_t{\mathbb E^{\bar \P}\left[\mathcal N\left(d_1(t,S_t,X_{t,T}^\tau) \right) \bigg{|}\widetilde \F_t\right]}, \label{term2}$$ with $$d_1(t,S_t,X_{t,T}^\tau) = d_2(t,S_t,X_{t,T}^\tau) + \sigma_S\sqrt{X_{t,T}^\tau}.$$ In the above computations, analogously to before, we have set $\bar Y_{t,T}:= \int_t^T\sqrt{P_{u-\tau}}{\mathrm d}\bar W_u$, for each $t \in [0,T)$. Consequently, we have that $\bar Y_{t,T}$ conditional on $\F_{T-\tau}^P$, is a Normally distributed random variable with mean $0$ and variance $X_{t,T}^\tau$, for each $t \in [0,T)$, since $Z$ is not affected by the change of measure from $\widehat \P$ to $\bar \P$. Indeed, by the change of numéraire theorem, we have that the probability measure $\bar \P$ turns out to be the minimal martingale measure corresponding to the choice of the BitCoin price process as benchmark. Further, by applying again the Bayes formula on the change of probability measure for conditional expectation, we get $$\begin{aligned} J_1 & = \widetilde S_t{\mathbb E^{\bar \P}\left[\mathcal N\left(d_1(t,S_t,X_{t,T}^\tau) \right)\bigg{|}\widetilde \F_t\right]}\nonumber\\ & = \widetilde S_t \frac{{\mathbb E^{\widehat \P}\left[\N\left(d_1(t,S_t,X_{t,T}^\tau) \right)e^{-\frac{\sigma_S^2}{2} \int_0^T P_{u-\tau}{\mathrm d}u + \sigma_S \int_0^T \sqrt{P_{u-\tau}}{\mathrm d}\widehat W_u}\bigg{|}\widetilde \F_t\right]}}{e^{-\frac{\sigma_S^2}{2} \int_0^t P_{u-\tau}{\mathrm d}u + \sigma_S \int_0^t \sqrt{P_{u-\tau}}{\mathrm d}\widehat W_u}}\nonumber\\ & = \widetilde S_t {\mathbb E^{\widehat \P}\left[\N\left(d_1(t,S_t,X_{t,T}^\tau) \right)e^{-\frac{\sigma_S^2}{2} X_{t,T}^\tau}{\mathbb E^{\widehat \P}\left[e^{\sigma_S Y_{t,T}} \bigg{|}\F_t^W \vee \F_{T-\tau}^P\right]}\bigg{|}\widetilde \F_t\right]}\nonumber\\ & = \widetilde S_t{\mathbb E^{\widehat \P}\left[\mathcal N\left(d_1(t,S_t,X_{t,T}^\tau) \right)\bigg{|}\widetilde \F_t\right]}, \label{term1}\end{aligned}$$ since the conditional Gaussian distribution of $Y_{t,T}$ gives $${\mathbb E^{\widehat \P}\left[e^{\sigma_S Y_{t,T}}\bigg{|}\F_t^W \vee \F_{T-\tau}^P\right]}= e^{\frac{\sigma_S^2}{2} X_{t,T}^\tau}.$$ Finally, gathering the two terms and , for every $t \in [0,T)$ we obtain $$\begin{aligned} C_t &= S_t{\mathbb E^{\widehat \P}\left[\mathcal N\left(d_1(t,S_t,X_{t,T}^\tau) \right)\bigg{|}\widetilde \F_t\right]}-K e^{-\int_t^T r(u) {\mathrm d}u}{\mathbb E^{\widehat \P}\left[\mathcal N\left( d_2(t,S_t,X_{t,T}^\tau) \right) \bigg{|}\widetilde \F_t\right]} \nonumber \\ &={\mathbb E^{\widehat \P}\left[C^{BS}(t,S_t,X_{t,T}^\tau)\bigg{|}\widetilde \F_t\right]}\nonumber \\ &= {\mathbb E^{\widehat \P}\left[C^{BS}(t,S_t,X_{t,T}^\tau)\bigg{|} S_t\right]}, \end{aligned}$$ where the last equality follows again from @pascucci2011pde [Lemma A.108], since for each $t \in [0,T)$, $X_{t,T}^\tau$ is independent of $\widetilde \F_t$ and $S_t$ is $\widetilde \F_t$-measurable. It is worth noticing that the option pricing formula only depends on the distribution of $X_{t,T}^\tau$ which is the same both under measure $\widehat \P$ and $\bar \P$. As observed in Remark \[rem:sigma\], formula evaluated in $S_t$ corresponds to the Black & Scholes price at time $t \in[0,T)$ of a European Call option written on $S$, with strike price $K$ and maturity $T$, in a market where the volatility parameter is given by $\sigma_S \sqrt{\frac{x}{T-t}}$. Then, for every $t \in [0,T)$ it may be written as: $$\label{eq:integr} C_t=\int_0^{+\infty} C^{BS}(t,S_t,x) f_{X_{t,T}^\tau}(x) {\mathrm d}x, $$ where $f_{X_{t,T}^\tau}(x)$ denotes the density function of $X_{t,T}^\tau$, for each $t \in [0,T)$ (if it exists). The price at time $t$ for a *plain vanilla* European option may also be written as a Black & Scholes style price: $$C_t=S_t Q_1-K e^{-\int_t^T r(u) {\mathrm d}s} Q_2,$$ where $$\begin{aligned} Q_1:={\mathbb E^{\widehat \P}\left[\mathcal N\left(d_1(t,S_t,X_{t,T}^\tau) \right)\bigg{|}S_t\right]} &= \int_0^{+\infty} \mathcal N\left( d_1(t,S_t,x) \right) f_{X_{t,T}^\tau}(x) {\mathrm d}x,\\\end{aligned}$$ and $$\begin{aligned} Q_2:={\mathbb E^{\widehat \P}\left[\mathcal N\left(d_2(t,S_t,X_{t,T}^\tau) \right)\bigg{|}S_t\right]} &= \int_0^{+\infty} \mathcal N\left( d_2(t,S_t,x) \right) f_{X_{t,T}^\tau}(x) {\mathrm d}x.\\\end{aligned}$$ To compute numerically derivative prices by the above formulas, we should compute the distribution of $X_{t,T}^\tau$, which is not an easy task. Similar formulas can be computed for other European style derivatives as for binary options which, indeed, are quoted in BitCoin markets. For the case of a Cash or Nothing Call, which is essentially a bet of $A$ on the exercise event, the risk-neutral pricing formula is given by $$\begin{aligned} C_t^{Bin} & =Ae^{-\int_t^T r(u) {\mathrm d}s}{\mathbb E^{\widehat \P}\left[\mathcal N\left( d_2(t,S_t,X_{t,T}^\tau) \right)\bigg{|} S_t\right]}\\ &= Ae^{-\int_t^T r(u) {\mathrm d}s} \int_0^{+\infty} \mathcal N\left( d_2(t,S_t,x) \right) f_{X_{t,T}^\tau}(x) {\mathrm d}x,\quad t \in [0,T). \label{binarysimple}\end{aligned}$$ By applying the Levy approximation, see @levy1992pricing, to $X_0^T$, the Call option pricing formula becomes $$C_0 = \int_0^{+\infty} C^{BS}(0,S_0,x) \mathcal{LN} pdf_{\alpha(T-\tau),\nu^2(T-\tau)} \left(x\right) {\mathrm d}x, \label{callexample}$$ which can be computed numerically, once parameters $\alpha(T-\tau),\nu(T-S)$ are obtained. Similarly, Binary Options with terminal value $A$ when in the money, are priced by computing numerically the following formula: $$C_0^{Bin}=A \int_0^{+\infty} \mathcal N\left( d_2(T-t,S_0,x) \right) \mathcal{LN} pdf_{\alpha(T-\tau),\nu^2(T-\tau)} \left(x\right){\mathrm d}x. \label{digitalexample}$$ Numerical exercise ------------------ In this subsection we compute European plain vanilla and binary option prices assuming that model parameter are known and considering several strike prices and expiration dates. Besides, we also let the initial sentiment and the delay values change in order to understand their contribution to the option price formation. Assume that model parameters are $\mu_P=0.03,\sigma_P=0.35, \sigma_S=0.04$, the riskless interest rate is $r=0.01$ and that the BitCoin price at time $t=0$ is $S_0=450$ (this is the price by October 2016). In Table \[tab:prices\], Call option prices are reported for $T=3$ months, $\tau$= 5 days. Rows correspond to different values of $P_0$ while columns to different values for the strike price. As expected, Call option prices are increasing with respect to initial sentiment for the BitCoin and decreasing with respect to strike prices. K 400 425 450 475 500 ------------ -------- -------- -------- ------- ------- $P_0=10$ 51.24 28.35 11.46 3.09 0.54 $P_0=100$ 64.12 48.05 34.94 24.69 16.97 $P_0=1000$ 128.68 117.75 107.77 98.66 90.35 : Call option prices against different strikes $K$ and for different values of $P_0$: $S_0=450,r=0.01,\mu_P=0.03,\sigma_P=0.35, \sigma_S=0.04$, $T=3$ months, $\tau$ = $1$ week (5 days).[]{data-label="tab:prices"} In Table \[tab:prices2\], Call option prices are summed up, for $P_0=100$, by letting the expiration date $T$ and the delay $\tau$ vary. Again as expected, for Plain Vanilla Calls the price increases with time to maturity. Increasing the delay reduces option prices; of course the spread is inversely related to the time to maturity of the option. \[htb\] K 400 425 450 475 500 ------------------------------ ------- ------- ------- ------- ------- $T$=1 month, $\tau$=1 week 52.85 33.09 18.27 8.81 3.71 $T$=1 month, $\tau$=2 weeks 51.58 30.62 15.18 6.13 2.00 $T$=3 months, $\tau$=1 week 64.12 48.05 34.94 24.69 16.97 $T$=3 months, $\tau$=2 weeks 62.95 46.65 33.42 23.18 15.60 : Call option prices against different Strikes $K$ and for different values of $T$ and $tau$: $S_0=450,r=0.01,\mu_P=0.03,\sigma_P=0.35, \sigma_S=0.04$ and $P_0=100$.[]{data-label="tab:prices2"} In Tables \[tab:prices3\] and \[tab:prices4\], analogous results are reported for Binary Options with outcome $A=100$; Table \[tab:prices3\] sums up Binary Cash-or-Nothing prices for $S_0=450$, $r=0.01$, $\mu_P=0.03$, $\sigma_P=0.35$, $\sigma_S=0.04$, $T=3$ months, $\tau= 1$ week (5 working days) against several strikes (in colums). Rows correspond to different values $P_0$ for the initial sentiment on BitCoins. As expected, prices are decreasing with respect to strike prices. Here, *in the money* (ITM) options values are decreasing with respect to $P_0$ while *out of the money* (OTM) ones are increasing. The difference in ITM and OTM prices is large for low values of $P_0$, while it is very small for a high level of the initial sentiment factor in BitCoins. This may be justified by the fact that, when the sentiment factor in the BitCoin is strong, all bets are worth, even the OTM ones, since the underlying value is expected to blow up. Binary Call prices decrease with respect to time to maturity for ITM options and increase for OTM options which become more likely to be exercised. The influence of the delay value is tiny, as for vanilla options, being larger for short time to maturities. K 400 425 450 475 500 ------------ ------- ------- ------- ------- ------- $P_0=10$ 97.17 82.77 50.31 18.87 4.24 $P_0=100$ 70.07 58.38 46.58 35.66 26.27 $P_0=1000$ 45.70 41.77 38.14 34.79 31.72 : Digital Cash or Nothing prices against different Strikes $K$ and for different values of $P_0$ on BitCoins. Market parameters are $S_0=450$, $r=0.01$, $\mu_P=0.03$, $\sigma_P=0.35$, $\sigma_S=0.04$, $T=3$ months, $\tau= 5$ days. The prize of the option is set to $A=100$.[]{data-label="tab:prices3"} \[htb\] K 400 425 450 475 500 ------------------------------ ------- ------- ------- ------- ------- $T$=1 month, $\tau$=1 week 86.93 69.97 48.27 28.11 13.83 $T$=1 month, $\tau$=2 weeks 91.50 74.23 48.69 24.84 9.80 $T$=3 months, $\tau$=1 week 70.07 58.38 46.58 35.66 26.27 $T$=3 months, $\tau$=2 weeks 71.21 59.10 46.77 35.36 25.62 : Digital Cash or Nothing prices against different Strikes $K$ and for different values of $T$ and $\tau$. Market parameters are $S_0=450$, $r=0.01$, $\mu_P=0.03$, $\sigma_P=0.35$, $\sigma_S=0.04$ and $P_0=100$. The prize of the option is set to $A=100$.[]{data-label="tab:prices4"} Model fitting on real data {#sec:fit} ========================== This section is devoted to the estimation of the model in (\[eq:S\])-(\[eq:BSdyn\]) on real data; the overall procedure is aimed at describing the dynamics of BitCoin price changes over time. In order to fit the model we need data for both the BitCoin price and the sentiment indicator. Several proxies have been suggested for the latter; traditional indicators of market sentiment on a stock asset such as the number and volume of transactions [@MainDrivers] as well as sentiment indicators such as the number of Google searches or Wikipedia requests in the period under investigation [@bukovina2016sentiment]. Internet-based proxies are particularly interesting for the BitCoin price formation, being BitCoin itself an internet-based asset. First we investigate which of the suggested proxies is consistent with the dynamics in and then we fit the full model to BitCoin prices. Daily data for BitCoin prices, volume and number of transactions are obtained through the website *http://blockchain.info* which provides a mean price among main exchanges trading on BitCoin and the total exchanged volume. Weekly data for the number of Google searches are downloaded from Google-Trends website (daily data are not available). Daily data for Wikipedia requests are obtained through the website *http://tools.wmflabs.org/pageviews*. Proxies for Sentiment {#sec:confind} --------------------- The univariate process $P_t$ is a Geometric Brownian motion; the corresponding discrete process of logarithmic returns is given by $X:={X_i,i\in \bN}$, where $X_i=\log\left(\frac{P_i}{P_{i-1}}\right)$, and it is well-known that these are independent an identically distributed with mean $\left(\mu_P-\frac{\sigma_P^2}{2}\right) \delta$ and variance $\sigma_P^2 \delta$. Hence, in order to choose a suitable proxy, based on discrete observations, for the process $P$, we simply perform a stationary test and a normality test on the corresponding realizations $\lbrace x_i\rbrace_i$ of $X$, using respectively the augmented Dickey–Fuller test [@Tsay:timeseries], and the one-sample Kolmogorov-Smirnov test [@Massey:kstest]. We consider the number and the volume of transactions as examples of traditional indicators and the number of Google searches and Wikipedia requests as examples of sentiment indicators. The first three series were investigated from 01/01/2012 to 31/03/2017 while Wikipedia requests are considered from 01/07/2015 to 31/03/2017. The tests are performed on the whole time series and on the sub-samples from 01/01/2015 to 31/03/2017. In Table \[tab:stattest\] and in Table \[tab:logtest\] we report the outcomes of the tests. \[tab:stattest\] ------------- ------------------ ------------------ ------------------ ------------------ Time series Num. trans. Vol. trans. Google searches Wiki requests All series 1.0000e-03\*\*\* 1.0000e-03\*\*\* 1.0000e-03\*\*\* Sub-sample 1.0000e-03\*\*\* 1.0000e-03\*\*\* 1.0000e-03\*\*\* 1.0000e-03\*\*\* ------------- ------------------ ------------------ ------------------ ------------------ : Augmented Dickey-Fuller test for $X$ with $\alpha=0.05$ \[tab:logtest\] ------------- -------------------- -------------------- -------------------- -------------------- Time series Num. trans. Vol. trans. Google searches Wiki requests All series 5.1011e-05\*\*\*\* 4.3576e-07\*\*\*\* 2.9246e-07\*\*\*\* Sub-sample 0.0035\*\* 0.2152 0.1012 1.3575e-11\*\*\*\* ------------- -------------------- -------------------- -------------------- -------------------- : Kolmogorov-Smirnov test for $X$ with $\alpha=0.05$ Non-stationarity is rejected for all proxies; besides, lognormality is not rejected, for the sub-sample from 01/01/2015 to 31/03/2017, both for the $Volume$ and the number of $Google\; searches$. In Figure \[fig:test\_norm\] we plot the histograms of the latter two proxies; it is evident that the log-normality fit for the $Volume$ time series is better. ![Histogram with normal distribution fit for $P=volume\; of\; transactions$ (on the left) and $P=Google\; searches$ (on the right).[]{data-label="fig:test_norm"}](test_norm_vol.png "fig:"){width=".45\textwidth"}![Histogram with normal distribution fit for $P=volume\; of\; transactions$ (on the left) and $P=Google\; searches$ (on the right).[]{data-label="fig:test_norm"}](test_norm_google.png "fig:"){width=".45\textwidth"} Estimation Results with QML method {#sec:est_res} ---------------------------------- According to the outcomes in Table \[tab:logtest\] we consider the daily time series of the volume of transactions and the weekly time series of the Google searches from 01/01/2015 to 31/03/2017 as suitable proxies for process $P$. We fit the model in and to the BitCoin price and the volume of transaction as well as to the BitCoin prices and number of Google searches by applying the Profile-Quasi-Likelihood (PML) method as describes in Section \[sec:fit\]. Google-trends provides a scaled time series for the number of searches so the the maximum value is $100$; in order to compare outcomes we do the same for the $Volume$ time series. In Figures \[fig:path\_goog\] and \[fig:path\_vol\] we plot the time series of BitCoin price with Google searches (weekly) and the time series of BitCoin price with volume of transactions (daily), respectively. ![Weekly time series of the Google searches (above) and BitCoin price (bottom) from 01/01/2015 to 31/03/2017.[]{data-label="fig:path_goog"}](path_goog.png) ![Daily time series of the volume of transactions (above) and BitCoin price (bottom) from 01/01/2015 to 31/03/2017.[]{data-label="fig:path_vol"}](path_vol.png) In what follows we assume that $\Delta$=1 week is the observation step for BitCoin log-returns. ### Sentiment maesured by Volume {#sec:voltrans} Given daily observations $ \lbrace P_i \rbrace _i$ of the volume of transactions we are able to compute the cumulative weekly sentiment $\lbrace A_i \rbrace_i$; for $\tau=0$ $A_i$ is simply the mean volume during the preceding week i.e. $$\label{eq:apprAi} A_i=\int_{\left(i-1\right)\Delta}^{i\Delta}P_{u} {\mathrm d}u =\sum_{j=1}^{7}\int_{\left(i-1\right)\Delta+\left(j-1\right)\delta}^{\left(i-1\right)\Delta+j\delta} P_{u} {\mathrm d}u =\sum_{j=1}^{7} P_{\left(i-1\right)\Delta+j\delta} \delta=\frac{\sum_{j=1}^{7} P_{\left(i-1\right)\Delta+j\delta}}{7} \Delta,$$ The generalization to $\tau>0$ is straightforward as soon as we assume $\tau=r\delta$ for some positive integer $r$; in which case $A_i$ would be the mean volume of the 7 days preceding time $i\Delta-r\delta$. By applying the profile quasi maximum likelihood we obtain $\tau=5$ days and the following estimates of other the parameters: \[tab:fitparvol2\] Variable Fitted value Std. error t-value P($>\lvert t \rvert$) ------------ -------------- ------------ --------- ----------------------- -- $\mu_P$ 1.0404 0.7373 1.4110 0.1610 $\sigma_P$ 1.1092 0.0725 15.2924 0.0000 $\mu_S$ 0.0153 0.0083 1.8434 0.0679 $\sigma_S$ 0.0830 0.0054 15.2403 0.0000 : Parameter fit with $\tau=5$ In order to asses parameters significance the $t$-stat is computed, for each parameter, under the null hypothesis that its value is zero; Table \[tab:fitparvol2\] shows that $\mu_P$ is not statistically significant and that $\mu_S$ is weakly significant. Finally we evaluate the confidence region for $\tau$ using the and $\tau\in \lbrace 0,1,2,\ldots,10 \rbrace$ days. We find that $\tau=2,3,4$ days belong to the confidence region. Estimates of other parameters are indeed very similar in any of such cases and analogous comments apply. ### Sentiment measured by Google Searches Assume now that Google searches are representative of sentiment about BitCoin. Since we have weekly data we should aggregate both Google searches and BitCoin returns to a coarser observation step; however this would reduce the time series length dramatically and corresponding estimates might be unreliable. Hence we assume that the available observations correspond to the cumulative sentiment time series $A_i$ and we assume it exist a non-negative integer $c$ such that $\tau=c\Delta$ i.e. in this case $\tau$ is on a weekly scale and not on a daily scale like in the previous case. By applying the profile quasi maximum likelihood we obtain $\tau=1$ week and the following estimates of other parameters: \[tab:fitpargoog1\] Variable Fitted value Std. error t-value P($>\lvert t \rvert$) ------------ -------------- ------------ --------- ----------------------- -- $\mu_P$ 0.9573 0.7315 1.3087 0.1935 $\sigma_P$ 1.0818 0.0714 15.1611 0.0000 $\mu_S$ 0.0181 0.0092 1.9534 0.0535 $\sigma_S$ 0.0867 0.0057 15.1774 0.0000 : Parameter fit with $\tau=1$ The $t$-value is reported for all parameters as in Section \[sec:voltrans\]. Again, $\mu_P$ is not statistically significant and that $\mu_S$ is weakly significant. Finally we evaluate the confidence regions for $\tau$ using the and $\tau\in \lbrace 0,1,2,\ldots,10 \rbrace$ weeks. We find that there are not other values of $\tau$ that belong to the confidence region. Model performance on market option prices {#sec:assessing} ========================================= Recently some online platforms have appeared where it is possible to trade on plain vanilla options on the BitCoin. A relevant platform where bid-ask quotes are publicly available is www.deribit.com; we will consider option prices on this website as “market prices”.  on July 28, 2017[]{data-label="fig:screen"}](screenshot01.png){width=".70\textwidth"} In Figure \[fig:screen\] the screenshot of the website on July 28, 2017; we will consider the mid-value of the Bid-Ask range as a benchmark for assessing model performance discarding options for which there was no transactions. Concerning the price of the underlying BitCoin, it was set as the price of the BitCoin index available from blockchain.info at the same time as the option data download, which also appears in the north-west corner of the screenshot. Every day two different expiration dates are available corresponding to a one month and two months maturity at issue. We are aware of possible synchronicity problems but as a first evaluation of the suggested pricing formula we intentionally neglect this friction. Model prices are then compared with corresponding market prices by computing the Root Mean Squared Error of the model across all the considered sample of options and of suitably chosen sub-samples. The same is done when prices are computed with the benchmark no-sentiment model chosen as Black and Scholes model. Of course the latter is estimated on the same time-series as those considered in Subsection \[sec:est\_res\]. Only the volatility estimation matters since we set $r=0$ for both models. In Tables \[tab:optprice28\]-\[tab:optprice28bis\] we report market data for July 28, 2017 as well as model option prices in the suggested model and for the Black and Scholes benchmark. On the chosen day available maturities were complete i.e. August 25 and September 29. \[tab:optprice28\] K Bid Ask Model (Vol.) Model (Google) BS ------ -------- -------- -------------- ---------------- -------- 2200 0.2200 0.2455 0.2125 0.2196 0.2110 2300 0.1956 0.2210 0.1820 0.1909 0.1799 2400 0.1600 0.1983 0.1538 0.1645 0.1510 2500 0.1280 0.1774 0.1281 0.1404 0.1248 2600 0.1050 0.1582 0.1052 0.1187 0.1015 2700 0.0850 0.1407 0.0867 0.0995 0.0812 2800 0.0703 0.1247 0.0696 0.0827 0.0640 2900 0.0540 0.1104 0.0482 0.0698 0.0497 0.0753 0.0395 0.0833 : Option Price with t=28 July and T=25 August \[tab:optprice28bis\] K Bid Ask Model (Vol.) Model (Google) BS ------ -------- -------- -------------- ---------------- -------- 2200 0.2108 0.2764 0.2350 0.2981 0.2297 2300 0.1881 0.2547 0.2091 0.2779 0.2029 2400 0.2004 0.2343 0.1851 0.2589 0.1781 2500 0.1700 0.2154 0.1631 0.2411 0.1555 2600 0.1650 0.1950 0.1431 0.2244 0.1349 2700 0.1168 0.1710 0.1249 0.2087 0.1164 2800 0.1044 0.1630 0.1086 0.1941 0.0999 2900 0.0934 0.1490 0.0940 0.1804 0.0853 0.0723 0.1535 0.0932 : Option Price with with t=28 July and T=29 September In Tables \[tab:RMSE\_vol2\]-\[tab:RMSE\_goog2\] we sum up the outcomes for the RMSE for All options and for subsamples obtained by considering the same expiration date respectively when sentiment is conveyed by the Volume and by Google searches. The same in Tables \[tab:RMSE\_vol1\]-\[tab:RMSE\_goog1\] where the subsample are obtained according to moneyness. Highlighted in the table (in bold) are cases where the plain vanilla Black and Scholes model does better than the model suggested here. Overall our pricing formula does much better than the benchmark in all cases. It is worth noticing that Google Searches tend to overprice long-term options while it is the very best for shorter term options as if this sentiment indicator is driven by enthusiasm giving a sudden impulse to options. \[tab:RMSE\_vol1\] Options Num. RMSE Model RMSE BS ------------- ------ ------------ ------------ All 144 0.3089 0.4879 Very Shorts 16 0.0898 **0.0817** 1 Months 32 0.1132 0.1448 2 Months 32 0.1306 0.1751 ITM 54 0.1536 0.2617 ATM 36 0.1687 0.2625 OTM 54 0.2082 0.3173 : RMSE Option Price with Volume of Transactions \[tab:RMSE\_vol2\] ------------ ------------ ------------ ------------ --------- ------------ --------- Date RMSE Model RMSE BS RMSE Model RMSE BS RMSE Model RMSE BS 20/07/2017 0.0712 **0.0648** 0.0627 0.1768 0.0949 0.1883 21/07/2017 0.0547 **0.0499** 0.0172 0.1428 0.0574 0.1512 22/07/2017 - - 0.0180 0.1394 0.0180 0.1394 23/07/2017 - - 0.0556 0.1475 0.0556 0.1475 24/07/2017 - - 0.0773 0.1420 0.0773 0.1420 25/07/2017 - - 0.1056 0.1445 0.1056 0.1445 26/07/2017 - - 0.1252 0.1519 0.1252 0.1519 27/07/2017 - - 0.1306 0.1508 0.1306 0.1508 28/07/2017 0.0753 0.0833 0.0723 0.0932 0.1044 0.1250 29/07/2017 0.0562 0.0717 0.0493 0.0919 0.0748 0.1165 30/07/2017 0.0531 0.0710 0.0705 0.0850 0.0883 0.1107 31/07/2017 0.0343 0.0621 0.0664 0.0795 0.0748 0.1009 ------------ ------------ ------------ ------------ --------- ------------ --------- : RMSE Option Price by Date with Volume of Transactions \[tab:RMSE\_goog1\] Options Num. RMSE Model RMSE BS ------------- ------ ------------ ------------ All 136 0.3380 0.4854 Very Shorts 8 0.0367 0.0648 1 Months 32 0.0621 0.1448 2 Months 32 0.2408 **0.1751** ITM 51 0.1912 0.2617 ATM 34 0.1613 0.2612 OTM 51 0.2273 0.3149 : RMSE Option Price with Google Searches \[tab:RMSE\_goog2\] ------------ ------------ --------- ------------ ------------ ------------ ------------ Date RMSE Model RMSE BS RMSE Model RMSE BS RMSE Model RMSE BS 20/07/2017 0.0367 0.0648 0.0578 0.1768 0.0684 0.1883 21/07/2017 - - 0.0158 0.1428 0.0158 0.1428 22/07/2017 - - 0.0253 0.1394 0.0253 0.1394 23/07/2017 - - 0.0891 0.1475 0.0891 0.1475 24/07/2017 - - 0.0935 0.1420 0.0935 0.1420 25/07/2017 - - 0.0961 0.1445 0.0961 0.1445 26/07/2017 - - 0.1018 0.1519 0.1018 0.1519 27/07/2017 - - 0.1028 0.1508 0.1028 0.1508 28/07/2017 0.0395 0.0833 0.1535 **0.0932** 0.1585 **0.1250** 29/07/2017 0.0275 0.0717 0.1601 **0.0919** 0.1625 **0.1165** 30/07/2017 0.0308 0.0710 0.0649 0.0850 0.0718 0.1107 31/07/2017 0.0243 0.0621 0.0677 0.0795 0.0720 0.1009 ------------ ------------ --------- ------------ ------------ ------------ ------------ : RMSE Option Price by Date with Google Searches Concluding remarks {#sec:remarks} ================== In this paper we borrow the idea, suggested in recent literature, that BitCoin prices are driven by sentiment on the BitCoin system and underlying technology. Main references in this area are @MainDrivers [@GoogleTrends; @SentimentAnalysis; @bukovina2016sentiment]. In order to account for such behavior we develop a model in continuous time which describes the dynamics of two factors, one representing the sentiment index on the BitCoin system and the other representing the BitCoin price itself, which is directly affected by the first factor; we also take into account a delay between the sentiment index and its delivered effect on the BitCoin price. We investigate statistical properties of the proposed model and we show its arbitrage-free property. Under our model assumption we derived a closed form approximation for the joint density of the discretely observed process and we proposed a statistical estimation for that model. By applying the classical risk-neutral evaluation we are able to derive a quasi-closed formula for European style derivatives on the BitCoin with special attention of Plain Vanilla and Binary options for which a market already exists (e.g. https://deribit.com, https://coinut.com ). Of course sentiment about BitCoin or, more generally, on cryptocurrencies or IT finance is not directly observed but several variables may be considered as indicators. Here, we analyzed the volume and more unconventional sentiment indicators such as the number of Google searches and the number of Wikipedia requests about the topic (as suggested @GoogleTrends [@SentimentAnalysis]). First of all, we investigated whether these proxies were consistent with the suggested model and we proved that both the volume of transactions and the number of Google searches give a good fit of the dynamics described in the model. Finally we fit the model using real data of BitCoin price with Volume of transactions and Google searches respectively and we provided the estimation results. Several open problems are left for future research. As a first issue would like to address a multivariate extension of the model in order to take into consideration the special feature of BitCoin being traded in different exchanges and related stylized facts and to investigate whether there are arbitrage opportunities between different BitCoin exchanges. Besides we would like to investigate whether the model is suitable to describe bubble effects which have been also evidenced for the BitCoin price dynamics. [**Acknowledgments**]{} The first and the second named authors are grateful to Banca d’Italia and Fondazione Cassa di Risparmio di Perugia for the financial support. Levy approximation {#appendix:levy} ================== In @levy1992pricing the author proves that the distribution of the mean integrated Brownian motion $\frac{1}{s} \int_0^s P_u {\mathrm d}u$ can be approximated with a log-normal distribution, at least for suitable values of the model parameters $\mu_P, \sigma_P$; the parameters of the approximating log-normal distribution are obtained by applying a moment matching technique. Set $$\label{def:IP} IP(s):=\int_0^s P_u {\mathrm d}u,\quad s>0.$$ Of course, the distribution of $IP(s)$ can also be approximated by a log-normal for $s >0$. By applying the moment matching technique the parameters of the corresponding log-Normal distribution for $IP(s)$ are given by $$\alpha(s)=\log\left(\frac{{\mathbb E\left[ IP(s)\right]}^2}{\sqrt{{\mathbb E\left[IP(s)^2\right]}}}\right),$$ $$\nu^2(s)=\log \left(\frac{{\mathbb E\left[IP(s)^2\right]}}{{\mathbb E\left[IP(s)\right]}^2}\right),$$ The approximate distribution density function of $IP(s)$ is thus given by $$f_{IP(s)}(x)= \mathcal{LN} pdf_{\alpha(s),\nu^2(s)} \left(x\right), \quad \mbox{ if } s > 0$$ where $ \mathcal{LN} pdf_{m,v} $ denotes the probability distribution function of a log-normal distribution with parameters $m$ and $v$, defined as $$\mathcal{LN} pdf_{m,v}(y)=\frac{1}{y\sqrt{2\pi v}} e^{-\frac{(log (y)-m)^2}{2v}}, \quad \forall\ y \in \R^+.$$ In the paper the above approximation is applied twice with completely different purposes. In Section \[sec:option-pricing\] it is applied to derive an approximate distribution for the integrated sentiment process starting at $t=0$ i.e. to $X_{0,T}^\tau=X_\tau^\tau +IP(T-\tau)$. Note that $X_{0,T}^\tau-X_\tau^\tau =IP(T-\tau)$ hence the derivations of its distribution is trivial once that of $IP(T-\tau)$ is known. In Section \[sec:fit\], once $\tau<\Delta$ is assigned, the Levy approximation is applied to derive the distribution of $A_1$ and of $A_i$ given $A_{i-1}$ where $A_1=X_\tau^\tau + IP(\Delta-\tau)$ and, for $i\geq 2$, $A_i=\int_{(i-1)\Delta-\tau}^{i\Delta-\tau} P_u du=\int_{-\tau}^{\Delta-\tau} P_{u+(i-1)\Delta} {\mathrm d}u =P_{(i-1)\Delta}\int_{-\tau}^{\Delta-\tau} P_{u} {\mathrm d}u =P_{(i-1)\Delta} \left( X_\tau^\tau +IP(\Delta-\tau) \right)$. Technical proofs {#appendix:technical} ================ In order to prove the Lemma let us first compute the mean and the variance of $IP(s)$ given in for each $s>0$. Fix $s >0$. Since $P_u > 0$ for each $u \in (0, s]$, by applying Fubini’s theorem we get $${\mathbb E\left[IP(s)\right]} = {\mathbb E\left[\int_{0}^{s}P_{u}{\mathrm d}u\right]}=\int_{0}^{s}{\mathbb E\left[P_{u}\right]}{\mathrm d}u,$$ where, for each $u \ge 0$, we have $$\begin{aligned} {\mathbb E\left[P_{u}\right]} & = \phi(0) e^{\left(\mu_{P}-\frac{\sigma_{P}^{2}}{2}\right)u} {\mathbb E\left[e^{\sigma_{P}Z_{u}}\right]} = \phi(0)e^{\mu_{P}u},\end{aligned}$$ since $P$ is a geometric Brownian motion with $P_0=\phi(0)$. Hence $${\mathbb E\left[IP(s)\right]}=\phi(0)\int_{0}^{s} e^{\mu_{P}u} {\mathrm d}u=\frac{\phi(0)}{\mu_{P}}\left(e^{\mu_{P}(s)}-1\right).$$ As for the variance of $IP(s)$, we have $$\begin{aligned} \mathbb V{\rm ar}[IP(s)] & = \mathbb V{\rm ar}\left[\int_0^{s} P_u {\mathrm d}u\right] = {\mathbb E\left[\left(\int_0^{s} P_u {\mathrm d}u\right)^2\right]} - {\mathbb E\left[\int_0^{s} P_u {\mathrm d}u\right]}^2,\end{aligned}$$ with $$\begin{aligned} {\mathbb E\left[\left(\int_0^{s} P_u {\mathrm d}u\right)^{2}\right]} & = 2{\mathbb E\left[\int_{0}^{s}P_{v}dv\int_{0}^{v}P_{u}{\mathrm d}u\right]} =2{\mathbb E\left[\int_{0}^{s}\int_{0}^{v}P_{u}P_{v}{\mathrm d}v{\mathrm d}u\right]}\\ & = 2\int_{0}^{s}\int_{0}^{v}{\mathbb E\left[P_{u}P_{v}\right]}{\mathrm d}v{\mathrm d}u, \label{eq:MP2}\end{aligned}$$ where the last equality again holds thanks to Fubini’s theorem. Moreover, by the independence property of the increments of Brownian motion, for $0 < u < v \leq s$, we get $$\begin{aligned} {\mathbb E\left[P_{u}P_{v}\right]} & = {\mathbb E\left[P_{u}^{2}e^{\left(\mu_{P}-\frac{\sigma_{P}^{2}}{2}\right)(v-u)+\sigma_{P}\left(Z_{v}-Z_{u}\right)}\right]}\\ & =e^{\left(\mu_{P}-\frac{\sigma_{P}^{2}}{2}\right)(v-u)} {\mathbb E\left[P_{u}^{2}{\mathbb E\left[e^{\sigma_{P}\left(Z_{v}-Z_{u}\right)}|\F_u^P\right]}\right]}\\ & =e^{\left(\mu_{P}-\frac{\sigma_{P}^{2}}{2}\right)(v-u)} {\mathbb E\left[P_{u}^{2}\right]}{\mathbb E\left[e^{\sigma_{P}\left(Z_{v}-Z_{u}\right)}\right]}\\ & =e^{\left(\mu_{P}-\frac{\sigma_{P}^{2}}{2}\right)(v-u)} {\mathbb E\left[P_{u}^{2}\right]}{\mathbb E\left[e^{\frac{\sigma_{P}^{2}\left(v-u)\right)}{2}}\right]}=e^{\mu_{P}(v-u)}{\mathbb E\left[P_{u}^{2}\right]}.\end{aligned}$$ Further, $$\begin{aligned} {\mathbb E\left[P_{u}^{2}\right]} & =\phi^2(0)e^{2\left(\mu_{P}-\frac{\sigma_{P}^{2}}{2}\right)u}{\mathbb E\left[e^{2\sigma_{P}Z_{u}}\right]} =\phi^2(0)e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)u}.\end{aligned}$$ Hence $$\label{eq:pp} {\mathbb E\left[P_{u}P_{v}\right]}=\phi^2(0)e^{\mu_{P}(v-u)}e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)u},$$ and by plugging into , we have $$\begin{aligned} {\mathbb E\left[IP(s)^{2}\right]} & = 2\phi^2(0)\int_{0}^{s}e^{\mu_{P}v} \int_{0}^{v}e^{\left(\mu_{P}+\sigma_{P}^{2}\right)u} {\mathrm d}u{\mathrm d}v\\ & = \frac{2\phi^{2}(0)}{\left(\mu_{P}+\sigma_{P}^{2}\right)\left(2\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)s} -1\right)-\frac{2\phi^{2}(0)}{\mu_{P}\left(\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\mu_{P}s} -1\right).\end{aligned}$$ Finally, gathering the results we get $$\begin{aligned} \mathbb V{\rm ar}[IP(s)] & = \frac{2\phi^{2}(0)}{\left(\mu_{P}+\sigma_{P}^{2}\right)\left(2\mu_{P}+\sigma_{P}^{2}\right)}\left[e^{\left(2\mu_{P}+\sigma_{P}^{2}\right)s} -1\right] \\ & \qquad - \frac{2\phi^{2}(0)}{\mu_{P}\left(\mu_{P}+\sigma_{P}^{2}\right)}\left(e^{\mu_{P}s} -1\right)-\left(\frac{\phi(0)}{\mu_{P}}\left(e^{\mu_{P} s} -1\right)\right)^2.\end{aligned}$$ Note that $X_t^\tau$, with $t\in [0,\tau]$, and $X_{t,T}^\tau$, with $t <T \leq \tau$, are fully deterministic and the computation is trivial. To prove points (i)-(iii), it suffices to observe that $$X_t^\tau=X_\tau^\tau+IP(t-\tau),\quad t > \tau,$$ $$X_{t,T}^\tau=\int_{t-\tau}^0 \phi(u){\mathrm d}u + IP(T-\tau),\quad t \leq \tau < T,$$ and the computation easily follows once those of $IP(s)$ are known for $s>0$. To prove point (ii), it is worth noticing that given $0 \leq v<s$ $$\begin{aligned} IP(s)-IP(v) & =\int_v^s P_u {\mathrm d}u = \int_v^{s} P_v e^{\left(\mu_P-\frac{\sigma_P^2}{2}\right)(u-v)+\sigma_P(Z_u-Z_v)} {\mathrm d}u\stackrel{({\rm law})}{=} P_v\int_v^{s} e^{\left(\mu_P-\frac{\sigma_P^2}{2}\right)(u-v)+\sigma_P Z_{u-v} } {\mathrm d}u \\ & = P_v\int_0^{s-v} e^{\left(\mu_P-\frac{\sigma_P^2}{2}\right)r+\sigma_P Z_r} {\mathrm d}r=P_v IP(s-v), \label{eq:u-t}\end{aligned}$$ where $r=u-v$. To obtain the desired result it suffices to note that, for $\tau\leq t<T$, $$\begin{aligned} X_{t,T}^\tau & =IP(T-\tau)-IP(t-\tau) $$ and apply . The computation of the mean and variance of the above difference is straightforward given the independence of Brownian increments. Firstly, we prove that formula defines a probability measure $\Q$ equivalent to $\P$ on $(\Omega,\F_T)$. This means we need to show that $L^\Q$ is an $(\bF,\P)$-martingale, that is, ${\mathbb E\left[L_T^\Q\right]}=1$. Since the $\bF$-progressively measurable process $\gamma$ can be suitably chosen, to prove this relation we can assume $\gamma \equiv 0$, without loss of generality. Set $$\label{def:alpha} \alpha_t := -\frac{\mu_S P_{t-\tau}-r(t)}{\sigma_S \sqrt{P_{t-\tau}}}, \quad t \in [0,T].$$ We observe that since $\phi(t) > 0$, for each $t \in [-L,0]$, in , by Theorem \[th:sol\], point (i), we have that $P_{t-\tau} > 0$, $\P$-a.s. for all $t \in [0,T]$, so that the process $\alpha=\{\alpha_t,\ t \in [0,T]\}$ given in is well-defined, as well as the random variable $L_T^\Q$. Clearly, $\alpha$ is an $\bF$-progressively measurable process. Moreover, since the trajectories of the process $P$ are continuous, then $P$ is almost surely bounded on $[0,T]$ and this implies that $\int_0^T|\alpha_u|^2 {\mathrm d}u < \infty$ $\P$-a.s.; on the other hand, the condition $\phi(t)>0$, for every $t \in [-L,0]$, implies that almost every path of $\left\{\frac{1}{\sigma_S \sqrt{P_{t-\tau}}},\ t \in [0,T]\right\}$ is bounded on the compact interval $[0,T]$. Set $\F_t^P:=\F_0^P=\{\Omega,\emptyset\}$, for $t \leq 0$. Then, $\alpha_u$, for every $u \in [0,T]$, is $\F_{T-\tau}^P$-measurable. Since $Z_{u-\tau}$ is independent of $W_u$, for every $u \in [\tau, T]$, the stochastic integral $\int_0^T \alpha_u {\mathrm d}W_u$ conditioned on $\F_{T-\tau}^P$ has a normal distribution with mean zero and variance $\int_0^T |\alpha_u|^2 {\mathrm d}u$. Consequently, the formula for the moment generating function of a normal distribution implies $${\mathbb E\left[e^{\int_0^T \alpha_u {\mathrm d}W_u}\bigg{|}\F_{T-\tau}^P\right]}=e^{\frac{1}{2}\int_0^T |\alpha_u|^2 {\mathrm d}u},$$ or equivalently $$\label{eq:gen} {\mathbb E\left[e^{\int_0^T \alpha_u {\mathrm d}W_u-\frac{1}{2}\int_0^T |\alpha_u|^2 {\mathrm d}u}\bigg{|}\F_{T-\tau}^P\right]}=1.$$ Taking the expectation of both sides of immediately yields ${\mathbb E\left[L_T^\Q\right]}=1$. Now, set $\widetilde S_t:= {\displaystyle}\frac{S_t}{B_t}$, for each $t \in [0,T]$. It remains to verify that the discounted BitCoin price process $\widetilde S=\{\widetilde S_t,\ t \in [0,T]\}$ is an $(\bF,\Q)$-martingale. By Girsanov’s theorem, under the change of measure from $\P$ to $\Q$, we have two independent $(\bF,\Q)$-Brownian motions $W^\Q=\{W_t^\Q,\ t \in [0,T]\}$ and $Z^\Q=\{Z_t^\Q,\ t \in [0,T]\}$ defined respectively by $$\begin{aligned} W_t^\Q & := W_t - \int_0^t\alpha_u {\mathrm d}u,\quad t \in [0,T],\\ Z_t^\Q & := Z_t + \int_0^t\gamma_s {\mathrm d}s, \quad t \in [0,T].\end{aligned}$$ Under the martingale measure $\Q$, the discounted BitCoin price process $\widetilde S$ satisfies the following dynamics $$\begin{aligned} {\mathrm d}\widetilde S_t & = \widetilde S_t\sigma_S\sqrt{P_{t-\tau}}{\mathrm d}W_t^\Q, \quad \widetilde S_0=s_0 \in \R_+,\end{aligned}$$ which implies that $\widetilde S$ is an $(\bF,\Q)$-local martingale. Finally, proceeding as above it is easy to check that $\widetilde S$ is a true $(\bF,\Q)$-martingale. By applying @levy1992pricing we have (see Appendix \[appendix:levy\]) that the distribution of $A_1-X_\tau^\tau$ can be approximated by a log-normal with parameters $$\begin{aligned} \alpha_1 & =\log\left(\frac{\mathbb{E}[IP(\Delta-\tau)]^2}{\sqrt{\mathbb{E}[IP(\Delta-\tau)^2]}} \right),\qquad \nu_1^2 = \log\left( \frac{\mathbb{E}[IP(\Delta-\tau)^2]}{\mathbb{E}[IP(\Delta-\tau)]^2} \right).\end{aligned}$$ By applying the outcomes of Lemma \[th:means\], we have $$\begin{aligned} \mathbb{E}[A_1] & = \mathbb{E} [P_0] \frac{e^{\mu_P (\Delta-\tau)}-1}{\mu_P} = P_0 \frac{e^{\mu_P (\Delta-\tau)}-1}{\mu_P}\\ \mathbb{E}[A_1^2] & = \mathbb{E} [P_0^2] \left( \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{\left(2\mu_P+\sigma_P^2\right)(\Delta-\tau)}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P(\Delta-\tau)}-1}{\mu_P} \right] \right)\\ & = P_0^2 \left( \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{\left(2\mu_P+\sigma_P^2\right)(\Delta-\tau)}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P(\Delta-\tau)}-1}{\mu_P} \right] \right).\end{aligned}$$ Hence $$\begin{aligned} \alpha_1 & = \log \left(\frac{P_0\left( \frac{e^{\mu_P \Delta}-1}{\mu_P} \right)^2}{\sqrt{ \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{\left(2\mu_P+\sigma_P^2\right)\Delta}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P\Delta}-1}{\mu_P} \right]}}\right) \\ \nu_1^2 & = \log\left(\frac{ \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{\left(2\mu_P+\sigma_P^2\right)\Delta}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P\Delta}-1}{\mu_P} \right]}{\left( \frac{e^{\mu_P \Delta}-1}{\mu_P} \right)^2}\right).\end{aligned}$$ By applying simple computation we get the outcomes for part (i). Moreover, $$\begin{aligned} \mathbb{E}[A_i] & = \mathbb{E} [P_{i-1}] \frac{e^{\mu_P \Delta}-1}{\mu_P} = \mathbb{E}[P_{i-2}] \; e^{\mu_P\Delta} \; \frac{e^{\mu_P \Delta}-1}{\mu_P} = \mathbb{E}[A_{i-1}]\; e^{\mu_P\Delta}\\ \mathbb{E}[A_i^2] & = \mathbb{E} [P_{i-1}^2] \left( \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{\left(2\mu_P+\sigma_P^2\right)\Delta}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P\Delta}-1}{\mu_P} \right] \right)\\ & = \mathbb{E}[P_{i-2}^2] e^{\left(2\mu_P+\sigma_P^2\right)\Delta} \left( \frac{2}{\mu_P+\sigma_P^2} \left[ \frac{e^{\left(2\mu_P+\sigma_P^2\right)\Delta}-1}{2\mu_P+\sigma_P^2}-\frac{e^{\mu_P\Delta}-1}{\mu_P} \right] \right)\\ & = \mathbb{E}[A_{i-1}^2]\; e^{\left(2\mu_P+\sigma_P^2\right)\Delta}.\end{aligned}$$ Then $$\begin{aligned} \alpha_i & = \log\left( \frac{\mathbb{E}[A_i]^2}{\sqrt{\mathbb{E}[A_i^2]}} \right) = \log\left( \frac{\mathbb{E}[A_{i-1}]^2 \left(e^{\mu_P\Delta}\right)^2}{\sqrt{\mathbb{E}[A_{i-1}^2] \; e^{\left(2\mu_P+\sigma_P^2\right)\Delta}}} \right) \\ & = \log\left( \frac{\mathbb{E}[A_{i-1}]^2}{\sqrt{\mathbb{E}[A_{i-1}^2]}} \right) + \log\left( \frac{\left(e^{\mu_P\Delta}\right)^2}{\sqrt{e^{\left(2\mu_P+\sigma_P^2\right)\Delta}}}\right) = \alpha_{i-1} + \left( \mu_P-\frac{\sigma_P^2}{2} \right)\Delta, \\ \nu_i^2 & = \log\left( \frac{\mathbb{E}[A_i^2]}{\sqrt{\mathbb{E}[A_i]^2}} \right) = \log\left( \frac{\mathbb{E}[A_{i-1}^2] \; e^{\left(2\mu_P+\sigma_P^2\right)\Delta}}{\mathbb{E}[A_{i-1}]^2 \; e^{(\mu_P\Delta)^2}} \right)\\ & =\log\left( \frac{\mathbb{E}[A_{i-1}^2]}{\sqrt{\mathbb{E}[A_{i-1}]^2}} \right) + \log\left( \frac{e^{\left(2\mu_P+\sigma_P^2\right)\Delta}}{e^{(\mu_P\Delta)^2}} \right) = \nu_{i-1}^2 + \sigma_P^2\Delta.\end{aligned}$$ Conditioning to $A_{i-1}$ $$\begin{aligned} \alpha_i & = \log\left(A_{i-1}\right) + \left( \mu_P-\frac{\sigma_P^2}{2}\right)\Delta, \\ \nu_i^2 & = \sigma_P^2\Delta,\end{aligned}$$ which gives part (ii).

--- abstract: '> To aid a variety of research studies, we propose [<span style="font-variant:small-caps;">TwiRole</span>]{}, a hybrid model for *role-related user classification on Twitter*, which detects male-related, female-related, and brand-related (i.e., organization or institution) users. [<span style="font-variant:small-caps;">TwiRole</span>]{} leverages features from tweet contents, user profiles, and profile images, and then applies our hybrid model to identify a user’s role. To evaluate it, we used two existing large datasets about Twitter users, and conducted *both intra- and inter-comparison experiments*. [<span style="font-variant:small-caps;">TwiRole</span>]{} outperforms existing methods and obtains more balanced results over the several roles. We also confirm that user names and profile images are good indicators for this task. Our research extends prior work that does not consider brand-related users, and is an aid to future evaluation efforts relative to investigations that rely upon self-labeled datasets.' author: - | Liuqing Li Ziqian Song Xuan Zhang Edward A. Fox\ \ Department of Computer Science, Virginia Tech\ Blacksburg, VA 24061, USA\ {liuqing, ziqian, xuancs, fox}@vt.edu bibliography: - 'aaai\_bib.bib' title: 'A Hybrid Model for Role-related User Classification on Twitter' --- Acknowledgements ================ We thank anonymous reviewers for their thorough comments on this manuscript. We thank XXX for funding the XXX project.

--- abstract: 'The main approach of traditional information retrieval (IR) is to examine how many words from a query appear in a document. A drawback of this approach, however, is that it may fail to detect relevant documents where no or only few words from a query are found. The semantic analysis methods such as LSA (latent semantic analysis) and LDA (latent Dirichlet allocation) have been proposed to address the issue, but their performance is not superior compared to common IR approaches. Here we present a query-document similarity measure motivated by the Word Mover’s Distance. Unlike other similarity measures, the proposed method relies on neural word embeddings to compute the distance between words. This process helps identify related words when no direct matches are found between a query and a document. Our method is efficient and straightforward to implement. The experimental results on TREC Genomics data show that our approach outperforms the BM25 ranking function by an average of 12% in mean average precision. Furthermore, for a real-world dataset collected from the PubMed^^ search logs, we combine the semantic measure with BM25 using a learning to rank method, which leads to improved ranking scores by up to 25%. This experiment demonstrates that the proposed approach and BM25 nicely complement each other and together produce superior performance.' author: - | Sun Kim, Nicolas Fiorini, W. John Wilbur and Zhiyong Lu\ National Center for Biotechnology Information\ National Library of Medicine, National Institutes of Health\ Bethesda, MD 20894, USA\ [{sun.kim,nicolas.fiorini,john.wilbur,zhiyong.lu}@nih.gov]{}\ bibliography: - 'arXiv\_2017.bib' title: 'Bridging the Gap: Incorporating a Semantic Similarity Measure for Effectively Mapping PubMed Queries to Documents' --- Introduction ============ In information retrieval (IR), queries and documents are typically represented by term vectors where each term is a content word and weighted by *tf-idf*, i.e. the product of the term frequency and the inverse document frequency, or other weighting schemes [@Salton1988]. The similarity of a query and a document is then determined as a dot product or cosine similarity. Although this works reasonably, the traditional IR scheme often fails to find relevant documents when synonymous or polysemous words are used in a dataset, e.g. a document including only “neoplasm" cannot be found when the word “cancer" is used in a query. One solution of this problem is to use query expansion [@Lu2009; @Carpineto2012; @Diaz2016; @Roy2016] or dictionaries, but these alternatives still depend on the same philosophy, i.e. queries and documents should share exactly the same words. While the term vector model computes similarities in a sparse and high-dimensional space, the semantic analysis methods such as latent semantic analysis (LSA) [@Deerwester1990; @Hofmann1999] and latent Dirichlet allocation (LDA) [@Blei2003] learn dense vector representations in a low-dimensional space. These methods choose a vector embedding for each term and estimate a similarity between terms by taking an inner product of their corresponding embeddings [@Sordoni2014]. Since the similarity is calculated in a latent (semantic) space based on context, the semantic analysis approaches do not require having common words between a query and documents. However, it has been shown that LSA and LDA methods do not produce superior results in various IR tasks [@Maas2011; @Baroni2014; @Pennington2014] and the classic ranking method, BM25 [@Robertson2009], usually outperforms those methods in document ranking [@Atreya2011; @Nalisnick2016]. Neural word embedding [@Bengio2003; @Mikolov2013] is similar to the semantic analysis methods described above. It learns low-dimensional word vectors from text, but while LSA and LDA utilize co-occurrences of words, neural word embedding learns word vectors to predict context words [@Baroni2014]. Moreover, training of semantic vectors is derived from neural networks. Both co-occurrence and neural word embedding approaches have been used for lexical semantic tasks such as semantic relatedness (e.g. king and queen), synonym detection (e.g. cancer and carcinoma) and concept categorization (e.g. banana and pineapple belong to fruits) [@Baroni2014; @Schnabel2015]. But, Baroni et al. showed that neural word embedding approaches generally performed better on such tasks with less effort required for parameter optimization. The neural word embedding models have also gained popularity in recent years due to their high performance in NLP tasks [@Levy2014]. Here we present a query-document similarity measure using a neural word embedding approach. This work is particularly motivated by the Word Mover’s Distance [@Kusner2015]. Unlike the common similarity measure taking query/document centroids of word embeddings, the proposed method evaluates a distance between individual words from a query and a document. Our first experiment was performed on the TREC 2006 and 2007 Genomics benchmark sets [@Hersh2006; @Hersh2007], and the experimental results showed that our approach was better than BM25 ranking. This was solely based on matching queries and documents by the semantic measure and no other feature was used for ranking documents. In general, conventional ranking models (e.g. BM25) rely on a manually designed ranking function and require heuristic optimization for parameters [@Liu2009; @Chapelle2011]. In the age of information explosion, this one-size-fits-all solution is no longer adequate. For instance, it is well known that links to a web page are an important source of information in web document search [@Brin1998], hence using the link information as well as the relevance between a query and a document is crucial for better ranking. In this regard, learning to rank [@Liu2009] has drawn much attention as a scheme to learn how to combine diverse features. Given feature vectors of documents and their relevance levels, a learning to rank approach learns an optimal way of weighting and combining multiple features. We argue that the single scores (or features) produced by BM25 and our proposed semantic measure complement each other, thus merging these two has a synergistic effect. To confirm this, we measured the impact on document ranking by combining BM25 and semantic scores using the learning to rank approach, LamdaMART [@Burges2008; @Burges2010]. Trained on PubMed user queries and their click-through data, we evaluated the search performance based on the most highly ranked 20 documents. As a result, we found that using our semantic measure further improved the performance of BM25. Taken together, we make the following important contributions in this work. First, to the best of our knowledge, this work represents the first investigation of query-document similarity for information retrieval using the recently proposed Word Mover’s Distance. Second, we modify the original Word Mover’s Distance algorithm so that it is computationally less expensive and thus more practical and scalable for real-world search scenarios (e.g. biomedical literature search). Third, we measure the actual impact of neural word embeddings in PubMed by utilizing user queries and relevance information derived from click-through data. Finally, on TREC and PubMed datasets, our proposed method achieves stronger performance than BM25. Methods ======= A common approach to computing similarity between texts (e.g. phrases, sentences or documents) is to take a centroid of word embeddings, and evaluate an inner product or cosine similarity between centroids[^1] [@Nalisnick2016; @Furnas1988]. This has found use in classification and clustering because they seek an overall topic of each document. However, taking a simple centroid is not a good approximator for calculating a distance between a query and a document [@Kusner2015]. This is mostly because queries tend to be short and finding the actual query words in documents is feasible and more accurate than comparing lossy centroids. Consistent with this, our approach here is to measure the distance between individual words, not the average distance between a query and a document. Word Mover’s Distance --------------------- Our work is based on the Word Mover’s Distance between text documents [@Kusner2015], which calculates the minimum cumulative distance that words from a document need to travel to match words from a second document. In this subsection, we outline the original Word Mover’s Distance algorithm, and our adapted model is described in Section 2.2. First, following Kusner et al. , documents are represented by normalized bag-of-words (BOW) vectors, i.e. if a word $w_i$ appears $tf_i$ times in a document, the weight is $$\begin{aligned} d_i = \frac{tf_i}{\sum_{i'=1}^n tf_{i'}},\end{aligned}$$ where $n$ is number of words in the document. The higher the weight, the more important the word. They assume a word embedding so that each word $w_i$ has an associated vector $\mathbf{x}_i$. The dissimilarity $c$ between $w_i$ and $w_j$ is then calculated by $$\begin{aligned} c(i,j)=\|\mathbf{x}_i-\mathbf{x}_j\|_2.\end{aligned}$$ The Word Mover’s Distance makes use of word importance and the relatedness of words as we now describe. Let $\mathbf{D}$ and $\mathbf{D'}$ be BOW representations of two documents $D$ and $D'$. Let $\mathbf{T} \in \mathcal{R}^{n \times n}$ be a flow matrix, where $\mathbf{T}_{ij} \geq 0$ denotes how much it costs to travel from $w_i$ in $D$ to $w_j$ in $D'$, and $n$ is the number of unique words appearing in $D$ and/or $D'$. To entirely transform $\mathbf{D}$ to $\mathbf{D'}$, we ensure that the entire outgoing flow from $w_i$ equals $d_i$ and the incoming flow to $w_j$ equals $d'_j$. The Word Mover’s Distance between $D$ and $D'$ is then defined as the minimum cumulative cost required to move all words from $D$ to $D'$ or vice versa, i.e. $$\begin{aligned} \min_{\mathbf{T} \geq 0} && \sum_{i,j=1}^n \mathbf{T}_{ij} c(i,j) \label{eq:01} \\ \mathrm{subject~to} && \sum_{j=1}^n \mathbf{T}_{ij} = d_i, \forall i \in \{1,...,n\} \nonumber \\ && \sum_{i=1}^n \mathbf{T}_{ij} = d'_j, \forall j \in \{1,...,n\}. \nonumber\end{aligned}$$ The solution is attained by finding $\mathbf{T}_{ij}$ that minimizes the expression in Eq. (\[eq:01\]). Kusner et al. applied this to obtain nearest neighbors for document classification, i.e. *k*-NN classification and it produced outstanding performance among other state-of-the-art approaches. What we have just described is the approach given in Kusner et al. We will modify the word weights and the measure of the relatedness of words to better suit our application. Our Query-Document Similarity Measure ------------------------------------- While the prior work gives a hint that the Word Mover’s Distance is a reasonable choice for evaluating a similarity between documents, it is uncertain how the same measure could be used for searching documents to satisfy a query. First, it is expensive to compute the Word Mover’s Distance. The time complexity of solving the distance problem is $O(n^3 \log n)$ [@Pele2009]. Second, the semantic space of queries is not the same as those of documents. A query consists of a small number of words in general, hence words in a query tend to be more ambiguous because of the restricted context. On the contrary, a text document is longer and more informational. Having this in mind, we realize that ideally two distinctive components could be employed for query-document search: 1) mapping queries to documents using a word embedding model trained on a document set and 2) mapping documents to queries using a word embedding model obtained from a query set. In this work, however, we aim to address the former, and the mapping of documents to queries remains as future work. For our purpose, we will change the word weight $d_i$ to incorporate inverse document frequency ($idf$), i.e. $$\begin{aligned} d_i = idf(i) \frac{tf_i}{\sum_{i'=1}^n tf_{i'}}, \label{eq:03}\end{aligned}$$ where $idf(i) = \log \frac{K-k_i+0.5}{k_i+0.5}$. $K$ is the size of a document set and $k_i$ is the number of documents that include the $i$th term. The rationale behind this is to weight words in such a way that common terms are given less importance. It is the *idf* factor normally used in *tf-idf* and BM25 [@Witten1999; @Wilbur2001]. In addition, our word embedding is a neural word embedding trained on the 25 million PubMed titles and abstracts. Let $\mathbf{Q}$ and $\mathbf{D}$ be BOW representations of a query $Q$ and a document $D$. $D$ and $D'$ in Section 2.1 are now replaced by $Q$ and $D$, respectively. Since we want to have a higher score for documents relevant to $Q$, $c(i,j)$ is redefined as a cosine similarity, i.e. $$\begin{aligned} c(i,j)=\frac{\mathbf{x}_i \cdot \mathbf{x}_j}{\|\mathbf{x}_i\|\|\mathbf{x}_j\|}.\end{aligned}$$ In addition, the problem we try to solve is the flow $Q \rightarrow D$. Hence, Eq. (\[eq:01\]) is rewritten as follows. $$\begin{aligned} \max_{\mathbf{T} \geq 0} && \sum_{i,j=1}^n \mathbf{T}_{ij} c(i,j) \label{eq:02} \\ \mathrm{subject~to} && \sum_{j=1}^n \mathbf{T}_{ij} = d_i, \forall i \in \{1,...,n\}, \nonumber\end{aligned}$$ where $d_i$ represents the word $w_i$ in $Q$. $idf(i)$ in Eq. (\[eq:03\]) is unknown for queries, therefore we compute $idf(i)$ based on the document collection. The optimal solution of the expression in Eq. (\[eq:02\]) is to map each word in $Q$ to the most similar word in $D$ based on word embeddings. The time complexity for getting the optimal solution is $O(mn)$, where $m$ is the number of unique query words and $n$ is the number of unique document words. In general, $m \ll n$ and evaluating the similarity between a query and a document can be implemented in parallel computation. Thus, the document ranking process can be quite efficient. Learning to Rank ---------------- In our study, we use learning to rank to merge two distinctive features, BM25 scores and our semantic measures. This approach is trained and evaluated on real-world PubMed user queries and their responses based on click-through data [@Joachims2002a]. While it is not common to use only two features for learning to rank, this approach is scalable and versatile. Adding more features subsequently should be straightforward and easy to implement. The performance result we obtain demonstrates the semantic measure is useful to rank documents according to users’ interests. We briefly outline learning to rank approaches [@Severyn2015; @Freund2003] in this subsection. For a list of retrieved documents, i.e. for a query $Q$ and a set of candidate documents, $D = \{ D_1, D_2, ..., D_m \}$, we are given their relevancy judgements $y = \{ y_1, y_2, ..., y_m \}$, where $y_i$ is a positive integer when the document $D_i$ is relevant and 0 otherwise. The goal of learning to rank is to build a model $h$ that can rank relevant documents near or at the top of the ranked retrieval list. To accomplish this, it is common to learn a function $h (\textbf{w}, \psi (Q,D))$, where $\textbf{w}$ is a weight vector applied to the feature vector $\psi (Q,D)$. A part of learning involves learning the weight vector but the form of $h$ may also require learning. For example, $h$ may involve learned decision trees as in our application. In particular, we use LambdaMART [@Burges2008; @Burges2010] for our experiments. LambdaMART is a pairwise learning to rank approach and is being used for PubMed relevance search. While the simplest approach (pointwise learning) is to train the function $h$ directly, pairwise approaches seek to train the model to place correct pairs higher than incorrect pairs, i.e. $ h (\textbf{w}, \psi (Q,D_i)) \geq h (\textbf{w}, \psi (Q,D_j)) + \epsilon $, where the document $D_i$ is relevant and $D_j$ is irrelevant. $\epsilon$ indicates a margin. LambdaMART is a boosted tree version of LambdaRank [@Burges2010]. An ensemble of LambdaMART, LambdaRank and logistic regression models won the Yahoo! learning to rank challenge [@Chapelle2011]. Results and Discussion ====================== Our resulting formula from the Word Mover’s Distance seeks to find the closest terms for each query word. Figure \[fig:01\] depicts an example with and without using our semantic matching. For the query, “negative pressure wound therapy", a traditional way of searching documents is to find those documents which include the words “negative", “pressure", “wound" and “therapy". As shown in the figure, the words, “pressure" and “therapy", cannot be found by perfect string match. On the other hand, within the same context, the semantic measure finds the closest words “NPWT" and “therapies" for “pressure" and “therapy", respectively. Identifying abbreviations and singular/plural would help match the same words, but this example is to give a general idea about the semantic matching process. Also note that using dictionaries such as synonyms and abbreviations requires an additional effort for manual annotation. In the following subsections, we describe the datasets and experiments, and discuss our results. Datasets -------- To evaluate our word embedding approach, we used two scientific literature datasets: TREC Genomics data and PubMed. Table \[tab:01\] shows the number of queries and documents in each dataset. TREC represents the benchmark sets created for the TREC 2006 and 2007 Genomics Tracks [@Hersh2006; @Hersh2007]. The original task is to retrieve passages relevant to topics (i.e. queries) from full-text articles, but the same set can be utilized for searching relevant PubMed documents. We consider a PubMed document relevant to a TREC query if and only if the full-text of the document contains a passage judged relevant to that query by the TREC judges. Our setup is more challenging because we only use PubMed abstracts, not full-text articles, to find evidence. -------------------------------------------- -- -- -- **Dataset & **\# Queries & **\# Documents\ TREC 2006 & 26 & 162,259\ TREC 2007 & 36 & 162,259\ PubMed & 27,870 & 27,098,629\ ****** -------------------------------------------- -- -- -- : Number of queries and documents for TREC and PubMed experiments. TREC 2006 includes 28 queries originally but two were removed because there were no relevant documents.\[tab:01\] Machine learning approaches, especially supervised ones such as learning to rank, are promising and popular nowadays. Nonetheless, they usually require a large set of training examples, and such datasets are particularly difficult to find in the biomedical domain. For this reason, we created a gold standard set based on real (anonymized) user queries and the actions users subsequently took, and named this the PubMed set. To build the PubMed set, we collected one year’s worth of search logs and restricted the set of queries to those where users requested the relevance order and which yielded at least 20 retrieved documents. This set contained many popular but duplicate queries. Therefore, we merged queries and summed up user actions for each of them. That is, for each document stored for each query, we counted the number of times it was clicked in the retrieved set (i.e. abstract click) and the number of times users requested full-text articles (i.e. full-text click). We considered the queries that appeared less than 10 times to be less informative because they were usually very specific, and we could not collect enough user actions for training. After this step, we further filtered out non-informational queries (e.g. author and journal names). As the result, 27,870 queries remained for the final set. The last step for producing the PubMed set was to assign relevance scores to documents for each query. We will do this based on user clicks. It is known that click-through data is a useful proxy for relevance judgments [@Joachims2002b; @Agrawal2009; @Xu2010]. Let $a(D,Q)$ be the number of clicks to the abstract of a document $D$ from the results page for the query $Q$. Let $f(D,Q)$ be the number of clicks from $D$’s abstract page to its full-text, which result from the query $Q$. Let $\lambda \in \mathbb{R}^+$ be the boost factor for documents without links to full-text articles. $FT (D)$ is the indicator function such that $FT (D) = 1$ if the document $D$ includes a link to full-text articles and $FT (D) = 0$ otherwise. We can then calculate the relevance, $y$, of a document for a given query: $$\begin{aligned} y(Q,D) = \mu \cdot a(Q,D) + (1-\mu) \cdot f(Q,D) + \nonumber \\ \frac{a(Q,D)}{\lambda} \cdot (1 - FT (D)),\end{aligned}$$ $\mu$ is the trade-off between the importance of abstract clicks and full-text clicks. The last term of the relevance function gives a slight boost to documents without full-text links, so that they get a better relevance (thus rank) than those for which full-text is available but never clicked, assuming they all have the same amount of abstract clicks. We manually tuned the parameters based on user behavior and log analyses, and used the settings, $\mu = 0.33$ and $\lambda = 15$. Compared to the TREC Genomics set, the full PubMed set is much larger, including all 27 million documents in PubMed. While the TREC and PubMed sets share essentially the same type of documents, the tested queries are quite different. The queries in TREC are a question type, e.g. “what is the role of MMS2 in cancer?" However, the PubMed set uses actual queries from PubMed users. In our experiments, the TREC set was used for evaluating BM25 and the semantic measure separately and the PubMed set was used for evaluating the learning to rank approach. We did not use the TREC set for learning to rank due to the small number of queries. Only 62 queries and 162,259 documents are available in TREC, whereas the PubMed set consists of many more queries and documents. Word Embeddings and Other Experimental Setup -------------------------------------------- We used the skip-gram model of *word2vec* [@Mikolov2013] to obtain word embeddings. The alternative models such as GloVe [@Pennington2014] and FastText [@Bojanowski2017] are available, but their performance varies depending on tasks and is comparable to *word2vec* overall [@Muneeb2015; @Cao2017]. *word2vec* was trained on titles and abstracts from over 25 million PubMed documents. Word vector size and window size were set to 100 and 10, respectively. These parameters were optimized to produce high recall for synonyms [@Yeganova2016]. Note that an independent set (i.e. synonyms) was used for tuning *word2vec* parameters, and the trained model is available online (<https://www.ncbi.nlm.nih.gov/IRET/DATASET>). For experiments, we removed stopwords from queries and documents. BM25 was chosen for performance comparison and the parameters were set to $k=1.9$ and $b=1.0$ [@Lin2007]. Among document ranking functions, BM25 shows a competitive performance [@Trotman2014]. It also outperforms co-occurrence based word embedding models [@Atreya2011; @Nalisnick2016]. For learning to rank approaches, 70% of the PubMed set was used for training and the rest for testing. The RankLib library (<https://sourceforge.net/p/lemur/wiki/RankLib>) was used for implementing LambdaMART and the PubMed experiments. TREC Experiments ---------------- Table \[tab:02\] presents the average precision of *tf-idf* (TFIDF), BM25, word vector centroid (CENTROID) and our embedding approach on the TREC dataset. Average precision [@Turpin2006] is the average of the precisions at the ranks where relevant documents appear. Relevance judgements in TREC are based on the pooling method [@Manning2008], i.e. relevance is manually assessed for top ranking documents returned by participating systems. Therefore, we only used the documents that annotators reviewed for our evaluation [@Lu2009]. --------------------------------------- -- -- **Method & **TREC 2006 & **TREC 2007\ TFIDF & 0.3018 & 0.2375\ BM25 & 0.3136 & 0.2463\ CENTROID & 0.2363 & 0.2459\ SEM & 0.3732 & 0.2601\ ****** --------------------------------------- -- -- : Mean average precision of *tf-idf* (TFIDF), BM25, word vector centroid (CENTROID) and our semantic approach (SEM) on the TREC set.\[tab:02\] As shown in Table \[tab:02\], BM25 performs better than TFIDF and CENTROID. CENTROID maps each query and document to a vector by taking a centroid of word embedding vectors, and the cosine similarity between two vectors is used for scoring and ranking documents. As mentioned earlier, this approach is not effective when multiple topics exist in a document. From the table, the embedding approach boosts the average precision of BM25 by 19% and 6% on TREC 2006 and 2007, respectively. However, CENTROID provides scores lower than BM25 and SEM approaches. Although our approach outperforms BM25 on TREC, we do not claim that BM25 and other traditional approaches can be completely replaced with the semantic method. We see the semantic approach as a means to narrow the gap between words in documents and those in queries (or users’ intentions). This leads to the next experiment using our semantic measure as a feature for ranking in learning to rank. PubMed Experiments ------------------ For the PubMed dataset, we used learning to rank to combine BM25 and our semantic measure. An advantage of using learning to rank is its flexibility to add more features and optimize performance by learning their importance. PubMed documents are semi-structured, consisting of title, abstract and many more fields. Since our interest lies in text, we only used titles and abstracts, and applied learning to rank in two different ways: 1) to find semantically closest words in titles (BM25 + SEM~Title~) and 2) to find semantically closest words in abstracts (BM25 + SEM~Abstract~). Although our semantic measure alone produces better ranking scores on the TREC set, this does not apply to user queries in PubMed. It is because user queries are often short, including around three words on average, and the semantic measure cannot differentiate documents when they include all query words. -------------------------------------------------------------------------- -- -- -- **Method & **NDCG@5 & **NDCG@10 & **NDCG@20\ BM25 & 0.0854 & 0.1145 & 0.1495\ BM25 + SEM~Title~ & 0.1048 (22.72%) & 0.1427 (24.59%) & 0.1839 (23.03%)\ BM25 + SEM~Abstract~ & 0.0917 (7.38%) & 0.1232 (7.57%) & 0.1592 (6.51%)\ ******** -------------------------------------------------------------------------- -- -- -- Table \[tab:03\] shows normalized discounted cumulative gain (NDCG) scores for top 5, 10 and 20 ranked documents for each approach. NDCG [@Burges2005] is a measure for ranking quality and it penalizes relevant documents appearing in lower ranks by adding a rank-based discount factor. In the table, reranking documents by learning to rank performs better than BM25 overall, however the larger gain is obtained from using titles (BM25 + SEM~Title~) by increasing NDCG@20 by 23%. NDCG@5 and NDCG@10 also perform better than BM25 by 23% and 25%, respectively. It is not surprising that SEM~Title~ produces better performance than SEM~Abstract~. The current PubMed search interface does not allow users to see abstracts on the results page, hence users click documents mostly based on titles. Nevertheless, it is clear that the abstract-based semantic distance helps achieve better performance. After our experiments for Table \[tab:03\], we also assessed the efficiency of learning to rank (BM25 + SEM~Title~) by measuring query processing speed in PubMed relevance search. Using 100 computing threads, 900 queries are processed per second, and for each query, the average processing time is 100 milliseconds, which is fast enough to be used in the production system. Conclusion ========== We presented a word embedding approach for measuring similarity between a query and a document. Starting from the Word Mover’s Distance, we reinterpreted the model for a query-document search problem. Even with the $Q \rightarrow D$ flow only, the word embedding approach is already efficient and effective. In this setup, the proposed approach cannot distinguish documents when they include all query words, but surprisingly, the word embedding approach shows remarkable performance on the TREC Genomics datasets. Moreover, applied to PubMed user queries and click-through data, our semantic measure allows to further improves BM25 ranking performance. This demonstrates that the semantic measure is an important feature for IR and is closely related to user clicks. While many deep learning solutions have been proposed recently, their slow training and lack of flexibility to adopt various features limit real-world use. However, our approach is more straightforward and can be easily added as a feature in the current PubMed relevance search framework. Proven by our PubMed search results, our semantic measure improves ranking performance without adding much overhead to the system. Acknowledgments {#acknowledgments .unnumbered} =============== This research was supported by the Intramural Research Program of the NIH, National Library of Medicine. [^1]: The implementation of *word2vec* also uses centroids of word vectors for calculating similarities (<https://code.google.com/archive/p/word2vec>).

--- abstract: 'We analyze the properties of a polytropic fluid which is radially accreted into a Schwarzschild black hole. The case where the adiabatic index $\gamma$ lies in the range $1 < \gamma \leq 5/3$ has been treated in previous work. In this article we analyze the complementary range $5/3 < \gamma \leq 2$. To this purpose, the problem is cast into an appropriate Hamiltonian dynamical system whose phase flow is analyzed. While for $1 < \gamma \leq 5/3$ the solutions are always characterized by the presence of a unique critical saddle point, we show that when $5/3 < \gamma \leq 2$, an additional critical point might appear which is a center point. For the parametrization used in this paper we prove that whenever this additional critical point appears, there is a homoclinic orbit.' author: - 'Eliana Chaverra$^{1,2}$, Patryk Mach$^3$, and Olivier Sarbach$^1$' bibliography: - 'refs\_accretion.bib' title: 'Michel accretion of a polytropic fluid with adiabatic index $\gamma > 5/3$: Global flows versus homoclinic orbits' --- Introduction ============ A spherically symmetric accretion model introduced by Bondi in [@hB52] belongs to classical textbook models of theoretical astrophysics. Its general-relativistic version was proposed by Michel [@fM72], who considered a spherically symmetric, purely radial, stationary flow of perfect fluid in the Schwarzschild spacetime. Since then, different variants of this model were studied in numerous works that took into account the self-gravity of the fluid, radiation transfer, electric charge, cosmological constant, non-zero angular momentum, etc. (see for instance [@eM99; @jKbKpMeMzS06; @pMeM08; @jKeMkRzS09; @eMtR10; @vDyE11; @jKeM13; @eBvDyE11; @sGsKaRtD07; @tDbC12; @eTsMjM12; @eTpTjM13; @fLmGfG14]). The standard way of parametrizing Bondi–Michel solutions is to prescribe the asymptotic values of the speed of sound and the density. By adopting such a parametrization one assumes [*eo ipso*]{} that the solution is global, i.e., it extends all the way from the horizon to infinite radii. Local solutions that cannot be extended to infinity were recently discovered in the cosmological context in [@pMeMjK13; @pMeM13; @pM15; @fF15] and, more surprisingly, in the standard Michel accretion (on the Schwarzschild background) of polytropic fluids with adiabatic indices $\gamma > 5/3$ [@Eliana-Master-thesis; @eCoS12; @eCoS15a]. Interestingly, these local solutions correspond to homoclinic orbits appearing in phase diagrams of the radial velocity vs. radius or the density vs. radius. Thus, from the mathematical point of view, the question of the existence of local vs. global solutions translates into the problem of the existence of homoclinic solutions in the appropriate phase diagrams. In [@pM15] this issue was investigated in the case of the Bondi–Michel accretion in Schwarzschild–anti-de Sitter spacetime for polytropic and isothermal (linear) equations of state. It was shown that for polytropic equations of state, with a contribution in the specific enthalpy due to the rest-mass (baryonic) density, only local, homoclinic solutions are allowed. In contrast to that, for isothermal equations of state of the form $p = k e$, where $p$ is the pressure, $e$ is the energy density, and $0 < k \leq 1$ is a constant, global solutions do exist for $k \ge 1/3$, while only local, homoclinic solutions are allowed for $k < 1/3$. The limiting value $k = 1/3$ corresponds exactly to the simple photon gas model. It is thus quite surprising to observe that in the classic case of a Schwarzschild black hole (vanishing cosmological constant) and polytropic fluids, both types of solutions are allowed for $\gamma > 5/3$. Unfortunately, it seems at present that it is difficult to formulate any simple and strict statement on the connection between the type of the equation of state and the existence of global or homoclinic solution. One should also note that the arguments given in [@pM15] are not as mathematically strict as one would expect them to be. For example, no general proof of the existence of homoclinic solutions is given. Their occurrence is rather observed on specific examples. In this paper, dealing with the relatively simple (yet probably the most important) case of accretion onto a Schwarzschild black hole, we are able to provide rigorous results concerning the existence of both homoclinic and global solutions, as well as the number and types of critical points on the appropriate phase diagrams. Although partial results hold for a quite general class of barotropic equations of state, we focus mainly on the polytropic Michel accretion with $5/3 < \gamma \le 2$. A preliminary analysis regarding the structure of the critical points in this model can be found in [@mB78], where it is assumed that the fluid particles are nonrelativistic at infinity. This choice corresponds to $L^2$ slightly larger than one in our notation below, and it excludes the existence of homoclinic orbits. The order of this paper is as follows. The equations governing the Michel accretion are collected in the next section. In Sec. \[sec\_dyn\_system\] we discuss different possibilities for defining a Hamiltonian dynamical system corresponding to the equations of the Michel model. Section \[Sec:Crit\] contains a characterization of the critical points, while in Sec. \[Sec:Global\] we state our main results on the topology of the orbits. Section \[Sec:Conclusions\] contains a brief summary and implications of our work. Michel accretion model ====================== In this section, we briefly review the relevant equations describing the Michel flow. We work in spherical polar coordinates $(t,r,\theta,\phi)$ and assume gravitational units with $c = G = 1$. A Schwarzschild black hole with mass $m$ is described by the metric $$g = -\left( 1 - \frac{2m}{r} \right) dt^2 + \left(1 - \frac{2m}{r} \right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2).$$ The accretion flow is governed by the conservation laws $$\label{aaa7} \nabla_\mu (\rho u^\mu) = 0, \quad \nabla_\mu T^{\mu \nu} = 0,$$ where $\rho$ is the baryonic (rest-mass) density of the gas, $u^\mu$ is its four-velocity, and $$T^{\mu \nu} = \rho h u^\mu u^\nu + p g^{\mu \nu}$$ denotes the energy-momentum tensor of the perfect-fluid. Here $h$ and $p$ are the specific enthaply and the pressure, respectively. In the following we restrict ourselves to barotropic equations of state, for which there is a functional relation $p = p(\rho)$, $h = h(\rho)$.[^1] Michel accretion solutions are obtained assuming that the flow of the gas is purely radial (the only non-vanishing components of the four-velocity are $u^t$ and $u^r$), spherically symmetric and stationary (all functions depend on the areal radius $r$ only). In this case Eqs. (\[aaa7\]) can be integrated yielding $$\label{aaa8} \rho r^2 u^r = \frac{j_n}{4\pi}, \quad h \sqrt{1 - \frac{2m}{r} + (u^r)^2} = L,$$ where $j_n$ and $L > 0$ are constants representing, respectively, the particle flux and the ratio between the energy and particle fluxes through a sphere of constant $r$. Assuming that $\rho > 0$, we have $j_n < 0$ for $u^r < 0$ (this situation corresponds to accretion) and $j_n > 0$ for $u^r > 0$ (the gas is flowing outwards; this situation is sometimes referred to as ‘wind’). In the following sections we work in dimensionless variables. We define $x = r/(2m)$, and $z = \rho/\rho_0$, where $\rho_0$ is a reference rest-mass density. In the main part of this paper we assume a polytropic equation of state of the form $p = K\rho^\gamma$, where $K$ and $\gamma$ are constants. In this case the specific enthalpy (per unit mass) can be expressed as $$h = 1 + \frac{\gamma}{\gamma - 1} K \rho^{\gamma - 1} = 1 + \frac{\gamma}{\gamma - 1} K \rho_0^{\gamma - 1} z^{\gamma - 1}.$$ Setting $\frac{\gamma}{\gamma - 1}K \rho_0^{\gamma - 1} = 1$, we get $$h =: f(z) = 1 + z^{\gamma - 1}.$$ Many of the results discussed in this paper are also valid for more general barotropic equations of state, provided they satisfy the following reasonable assumptions on the function $f: [0,\infty]\to {\mathbb{R}}$: First, we require $f$ to be twice differentiable, positive and monotonously increasing, such that the sound speed $$\nu(z) := \sqrt{ \frac{z}{f(z)}\frac{df}{dz}(z)Ê}$$ is a well-defined differentiable function of $z > 0$. Further, we assume: 1. $0 < \nu(z) < 1$ for all $z > 0$, 2. $f(z)\to 1$ and $\nu(z)\to 0$ as $z\to 0$, 3. $f(z)\to \infty$ as $z\to \infty$. Condition (i) restricts the equations of state to those having subluminal sound speeds. Condition (ii) implies that the internal energy of the gas vanishes in the limit of vanishing density, while (iii) means that it diverges as $z\to\infty$. Note that the polytropic equation of state satisfies all three conditions provided $1 < \gamma\leq 2$. In the following, we will require the satisfaction of conditions (i–iii), and when we specialize to the polytropic case, this will be stated explicitly. In terms of the above dimensionless variables, Eqs. (\[aaa8\]) can be written as $$\begin{aligned} \label{fff1} x^2 z u &=& \mu = \mathrm{const},\\ f(z)^2\left( 1 - \frac{1}{x} + u^2 \right) &=& L^2 = \mathrm{const}, \label{fff2}\end{aligned}$$ where for simplicity we denoted $u := u^r$, and $\mu := j_n/(16\pi\rho_0 m^2)$. The same notation is also used in [@eCoS15a]. Michel accretion as a Hamiltonian dynamical system {#sec_dyn_system} ================================================== For what follows, it is convenient to introduce a (fictitious) dynamical system, whose orbits consist of graphs of solutions of system (\[fff1\]–\[fff2\]). Such a dynamical system can be defined conveniently either in terms of $x$ and $u$ (as in [@pM15]), or in terms of $x$ and $z$. The latter convention was adopted in [@eCoS15a], and we use it here as well. Differentiating Eqs. (\[fff1\]) and (\[fff2\]) with respect to $x$ one obtains $$\frac{dz}{dx} = \frac{\frac{1}{x}\left( 4 u^2 - \frac{1}{x} \right)}{\frac{2}{z} \left[ \nu(z)^2 \left(1 - \frac{1}{x} + u^2 \right) - u^2 \right]},$$ where we have used the relation $df/dz(z) = f(z)\nu(z)^2/z$. Then, choosing a parameter $l$ so that $$\label{fff3} \frac{dx}{dl} = \frac{2f(z)^2}{z} \left[ \nu(z)^2 \left(1 - \frac{1}{x} + u^2 \right) - u^2 \right],$$ one can write $$\label{fff4} \frac{dz}{dl} = \frac{f(z)^2}{x}\left( 4 u^2 - \frac{1}{x} \right).$$ Equations (\[fff3\]) and (\[fff4\]) define a two-dimensional dynamical system, provided that the terms containing $u^2$ can be reexpressed in terms of $x$ and $z$. This can be achieved by using either Eq. (\[fff1\]) or Eq. (\[fff2\]). In the first case one obtains a dynamical system which depends on the parameter $\mu$. Hence, all orbits correspond to the same value of $\mu$, but, as we will see, they have different values of $L$ associated to them. In the second case, one obtains a dynamical system with $L$ being a parameter. Accordingly, all orbits correspond to the same value of $L$, and they are associated with different values of $\mu$. In both cases the resulting dynamical system is Hamiltonian. The first option was used in [@eCoS15a] and corresponds to the choice $$F_\mu (x,z) := \left. f(z)^2 \left( 1 - \frac{1}{x} + u^2 \right) \right|_{u = \frac{\mu}{x^2 z}} = L^2$$ for the Hamiltonian, with associated equations of motion $$\begin{aligned} \label{fff5} \frac{dx}{dl} & = & \frac{\partial F_\mu}{\partial z}(x,z) = \left. \frac{2 f(z)^2}{z} \left[ \nu(z)^2 \left(1 - \frac{1}{x} + u^2 \right) - u^2 \right] \right|_{u = \frac{\mu}{x^2 z}}, \\ \frac{dz}{dl} & = & - \frac{\partial F_\mu}{\partial x}(x,z) = \left. \frac{f(z)^2}{x} \left(4 u^2 - \frac{1}{x} \right) \right|_{u = \frac{\mu}{x^2 z}}, \label{fff6}\end{aligned}$$ where $L$ parametrizes the level sets of the Hamiltonian $F_\mu$, and all the orbits have the same value of $\mu$, as explained above. The second possibility, which we find more convenient for the purpose of the present article, is to assume that the value of $L$ is fixed in the entire phase portrait. Accordingly, we define a new Hamiltonian as $$H_L(x,z) := x^4 z^2\left( \frac{L^2}{f(z)^2} - 1 + \frac{1}{x} \right) = \mu^2. \label{Eq:HLDef}$$ The corresponding dynamical system reads now $$\begin{aligned} \label{fff7} \frac{dx}{d\tilde l} & = & \frac{\partial H_L}{\partial z}(x,z) = 2x^4 z\left[ \frac{L^2}{f(z)^2}(1 - \nu(z)^2) - 1 + \frac{1}{x} \right], \\ \frac{dz}{d\tilde l} & = & - \frac{\partial H_L}{\partial x}(x,z) = - x^3 z^2\left( \frac{4L^2}{f(z)^2} - 4 + \frac{3}{x} \right). \label{fff8}\end{aligned}$$ In this description, it is $\mu$ that parametrizes the level sets of the Hamiltonian, while all the orbits have the same value of $L$. The above system is equivalent to the one given by Eqs. (\[fff3\]) and (\[fff4\]) with the substitution $$u^2 = \frac{L^2}{f(z)^2} - 1 + \frac{1}{x},$$ up to the ‘time’ reparametrization of the orbits (a replacement of $l$ by a new parameter $\tilde l$). The general relation between two functionally related Hamiltonians and the corresponding dynamical system is discussed in the Appendix. Note that in the second formulation the value of the Hamiltonian is directly related to the mass accretion rate $\mu$. As we will show below, the value of the parameter $L$ plays a decisive role in the nature of the transonic flow solutions. In order to illustrate this fact, suppose there exists a solution $z(x)$ of $H_L(x,z(x)) = \mu^2$ which extends all the way to the asymptotic region $x\to \infty$ and has $z_\infty = \lim\limits_{x\to \infty} z(x) > 0$. Then, it follows from Eq. (\[Eq:HLDef\]) that $f(z_\infty) = L$. Since $f\geq 1$, this means that a necessary condition for a globally-defined solution to exist is that $L\geq 1$. It should be stressed that each orbit (or a solution of the accretion problem) can be equivalently represented on phase portraits of systems (\[fff5\]–\[fff6\]) or (\[fff7\]–\[fff8\]). On the other hand, the phase portraits of systems (\[fff5\]–\[fff6\]) and (\[fff7\]–\[fff8\]) are composed of different families of orbits; thus in some cases they can look differently. In the next two sections we work with system (\[fff7\]–\[fff8\]), which allows for a more convenient characterization of both critical points and the types of the orbits. For clarity, we decided to omit an analogous discussion concerning system (\[fff5\]–\[fff6\]), except for a few remarks that we make at the end of Sec. \[Sec:Global\]. Critical points {#Sec:Crit} =============== In this section we analyze the critical points of the Hamiltonian system (\[fff7\]–\[fff8\]), which are determined by the zeros of the gradient of $H_L$. Since $$\begin{aligned} \frac{\partial H_L}{\partial x}(x,z) &=& x^3 z^2\left( \frac{4L^2}{f(z)^2} - 4 + \frac{3}{x} \right), \label{Eq:HLx}\\ \frac{\partial H_L}{\partial z}(x,z) &=& 2x^4 z\left[ \frac{L^2}{f(z)^2}(1 - \nu(z)^2) - 1 + \frac{1}{x} \right], \label{Eq:HLz}\end{aligned}$$ we find the conditions $$x = \frac{3}{4} + \frac{1}{4\nu(z)^2},\qquad \frac{f(z)^2}{1 + 3\nu(z)^2} = L^2 \label{Eq:xzcrit}$$ for a critical point. Since $0\leq\nu(z) < 1$, it follows that any critical point $(x,z)$ must lie outside the event horizon. The number of critical points depends on the behavior of the function ${\cal L}: [0,\infty) \to {\mathbb{R}}$ defined by $${\cal L}(z) := \frac{f(z)^2}{1 + 3\nu(z)^2},\qquad z\geq 0. \label{Eq:LDef}$$ Note that our assumptions on $f(z)$ and $\nu(z)$ imply that ${\cal L}(0) = 1$ and $\lim\limits_{z\to\infty} {\cal L}(z) = \infty$, and thus there always exists at least one critical point when $L^2 > 1$. If ${\cal L}$ is strictly monotonous, there is precisely one critical point for $L^2 > 1$ and none for $L^2\leq 1$. The following lemma describes the number of critical points as a function of $L$ for the particular case of the polytropic equation of state. ![\[Fig:LBehavior\] Typical qualitative behavior of the function ${\cal L}(z)$, determining the critical points of the system. In this case, there are two critical points for $L_{min}^2 < L^2 < 1$ and one for $L^2 > 1$.](MixedEqnState2.pdf){width="45.00000%"} \[Lem:Crit\] Consider a polytropic equation of state, for which $$f(z) = 1 + z^{\gamma-1},\qquad \nu(z) = \sqrt{(\gamma-1)\frac{z^{\gamma-1}}{1 + z^{\gamma-1}}},\qquad z \geq 0,$$ with adiabatic index $1 < \gamma \leq 2$. Let $$L_{min} := \frac{3}{2}\frac{\gamma-1}{\left(\gamma - \frac{2}{3} \right)^{3/2}}.$$ Then, system (\[Eq:HLDef\]) has a unique critical point for each $L^2 > 1$. When $1 < \gamma\leq 5/3$, there are no critical points for $L^2 < 1$, while for $5/3 < \gamma\leq 2$ there are no critical points for $L^2 < L_{min}^2$ and precisely two critical points for each $L_{min}^2 < L^2 < 1$. [[**Proof.** ]{}]{}Setting $y := \nu(z)^2$, one can rewrite the condition ${\cal L}(z) = L^2$ as $$G(y) := \frac{1}{1 + 3y}\frac{1}{\left( 1 - \frac{y}{\gamma-1} \right)^2} = L^2. \label{Eq:GDef}$$ Note that $y = \nu(z)^2$ lies in the interval $[0,\gamma-1)$. The function $G: [0,\gamma-1)\to{\mathbb{R}}$ defined above satisfies $G(0) = 1$ and $\lim\limits_{y\to \gamma-1} G(y) = \infty$, and it has derivative $$\frac{dG}{dy}(y) = \frac{1}{(1 + 3y)^2}\frac{1}{\left( 1 - \frac{y}{\gamma-1} \right)^3} \frac{9y - (3\gamma - 5)}{\gamma-1}.$$ When $1 < \gamma \leq 5/3$ it follows that $G$ is strictly monotonically increasing, and hence there exists a unique critical point for each $L^2 > 1$. In contrast to this, when $5/3 < \gamma\leq 2$, $G$ has a global minimum at $y = y_m := (3\gamma-5)/9$, where $G(y_m) = L_{min}^2 < 1$, and the Lemma follows. [$\fbox{\hspace{0.3mm}}$ ]{} [**Remarks**]{}: 1. $L_{min}^2$ decreases monotonically from $1$ to $3^5/4^4 = 243/256$, as $\gamma$ increases from $5/3$ to $2$.\ 2. It is easy to verify that when $5/3 < \gamma \leq 2$, the function $G(y)$ crosses the value $L^2 = 1$ at $y = 0$ and $$y = y_* := \gamma - 1 - \frac{1}{6}\left[ 1 + \sqrt{12\gamma-11} \right].$$ Therefore, when $L_{min}^2 < L^2 < 1$, the two critical points $(x_1,z_1)$ and $(x_2,z_2)$ satisfy $$0 < y_1 < y_m,\qquad y_m < y_2 < y_*,$$ and when $L^2 > 1$ the unique critical point $(x_c,z_c)$ lies in the interval $y_* < y_c < y_0 := \gamma-1$. Since $x = 3/4 + 1/(4y)$ the corresponding intervals for the location of the critical point are $$x_1 > x_m,\qquad x_m > x_2 > x_*,$$ and $x_* > x_c > x_0$, where $$x_0 := \frac{1}{4}\frac{3\gamma-2}{\gamma-1},\qquad x_* := \frac{3}{4} + \frac{1}{4y_*},\qquad x_m := \frac{1}{4}\frac{3\gamma-2}{\gamma - 5/3}.$$ It follows from the behavior of the function $G(y)$ that as $L^2$ increases from $L_{min}^2$ to $1$, $y_1$ decreases from $y_m$ to $0$ while $y_2$ increases from $y_m$ to $y_*$. Correspondingly, $x_1$ increases from $x_m$ to $\infty$, while $x_2$ decreases from $x_m$ to $x_*$. Furthermore, as $L^2$ increases from $1$ to $\infty$, $y_c$ increases from $y_*$ to $\gamma - 1$, and $x_c$ decreases from $x_*$ to $x_0$. 3. Explicit expressions for the values of $y_1$ and $y_2$ can be given using Cardano’s formulae. For $5/3 < \gamma < 2$ and $L_{min}^2 < L^2 < 1$, one has $$\begin{aligned} y_1 & = & \frac{1}{9} \left\{ 6 \gamma - 7 - 2(3 \gamma - 2) \cos \left[ \frac{\pi}{3} - \frac{1}{3} \mathrm{arccos} \left( \frac{2 L_{min}^2}{L^2} - 1 \right) \right] \right\}, \\ y_2 & = & \frac{1}{9} \left\{ 6 \gamma - 7 - 2(3 \gamma - 2) \cos \left[ \frac{\pi}{3} + \frac{1}{3} \mathrm{arccos} \left( \frac{2 L_{min}^2}{L^2} - 1 \right) \right] \right\}. \label{cardanoy2}\end{aligned}$$ For $L^2 > 1$, the unique critical point corresponds to the square of the speed of sound given by Eq. (\[cardanoy2\]) (formula for $y_2$). Next, we analyze whether the critical points are saddle points or extrema of the Hamiltonian $H_L$. For this, we evaluate the Hessian of $H_L$ at a critical point $(x_c,z_c)$: $$D^2 H_L(x_c,z_c) = \left. \left( \begin{array}{cc} \frac{\partial^2 H_L}{\partial x^2} & \frac{\partial^2 H_L}{\partial x\partial z} \\ \frac{\partial^2 H_L}{\partial z\partial x} & \frac{\partial^2 H_L}{\partial z^2} \end{array} \right) \right|_{(x_c,z_c)} = -x_c^3\left( \begin{array}{cc} 3\frac{z_c^2}{x_c^2} & 2\frac{z_c}{x_c} \\ 2\frac{z_c}{x_c} & 1 - \nu_c^2 + w_c \end{array} \right),$$ where we have defined $$\nu_c := \nu(z_c),\qquad w_c := \frac{\partial\log\nu}{\partial\log z}(z_c).$$ The determinant of the Hessian is $$\det\left[ D^2 H_L(x_c,z_c) \right] = -x_c^4 z_c^2(1 + 3\nu_c^2 - 3w_c),$$ and hence the critical point is a saddle point, if $1 + 3\nu_c^2 - 3w_c > 0$, and a local extremum, if $1 + 3\nu_c^2 - 3w_c < 0$. As it turns out, the sign of the determinant is related to the sign of the slope of the function ${\cal L}$ defined in Eq. (\[Eq:LDef\]) at the critical point. Indeed, a short calculation reveals that $$\frac{d}{dz}{\cal L}(z_c) = \frac{L^2}{2 x_c z_c}(1 + 3\nu_c^2 - 3w_c) = -\frac{L^2}{2x_c^5 z_c^3}\det\left[ D^2 H_L(x_c,z_c) \right].$$ Therefore, the critical point is a saddle whenever the function ${\cal L}(z)$ crosses the value $L^2$ from below, and an extremum whenever the function ${\cal L}(z)$ crosses $L^2$ from above. The next lemma describes the situation for a polytropic fluid and is a direct consequence of Lemma \[Lem:Crit\] and the above remarks. \[Lem:Crit2\] For the particular case of the polytropic equation of state with adiabatic index $1 < \gamma\leq 2$, we have the following: 1. For $L^2 > 1$, the unique critical point is a saddle point lying in the interval $x_c = x_{saddle}\in (x_0,\infty)$. 2. When $5/3 < \gamma \leq 2$ and $L_{min}^2 < L^2 < 1$ there is one saddle critical point lying in the interval $x_2 = x_{saddle}\in (x_*,x_m)$ and one center critical point lying in the interval $x_1 = x_{center}\in (x_m,\infty)$. As $L^2$ increases from $L_{min}^2$ to $1$, $x_1 = x_{center}$ increases from $x_m$ to $\infty$, and $x_2 = x_{saddle}$ decreases from $x_m$ to $x_*$. [**Remark**]{}: In fact, the statement of the lemma holds true as long as the function ${\cal L}(z)$ defined in Eq. (\[Eq:LDef\]) has a qualitative behavior similar to the one shown in Fig. \[Fig:LBehavior\]. For later use we will also need the slopes of the two level curves of $H_L$ crossing a critical saddle point. For this, let $(x(l),z(l))$ be such a level curve. Differentiating the relation $$H_L(x(l),z(l)) = \mu^2$$ twice with respect to the parameter $l$ and evaluating the result at a saddle point $(x_c,z_c)$ we obtain the quadratic equation $$A + 2B \frac{dz}{dx}(x_c) + C \left[ \frac{dz}{dx}(x_c) \right]^2 = 0$$ for the slope $dz/dx(x_c)$ of the curve at $x = x_c$, where $$A := \frac{\partial^2 H_L}{\partial x^2}(x_c,z_c) = -3x_c z_c^2,\quad B := \frac{\partial^2 H_L}{\partial x\partial z}(x_c,z_c) = -2x_c^2 z_c,\quad C := \frac{\partial^2 H_L}{\partial z^2}(x_c,z_c) = -x_c^3(1 - \nu_c^2 + w_c). \label{Eq:DefABC}$$ Therefore, for $C < 0$, the slope of the two level curves $\Gamma_\pm(x_c,z_c)$ through $(x_c,z_c)$ is given by $$\frac{dz_{\pm}}{dx}(x_c) = \frac{-B \mp \sqrt{B^2 - AC}}{C} = \frac{z_c}{x_c}\frac{-2\pm \sqrt{1 + 3\nu_c^2 - 3w_c}}{1 - \nu_c^2 + w_c}. \label{Eq:GammaPMSlope}$$ Global extensions {#Sec:Global} ================= The main goal of this section is to analyze the global behavior of the phase flow defined by the Hamiltonian system (\[Eq:HLDef\]). We are particularly interested in the global structure of the local one-dimensional stable ($\Gamma_-$) and unstable ($\Gamma_+$) manifolds associated with the saddle critical point. In [@eCoS15a] two of us treated the case of a perfect fluid accreted by a static black hole satisfying certain assumptions. In particular, the results in [@eCoS15a] cover the case of a polytropic fluid with adiabatic index $1 < \gamma \leq 5/3$ accreted by a Schwarzschild black hole, and it was proven that in this case the unstable manifold $\Gamma_+$ associated with the saddle critical point extends all the way from the event horizon to the asymptotic region $x\to\infty$, where the particle density converges to a finite, positive value. In this section, we discuss the complementary case, that is, the case of a polytropic fluid with adiabatic index $5/3 < \gamma\leq 2$. Our main result shows that when $L^2 > 1$ (the case where there exists a unique critical saddle point) the extension of $\Gamma_-$ has exactly the same qualitative properties as in the case $1 < \gamma\leq 5/3$ and describes a global accretion flow extending from the horizon to the asymptotic region. In contrast to this, when $L_{min}^2 < L^2 < 1$, we prove that $\Gamma_-$ and $\Gamma_+$ connect to each other and form a homoclinic orbit. We base our global analysis on the two curves $c_1$, $c_2$ in phase space $\Omega := \{ (x,z) : x > 0, z > 0 \}$, corresponding to those points $(x,z)$ with vanishing partial derivative of $H_L$ with respect to $x$ and $z$, respectively.[^2] These curves can be parametrized as follows (cf. Eqs. (\[Eq:HLx\],\[Eq:HLz\])): $$\begin{aligned} c_1 &:& x_1(z) = \frac{3}{4}\left[ 1 - \frac{L^2}{f(z)^2} \right]^{-1},\qquad f(z) > L, \label{Eq:Gamma1Def}\\ c_2 &:& x_2(z) = \left[ 1 - \frac{L^2}{f(z)^2}(1 - \nu(z)^2) \right]^{-1},\qquad f(z) > L\sqrt{1 - \nu(z)^2}. \label{Eq:Gamma2Def}\end{aligned}$$ By definition, the curves $c_1$ and $c_2$ intersect each other precisely at the critical points of the system, and further they divide the phase space into different regions, the components of the Hamiltonian vector field associated with $H_L$ having a fixed sign in each of these regions. Before we focus our attention on the polytropic case, let us make a few general remarks about the qualitative behavior of the curves $c_1$ and $c_2$. First, a short calculation reveals that $$\frac{dx_1}{dz}(z) = -\frac{8L^2\nu(z)^2}{3zf(z)^2} x_1(z)^2 < 0, \label{Eq:dx1dz}$$ and $$\frac{dx_2}{dz}(z) = -\frac{2L^2\nu(z)^2}{z f(z)^2} x_2(z)^2\left[ 1 - \nu(z)^2 + w(z) \right],$$ which is negative as long as $1-\nu^2 + w > 0$.[^3] Consequently, both functions $x_1(z)$ and $x_2(z)$ are monotonically decreasing in $z$. Furthermore, as $z\to \infty$, our general conditions (i) and (iii) imply that $$\lim\limits_{z\to\infty} x_1(z) = \frac{3}{4} < \lim\limits_{z\to\infty} x_2(z) = 1.$$ When $z\to 0$, we have $f(z)\to 1$ and $\nu(z)\to 0$ by assumption (ii), and thus the conditions $f(z) > L$ and $f(z) > L\sqrt{1 - \nu(z)^2}$ are only satisfied for all $z\geq 0$, if $L^2 < 1$. In this case $$\lim\limits_{z\to 0} x_1(z) = \frac{3}{4}\frac{1}{1 - L^2} < \lim\limits_{z\to 0} x_2(z) = \frac{1}{1 - L^2}.$$ However, when $L > 1$, there are minimal values $\tilde{z}_1 > 0$ and $\tilde{z}_2 > 0$ such that $f(\tilde{z}_1) = L$ and $f(z) > L$ for all $z > \tilde{z}_1$, while $f(\tilde{z}_2) = L\sqrt{1 - \nu(\tilde{z}_2)^2}$ and $f(z) > L\sqrt{1 - \nu(z)^2}$ for all $z > \tilde{z}_2$. In this case, both $x_1(z)$ and $x_2(z)$ tend to infinity as $z$ decreases to $\tilde{z}_1$ and $\tilde{z}_2$, respectively. Finally, let us compare the slopes of the curves $c_{1,2}$ to those of the level curves $\Gamma_\pm$ of $H_L$ at a critical saddle point. Using the fact that along $c_1$ we have $$\frac{\partial H_L}{\partial x}(x_1(z), z) = 0,$$ differentiation with respect to $z$ and evaluation at the critical point yields $$A\frac{dx_1}{dz}(z_c) + B = 0,$$ where the coefficients $A,B$ are defined in Eq. (\[Eq:DefABC\]). Consequently, we find that $$\left( \frac{dz}{dx} \right)_1(x_c) = -\frac{A}{B},$$ where the subindex $1$ means that the differentiation is taken along the curve $c_1$. Similarly, the slope of $c_2$ at the critical point is given by $$\left( \frac{dz}{dx} \right)_2(x_c) = -\frac{B}{C}.$$ Comparing these expressions to the ones given in Eq. (\[Eq:GammaPMSlope\]) we find, since $A,B,C < 0$ and $B^2 - A C > 0$, $$\frac{dz_-}{dx}(x_c) < \left( \frac{dz}{dx} \right)_2(x_c) < \left( \frac{dz}{dx} \right)_1(x_c) < \frac{dz_+}{dx}(x_c) < 0.$$ In the following, we focus on the polytropic case and analyze the phase space for the two cases $L > 1$ and $L_{min} < L < 1$ separately. Case I: $L > 1$ --------------- According to Lemma \[Lem:Crit2\] there is a unique saddle point in this case, and according to the remarks above the qualitative behavior of the curves $c_{1,2}$ and $\Gamma_\pm$ are as shown in Fig. \[Fig:FaseCaseI\]. ![\[Fig:FaseCaseI\] A sketch of the phase space, showing the saddle critical point $(x_c,z_c)$ with the associated stable ($\Gamma_-$) and unstable ($\Gamma_+$) manifolds, and the special curves $c_1$ and $c_2$ defined in Eqs. (\[Eq:Gamma1Def\]) and (\[Eq:Gamma2Def\]). Also shown is the direction of the flow in each of the regions and along the curves $c_1$ and $c_2$. The flow’s radial velocity measured by static observers is subsonic in the region above the curve $c_2$ and supersonic in the region below $c_2$.](Esfase.pdf){width="60.00000%"} Following the arguments described in Sec. IIIC of [@eCoS15a], one can show that the unstable manifold $\Gamma_+$ extends to the horizon $x = 1$ on one side of $x_c$, and that it must extend to $x\to \infty$ above the curve $c_1$ on the other side of the critical saddle point. Furthermore, the stable manifold $\Gamma_-$ asymptotes to the horizon $x=1$ with $z\to \infty$ on one side of $x_c$, while it must extend to $x\to \infty$ below the curve $c_2$ on the other side of $x_c$ (see Fig. \[Fig:FaseCaseI\]). Case II: $L_{min} < L < 1$ and $5/3 < \gamma \leq 2$ ---------------------------------------------------- In this case, it follows from Lemmas \[Lem:Crit\] and \[Lem:Crit2\] that there are two critical points, one saddle and one center (see Fig. \[Fig:FaseCaseII\] for a sketch of the phase diagram). ![\[Fig:FaseCaseII\] A sketch of the phase space, showing the saddle critical point $(x_1,z_1)$ with the associated stable ($\Gamma_-$) and unstable ($\Gamma_+$) manifolds, the center critical point $(x_2,z_2)$, and the curves $c_1$ and $c_2$ defined in Eqs. (\[Eq:Gamma1Def\]) and (\[Eq:Gamma2Def\]). Also shown is the direction of the flow in each of the regions and along the curves $c_1$ and $c_2$. As proven in the text, in this case $\Gamma_+$ and $\Gamma_-$ connect to each other and form a homoclinic orbit.](EsfaseTwoCrit.pdf){width="60.00000%"} The extension of the stable and unstable manifolds $\Gamma_\pm$ towards the horizon is qualitatively the same as in the previous case. Furthermore, $\Gamma_+$ extends to $x = x_2$ above the curve $c_1$ and likewise $\Gamma_-$ extends to $x = x_2$ below the curve $c_2$. However, as we show now, the behavior of the extensions of $\Gamma_\pm$ beyond $x = x_2$ changes radically when compared to the previous case. First, we notice that $H_L(x,z) = \mu^2 > 0$ implies that $$\frac{1}{x} \geq 1 - \frac{L^2}{f(z)^2} \geq 1 - L^2,$$ which is positive when $L^2 < 1$. Therefore, when $L^2 < 1$, $x$ cannot extend to infinity along the level curves $\Gamma_\pm$ and is bounded by $1/(1-L^2)$. Since along $\Gamma_+$ the coordinate $z$ decreases and $x$ increases as one moves away from the critical saddle point $x_1$, and since $x$ is bounded, it follows that $\Gamma_+$ must intersect the curve $c_2$ at some point $x_{max} > x_2$ (see Fig. \[Fig:FaseCaseII\]). Extending further $\Gamma_+$ in the region between $c_2$ and $c_1$ we see that in fact $\Gamma_+$ has to intersect the curve $c_1$ as well. Let us denote this point by $P_+$. Using a similar argument, it follows that $\Gamma_-$ must intersect the curve $c_1$ at some point $P_-$, say. We now claim that $P_+ = P_-$, implying that the two curves $\Gamma_\pm$ connect to each other and form a homoclinic orbit. For this, we use the following lemma. Consider the function $J(z) := H_L(x_1(z),z)$, $f(z) > L$, which represents the value of the Hamiltonian $H_L$ along the curve $c_1$. Then, $J$ is strictly monotonously increasing on the intervals $(0,z_2)$ and $(z_1,\infty)$ and strictly monotonously decreasing on the interval $(z_1,z_2)$. [[**Proof.** ]{}]{}Differentiating both sides of $J(z) = H_L(x_1(z),z)$ with respect to $z$ we find $$\frac{dJ}{dz}(z) = \frac{\partial H_L}{\partial x}(x_1(z),z)\frac{dx_1}{dz}(z) + \frac{\partial H_L}{\partial z}(x_1(z),z).$$ The first term on the right-hand side is zero because by definition the partial derivative of $H_L$ with respect to $x$ is zero on $c_1$. For the second term, we use Eqs. (\[Eq:HLz\],\[Eq:Gamma1Def\]) and find $$\frac{dJ}{dz}(z) = \frac{2}{3} z x_1(z)^4\left[ 1 - \frac{L^2}{{\cal L}(z)} \right],$$ with ${\cal L}(z)$ the function defined in Eq. (\[Eq:LDef\]). As a consequence of the proof of Lemma \[Lem:Crit\], we have ${\cal L}(z) > L^2$ for $0 < z < z_1$ or $z_2 < z < \infty$ and ${\cal L}(z) < L^2$ for $z_1 < z < z_2$, and the lemma follows. [$\fbox{\hspace{0.3mm}}$ ]{} As a consequence of the previous lemma, the value of $H_L$ must increase as one moves along the curve $c_1$ away from $x = x_2$ with increasing $x$. Since the value of $H_L$ is the same constant along $\Gamma_\pm$, it follows that $P_+ = P_-$ and we have a homoclinic orbit. Note that the results of this section do not make explicit use of the polytropic equation of state. In fact, they remain valid for any equation of state for which the function ${\cal L}(z)$ defined in Eq. (\[Eq:LDef\]) has the same qualitative behavior as the one shown in Fig. \[Fig:LBehavior\] and for which the quantity $1 - \nu(z)^2 + w(z)$ is positive. A comment on the properties of system (\[fff5\]–\[fff6\]) --------------------------------------------------------- A similar analysis to that presented in the preceding sections can be also carried out for system (\[fff5\]–\[fff6\]) (defined by the Hamiltonian $F_\mu$), but it is difficult to formulate a simple characterization of the behavior of its orbits that would depend on the parameters $\gamma$ and $\mu$ only. The asymptotic behavior of a given orbit depends mainly on the value of $L$ associated with it, and hence a much more clear characterization can be given for system (\[fff7\]–\[fff8\]), defined by the Hamiltonian $H_L$, where $L$ is a parameter. In addition, contrary to the situation described in the preceding sections (for the system with the Hamiltonian $H_L$), system (\[fff5\]–\[fff6\]) admits phase portraits with two critical points (a saddle and a center-type point) and no homoclinic orbit, as well as phase portraits with a homoclinic orbit present. The former case appears when the value of $F_\mu = L^2$ at the saddle-type critical point is less than one. Conclusions {#Sec:Conclusions} =========== The analysis presented in this paper was motivated by a recent discovery (in [@Eliana-Master-thesis; @eCoS12; @eCoS15a]) of homoclinic solutions appearing in the Michel model of relativistic steady accretion. They can be found assuming a Schwarzschild background geometry and a standard polytropic equation of state with adiabatic index $5/3 < \gamma \leq 2$. The assumption of a polytropic equation of state leads to the equations of the model that are just simple enough to allow for a precise characterization of the critical points and general properties of solutions, although (with the exception of the case $\gamma = 2$, for which Eqs. (\[fff1\]) and (\[fff2\]) lead to a fourth–order polynomial equation for $z$) they are still sufficiently complex not to allow to be solved exactly. The key concept of the analysis of this paper is the correspondence between the equations describing the accretion flow and a (fictitious) dynamical system, that allows to use the terminology (and methods) originating in the theory of dynamical systems. This dynamical system can be defined in many ways (see the Appendix), and the qualitative behavior of the phase portraits can depend on the parametrization that one uses. Here we chose a version for which the complete characterization of the critical points and the types of the orbits seems to be the simplest. In this formulation there is a homoclinic orbit, whenever two critical points (a saddle and a center-type one) are present in the phase diagram and a unique global solution, which extends from the horizon to the asymptotic region whenever there is a unique critical point. The physical implication of the fact that a graph of a given solution belongs to a homoclinic orbit is that this particular solution is local—it cannot be extended to arbitrarily large radii. Consequently, such solutions are incompatible with the asymptotic boundary conditions that are imposed on the values of a solution at infinity. Homoclinic accretion solutions were discovered also for models with a negative cosmological constant and adiabatic indices $\gamma < 5/3$, as well as for isothermal equations of state with the square of the speed of sound less than $1/3$ [@pMeMjK13; @pMeM13; @pM15]. An account for the case including charged black holes can be found in [@fF15]. The analysis presented in this paper seems to be by far the most complete, but of course it provides only a partial answer to the question of the subtle interplay between the assumed equation of state, the form of the metric, and the existence of homoclinic solutions. It is however worth noticing that the analysis of Sec. \[Sec:Global\] only weakly depends on the equation of state. PM acknowledges discussions with Piotr Bizoń, Edward Malec and Zdzis[ł]{}aw Golda. This research was supported in part by CONACyT Grants No. 238758, by the Polish Ministry of Science and Higher Education grant IP2012 000172, by the NCN grant DEC-2012/06/A/ST2/00397 and by a CIC Grant to Universidad Michoacana. In this appendix we show that two Hamiltonians related functionally as in Sec. \[sec\_dyn\_system\] define the same orbits. Consider two differentiable functions $H_1$ and $H_2$ of two variables $x$ and $z$. Suppose we have the following relation between $x$, $z$, $H_1(x,z)$, and $H_2(x,z)$: $$\label{app0} G(x, z, H_1(x,z), H_2(x,z)) = 0,$$ with a function $G$ which is partially differentiable in all its arguments and which satisfies $$\frac{\partial G}{\partial H_1} \neq 0, \qquad \frac{\partial G}{\partial H_2}\neq 0.$$ For example, for the two Hamiltonians introduced Sec. \[sec\_dyn\_system\], we have $H_1(x,z) = F_\mu(x,z)$ and $H_2(x,z) = H_L(x,z)$, and they are functionally related by $$G(x,z,H_1,H_2) := x^4 z^2 H_1 - f(z)^2\left[ x^4\left( 1 - \frac{1}{x} \right)z^2 + H_2 \right] = 0.$$ Differentiating Eq. (\[app0\]) with respect to $x$ and $z$ we see that $$\label{app1} \frac{\partial G}{\partial x} + \frac{\partial G}{\partial H_1} \frac{\partial H_1}{\partial x} + \frac{\partial G}{\partial H_2} \frac{\partial H_2}{\partial x} = 0, \quad \frac{\partial G}{\partial z} + \frac{\partial G}{\partial H_1} \frac{\partial H_1}{\partial z} + \frac{\partial G}{\partial H_2} \frac{\partial H_2}{\partial z} = 0.$$ The two dynamical systems that we consider in Sec. \[sec\_dyn\_system\] are defined by two Hamiltonians, denoted here by $H_1$ and $H_2$, with the following property: For the system defined by the Hamiltonian $H_1$, the function $H_2(x,z) = \mu^2$ is a constant, and it is treated as a parameter. Conversely, for the system with the Hamiltonian given by $H_2$, the function $H_1(x,z) = L^2$ is a constant parameter. Consider now an orbit in the system with the Hamiltonian $H_1(x,z)$ and $H_2 = \mathrm{const}$. It is given as a solution of the system of equations $$\frac{dx}{dl} = \frac{\partial H_1(x,z)}{\partial z} = - \frac{\frac{\partial G(x,z,H_1(x,z),H_2)}{\partial z}}{\frac{\partial G(x,z,H_1(x,z),H_2)}{\partial H_1}}, \quad \frac{dz}{dl} = -\frac{\partial H_1(x,z)}{\partial x} = \frac{\frac{\partial G(x,z,H_1(x,z),H_2)}{\partial x}}{\frac{\partial G(x,z,H_1(x,z),H_2)}{\partial H_1}},$$ where we have used Eq. (\[app1\]). Conversely, for a system defined by the Hamiltonian $H_2(x,z)$ with $H_1 = \mathrm{const}$ one obtains $$\frac{dx}{d \tilde l} = \frac{\partial H_2(x,z)}{\partial z} = - \frac{\frac{\partial G(x,z,H_1,H_2(x,z))}{\partial z}}{\frac{\partial G(x,z,H_1,H_2(x,z))}{\partial H_2}}, \quad \frac{dz}{d \tilde l} = -\frac{\partial H_2(x,z)}{\partial x} = \frac{\frac{\partial G(x,z,H_1,H_2(x,z))}{\partial x}}{\frac{\partial G(x,z,H_1,H_2(x,z))}{\partial H_2}}.$$ Because for an autonomous system, the Hamiltonian is a constant of motion, we easily see that for a given orbit (described in both systems) one has $$\frac{dx}{dl} = \frac{\frac{\partial G (x,z,H_1,H_2)}{\partial H_2}}{\frac{\partial G (x,z,H_1,H_2)}{\partial H_1}} \frac{d x}{d \tilde l}, \quad \frac{dz}{dl} = \frac{\frac{\partial G (x,z,H_1,H_2)}{\partial H_2}}{\frac{\partial G (x,z,H_1,H_2)}{\partial H_1}} \frac{d z}{d \tilde l}.$$ This means that the solutions in both Hamiltonian systems are equivalent, up to a reparametrization of the ‘time’ parameter $l$ or $\tilde l$, as claimed. The above result does not mean, however, that the phase portraits would look the same in both cases. The reason behind this fact is that, in general, they are composed of different orbits. Systems (\[fff5\]–\[fff6\]) and (\[fff7\]–\[fff8\]) considered in this paper provide a good illustration of this last fact. While for system (\[fff7\]–\[fff8\]) we have already proved that the existence of a center-type critical point implies the existence of a homoclinic orbit, one can have a phase portrait of system (\[fff5\]–\[fff6\]) with a center-type critical point and no homoclinic orbit. [^1]: As shown in [@eCoS15a], for Michel accretion this condition follows automatically from the assumptions that the accretion flow is smooth, spherically symmetric and steady-state. [^2]: Note that the curve $c_2$ has the following physical interpretation: it divides $\Omega$ into a region (above $c_2$) where the flow’s radial velocity, $$v = -\sqrt{1 - \frac{f(z)^2}{L^2}\left(1 - \frac{1}{x} \right)},$$ as measured by static observers is subsonic and a region (below $c_2$) where this radial velocity is supersonic. [^3]: Note that since $\nu^2 < 1$, this condition is automatically satisfied, if $w\geq 0$. In particular, it is satisfied in the polytropic case as long as $1 < \gamma \leq 2$.

--- abstract: 'The isotope ratio, $^{85}$Rb/$^{87}$Rb, places constraints on models of the nucleosynthesis of heavy elements, but there is no precise determination of the ratio for material beyond the Solar System. We report the first measurement of the interstellar Rb isotope ratio. Our measurement of the Rb [I]{} line at 7800 Å  for the diffuse gas toward $\rho$ Oph A yields a value of $1.21 \pm 0.30$ (1-$\sigma$) that differs significantly from the meteoritic value of 2.59. The Rb/K elemental abundance ratio for the cloud also is lower than that seen in meteorites. Comparison of the $^{85}$Rb/K and $^{87}$Rb/K ratios with meteoritic values indicates that the interstellar $^{85}$Rb abundance in this direction is lower than the Solar System abundance. We attribute the lower abundance to a reduced contribution from the $r$-process. Interstellar abundances for Kr, Cd, and Sn are consistent with much less $r$-process synthesis for the solar neighborhood compared to the amount inferred for the Solar System.' author: - 'S.R. Federman$^,$, David C. Knauth, and David L. Lambert' title: The Interstellar Rubidium Isotope Ratio toward Rho Ophiuchi A --- Introduction ============ Studies on rubidium in stellar atmospheres and the interstellar medium (ISM) are few, but it is not for a lack of interest in its abundance. The production of Rb involves the neutron capture $s$- and $r$-processes. Rubidium production by the $s$-process occurs through the ‘weak’ process in the He- and C-burning layers of massive stars and through the ‘main’ process in the He-shell of low-mass AGB stars. Supernovae from core collapse of massive stars are the likely source of the $r$-process. Analysis of the Solar System abundances provides estimates of the fractional responsibility of these neutron capture processes for the Rb isotopes $^{85}$Rb and $^{87}$Rb. Our estimates drawn from averaging the results of Beer, Walter, & Käppeler (1992), Raiteri et al. (1993), and Arlandini et al. (1999) are that $^{85}$Rb is about 35% $s$- and 65% $r$-process in origin with the weak $s$-process being one third of the $s$-process contribution. For $^{87}$Rb, the fractions are about 70% from the $s$- and 30% from the $r$-process with a minor contribution from the weak $s$-process. As a result of the different relative contributions of the $s$- and $r$-processes to the two isotopes, a measurement of the Rb isotope ratio is a clue to the history of heavy element nucleosynthesis. Unfortunately, rubidium is a difficult element to measure. It is potentially measureable only in cool stars. Resonance lines of Rb [I]{} are detectable at 7800 and 7947 Å, but blending with atomic and/or molecular lines adds an unwelcome difficulty to an abundance analysis. Rubidium abundances for unevolved stars were reported by Gratton & Sneden (1994) and Tomkin & Lambert (1989). Measurements of the abundance for giant stars were obtained by Lambert et al. (1995), Abia & Wallerstein (1998), and Abia et al. (2001). The primary aim of these latter studies was to exploit the use of Rb as a neutron densitometer for the $s$-process site (Lambert et al. 1995). This use arises from the role of $^{85}$Kr as a branch in the $s$-process path. The branch directly affects the isotope ratio, $^{85}$Rb/$^{87}$Rb, but it is not directly measureable from even very high resolution stellar spectra (Lambert & Luck 1976) because the lines are too broad. Earlier interstellar measurements yielded upper limits (Federman et al. 1985) because the Rb abundance is quite small ($2.5 \times 10^{-10}$ in meteorites; Anders & Grevesse 1989), and it is derived from Rb [I]{}, not the dominant form Rb [II]{}. The only interstellar detection so far comes from observations on two heavily reddened stars in Cygnus (Gredel, Black, & Yan 2001). \[An earlier reported detection by Jura & Smith (1981) could not be confirmed by Federman et al. (1985).\] Here, we present high-resolution spectra revealing not only another interstellar detection, but also the first measurement determining the $^{85}$Rb/$^{87}$Rb isotope ratio for extrasolar gas. Observations and Analysis ========================= The star $\rho$ Oph A is an ideal target for a study on Rb isotopes. It is relatively bright ($V$ $=$ 5.0), moderately reddened with $E$($B-V$) $=$ 0.47, and has one main interstellar component (e.g., Lemoine et al. 1993). The weak lines of Li [I]{} (Lemoine et al. 1993) and K [I]{} $\lambda$4044 (Crutcher 1978) toward this star also show more absorption than is typical for a diffuse cloud. The data on Rb [I]{} $\lambda$7800 were acquired with the 2dcoudé spectrograph (Tull et al. 1995) on the Harlan J. Smith 2.7 m telescope at McDonald Observatory in 2003 May. We used the high-resolution mode with echelle grating E2, centered on 7444 Å in order 46. The spectra were imaged onto a $2048 \times 2048$ Tektronics CCD (TK3). The 145 $\mu$m slit provided a resolving power of 175,000 as determined from widths of lines in the Th-Ar comparison spectrum, sufficient to discern broadening of the interstellar lines. We obtained calibration exposures for dark current the first night of the run, bias correction and flat fielding each night, and Th-Ar spectra every 2 to 3 hours during the night. Our primary target, $\rho$ Oph A, was observed for a total of 8$^h$, with individual exposures of 30$^m$ per frame; this procedure minimized the deleterious effects caused by cosmic rays. In addition, the unreddened star $\alpha$ Vir was observed for a total of 100$^m$ centered on the slit to check for contamination from weak telluric lines and CCD blemishes not removed during the flat-fielding process. No contamination is present. An additional 60$^m$ was utilized to trail $\alpha$ Vir along the slit to act as a stellar flat field. The analysis described below utilized spectra of the K [I]{} line at 4044 Å that were acquired for another project (Knauth, Federman, & Lambert 2004). While details will be presented in Knauth et al. (2004), we note that the instrumentation and observing procedure were basically the same as those for Rb [I]{}, except echelle grating E1 was used. There is at most a 0.1 km s$^{-1}$ difference in line velocities between the two setups according to the dispersion solutions found from Th-Ar spectra. Standard routines within the IRAF environment were used to extract one-dimensional spectra that were Doppler-corrected and normalized to unity. The Rb [I]{} spectrum for the interstellar gas toward $\rho$ Oph A is displayed in Fig. 1. The appearance of two ‘features’ arises from the 62 mÅ  hyperfine splitting in $^{85}$Rb combined with the stronger (redder) hyperfine component in $^{87}$Rb. The rms deviations in the stellar continuum yield a signal-to-noise ratio per pixel of 1200 for $\rho$ Oph A; there are 2.9 pixels per resolution element. As in our earlier work on the Li isotope ratio acquired at the same resolution (Knauth et al. 2000), the Rb [I]{} and K [I]{} lines were fitted to extract column densities for neutral $^{85}$Rb, $^{87}$Rb, and K. The relevant atomic data used for input (Morton 1991, 2000) and the resulting equivalent widths ($W_{\lambda}$) and column densities ($N$) are given in Table 1. There is one main component at $V_{LSR}$ of 1.9 km s$^{-1}$ (Lemoine et al. 1993; Pan et al. 2003), which at ultra-high-resolution is seen as two in K [I]{} $\lambda$7699 spectra (Welty & Hobbs 2001). Another component at 3.5 km s$^{-1}$ appears in K [I]{} $\lambda$7699 (Welty & Hobbs 2001; Pan et al. 2003), having $\approx$ 20% of the column of the main component. This second component is marginally seen (at the 2-$\sigma$ level) in our K [I]{} $\lambda$4044 spectrum, but there is no evidence for it in the Rb [I]{} line. The uncertainties in column densities were inferred from a map of chi-squared confidence levels (Fig. 2). While optical depth effects are not important for these very weak lines, the profile fitting code (see Knauth et al. 2003) determines the $b$-value as well (from the difference between measured line width and the instrumental width determined from Th-Ar lines). We obtained $b$-values of 1.0 and 0.8 km s$^{-1}$ for Rb [I]{} and K [I]{}; the slight difference is not significant at our spectral resolution. Results ======= The primary result of our study is the determination of a Rb isotope ratio for the main cloud toward $\rho$ Oph A. The best fit (with a reduced chi squared of 1.43 for 47 degrees of freedom) yields a $^{85}$Rb/$^{87}$Rb ratio of $1.21 \pm 0.30$ – see top panel of Fig. 1. This differs significantly from the meteoritic value of 2.59 (Anders & Grevesse 1989). We attempted to fit our data with the Solar System ratio as well (bottom panel of Fig. 1). The reduced chi squared is worse (2.08) and an F-test shows that there is 50% confidence in this fit. Examination of the lower panel reveals where the difference lies: Neither the blue nor red shoulders of the lines are fit well using a Solar System ratio. Furthermore, the goodness of fit can be judged by variations in the residuals (data minus fit) outside and inside the absorption profile. The variation in the lower panel is not as consistent across the spectrum. Other syntheses with the Solar System ratio were attempted without success. Because both shoulders show more absorption than is predicted by this ratio, addition of the redder 3.5 km s$^{-1}$ component would not improve the fit. Broader lines could fill in the shoulders of the observed Rb [I]{} profile, as could a two-component fit upon shifting the profile by about 1 resolution element. \[A $b$-value as large as 1.8 km s$^{-1}$ is possible from our $\chi^2$ analysis, though it is not consistent with other data (e.g., Welty & Hobbs 2001).\] However, the $^{85}$Rb hyperfine components are broadened as well, yielding a chi squared that is almost double the $\chi^2$ of the best fit in both cases. Finally, we note that the lines representing different values of $^{85}$Rb/$^{87}$Rb in Fig. 2 favor a ratio between 1.0 and 1.5. The Solar System ratio of 2.6 differs by 2- to 3-$\sigma$ in each quantity (column of Rb isotope), consistent with the 4.5-$\sigma$ difference seen in the uncertainty listed for the $^{85}$Rb/$^{87}$Rb ratio. The Rb/K elemental ratio provides a means to determine for this interstellar cloud whether the $^{85}$Rb abundance is lower or the $^{87}$Rb abundance is higher relative to the abundances in meteorites. Since Rb [I]{} and K [I]{} are minor interstellar species, ionization balance is required to extract an elemental abundance. The elemental ratio does not depend on electron density, and we assume that the depletion levels for alkalis are the same (see Welty & Hobbs 2001; Knauth et al. 2003). Then for the Rb/K ratio, we have $$\frac{A_g({\rm Rb})}{A_g({\rm K})} = \left[\frac{N({\rm Rb~{\small I}})}{N({\rm K~{\small I}})}\right] \left[\frac{G({\rm Rb~{\small I}})}{G({\rm K~{\small I}})}\right] \left[\frac{\alpha({\rm K~{\small I}})}{\alpha({\rm Rb~{\small I}})}\right],$$ where $G$(X) is the photoionization rate corrected for grain attentuation and $\alpha$(X) is the rate coefficient for radiative recombination. Comparison of the theoretical calculations for radiative recombination of Rb and K (Wane 1985; Péquignot & Aldrovandi 1986; Wane & Aymar 1987) reveals that the coefficients are the same to within about 10%, confirming the assumption made in other interstellar studies (Jura & Smith 1981; Federman et al. 1985; Gredel et al. 2001). The photoionization rates for Rb [I]{} and K [I]{} are $3.42 \times 10^{-12}$ and $8.67 \times 10^{-12}$ s$^{-1}$, respectively. The cross sections for Rb [I]{} are from the theoretical calculations of Weisheit (1972) for 1150 to 1250 Å, scaled to the measurements of Marr & Creek (1968) at longer wavelengths, while those for K [I]{} are from the measurements of Hudson & Carter (1965, 1967), Marr & Creek (1968), and Sandner et al. (1981). Since the ionization potentials for Rb [I]{} and K [I]{} are similar, 4.18 and 4.34 eV, there is no appreciable differential attenuation. We use this fact to account for Rb [I]{} absorption below 1150 Å; based on the total rate for K [I]{}, we applied a 13% ‘correction’ to $G$(Rb [I]{}). The result is a Rb/K ratio of $(1.3 \pm 0.3) \times 10^{-3}$ compared to the Solar System ratio of $(1.9 \pm 0.2) \times 10^{-3}$ (Anders & Grevesse 1989). Limits or values of the Rb/K ratio are available from earlier studies of interstellar Rb as well. For the clouds toward $o$ Per, $\zeta$ Per, and $\zeta$ Oph, the most conservative limits on Rb [I]{} absorption (Federman et al. 1985) are used with our more recent determinations of the K [I]{} column density from the weak line at 4044 Å (Knauth et al. 2000, 2003). The 3-$\sigma$ upper limits for $N$(Rb [I]{}) are $\le 2.8 \times 10^9$, $\le 1.9 \times 10^9$, and $\le 3.7 \times 10^9$ cm$^{-2}$, respectively. The limits on the Rb/K ratio then become $\le 1.2 \times 10^{-3}$, $\le 1.0 \times 10^{-3}$, and $\le 1.8 \times 10^{-3}$. The Rb [I]{} detections toward Cyg OB2 Nos. 5 and 12 (Gredel et al. 2001) can be combined with results from high-resolution K [I]{} $\lambda$7699 spectra (McCall et al. 2002). For the three main molecular components ($+4.0$, $+6.5$, and $+12.3$ km s$^{-1}$), we derive values for $N$(K [I]{}) of $9.4(7.7) \times 10^{11}$, $8.1(12) \times 10^{11}$, and $6.9(16) \times 10^{11}$ cm$^{-2}$ for the gas toward No. 5 (No. 12) when adopting a $b$-value of 1 km s$^{-1}$. The respective Rb/K ratios are $1.4 \times 10^{-3}$ and $1.2 \times 10^{-3}$. (The uncertainties for these ratios are hard to quantify because the Rb [I]{} component structure is not known and because the K [I]{} column densities from $\lambda$7699 are very susceptible to small changes in adopted $b$-value.) These comparisons suggest that the interstellar Rb/K ratio may be lower than the meteoritic abundance throughout the solar neighborhood. Discussion ========== Insight into the nucleosynthesis of Rb is obtainable from the ratios $^{85}$Rb/K and $^{87}$Rb/K for the main 1.9 km s$^{-1}$ component toward $\rho$ Oph A. We find $^{85}$Rb/K = $(0.73 \pm 0.12) \times 10^{-3}$ and $^{87}$Rb/K = $(0.60 \pm 0.13) \times 10^{-3}$, assuming that $G$(Rb [I]{}) and $\alpha$(Rb [I]{}) are not dependent on isotope. The Solar System ratios are $^{85}$Rb/K = $(1.37 \pm 0.14) \times 10^{-3}$ and $^{87}$Rb/K = $(0.53 \pm 0.06) \times 10^{-3}$. These ratios show that $^{85}$Rb is underabundant in the gas by a factor of about two, but $^{87}$Rb has about the Solar System abundance. In other words, the $^{85}$Rb/$^{87}$Rb and Rb/K ratios toward $\rho$ Oph A are low because there is less $^{85}$Rb. The ‘missing’ $^{85}$Rb is comparable to predictions for the $r$-process component in stellar models (Beer et al. 1992; Raiteri et al. 1993), $\sim$ 65%. Moreover, the models (Beer et al. 1992; Raiteri et al. 1993; Arlandini et al. 1999) indicate that $^{87}$Rb arises mainly from the $s$-process. We are led to believe that the lower interstellar abundance for $^{85}$Rb is due to less $r$-process synthesis. It is widely assumed that the site of the $r$-process is the very deep interior of a Type II supernova. Potassium is also a product of a SN II; it is synthesized in explosive oxygen burning (e.g., Clayton 2003). These different sites and quite different synthesis mechanisms make it likely that the $r$-process yield of a Rb isotope relative to the yield of K can vary; it may depend for example on the mass, metallicity, and rotation of a SN II’s progenitor. The proposal that local interstellar gas may be deficient in $r$-process products relative to Solar System material is curiously supported by interstellar abundances of other lightly depleted or undepleted elements with a $r$-process contribution. We emphasize directions probing material toward Sco OB2, which includes $\rho$ Oph A. For Solar System material, about 50% of Kr, 50% of Cd, and 30% of Sn are attributable to the $r$-process (Beer et al. 1992; Raiteri et al. 1993; Arlandini et al. 1999). The interstellar Kr abundance within a few hundred parsecs of the Sun is constant at 60% of the solar-wind value (Cartledge et al. 2001, 2003). Though there are fewer determinations of the interstellar Cd abundance, it is constant at 80% of the Solar System value (Sofia, Meyer, & Cardelli 1999). Sofia et al. find a Sn abundance that is essentially solar, if not slightly greater than solar, for sight lines showing little depletion in other elements. Arsenic, which is mostly $r$-process (Beer et al. 1992; Raiteri et al. 1993) and has only 1 isotope, is seen currently only in three reddened directions (Cardelli et al. 1993; Federman et al. 2003). It has a higher condensation temperature (see Cardelli et al. 1993) and thus it has more depletion, especially in reddened directions. Measurements of As [II]{} in lightly reddened sight lines, where little depletion is expected, appear necessary to confirm our hypothesis. Variations in space and time of the mix of $s$- and $r$-process products in the Galaxy’s interstellar medium should be reflected in the abundances of the heavy elements in unevolved stars. Relative abundances of elements such as Sr, Y, and Zr and heavier elements such as Ba, Ce, and Eu for dwarfs of the thin disk do not vary from star to star by more than the measurement errors (Edvardsson et al. 1993; Chen et al. 2000; Reddy et al. 2003). These measurements would seem to exclude the larger variations expected for stars formed from gas with a large amplitude in the ratio of $s$- to $r$-process products. Our interstellar results suggest that differences in $n$-capture nucleosynthesis may be local to the solar neighborhood. Concluding Remarks ================== Our estimate of the Rb isotope and the Rb/K ratios combined with the interstellar abundances of Kr, Cd, and Sn raise the intriguing possibility that the $r$-process products for such light $n$-capture elements are relatively underrepresented in the local interstellar medium relative to their contribution in the 4.5 Gyr older Solar System material. While models of the $s$-process are becoming quite sophisticated and generally give similar results, differences limit the ability to make definitive statements. For these rarer elements, the Solar System abundance is sometimes a matter of debate; Kr is an example. The precision of the atomic data is another factor. For Sn [II]{}, the $f$-values adopted by Sofia et al. (1999) are now known to be valid (Schectman et al. 2000; Alonso-Medina, Colón, & Matińez 2003). A key test of our hypothesis is the measurement of accurate interstellar abundances for additional elements, particularly those derived mainly from the $r$-process. We thank Dr. B. McCall, who kindly provided us with his K [I]{} spectra, and Dr. D. York for helpful suggestions. This work was supported in part by NASA LTSA grant NAG5-4957 to the Univ. of Toledo. Abia, C., Busso, M., Gallino, R., Dominguez, I., Straniero, O., & Isern, J. 2001, , 559, 1117 Abia, C., & Wallerstein, G. 1998, MNRAS, 293, 89 Alonso-Medina, A., Colón, C., & Martińez, C.H. 2003, , 595, 550 Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197 Arlandini, C., Käppeler, F., Wisshak, K., Gallino, R., Lugaro, M., Busso, M., & Straniero, O. 1999, , 525, 886 Beer, H., Walter, G., & Käppeler, F. 1992, , 389, 784 Cardelli, J.A., Federman, S.R., Lambert, D.L., & Theodosiou, C.E. 1993, , 416, L41 Cartledge, S.I.B., Meyer, D.M., & Lauroesch, J.T. 2003, , 597, 408 Cartledge, S.I.B., Meyer, D.M., Lauroesch, J.T., & Sofia, U.J. 2001, , 562, 394 Chen, Y.Q., Nissen, P.E., Zhao, G., Zhang, H.W., & Benoni, T. 2000, A&AS, 141, 491 Clayton, D. 2003, Handbook of Isotopes in the Cosmos – Hydrogen to Gallium (Cambridge: Cambridge Univ. Press) Crutcher, R.M. 1978, , 219, 72 Edvardsson, B., Andersen, J., Gustafsson, B., Lambert, D.L., Nissen, P.E., & Tomkin, J. 1993, A&A 275, 101 Federman, S.R., Lambert, D.L., Sheffer, Y., Cardelli, J.A., Andersson, B.-G., van Dishoeck, E.F., & Zsargó, J. 2003, , 591, 986 Federman, S.R., Sneden, C., Schempp, W.V., & Smith, W.H. 1985, , 290, L55 Gratton, R.G., & Sneden, C. 1994, A&A, 287, 927 Gredel, R., Black, J.H., & Yan, M. 2001, A&A, 375, 553 Hudson, R.D., & Carter, V.L. 1965, Phys. Rev., 139, A1426 ------------------------------------------------------------------------ . 1967, J. Opt. Soc. Am., 57, 1471 Jura, M., & Smith, W.H. 1981, , 251, L43 Käppeler, F., Beer, H., & Wisshak, K. 1989, Rep. Prog. Phys., 52, 945 Knauth, D.C., Federman, S.R., & Lambert, D.L. 2003, , 586, 268; erratum, 594, 664 Knauth, D.C., Federman, S.R., & Lambert, D.L. 2004, in preparation Knauth, D.C., Federman, S.R., Lambert, D.L., & Crane, P. 2000, Nature, 405, 656 Lambert, D.L., & Luck, R.E. 1976, Observatory, 96, 100 Lambert, D.L. Smith, V.V., Busso, M., Gallino, R., & Straniero, O. 1995, , 450, 302 Lemoine, M., Ferlet, R., Vidal-Madjar, A., Emerich, C., & Bertin, P. 1993, A&A, 269, 469 Marr, G.V., & Creek, D.M. 1968, Proc. R. Soc. London A, 304, 233 McCall, B.J., et al. 2002, , 567, 391 Morton, D.C. 1991, ApJS, 77, 119 ------------------------------------------------------------------------ . 2000, ApJS, 130, 403 Pan, K., Federman, S.R., Cunha, K., Smith, V.V., & Welty, D.E. 2003, submitted to Péquignot, D., & Aldrovandi, S.M.V. 1986, A&A, 161, 169 Raiteri, C.M., Gallino, R., Busso, M., Neuberger, D., Käppeler, F. 1993, , 419, 207 Reddy, B.E., Tomkin, J., Lambert, D.L., & Allende Prieto, C. 2003, MNRAS, 340, 304 Sandner, W., Gallagher, T.F., Safinya, K.A., & Gounand, F. 1981, Phys. Rev. A, 23, 2732 Schectman, R.M., Cheng, S., Curtis, L.J., Federman, S.R., Fritts, M.C., & Irving, R.C. 2000, , 542, 400 Sofia, U.J., Meyer, D.M., & Cardelli, J.A. 1999, , 522, L137 Tomkin, J., & Lambert, D.L. 1999, , 523, 234 Tull, R.G., MacQueen, P.J., Sneden, C., & Lambert, D.L. 1995, PASP, 107, 251 Wane, S. 1985, J. Phys. B, 18, 3881 Wane, S., & Aymar, M. 1987, J. Phys. B, 20, 2657 Weisheit, J.C. 1972, Phys. Rev. A, 5, 1621 Welty, D.E., & Hobbs, L.M. 2001, ApJS, 133, 245 [lccccc]{}\ Species & $\lambda$ & $F_l \rightarrow F_u$ & $f$-value & $W_{\lambda}$ & $N$\ & (Å) & & & (mÅ) & (cm$^{-2}$)\ $^{85}$Rb [I]{} & 7800.232 $^a$ & $2 \rightarrow 4,3,2,1$ $^a$ & $2.90 \times 10^{-1}$ $^a$ & $0.40 \pm 0.04(\rm obs) \pm 0.07 (\rm sys)$ & $(2.56 \pm 0.38) \times 10^9$\ & 7800.294 $^a$ & $3 \rightarrow 4,3,2,1$ $^a$ & $4.06 \times 10^{-1}$ $^a$ & $0.56 \pm 0.04(\rm obs) \pm 0.07 (\rm sys)$ &\ $^{87}$Rb [I]{} & 7800.183 $^a$ & $1 \rightarrow 3,2,1$ $^a$ & $2.61 \times 10^{-1}$ $^a$ & $0.30 \pm 0.04(\rm obs) \pm 0.07 (\rm sys)$ & $(2.12 \pm 0.43) \times 10^9$\ & 7800.321 $^a$ & $2 \rightarrow 3,2,1$ $^a$ & $4.35 \times 10^{-1}$ $^a$ & $0.40 \pm 0.04(\rm obs) \pm 0.07 (\rm sys)$ &\ K [I]{} & 4044.143 $^b$ & $^c$ & $6.09 \times 10^{-3}$ $^b$ & $1.19 \pm 0.09(\rm obs)$ $^d$ & $(1.38 \times 0.11) \times 10^{12}$ $^d$\

--- abstract: 'Greyscale image colorization for applications in image restoration has seen significant improvements in recent years. Many of these techniques that use learning-based methods struggle to effectively colorize sparse inputs. With the consistent growth of the anime industry, the ability to colorize sparse input such as line art can reduce significant cost and redundant work for production studios by eliminating the in-between frame colorization process. Simply using existing methods yields inconsistent colors between related frames resulting in a flicker effect in the final video. In order to successfully automate key areas of large-scale anime production, the colorization of line arts must be temporally consistent between frames. This paper proposes a method to colorize line art frames in an adversarial setting, to create temporally coherent video of large anime by improving existing image to image translation methods. We show that by adding an extra condition to the generator and discriminator, we can effectively create temporally consistent video sequences from anime line arts.' author: - title: Automatic Temporally Coherent Video Colorization --- Introduction ============ In recent years the popularity of anime, Japanese books and cartoons, has risen significantly. The Japanese animation market has recorded major growth for seven consecutive years crossing 18 billion dollars in revenue with the largest growth occurring in movies and internet distribution in 2017 [@Anime:Industry]. The production of anime is a multistaged process that requires time and effort from multiple teams of artists. The drawing of key frames that define major character movements are done by lead artists while in-between frames that fill in these motions are animated by inexperienced artists. The colorization of line art sketches from key frames and in-bewteen frames is considered to be tedious, repetitive, and low pay work. Thus, finding an automatic pipeline for consistently producing thousands of colored frames per episode from line art frames can save significant expenses for anime production studios while simultaneously expediting the animation process. Several approaches exist that address the problem of image colorization. Non learning-based methods often use reference images to match luminance or color histograms [@Colorizationnondeep; @Colorizationnondeep2] while learning-based methods use Convolutional Neural Networks (CNNs) and treat the problem as a classification task at the pixel level. These techniques take greyscale images as input and do not consider sparse inputs such as anime line art drawings. In addition, when used to form stitched-together video from individual frames, the video lacks consistency between frames. Creating a method to colorize frames to coherent video sequences is crucial for large-scale anime production. Methods that target anime line art colorization rely heavily on learned priors, or color hints to generate a single colored image [@Anime:User-guided; @Anime:Style-Transfer]. These methods do not scale to colorization of frame sequences and instead target the colorization of generalized single line art drawings to cater to a broad audience of artists and enthusiasts. ![(Left) Ground truth colored image. (Center) Computed synthetic line art. (Right) Generated colorized frame from line art. []{data-label="fig:my_label"}](fig2.jpg){width="\textwidth"} The image to image translation method presented by Isola [*et al*. ]{}[@Pix2pix] uses Generative Adversarial Networks (GANs) conditioned on some input to learn a mapping from input to target image by minimizing a loss function. The mapping learned from the proposed technique works with some sparse inputs and has been used in applications such as converting segmentation labels to street view images. We aim to extend this model to work with synthetic line arts. Our proposed model takes temporal information into account to encourage consistency between colorized frames in video form. Our source code is available at: <https://github.com/Harry-Thasarathan/TCVC> Related Works ============= Image to image translation using conditional GANs [@goodfellow2014generative; @odena2018generator] is especially effective in comparison to CNN-based models for colorization tasks [@Nazeri:CGAN]. This model successfully maps a high dimensional input to a high dimensional output using a U-Net [@U-Net] based generator and patch-based discriminator [@Pix2pix]. The closer the input image is to the target, the better the learned mapping is. As a result, this technique is particularly suited to colorization tasks. The U-Net architecture acts as an encoder decoder to produce images conditioned on some input. The limitation with U-Net is the information bottleneck that results from downsampling and then upsampling an input image. Skip connections copy mirrored layers in the encoder to the decoder, but downsampling to 2x2 can lose information. This is especially relevant when considering the sparse nature of line art in comparison to greyscale images. Downsampling input data that is already sparse to that extent should be avoided in the context of anime line art colorization due to the risk of data loss. The neural algorithm for artistic style presented by Gatys [*et al*. ]{}[@Style:Gatys] provides a method for the creation of artistic imagery. This is highly relevant as it demonstrates a way to learn representations of both content and style between two images using the pretrained VGG network [@simonyan2014very], then transfer that learned representation with iterative updates on a target image. Johnson [*et al*. ]{}[@Style:Johnson] showed that this model is optimal for transferring a learned representation of style from a painting which includes encoding texture information to an input photo. Although this method alone is not effective in transferring color to inputs like line art for our specific task, the ability to learn and differentiate style and content using a pretrained network is highly useful [@gondal2018unreasonable]. There are very few existing methods for anime line art colorization. Existing methods are made to be highly generalizable for use by different artists [@Anime:Paintschainer]. As a result, they rely on user provided color hints to colorize individual line art drawings that are not related. For large-scale anime production, providing color hints for 3000 frames of an episode should be avoided as it will be monotonous work requiring significant user intervention. Additionally, learning a representation for a show based on previous already colored episodes will be more useful for building a method to create temporally coherent frames for future episodes and seasons. Method ====== Our method attempts to colorize line art frames for large-scale anime production settings by taking into consideration temporal information to account for consistency between frames. We also try to minimize the amount of user intervention required to create colorized frames to mitigate the monotonous workload otherwise required by artists. Loss Objective -------------- Let $G$ and $D$ represent the generator and discriminator of our colorization network. Our generator takes a greyscale or line art image $\mathbf{I}_{line}$ as input conditioned on the previous color frame $\mathbf{F}_{prev}$ and returns a color prediction temporally consistent to the previously colored frame $$\mathbf{F}_{pred} = G(\mathbf{I}_{line},\mathbf{F}_{prev}).$$ A joint loss is used to train the network that takes advantage of both conditional GAN and neural style algorithm that consists of an adversarial loss, style loss, content loss, and $l_{1}$ loss. The adversarial loss is defined in equation (\[eq:g\_adv\]) and we include $\mathbf{F}_{prev}$ as an additional condition to encourage temporal consistency. $$\begin{gathered} \mathcal{L}_{adv} =\mathbb{E}_{(\mathbf{I}_{line},\mathbf{F}_{prev})} \left[ \log D(\mathbf{I}_{line},\mathbf{F}_{prev}) \right] \\ + \mathbb{E}_{\mathbf{F}_{prev}} \log \left[ 1 - D(\mathbf{F}_{pred}, \mathbf{F}_{prev}) \right]. \label{eq:g_adv}\end{gathered}$$ We incorporate content and style loss described in [@Style:Gatys; @Style:Johnson] to further supplement the training of our colorization network. Content loss encourages perceptual similarity while style loss encourages texture similarities between predicted and ground truth color frames. Perceptual similarity is accomplished by minimizing the Manhattan distance between feature maps generated by intermediate layers of VGG-19. Content loss is defined by equation (\[eq:l\_content\]) where $\phi_{i}$ represents the activation map at a given layer $i$ of VGG-19, $\mathbf{F}_{gt}$ is the current ground truth color frame, and $\mathbf{F}_{pred}$ is the generated frame. $N_i$ represents the number of elements in the $i^{th}$ activation layer of VGG-19. For our work, we use activation maps from layers $\tt{relu1\_1, ~relu2\_1, ~relu3\_1, ~relu4\_1}$ and $\tt{relu5\_1}$ $$\mathcal{L}_{content} = \mathbb{E}_i \left[\frac{1}{N_i} \left\lVert \phi_i (\mathbf{F}_{gt}) - \phi_i (\mathbf{F}_{pred}) \right \rVert_1 \right]. \label{eq:l_content}$$ Style loss is calculated in a similar manner but rather than calculating the Manhattan distance between feature maps, we calculate the distance between the gram matrices of the feature maps $$\mathcal{L}_{style} = \mathbb{E}_j \left[ \lVert G_j^{\phi} ({\mathbf{F}}_{pred}) - G_j^{\phi} ({\mathbf{F}}_{gt}) \rVert_1 \right]. \label{eq:l_style}$$ The Gram matrix of feature map $\phi$ is defined by $G^{\phi}_{j}$ in equation (\[eq:l\_style\]) which distributes spatial information containing non localized information such as texture, shape, and style. We also add an $l_{1}$ term to our overall loss objective to preserve structure and encourage the generator to produce results similar to our ground truth. We use adversarial loss with $l_{1}$ to produce sharper generated outputs. The resulting final loss objective is the following $$\mathcal{L} = \lambda_{adv}\mathcal{L}_{adv} + \lambda_{content}\mathcal{L}_{content}+ \lambda_{style}\mathcal{L}_{style}+ \lambda{l_{1}}\mathcal{L}_{l_{1}}.$$ For our experiments $\lambda_{adv}$ = $\lambda_{content}$ = 1, $\lambda_{style}$ = 1000 and $\lambda_{l1}$ = 10. We do not incorporate content loss for greyscale experiments since the content of a greyscale and colored image are the same. {height=".14\textheight"} Network Architecture -------------------- Our generator is comprised of 2 downsampling layers and 8 residual blocks [@residualblocks] that circumvent the bottle-neck issue raised when using a U-Net-based architecture. U-Net downsamples the input image to 2x2 which adversely affects already sparse input data. On the other hand, residual blocks reduce the need to downsample, instead allowing layers to be skipped in the training process. A simpler residual function is learned versus a function that directly maps input line art to colored frames, where each frame is a single image. Instance normalization [@Norm:Instance] is used in place of batch normalization since smaller batch sizes are used for memory requirements when conditioning two images. With smaller batches, it is more effective to normalize across the spatial dimension alone versus the spatial and batch dimension for all images. The output of the residual blocks is then upsampled to the original input size. Our discriminator architecture uses a 70x70 patch GAN as in Isola [*et al*. ]{}[@Pix2pix] who showed mapping generator predictions to a scale of $N\times{N}$ outputs and classifying each patch as real or fake can be more effective than regular discriminators that map to a single scalar. Our discriminator also takes advantage of spectral normalization to stabilize training as shown in [@Norm:Spectral]. By restricting the Lipschitz constant of the discriminator to 1, we prevent the discriminator from learning a representation that perfectly differentiates real from fake generated images. Instead the discriminator is encouraged to find the intended optimal solution. In Chan [*et al*. ]{}[@Dance:Effros] temporal information is introduced in a cGAN architecture by adding extra conditions on their generator and discriminator. The current input and previously generated image are conditions for their generator. The discriminator is then conditioned on the current and previous input, generated image, and ground truth. We take inspiration from this method and incorporate temporal information in our network with added conditions to our generator and discriminator. The conditions used by Chan [*et al*. ]{}[@Dance:Effros] can be computationally expensive for our resources so we choose to use the line art frame and the previous ground truth color frame as conditions to our generator. The first frame of the training set does not have a previous frame, so we pass a blank image as the condition to the network. In every other case we randomly sample either the blank image or the corresponding previous color frame according to a Bernoulli distribution. Thus there is a 50 percent chance that any given condition $\mathbf{F}_{prev}$ is a blank image. Experiments ----------- ### Implementation Strategy In order to show viability for large-scale productions, we choose to make our dataset from the anime Dragonball which has nearly 300 episodes spread over 9 seasons. Our dataset are 2 full seasons obtained from legal sources. In order to extract frames, we use OpenCV to save frame sequences. We thereby create a training set of 84,000 frames and a test and validation set of just over 15,000 frames that make up multiple episodes previously unseen by our network at inference. Our input pipeline converts RGB frames to both greyscale and line art for separate experiments. We convert the colored ground truth to a single channel greyscale input and mimic line art using Canny edge detection [@canny1986computational] with gaussian filter of standard deviation 1. Our proposed method is implemented in PyTorch with input frames resized to $256\times{256}$. For every experiment we feed our network batch sizes of 16 for 35 epochs. The generator is trained with a learning rate of $10^{-4}$ while the discriminator trains with a one tenth learning rate of the generator using the Adam optimizer algorithm [@Adam]. By doing so, the generator is given the opportunity to learn a mapping before the discriminator becomes too strong which prevents any useful training. Our baseline for experiments was a pix2pix U-Net-based conditional GAN that was only conditioned on line art without considering temporal information, style, and content loss. For both our method and baseline we conduct two separate experiments, one on colorizing greyscale frames and the other on colorizing line art frames. Greyscale images are the most often used medium for colorization related tasks so we include them in our experiments. Being able to colorize greyscale images effectively can also be useful for Japanese comics known as manga. Manga are traditionally greyscale because it is too expensive to pay artists of a popular series to color thousands of images per book that run more than 300 issues. Although manga take the form of books, being able to account for temporal information is still highly relevant. Every page of a manga can consist of small frames that are correlated with each other. Characters are often seen in multiple frames repeated on a page as they move and interact. Thus maintaining consistency even between images that do not make up a video is necessary for automatic colorization. ### Validation Metrics In order to validate results from our experiments, we employ three different image quality metrics, namely Fréchet inception distance (FID) [@zhang2018unreasonable], structural similarity index (SSIM) [@wang2004image], and peak signal to noise ratio (PSNR). Evaluation metrics that assume pixel wise independence like PSNR and SSIM can assign favorable scores to perceptually inaccurate results or penalize scores for visibly large differences that do not necessarily imply low perceptual quality [@zhang2018unreasonable]. Especially for the case of anime colorization, differences that do not affect human perception are still favorable results. For this reason, we include FID which has been shown to correlate closely with human perception [@heusel2017gans]. FID measures the distance between activations of Inception-v3 [@inception] trained on ImageNet [@russakovsky2015imagenet] and accounts for both the diversity and visual fidelity of generated samples. Results ======= Qualitative Comparison ---------------------- Results obtained from our baseline model using U-Net suffered from the checkerboard effect [@odena2016deconvolution] especially with line art input as presented in figure \[LAImg\] likely due to its sparse nature. The greyscale colorization was perceptually natural apart from the checkerboard on some images, however when combining multiple consecutive frames to form a video, the flicker effect from slight variations in colors between frames was very apparent. Inconsistencies between frames resulting in color variations is even more apparent when using line art input in our baseline. Figure \[seq2\] shows these color inconsistencies that lead to flickering when the frames are stitched together to form a video. With our model, the checkerboard effect was non existent for line art generated frames even when compared to using greyscale input with our baseline in figure \[GSImg\]. The content and style loss contributed to learning texture information while reducing unwanted textures such as the checkerboard effect. Additionally, the generated frames of our model from both line art and greyscale created a more coherent video. This is shown both in figure \[seq1\] and \[seq2\] where our model produces consecutive frames with less variation which in return reduces flickering in the final video. The flicker effect still exists but is not as significant as our baseline model. [.3]{} ![Ours[]{data-label="fig:image3"}](Roshi_Grey.png "fig:"){width="\textwidth"} [.3]{} ![Ours[]{data-label="fig:image3"}](Roshi_Eval_Base_GS.png "fig:"){width="\textwidth"} [.3]{} ![Ours[]{data-label="fig:image3"}](Roshi_temporal_GS.jpg "fig:"){width="\textwidth"} [.3]{} ![Ours[]{data-label="fig:image"}](Roshisig1.jpg "fig:"){width="\textwidth"} [.3]{} ![Ours[]{data-label="fig:image"}](Roshi_Eval_Base_LA.png "fig:"){width="\textwidth"} [.3]{} ![Ours[]{data-label="fig:image"}](RoshiLA2.jpg "fig:"){width="\textwidth"} {height=".21\textheight"} ![Colorized sequence of frames from line art frames. The first frame is conditioned on a blank image with each successive frame being conditioned on the previously generated frame. (Left) Synthesized line art. (Center) Baseline results. (Right) Our results.[]{data-label="seq2"}](fig4.jpg){height=".8\textheight"} Quantitative Comparison ----------------------- Quantitative comparisons are reflected in table \[tab1\] using PSNR, SSIM, and FID of both models over the validation set. PSNR is higher in our model while the structural similarity index shows our model is closer to the ground truth colored images than our baseline. In terms of human perceptual clarity, the Fréchet inception distance between our model that leverages residual blocks with style and content loss is more effective in producing realistic results than the baseline model. ------ ----------- ---------- ----------- ---------- Greyscale Line art Greyscale Line art FID 20.69 32.46 9.29 19.12 SSIM 0.72 0.36 0.78 0.57 PSNR 14.15 8.74 17.38 16.03 ------ ----------- ---------- ----------- ---------- : Quantitative results SSIM, PSNR, and FID between baseline and our model[]{data-label="tab1"} Discussion and Future Work ========================== Using learning-based methods to map sparse inputs to colored outputs is challenging, however it is evident from our results that it is possible specifically in the domain of anime production. If artists make detailed sketches, they can circumvent the colorization process of key frames and in-between frames thereby expediting the production process while reducing the overall cost. Rather than require color artists to perform the task, the coloring workflow can be adapted such that the number of artists can be reduced, requiring only a few artists to fix regions colored incorrectly. This is true for both anime and manga production. For manga, which is already greyscale, even less effort is needed to automatically colorize frames in each page. Greyscale frames produce results requiring less color correction from artists since more information is present in the input compared to line art. For minimal extra cost, studios can publish colored versions of their most popular manga. The main issue that still needs to be addressed is what to do when new characters are introduced. In our case, the validation set consisted of the last four episodes of season 2 in which a new villain is introduced to set up the next season. With our existing model, when episodes that introduce new characters are added, the incorrectly colored characters can be manually colored by artists. When the final episode is finished, they can be added back to the dataset to be trained on the network again. By consistently retraining the model on previously colorized frames adjusted by artists for new characters, future episodes can be correctly colored. Thus, the model can be incorporated into a workflow that circumvents the issue. Canny edge detection sometimes misses the dots used for eyes or the lines for mouths in scenes where characters are distant in the frame. Finding methods to more effectively simulate line art will greatly improve the network in coloring line art frames. Acknowledgment {#acknowledgment .unnumbered} ============== This research was supported in part by an NSERC Discovery Grant. The authors gratefully acknowledge the support of NVIDIA Corporation for donation of GPUs through its Academic Grant Program. [1]{} H. Masuda, T. Sudo, K. Rikukawa, Y. Mori, N. Ito, and Y. Kameyama. Anime Industry Report 2017. *The Association of Japanese Animations*, 2017. T. Welsh, M. Ashikhmin, and K.  Mueller. Transferring color to greyscale images. *In ACM TOG*, volume 21, pages 277–280, 2002. A. Levin, D. Lischinski, and Y. Weiss. Colorization using optimization. In *ACM transactions on graphics (tog)*, volume 23, pages 689–694. ACM, 2004. L. A. Gatys, A. S. Ecker, and M. Bethge. A Neural Algorithm of Artistic Style. In *Journal of Vision*, page 326, 2016. I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In [*Advances in neural information processing systems*]{}, pages 2672–2680, 2014. K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*]{}, pages 770–778, 2016. L. Gatys, A. S. Ecker, and M. Bethge. Texture synthesis using convolutional neural networks. In [*Advances in Neural Information Processing Systems*]{}, pages 262–270, 2015. M. W. Gondal, B. Sch[ö]{}lkopf, and M. Hirsch. The unreasonable effectiveness of texture transfer for single image super-resolution. In [*Workshop and Challenge on Perceptual Image Restoration and Manipulation (PIRM) at the 15th European Conference on Computer Vision (ECCV)*]{}, 2018. J. Canny. A computational approach to edge detection. , pages 679–698, 1986. J. Johnson, A. Alahi, and L. Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In [*European Conference on Computer Vision (ECCV)*]{}, pages 694–711. Springer, 2016. P. Isola, J.-Y. Zhu, T. Zhou, and A. A. Efros. Image-to-image translation with conditional adversarial networks. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*]{}, 2017. A. Odena, J. Buckman, C. Olsson, T. B. Brown, C. Olah, C. Raffel, and I. Goodfellow. Is generator conditioning causally related to gan performance? In [*Proceedings of the 35th International Conference on Machine Learning*]{}, 2018. A. Odena, V. Dumoulin, and C. Olah. Deconvolution and checkerboard artifacts. , 1(10):e3, 2016. O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, et al. Imagenet large scale visual recognition challenge. , 115(3):211–252, 2015. C. Chan, S. Ginosar, T. Zhou and A. A. Effros. Everybody Dance Now. *arXiv preprint arXiv arXiv:1808.07371*, 2018. D. P. Kingma,J. Ba. Adam: A Method for Stochastic Optimization. In [*International Conference on Learning Representations*]{}, 2015. K. Nazeri, E. Ng, and M. Ebrahimi. Image Colorization Using Generative Adversarial Networks. *Articulated Motion and Deformable Objects Lecture Notes in Computer Science*, 2018. K. Nazeri, E. Ng, T. Joseph, F. Z. Qureshi, and M. Ebrahimi. EdgeConnect: Generative Image Inpainting with Adversarial Edge Learning. *arXiv preprint arXiv arXiv:1901.00212*, 2019. R. Zhang, P. Isola, A. A. Efros, E. Shechtman, and O. Wang. The unreasonable effectiveness of deep features as a perceptual metric. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*]{}, 2018. Y. Ci, X. Ma, Z. Wang, H. Li and Z. Luo. User-Guided Deep Anime Line Art Colorization with Conditional Adversarial Networks. *ACM Multimedia Conference*, 2018, Soeul South Korea. L. Zhang, Y. Ji and X. Lin. Style Transfer for Anime Sketches with Enhanced Residual U-Net and Auxiliary Classifier GAN. *Asian Conference on Pattern Recognition (ACPR)*, 2017. T. Miyato, T. Kataoka, M. Koyama, and Y. Yoshida. Spectral normalization for generative adversarial networks. In [*International Conference on Learning Representations*]{}, 2018. D. Ulyanov, A. Vedaldi, and V.  Lempitsky. Improved texture networks: Maximizing quality and diversity in feedforward stylization and texture synthesis. In *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, 2017. M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In [*Advances in Neural Information Processing Systems*]{}, pages 6626–6637, 2017. O. Ronneberger, P. Fischer, T. Brox. U-Net: Convolutional Networks for Biomedical Image Segmentation. In [ *Lecture Notes in Computer Science Medical Image Computing and Computer-Assisted Intervention – MICCAI*]{}, pages 234–-241, 2015. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality assessment: from error visibility to structural similarity. , 13(4):600–612, 2004. K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. , abs/1409.1556, 2014. C. Szegedy, V. Vanhoucke, C. Ioffe, J. Shlens, and Z. Wojna. Rethinking the Inception Architecture for Computer Vision. *2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, 2016. T. Yonetsuji. Paintschainer. *github.com/pfnet/PaintsChainer*, 2017.

--- abstract: | We present a 3D dynamical model of the orbital induced curvature of the wind-wind collision region in binary star systems. Momentum balance equations are used to determine the position and shape of the contact discontinuity between the stars, while further downstream the gas is assumed to behave ballistically. An archimedean spiral structure is formed by the motion of the stars, with clear resemblance to high resolution images of the so-called “pinwheel nebulae”. A key advantage of this approach over grid or smoothed particle hydrodynamic models is its significantly reduced computational cost, while it also allows the study of the structure obtained in an eccentric orbit. The model is relevant to symbiotic systems and $\gamma$-ray binaries, as well as systems with O-type and Wolf-Rayet stars. As an example application, we simulate the X-ray emission from hypothetical O+O and WR+O star binaries, and describe a method of ray tracing through the 3D spiral structure to account for absorption by the circumstellar material in the system. Such calculations may be easily adapted to study observations at wavelengths ranging from the radio to $\gamma$-ray. author: - | E. R. Parkin[^1] & J. M. Pittard\ School of Physics and Astronomy, The University of Leeds, Leeds LS2 9JT, UK date: 'Accepted 2008 May 21. Received 2008 May 20; in original form 2008 April 22' title: A 3D dynamical model of the colliding winds in binary systems --- \[firstpage\] hydrodynamics - methods:numerical - stars:early-type - X-rays:stars - stars:binaries - stars:winds Introduction {#sec:intro} ============ Colliding winds occur in various types of stellar binaries, including those with massive OB and Wolf-Rayet (WR) stars, lower mass eruptive symbiotic systems containing a white dwarf and red giant star which undergo a “slow nova” outburst, and binary systems which contain one or two pulsars blowing a pulsar wind(s). High-spatial-resolution observations are revealing many interesting features in such systems. In massive O+O and WR+O binaries, radio interferometry has spatially-resolved emission from non-thermal electrons at the apex of the wind-wind collision [e.g., @Williams:1997; @Dougherty:2000; @Dougherty:2005; @Contreras:2004]. Beautiful “pinwheel” structures which trace dust emission can also be observed [e.g., @Tuthill:1999; @Tuthill:2006; @Tuthill:2008; @Monnier:1999; @Marchenko:2002]. The shape of these structures can be described by archimedean spirals which are believed to follow the wind-wind collision region in systems where the winds are of very unequal momentum. Colliding winds also play a key role in eruptive symbiotic systems, where a hot, fast, diffuse wind from a white dwarf companion interacts with a slow massive wind from a Mira type primary star[^2]. The class of eruptive symbiotics can be divided into two further subtypes: classical symbiotics, in which the bolometric luminosity remains constant and outbursts typically last about 100 days (Z And is an example), and the more powerful eruptions known as symbiotic novae, where the bolometric luminosity increases by a factor of order 10-100 on a timescale of about a year, and the system stays in an active state for $\gtsimm 10\;$yrs (well-known examples are V1016 Cyg, HM Sge, and AG Pegasi). Colliding winds may also play a key role in the newly discovered class of systems called $\gamma$-ray binaries [@Aharonian:2005a; @Aharonian:2005b; @Albert:2006]. The nature of these systems is still controversial, though in the case of PSR B1259-63, it is clear that a relativistic wind from a pulsar collides with the stellar wind from a Be star. The orbit is highly eccentric ($e = 0.87$), and has a period of 3.4 yr [@Johnston:2005]. The TeV $\gamma$-ray emission arises from the Inverse Compton cooling of ultra-relativistic electrons accelerated at the pulsar wind termination shock [e.g. @Khangulyan:2007]. In contrast, the nature of the sources LS5039 and LS I +61 303 is less clear, since the type of compact object has not been established beyond doubt [@Romero:2007; @Dubus:2008; @Khangulyan:2008]. While there has been much progress in modelling the dynamical structure of the colliding winds in early-type binary systems, the majority of work has been limited to 2D [e.g., @Stevens:1992; @Gayley:1997; @Pittard:1997; @Pittard:1998; @Pittard:2007; @Zhekov:2007]. 3-dimensional hydrodynamical calculations have been performed by @Pittard:1999, @Walder:2002, and @Lemaster:2007, while a ballistic model was presented by @Harries:2004. An SPH model has recently been computed by @Okazaki:2008. Dynamical models for symbiotic novae have been presented by @Girard:1987 and @Kenny:2005 [@Kenny:2007], while 3D hydrodynamical models have been presented by @Walder:2000. Models of the wind-wind collision in classical symbiotics have been presented by @Mitsumoto:2005 and @Bisikalo:2006. Relativistic hydrodynamics [@Bogovalov:2008] and SPH [@Romero:2007] models have been used to investigate the wind-wind collision in pulsar wind binary systems. Although dramatic improvements in computational power and techniques in recent years have spurred the development of 3D models of colliding winds, such work remains computationally expensive, and it is still difficult to perform simulations of CWB’s even on high performance parallel machines when the orbital eccentricity is high. We therefore present a new method which captures the flow dynamics while requiring less computational resources. At its heart, our approach adopts the equations for the ram pressure balance between the two winds as detailed by @Canto:1996. In this work it is assumed that both winds are highly radiative, rapidly cool, and fully mix. While these assumptions are only relevant in close binaries, it provides a convenient starting point and the position of the contact discontinuity is unlikely to drastically change even if the wind-wind collision is essentially adiabatic. Then, at some distance downstream of the apex of the wind-wind collision region (WCR) the flow is assumed to reach a terminal speed and to thereafter flow ballistically (i.e. no net force acting upon it). This ballistic treatment has similarities to many previous works [e.g., @Girard:1987; @Harries:2004; @Kenny:2007]. The derivation in @Canto:1996 has also been widely used to model observable properties [e.g., @Foellmi:2008; @Henley:2008]. This paper is organised as follows. In § \[sec:shkconemodel\] we explain the steps necessary to construct our dynamical model of the wind-wind collision. § \[subsec:emission\] shows how it can be used to simulate the X-ray emission and circumstellar absorption arising from the WCR in early-type binary systems, though this is but one example of the potential use of such a model. In § \[sec:conclusions\] we summarize and conclude our findings, and outline possible future directions. The dynamical model {#sec:shkconemodel} =================== Overview {#subsec:shkconeoverview} -------- In the model the orbit is calculated in the frame of one of the stars (herafter referred to as the primary star). The winds are assumed to reach their terminal speeds before they collide. The contact discontinuity (CD) is split into two sections, to account for the effect of orbital motion: i\) A region close to the apex of the WCR where the flow from the stagnation point is accelerating along the CD (hereafter called the “shock cap”). The shock cap is terminated where the flow is assumed to become ballistic (the “ballistic point”), the exact point being calibrated against hydrodynamical models (see § \[subsec:bcd\]). While the properties of the shock cap are assumed to be axisymmetric, orbital motion introduces an aberration angle which means that the symmetry axis and the line of centres of the stars are not colinear (see § \[subsec:shockcap\]). ii\) A region beyond the ballistic point where the flow along the contact discontinuity is unaffected by the primary and secondary stars’ gravity, ram pressure from the winds, or thermal pressure in the WCR. If the stellar winds have differing speeds, the flow in this region is assumed to move with the speed of the slower wind, since this is the wind which responds least to the orbital motion of the stars, and dominates the absorption in the system (in the models presented in this paper, both winds have the same speed of $2000 \thinspace \rm{kms}^{-1}$ - see Table \[tab:models\]). This region is termed the “ballistic CD”. By separating the CD into these two sections we can model the effect of the winding of the CD around the stars and the subsequent absorption by the un-shocked winds. We do not attempt to model the shocks which bound either side of the CD in this work, as in many circumstances the shocked gas efficiently cools and is compressed by the ram pressure of the pre-shock wind into a thin dense sheet coincident with the CD. For instance, in symbiotic novae, the hot wind is likely to be strongly radiative [see Fig. 4 in @Kenny:2005], as of course is the cool wind, and our model therefore gives the position of the shocked gas and the dense spiral shells which subsequently form. Strong radiative cooling is also a feature of the WCR in many massive binaries. In , for example, the primary LBV wind is so dense (and slow) that it is strongly radiative around the entire orbit [@Pittard:1998b]. The importance of cooling in the WCR can be quantified using the cooling parameter [@Stevens:1992], $$\chi = \frac{t_{\rm{cool}}}{t_{\rm{esc}}} = \frac{v^4 _8 d_{12}}{\dot{M} _{-7}},$$ where $v_8$ is the wind velocity in units of $1000 \thinspace\rm{km\thinspace s}^{-1}$, $d_{12}$ is the separation of the stars in units of $10^{12}\rm{cm}$, $\dot{M}_{7}$ is the mass-loss rate of the star in units of $10^{-7} \Msolpyr$, $t_{\rm{cool}}$ is the cooling time, and $t_{\rm{esc}}$ ($=d / c_{\rm s}, \thinspace c_{\rm s}$ is the postshock sound speed) is the characteristic time for hot gas near the apex of the WCR to flow downstream. In practice, hydrodynamical simulations show that the wind collision region (WCR) is adiabatic for $\chi \gtsimm 3$, whereas for $\chi \ltsimm 3$ it cools rapidly (Fig. \[fig:wcr\]). Fig. \[fig:chi\] shows the value of $\chi$ as a function of orbital period for each shocked wind in a hypothetical O+O star binary with a circular orbit, wind speeds of $2000\;\kmps$, and mass-loss rates of $10^{-6}\;\Msolpyr$ and $2 \times 10^{-7}\;\Msolpyr$ for the primary and secondary star of masses 50 and 30 respectively. Clearly both of the shocked winds are largely adiabatic, even down to an orbital period of $10\;$days (in shorter period systems the stars are close enough together that acceleration/deceleration of the winds needs to be considered). However, in a hypothetical WR+O system where the mass-loss rates of the primary and secondary stars are now $2 \times 10^{-5}\;\Msolpyr$ and $10^{-6}\Msolpyr$, and both stars have masses of $50\Msol$ and wind speeds of $2000\thinspace \rm{kms}^{-1}$, cooling is important for orbital periods $\ltsimm 1\;$yr. If the WR star is a WC subtype, cooling is important for periods up to several years, since cooling is more efficient with such abundances [see, e.g., @Stevens:1992]. Thus, Fig. \[fig:chi\] shows that the denser winds from WR stars are likely to produce radiative shocks in many instances, though the O+O systems will usually be adiabatic unless the orbital period, $P \ltsimm 10\;$d, or the winds are slower and/or denser than assumed above. The postshock winds of both the primary and secondary stars in the simulations discussed in § \[subsec:emission\] are largely adiabatic. In such cases, the temperature of the hot gas in the WCR as a function of distance downstream from the stagnation point at the apex of the WCR has been determined by @Kenny:2005. With this information it is possible to derive the width of the post-shock layer, and hence the position of the shocks, as a function of downstream distance. However, this is beyond the scope of the present work. In the following sections we detail the modelling of the shock cap and ballistic CD. The shock cap {#subsec:shockcap} ------------- The shape of the shock cap is determined from momentum balance requirements. The surface density and velocity of the flow along the shock cap are obtained from Eqs. 29 and 30 of [@Canto:1996] (the latter scaled to the speed of the slower wind). Assuming the winds are already at their terminal velocity when they reach the shocks, the locus of the CD is $R(\theta_{\rm c1})$, where $R$ is the distance from the centre of the primary star and $\theta_{\rm c1}$ is the angle between the vector to the primary star and the line-of-centres (see Fig. \[fig:cantofig1\]). The ratio of the wind momenta is given by $$\eta \equiv \frac{\dot{M}_{2}v_{\infty2}}{\dot{M}_{1}v_{\infty1}}, \label{eqn:eta}$$ where $\dot{M_{1}}$, $v_{\infty1}$, $\dot{M_{2}}$, and $v_{\infty2}$ are the mass-loss rates and terminal velocities of the primary and secondary stars respectively. The shock cap is symmetrical about the line of centres before the effects of orbital motion are introduced. The 2D ($r,z$) coordinates of points on the shock cap in units of the stellar separation, $d_{\rm sep}$, are $$z = \frac{\tan\theta_{c_1}}{\tan\theta_{\rm c2}+\tan\theta_{\rm c1}},\\ \label{eqn:zandr1}$$ $$r = z\tan\theta_{\rm c2}. \label{eqn:zandr2}$$ To determine the coordinates in 3D, the 2D arms of the WCR can be rotated azimuthally. The $x, y$, and $z$ vectors ($x_{\rm{cap}}$, $y_{\rm{cap}}$, and $z_{\rm{cap}}$ respectively) from the center of the primary star to coordinates on the shock cap are then $$\begin{aligned} x_{\rm{cap}} = & d_{\rm{sep}}(z\cos\omega - r\sin\omega\cos\zeta), \\ y_{\rm{cap}} = & d_{\rm{sep}}(z\sin\omega + r\cos\omega\cos\zeta), \\ z_{\rm{cap}} = & d_{\rm{sep}}(r\sin\zeta),\end{aligned}$$ where $\omega$ is the true anomaly of the orbit and $\zeta$ is the azimuthal angle subtended between a coordinate on the surface of the shock cap, the line of centres, and the $xy$ (orbital) plane. The number of coordinate points on the shock cap is determined by the values of $\theta_{\rm c1\infty}$, $\delta\theta_{c1}$ and $\delta\zeta$. With $\delta\theta_{\rm c1} = 1^{\circ}$ and $\delta\zeta = 18^{\circ}$ (i.e. 20 azimuthal points per 2D $rz$ value), the shock cap consists of $\sim 10^{3}$ separate coordinate points. Eq. 29 of [@Canto:1996] is used to determine the tangential velocity along the CD, and thus the position of the ballistic point in 2D axisymmetry. Since the size of the wind-wind collision scales with the orbital separation, dramatic variations occur in systems with highly eccentric orbits, as shown in Fig. \[fig:shkcones\] where $e=0.9$; the high eccentricity means that the shock cap at periastron has a linear scale which is 20 times smaller than that at apastron. Such high eccentricities occur in two of the most well-known colliding winds systems, and WR140, and also in PSR B1259-63, one of the $\gamma$-ray binaries. Another effect resulting from orbital motion is the aberration (skew) of the apex of the WCR due to the net velocity vector of the orbit (i.e. the motion of the secondary star relative to the primary star). The skew angle, $\mu$, which is the angle between the symmetry axis of the shock cap and the line of centres of the stars is approximated by $$\tan\mu = \frac{v_{\rm{orb}}}{v_{\infty}}, \label{eqn:coriolis}$$ where the speed of the slower wind is used. In the frame of the primary star, $$v_{\rm{orb}} = \left[G(M_{1} + M_{2})\left(\frac{2}{d_{\rm{sep}}}-\frac{1}{a}\right)\right]^{1/2}, \label{eqn:coriolis2}$$ for stars of mass $M_{1}$ and $M_{2}$ and an orbital semi-major axis, $a$. The aberration is significant in symbiotic novae because of the low wind speed of the cool star (for instance, a symbiotic system with $e = 0.0$, $M_{1}+M_{2}=2.5\;\Msol$, and $d_{\rm sep}=10\;$au has $v_{\rm orb} = 15\;\kmps$, which is comparable to the speed of the cool wind). In contrast, the aberration is small in early-type binaries (for instance, an O+O binary with $e=0.0$, $M_{1}+M_{2}=80\;\Msol$, and $d_{\rm sep}=4.3\;$au has $v_{\rm orb} = 130\;\kmps$, which is typically much smaller than the wind speeds), unless the orbit has high eccentricity. In such cases the magnitude of the skew varies throughout the orbit. Fig. \[fig:skew\_angle\] shows how $\mu$ varies throughout the orbit for the Model A O+O star binary with parameters as in Table \[tab:models\] and with $e=0.3$ and 0.9. A peak value is reached at periastron passage ($\phi = 0.0$) when the relative orbital speed of the stars reaches it’s highest value, and the lowest value of $\mu$ occurs when the stars are at apastron ($\phi = 0.5$) and the relative orbital velocity is a minimum. The variation of $\mu$ between apastron and periastron increases with the eccentricity of the orbit. The skew angle $\mu$ can affect the proximity of regions of the shock cap to the primary star around periastron (Fig. \[fig:shkcap\_skew\]), and the resulting level of occultation and attenuation. The ballistic CD {#subsec:bcd} ---------------- To construct the large-scale 3D structure of the WCR, gas packets are released from the endpoints of the shock cap at specific phase intervals with a velocity equal to the slower wind, $v_{\rm sl}$. The $x, y$, and $z$ components of the velocity of gas leaving the end of the shock cap at a specific orbital phase are given by: $$\begin{aligned} \hat{v}_{x} = & v_{\rm sl}\cos(\omega -\mu + \lambda\cos\zeta),\\ \hat{v}_{y} = & v_{\rm sl}\sin(\omega -\mu + \lambda\cos\zeta),\\ \hat{v}_{z} = & v_{\rm sl}\sin\zeta \\\end{aligned}$$ where $\lambda$ = $\theta_{\rm c1\infty}$ is the asymptotic half-opening angle of the contact discontinuity viewed from the star with the stronger wind. The ballistic part of the CD is then constructed by considering a sequence of previous positions of the ballistic points at the termination of the shock cap, and the current position of the gas flow from these points given that they move along linear trajectories (see Fig. \[fig:cdcoords\_howto\]). The position of points on the ballistic part of the CD ($x_{\rm{CD}}, y_{\rm{CD}}$, and $z_{\rm{CD}}$) at the time $t$ is given by their position at the time they were emitted from the end of the shock cap ($x_{\rm{cap}}, y_{\rm{cap}}$, and $z_{\rm{cap}}$) plus the distance they have since travelled at velocity $\hat{v}_{\rm{x}},\hat{v}_{\rm{y}},\hat{v}_{\rm{z}}$, i.e. $$\begin{aligned} x_{\rm{CD}}(t) = & x_{\rm{cap}}(t-T) + \hat{v}_{\rm{x}}(t-T)T,\\ y_{\rm{CD}}(t) = & y_{\rm{cap}}(t-T) + \hat{v}_{\rm{y}}(t-T)T,\\ z_{\rm{CD}}(t) = & z_{\rm{cap}}(t-T) + \hat{v}_{\rm{z}}(t-T)T,\end{aligned}$$ where $T$ is the time elapsed since the flow left the end of the shock cap. [c]{}\ \ -- -- -- -- The number of coordinates in the ballistic CD is dependant on the number of phase steps around the orbit, the number of orbital revolutions followed, and the number of azimuthal steps (i.e. $\delta\zeta$). In this work, the ballistic CD consists of 2000 coordinate positions along each azimuthal trajectory (1000 per orbit traced). Tests performed using a 3D hydrodynamics code confirm that the Coriolis force, which causes the curvature to the WCR, becomes significant once the flow from the stagnation point is accelerated to 70% - 90% of the terminal speed of the slower wind (Fig. \[fig:comparison\]), and the gas is at a distance from the stars of order the stellar separation. Both of these conditions are satisfied by the 85% cut-off attained via calibration of the dynamic model against hydrodynamic models. Interestingly, varying the value of the cut-off percentage has the effect of improving the fit to one spiral arm but reducing the quality of the fit to the other arm. Using the Model A paramters (Table \[tab:models\]) the off-axis distance of the ballistic point from the line of centres, $r_{\rm{max}}$, increases by a factor of $\sim 3$ between 70 % and 90 % (Table \[tab:cutoffs\]), whereas the opening angle of the shock increases by roughly a half with a more linear relation. In § \[sec:results\] we show that there is little difference in the X-ray lightcurves when this percentage is varied slightly. ----------------- ------------------ ------------------------- Cutoff $r_{\rm{max}}$ $\theta_{\rm c1\infty}$ $(\%)$ $(d_{\rm{sep}})$ $(^{\circ})$ 70 0.53 33 80 0.78 40 85 0.99 44 90 1.40 49 \[tab:cutoffs\] ----------------- ------------------ ------------------------- : Transition points between the shock cap and the ballistic CD for varying percentages of the postshock flow velocity. $r_{\rm{max}}$ is calculated using Eq \[eqn:zandr2\]. $\theta_{\rm c1\infty}$ is the opening angle of the shock at the transition point measured from the primary star (see Fig. \[fig:cantofig1\]). Fig. \[fig:cdcoords\_phi\] shows the effect of the motion of the stars on the ballistic CD on scales of the order of the semi-major axis. The curvature of the CD close to the end of the shock cap is greatest when the relative orbital velocity of the stars is high. The smooth connection of the ballistic CD to the shock cap indicates that the assumptions inherent in the model are good at this level. -- -- -- -- The structure of the ballistic CD at large scales is shown in Fig. \[fig:cdcoords\_ecc\] for a range of orbital eccentricities. At low orbital eccentricities, the spiral structures resemble the 3D hydrodynamical models of [@Walder:2000; @Walder:2002] and [-@Lemaster:2007], the dust spiral models of the pinwheel nebula WR104 by [@Harries:2004] and [@Tuthill:2008], and the CWo model for symbiotics developed by [@Kenny:2007]. Note, however, that this figure shows the projection of the CD onto the orbital plane, and not the position of the shocks either side of it. If the shocked region were largely adiabatic, the shocks would stand off from the CD and the width of the spiral structure on the orbital plane would be somewhat greater. At $e=0.9$ the secondary star moves very quickly through periastron, resulting in the projected CD (which encompasses the region of unshocked secondary wind) thinning to the left of the stars. In contrast, there exists a large region of unshocked secondary wind to the right of the stars, as the secondary star moves slowly around apastron. This creates a low density cavity in the primary wind. The X-ray attenuation in such systems will depend on the orbital phase, as well as being sensitive to the position of the observer, and in principle may vary widely. For instance, in a system like $\eta\;$Car, the primary wind is very dense and much more strongly absorbing than the secondary wind. An observer located on the positive $x$-axis at infinity will predominantly view through the low density unshocked wind of the secondary star, whereas an observer on the negative $x$-axis will predominantly view through the high density unshocked wind of the primary star. As the column density scales directly with the density of the gas, these observers will see significantly different X-ray lightcurves. On the other hand, if the primary wind is more rarefied than the secondary wind, this behaviour reverses. Finally, we note that in systems with highly eccentric orbits, the amount of attenuation at phases around apastron may depend on the skew angle of the shock cap which occurs around periastron. This is because the skew angle of the shock cap affects the duration and phase where primary/secondary wind material is emitted in a certain direction. Depending on the viewing angle into the system, the inclusion of aberration effects may result in a variation in the attenuation to emission concentrated near the apex of the shock cap due to the alteration in path length through the more strongly absorbing wind. ------- --------- --------- -------------------- -------------------- -------- ------------------------- --------- ------ ------------ ------------ Model $M_{1}$ $M_{2}$ $\dot{M}_{1}$ $\dot{M}_{2}$ $\eta$ $\theta_{\rm c1\infty}$ $P$ $a$ $\chi_{1}$ $\chi_{2}$ () () () () ($^{\circ}$) (au) A 50 30 $1.0\times10^{-6}$ $2.0\times10^{-7}$ 0.20 62.6 1 yr 4.3 100 500 B 50 30 $1.0\times10^{-6}$ $2.0\times10^{-7}$ 0.20 62.6 1 month 0.81 20 100 C 50 50 $2.0\times10^{-5}$ $1.0\times10^{-6}$ 0.05 41.0 1 yr 4.3 5 100 ------- --------- --------- -------------------- -------------------- -------- ------------------------- --------- ------ ------------ ------------ An example application - X-ray emission and absorption in an early-type binary {#subsec:emission} ============================================================================== As an example application of the model described in § \[sec:shkconemodel\], we consider the X-ray emission from hypothetical O+O and WR+O-star colliding wind binaries. Model A is an O+O binary with an orbital period of $1\;$yr and semi-major axis $a=4.3\;$au. Model B examines the increasing effects of absorption as the orbital period is reduced to $1\;$month. The third model system (Model C) consists of a WR star with a mass-loss rate of $2\times10^{-5}\Msolpyr$. The high velocities of the stellar winds are sufficient to cause the postshock gas to emit at X-ray wavelengths, and both winds are essentially adiabatic in the systems considered (see Table \[tab:models\]). For the three models considered we use a distance of 1 kpc, ISM column of $5\times10^{21}\;\rm{cm}^{-2}$, and orbital eccentricity, $e=0.3$. The X-ray emission ------------------ The X-ray emission from the WCR is a function of the gas temperature and density. Since the dynamical model discussed in the previous section does not contain such information, we use a grid-based, 2D hydrodynamical calculation of an axis-symmetric WCR to obtain this. The numerical code is second-order accurate in time and space [@Falle:1996; @Falle:1998]. The resulting emission as a function of off-axis distance is then mapped onto the coordinate positions in the 3D dynamical model. In this way we obtain the benefit of effectively modelling the thermodynamic and hydrodynamic behaviour responsible for the production of the X-ray emission, while simultaneously accounting for the effect of the motion of the stars on the large-scale structure of the WCR and the subsequent wind attenuation. Since the hydrodynamic calculation is 2D, the computational requirements remain low. The X-ray emission calculated from each hydrodynamic cell in the WCR is $\Gamma(E)=n^{2}V\Lambda(E,T)$, where $n$ is the gas number density ($\cm^{-3}$), $V$ is the cell volume ($\cm^{3}$), and $\Lambda(E,T)$ is the emissivity as a function of energy $E$ and temperature $T$ for optically thin gas in collisional ionization equilibrium ($\rm{erg\; cm}^{3}s^{-1}$). $\Lambda(E,T)$ is obtained from look-up tables calculated from the *MEKAL* plasma code [@Leidahl:1995 and references therein] containing 200 logarithmically spaced energy bins in the range 0.1-10.0 keV, and 101 logarithmically spaced temperatures from $10^{4}$ to $10^{9} \;\rm{K}$. Solar abundances are assumed for the O-star winds and the WR wind is assumed to have WN8 abundances (mass fractions of: H/He=0, C/He = $1.7\times10^{-4}$, N/He = $5\times10^{-3}$, and O/He = $1\times10^{-4}$). The emissivity of solar abundance gas is shown in Fig. \[fig:emiss\_spec\] and the corresponding opacity is shown in Fig. \[fig:abs\_spec\]. The WN8 emissivities are very similar to those at solar abundance. Opacity values are also similar for solar and WN8 abundances, with the most significant difference being a factor of 2 increase at $10^{4}\;$K at energies below $\sim 1\:$ keV. [l]{}\ [l]{}\ The emission values are then appropriately scaled for the changing stellar separation around the orbit [$L_{\rm{X}} \propto d_{\rm sep}^{-1}$ in the adiabatic limit, @Stevens:1992] and placed onto the 3D shock cap and ballistic CD. Emission values are assigned to points within $3\;d_{\rm sep}$ of the apex of the WCR. This accounts for $\sim 90$ per cent of the 0.1-10 keV emission and $> 99$ per cent of the 2-10 keV emission. The attenuation {#sec:absorption} --------------- To compute X-ray lightcurves, the orientation of the observer relative to the system must be specified. Since the model assumes the orbit of the stars is in the $xy$ plane, viewing angles into the system can be described by the inclination angle that the line-of-sight makes with the $z$ axis, $i$, and the angle the projected line-of-sight makes with the major axis of the orbit, $\theta$. Positive values of $\theta$ correspond to projected lines of sight in the prograde direction from the positive $x$ axis. The components of the unit vector along the line-of-sight, $\underline{\hat{u}}$, are thus $$\begin{aligned} u_{x} & = & \cos\theta\sin i, \\ u_{y} & = & \sin\theta\sin i, \\ u_{z} & = & \cos i. \label{eqn:losunitvectors}\end{aligned}$$ There are 3 ways in which the intrinsic X-ray emission can be attenuated. First, it can be occulted by the stars (this effect, of course, is greatest in short period systems). Second, there will be some absorption through the un-shocked stellar winds. Finally, there will be attenuation through the shocked gas in the WCR. The latter is only important in systems where the shocked gas of at least one of the winds is strongly radiative, otherwise the gas in the WCR remains hot and the attenuation through it is small. But if there is significant cooling, as for example occurs when the cool wind in symbiotic systems is shocked, a thin, dense, and cold layer of gas is formed at the CD, which will be a significant source of attenuation in the system. Significant cooling of the WCR can also occur in early-type binary systems of which is an example. We now describe how attenuation by each of the above-mentioned methods is calculated in our model. -- -- -- -- ### Occultation by the stars {#subsec:occultation} An important line-of-sight effect in binary star systems is occultation, particularly in the case of eclipsing binaries. To calculate this effect in our model, a line-of-sight is traced from each emitting region on the shock cap and ballistic CD, and its distance of closest approach to the centre of each star is calculated. If this distance is less than the radius of the star, and the star is in front of the emitting region, then occultation occurs, and none of the emission from the emitting region being considered reaches the observer. A visual representation of the occultation of the WCR by the primary star is shown in Fig. \[fig:occ\_shkcaps\]. The degree of occultation can be reduced by reducing the inclination angle $i$ (since the strongest X-ray emission occurs at the apex of the WCR). Although not shown, the phase at which the maximum occultation occurs can be altered by changing the value of $\theta$. Occultation causes little change to the observed luminosity over the entire orbit for the Model A system. This is due to the relatively small size of the stars in comparison to the extended emitting region for the 0.1-10.0 keV X-rays. Occultation effects become more noticeable in shorter period systems, and/or those with highly eccentric orbits (since the linear size of the shock cap is $\propto 1/d_{\rm{sep}}$). Occultation is also favoured where one (or both) of the stars has a large stellar radius (e.g. $\eta\;$Car, Parkin et al., in preparation), and when $i$ is large. For instance, the $e=0.9$ lightcurve in Fig. \[fig:int\_lc\] shows a pronounced occultation effect at orbital phase $\phi \simeq 1.00$, during which the emission falls sharply by a factor of 2. The width of the minimum due to occultation effects is very narrow as the high eccentricity means that the stars move rapidly through periastron, but the depth of the minimum is large ($\sim 75$% of the intrinsic 2-10 keV emission is occulted). ### Absorption by the un-shocked stellar winds {#subsec:windattenuation} For inclinations, $| i | \geq \pi/2 - \theta_{\rm c1\infty}$, the line-of-sight from emitting regions near the apex of the WCR will intersect the CD numerous times as it spirals out, and thus traverses first through one wind and then the other, etc. The total column density along a line-of-sight is then the sum of the individual column densities along the specific distances travelled in each wind. Accurate knowledge of where the line-of-sight intersects the CD, and the density of the gas at any point in space is therefore required if the total column density along a given sight line is to be calculated. To determine if and where an intersection through the CD occurs, the shock cap and ballistic CD are tesselated into a sequence of triangular planar facets constructed between three neighbouring coordinates ($\underline{P}_{\rm{a}}$, $\underline{P}_{\rm{b}}$, and $\underline{P}_{\rm{c}}$). To determine if the line-of-sight intersects a given triangle the normal to the plane in which the triangle lies, $\underline{\hat{n}}$, is calculated from $$\underline{\hat{n}} = (\underline{P}_{\rm{b}}-\underline{P}_{\rm{a}})\times(\underline{P}_{\rm{c}}-\underline{P}_{\rm{a}}). \label{eqn:intnormal}$$ The dot product of $\underline{\hat{n}}$ with the line-of-sight vector gives the angle between the line-of-sight and the plane. If the resultant angle is non-zero the line-of-sight vector will intersect the plane in which the triangular facet lies at some point in space. The component vectors to the intersection point ($x _{\inter}$, $y _{\inter}$, and $z _{\inter}$) are found by substituting the line parameter at the intersection point, $$\kappa = \frac{n_{\rm{x}}x_{\cap} + n_{\rm{y}}y_{\cap} + n_{\rm{z}}z_{\cap} + n_{\rm{const}}}{n_{\rm{x}}u_{\rm{x}} + n_{\rm{y}}u_{\rm{y}} + n_{\rm{z}}{u_{\rm{z}}}}, \label{eqn:intlinepar}$$ into the equations $$\begin{aligned} x_{\inter} = & x_{\cap} + \kappa u _{\rm{x}},\\ y_{\inter} = & y_{\cap} + \kappa u _{\rm{y}},\\ z_{\inter} = & z_{\cap} + \kappa u _{\rm{z}},\end{aligned}$$ where the equation of the plane with normal $\underline{\hat{n}}$ and vector components $n_{\rm{x}}$, $n_{\rm{y}}$, and $n_{\rm{z}}$ is $$n_{\rm{x}} \underline{x} + n_{\rm{y}} \underline{y} + n_{\rm{z}} \underline{z} + n_{\rm{const}} = 0 \label{eqn:plane}$$ In general, the intersection occurs outside of the triangular facet. Unit vectors are constructed between the corner points of the facet and the intersection point to determine whether the intersection occurs within its boundaries. The three dot products between these three unit vectors gives the angles $\theta_{a}$, $\theta_{b}$, and $\theta_{c}$. Only if the intersection point lies within the boundaries of the triangular facet will the equation $\theta_{a}+\theta_{b}+\theta_{c}=2\pi $ be satisfied (see Fig. \[fig:triangles\]). By looping over the entire sequence of triangles, every possible intersection of the line-of-sight with the CD is determined. With the coordinates of the intersection points ($x_{\inter}$, $y_{\inter}$, and $z_{\inter}$), it is a simple task to calculate the column density through the unshocked winds, $\sigma_{\rm{w}}$. Lines-of-sight which pass very close to the stars sample the acceleration region of the wind. Therefore, we use a $\beta$-velocity law of the form $$v(r) = v_{\infty}\left(1 - \frac{R_{\ast}}{r}\right)^{\beta} \label{eqn:betavellaw}$$ to determine the density of the wind at radius $r$ from the star. $\beta$ describes the acceleration of the wind with $\beta=0.8$ appropriate for O star winds [@Lamers:1999]. Because the width of the WCR is not considered in our model, the volume of unshocked wind and the resulting attenuation are overestimated, though this approximation will not have a signifcant impact on our results. ### Absorption by the shocked stellar winds {#subsec:sdattenuation} As already mentioned, the attenuation of X-rays through the shocked wind(s) needs to be considered if one or both winds strongly cool. In the O+O and WR+O-star binaries considered in this section, the shocked gas is largely adiabatic. However, for completeness, we discuss here a methodology for calculating the absorption due to X-rays intersecting a cold dense layer of postshock gas at the CD. This is applied to models of in Parkin et al. (in preparation). In Figs. \[fig:cutoff\_lc\], \[fig:inc\_lc\], \[fig:los\_lc\_1month\], \[fig:inc\_lc\_wr\], \[fig:los\_lc\_wr\], \[fig:all\_col\] and \[fig:specs\] this effect does not need to be considered. The surface density, $\sigma_{s}$, of the postshock gas along the CD, when both winds have $\chi \ltsimm 1$, has been computed by @Girard:1987, @Canto:1996, and @Kenny:2005. In each of these works, turbulence in the postshock flow is assumed to fully mix the material from both winds and the surface density calculated is for shocked gas from both winds. Alternatively, if only one of the winds is radiative (i.e. the other remains largely adiabatic), or the postshock flow is assumed not to mix, then the surface density can be calculated from considering conservation of mass flux [e.g., @Antokhin:2004]. To calculate the surface densities in Figs \[fig:abs\_lc\] and \[fig:abs\_col\] we have used Eq.(30) of [@Canto:1996]. In our model, the ballistic part of the WCR is asymmetric due to orbital motion. Since the pre-shock flow is practically tangential to the CD at this point, we calculate the total surface density of the postshock winds (which in this subsection are assumed to cool) in this region by considering conservation of mass flux. The surface density of the postshock gas close to the apex of the WCR varies by over an order of magnitude between periastron and apastron when $e=0.9$. Since the width of the cool dense layer of gas alongside the CD is not infinitely thin, the degree of absorption through it depends on the angle subtended between the line-of-sight and the normal to its (i.e. the CD’s) surface, $\gamma$. The column density intersected by the line-of-sight is therefore $$\sigma'_{\rm{s}} = \frac{\sigma_{\rm{s}}}{\cos \gamma}, \label{eqn:mod_sigma}$$ where $\sigma_{s}$ is the actual surface density of the cooled layer. When the line-of-sight becomes closely tangential to the CD, $\sigma'_{\rm{s}}$ can become large, even if $\sigma_{\rm{s}}$ itself is not particularly large. The maximum value of $\sigma'_{\rm{s}}$ is constrained by the curvature of the WCR and the finite path length through the shocked gas. To determine the maximum path length requires knowledge of the width of the cooled layer and its radius of curvature at the point of interest on the CD. On the shock cap the density of the cooled postshock region, $\rho_{\rm{ps}}$, can be determined by equating the ram pressure of the preshock gas with the thermal pressure of the postshock gas [@Kashi:2007], $$\rho_{\rm{ps}} = \frac{m_{\rm{H}} \rho_1(r) (v_{\infty1} \sin\xi)^2}{k_{\rm{B}} T_{\rm{ps}}}, \label{eqn:rhops}$$ where $T_{\rm{ps}}$ is the temperature of the cooled postshock gas ($T_{\rm{ps}}$ is taken to be $10^4\;$K), $\rho(r)$ is the preshock gas density as a function of distance from the respective star, $\xi$ is the angle between the preshock velocity vector and the tangent to the shock surface, and $m_{\rm{H}}$ and $k_{\rm{B}}$ are the mass of a hydrogen nucleus and Boltzmann’s constant respectively. The thickness of the cooled layer, $l_{\rm{shock}}$, is then $$l_{\rm{shock}} = \frac{\sigma_{\rm{s}}}{\rho_{\rm{ps}}}.$$ The thickness of the cooled layer in the ballistic CD region cannot be calculated in this manner because the shocks are now fully oblique. Therefore, a linear extrapolation is used to determine the downstream thickness. The radius of curvature at a point on the shock cap is $$\Lambda = \left|\frac{ds}{d\hat{t}}\right|,$$ where $ds$ is the distance between two points on the WCR and $d\hat{t}$ is the difference in the unit vectors tangent to the WCR at those two points. Consideration of the maximum path length through the cool dense layer, $d_{\rm{max}}$, then gives the maximum value for $\frac{\sigma'_{\rm{s}}}{\sigma_{\rm{s}}}$ as $$\left|\frac{\sigma'_{\rm{s}}}{\sigma_{\rm{s}}}\right|_{\rm{max}} \simeq \frac{d_{\rm{max}}}{l_{\rm{shock}}} = \sqrt{4+\frac{8\Lambda}{l_{\rm{shock}}}}. \label{eqn:maxsigmamod}$$ The skewing of the shock cap due to orbital motion will evoke an asymmetry in the postshock gas density [@Lemaster:2007]. This is naturally accounted for in Eqs. \[eqn:rhops\] and \[eqn:maxsigmamod\]. ### The observed emission {#subsec:observedemission} [l]{}\ [l]{}\ In the hypothetical binary systems considered in this paper, the shocked gas in the WCR remains hot as it flows out of the system and thus contributes insignificantly to the absorption. Hence the total column density along a given line-of-sight is the sum of the column densities through the unshocked winds. The attenuation declines as the line-of-sight leaves the system, and is negligible at the distances which our model extends to (the distance the wind flows over two orbits). Absorption cross-sections for solar abundance gas at $10^4\;$K are used to obtain the optical depth, $\tau$, along specific lines-of-sight in 200 logarithmically spaced bins over the energy range 0.1-10.0 keV. The observed attenuated emission, $I_{\rm{obs}} = I_{0} e^{-\tau}$, where $I_{0}$ is the intrinsic emission. Fig. \[fig:abs\_lc\] demonstrates the effect of including the various attenuation mechanisms on the resultant emission. As previously mentioned, occultation causes little reduction in emission because of the minute size of the stars compared to the extended WCR (Fig. \[fig:occ\_shkcaps\]). For the assumed position of the observer, absorption by the unshocked winds increases as the stars approach each other and reaches a maximum at periastron. For illustrative purposes we also show the attenuation that occurs if the postshock gas cools and forms a thin dense layer along the CD (this does not occur in the systems considered since the shocked gas remains largely adiabatic as it flows out of the system). When the line-of-sight becomes closely tangential to the WCR the path length of X-rays through the shocked gas and the subsequent attenuation via this mechanism reaches a maximum. This can be seen in the small dips in the lightcurve at orbital phases 0.18 and 0.82, with corresponding peaks in the emission weighted column shown in Fig. \[fig:abs\_col\]. The emission weighted column density is calculated as $\Sigma \sigma_{\rm{tot}} I_{0} / \Sigma I_{0}$, where the summation is over all sightlines to emitting regions and $\sigma_{\rm{tot}}$ is the total column density (cm$^{-2}$) along each sightline. This weighting is more informative than the column densities presented in [@Lemaster:2007] which were only calculated along a single sight-line into the system. Resolution tests have determined that the minimum number of phase steps required for convergence of the attenuated X-ray lightcurves is dependant on the ratio $v_{\rm{orb}} / v_{\rm sl}$, with of order 1000 phase steps required for an orbit with $e=0.9$ and $v_{\rm{orb}} / v_{\rm sl} \approx 1$. Results {#sec:results} ------- ### The X-ray lightcurve {#subsec:lightcurve} In this section we compute X-ray lightcurves for the hypothetical systems considered. For the O+O systems we use emissivity and opacity data calculated assuming solar abundances (Figs. \[fig:emiss\_spec\] and  \[fig:abs\_spec\] respectively). For the WR wind we use data appropriate for WN8 abundances. [l]{}\ -- -- -- -- Fig. \[fig:cutoff\_lc\] shows the synthetic lightcurves produced for models where the transition between the shock cap and the ballistic CD is varied. There is a maximum divergence of $\sim 6\%$ between cases where the transition occurs at a velocity cut-off of 70 % and 90 % of the speed of the slower wind, which shows that the resulting lightcurves are not very sensitive to this assumption. Varying the orbital inclination angle changes the amount of attenuation that the intrinsic emission suffers on its way to the observer. However, there is little circumstellar attenuation for Model A (Fig. \[fig:inc\_lc\], left panel), and the synthetic lightcurves are almost identical over the entire orbital period. This is because the emitting volume is large (so occultation by the stars is negligible), and because the stellar separation is wide, so that the circumstellar density at the WCR is relatively low. Attenuation effects become more prominent if the orbital period is reduced to $1\;$month (Fig. \[fig:inc\_lc\], right panel), and distinct differences in the lightcurves occur around periastron. The $i=0^{\circ}$ lightcurve is smooth, and reflects the fact that the increase in the intrinsic emission due to the changing orbital separation ($L_{\rm x} \propto 1/d_{\rm sep}$) more than offsets the peak in attenuation through the primary wind at periastron. Increasing the inclination enhances the attenuation around periastron. The dip seen in both the $i=60^{\circ}$ and $i=90^{\circ}$ curves is offset from the time of periastron ($\phi = 1.0$) because of the skew to the WCR caused by orbital motion. As already mentioned in § \[subsec:occultation\], the stars fail to provide any significant eclipse of the emitting region. Fig. \[fig:los\_lc\_1month\] examines the dependence of the observed emission on the angle subtended between the line-of-sight and the semi-major axis. The $\theta = 90^{\circ}$ and $-90^{\circ}$ curves appear to be almost identical reflected copies around orbital phase $\phi \simeq 0.5$, with the differences around periastron being due to the aberration of the WCR. Absorption does not strongly affect the observed emission (even if the orbital period is reduced to 1 month) as the density contrast between the O-star winds is not very large. The largest difference between the model results ($\sim 25\%$) occurs at periastron. A comparison between current observational data and such models may allow constraints to be placed on the orientation of specific O+O-star systems. [l]{}\ The higher primary mass-loss rate in the WR+O system leads to a greater depenence of the observed emission on the line-of-sight (Figs. \[fig:inc\_lc\_wr\] and  \[fig:los\_lc\_wr\]), as well as higher X-ray luminosities. The minimums in the curves close to periastron in Fig. \[fig:inc\_lc\_wr\], especially in the case of the $i=90^{\circ}$ curve, are the result of the the X-rays passing through the dense WR wind. However, there is again little difference between the $i=60^{\circ}$ and $i=90^{\circ}$ curves at apastron as the WCR is viewed predominantly through the less dense O-star wind, though the attenuation at lower inclinations is slightly higher as the apex of the WCR is viewed through the denser wind from the WR star. Rotating the line-of-sight within the orbital plane again causes significant alterations to the observed emission (Fig. \[fig:los\_lc\_wr\]). The $\theta=0^{\circ}$ curve sees the largest degree of attenuation around periastron and the lowest around apastron, with the opposite being true for the $\theta=180^{\circ}$ curve. As was also the case in Fig. \[fig:los\_lc\_1month\], the $\theta = 90^{\circ}$ and $-90^{\circ}$ lightcurves appear to be almost perfect reflected copies of each other. The dip seen in the $\theta = 90^{\circ}$ curve at orbital phase $\phi \simeq 0.17$ marks opposition. Features such as these could be particularly useful for constraining the orientation of systems. [l]{}\ [l]{}\ Fig. \[fig:all\_col\] shows the variation with phase of the emission weighted column density for the 3 hypothetical systems considered. The column density is highest when viewed through the primary wind, and lowest when viewing through the secondary wind. It is lowest for Model A, and is approximately $5 \times$ higher when the period is reduced to $1\;$month (Model B). This simply reflects the $\simeq 5 \times$ smaller separation and the $\simeq 30 \times$ higher densities. The $\sim 10 \times$ changes in the column density between apastron and periastron in Model A curve reflects the $1.86 \times$ change in stellar separation and $5\times$ change in wind density (or stellar mass-loss rate) as the line-of-sight switches from the secondary wind into the primary wind. The different slopes of the column density either side of periastron are caused by the asymmetry of the WCR. The rise in column density at $\phi \sim 0.8$ begins when the shock cap rotates and lines-of-sight start to see the emission through the denser primary wind. The rise occurs at an earlier phase for the WR+O system because of the lower value of the wind momentum ratio and the narrowing of the opening angle of the WCR. At $\phi = 0.964$ and 0.055 the slope in the emission weighted column is reduced, and this feature marks the point where the bow shock arms are tangential with the line-of-sight. When the emission weighted column density is plotted alongside the average column density the change in slope occurs at a point where the two curves intersect. The average column density has a continual rise and a peak at periastron. This tells us that the column density to the entire emission region reaches a maximum at periastron, whereas attenuation to the points with highest intrinsic emission remains roughly constant for a short period. It is also interesting that both the O+O and WR+O systems with $P = 1\;$year (Model A and C respectively) have flatter profiles at maximum column density. This indicates that the shape of the column density curve is sensitive to the aberration and orbital induced curvature of the WCR, and thus to the orbital period. [l]{}\ ### X-ray spectra {#subsec:spectra} Fig. \[fig:specs\] shows synthetic spectra at periastron and apastron for the simulations performed. The slope of the spectra at high energies is the same for the O+O systems since the preshock velocities, and therefore postshock gas temperatures, do not change. In all cases spectra at apastron show lower flux in the 2.0-10.0 keV energy band, although flux below 1 keV is higher. This is because the intrinsic emission scales as $1 / d_{\rm{sep}}$, but the is weaker when viewed through the companion’s wind. The low energy turnover in the periastron spectrum extends to higher energies for the WR+O system due to the higher mass-loss rate and absorption of the WR wind. [l]{}\ \ \ Conclusions {#sec:conclusions} =========== We have presented a 3D dynamical model of the colliding winds in binary systems where both stars drive a significant wind. In circular systems, the WCR adopts a spiral shaped structure similar to those observed in massive binary star systems. In systems with eccentric orbits, the shape of the WCR becomes increasingly deformed as the eccentricity increases, with the winds increasingly being channeled into a specific direction. A major advantage of the model is its low computational cost and the fact that it can be easily adapted to model a wide variety of observational data (from the radio to $\gamma$-ray) and systems (from early type binaries, to $\gamma$-ray binaries with a pulsar wind, to symbiotic novae). As an example exercise, the X-ray emission from hypothetical O+O and WR+O-star systems was modelled. The intrinsic emission was computed from a 2D grid-based hydrodynamical model of the WCR, and then mapped onto the surface separating the winds in the 3D dynamical model. Absorption due to the unshocked stellar winds (and also cooled postshock material) can be considered, although in the hypothetical systems that were modelled the gas in the WCR remains largely adiabatic as it flows out of the system so that only the former is calculated. Ray-tracing through the 3D spiral structures then gives the attenuated emission, and synthetic spectra and lightcurves are produced. The lightcurves and spectra show that observational characteristics of the X-ray emission from early-type binaries can be reproduced. For instance, the model with a 1 year orbit (Model A) is representative of wide O+O binaries such as HD15558 [@DeBecker:2006a], and in this particular system could be useful in determining whether there are two or three counterparts. The results from the 1 month orbit simulation (Model B) are instead most applicable to X-ray observations of close O+O binaries such as $\iota$ Orionis [@Pittard:2000], CygOB2\#8A [@DeBecker:2006b], and HD 93403 [@Rauw:2000], to name but a few. The model can also tackle systems with different abundances for each wind such as WR+O-star systems. Our WR+O star model with a $1\;$ year orbital period (Model C) is applicable to systems like WR25, WR108, WR133, WR138 [@vanderHucht:2001], and among the WN stars and WR19, WR125, WR137, WR98a, WR104, and WR140 [@Pollock:2005; @Pittard:2006] among the WC stars. Mass-loss rate determinations can be made from comparison of the predicted magnitude of the X-ray flux with observations [@Stevens:1996; @Pittard:2002]. In principle it is possible to use the shape of the X-ray lightcurve to constrain the inclination and orientation of the system. Our results reveal that for wind momentum ratios of order 0.2, the lack of significant absorption means that this will be very difficult if applied to O+O-star systems with periods of order one year, but becomes possible for orbital periods of order one month. The variation in absorption is much more significant when the wind momentum ratio is lower and the density of the winds is more disparate. This is the case for WR+O, LBV+O, and LBV+WR systems. In future work we will apply the dynamical model to the X-ray and forbidden line emission from $\eta\;$Car, the X-ray lightcurve of WR140, and emission line profiles of colliding wind binaries. Acknowledgements {#acknowledgements .unnumbered} ---------------- We would like to thank Perry Williams for the 2D code which was the basis for the 3D model in this work. ERP thanks the University of Leeds for funding. JMP gratefully acknowledges funding from the Royal Society. [99]{} Aharonian et al., 2005, A&A, 442, 1 Aharonian et al., 2005, Science, 309, 746 Albert et al., 2006, Science, 312, 1771 Antokhin I. I., Owocki S. P., Brown J. C., 2004, ApJ, 611, 434 Bednarek W., 2006, MNRAS, 371, 1737 Bisikalo D. V., Boyarchuk A. A., Kilpio E. Yu., Tomov N. A., Tomova M. T., 2006, ARep, 50, 722 Bogovalov S. V., Khangulyan D., Koldoba A. V., Ustyugova G. V., Aharonian F. A., 2008, arXiv:0710.1961v1 Canto J., Raga A. C., Wilkin F. P.,1996, ApJ, 469, 729 Contreras M. E., Montes G., Wilkin F. P., 2004, RMxAA, 40, 53 De Becker M., Rauw G., Manfroid J., Eenens P., 2006, A&A, 456, 1121 De Becker et al., 2006, MNRAS, 371, 1280 Dougherty S. M., Williams P. M., Pollacco D. L., 2000, MNRAS, 316, 143 Dougherty S. M., Beasley A. J., Claussen M. J., Zauderer B. A., Bolingbroke N. J., 2005, ApJ, 623, 477 Dubus G., Cerutti B., Henri G., A&A, 2008, 477, 691 Eichler D., Usov V., 1993, ApJ, 402, 271 Falle S. A. E. G., Komissarov S. S., 1996, MNRAS, 278, 586 Falle S. A. E. G., Komissarov S. S., Joarder P., 1998, MNRAS, 297, 265 Foellmi et al., 2008, RMxAA, 44, 3 Gawryszczak A. J., Mikolajewska J., Rozyczka M., 2002, A&A, 385, 205 Gayley K. G., Owocki S. P., Cranmer S. R., 1997, ApJ, 475, 786 Girard T., Willson L. A., 1987, A&A, 183, 247 Harries T. J., Monnier J. D., Symington N. H., Kurosawa R., 2004, MNRAS, 350, 565 He J. H., 2007, A&A, 467, 1081 Henley D. B., Corcoran M. F., Pittard J. M., Stevens I. R., Hamaguchi K., Gull T. R., 2008, arXiv:0801.4779v2 Kashi A., Soker N., 2007, MNRAS, 378, 1609 Kenny H. T., Taylor A. R., 2005, ApJ, 619, 527 Kenny H. T., Taylor A. R., 2007, ApJ, 662, 1231 Khangulyan D., Hnatic S., Aharonian F., Bogovalov S., 2007, MNRAS, 380, 320 Khangulyan D., Aharonian F., Bosch-Ramon V., 2008, MNRAS, 383, 467 Johnston S., Ball L., Wang N., Manchester R. N., 2005, MNRAS, 358, 1069 Lamers H. J. G. L. M., Cassinelli J. P., 1999, “Introduction to Stellar Winds”, CUP Leidahl D. A., Osterheld A. L., Goldstein W. H., 1995, ApJ, 438, L115 Lemaster M. N., Stone J. M., Gardiner T. A., 2007, ApJ, 662, 582 Marchenko S. V., Moffat A. F. J., Vacca W. D., Côté S., Doyon R., 2002, ApJ, 565, L59 Mastrodemos N., Morris M., 1999, ApJ, 523, 357 Mauron N., Huggins P. J., 2006, A&A, 452, 257 Mitsumoto M., et al., 2005, ARep, 49, 884 Monnier J. D., Tuthill P. G., Danchi W. C., 1999, ApJ, 525, L97 Okazaki A. T., Owocki S. P., Russell C. M. P., Corcoran M. F., 2008, arXiv:0803.3977v1 Pittard J. M., 1998, MNRAS, 300, 479 Pittard J. M., 1999, in “Wolf-Rayet Phenomena in Massive Stars and Starburst Galaxies”, eds. K. A. van der Hucht, G. Koenigsberger, P. R. J. Eenens, IAU Symp. No. 193, 386 Pittard J. M., 2007, ApJ, 660, 141P Pittard J. M., Corcoran M. F., 2002, A&A, 383, 636 Pittard J. M., Dougherty S. M., 2006, MNRAS, 372, 801 Pittard J. M., Stevens I. R., 1997, MNRAS, 292, 298 Pittard J. M., Stevens I. R., Corcoran M. F., Gayley K. G., Marchenko S. V., Rauw G., 2000, MNRAS, 319, 137 Pittard J. M., Stevens I. R., Corcoran M. F., Ishibashi K., 1998, MNRAS, 299, L5 Pollock et al., 2005, ApJ, 629, 482 Reimer A., Pohl M., Reimer O., 2006, ApJ, 644, 1118 Rauw G., Sana H., Gosset E., Vreux J. -M., Jehin E., Parmentier G., 2000, A&A, 360, 1003 Romero G. E., Okazaki A. T., Orellana M., Owocki S. P., 2007, A&A, 474, 15 Stevens I. R., 1996, RMxAC, 5, 61 Stevens I. R., Blondin J. M., Pollock A. M. T., 1992, ApJ, 386, 265 Tuthill P. G., Monnier J. D., Danchi W. C., 1999, Nature, 398, 487 Tuthill P. G., Monnier J. D., Tanner A., Figer D., Ghez A., Danchi W., 2006, Science, 316, 247 Tuthill P. G., Monnier J. D. Lawrance N., Danchi W. C. Owocki S. P., Gayley K. G., 2008, ApJ, 675, 698 van der Hucht K. A., 2001, NewAR, 45, 135 Walder R., Folini D., 2000, in “Thermal and Ionization Aspects of Flows from Hot Stars: Observations and Theory”, eds. H. J. G. L. M. Lamers, A. Sapar, ASP Conf. Ser., 204, 331 Walder R., Folini D., 2002, in “A Massive Star Odyssey, from Main Sequence to Supernova”, eds. K. A. van der Hucht, A. Herrero, C. Esteban, IAU Symp. No. 212, 139 Williams P. M., Dougherty S. M., Davis R. J., van der Hucht K. A., Bode M. F., Setia Gunawan D. Y. A., 1997, MNRAS, 289, 10 Zhekov S. A., 2007, MNRAS, 382, 886 \[lastpage\] [^1]: E-mail: phy1erp@leeds.ac.uk [^2]: Such systems are to be distinguished from the interacting wind phenomenon which occurs in AGB binaries where density structures in the AGB wind are created either due to the reflex-action of the evolved star around the centre-of-mass of the system [@Mastrodemos:1999; @Mauron:2006; @He:2007], or due to gravitational focusing [@Gawryszczak:2002]. Here we focus exclusively on systems involving the interaction of winds from separate stars.

--- author: - 'Hidekatsu <span style="font-variant:small-caps;">Nemura</span>,$^{1,2}$ Sinya <span style="font-variant:small-caps;">Aoki</span>,$^{1,2,3}$ Takumi <span style="font-variant:small-caps;">Doi</span>,$^2$ Shinya <span style="font-variant:small-caps;">Gongyo</span>,$^{2,3}$ Tetsuo <span style="font-variant:small-caps;">Hatsuda</span>,$^{2,4}$ Yoichi <span style="font-variant:small-caps;">Ikeda</span>,$^2$ Takashi <span style="font-variant:small-caps;">Inoue</span>,$^{2,5}$ Takumi <span style="font-variant:small-caps;">Iritani</span>,$^{2,6}$ Noriyoshi <span style="font-variant:small-caps;">Ishii</span>,$^{2,7}$ Takaya <span style="font-variant:small-caps;">Miyamoto</span>,$^{2,3}$ Keiko <span style="font-variant:small-caps;">Murano</span>,$^{2,7}$ and Kenji <span style="font-variant:small-caps;">Sasaki</span>$^{1,2}$' title: A Fast Algorithm for Lattice Hyperonic Potentials ---  UTCCS-P-89, RIKEN-QHP-230, YITP-16-67 Introduction ============ Nuclear force and hyperonic nuclear forces provide a useful starting point to figure out how the hypernuclear systems are bound, in which hyperons (or strange quarks) are embedded in normal nuclear systems as “impurities.”[@Hashimoto:2006aw] From studies of few-body systems for $s$-shell $\Lambda$ hypernuclei, for example, it has been pointed out that the coupled-channel $\Lambda N-\Sigma N$ interaction plays a vital role to make a hypernucleus being bounded[@Nemura:2002fu] although phenomenological hyperon-nucleon potentials are not well constrained from experimental data. In the past several years, a new approach to study the hadronic forces from the lattice QCD has been proposed[@Ishii:2006ec; @Aoki:2009ji]. In this approach, the interhadron potential is obtained by means of the lattice QCD measurement of the Nambu-Bethe-Salpeter (NBS) wave function. The observables such as the phase shifts and the binding energies are calculated through the resultant potential[@Aoki:2012tk]. This approach has been further extended and applied to various problems. See Ref.[@Sasaki:2015ifa] and references therein for the state-of-the-art outcomes. In addition, a large scale lattice QCD calculation is now in progress[@DoiIshiiSasaki2015HYP] to study the baryon interactions from $NN$ to $\Xi\Xi$ by measuring the NBS wave functions for 52 channels from the $2+1$ flavor lattice QCD, which is founded on inconspicuous but vital work[@Nemura:2015yha] that reveals a beneficial algorithm for computing a large number of NBS wave functions simultaneously; see Eqs. (34)-(38) in Ref. [@Nemura:2015yha] for the specific channels of the above 52 NBS wave functions. The purpose of this report is to present a key aspect of the algorithm that performs efficiently a concurrent computation of such a lot of NBS wave functions for various baryon channels. As a preliminary snapshot of the ongoing work, effective $\Lambda N$ potential at almost physical quark masses corresponding to ($m_{\pi}$,$m_{K}$)$\approx$(146,525)MeV is also presented, which is obtained from the single correlator $\langle p\Lambda\overline{p\Lambda}\rangle$ by adopting a recipe in which the effects from the $\Sigma N$ channel are effectively included. Effective Baryon Block Algorithm ================================ In the HAL QCD method, the interaction is obtained through the four-point correlator defined by $$\begin{array}{c} { F}_{\alpha_{1}\alpha_{2},\alpha_{3}\alpha_{4}} ^{\langle B_1B_2\overline{B_3B_4}\rangle}(\vec{r},t-t_0) = % \sum_{\vec{X}} \left\langle 0 \left| % B_{1,{\alpha_1}}(\vec{X}+\vec{r},t) B_{2,{\alpha_2}}(\vec{X},t) % \overline{{\cal J}_{B_{3,\alpha_{3}} B_{4,\alpha_{4}}} (t_0)} % \right| 0 \right\rangle, \end{array}$$ where the summation over $\vec{X}$ selects states with zero total momentum. The $B_{1,\alpha_1}(x)$ and $B_{2,\alpha_2}(y)$ denote the interpolating fields of the baryons such as $$\!\!\! \begin{array}{llll} p \! = \! \varepsilon_{abc} \left( u_a C\gamma_5 d_b \right) u_c,\! % & n \! = \! - \varepsilon_{abc} \left( u_a C\gamma_5 d_b \right) d_c,\! & \Sigma^{+} \! = \! - \varepsilon_{abc} \left( u_a C\gamma_5 s_b \right) u_c,\! % & \Sigma^{-} \! = \! - \varepsilon_{abc} \left( d_a C\gamma_5 s_b \right) d_c,\! \\ \Sigma^{0} \! = \! {1\over\sqrt{2}} \left( X_u \! - \! X_d \right),\! % & \Lambda \! = \! {1\over \sqrt{6}} \left( X_u \! + \! X_d \! - \! 2 X_s \right),\! & \Xi^{0} \! = \! \varepsilon_{abc} \left( u_a C\gamma_5 s_b \right) s_{c},\! % & \Xi^{-} \! = \! - \varepsilon_{abc} \left( d_a C\gamma_5 s_b \right) s_{c},\! \\ \mbox{where} & X_u = \varepsilon_{abc} \left( d_a C\gamma_5 s_b \right) u_c, & X_d = \varepsilon_{abc} \left( s_a C\gamma_5 u_b \right) d_c, & X_s = \varepsilon_{abc} \left( u_a C\gamma_5 d_b \right) s_c. \end{array} \label{BaryonOperatorsOctet}$$ For simplicity, we have suppressed the explicit spinor indices and spatial coordinates in Eq. (\[BaryonOperatorsOctet\]). $\overline{{\cal J}_{B_{3,\alpha_{3}}B_{4,\alpha_{4}}} (t_0)}$ is the source operator which creates $B_{3}B_{4}$ states. Hereafter, the explicit time dependence is suppressed for simplicity. The 4pt correlator is evaluated through considering the Wick’s contraction together with defining the baryon blocks $[B_{1,\alpha_{1}}^{(0)}](\vec{x};~\bm{\xi}_{P_{1}P_{2}P_{3}}^\prime)$ and $[B_{2,\alpha_{2}}^{(0)}](\vec{y};~\bm{\xi}_{P_{4}P_{5}P_{6}}^\prime)$, $$\begin{aligned} { F} ^{\langle B_1B_2\overline{B_3B_4}\rangle} _{{\alpha_1}{\alpha_2},{\alpha_3}{\alpha_4}}(\vec{r}% ) % &=& \sum_{\vec{X}} \sum_{P} \sigma_{P} ~ [B_{1,\alpha_1}^{(0)}](\vec{X}+\vec{r}; ~ \xi^{\prime}_{P_{1}}, \xi^{\prime}_{P_{2}}, \xi^{\prime}_{P_{3}} ) ~ [B_{2,\alpha_2}^{(0)}](\vec{X}; ~ \xi^{\prime}_{P_{4}}, \xi^{\prime}_{P_{5}}, \xi^{\prime}_{P_{6}} ) \nonumber \\ && \qquad \times \varepsilon_{c_{{1}}^{\prime} c_{{2}}^{\prime} c_{{3}}^{\prime}} \varepsilon_{c_{{4}}^{\prime} c_{{5}}^{\prime} c_{{6}}^{\prime}} (C\gamma_5)_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}} (C\gamma_5)_{\alpha_{4}^{\prime}\alpha_{5}^{\prime}} \delta_{\alpha_{3}^{\prime}\alpha_{3}} \delta_{\alpha_{6}^{\prime}\alpha_{4}}, \label{NaiveBaryonBlocks} %\end{aligned}$$ $$\begin{array}{ll} \!\!\!\mbox{with} & ~[B_{1,\alpha_{1}}^{(0)}](\vec{x};~ \bm{\xi}_{P_{1}P_{2}P_{3}}^\prime) = [B_{1,\alpha_{1}}^{(0)}](\vec{x};~ \xi_{P_{1}}^{\prime}, \xi_{P_{2}}^{\prime}, \xi_{P_{3}}^{\prime} ) = \left\langle % B_{1,\alpha_{1}}(\vec{x}) % ~ \bar{q}_{B_{1},3}^{\prime}(\xi_{P_{3}}^{\prime}) \bar{q}_{B_{1},2}^{\prime}(\xi_{P_{2}}^{\prime}) \bar{q}_{B_{1},1}^{\prime}(\xi_{P_{1}}^{\prime}) % \right\rangle, \quad \mbox{and} % \\ % & ~ [B_{2,\alpha_{2}}^{(0)}](\vec{y};~ \bm{\xi}_{P_{4}P_{5}P_{6}}^\prime) = [B_{2,\alpha_{2}}^{(0)}](\vec{y};~ \xi_{P_{4}}^{\prime}, \xi_{P_{5}}^{\prime}, \xi_{P_{6}}^{\prime} ) = \left\langle % B_{2,\alpha_{2}}(\vec{y}) % ~ \bar{q}_{B_{2},6}^{\prime}(\xi_{P_{6}}^{\prime}) \bar{q}_{B_{2},5}^{\prime}(\xi_{P_{5}}^{\prime}) \bar{q}_{B_{2},4}^{\prime}(\xi_{P_{4}}^{\prime}) % \right\rangle, % \end{array}$$ where $\sigma_{P}$ and $\{\xi_{P_{1}}^{\prime},\cdots,\xi_{P_{6}}^{\prime}\}$ are the sign factor and the set of permutated spin-color-space-time coordinates for each permutation $P$ due to the Wick’s contraction, respectively. Both 3-tuple sets of the quark fields $\{\bar{q}_{B_{1},1}^{\prime},\bar{q}_{B_{1},2}^{\prime}, \bar{q}_{B_{1},3}^{\prime}\}$ and $\{\bar{q}_{B_{2},4}^{\prime},\bar{q}_{B_{2},5}^{\prime}, \bar{q}_{B_{2},6}^{\prime}\}$ are ordered so as to create the $B_{1}$ and $B_{2}$ states properly. Taking the expression in Eq. (\[NaiveBaryonBlocks\]), the number of the iterations to obtain a $ { F} ^{\langle B_1B_2\overline{B_3B_4}\rangle} _{{\alpha_1}{\alpha_2},{\alpha_3}{\alpha_4}}(\vec{r}% )$ except the spatial degrees of freedom reduces to $(N_{c}!N_{\alpha})^{B}\times N_{u}!N_{d}!N_{s}!\times 2^{N_{\Lambda}+N_{\Sigma^{0}}-B}$ from the number of iterations in naive counting that is $(N_{c}!N_{\alpha})^{2B}\times N_{u}!N_{d}!N_{s}!$, where $N_{c}=3, N_{\alpha}=4$ and $N_{\Lambda}$, $N_{\Sigma^{0}}$, $N_{u},N_{d},N_{s}$ and $B$ are the numbers of $\Lambda,\Sigma^{0}$, up-quark, down-quark, strange-quark and the baryons (i.e., always $B=2$ in the present study), respectively. Through the employment of Fast-Fourier-Transform (FFT)[@Ishii:2009zr; @Nemura:2009kc] we attain the expression in terms of the effective baryon blocks[@Nemura:2015yha] $${ F} ^{\langle B_1B_2\overline{B_3B_4}\rangle} _{{\alpha_1}{\alpha_2},{\alpha_3}{\alpha_4}}\!(\vec{r}% ) \! =\!\!\! \sum_{P} \! \sigma_{P} \!\! \sum_{\vec{X}} \! \left( [\!B_{1,\alpha_1}^{(P)}\!](\vec{X}\!\!+\!\!\vec{r}) \! \times \! [B_{2,\alpha_2}^{(P)}](\vec{X}) \right)_{\alpha_3\alpha_4% } \nonumber % \!\! =\!\! {1\over L^3} \!\! \sum_{\vec{q}} \left( \! \sum_{P} \! \sigma_{P} \! \left( [\widetilde{B_{1,\alpha_1}^{(P)}}]( \vec{q}) \! \times \! [\widetilde{B_{2,\alpha_2}^{(P)}}](-\vec{q}) \! \right)_{\alpha_3\alpha_4} \! \right) \! {\rm e}^{i\vec{q}\cdot\vec{r}}. \label{B1B2.B1B2B4B3.FFT}$$ The effective baryon blocks, $ \left( [\widetilde{B_{1,\alpha_1}^{(P)}}]( \vec{q}) \times [\widetilde{B_{2,\alpha_2}^{(P)}}](-\vec{q}) \right)_{\alpha_{3}\alpha_{4}}, $ are obtained by multiplying tensorial factors with the normal baryon blocks; for example, the specific form of the 4pt correlator ${F}^{\langle p \Lambda\overline{pX_{u}}\rangle} _{{\alpha_1}{\alpha_2},{\alpha_{3}}{\alpha_{4}}}(\vec{r})$ of the $\langle p \Lambda\overline{pX_{u}}\rangle$ channel is given by [@Nemura:2015yha], $$\begin{aligned} {F}^{\langle p \Lambda\overline{pX_{u}}\rangle} _{{\alpha_1}{\alpha_2},{\alpha_3}{\alpha_4}}(\vec{r}) % &=& {1\over L^3} \sum_{\vec{q}} \left( %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [\widetilde{ p}_{\alpha_1\alpha_3}^{(1)}]( \vec{q}) [\widetilde{\Lambda}_{\alpha_2\alpha_4}^{(1)}](-\vec{q}) - [\widetilde{ p}_{\alpha_1\alpha_4}^{(2)}]_{c_{3}^{\prime},c_{6}^{\prime}}( \vec{q}) [\widetilde{\Lambda}_{\alpha_2\alpha_3}^{(2)}]_{c_{3}^{\prime},c_{6}^{\prime}}(-\vec{q}) %%%%%%%%%%%%%%%%%% \right. \nonumber \\ && % \left. %%%%%%%%%%%%%%%%%% - [\widetilde{ p}_{\alpha_1\alpha_3}^{(3)}]_{c_{2}^{\prime},\alpha_{2}^{\prime},c_{4}^{\prime},\alpha_{4}^{\prime}}( \vec{q}) [\widetilde{\Lambda}_{\alpha_2\alpha_4}^{(3)}]_{c_{2}^{\prime},\alpha_{2}^{\prime},c_{4}^{\prime},\alpha_{4}^{\prime}}(-\vec{q}) %%%%%%%%%%%%%%%%%% \right. % \left. %%%%%%%%%%%%%%%%%% + [\widetilde{ p}_{\alpha_1\alpha_4}^{(4)}]_{c_{1}^{\prime},\alpha_{1}^{\prime},c_{5}^{\prime},\alpha_{5}^{\prime}}( \vec{q}) [\widetilde{\Lambda}_{\alpha_2\alpha_3}^{(4)}]_{c_{1}^{\prime},\alpha_{1}^{\prime},c_{5}^{\prime},\alpha_{5}^{\prime}}(-\vec{q}) %%%%%%%%%%%%%%%%%% \right. \nonumber \\ && % \left. %%%%%%%%%%%%%%%%%% + [\widetilde{ p}_{\alpha_1\alpha_3\alpha_4}^{(5)}]_{c_{1}^{\prime},\alpha_{1}^{\prime},c_{6}^{\prime}}( \vec{q}) [\widetilde{\Lambda}_{\alpha_2 }^{(5)}]_{c_{1}^{\prime},\alpha_{1}^{\prime},c_{6}^{\prime}}(-\vec{q}) %%%%%%%%%%%%%%%%%% \right. % \left. %%%%%%%%%%%%%%%%%% - [\widetilde{ p}_{\alpha_1\alpha_3\alpha_4}^{(6)}]_{c_{3}^{\prime},c_{5}^{\prime},\alpha_{5}^{\prime}}( \vec{q}) [\widetilde{\Lambda}_{\alpha_2 }^{(6)}]_{c_{3}^{\prime},c_{5}^{\prime},\alpha_{5}^{\prime}}(-\vec{q}) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \right) {\rm e}^{i\vec{q}\cdot\vec{r}}, \label{pL.pLXup.FFT}\end{aligned}$$ where $$\begin{aligned} && \begin{array}{l} ~[\widetilde{ p}_{\alpha_1\alpha_3}^{(1)}]( \vec{q}) = [\widetilde{ p}_{\alpha_{1}}^{(0)}](\vec{q};~\bm{\xi}_{{1}{2}{3}}^\prime) \varepsilon_{c_{{1}}^{\prime} c_{{2}}^{\prime} c_{{3}}^{\prime}} (C\gamma_5)_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}} \delta_{\alpha_{3}^{\prime}\alpha_{3}}, \\ ~[\widetilde{\Lambda}_{\alpha_2\alpha_4}^{(1)}](-\vec{q}) = [\widetilde{\Lambda}_{\alpha_{2}}^{(0)}](-\vec{q};\bm{\xi}_{{4}{5}{6}}^\prime) \varepsilon_{c_{{4}}^{\prime} c_{{5}}^{\prime} c_{{6}}^{\prime}} (C\gamma_5)_{\alpha_{4}^{\prime}\alpha_{5}^{\prime}} \delta_{\alpha_{6}^{\prime}\alpha_{4}}, \end{array} \\ %%%%%%%%%%%%%%%%%% && \begin{array}{l} ~[\widetilde{ p}_{\alpha_1\alpha_4}^{(2)}]_{c_{3}^{\prime}c_{6}^{\prime}}( \vec{q}) = [\widetilde{ p}_{\alpha_{1}}^{(0)}](\vec{q};~\bm{\xi}_{{1}{2}{6}}^\prime) \varepsilon_{c_{{1}}^{\prime} c_{{2}}^{\prime} c_{{3}}^{\prime}} (C\gamma_5)_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}} \delta_{\alpha_{6}^{\prime}\alpha_{4}}, \\ ~[\widetilde{\Lambda}_{\alpha_2\alpha_3}^{(2)}]_{c_{3}^{\prime}c_{6}^{\prime}}(-\vec{q}) = [\widetilde{\Lambda}_{\alpha_{2}}^{(0)}](-\vec{q};\bm{\xi}_{{4}{5}{3}}^\prime) \varepsilon_{c_{{4}}^{\prime} c_{{5}}^{\prime} c_{{6}}^{\prime}} (C\gamma_5)_{\alpha_{4}^{\prime}\alpha_{5}^{\prime}} \delta_{\alpha_{3}^{\prime}\alpha_{3}}, \end{array} \\ %%%%%%%%%%%%%%%%%% && \begin{array}{l} ~[\widetilde{ p}_{\alpha_1\alpha_3}^{(3)}]_{c_{2}^{\prime}\alpha_{2}^{\prime}c_{4}^{\prime}\alpha_{4}^{\prime}}( \vec{q}) = [\widetilde{ p}_{\alpha_{1}}^{(0)}](\vec{q};~\bm{\xi}_{{1}{4}{3}}^\prime) \varepsilon_{c_{{1}}^{\prime} c_{{2}}^{\prime} c_{{3}}^{\prime}} (C\gamma_5)_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}} \delta_{\alpha_{3}^{\prime}\alpha_{3}}, \\ ~[\widetilde{\Lambda}_{\alpha_2\alpha_4}^{(3)}]_{c_{2}^{\prime}\alpha_{2}^{\prime}c_{4}^{\prime}\alpha_{4}^{\prime}}(-\vec{q}) = [\widetilde{\Lambda}_{\alpha_{2}}^{(0)}](-\vec{q};\bm{\xi}_{{2}{5}{6}}^\prime) \varepsilon_{c_{{4}}^{\prime} c_{{5}}^{\prime} c_{{6}}^{\prime}} (C\gamma_5)_{\alpha_{4}^{\prime}\alpha_{5}^{\prime}} \delta_{\alpha_{6}^{\prime}\alpha_{4}}, \end{array} \\ %%%%%%%%%%%%%%%%%% && \begin{array}{l} ~[\widetilde{ p}_{\alpha_1\alpha_4}^{(4)}]_{c_{1}^{\prime}\alpha_{1}^{\prime}c_{5}^{\prime}\alpha_{5}^{\prime}}( \vec{q}) = [\widetilde{ p}_{\alpha_{1}}^{(0)}](\vec{q};~\bm{\xi}_{{1}{4}{6}}^\prime) \varepsilon_{c_{{4}}^{\prime} c_{{5}}^{\prime} c_{{6}}^{\prime}} (C\gamma_5)_{\alpha_{4}^{\prime}\alpha_{5}^{\prime}} \delta_{\alpha_{6}^{\prime}\alpha_{4}}, \\ ~[\widetilde{\Lambda}_{\alpha_2\alpha_3}^{(4)}]_{c_{1}^{\prime}\alpha_{1}^{\prime}c_{5}^{\prime}\alpha_{5}^{\prime}}(-\vec{q}) = [\widetilde{\Lambda}_{\alpha_{2}}^{(0)}](-\vec{q};\bm{\xi}_{{2}{5}{3}}^\prime) \varepsilon_{c_{{1}}^{\prime} c_{{2}}^{\prime} c_{{3}}^{\prime}} (C\gamma_5)_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}} \delta_{\alpha_{3}^{\prime}\alpha_{3}}, \end{array} \\ %%%%%%%%%%%%%%%%%% && \begin{array}{l} ~[\widetilde{ p}_{\alpha_1\alpha_3\alpha_4}^{(5)}]_{c_{1}^{\prime}\alpha_{1}^{\prime}c_{6}^{\prime}}( \vec{q}) = [\widetilde{ p}_{\alpha_{1}}^{(0)}](\vec{q};~\bm{\xi}_{{3}{2}{6}}^\prime) \varepsilon_{c_{{1}}^{\prime} c_{{2}}^{\prime} c_{{3}}^{\prime}} (C\gamma_5)_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}} \delta_{\alpha_{3}^{\prime}\alpha_{3}} \delta_{\alpha_{6}^{\prime}\alpha_{4}}, \\ ~[\widetilde{\Lambda}_{\alpha_2 }^{(5)}]_{c_{1}^{\prime}\alpha_{1}^{\prime}c_{6}^{\prime}}(-\vec{q}) = [\widetilde{\Lambda}_{\alpha_{2}}^{(0)}](-\vec{q};\bm{\xi}_{{4}{5}{1}}^\prime) \varepsilon_{c_{{4}}^{\prime} c_{{5}}^{\prime} c_{{6}}^{\prime}} (C\gamma_5)_{\alpha_{4}^{\prime}\alpha_{5}^{\prime}}, \end{array} \\ %%%%%%%%%%%%%%%%%% && \begin{array}{l} ~[\widetilde{ p}_{\alpha_1\alpha_3\alpha_4}^{(6)}]_{c_{3}^{\prime}c_{5}^{\prime}\alpha_{5}^{\prime}}( \vec{q}) = [\widetilde{ p}_{\alpha_{1}}^{(0)}](\vec{q};~\bm{\xi}_{{3}{4}{6}}^\prime) \varepsilon_{c_{{4}}^{\prime} c_{{5}}^{\prime} c_{{6}}^{\prime}} (C\gamma_5)_{\alpha_{4}^{\prime}\alpha_{5}^{\prime}} \delta_{\alpha_{3}^{\prime}\alpha_{3}} \delta_{\alpha_{6}^{\prime}\alpha_{4}}, \\ ~[\widetilde{\Lambda}_{\alpha_2 }^{(6)}]_{c_{3}^{\prime}c_{5}^{\prime}\alpha_{5}^{\prime}}(-\vec{q}) = [\widetilde{\Lambda}_{\alpha_{2}}^{(0)}](-\vec{q};\bm{\xi}_{{2}{5}{1}}^\prime) \varepsilon_{c_{{1}}^{\prime} c_{{2}}^{\prime} c_{{3}}^{\prime}} (C\gamma_5)_{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}, \end{array} \label{EffectiveBaryonBlocksProtonLambdaStructuredForm}\end{aligned}$$ $$\begin{aligned} ~[p_{\alpha_{1}}^{(0)}](\vec{x};~\bm{\xi}_{{1}{2}{3}}^\prime) % &=& \varepsilon_{b_{{1}} b_{{2}} b_{{3}}} (C\gamma_5)_{\beta_{1}\beta_{2}} \delta_{\beta_{3}\alpha_{1}} \det\left| \begin{array}{cc} \langle u(\zeta_{1}) \bar{u}(\xi_{1}^{\prime}) \rangle & \langle u(\zeta_{1}) \bar{u}(\xi_{3}^{\prime}) \rangle \\ \langle u(\zeta_{3}) \bar{u}(\xi_{1}^{\prime}) \rangle & \langle u(\zeta_{3}) \bar{u}(\xi_{3}^{\prime}) \rangle \end{array} \right| \langle d(\zeta_{2}) \bar{d}(\xi_{2}^{\prime}) \rangle, % % \\ % ~ [\Lambda_{\alpha_{2}}^{(0)}](\vec{y};~\bm{\xi}_{{4}{5}{6}}^\prime) % &=& {1\over \sqrt{6}} \varepsilon_{b_{4} b_{5} b_{6}} \left\{ (C\gamma_5)_{\beta_{4}\beta_{5}} \delta_{\beta_{6}\alpha_{2}} + (C\gamma_5)_{\beta_{5}\beta_{6}} \delta_{\beta_{4}\alpha_{2}} -2 (C\gamma_5)_{\beta_{6}\beta_{4}} \delta_{\beta_{5}\alpha_{2}} \right\} \nonumber \\ && \qquad \times {\langle u(\zeta_{6}) \bar{u}(\xi_{6}^{\prime}) \rangle} \langle d(\zeta_{4}) \bar{d}(\xi_{4}^{\prime}) \rangle \langle s(\zeta_{5}) \bar{s}(\xi_{5}^{\prime}) \rangle.\end{aligned}$$ NB A roll of coordinates, $\{\bm{\xi}_{{1}{2}{3}}^\prime, \bm{\xi}_{{4}{5}{6}}^\prime \}$, is now called while Ref. [@Nemura:2015yha] has used $\{\bm{\xi}_{{1}{4}{2}}^\prime, \bm{\xi}_{{5}{6}{3}}^\prime \}$; this is advantageous for generalizing the algorithm toward various (e.g., 52) baryon channels. By employing the effective block algorithm, the number of iterations to evaluate the r.h.s. of Eq. (\[pL.pLXup.FFT\]) except the momentum space degrees of freedom becomes $1+N_{c}^{2}+N_{c}^{2}N_{\alpha}^{2}+N_{c}^{2}N_{\alpha}^{2}+ N_{c}^{2}N_{\alpha}+N_{c}^{2}N_{\alpha}=370$, which is remarkably smaller than the number $(N_{c}!N_{\alpha})^{B}\times N_{u}!N_{d}!N_{s}!\times 2^{N_{\Lambda}+N_{\Sigma^{0}}-B} = 3456$ that is seen in Eq. (\[NaiveBaryonBlocks\]). The manipulation for the expression of Eq. (\[B1B2.B1B2B4B3.FFT\]) can be automatically done once the set of the interpolating fields (i.e., the quantum numbers) of both sink and source parts is given. Results ======= A large number of baryon-baryon potentials including the channels from $NN$ to $\Xi\Xi$ are studied through the nearly physical point lattice QCD calculation by means of HAL QCD method with $N_{f}=2+1$ dynamical clover fermion gauge configurations generated on a $L^4=96^4$ lattice using K computer, where the actual computing jobs are launched with the unified contraction algorithm (UCA)[@Doi:2012xd]; see also Ref.[@Nemura:2015yha] for the thoroughgoing consistency check in the numerical outputs and comparison at various occasions between the UCA and the present algorithm. Preliminary studies indicate that the physical volume is $(aL)^4\approx$(8.2fm)$^4$ with the lattice spacing $a\approx 0.085$fm and $(m_{\pi},m_{K})\approx(146,525)$MeV. See Ref.[@Ishikawa:2015rho] for details on the generation of the gauge configuration. Figure \[Fig\_LNpots\] shows the preliminary snapshots for the $\Lambda N$ potential which implicitly includes effects from the $\Sigma N$. The statistics in the figure is only a fraction about $1/10$ of the entire schedule in FY2015. The left (center) panel shows the central potential in $^1S_0$ ($^3S_1$-$^3D_1$) channel. Short-range repulsion and medium-to-long-range attraction are found in the range $t-t_{0}=7-9$ for both central potentials. The right panel shows the tensor potential in the $^3S_1$-$^3D_1$ channel. The results in the range $t-t_{0}=7-9$ show weak tensor force in the $\Lambda N-\Lambda N$ diagonal channel. More comprehensive study which explicitly considers the effects from the $\Sigma N$ with increasing the statistics will be presented near future. ![Left: $\Lambda N$ central potential in the $^1S_0$ channel calculated with nearly physical point lattice QCD calculation on a volume $(96a)^4\approx$(8.2fm)$^4$ with the lattice spacing $a\approx 0.085$fm and $(m_{\pi},m_{K})\approx(146,525)$MeV. Center: $\Lambda N$ central potential in the $^3S_1$-$^3D_1$ channel. Right: $\Lambda N$ tensor potential in the $^3S_1$-$^3D_1$ channel. \[Fig\_LNpots\]](fig1left.eps){width="99.00000%"}   ![Left: $\Lambda N$ central potential in the $^1S_0$ channel calculated with nearly physical point lattice QCD calculation on a volume $(96a)^4\approx$(8.2fm)$^4$ with the lattice spacing $a\approx 0.085$fm and $(m_{\pi},m_{K})\approx(146,525)$MeV. Center: $\Lambda N$ central potential in the $^3S_1$-$^3D_1$ channel. Right: $\Lambda N$ tensor potential in the $^3S_1$-$^3D_1$ channel. \[Fig\_LNpots\]](fig1center.eps){width="99.00000%"}   ![Left: $\Lambda N$ central potential in the $^1S_0$ channel calculated with nearly physical point lattice QCD calculation on a volume $(96a)^4\approx$(8.2fm)$^4$ with the lattice spacing $a\approx 0.085$fm and $(m_{\pi},m_{K})\approx(146,525)$MeV. Center: $\Lambda N$ central potential in the $^3S_1$-$^3D_1$ channel. Right: $\Lambda N$ tensor potential in the $^3S_1$-$^3D_1$ channel. \[Fig\_LNpots\]](fig1right.eps){width="99.00000%"} Summary ======= In this paper, we present an efficient algorithm to calculate a large number of NBS wave functions simultaneously. The effective block algorithm significantly reduces the numerical costs for the calculation of 52 two-baryon channels and the computer program implemented lays the foundation for the nearly physical point lattice QCD calculation with volume size $(96a)^4\approx$($8.2$fm)$^4$. The preliminary snapshots of the $\Lambda N$ potential are presented. Central potentials in both $^1S_0$ and $^3S_1$-$^3D_1$ channels show the short-range repulsion and medium-to-long-range attraction. The tensor potential of the $\Lambda N-\Lambda N$ diagonal channel seems to be weaker than the tensor potential of $NN$ interaction. Further studies explicitly considering $\Sigma N$ channels together with employing larger statistics are under progress and will be reported in the near future. Acknowledgments {#acknowledgments .unnumbered} =============== The lattice QCD calculations have been performed on the K computer at RIKEN, AICS (Nos. hp120281, hp130023, hp140209, hp150223), HOKUSAI FX100 computer at RIKEN, Wako (No. G15023) and HA-PACS at University of Tsukuba (Nos. 12b-13, 13a-25, 14a-25, 15a-33, 14a-20, 15a-30). We thank ILDG/JLDG [@ILDGJLDG] which serves as an essential infrastructure in this study. This work is supported in part by MEXT Grant-in-Aid for Scientific Research (25105505, 15K17667, 25287046, 26400281), and SPIRE (Strategic Program for Innovative REsearch) Field 5 project. We thank all collaborators in this project. [99]{} Reviewed in O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys.  [**57**]{}, 564 (2006). H. Nemura, Y. Akaishi and Y. Suzuki, Phys. Rev. Lett.  [**89**]{}, 142504 (2002) \[arXiv:nucl-th/0203013\]. N. Ishii, S. Aoki, T. Hatsuda, Phys. Rev. Lett. [**99**]{}, 022001 (2007). S. Aoki, T. Hatsuda and N. Ishii, Prog. Theor. Phys.  [**123**]{} (2010) 89. S. Aoki [*et al.*]{} \[HAL QCD Collaboration\], PTEP [**2012**]{}, 01A105 (2012). K. Sasaki [*et al.*]{} \[HAL QCD Collaboration\], PTEP [**2015**]{}, no. 11, 113B01 (2015). T. Doi [*et al.*]{}, in these proceedings; arXiv:1512.04199 \[hep-lat\]; N. Ishii [*et al.*]{}, in these proceedings; K. Sasaki [*et al.*]{}, in these proceedings; H. Nemura, arXiv:1510.00903 \[hep-lat\]. N. Ishii, S. Aoki and T. Hatsuda, PoS LATTICE [**2008**]{}, 155 (2008) \[arXiv:0903.5497 \[hep-lat\]\]. H. Nemura, N. Ishii, S. Aoki and T. Hatsuda \[PACS-CS Collaboration\], PoS [**LATTICE2008**]{}, 156 (2008) \[arXiv:0902.1251 \[hep-lat\]\]. T. Doi and M. G. Endres, Comput. Phys. Commun.  [**184**]{}, 117 (2013). K.-I. Ishikawa [*et al.*]{}, PoS LATTICE [**2015**]{}, 075 (2015) \[arXiv:1511.09222 \[hep-lat\]\]. See `http://www.lqcd.org/ildg` and `http://www.jldg.org`

--- abstract: 'Let $\Omega \subseteq \{1,\dots,m\} \times \{1,\dots,n\}$. We consider fibers of coordinate projections $\pi_\Omega : \mathscr{M}_k(r,m \times n) \rightarrow k^{\# \Omega}$ from the algebraic variety of $m \times n$ matrices of rank at most $r$ over an infinite field $k$. For $\#\Omega = \dim \mathscr{M}_k(r,m \times n)$ we describe a class of $\Omega$’s for which there exist non-empty Zariski open sets $\mathscr{U}_\Omega \subset \mathscr{M}_k(r,m \times n)$ such that $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big) \cap \mathscr{U}_\Omega$ is a finite set $\forall X \in \mathscr{U}_\Omega$. For this we interpret matrix completion from a point of view of hyperplane sections on the Grassmannian $\operatorname{Gr}(r,m)$. Crucial is a description by Sturmfels $\&$ Zelevinsky [@SZ93] of classes of local coordinates on $\operatorname{Gr}(r,m)$ induced by vertices of the Newton polytope of the product of maximal minors of an $m \times (m-r)$ matrix of variables.' address: - 'School of Information Science and Technology, ShanghaiTech University, No.393 Huaxia Middle Road, Pudong Area, Shanghai, China' - 'Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy' author: - 'Manolis C. Tsakiris' title: 'Finiteness of fibers in matrix completion via Pl[ü]{}cker coordinates' --- Introduction ============ Matrix completion problems are concerned with fibers of coordinate projections $\pi_\Omega: \mathscr{M} \rightarrow k^{\# \Omega}$. Here $k$ is a field, $\mathscr{M}$ a subset of $k^{m \times n}$ and $\Omega$ a subset of $ [m] \times [n]$ with $[s]=\{1,\dots,s\}$ whenever $s$ is a positive integer. With that $\pi_\Omega$ is defined by $(X_{ij})_{(i,j) \in [m] \times [n]} \mapsto (X_{ij})_{(i,j) \in \Omega}$. For $y \in k^{\# \Omega}$ an important question is whether the fiber $\pi_\Omega^{-1}(y)$ over $y$ is non-empty. Positive semi-definite and Euclidean distance completions or completion ranks are problems that fall into this category, e.g., see [@L09], [@BBS20]. In this paper we are interested in fibers of the form $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big)$ for $X \in \mathscr{M}$, for the case where $\mathscr{M}=\mathscr{M}_k(r,m \times n)$ is the algebraic variety of $m\times n$ matrices with entries in $k$ of rank at most $r$. Computing an element of the fiber $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big)$ is often known as low-rank matrix completion, a fundamental problem in data science and machine learning. A natural approach for such computation is via Gr[ö]{}bner basis methods, which however entail in general a doubly exponential complexity in $\max\{m,n\}$. Efforts to circumvent this difficulty led to a seminal paper by Cand[è]{}s $\&$ Recht [@CR09] where they proved that for $k=\mathbb{R}$ and $\Omega$ sampled uniformly at random with $\#\Omega \sim \mathcal{O}(r \max\{m,n\}^{6/5} \log \max\{m,n\})$ an incoherent[^1] $X$ can be recovered from $\pi_\Omega(X)$ with high probability as the unique global optimum of a convex optimization problem solvable in polynomial time. Shortly after, in another famous paper [@CT10] Cand[è]{}s $\&$ Tao employed long and complicated combinatorial arguments to remove the exponent $6/5$ and reach a similar conclusion for projection ranks $\#\Omega$ of the order $\dim \mathscr{M}_k(r,m \times n) = r(m+n-r)$[^2] up to a poly-logarithmic factor. Recht [@R11] simplified the proof by using results of Gross in quantum information theory, who in turn generalized the theorems from coordinate to arbitrary linear projections [@G11]. In spite of the attention that the aforementioned work of Cand[è]{}s, Recht, Tao and Gross has ever since been drawing, the following fundamental geometric analogue of the question they answered remains, to the best of the author’s knowledge, open: \[question\] What is a characterization of $\Omega$ with $\# \Omega=\dim \mathscr{M}_k(r,m \times n)$ for which there is a dense set $\mathscr{U}_\Omega$ in the Zariski topology of $k^{m \times n}$ with $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big)$ finite $\forall X \in \mathscr{U}_\Omega$? Question \[question\] can be equivalently formulated as characterizing the maximal independent sets of the algebraic matroid associated to $\mathscr{M}_k(r,m \times n)$. These are the maximal algebraically independent over $k$ subsets of generating variables in the quotient ring of $\mathscr{M}_k(r,m \times n)$. Using the tropicalization of the Grassmannian $\operatorname{Gr}(2,m)$ D.I. Bernstein [@B17] was able to give such a characterization for $r=2$. Earlier steps in this direction were by Kiraly, Theran $\&$ Tomioka [@KT12], [@KTT15] who drew various connections between algebraic geometry, matroid theory and matrix completion, as well as by Pimentel-Alarc[ó]{}n, Boston $\&$ Nowak who for $n\ge r(m-r)$ suggested an intriguing answer to Question \[question\] for which a sufficient justification remains to be given[^3]. Recently, replacing $\pi_\Omega$ by $X \stackrel{\pi_F}{\longmapsto} \big(f_i(X): \, i \in [N]\big)$ for an arbitrary collection of linear functionals $F = \big(f_i \in \operatorname{Hom}_k\big(k^{m \times n},k\big): \, i \in [N] \big)$ Rong, Wang $\&$ Xu [@RWX19] proved that for general [^4] $F$ the map $\pi_F$ is injective on a dense subset as long as $N > \dim \mathscr{M}_k(r,m \times n)$. They further conjectured the existence of special $F$’s with $N = \dim \mathscr{M}_k(r,m \times n)$ that allow the same conclusion. Related to this problem is the problem of phase retrieval for which an algebraic-geometric characterization of injectivity was given in [@CEHV15]. In this paper we give a partial answer to Question \[question\] by providing a class of $\Omega$’s with $\#\Omega=\dim \mathscr{M}_k(r,m \times n)$ for which non-empty Zariski open sets $\mathscr{U}_\Omega$’s exist such that $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big) \cap \mathscr{U}_\Omega$ is a finite set for every $X \in \mathscr{U}_\Omega$. There are two main insights involved. The first is interpreting matrix completion from a point of view of hyperplane sections on the Grassmannian $\operatorname{Gr}(r,m)$. For $\omega \subset [m]$ define the coordinate projection $\pi_\omega: k^m \rightarrow k^{\#\omega}$ by $(\xi_i)_{i \in [m]} \mapsto (\xi_i)_{i \in \omega}$. Write $\Omega = \bigcup_{j \in [n]} \omega_j \times \{j\}$ where the $\omega_j$’s are subsets of $[m]$. Let $X=[x_1 \cdots x_n] \in \mathscr{M}_k(r,m \times n)$ and denote by $\mathfrak{c}(X)$ the column-space of $X$. Given $\pi_\Omega(X)$ if $\mathfrak{c}(X)$ were known and $\dim \pi_{\omega_j}\big(\mathfrak{c}(X)\big)=r$ for every $j \in [n]$ then one could uniquely obtain $X$ from $\pi_\Omega(X)$ and $\mathfrak{c}(X)$ by solving $n$ systems of linear equations. Write $\Omega_j$ for the set of all subsets of $\omega_j$ of cardinality $r+1$. Suppose $\Omega_{j} \neq \emptyset $ for some $j \in [n]$ and let $\phi_j \in \Omega_j$. For $S \in \operatorname{Gr}(r,m)$ if $\pi_{\omega_j}(x_j) \in \pi_{\omega_j}(S)$ then certainly $\pi_{\phi_j}(x_j) \in \pi_{\phi_j}(S)$ which with the additional hypothesis $\dim \pi_{\phi_j}\big(S\big)=r$ is equivalent to $$\begin{aligned} \sum_{i \in [r+1]} (-1)^{i-1} x_{\phi_{ij} j} [\phi_j \setminus \{\phi_{ij}\}]_{S} = 0 \, \, \, \text{and not all $[\phi_j \setminus \{\phi_{ij}\}]_{S}$'s are zero} \label{eq:Sections}\end{aligned}$$ where $\phi_{ij}$ is the $i$-th element of $\phi_j$, $x_{ij}$ the $i$-th coordinate of $x_j$ and $[\phi_j \setminus \{\phi_{ij}\}]_{S}$ denotes the Pl[ü]{}cker coordinate of $S$ corresponding to the ordered indices $\phi_j \setminus \{\phi_{ij}\}$. The question then becomes whether all such hyperplane sections are sufficiently general to cut the open locus $\dim \pi_{\phi}\big(S\big)=r, \, \phi \in \Omega_j, \, j \in [n]$ of $\operatorname{Gr}(r,m)$ into a finite number of points. Our second insight is to answer this question by making use of classes of local coordinates on $\operatorname{Gr}(r,m)$ induced by *supports of linkage matching fields*, or SLMF’s for brevity. These local coordinates were already known by Gelfand, Graev $\&$ Retakh [@GGR90] from an analytic point of view in the context of hypergeometric functions. In their effort to prove the universal Gr[ö]{}bner basis property of maximal minors[^5] Sturmfels $\&$ Zelevinsky [@SZ93] introduced LMF’s and proved that their supports induce the same classes of local coordinates as in [@GGR90]. SLMF’s of size $(m,m-r)$ are related to the Newton polytope $\Pi_{m,m-r}$ of the product of maximal minors of an $m \times (m-r)$ matrix of variables. The vertices of $\Pi_{m,m-r}$ encode all distinct initial ideals of the ideal of maximal minors for all possible monomial orders. Sturmfels $\&$ Zelevinsky proved that a set is the support set of a vertex of $\Pi_{m,m-r}$ if and only if it is an SLMF. LMF’s have recently found applications in tropical geometry, e.g., see [@FR15], [@LS18]. A combinatorial characterization is as follows [@SZ93]. The SLMF’s of size $(m,m-r)$ are the subsets $\Phi$ of $[m] \times [m-r]$ written as $\Phi = \bigcup_{j \in [m-r]} \phi_j \times \{j\}$ with the $\phi_j$’s subsets of $[m]$ satisfying[^6]. $$\begin{aligned} \#\phi_j = r+1, \, j \in [m-r] \, \, \, \, \, \, \, \, \, \text{and} \, \, \, \, \, \, \, \, \, \# \bigcup_{j \in \mathcal{T}} \phi_j \ge \# \mathcal{T} + r, \, \, \, \mathcal{T} \subseteq [m-r]. \label{eq:SLMF}\end{aligned}$$ Recall that $\Omega_j$ is the set of all subsets of $\omega_j$ of cardinality $r+1$. We prove: \[thm:main\] Suppose $k$ is infinite, $\#\Omega = \dim \mathscr{M}_k(r,m \times n), \, \#\omega_j\ge r \, \, \forall j \in [n]$ and that there is a partition $[n] = \bigcup_{\ell \in [r]} \mathcal{T}_\ell$ such that for $\ell \in [r]$ there are $\phi_{j}^\ell \in \bigcup_{j' \in \mathcal{T}_\ell} \Omega_{j'}, \, j \in [m-r]$ with each $\Phi_\ell = \bigcup_{j \in [m-r]} \phi_{j}^\ell \times \{j\}$ an SLMF of size $(m,m-r)$. Then there is a non-empty Zariski open set $\mathscr{U}_\Omega \subseteq \mathscr{M}_k(r,m \times n)$ with $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big) \cap \mathscr{U}_\Omega$ finite for $X \in \mathscr{U}_\Omega$. In general $\pi_\Omega^{-1}(\pi_\Omega(X)) \cap \mathscr{U}_\Omega$ in Theorem \[thm:main\] may consist of multiple points. However we have the following injectivity statement, which also answers in the affirmative the afforementioned conjecture of Rong, Wang $\&$ Xu [@RWX19]. \[thm:injective\] In addition to the hypothesis of Theorem \[thm:main\] on $\Omega$ suppose that all $\Phi_\ell$’s are equal to the same SLMF $\Phi$. Then there exists a non-empty Zariski open set $\mathscr{U}_\Omega \subseteq \mathscr{M}_k(r,m \times n)$ such that $\pi_\Omega$ is injective on $\mathscr{U}_\Omega$. Conversely for any SLMF $\Phi$ of size $(m,m-r)$ there exists an $\Omega$ that satisfies the hypothesis and $\Phi_\ell = \Phi$ for every $\ell \in [r]$. We discuss preliminaries in §\[section:Preliminaries\], give the proofs of Theorems \[thm:main\] and \[thm:injective\] in §\[section:Proofs\] and conclude with some examples in §\[section:Examples\]. The author thanks Prof. Aldo Conca for many inspiring discussions on the subject. Preliminaries {#section:Preliminaries} ============= We recall the beautiful relationship between supports of linkage matching fields of size $(m,m-r)$ and local coordinates on $\operatorname{Gr}(r,m)$ described by Sturmfels $\&$ Zelevinsky [@SZ93]. We tailor the exposition to our needs with Lemma \[lem:SZ93\] and equation the main ingredients needed from [@SZ93], while we make two observations, Lemma \[lem:NonDimensionDrop\] and Lemma \[lem:PointProjection\], of fundamental role in §\[subsection:Proof1\]. Example \[ex:SZ93\] in §\[section:Examples\] illustrates some of the main ideas here. Let $S \in \operatorname{Gr}(r,m)$ and denote by $S^\perp \in \operatorname{Gr}(m-r,m)$ the orthogonal complement of $S$. Working with the standard basis of $k^m$ the canonical isomorphism $\operatorname{Gr}(r-m,m) \rightarrow \operatorname{Gr}(r,m)$ sends the Pl[ü]{}cker coordinate $[\psi]_{S^\perp}$ to $\sigma(\psi,[m] \setminus \psi) \, \big[[m] \setminus \psi \big]_{S}$, where $\psi$ is any subset of $[m]$ of cardinality $m-r$ and $\sigma(\psi,[m] \setminus \psi)$ is $-1$ raised to the number of elements $(a,b) \in \psi \times ([m] \setminus \psi)$ with $a>b$[^7]. Let $A \in k^{m \times (m-r)}$ contain a basis of $S^\perp$ in its columns. Let $\Phi = \bigcup_{j \in [m-r]} \phi_j \times \{j\}$ be an SLMF of size $(m,m-r)$. For $j \in [m-r]$ denote by $H_j$ the $k$-subspace of $k^{m-r}$ spanned by these rows of $A$ indexed by $[m]\setminus \phi_j$. The locus $V_\Phi$ of $\operatorname{Gr}(r,m)$ where the $H_j$’s have codimension $1$ and $\bigcap_{j \in [m-r]} H_j = 0$ is open. Suppose $S \in V_\Phi$. Then there is an automorphism of $k^{m-r}$ that takes $H_j$ to the hyperplane with normal vector $e_j$, the latter having zeros everywhere and a $1$ at position $j$. Changing the basis we see that $S$ can also be represented by some $\tilde{A} \in k^{m \times (m-r)}$ which is sparse with support on $\Phi$. Let $\mathfrak{m}_{j j'}$ be the minor of $A$ corresponding to row indices in $[m] \setminus \phi_j$ and column indices $[m-r] \setminus \{j'\}$ and set $M=(\mathfrak{m}_{j j'}: \, j, j' \in [m-r])$. Use respective notations $\tilde{\mathfrak{m}}_{j j'}$ and $\tilde{M}=(\tilde{\mathfrak{m}}_{j j'}: \, j, j' \in [m-r])$ for $\tilde{A}$. By definition of $V_\Phi$ all $\tilde{\mathfrak{m}}_{j j}$’s are non-zero thus, viewed as an element of $\mathbb{P}^{m-1}$, the $j$-th column $\tilde{a}_j$ of $\tilde{A}$ satisfies $$\begin{aligned} \tilde{a}_{\phi_{ij} j} = (-1)^{i-1}\big[ \phi_j \setminus \{\phi_{ij}\}\big]_{S}\, \, \, \text{for} \, \, \, i \in [r+1] \, \, \, \, \, \, \text{and} \, \, \, \, \, \, \tilde{a}_{ij } = 0 \, \, \, \text{for} \, \, \, i \not\in \phi_j \nonumber$$ Since $\tilde{A}$ has full rank this gives: \[lem:NonDimensionDrop\] If $S \in V_\Phi$ then $\dim \pi_{\phi_j}(S) = r$ for every $j \in [m-r]$. Next consider the rational maps $\gamma_{\phi_j}:\operatorname{Gr}(r,m) \dashedrightarrow \mathbb{P}^r$ and $\gamma_\Phi: \operatorname{Gr}(r,m) \dashedrightarrow \prod_{j \in [m-r]} \mathbb{P}^r$ $$\begin{aligned} S \stackrel{\gamma_{\phi_j}}{\longmapsto} & \big((-1)^{i-1} [\phi_j \setminus \{\phi_{ij}\}]_S:\, i \in [r+1]\big) \nonumber \\ S \stackrel{\gamma_\Phi}{\longmapsto}& \big(\gamma_{\phi_j}(S): \, j \in [m-r]\big) \nonumber\end{aligned}$$ \[lem:SZ93\] The rational map $\gamma_\Phi$ is an open embedding on $V_\Phi$. In particular, for $S \in V_\Phi$ the matrix representation $\tilde{A}$ of $S$ is unique up to an arbitrary scaling of each column. Let $T=k\big[[\psi]: \, \psi \subset [m], \, \#\psi=r\big]$ be a polynomial ring generated by variables $[\psi]$’s associated with the Pl[ü]{}cker embedding of $\operatorname{Gr}(r,m)$. By computing the normal vectors of the $H_j$’s in terms of the $\mathfrak{m}_{j j'}$’s it follows that $S \in V_\Phi$ if and only if $\det(M) \neq 0$. Since $\tilde{M} = M C$ where $C$ is an invertible matrix[^8] we see that $V_\Phi$ is defined by the non-vanishing of the polynomial $\prod_{j \in [m-r]} \tilde{\mathfrak{m}}_{jj}$. In turn this gives us the equation of this hypersurface in terms of Pl[ü]{}cker coordinates [^9]: $$\begin{aligned} p_\Phi = \det\Big(\big[\phi_\alpha \setminus \{ \beta \} \big]: \, \alpha \in [m-r]\setminus 1, \, \beta \in [m] \setminus \phi_1\Big) \, \, \, \in T \label{eq:Vphi}\end{aligned}$$ We close this section with an important observation for the sequel. \[lem:PointProjection\] Let $S \in \operatorname{Gr}(r,m)$ and $x \in k^m$. Suppose that $\pi_{\phi_j}(x) \in \pi_{\phi_j}(S)$ for every $j \in [m-r]$. If $S \in V_\Phi$ then $x \in S$. Under the hypothesis Lemma \[lem:NonDimensionDrop\] implies that the relation $\pi_{\phi_j}(x) \in \pi_{\phi_j}(S)$ is equivalent to the relation . In turn is equivalent to the linear form defined by the $j$-th column of $\tilde{A}$ vanishing at $x$. Since this is true for every $j \in [m-r]$ and since the columns of $\tilde{A}$ form a basis for $S^\perp$ this is the same as $x \in S$. Proofs {#section:Proofs} ====== Proof of Theorem \[thm:main\] {#subsection:Proof1} ----------------------------- For clarity we reserve the symbol $\mathscr{M}_k(r,m \times n)$ for the set of $m \times n$ matrices with entries in $k$ of rank at most $r$ and $\pi_{\Omega}: \mathscr{M}_k(r,m \times n) \rightarrow k^{\#\Omega}$ for the coordinate projection as above. Instead we let $\operatorname{M}_k(r,m \times n) = \operatorname{Spec}(T'/I')$ with $T'=k\big[z_{ij}: \, (i,j) \in [m] \times [n]\big]$ a polynomial ring in the $z_{ij}$’s and $I'$ the ideal generated by all $(r+1)\times(r+1)$ minors of the matrix $Z = \big(z_{ij}: \, (i,j) \in [m] \times [n]\big)$. We let $\Pi_{\Omega,k}: \operatorname{M}_k(r,m \times n) \rightarrow \mathbb{A}^{\#\Omega}_k$ be the corresponding morphism of affine schemes induced by the ring homomorphism $k[z_{ij}: \, (i,j) \in \Omega] \rightarrow T'/I'$ given by $z_{ij} \mapsto z_{ij}+I'$. Let $K$ be the algebraic closure of $k$. It is enough to prove that there is a non-empty open set $U_\Omega$ of $\operatorname{M}_K(r,m \times n)$ with $\Pi_{\Omega,K}^{-1}\big(\Pi_{\Omega,K}(X)\big) \cap U_\Omega$ a $0$-dimensional $K$-scheme $\forall X \in U_\Omega$. For then let $q_1,\dots,q_s \in T' \otimes_k K$ be the defining polynomials of the complement of $U_\Omega$. When $k \neq K$ expanding their coefficients using a (possibly infinite) basis of $K$ over $k$ gives finitely many non-zero polynomials in $T'$. Since $k$ is infinite $\mathscr{M}_k(r,m \times n)$ is irreducible so the complement of their vanishing locus is a non-empty open set $\mathscr{U}_\Omega$ of $\mathscr{M}_k(r,m \times n)$. Then we are done because any point $X \in \mathscr{U}_\Omega$ corresponds to a closed point $X \in U_\Omega$ and there is a set-theoretic inclusion of fibers $\pi_{\Omega}^{-1}\big(\pi_{\Omega}(X)\big) \cap \mathscr{U}_\Omega \subset \Pi_{\Omega,K}^{-1}\big(\Pi_{\Omega,K}(X)\big) \cap U_\Omega$. Hence in what follows we assume $k=K$ and drop the subscript $k$. With $I$ the ideal of $T$ generated by the Pl[ü]{}cker relations and the standard $\mathbb{Z}_{\ge 0}$-grading of $T$ we have $\operatorname{Gr}(r,m) = \operatorname{Proj}(T/I)$. Let $T'' = T \otimes_k T'$ and $I'' = IT'' + I' T''$. We view $T''$ as a $\mathbb{Z}_{\ge 0}$-graded ring with its graded structure inherited from that of $T$. Then $\operatorname{Proj}(T''/I'')=\operatorname{Gr}(r,m) \times_{\operatorname{Spec}(k)} \operatorname{M}(r,m \times n)$. Let $\Phi_\ell=\bigcup_{j \in [m-r]} \phi_{j}^\ell \times \{j\}, \, \ell \in [r]$ be the SLMF’s that appear in the hypothesis and set $p$ the product of all $p_{\Phi_\ell}$’s (see eq. ). Since $\operatorname{Proj}\big(T/I + (p)\big)$ is a union of $r$ hypersurfaces of $\operatorname{Gr}(r,m)$ we have that $\operatorname{Spec}(T''/I'')_{(p)}$ is a non-empty affine open subscheme of $\operatorname{Proj}(T''/I'')$, where $(T''/I'')_{(p)}$ is the homogeneous localization of $T''/I''$ at the multiplicatively closed set $\{1,\bar{p},\bar{p}^2,\dots\}$, with $\bar{p}$ the class of $p$ in $T''/I''$. Next we define an incidence correspondence on $\operatorname{Spec}(T'' / I'')_{(p)}$. Suppose $\phi_{j}^\ell \in \Omega_{j'}$ for some $j' \in \mathcal{T}_\ell$, then $\phi_{j}^\ell$ induces the following linear form of $T''$ $$\sum_{i \in [r+1]} (-1)^{i-1} z_{\phi_{ij}^\ell j'} [\phi_{j}^\ell \setminus \{\phi_{ij}^\ell\}]$$ Denote by $J$ the ideal of $T''$ generated by the $r(m-r)$ linear forms induced by all $\phi_{j}^{\ell}$’s. Consider the incidence correspondence $\operatorname{Proj}(T'' / I''+J)$ and its open locus $\operatorname{Spec}(T'' / I''+J)_{(p)}$. By Lemma \[lem:NonDimensionDrop\] and relation the closed points $(S,X)$ of $\operatorname{Spec}(T'' / I''+J)_{(p)}$ satisfy $$\begin{aligned} \pi_{\phi_j^\ell}(x_j) \in \pi_{\phi_j^\ell}(S) \, \, \, \, j \in \mathcal{T}_\ell, \, \, \, \, \, \, \dim \pi_{\phi_j^\ell}(S) = r \label{eq:GeometricIncidences}\end{aligned}$$ Let $\rho: \operatorname{Spec}(T'' / I''+J)_{(p)} \rightarrow \operatorname{M}(r,m \times n)$ be the projection onto the second factor. \[lem:Example\] There exists a closed point $Q=(S,X) \in \operatorname{Spec}(T'' / I''+J)_{(p)}$ for which we have an equality of sets $\rho^{-1}(\rho(Q))=\{Q\}$, in particular $\dim \rho^{-1}(\rho(Q)) = 0$. Let $S$ be any closed point of $\operatorname{Spec}(T/I)_{(p)}$ and let $s_\ell, \, \ell \in [r]$ be a $k$-basis for $S$. Define a closed point $X$ of $\operatorname{M}(r,m \times n)$ by setting $x_j = s_\ell$ whenever $j \in \mathcal{T}_\ell$. Clearly $Q=(S,X) \in \operatorname{Spec}(T'' / I''+J)_{(p)}$. Let $(S',X)$ be a closed point of $\rho^{-1}(\rho(Q))$. By and the definition of $Q$ we have $\pi_{\phi_j^\ell}(s_\ell) \in \pi_{\phi_j^\ell}(S')$ for every $j \in [m-r]$. Lemma \[lem:PointProjection\] and the definition of $p$ give $s_\ell \in S'$. This is true for every $\ell \in [r]$ so that $S' =\operatorname{Span}(s_\ell: \, \ell \in [r])= S$. Hence $Q$ is the only closed point of $\rho^{-1}(\rho(Q))$. Now $\rho^{-1}(\rho(Q)) = \operatorname{Spec}\big(T/I+J|_{X} \big)_{(p)}$ with $J|_{X}$ obtained from $J$ under $Z \mapsto X$. But $\operatorname{Spec}\big(T/I+J|_{X} \big)_{(p)}$ is a Jacobson space [@S20] and as such it is equal to the closure of $Q$. \[lemma:open\] There is a non-empty open set $U \subset \operatorname{M}(r,m \times n)$ with $\dim \rho^{-1}(X)=0, \, \forall X \in U$. Let $\bar{T}'' = T''/I''$ and $\bar{J}$ the class of $J$ in $\bar{T}''$. Note that $\bar{J}$ is generated by $r(m-r) = \dim \operatorname{Gr}(r,m)$ elements. Hence by Krull’s height theorem every minimal prime $\bar{P}$ over $\bar{J}$ will have height at most $r(m-r)$. Since $k$ is algebraically closed $\bar{T}''$ is an affine domain. Hence for $\operatorname{Proj}(\bar{T}'' / \bar{P})$ any irreducible component of $\operatorname{Proj}(\bar{T}'' / \bar{J})$ we have $$\dim \operatorname{Proj}(\bar{T}'' / \bar{P}) \ge \dim \operatorname{Proj}(\bar{T}'') - r(m-r) = \dim \operatorname{M}(r,m \times n)$$ Now every irreducible component $Y$ of $\operatorname{Spec}(\bar{T}'' / \bar{J})_{(p)}$ has the form $\operatorname{Spec}(\bar{T}'' / \bar{P})_{(p)}$ where $\operatorname{Proj}(\bar{T}'' / \bar{P})$ is an irreducible component of $\operatorname{Proj}(\bar{T}'' / \bar{J})$. As such $\dim Y \ge \dim \operatorname{M}(r,m \times n)$. By the upper semi-continuity of the fiber dimension on the source the locus $V_{\min} \subset \operatorname{Spec}(\bar{T}'' / \bar{J})_{(p)}$ on which the fiber dimension is $0$ is open. By Lemma \[lem:Example\] $V_{\min}$ is not empty. Let $Y$ be an irreducible component of $\operatorname{Spec}(\bar{T}'' / \bar{J})_{(p)}$ that intersects $V_{\min}$. Consider $\rho|_{Y \cap V_{\min}} : Y \cap V_{\min} \rightarrow \operatorname{M}(r, m \times n)$. Since all fibers of $\rho|_{Y \cap V_{\min}}$ have dimension $0$ it must be that the dimension of the image of $\rho|_{Y \cap V_{\min}}$ coincides with $\dim Y \cap V_{\min}$. Thus $$\dim \operatorname{M}(r, m \times n) \ge \dim \rho|_{Y \cap V_{\min}}(Y \cap V_{\min}) = \dim Y \cap V_{\min} = \dim Y \ge \dim \operatorname{M}(r,m \times n)$$ that is $\rho|_{Y \cap V_{\min}}$ is dominant. By Chevalley’s theorem the image of $\rho|_{Y \cap V_{\min}}$ is constructible, say $\rho|_{Y \cap V_{\min}}(Y \cap V_{\min}) = \bigcup_{\alpha \in \mathcal{A}} Y_\alpha \cap U_\alpha$ where $\mathcal{A}$ is finite and $Y_\alpha,U_\alpha$ are closed and open respectively in $\operatorname{M}(r,m \times n)$. Since $\rho|_{Y \cap V_{\min}}$ is dominant there is $\beta \in \mathcal{A}$ with $Y_\beta =\operatorname{M}(r,m \times n)$. That is $\rho|_{Y \cap V_{\min}}(Y \cap V_{\min})$ contains a non-empty open set $U = U_\beta$. Intersect the $U$ of Lemma \[lemma:open\] with the open locus of $\operatorname{M}(r,m\times n)$ where $\dim \pi_{\psi} \big(\mathfrak{c}(X)\big)=r$ for every $\psi \subset \omega_j, \, j \in [n]$ with $\# \psi = r$ and $\mathfrak{c}(X) \in \bigcap_{\ell \in [r]} V_{\Phi_\ell}$, and call the result $U_\Omega$. For a closed point $X \in U_\Omega$ the fiber $\Pi_{\Omega}^{-1}\big((\Pi_{\Omega}(X)\big) \cap U_\Omega$ has finitely many closed points $X'$ because the map $X' \mapsto (\mathfrak{c}(X'),X)$ injects them into closed points of $\rho^{-1}(X)$. Since $\Pi_{\Omega}^{-1}\big((\Pi_{\Omega}(X)\big) \cap U_\Omega$ is a Jacobson space it must be a $0$-dimensional scheme. Proof of Theorem \[thm:injective\] {#subsection:Proof2} ---------------------------------- Write $\Phi_\ell = \Phi = \bigcup_{j \in [m-r]} \phi_j \times \{j\}$ for every $\ell$. Then for every $\alpha \in [m-r]$ there is a subset $\mathcal{L}_\alpha \subset [n]$ of cardinality $r$ such that $\phi_\alpha \subseteq \omega_{j}, \, \forall j \in \mathcal{L}_\alpha$. Call $\mathscr{U}_\Omega$ the non-empty open set of $\mathscr{M}_k(r,m \times n)$ on which $\mathfrak{c}(X) \in V_\Phi$, none of the Pl[ü]{}cker coordinates of $\mathfrak{c}(X)$ vanishes and the $\{x_j: \, j \in \mathcal{L}_\alpha\}$ are linearly independent for every $\alpha \in [m-r]$. Since $\operatorname{Span}(x_j: \, j \in \mathcal{L}_\alpha)$ is the same as $\mathfrak{c}(X)$ so will be their projections under $\pi_{\phi_\alpha}$. Lemma \[lem:SZ93\] asserts that the data $\pi_{\phi_\alpha}\big(\mathfrak{c}(X)\big), \, \alpha \in [m-r]$ uniquely determine $\mathfrak{c}(X)$ on $V_\Phi$. Since $\# \omega_j \ge r, \, \forall j \in [n]$ the following data uniquely determine $X$ $$\pi_{\phi_\alpha}\big(\operatorname{Span}(x_j: \, j \in \mathcal{L}_\alpha)\big), \, \alpha \in [m-r] \, \, \, \, \, \, \text{and} \, \, \, \, \, \, \pi_{\Omega}(X)$$ Let $\Phi \subset [m] \times [m-r]$ be any SLMF of size $(m,m-r)$. We prove the existence of an $\Omega \subset [m] \times [n]$ such that 1) $\# \Omega = \dim \mathscr{M}_k(r,m \times n)$, 2) $\# \omega_j \ge r$ and 3) for every $\alpha \in [m-r]$ there is a subset $\mathcal{L}_\alpha \subset [n]$ of cardinality $r$ with $\phi_\alpha \subseteq \omega_{j}, \, \forall j \in \mathcal{L}_\alpha$. We argue by induction on $n$. For $n=r$ take $\Omega = [m] \times [n]$. Suppose $n>r$. By induction there is an $\Omega' \subset [m] \times [n-1]$ with the required as above properties. Then take $\Omega = \Omega' \cup \big([r] \times \{n\}\big)$. Examples {#section:Examples} ======== \[ex:SZ93\] Let $r=2, m=6$ and consider the following $\Phi=\bigcup_{j \in [4]} \phi_j \times \{j\} \subset [m] \times [m-r]$: $$\Phi = \{2,4,6\} \times \{1\} \, \, \cup \, \, \{1,2,4\} \times \{2\} \, \, \cup \, \, \{1,2,5\} \times \{3\} \, \, \cup \, \, \{1,3,5\} \times \{4\}$$ and its representation by its indicator matrix: $$\begin{bmatrix} 0& 1 & 1 & 1 \\ 1& 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1& 0& 0& 0 \end{bmatrix}$$ This is a linkage matching field since it satisfies condition . It defines an open set $V_\Phi$ in $\operatorname{Gr}(2,6)$ on which the rational map $\operatorname{Gr}(2,6) \rightarrow \mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2$ given by $$S \mapsto \left(\begin{bmatrix} \begin{array}{r} [46]_S\\ -[26]_S \\ [24]_S \end{array} \end{bmatrix}, \begin{bmatrix} \begin{array}{r} [24]_S\\ -[14]_S \\ [12]_S \end{array}\end{bmatrix}, \begin{bmatrix} \begin{array}{r} [25]_S\\ -[15]_S \\ [12]_S \end{array} \end{bmatrix}, \begin{bmatrix} \begin{array}{r} [35]_S\\ -[15]_S \\ [13]_S \end{array} \end{bmatrix} \right)$$ is injective. These $4$ elements of $\mathbb{P}^2$ are precisely the normal vectors of the $4$ planes in $k^3$ that one gets by projecting a general $2$-dimensional subspace $S$ in $k^6$ onto the $3$ coordinates indicated by each of the $\phi_j$’s. For each $S \in V_\Phi$ there is a unique up to a scaling of its columns $6 \times 4$ matrix with the same sparsity pattern as $\Phi$ whose column-space is $S^\perp$: $$\begin{bmatrix} \begin{array}{rrrr} 0 \, \, \, \, \, \, \,& [24]_S & [25]_S & [35]_S \\ [46]_S & -[14]_S & -[15]_S & 0 \, \, \, \, \, \, \, \\ 0 \, \, \, \, \, \, \, & 0 \, \, \, \, \, \, \, & 0 \, \, \, \, \, \, \, & -[15]_S \\ -[26]_S & [12]_S & 0 \, \, \, \, \, \, \, & 0 \, \, \, \, \, \, \, \\ 0 \, \, \, \, \, \, \,& 0 \, \, \, \, \, \, \,& [12]_S & [13]_S \\ [24]_S & 0 \, \, \, \, \, \, \,& 0 \, \, \, \, \, \, \, & 0 \, \, \, \, \, \, \, \\ \end{array} \end{bmatrix}$$ The polynomial that defines the complement of $V_\Phi$ is $$p_\Phi = \det\left( \begin{bmatrix} \begin{array}{rrrr} [24]_S & [25]_S & [35]_S \\ 0 \, \, \, \, \, \, \, & 0 \, \, \, \, \, \, \, & -[15]_S \\ 0 \, \, \, \, \, \, \,& [12]_S & [13]_S \\ \end{array} \end{bmatrix} \right) = [12]_S [24]_S [15]_S$$ The following two examples illustrate Theorem \[thm:main\]. \[ex:main\] Let $r=2, \, m=6, \, n=5$ and consider the following $\Omega \subset [6] \times [5]$ with $\# \Omega = 18 = \dim \mathscr{M}_k(2,6 \times 5)$ represented by its indicator matrix: $$\Omega = \begin{bmatrix} 1 & 0 & 0 & 1 & 1 \\ 1& 0 & 1 & 1 & 0 \\ 1& 0 & 0& 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1\\ 0 & 1 & 1 & 0 & 0 \end{bmatrix}$$ Consider the partition $[5] = \mathcal{T}_1 \cup \mathcal{T}_2$ with $\mathcal{T}_1=\{1,2\}$ and $\mathcal{T}_2=\{3,4,5\}$. Now take $$\Phi_1 = \begin{bmatrix} 1 & 1 & 1 & 0 \\ 1& 1 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0& 0& 0& 1 \end{bmatrix}, \, \, \, \Phi_2 = \begin{bmatrix} 0& 1 & 1 & 1 \\ 1& 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1& 0& 0& 0 \end{bmatrix}$$ $\Phi_1, \Phi_2$ are SLMF’s since they satisfy . $\Phi_1$ is associated with the first $2$ columns of $\Omega$ ($\mathcal{T}_1$), while $\Phi_2$ with the last $3$ columns of $\Omega $ ($\mathcal{T}_2$). A computation with `Macaulay2` suggests that $\operatorname{Spec}\big(T/I+J|_{X} \big)_{(p)}$ (hence also $\pi_\Omega^{-1}(\pi_\Omega(X))$) consists only of $X$, for general $X$. \[ex:main-multiplePoints\] Let $r=2, m=6, n=8$ and $\Omega$ with $\#\Omega=24=\dim \mathscr{M}_k(2,6 \times 8)$ given by $$\Omega = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \end{bmatrix}$$ With $\mathcal{T}_1=\{1,2,3,4\}, \mathcal{T}_2=\{5,6,7,8\}$, $\Phi_1, \Phi_2$ are the leftmost and rightmost respectively blocks of $\Omega$ and both satisfy . A computation with `Macaulay2` suggests that $\operatorname{Spec}\big(T/I+J|_{X} \big)_{(p)}$ consists of $2$ points over $k$ non-algebraically closed and $4$ points otherwise. Bernstein, D.I., 2017. *Completion of tree metrics and rank 2 matrices.* Linear Algebra and its Applications, 533, pp.1-13. Bernstein, D.I., Blekherman, G. and Sinn, R., 2020. *Typical and generic ranks in matrix completion.* Linear Algebra and its Applications, 585, pp.71-104. Bernstein, D. and Zelevinsky, A., 1993. *Combinatorics of maximal minors.* Journal of Algebraic Combinatorics, 2(2), pp.111-121. Boocher, A., 2012. *Free resolutions and sparse determinantal ideals*. Mathematical Research Letters, 19(4), pp.805-821. Bruns, W. and Herzog, H.J., 1998. *Cohen-Macaulay rings* (No. 39). Cambridge university press. Cand[è]{}s, E.J. and Recht, B., 2009. *Exact matrix completion via convex optimization.* Foundations of Computational mathematics, 9(6), p.717. Cand[è]{}s, E.J. and Tao, T., 2010. *The power of convex relaxation: Near-optimal matrix completion.* IEEE Transactions on Information Theory, 56(5), pp.2053-2080. Conca, A., De Negri, E. and Gorla, E., 2014. *Universal Gr[ö]{}bner bases for maximal minors.* International Mathematics Research Notices, 2015(11), pp.3245-3262. Conca, A., De Negri, E. and Gorla, E., *Universal Gr[ö]{}bner Bases and Cartwright–Sturmfels Ideals*. International Mathematics Research Notices *to appear*. Conca, A., Edidin, D., Hering, M. and Vinzant, C., 2015. *An algebraic characterization of injectivity in phase retrieval.* Applied and Computational Harmonic Analysis, 38(2), pp.346-356. Fink, A. and Rinc[ó]{}n, F., 2015. *Stiefel tropical linear spaces.* Journal of Combinatorial Theory, Series A, 135, pp.291-331. Gelfand, I.M., Graev, M.I. and Retakh, V.S., 1990. *$\Gamma$-series and general hypergeometric functions on the manifold of $k× \times m$ matrices.* Preprint IPM, (64). Gross, D., 2011. *Recovering low-rank matrices from few coefficients in any basis.* IEEE Transactions on Information Theory, 57(3), pp.1548-1566. Harris, J., 2013. *Algebraic geometry: a first course* (Vol. 133). Springer Science $\&$ Business Media. Kalinin, M.Y., 2009. *Universal and comprehensive Gr[ö]{}bner bases of the classical determinantal ideal.* Zapiski Nauchnykh Seminarov POMI, 373, pp.134-143. Kir[á]{}ly, F.J., Theran, L. and Tomioka, R., 2015. *The algebraic combinatorial approach for low-rank matrix completion.* The Journal of Machine Learning Research, 16(1), pp.1391-1436. Kir[á]{}ly, F. and Tomioka, R., 2012, June. *A combinatorial algebraic approach for the identifiability of low-rank matrix completion.* 29th International Conference on Machine Learning (pp. 755-762). Laurent, M., 2009. *Matrix completion problems.* Encyclopedia of Optimization, pp.1967-1975. Loho, G. and Smith, B., 2018. *Matching fields and lattice points of simplices.* arXiv preprint arXiv:1804.01595. Pimentel-Alarc[ó]{}n, D.L., Boston, N. and Nowak, R.D., 2016. *A characterization of deterministic sampling patterns for low-rank matrix completion.* IEEE Journal of Selected Topics in Signal Processing, 10(4), pp.623-636. Pimentel-Alarc[ó]{}n, D.L., Boston, N. and Nowak, R.D., 2016. *Corrections to “A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion”.* IEEE Journal of Selected Topics in Signal Processing, 10(8), pp.1567-1567. Pimentel-Alarc[ó]{}n, D.L., Boston, N. and Nowak, R.D., 2015, June. *Deterministic conditions for subspace identifiability from incomplete sampling.* IEEE International Symposium on Information Theory (ISIT), pp. 2191-2195. Recht, B., 2011. *A simpler approach to matrix completion.* Journal of Machine Learning Research, 12(Dec), pp.3413-3430. Rong, Y., Wang, Y. and Xu, Z., 2019. *Almost everywhere injectivity conditions for the matrix recovery problem.* Applied and Computational Harmonic Analysis. *The Stacks project* [Section 005T](https://stacks.math.columbia.edu/tag/005T) Sturmfels, B. and Zelevinsky, A., 1993. *Maximal minors and their leading terms.* Advances in mathematics, 98(1), pp.65-112. [^1]: The reader is referred to [@CR09] for the notion of incoherence. [^2]: This is the dimension of $\mathscr{M}_k(r,m \times n)$ as an affine variety. [^3]: In the proof of Theorem 1 in [@PABN16a] it is not enough to show that the family $\mathcal{F}$ contains $r(d-r)$ algebraically independent elements. The remark also applies to [@PABN16b]. [^4]: Here general is meant in the algebraic geometry sense, e.g., see p.54 in Harris [@H13] and note the distinction with *generic*. [^5]: The universal Gr[ö]{}bner basis property of maximal minors was instead proved in [@BZ93] using LMF’s, in [@K09], and was later generalized in [@B12], [@CDG15], [@CDG19]. [^6]: The same formula appeared in [@PABN15] where its relation to local coordinates on $\operatorname{Gr}(r,m)$ was suggested independently of [@SZ93]. In the former the proof of Lemma 4 appears to be incomplete. [^7]: E.g., see §1.6 in Bruns $\&$ Herzog [@BH98]. [^8]: This follows from the functoriality of $\wedge^{m-r-1}$. [^9]: If $\beta \not\in \phi_a$ then $\big[\phi_\alpha \setminus \{ \beta \} \big] = 0$. Only the sign may change if one replaces $1$ by any $j \in [m-r]$ in .

--- abstract: 'Ab initio calculations of surface-state mediated interactions between Cu adatoms on transition metal surfaces are presented. We concentrate on Co/Cu(111) and Co(0001) substrates and compare results with our calculations for Cu(111). Our studies show that surface states of Co/Cu(111) and Co(0001) are spin-polarized. We reveal that long-range interactions between adatoms are mainly determined by $sp$-majority states. In contrast to Cu(111) and Co/Cu(111), the interaction between adatoms on Co(0001) is strongly suppressed at large adsorbate separations.' address: - 'Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany ' - 'Fachbereich Physik, Martin-Luther-Universität, Halle-Wittenberg, Friedemann-Bach-Platz 6, D-06099 Halle, Germany' author: - 'V.S. Stepanyuk' - 'L. Niebergall' - 'A.N. Baranov' - 'W. Hergert' - 'P. Bruno' title: 'Long-range electronic interactions between adatoms on transition metal surfaces' --- , , , , Ab initio calculations ,spin-polarized surface states ,long-range interactions 71.15.Mb ,71.20.Be ,73.20.At In the last few years there has been a renewed interest in the study of indirect adsorbate interactions on metal surfaces predicted by Lau and Kohn in 1978[@c1]. This is at least in part due to the possibility to probe directly such interactions by scanning tunnelling microscopy (STM). A low temperature STM has allowed to resolve adsorbate interactions up to 80 Å[@c2; @c3]. It has been shown that oscillations of the electron density around adsorbates in a two-dimensional (2D) electron system can lead to a long-range, oscillatory, Friedel-type adsorbate-adsorbate interaction which decays as $1/d^2$. The required 2D nearly free electron gas could be realized in Shockley type surface states of metal surfaces. For example, surface-state electrons on the (111) surfaces of noble metals form a 2D nearly free electron gas. Several theoretical investigations have demonstrated that despite the fact that indirect adsorbate interactions are small (a few meV), they can significantly influence the growth of nanostructures[@c4; @c5; @c6]. Very recent STM experiments and ab initio calculations have revealed that surface states of transition metal nanostructures can be spin-polarized[@c7; @c8]. In particular, it has been shown that the electronic states of fcc Co monolayers and Co islands on Cu(111) are spin-polarized. Similar results have been reported for surface states of hcp Co(0001)[@c9]. In this paper, we present the first ab initio calculations of adsorbate-adsorbate interactions on transition metal surfaces. We concentrate on the interaction between Cu adatoms on the fcc Co/Cu(111) and the hcp Co(0001) substrates. Results are compared with the interaction on Cu(111). We demonstrate that the substrate-mediated interactions on magnetic substrates are mainly determined by $sp$-majority states. Our results show that adsorbate interactions on Co/Cu(111) are long-ranged and oscillatory. However, for Co(0001) we find that the substrate-mediated interactions are strongly suppressed at large distances. Our calculations are based on the density functional theory and multiple-scattering approach using the Korringa-Kohn-Rostoker Green’s functionmethod [@c6; @c7; @c10]. We treat the surface as an infinite two-dimensional perturbation of the bulk. Taking into account the 2D periodicity of the ideal surface, we calculate the structural Green’s function by solving a Dyson equation self-consistently. The consideration of adsorbate atoms on the surface destroyes the translation symmetry. Therefore the Green’s function of the adsorbate adatom on the surface is calculated in a real space formulation. The structural Green’s function of the ideal surface in real space representation is then used as the reference Green’s function for the calculation of the adatom-surface system. Details of the method and its first applications for calculations of surface-state electrons and adsorbate-adsorbate interactions can be found in our previous work[@c6; @c7; @c10]. Calculations for the long-range interactions have been performed for the relaxed and unrelaxed positions of adatoms. However, we have found that the substrate-mediated interactions at large distances are essentially unmodified by the inclusion of the relaxation. First, we present the results for the interaction between Cu adatoms on Cu(111). Our calculation for the Cu(111) surface gives a surface-state Fermi wavelength $\lambda_F=29$ Å and a surface-state band edge at E$_0$=-0.5 eV below the Fermi level. Due to the long Fermi-wavelength the confinement property of the Cu adatom should exist for large distances around the Cu adsorbate. The scattering of surface state electrons by Cu adatoms leads to quantum interference patterns and to the long-range interactions between the two Cu adatoms[@c6]. Our calculations for the interaction energy between Cu adatoms on Cu(111) are presented in Fig.1.\ ![Calculated interaction energies between two Cu adatoms on Cu(111). Inset shows the long-range oscillatory interaction at large distances.[]{data-label="fig1"}](fig1.ps){width="10cm"} These results show that the interaction energy is oscillatory with a period of about 15 Å. The envelope of the magnitude decays as $1/d^2$ in agreement with the prediction of Lau and Kohn[@c1]. In recent STM experiments on large Co islands supported on Cu(111) the standing wave patterns in the local density of states (LDOS) due to the quantum interference of surface-state electrons have been observed[@c7; @c8]. Ab initio calculations[@c7] have revealed that a majority free-electron like surface states give rise to LDOS oscillations on Co islands. In Fig.2, as an example we show the contour plot for majority and minority spectral densities of two monolayers of Co on Cu(111). {width="14cm"} We find that the majority $sp$ states have a parabolic dispersion relation, described by an onset below the Fermi level at $E_0$=0.28 eV and an effective mass $m^*=0.50m_e$. Compared to Cu(111), a surface-state Fermi wavelength for 2Co/Cu(111) is found to be significantly larger ($\lambda_F=48$ Å). In the minority channel the $d$ contribution is found to be dominant. The strong feature below the Fermi level (cf. Fig.2b) is mainly determined by the $d$-states with a small $sp$ contribution. These states lead to a strong localized peak at about 0.4 eV below the Fermi energy observed in experiments[@c7; @c8]. Our calculations show that the LDOS around a single Cu adatom on 2Co/Cu(001) displays the long-range Friedel oscillations caused by the quantum interference of the $sp$-majority surface-state electrons. These oscillations lead to a long-range interaction between adatoms. Results presented in Fig.3. {width="10cm"} clearly demonstrate that the interaction between Cu adatoms is oscillatory with a period of about 24 Å corresponding to the Fermi wavelength of the majority $sp$ states of fcc Co (cf. Fig.2a). It is important to note that our studies for different number of Co layers on Cu(111) have shown that the surface-state band edge of the majority electrons increases with coverage[@c7]. Therefore, there is the possibility of tailoring the long-range interactions on Co/Cu(111) by variation of the coverage. Our results for different number of Co monolayers on Cu(111) will be presented elsewhere. Finally, we discuss our results for the hcp Co(0001) substrate. Recent spin-polarized STM experiments of Okuno et al.,[@c9] have found a spin-polarized surface state at -0.43 eV relative to the Fermi energy. Our calculations of the spectral densities for majority and minority states are shown in Fig.4. {width="14cm"} \[fig4\] One can see that a minority surface state exists and the energy of its minimum agrees well with the experiment[@c9]. Our analysis shows that this state has $d$ character similar to the minority state of fcc Co (cf. Fig.2b). However, in contrast to the fcc Co, the $sp$-majority states of the hcp Co(0001) are strongly shifted to higher energies and become unoccupied. These results suggest that the substrate-mediated interaction between adatoms on Co(0001) could be very much different from that on Cu(111) and fcc Co substrates. Our calculations presented in Fig.5 show that the interaction energy between Cu adatoms on Co(0001) decays very fast and practically vanishes for adatom-adatom separation larger than 15 Å. ![Calculated interaction energies between two Cu adatoms on the hcp Co(0001). Inset shows the interaction at large distances. []{data-label="fig5"}](fig5.ps){width="10cm"} In summary, we have performed first ab initio calculations for the substrate mediated adsorbate-adsorbate interactions on transition metal surfaces. We have found that the surface states on the fcc Co and the hcp Co substrates are spin-polarized. Our results predict that the spin-polarization of surface-state electrons can strongly affect interactions between adatoms. We reveal that mainly majority $sp$-states determine the interaction at large adatom-adatom separations. [00]{} K.H. Lau and W.  Kohn, Surf. Sci [**75**]{}, 69 (1978). J. Repp, F. Moresco, G. Meyer, K.-H. Rieder, P. Hyldgaard, and M. Persson, Phys. Rev. Lett. [**85**]{}, 2981 (2000); N. Knorr, H. Brune, M. Epple, A. Hirstein, M.A. Schneider, and K. Kern, Phys. Rev. B [**65**]{}, 115420 (2002). A. Bogicevic, S. Ovesson, P. Hyldgaard, B.I. Lundqvist, H.  Brune, and D.R.  Jennison, Phys. Rev. Lett. [**85**]{}, 1910 (2000). K.A.  Fichthorn and M. Scheffler, Phys. Rev. Lett. [**84**]{}, 5371 (2000). V.S. Stepanyuk, A.N. Baranov, D.V. Tsivlin, W. Hergert, P. Bruno, N. Knorr, M.A. Schneider, and K. Kern, Phys. Rev. B [**68**]{}, 205410 (2003). L. Diekhöner, M.A. Schneider, A.N. Baranov, V.S. Stepanyuk, P. Bruno, and K. Kern, Phys. Rev. Lett.[**90**]{}, 236801 (2003). O. Pietzsch, A. Kubetzka, M. Bode, and R. Wiesendanger, Phys. Rev. Lett. [**92**]{}, 057202 (2004). S.N. Okuno, T. Kishi, and K. Tanaka, Phys. Rev. Lett. [**88**]{}, 066803 (2002). K. Wildberger,V.S. Stepanyuk, P. Lang, R. Zeller, and P.H. Dederichs, Phys. Rev. Lett. [**75**]{}, 509 (1995).

--- abstract: 'The COVID-19 pandemic has posed a unique challenge for the world to find solutions, ranging from vaccines to ICT solutions to slow down the virus spreading. Due to the highly contagious nature of the virus, *social distancing* is one fundamental measure which has already adopted by many countries. At the technical level, this prioritises *contact tracing* solutions, which can alert the users who have been in close contact with the infected persons and meanwhile allow heath authorities to take proper actions. In this paper, we examine several existing privacy-aware contact tracing solutions and analyse their (dis)advantages. At the end, we describe several major observations and outline an interdisciplinary research agenda towards more comprehensive and effective privacy-aware contact tracing solutions.' author: - Qiang Tang bibliography: - 'ConTrack.bib' title: 'Privacy-Preserving Contact Tracing: current solutions and open questions' --- Introduction ============ The Coronavirus disease (COVID-19) pandemic, caused by the SARS-CoV-2 virus, has put the world into a panic mode. Up to now, the virus has contracted more than 2 million victims, and more than 120 thousands victims have lost their lives[^1]. Among the survivors, many of them are in critical conditions and heavily rely on medical equipments such as ventilators to survive. It is reported that many victims have tragically died due to the shortage of such equipments[^2]. While these numbers are still increasing on a daily basis, the world has united unprecedentedly to find solutions to suppress the pandemic. According to the World Health Organization (WHO), on 31 December 2019, the WHO China Country Office was informed of a pneumonia of unknown cause, detected in the city of Wuhan in Hubei province, China. Soon, similar cases, later being attributed to COVID-19, appeared in Wuhan and Hubei province rapidly. To combat the epidemic, the Chinese government adopted a variety of extreme measures, e.g. completely close the borders of villages, cities, and provinces; track and then quarantine all close contacts of COVID-19 victims; make wearing masks mandatory. From the public information, the epidemic has been successfully controlled in China and only dozens of new cases are identified daily now. In a sharp contrast, the numbers are still rapidly growing in the western democracies, where some of these extreme measures cannot be easily enforced due to the potential violation of fundamental rights. Nevertheless, in order to slow down the virus spreading and reduce the pressure on medical systems, *social distancing* is widely promoted and enforced, furthermore many countries are investigating the concept of *contact tracing*. Without careful considerations, *contact tracing* can turn into a massive surveillance tool so that individual’s privacy can be seriously damaged, see the case analysis for China [^3] and for South Korea [^4]. What makes things harder is that the perception of privacy heavily depends on the political regime and the underlying culture, see the analysis by Asghar et al. towards Singapore’s *TraceTogether* app[^5]. It remains an open problem to design privacy-aware *contact tracing* solutions which also satisfy the requirements in other dimensions. SoTA of Privacy-aware *contact tracing* --------------------------------------- Up to now, a number of *contact tracing* solutions have been introduced. For instance, China and South Korea have begun tracing COVID-19 victims and their contacts from the very early stage of the epidemic, via smartphones as well as face recognition technologies. In addition, the Korean government even made a lot of collected data public[^6]. On one hand, these mandatory tracing solutions greatly facilitate the containment of the virus, but on the other hand it also raises serious privacy concerns[^7] [@Ferretti20]. Recently, the Singapore government published an app, named *TraceTogether* [^8], which exploits some cryptographic primitives for privacy protection. This app has attracted the attention from some other countries, such as Australia, which are currently evaluating its privacy guarantee according to their own privacy regulations, see [@aus2020]. At the end of March 2020, Israel passed an emergency law to launch a smartphone app to reveal if a user was, over the previous 14 days, in close proximity to anyone who has contracted the virus. Besides the efforts at the national level, initiatives have also been started by the general public. For instance, a Pan-European Privacy-Preserving Proximity Tracing (PEPP-PT) project has been kicked off with both public and private partners from several EU countries[^9], MIT is collaborating with WHO to advocate its app named *Private Kit: Safe Paths*[^10], Google and Apple are also collaborating on new solutions[^11]. There are also more theoretical proposals, e.g. those from [@icc18; @Brack2020; @dp3t2020; @iacr2020]. In the meantime, the concept of *contact tracing* and proposed solutions have been scrutinized by commenters and researchers. For instance, Anderson provide very insightful analysis on the practical aspects of *contact tracing*[^12]. Wang gave some interesting remarks on current *contact tracing* solutions[^13], including that from Google and Apple solutions[^14]. Asghar et al. analysed Singapore’s *TraceTogether* app[^15], and Vaudenay [@Vaudenay2020] provided a detailed analysis of the DP-3T solution by Troncoso et al. [@dp3t2020]. Contribution ------------ In this paper, we aim at a deeper understanding about the utility and privacy issues associated with emerging *contact tracing* solutions, particularly those related to respiratory system diseases such as COVID-19. Our contribution lies in three aspects. 1. We analyse the application context in this COVID-19 pandemic and identify a broad set of utility and security requirements. 2. We examine several existing privacy-aware *contact tracing* solutions and analyse their (dis)advantages. These solutions include the *TraceTogether* app from Singapore[^16] and three cryptographic solutions by Reichert et al. [@iacr2020] and Altuwaiyan et al. [@icc18] and Troncoso et al. [@dp3t2020]. It is worth noting that many similar solutions exist and more apps are being developed, and we wish our analysis can be extended to them. 3. We summarize our findings into several major observations and outline an interdisciplinary research agenda towards more comprehensive and effective privacy-aware *contact tracing* solutions. Note that many of the above works have not been formally published, so that they might be updated by their authors from time to time. Nevertheless, we base our analysis on some specific versions which were available when this paper has been written, as indicated in the references. A Closer Look at Contact Tracing ================================ In the public health domain, *contact tracing* refers to the process of identification of contacts who may have come into contact with an infected victim and subsequent collection of further information about these contacts. By tracing the contacts of infected individuals, testing them for infection, treating the infected and tracing their contacts in turn, public health aims to reduce infections in the population[^17]. In practice, *contact tracing* is widely performed for diseases like sexually transmitted infections (including HIV) and virus infections (e.g. SARS-CoV and SARS-CoV-2/COVID-19). Despite some pioneering attempts in applying advanced ICT technologies, e.g. the *FluPhone* [@Yoneki11], *contact tracing* has mainly been implemented manually by medical personnel. Regardless, this approach has been proven effective in combating contiguous diseases because it can at least (1) interrupt ongoing transmission and reduce spread, alert contacts to the possibility of infection and offer preventive counseling or prophylactic care and (2) allow the medical professionals to learn about the epidemiology in a particular population. ![Quarantine Plan (taken from [@Ferretti20])[]{data-label="quarantine"}](flow) Ferretti et al. [@Ferretti20] investigated the key parameters of epidemic spread for COVID-19 and concluded that viral spread is too fast to be contained by manual *contact tracing* (see an illustration in Fig. \[quarantine\]) but could be controlled if this process was faster, more efficient and happened at scale. This necessitates digital *contact tracing* solutions, such as those based on mobile apps, which can efficiently achieve epidemic control in large scale. Interestingly, this coincides with the empirical analysis of the control measures in China, by Tian et al. [@Tian20]. In the following-up discussions, we focus on digital *contact tracing* solutions, by cautiously referring to their manual ancestors in the aspect of functional requirements. For simplicity, we omit the term “digital" in the rest of the paper. Preliminary on *contact tracing* -------------------------------- Overall, the purpose of a *contact tracing* solution is to prevent an epidemic or a pandemic caused by a contagious virus or something similar. Depending on the standing point, it may serve (at least) two purposes. - At a global level, it helps medical personnel to trace the pattern of virus spreading, produce transmission graphs, trace the origin of the virus, and so on. With adequate knowledge of the virus, the authority can take appropriate actions (e.g. disinfecting a facility) and make appropriate plans (e.g. enforcing *social distancing*) to fight against the virus and prevent future similar epidemic or pandemic. - At an individual level, it helps the medical personnel to alert individuals who might have been infected. Alternatively, it may allow an individual to evaluate his/her risks of being infected and take further actions. As a quick remark, most cryptographic solutions focus on the individual-level *contact tracing* and devoted their attention to privacy protection for individuals. For many diseases, such as COVID-19, an individual might contract the virus with either direct or indirect contact to the infected person. In case of *direct contact*, we can imagine that the droplets containing virus from the infected can fly to his mouth, nose, or eyes. In addition, the droplets might also attach to his clothes. In case of *indirect contact*, we can imagine that the infected leaves virus-droplets on a book, a chair, or any physical objects, and later on an individual might tough the object and contract the virus. Taking the COVID-19 as an example, the virus can be transmitted through either direct or indirect contacts, so that it makes comprehensive *contact tracing* a very hard problem. To design a *contact tracing* solution, the main anchor is location data. Intuitively, if two persons are located in close locations at a certain point of time then we can informally assume that they have close contact with each other[^18]. In reality, other anchoring technologies might be employed, such as face recognition and other AI-based tracking technologies. However, we do not consider them because such technologies are only deployed in very limited regions of the world. With the abundance of electronic gadgets, location data can be generated and collected in many ways, e.g. GPS, WIFI, Telcom Cell Towers, Bluetooth beacons. Broadly, we can categorize location data into two categories. - One is *absolute location* data. In this category, we can think of GPS location, location with respect to static WIFI access points and Telcom cell towers. A location data point can often be written in the form of geolocation coordinate pair. - The other is *relative location* data. In this category, we can think about the pairing of two Bluetooth-enabled smart devices, the boarding on a transportation tool such as planes, buses, cars, or ships. In this case, we can have some reference description about the location, for example both persons are on the same flight on the day XYZ. Besides the difference in collection and management, when being applied in *contact tracing* solutions, they also have very different precision and security implications. With respect to precision, *absolute location* data is often generated by external infrastructures and might not be precise enough to define “close contact" in a epidemic and pandemic. On the other hand, such location data can be collected constantly and will provide a big picture on the mobility patterns of individuals. In contrast, *relative location* could be more precise. But in order to collect such data, we need to assume a large potion of the population will install the same app to support the service. Another drawback of this type of location data is that it might be ad hoc and will not be able to provide a comprehensive view of individuals’ mobility history. Moreover, it is hard to use such data alone to study the transmission pattern of a disease in a population. It seems that, in order to facilitate the global and individual level objectives, a *contact tracing* solution should be based on fusing location data of both categories. With respect to security, two aspects are of crucial importance. One aspect is data authenticity. In some scenarios, *absolute location* data could be more authentic because a third party could offer some sort of attestation. For instance, a Telcom operator can attest the location of an individual’s smartphone. In contrast, it is harder to find attestations on an individual’s *relative location* data. However, there are exceptions. For instance, if a user presented a boarding pass for a flight, then it can be an attestation that this user is on the plane during a certain period of time. The other aspect is privacy. Naturally, we can imagine that authentic data has tight bound to the identity of an individual so that it will be more privacy invasive if disclosed. It is a difficult task to balance the authenticity and privacy of location data and some tradeoff might be inevitable. System Architecture and Requirements ------------------------------------ With respect to the existing *contact tracing* solutions, there is neither a uniform system architecture nor a defined set of participants. Nonetheless, all the potential participants can be divided into two groups. In one group, there are individual users, who either have been confirmed with infection or have the risk of being infected when in close contact with the infected. In the other group, there are third parties, which vary in specific solutions. For instance, one potential player in this group is the health authority and medical personnel, who need to evaluate the situation and help the individual users if necessary. If a *contact tracing* solution relies on smartphone apps, then the developer of this app may also be involved. In case that individual users needs to communicate with each other, then a server may be required to facilitate the communication. Without loss of generality, the system architecture can be illustrated as in Fig. \[architecture\], where (, , ,) indicate the four phases in the workflow of a *contact tracing* solution, i.e. (*initialisation*, *sensing*, *reporting*, *tracing*). ![System Architecture[]{data-label="architecture"}](ct) - In the *initialisation* phase, individual users and relevant third parties need to set up the system to enable the operations in other phases. For example, every individual user might be required to have a smart phone and download an app from a third party. In addition, cryptographic credentials may need to be generated and distributed. - In the *sensing* phase, individual users will record their own location trails and also collect location data from their close contacts. - In the *reporting* phase, if an individual user is confirmed to be infected then she needs to collaborate with some third parties to make her relevant location data available for the further uses. - In the *tracing* phase, some third parties could collect and aggregate the location data from infected individuals for any possible legitimate purposes. For instance, a third party can communicate with the close contacts of the infected, or let the uninfected individuals to evaluate the risks of being infected on their own. In comparison to other digital solutions (e.g. a general-purpose social app), a *contact tracing* solution is expected to provide stricter guarantees towards utility and security. As to utility, the solution should provide fine-grained and accurate measurement of the distance between a contact and the infected victim. The “fine-grained" requirement refers to a precise description of the occurring time and the duration of the contacting event, while the “accurate" requirement means that the error in the distance measurement should be as small as possible. Take the COVID-19 as an example, it is commonly considered that there is a risk when a contact and the infected victim stay together within 2 meters. In this case, if a GPS system has the error around 5 meters, then it should not be used alone in any solution. The security guarantee imposes requirements on the protection for both authenticity and privacy. - The “authenticity" requirement means that the reported location data from a user must be real and should not have been forged or modified by anybody including the user herself. Lacking of authenticity can cause a lot of serious issues, as happened in China and South Korea. For example, an infected victim can blackmail a shop by claiming that she has been a visitor, an infected victim can cause a social panic by claiming that she has visited some heavily populated areas such as shopping malls or train stations, a user can forge location data in order to probe the location data of infected victims and identify them given that some matching service is offered, and fake location data from infected users can mislead medical personnel in their professional activities. In addition, the “authenticity" requirement can be extended to the binding property between the location data to an individual. Quite often, smart devices are used to collect and manage location data, while such devices can be shared by several individuals. Lacking of binding can lead to some fraud activities. For example, an individual user can present another user’s device and location data to get a priority in virus testing or avoid going to work by triggering some quarantine policy on purpose. - The “privacy" requirement applies to both infected victims and other users. Except for disclosing the necessary information to the authorities, an infected victim might want to prevent any further disclosure to avoid social embarrassment or discrimination. An individual user may want to check his risk of being infected by matching his location data with that from infected, but he may not want to disclose his data to the third parties or the infected for any other purpose. Note that the DP-3T consortium recently published a summary of privacy and security threats[^19]. Analysis of Existing Solutions ============================== In this section, we provide some high-level analysis of several solutions against our formulation in Section 2. The common issues with these solutions are summarized into the major observations in Section 4. Singapore’s *TraceTogether* --------------------------- The *TraceTogether* protocol from Singapore, as recapped in [@aus2020], has two types of entities, namely users ($1 \leq i \leq N$) and the Ministry of Health (MoH) of the Singapore government. It is assumed that all users trust the MoH to protect their information. Note that, as shown below, the users are not required to share everything with MoH if they have not been in close contact with any confirmed COVID-19 victim. The protocol is elaborated below. - In the *initialisation* phase, a user $i$ downloads the *TraceTogether* app and install it on her smartphone. The app sends the phone number $NUM_i$ to MoH and receives a pseudonym $ID_i$. MoH stores the ($NUM_i, ID_i$) pair in its database. MoH generates a secret key $K$ and selects an encryption algorithm $\mathsf{Enc}$. At the beginning of the app launch, MoH decides some time intervals $[t_0, t_1, \cdots]$, which will end when the pandemic is over. For the user $i$, MoH pushes $TID_{i,x} = \mathsf{Enc}(ID_i, t_x; K)$ to user $i$’s app at the beginning of $t_x$, for $x \geq 0$. - In the *sensing* phase, user $i$ broadcasts $TID_{i,x}$ at the time interval $[t_x, t_{x+1})$ for all $x \geq 0$. For example, when user $i$ and user $j$ come into a range of Bluetooth communication at the interval $[t_x, t_{x+1})$, then they will exchange $TID_{i,x}$ and $TID_{j,x}$. They will store a $(TID_{i,x}, TID_{j,x}, Sigstren)$ locally in their smartphones, respectively. The parameter $Sigstren$ indicates the Bluetooth signal strength between their devices. - In the *reporting* phase, suppose that user $i$ has been tested positive for COVID-19, then she is obliged to share with MoH the locally-stored pairs $(TID_{i,x}, TID_{j,x})$ for all relevant $j$ and $x$. - In the *tracing* phase, after receiving the pairs from user $i$, MoH decrypts every $TID_{j,x}$ and obtains $ID_j$. Based on $ID_j$, MoH can looks up $NUM_j$ and then contact user $j$ for further instructions. Whether or not to install the *TraceTogether* app is a voluntary choice for the Singapore residents, and it is not clear how many installations have been made until this moment. So far, we have not found any official information showing how much this app has contributed to Singapore’s war against COVID-19. Soon after its launch, Asghar et al. pointed out some privacy concerns[^20]. We have the following additional comments. With *TraceTogether*, individual users are required to put more trust on the third party - MoH, than in other solutions. Note that this might be a result of the political regime and cultural status of Singapore. To prevent a curious MoH from learning unnecessary mobility information, the mobility data of low-risk individuals are not required to be uploaded to MoH. However, things will change when more and more individual users are infected and diagnosed. By then, MoH will have decrypted data for most of the individuals and learned their *relative location* data at certain points of time. If a big portion of the Singapore population has deployed the app, and then the location data on their smartphones would provide MoH a very clear mobility view of the whole Singapore population. Without using secure hardware or other trusted computing technology, a malicious user can potentially manipulate the location data collected by the app, e.g. delete or add entries. Moreover, an attacker can mount relay attacks, e.g. to relay the Bluetooth signal from Alice’s smartphone to Bob’s smartphone even when they are far from each other. To this end, some other attacks, demonstrated in Vaudenay’s analysis [@Vaudenay2020] against DP-3T solution by Troncoso et al. [@dp3t2020], can also apply. Regarding the solutions employing Bluetooth for distance measurement, one attack deserves special attention. An attacker can place a *Bluetooth range extender* in a relatively populated place, such as a city square, and as a result it will make any pair of users a close contact to each other. Such an attack is easy to mount and will seriously distort the functionality of the underlying solution. In comparison to other solutions, one advantage of *TraceTogether* is that it offers MoH the ability to draw the transmission graph of COVID-19 in the population which has installed the app. It can be easily done based on the fact that the temporary identifiers are linked to the phone numbers, which can help MoH identify the individual users. This advantage is an outcome of the tradeoff between privacy and utility. Reichert et al.’s MPC Solution ------------------------------ In the solution proposed by Reichert et al. [@iacr2020], it is assumed that user $i$ $(1 \leq i \leq N)$ possesses a smart device that can collect and store geolocation data. In addition, a Health Authority (HA) will collect the geolocation history of all COVID-19 victims and offers data matching as a service. - In the *initialisation* phase, HA prepares the cryptographic key materials for generating garbled circuits for later use. - In the *sensing* phase, user $i$ records her geolocation data points on the fly. Let the time intervals be denoted by $[t_0, t_1, \cdots]$. At time $t_x$, user $i$ generates and stores a tuple $(t_x, l_{x,u}, l_{x,v})$, where $l_{x,u}$ and $l_{x,v}$ represent the latitude and longitude of the location, respectively. - In the *reporting* phase, if user $i$ has been tested positive for COVID-19, then she shares with HA her $(t_x, l_{x,u}, l_{x,v})$ for all relevant $x$. - In the *tracing* phase, suppose user $j$ wants to check whether he has been in close contact with any COVID-19 victim. For any COVID-19 victim user $i$, HA constructs a garbled circuit based on her geolocation data points and shares the circuit with user $j$, who can then retrieve some key materials from the HA and privately evaluate the circuit based on his own geolocation data points. Note that user $j$ is assumed to have high risk if, for some of her data point(s), both the time stamps and the geolocation coordinates are close to one of that from user $i$. This is a theoretical cryptographic solution, which covers the matching between an infected and other individual users while ignoring other aspects of a *contact tracing* solution. With respect to the cryptographic design, scalability will be a bottleneck. For any user $j$, HA will need to prepare garbled circuits for all the infected victims, and interact with user $j$ in the *tracing* phase. When the sizes of infected populations and the requesters become large, the complexity for HA will be formidable. In addition, malicious users might leverage this to mount (D)DoS attacks, unless proper countermeasures are deployed. Another issue is the lacking of details on the computation of infection risks, which are based on the proximity of both stamps and the geolocation coordinates. It is unclear how to set a threshold on the time stamps, considering there are a variety of mobility patterns between the concerned users. Altuwaiyan et al.’s Matching Solution ------------------------------------- In the solution proposed by Altuwaiyan et al. [@icc18], it is assumed that user $i$ $(1 \leq i \leq N)$ possesses a smart device that can exchange information (e.g. Bluetooth messages) with similar devices nearby. In addition, a server will collect the data of all infected victims and offer data matching as a service. - In the *initialisation* phase, there is no special setup for the server and users. - In the *sensing* phase, user $i$ exchanges information with similar devices nearby. Let the time intervals be denoted by $[t_0, t_1, \cdots]$. At time $t_x$, user $i$ generates and stores a tuple of data points $(t_x, (m_{i,1}, r_{i,1}, p_{i,1}), \cdots, (m_{i,n_{i,x}}, r_{i,n_{i,x}}, p_{i,n_{i,x}}))$, where $(m_{i,1}, r_{i,1}, p_{i,1})$ records information about the first encountered device, where $m_{i,1}$ is the hashed identifier of this device, $r_{i,1}$ is the detected signal strength and $p_{i,1}$ is the type of device, and so on. - In the *reporting* phase, if user $i$ has been tested positive, then she shares with the server her data $(t_x, (m_{i,1}, r_{i,1}, p_{i,1}), \cdots, (m_{i,n_{i,x}}, r_{i,n_{i,x}}, p_{i,n_{i,x}})$ for all relevant $x$. - In the *tracing* phase, suppose user $j$ wants to check whether he has been in close contact with some infected victim, he generates a public/private key pair $(pk,sk)$ for a homomorphic encryption scheme. Then, user $j$ sends the timestamps in his data to the server, which will find all the infected users who have some overlapped timestamps. For each such infected user $i$ and overlapped timestamp $t_x$, the server and user $j$ perform the following protocol. 1. User $j$ sends $(\mathsf{Enc}(m_{j,1}, pk), \cdots, \mathsf{Enc}(m_{j,n_{j,x}}, pk))$ to the server. 2. The server computes the following matrix and sends it to user $j$. $\mathsf{Rand}(\mathsf{Enc}(m_{j,1}-m_{i,1}, pk))$ $\cdots$ $\mathsf{Rand}(\mathsf{Enc}(m_{j,n_{j,x}}-m_{i,1}, pk))$ ---------------------------------------------------------- ---------- ---------------------------------------------------------------- $\vdots$ $\vdots$ $\vdots$ $\mathsf{Rand}(\mathsf{Enc}(m_{j,1}-m_{i,n_{i,x}}, pk))$ $\cdots$ $\mathsf{Rand}(\mathsf{Enc}(m_{j,n_{j,x}}-m_{i,n_{j,x}}, pk))$ In the table, $\mathsf{Rand}$ is a ciphertext randomization function. If the ciphertext encrypts 0 then the randomized ciphertext still encrypts 0, otherwise the randomized ciphertext will encrypt a random number. 3. User $j$ decrypts every ciphertext in the received matrix, and obtains the $m_{j,y}$, where $1 \leq y \leq n_{j,x}$, which is overlapped with that from user $i$. Then, user $j$ sends all the matched $m_{j,y}$ together with $r_{j,y}$ to the server. 4. Based on the data from user $j$, the server computes the distance between user $i$ and user $j$, and then acts accordingly. We note that, in addition to the aforementioned privacy-aware matching protocol, the other contribution of Altuwaiyan et al. [@icc18] is a new method to measure the distance between two smart devices. In comparison to other solutions where the distance is measured based on the perceived signal strength of the peer device, the method leverages on the signal strengths of more devices so that it is more accurate in practice. Regarding privacy, the infected victims reveal the location data to the server, where the location data contain hashed network identifiers such as those of WIFI access points. Give that network identifiers are often static, this allows the server to recover the *absolute location* data points of the infected users. If a match has been found in the *tracing* phase, then user $j$’s *absolute location* at some time stamps are also revealed to the server. In addition, in order to improve computational efficiency, time stamps are always disclosed to the server. The revelation of time stamps and *absolute location* implies serious privacy leakage, and should be avoided. DP-3T Solution -------------- In the solution proposed by Troncoso et al. [@dp3t2020], it is assumed that user $i$ $(1 \leq i \leq N)$ possesses a smart device that can collect and store data. In addition, there is a backend server, and the Health Authority (HA). Note that the backend server acts as a communication platform to facilitate the matching activities among the users. Let $\mathsf{H}$, $\mathsf{PRG}$ and $\mathsf{PRF}$ denote a cryptographic hash function, a pseudorandom number generator and a pseudorandom function, respectively. - In the *initialisation* phase, user $i$ generates a random initial daily key $SK_{i,0}$, and computes the following-up daily keys based on a chain of hashes: i.e. the key for day 1 is $SK_{i,1}=\mathsf{H}(SK_{i,0})$ and the key for day $x$ is $SK_{i,x}=\mathsf{H}(SK_{i,x-1})$. Suppose $n$ ephemeral identifiers are required in one day, then the identifiers for user $i$ on the day $x$ are generated as follows: $$EphID_{i, x, 1}||\cdots||EphID_{i, x, n}= \mathsf{PRG}(\mathsf{PRF}(SK_{i,x}, ``broadcast key"))$$ - In the *sensing* phase, on the day $x$, user $i$ broadcasts the ephemeral identifiers $\{EphID_{i, x, 1}, \cdots, EphID_{i, x, n}\}$ in a random order. At the same time, her smart device stores the received ephemeral identifiers together with the corresponding proximity (based on signal strength), duration, and other auxiliary data, and a coarse time indication (e.g., “The morning of April 2”). - In the *reporting* phase, if user $i$ has been tested positive for COVID-19, then HA will instruct her to send $SK_{i,x}$ to the backend server, where $x$ is the first day that user $i$ becomes infectious. After sending the $SK_{i,x}$ to the backend server, user $i$ chooses a new daily key $SK_{i,y}$ depending on the day when this event occurs. While not mentioned in [@dp3t2020], we believe this new key should also be sent to the backend server, as user $i$ might continue to be infectious. - In the *tracing* phase, periodically, the backend server broadcasts $SK_{i,x}$ after user $i$ has been confirmed with the infection. On receiving $SK_{i,x}$, user $j$ can recompute the ephemeral identifiers for day $x$ as follows $$\mathsf{PRG}(\mathsf{PRF}(SK_{i,x}, ``broadcast key")).$$ Similarly, user $j$ can compute the identifiers for day $x+1$ and so on. With the ephemeral identifiers, user $j$ can check whether any of the computed identifiers appears in her local storage. Based on the associated information, namely “proximity, duration, and other auxiliary data, and a coarse time indication", user $j$ can act accordingly. Vaudenay [@Vaudenay2020] provided detailed security and privacy analysis against this solution. It argues that decentralisation introduces new attack vectors against privacy, contrasting to the common belief that decentralisation helps solve the privacy concerns in centralised systems. One of the conclusions from [@Vaudenay2020] is that trusted computing technology seems unavoidable in order to prevent all the identified attacks. As noted by Vaudenay [@Vaudenay2020], a lot of practical details are missing from the whitepaper [@dp3t2020]. For instance, it is unclear how to instantiate the backend server. With regard to its pan European ambition, it is not clear how HAs and backend servers from different countries can efficiently coordinate with each other to take prompt responses. It also remains open how users’ privacy will be infected if the credentials of some HAs and backend servers are compromised. We want to point out that the hashchain-based method of generating daily keys, see more information in the *initialisation* phase, brings linkability risks when an individual’s credential is leaked to an attacker, e.g. via malware. For example, if the attacker learns $SK_{i,x}$ then it can compute identifiers for day $x+1$ and so on. Then, the attacker can link user $i$ to his broadcasted identifiers, either collected by the attacker itself or or bought from other attackers. This could lead to disclosure of *absolute location* data points of user $i$. By design, the DP-3T protocol favors more the privacy for user $j$ than the infected user $i$. Referring to the *tracing* phase description, if an identifier $EphID^*$ appears in his local storage user $j$ learns the exact time stamp when the perceived risk occurs. This can very likely enable user $j$ to link $EphID^*$ to the real person, whose device has broadcasted $EphID^*$, due to the fact that user $j$ can be close to very few users at a specific time stamp. Moreover, this also raises a concern of *targeted identification attack*. If an attacker wants to find out whether or not some users have been infected, then he can simply get close enough to them and make their devices exchange identifiers (of course he should record the time stamps), and finally he can make a decision if identifiers have been matched at these time stamps. In contrast, the *TraceTogether* avoids this problem by letting MoH calculate the infection risk for encountered users. Overall, the DP-3T protocol leaks too much unnecessary information about the infected users to the public and raises serious privacy concerns without deploying any further countermeasure. One possible solution is to deploy a two party secure computation protocol between the backend server and user $j$, for the latter to only learn a risk score. How to scale this up is a very practical problem to be addressed. Discussion and a Future Research Roadmap ======================================== After examining many privacy-aware *contact tracing* solutions, we feel that the situation is a bit chaotic at this moment. There is a rush for the academia to propose new technical solutions and for the industry to launch new apps, and new initiatives pop up every day. These efforts will somehow contribute to the fight against the pandemic, at least raise the awareness and help start discussions. Unfortunately, it is a pity that many solutions are based on assumptions which are unrealistic and hamper their usefulness and massive adoption. In this section, we first discuss our major observations, partly based on the analysis in Section 3, and then put forward a roadmap which foresees an interdisciplinary research agenda for the future. The first major observation is that many solutions only focused on the *direct contact* scenario and paid little attention to the precision of location data and the accuracy of distance measurement (e.g. those based on Bluetooth signal strength). In addition, a number of exceptional situations have not been taken into account. For instance, the neighbours of an infected and quarantined victim can be in very close range and be classified to be of high risk, while the truth is that they are in different apartments or houses so that the risk of being infected is low. Inaccurate distance estimation and false risk alerts will cause unnecessary panic in the population and also waste a lot of resources in the healthcare system to address the fake suspects. Moreover, the scenario of *indirect contact* has been ignored most of the time. Due to the fact that COVID-19 can be transmitted via indirect contact. Ignoring this scenario makes the existing *contact tracing* solutions less practical and effective than expected. The second major observation is that the scope of *contact tracing* in existing solutions is very narrow. It is mostly about an individual evaluates his risk of being infected based on his contact history with the infected victims. However, as we have described in the beginning of Section 2 and Section 2.1, *contact tracing* is supposed to enable health authority and medical personnel not only to study the epidemic or pandemic at the global level but also to help the individual at the individual level. This raises an open question how the health authority and medical personnel can collect the necessary information and perform their normal duties. Inevitably, new privacy-aware solutions need to be designed and implemented to fulfill the objectives of a comprehensive *contact tracing* yet minimizes information disclosure. One root problem, facing both existing and new solutions, is that a consensus on the trust relationship between the different players in the scope of *contact tracing* is missing. When an app is deployed, more practical questions can occur. For example, how the results from the app of Google and Apple[^21] should be interpreted, how the liability is distributed, and can an individual interact with the health authority and medical personnel on the basis of these results? The third major observation is that most solutions only emphasized the privacy concerns in *contact tracing* while paying no attention to the authenticity aspect (elaborated in Section 2.2). It implies that individual users can potentially forge location data for herself and for others as well. Consequently, the results of *contact tracing* could have been manipulated to a large extent, and enable the dishonest users and attackers to disrupt the service and commit various fraud activities. Of particular importance is that the lacking of authenticity can also lead to the breach of privacy, as demonstrated by Vaudenay [@Vaudenay2020]. How to balance authenticity and privacy seems to be the biggest challenge in designing a privacy-aware and effective *contact tracing* solution. The fourth major observation is that most theoretical solutions have not provided all the technical details to facilitate an implementation. In many solutions, an un-trusted backend server is required. It is unclear how this server can be chosen in practice and how to incentivize it to perform in the same way as what has specified. Furthermore, a health authority is often involved and required to perform some cryptographic operations. This seems to be an unrealistic requirement for an governmental authority in many countries. At least, it will not be easily done in a short period of time. To fight against a pandemic, like COVID-19, every technology and solution could matter. Nevertheless, we believe it is also important to set an interdisciplinary agenda to comprehensively formulate the problem, identify the requirements, and search for the opportunities. We foresee the following key research elements on the roadmap. - Bridge the gap between health authority and solution designers. This essentially requires the solution designers to figure out what an effective *contact tracing* solution needs to generate, for both individual users, health authority and the medical personnel. Accordingly, different players’ roles should be clearly defined. Decisions should be drawn on the basis of regulations (at least) related to healthcare and privacy. - Evaluate and model the privacy and other security risks. This will need to clarify the trust relationships among the players and result in a set of security requirements, which should be satisfy by a solution. It should also reflect the accountability and liability configuration among the players. Various tradeoffs could be inevitable among privacy, utility, efficiency and other aspects. - Build incentive mechanisms into solutions from the beginning. Instead of simply providing a dichotomy choice through “opt-in" and “opt-out", it is important to incentivize the participation of individual users and other players, e.g. by employing technologies such as DLT or Blockchain. It is also important to deploy mechanisms to deter dishonest and malicious behaviours and encourage honest behaviour for the society good. In particular, it should prevent the solutions from being used in any manner as a surveillance tool for either political or economic purposes. Acknowledgement {#acknowledgement .unnumbered} =============== This work is partially funded by the European Unions Horizon 2020 SPARTA project, under grant agreement No 830892. [^1]: <https://www.worldometers.info/coronavirus/> [^2]: <https://en.wikipedia.org/wiki/2019-20_coronavirus_pandemic> [^3]: <https://www.nytimes.com/2020/03/01/business/china-coronavirus-surveillance.html> [^4]: <https://www.nature.com/articles/d41586-020-00740-y> [^5]: <http://tiny.cc/fljqmz> [^6]: <https://coronamap.site/> [^7]: <https://www.nature.com/articles/d41586-020-00740-y> [^8]: <http://tiny.cc/onlqmz> [^9]: <https://www.pepp-pt.org/> [^10]: <http://safepaths.mit.edu/> [^11]: <https://www.apple.com/covid19/contacttracing/> [^12]: [<http://tiny.cc/2z0zmz>]{} [^13]: [http://tiny.cc/tx0zmz]{} [^14]: <https://www.apple.com/covid19/contacttracing/> [^15]: <http://tiny.cc/fljqmz> [^16]: <http://tiny.cc/onlqmz> [^17]: <https://en.wikipedia.org/wiki/Contact_tracing> [^18]: Note that there are exceptions though. [^19]: <https://github.com/DP-3T/documents> [^20]: <http://tiny.cc/fljqmz> [^21]: <https://www.apple.com/covid19/contacttracing/>

--- abstract: 'Curves in ${\mathbb R}^n$ for which the ratios between two consecutive curvatures are constant are characterized by the fact that their tangent indicatrix is a geodesic in a flat torus. For $n= 3,4$, spherical curves of this kind are also studied and compared with intrinsic helices in the sphere.' address: 'Dep. de Geometria i Topologia, Universitat de València, Avd. Vicent Andrés Estellés, 1, E-46100-Burjassot (València), Spain' author: - 'J. Monterde' date: 'December 14, 2004' title: Curves with constant curvature ratios --- [^1] Introduction ============ The notion of a generalized helix in ${\mathbb R}^3$, a curve making a constant angle with a fixed direction, can be generalized to higher dimensions in many ways. In [@RS] the same definition is proposed but in ${\mathbb R}^n$. In [@Ha] the definition is more restrictive: the fixed direction makes a constant angle with all the vectors of the Frenet frame. It is easy to check that this definition only works in the odd dimensional case. Moreover, in the same reference, it is proven that the definition is equivalent to the fact that the ratios $\frac{k_2}{k_1},\frac{k_4}{k_3}, \dots $, $k_i$ being the curvatures, are constant. This statement is related with the Lancret Theorem for generalized helices in ${\mathbb R}^3$ (the ratio of torsion to curvature is constant). Finally, in [@Ba] the author proposes a definition of a general helix in a $3$-dimensional real-space-form substituting the fixed direction in the usual definition of generalized helix by a Killing vector field along the curve. In this paper we study the curves in ${\mathbb R}^n$ for which all the ratios $\frac{k_2}{k_1},\frac{k_3}{k_2},\frac{k_4}{k_3}, \dots $ are constant. We call them curves with constant curvature ratios or ccr-curves. The main result is that, in the even dimensional case, a curve has constant curvature ratios if and only if its tangent indicatrix is a geodesic in the flat torus. In the odd case, a constant must be added as the new coordinate function. In the last section we show that a ccr-curve in $S^3$ is a general helix in the sense of [@Ba] if and only if it has constant curvatures. To achieve this result, we have obtained the characterization of spherical curves in ${\mathbb R}^4$ in terms of the curvatures. Moreover, we have also found explicit examples of spherical ccr-curves with non-constant curvatures. Frenet’s elements for a curve in ${\mathbb R}^n$ ================================================ Let us recall from [@Kl] the definition of the Frenet frame and curvatures. For $C^{n-1}$ curves, $\alpha$, which have linearly independent derivatives up to order $n-1$, the moving Frenet frame is constructed as it were in usual space using the Gram-Schmidt process. Orthonormal vectors $\{\overrightarrow{{\bf e_1}},\overrightarrow{{\bf e_2}},\dots, \overrightarrow{{\bf e_{n-1}}}\}$ are obtained and the last vector is added as the unit vector in ${\mathbb R}^n$ such that $\{\overrightarrow{{\bf e_1}},\overrightarrow{{\bf e_2}},\dots, \overrightarrow{{\bf e_{n}}}\}$ is an orthonormal basis with positive orientation. The $i$th curvature is defined as $$k_i= \frac{<\dot{\overrightarrow{{\bf e_i}}},\overrightarrow{{\bf e_{i+1}}}>}{||\alpha'||},$$ for $i = 1 ,\dots, n-1$. Frenet’s formulae in $n$-space can be written as $$\label{formules-Frenet} \begin{pmatrix} \dot{\overrightarrow{{\bf e_1}}}(s)\\ \dot{\overrightarrow{{\bf e_2}}}(s)\\ \dot{\overrightarrow{{\bf e_3}}}(s)\\ \vdots \\ \dot{\overrightarrow{{\bf e_{n-1}}}}(s) \\ \dot{\overrightarrow{{\bf e_{n}}}}(s) \end{pmatrix} = \begin{pmatrix} 0 & k_1 & 0 & 0 & \dots & 0 & 0 \\ -k_1 & 0 & k_2& 0 & \dots & 0 & 0 \\ 0 & -k_2 & 0 & k_3 & \dots & 0 & 0 \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0 & 0 & 0 & 0 & \dots & 0 & k_{n-1} \\ 0 & 0 & 0 & 0 & \dots & -k_{n-1} & 0 \\ \end{pmatrix} \begin{pmatrix} {\overrightarrow{{\bf e_1}}}(s)\\ {\overrightarrow{{\bf e_2}}}(s)\\ {\overrightarrow{{\bf e_3}}}(s)\\ \vdots \\ {\overrightarrow{{\bf e_{n-1}}}}(s) \\ {\overrightarrow{{\bf e_{n}}}}(s) \end{pmatrix}.$$ In accordance with [@RS] we will say that a curve is twisted if its last curvature, $k_{n-1}$ is not zero. Sometimes, we will also say that the curve is not regular. ccr-curves ========== Instead of looking for curves making a constant angle with a fixed direction as in [@Ha] or [@RS], we will study another way of generalizing the notion of helix. A curve $\alpha:I\to {\mathbb R}^n$ is said to have constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients $\frac{k_{i+1}}{k_i}$ are constant. As is well known, generalized helices in ${\mathbb R}^3$ are characterized by the fact that the quotient $\frac\tau\kappa$ is constant (Lancret’s theorem). It is in this sense that ccr-curves are a generalization to ${\mathbb R}^n$ of generalized helices in ${\mathbb R}^3$. In [@Ha] the author defines a generalized helix in the n-dimensional space (n odd) as a curve satisfying that the ratios $\frac{k_2}{k_1},\frac{k_4}{k_3}, \dots $ are constant. It is also proven that a curve is a generalized helix if and only if there exists a fixed direction which makes constant angles with all the vectors of the Frenet frame. Obviously, ccr-curves are a subset of generalized helices in the sense of [@Ha]. Examples -------- ### Example with constant curvatures The subset of ${\mathbb R}^{2n}$ parametrized by $\overrightarrow{\bf x}(u_1,u_2,\dots,u_n) = $ $$= (r_1\cos(u_1),r_1\sin(u_1), r_2\cos(u_2),r_2\sin(u_2),\dots,r_n\cos(u_n),r_n\sin(u_n))$$ where $u_i\in{\mathbb R}$ is called a flat torus in ${\mathbb R}^{2n}$. By analogy, the subset of ${\mathbb R}^{2n+1}$ parametrized by $\overrightarrow{\bf x}(u_1,u_2,\dots,u_n) = $ $$= (r_1\cos(u_1),r_1\sin(u_1), r_2\cos(u_2),r_2\sin(u_2),\dots,r_n\cos(u_n),r_n\sin(u_n),a)$$ where $u_i\in{\mathbb R}$ and $a$ is a real constant, will be called a flat torus in ${\mathbb R}^{2n+1}$. It is just a matter of computation to show that any curve in a flat torus of the kind $$\alpha(t) = \overrightarrow{\bf x}(m_1 t,m_2 t,\dots,m_n t)$$ has all its curvatures constant (see [@Ro]). These curves are the geodesics of the flat tori, and it is proven in the cited paper that they are twisted curves if and only if the constants $m_i\ne m_j$ for all $i\ne j$. ### Example with non-constant curvatures Now, let $k(s)$ be a positive function. Let us define $g(s) = \int_0^sk(u) du$. If $\alpha$ is a curve parametrized by its arc-length and with constant curvatures, $a_1,a_2,\dots,a_{n-1}$, then the curve $\beta(s) = \int_0^s \overrightarrow{\bf e_1}^\alpha(g(u)) du$ is a curve whose curvatures are $k_i(s) = a_i k(s)$. Note that $\dot{\beta}(s) = \overrightarrow{\bf e_1}^\alpha(g(s))$. This implies that $\overrightarrow{\bf e_1}^\beta(s) = \overrightarrow{\bf e_1}^\alpha(g(s))$. Taking derivatives $k_1^\beta(s) \overrightarrow{\bf e_2}^\beta(s) = k_1^\alpha(g(s)) \overrightarrow{\bf e_2}^\alpha(g(s))k(s)$. Therefore, $$\overrightarrow{\bf e_2}^\beta(s) = \overrightarrow{\bf e_2}^\alpha(g(s)), \qquad \text{and}\qquad k_1^\beta(s) = a_1 k(s).$$ By similar arguments it is possible to show that $k_i^\beta(s) = a_i k(s)$ for any $i =1 ,\dots, n-1$. Therefore, $\beta$ is a ccr-curve with non-constant curvatures. In the next section we will show that any ccr-curve is of this kind. Solving the natural equations for ccr-curves ============================================ The Frenet formulae can be explicitly integrated only for some particular cases. Ccr-curves are one of these. In fact, Frenet’s formulae are $$\begin{pmatrix} \dot{\overrightarrow{{\bf e_1}}}(s)\\ \dot{\overrightarrow{{\bf e_2}}}(s)\\ \dot{\overrightarrow{{\bf e_3}}}(s)\\ \vdots \\ \dot{\overrightarrow{{\bf e_{n-1}}}}(s) \\ \dot{\overrightarrow{{\bf e_{n}}}}(s) \end{pmatrix} = k_1(s) \begin{pmatrix} 0 & 1 & 0 & 0 & \dots & 0 & 0 \\ 1 & 0 & c_2& 0 & \dots & 0 & 0 \\ 0 & -c_2 & 0 & c_3 & \dots & 0 & 0 \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0 & 0 & 0 & 0 & \dots & 0 & c_{n-1} \\ 0 & 0 & 0 & 0 & \dots & -c_{n-1} & 0 \\ \end{pmatrix} \begin{pmatrix} {\overrightarrow{{\bf e_1}}}(s)\\ {\overrightarrow{{\bf e_2}}}(s)\\ {\overrightarrow{{\bf e_3}}}(s)\\ \vdots \\ {\overrightarrow{{\bf e_{n-1}}}}(s) \\ {\overrightarrow{{\bf e_{n}}}}(s) \end{pmatrix},$$ for some constants, $c_2,\dots, c_{n-1}$. Reparametrization of the curve allows that system to be reduced to an easier one. The reparametrization is given by the inverse function of $$g(s) = \int_0^s k_1(u)du.$$ Note that $t=g(s)$ is a reparametrization because $k_1$ is a positive function. The reparametrization we need is the inverse function $s = g^{-1}(t)$. It is a simple matter to verify that, with respect to parameter $t$, the Frenet’s formulae are reduced to a linear system of first order differential equations with constant coefficients $$\label{eqs-coefs-const} \begin{pmatrix} \overrightarrow{\bf e_1}'(t)\\ \overrightarrow{\bf e_2}'(t)\\ \overrightarrow{\bf e_3}'(t)\\ \vdots \\ \overrightarrow{\bf e_{n-1}}'(t) \\ \overrightarrow{\bf e_n}'(t) \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 & 0 & \dots & 0 & 0 \\ 1 & 0 & c_2& 0 & \dots & 0 & 0 \\ 0 & -c_2 & 0 & c_3 & \dots & 0 & 0 \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0 & 0 & 0 & 0 & \dots & 0 & c_{n-1} \\ 0 & 0 & 0 & 0 & \dots & -c_{n-1} & 0 \\ \end{pmatrix} \begin{pmatrix} {\overrightarrow{{\bf e_1}}}(t)\\ {\overrightarrow{{\bf e_2}}}(t)\\ {\overrightarrow{{\bf e_3}}}(t)\\ \vdots \\ {\overrightarrow{{\bf e_{n-1}}}}(t) \\ {\overrightarrow{{\bf e_{n}}}}(t) \end{pmatrix}.$$ We can apply the well-known methods of integration of systems of linear equations with constant coefficients. Let $F_n$ be the matrix of constant coefficients of this system. Eigenvalues and their multiplicity ---------------------------------- The first thing we have to do is to compute the eigenvalues of the coefficient matrix. Due to the skewsymmetry of the matrix, it can have not real eigenvalues other than zero. Due to the fact that the determinant of $F_n$ vanishes only for odd $n$, we can say that for odd dimensions, $0$ is an eigenvalue, whereas for even dimensions, $0$ is an eigenvalue only if $k_{n-1} = 0$. By definition, we have that constants $c_2,c_3,\dots,c_{n-2}$ are not zero. If the last constant, $c_{n-1}$, vanishes, then the same happens with the last curvature function $k_{n-1}$. In this case the curve is included in a hyperspace, so we can consider it to be a curve in an $n-1$ dimensional space. Therefore, from now on, we shall consider that all the curvatures, and then all the constants $c_i$, are not zero. Note that, in this case, for any $x\in {\mathbb C}$, the rank (in ${\mathbb C}$) of the matrix $$\begin{pmatrix} x & 1 & 0 & 0 & \dots & 0 & 0 \\ 1 & x & c_2& 0 & \dots & 0 & 0 \\ 0 & -c_2 & x & c_3 & \dots & 0 & 0 \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0 & 0 & 0 & 0 & \dots & x & c_{n-1} \\ 0 & 0 & 0 & 0 & \dots & -c_{n-1} & x \\ \end{pmatrix}$$ is at least $n-1$. Therefore, their eigenvalues are all of multiplicity $1$. Canonical Jordan form --------------------- Let $a_\ell\pm {\bf i} b_\ell$, $\ell = 1, \dots, [\frac n2]$, with $a_\ell,b_\ell\in{\mathbb R}$, be the non-zero eigenvalues of the coefficient matrix. Therefore, for $n= 2k$, the associated canonical Jordan form is of the kind $$\begin{pmatrix} J_1 & 0 & \dots & 0 \\ 0 & J_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \dots &J_k \\ \end{pmatrix}$$ where $J_\ell = \begin{pmatrix} a_\ell & -b_\ell \\ b_\ell & a_\ell \end{pmatrix}$. The matrix can be diagonalized because all the eigenvalues are of multiplicity one. Therefore, there is a orthogonal matrix, $S$, such that if $C$ is the matrix of constant coefficients, then $$C = S^{-1}JS.$$ Therefore, the general solution of the system for the first vector is $${\overrightarrow{\bf e}_1}(u) := \sum_{\ell=1}^{k}\overrightarrow{A_\ell}\ e^{a_\ell u}\cos(b_\ell\ u) +\overrightarrow{B_\ell}\ e^{a_\ell u} \sin(b_\ell\ u),$$ where $\{\overrightarrow{A_\ell},\overrightarrow{B_\ell}\}_{\ell = 1}^k$ is a family of orthogonal vectors. For $n= 2k+1$, the associated canonical Jordan form is of the kind $$\begin{pmatrix} 0 & 0 & 0 & \dots & 0 \\ 0 & J_1 & 0 & \dots & 0 \\ 0 & 0 & J_2 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \dots &J_k \\ \end{pmatrix}$$ Now, the general solution of the system for the first vector is $${\overrightarrow{\bf e}_1}(u) := \overrightarrow{A_0}+ \sum_{\ell=1}^{k}\overrightarrow{A_\ell}\ e^{a_\ell u}\cos(b_\ell\ u) +\overrightarrow{B_\ell}\ e^{a_\ell u}\sin(b_\ell\ u),$$ where $\{\overrightarrow{A_0}\}\cup \{\overrightarrow{A_\ell},\overrightarrow{B_\ell}\}_{\ell = 1}^k$ is a family of orthogonal vectors. The eigenvalues are pure imaginaries ------------------------------------ Condition $||{\overrightarrow{\bf e}_1}(u)|| = 1$ for all $u$ implies that all the real parts of the eigenvalues are zero. Indeed, if, for example, $a_1 \ne 0$, then let $m$ be a non-zero coordinate of $\overrightarrow{A_1}$. Bearing in mind that $$|m|\ e^{a_1 u}\ |\cos(b_1 u)| \le ||{\overrightarrow{\bf e}_1}(u)||,$$ and that the left-hand member is an unbounded function, then $||{\overrightarrow{\bf e}_1}(u)||\ne 1$. Therefore, all the real parts of the eigenvalues are zero and the general solution (in the even case) of the system for the first vector is $${\overrightarrow{\bf e}_1}(u) := \sum_{\ell=1}^{k}\overrightarrow{A_\ell} \cos(b_\ell\ u) +\overrightarrow{B_\ell} \sin(b_\ell\ u).$$ Analogously for the odd case. Moreover, let us recall that the vectors $\{\overrightarrow{A_i},\overrightarrow{B_i}\}_{i=1}^k$ are an orthogonal base of ${\Bbb R}^n$ associated to the canonical Jordan form. The main result --------------- Finally, an isometry of ${\mathbb R}^n$ allows us to state the next result. A curve has constant curvature ratios if and only if its tangent indicatrix is a twisted geodesic on a flat torus. Note that in the odd dimensional case this result implies that the last coordinate of the tangent indicatrix is a constant. So, there is a direction making a constant angle with the curve. Nevertheless, this is not the case in the even dimensional case. There are no fixed directions making a constant angle with the tangent vector. When all the curvatures are constant, then the curve is also a ccr-curve and its tangent indicatrix is of the kind described in the previous statement. Moreover, the reparametrization $g(s) = \int_0^s k_1(u)du$ is just the product by a constant. Since the integration of a geodesic on a flat torus in ${\mathbb R}^{2k}$ with respect to its parameter is again a curve of the same kind, we get the following corollary: A curve has constant curvatures if and only if it is 1. a twisted geodesic on a flat torus, in the even dimensional case, or 2. a twisted geodesic on a flat torus times a linear function of the parameter, in the odd dimensional case. $n = 3$ ------- The eigenvalues of the matrix of coefficients are $0$ and $\pm \sqrt{1+c^2}\ {\bf i}$ ($c= c_2$, to simplify). Therefore, the general solution of the system for the first vector is $${\overrightarrow{{\bf e_1}}}(u) =\overrightarrow{A_1} +\overrightarrow{A_2} \cos(\sqrt{1+c^2} u) +\overrightarrow{A_3} \sin(\sqrt{1+c^2} u),$$ where $\overrightarrow{A_i}, i = 1,2,3$ are constant vectors. Once we have the tangent vector, we only have to undo the reparametrization and to integrate to obtain the curve $$\alpha(s) = x_0+ \overrightarrow{c_1} s +\overrightarrow{c_2}\int_0^s \cos(\sqrt{1+c^2}g(v))dv + \overrightarrow{c_3} \int_0^s\sin(\sqrt{1+c^2}g(v))dv.$$ $n = 4$ ------- The eigenvalues are $$\pm \frac {\bf i}{\sqrt{2}} \sqrt{ 1+ c_2^2 + c_3^2 \pm \sqrt{(1 + c_2^2 + c_3^2)^2-4 c_3^2}}.$$ Therefore, the general solution of the system for the first vector is $${\overrightarrow{\bf e}_1}(u) := \overrightarrow{A_1} \cos(m_{+} u) +\overrightarrow{B_1} \sin(m_{+} u)+\overrightarrow{A_2} \cos(m_{-} u) +\overrightarrow{B_2} \sin(m_{-} u),$$ where $$m_{\pm} = \frac 1{\sqrt{2}} \sqrt{ 1+ c_2^2 + c_3^2 \pm \sqrt{(1 + c_2^2 + c_3^2)^2-4 c_3^2}}$$ and where $\overrightarrow{A_i},\overrightarrow{B_i}, i = 1,2$ are constant vectors. Spherical ccr-curves ==================== In order to compare ccr-curves with the definition of generalized helices given in [@Ba], we will try to determine which ccr-curves are included in a sphere. A curve $\alpha:I\to {\mathbb R}^4$ is spherical, i.e., it is contained in a sphere of radius $R$, if and only if $$\label{spherical-eq-n=4} \frac 1{k_1^2} + \left(\frac{\dot{k_1}}{k_1^2k_2} \right)^2 + \frac 1{k_3^2}\left(\left(\frac{\dot{k_1}}{k_1^2k_2}\right)^{\dot{\ }}-\frac {k_2}{k_1} \right)^2 = R^2.$$ The proof here is similar to that for spherical curves in ${\mathbb R}^3$. It consists in obtaining information thanks to successive derivatives of the expression $<\alpha(s)-m,\alpha(s)-m> = R^2$, where $m$ is the center of the sphere. In particular, what can be proven is that spherical curves can be decomposed as $$\label{spherical-curve} \alpha(s) = m - \frac R{k_1} {\overrightarrow{{\bf e_2}}}(s) +R \frac{\dot{k_1}}{k_1^2k_2} {\overrightarrow{{\bf e_3}}}(s) +R\frac 1{k_3}\left(\left(\frac{\dot{k_1}}{k_1^2k_2}\right)^\cdot +\frac {k_2}{k_1} \right){\overrightarrow{{\bf e_4}}}(s).$$ As a corollary we obtain the classical result for spherical three-dimensional curves: A curve $\alpha:I\to {\mathbb R}^3$ is spherical, i.e., it is contained in a sphere of radius $R$, if and only if $$\label{spherical-eq-n=3} \frac 1{k_1^2} + \left(\frac{\dot{k_1}}{k_1^2k_2} \right)^2 = R^2.$$ From now on, we shall suppose that $m= 0$ and $R= 1$. Spherical ccr-curves in ${\mathbb R}^3$ --------------------------------------- In this case, we can rewrite Eq. \[spherical-eq-n=3\] in terms of curvature, $k_1 = \kappa$, and torsion $k_2 = \tau = c \kappa$, $c$ being a constant. $$\frac {\dot\kappa}{\kappa^2\sqrt{\kappa^2-1}} = \pm c.$$ Let us consider just the positive sign. This differential equation can be integrated and the solution is $$\kappa(s) = \frac 1{\sqrt{1-(cs+s_0)^2}}.$$ Thanks to a shift of the parameter we get that the curvature and torsion of a spherical generalized helix are given by $$\kappa(s) = \frac 1{\sqrt{1-c^2s^2}},\qquad \tau(s) = \frac c{\sqrt{1-c^2s^2}}.$$ We now need to compute the reparametrization $$u=g(s) = \int_{0}^s \kappa(t)dt = \frac1c \arcsin(cs).$$ With the appropriate initial conditions, the generalized spherical helix is $$\begin{array}{rcl} \alpha_c(s) &=& (\sqrt{1-c^2 s^2} \cos(\frac{\sqrt{1+c^2}\arcsin(cs)}{c})+ \frac{c^2 s}{\sqrt{1+c^2}}\sin(\frac{\sqrt{1+c^2}\arcsin(cs)}{c}),\\[2mm] &&-\sqrt{1-c^2 s^2} \sin(\frac{\sqrt{1+c^2}\arcsin(cs)}{c})+ \frac{c^2 s}{\sqrt{1+c^2}}\cos(\frac{\sqrt{1+c^2}\arcsin(cs)}{c}),\\[2mm] && \frac{c s}{\sqrt{1+c^2}}) \end{array}$$ Note that the curve $\alpha_c$ is defined in the interval $]-\frac 1c,\frac 1c[$. If we change the parameter in accordance with $s = \frac 1c \sin t$, the spherical helix is now parametrized as $$\begin{array}{rcl} \beta_c(t) &=& (\cos t \cos(\frac{\sqrt{1+c^2}}{c}t)+ \frac{c}{\sqrt{1+c^2}}\sin t\sin(\frac{\sqrt{1+c^2}}{c}t),\\[2mm] &&-\cos t \sin(\frac{\sqrt{1+c^2}}{c}t)+ \frac{c}{\sqrt{1+c^2}}\sin t\cos(\frac{\sqrt{1+c^2}}{c}t),\frac{\sin t}{\sqrt{1+c^2}}) \end{array}$$ Now, it is clear that the projection of these curves on the plane $xy$ are arcs of epicycloids. This result was known by W. Blaschke, as is mentioned in [@St], where it is also proven by different methods. Spherical ccr-curves in ${\mathbb R}^4$ --------------------------------------- ### The constant curvatures case. The curve $$\alpha(s) = \frac 1{\sqrt{r_1^2+r_2^2}} (\frac{r_1}{m_1} \sin(m_1 s),-\frac{r_1}{m_1} \cos(m_1 s),\frac{r_2}{m_2} \sin(m_2 s),-\frac{r_2}{m_2} \cos(m_2 s))$$ is a spherical curve (with radius $1$), if and only if $$r_1^2m_2^2+r_2^2m_1^2 = m_1^2m_2^2(r_1^2+r_2^2).$$ ### The non-constant case. In this case, we can rewrite Eq. \[spherical-eq-n=4\] in terms of curvature, $k_1$, $k_2 = c_2 k_1$ and $k_3 = c_3 k_1$, where $c_2, c_3$ are constants. $$\frac 1{k_1^2} + \left(\frac{\dot{k_1}}{c_2k_1^3} \right)^2 + \frac 1{c_3^2k_1^2}\left(\left(\frac{\dot{k_1}}{c_2k_1^3}\right)^\cdot +c_2 \right)^2 = 1.$$ By changing $f = \frac 1{k_1^2}$ the equation is reduced to $$\label{reduced} f + \frac 1{4c_2^2} \dot{f}^2 +\frac 1{c_3^2} f(-\frac 1{2c_2} \ddot{f} + c_2)^2 = 1.$$ Computation of the general solution seems to be a difficult task. Instead, we can try to compute some particular solutions. For instance, the constant solution $f(s) = \frac{c_3^2}{c_2^2 + c_3^2}$ or the polynomial solutions of degree $2$ $$f(s) = \frac{-2 c_2^2 + c_3^2 - c_3\sqrt{-8 c_2^2 + c_3^2}}{2(c_2^2 + c_3^2)} + \frac 12 \left(2 c_2^2 - c_3^2 - c_3\sqrt{-8 c_2^2 + c_3^2}\right) s^2,$$ $$f(s) = 2 c_2 s +\frac 12 \left(2 c_2^2 - c_3^2 - c_3\sqrt{-8 c_2^2 + c_3^2}\right) s^2.$$ For these three particular solutions the reparametrization $g$, where $g(s) = \int_0^s k_1(t)dt = \int_0^s \frac 1{\sqrt{f(t)}}dt,$ can be computed explicitly. We can thus obtain explicit examples of ccr-curves in $S^3$ with non-constant curvatures. A particular case. With $c_2= \frac 12, c_3:=\frac{\sqrt{3}}2$, then $m_1= \sqrt{\frac 32}, m_2 = \frac 1{\sqrt{2}}$ and $r_1= r_2 = \frac 1{\sqrt{2}}$. The function $f(s) = \frac 12 - 2s^2$ is a solution of Eq. \[reduced\]. Therefore, $k_1(s) = \frac 2{\sqrt{1-4s^2}}$, and $g(s) = \int_0^s \frac 2{\sqrt{1-4t^2}}dt = \arcsin(2s)$. If $${\overrightarrow{{\bf e_1}}}(t) = \frac 1{\sqrt{2}} (\cos(\sqrt{\frac 32}t), \sin(\sqrt{\frac 32}t),\cos(\frac 1{\sqrt{2}}t),\sin(\frac 1{\sqrt{2}}t)),$$ then $$\alpha(s) = (0, -\frac{\sqrt{3}}2, 0, \frac 12)+\int_0^s {\overrightarrow{{\bf e_1}}}(\arcsin(2u))du, \quad s\in\ ]-\frac 12,\frac 12[$$ is a spherical ccr-curve with center at the origin of coordinates, with radius $1$ and with non-constant curvatures. Intrinsic generalized helices ============================= In [@Ba] the author proposes a definition of general helix on a $3$-dimensional real-space-form substituting the fixed direction in the usual definition of generalized helix by a Killing vector field along the curve. Let $\alpha:I\to M$ be an immersed curve in a $3$-dimensional real-space-form $M$. Let us denote the intrinsic Frenet frame by $\{\overrightarrow{{\bf t}}, \overrightarrow{{\bf n}},\overrightarrow{{\bf b}}\}$. The intrinsic Frenet’s formulae are $$\label{intrinsic-Frenet-formulae} \left\{\aligned \nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf t}} &= \kappa \overrightarrow{{\bf n}},\\ \nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf n}} &= -\kappa \overrightarrow{{\bf t}}+ \tau\overrightarrow{{\bf b}},\\ \nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf b}} &= -\tau \overrightarrow{{\bf n}}, \endaligned \right.$$ where $\nabla$ is the Levi-Civita connection of $M$ and where $\kappa$ and $\tau$ are called the intrinsic curvature and torsion functions of curve $\alpha$, respectively. From now on we shall suppose that $M= S^3$. Therefore, any curve on $S^3$ can also be considered to be a curve in ${\mathbb R}^4$. We shall try to obtain the relationship between the Frenet elements, $\{\overrightarrow{{\bf e_1}},\overrightarrow{{\bf e_2}},\overrightarrow{{\bf e_3}},\overrightarrow{{\bf e_4}}, k_1, k_2,k_3\}$, of the curve as a curve in $4$-dimensional Euclidian space and the intrinsic Frenet elements $\{\overrightarrow{{\bf t}}, \overrightarrow{{\bf n}},\overrightarrow{{\bf b}}, \kappa,\tau\}$. Note first that $ \overrightarrow{{\bf t}}= \overrightarrow{{\bf e_1}}.$ Then $$\nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf t}} = \dot{\overrightarrow{\bf e_1}}- <\dot{\overrightarrow{\bf e_1}},\alpha>\alpha = k_1(\overrightarrow{\bf e_2}- <{\overrightarrow{\bf e_2}},\alpha>\alpha),$$ where we have used as the Gauss map of the sphere the identity map. Therefore $$\label{intrinsic-normal} \overrightarrow{\bf n} = \frac{\nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf t}}}{||\nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf t}}||}=\frac 1{\sqrt{1-<{\overrightarrow{\bf e_2}},\alpha>^2}} (\overrightarrow{\bf e_2}- <{\overrightarrow{\bf e_2}},\alpha>\alpha),$$ and $$\kappa = <\nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf t}},\overrightarrow{\bf n}> = k_1 \sqrt{1-<{\overrightarrow{\bf e_2}},\alpha>^2}= \sqrt{k_1^2-1},$$ which were obtained using Eq. \[spherical-curve\]. The intrinsic binormal vector is the only vector such that $\{\overrightarrow{{\bf t}}, \overrightarrow{{\bf n}},\overrightarrow{{\bf b}}, \alpha\}$ is an orthonormal basis of ${\mathbb R}^4$ with positive orientation. Then $$\overrightarrow{{\bf b}} = \alpha\wedge \overrightarrow{{\bf t}}\wedge \overrightarrow{{\bf n}}.$$ Now, by replacing the intrinsic tangent and normal with $ \overrightarrow{{\bf t}}= \overrightarrow{{\bf e_1}}$ and \[intrinsic-normal\], we get $$\overrightarrow{{\bf b}} = \frac {k_1}{\sqrt{k_1^2-1}}\ \alpha\wedge \overrightarrow{{\bf e_1}}\wedge \overrightarrow{{\bf e_2}} = \frac {1}{\sqrt{1-(\frac{1}{k_1})^2}}\ \alpha\wedge \overrightarrow{{\bf e_1}}\wedge \overrightarrow{{\bf e_2}}.$$ Therefore $$\dot{\overrightarrow{\bf b}} = \left(\frac {1}{\sqrt{1-(\frac{1}{k_1})^2}}\right)^\cdot\ \alpha\wedge \overrightarrow{{\bf e_1}}\wedge \overrightarrow{{\bf e_2}}+ \frac {1}{\sqrt{1-(\frac{1}{k_1})^2}}\ \alpha\wedge \overrightarrow{{\bf e_1}}\wedge k_2\overrightarrow{{\bf e_3}}.$$ A consequence of this computation is that $<\dot{\overrightarrow{\bf b}}, \alpha> = 0$, and therefore, $\nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf b}}= \dot{\overrightarrow{\bf b}}$. Finally, $$\begin{array}{rcl} \tau &=& - <\nabla_{\overrightarrow{{\bf t}}}\overrightarrow{{\bf b}},\overrightarrow{{\bf n}}> \\[3mm] &=& -<\frac {1}{\sqrt{1-(\frac{1}{k_1})^2}}\ \alpha\wedge \overrightarrow{{\bf e_1}}\wedge k_2\overrightarrow{{\bf e_3}},\frac 1{\sqrt{1-(\frac{1}{k_1})^2}} \overrightarrow{\bf e_2}>\\[5mm] &=& -\frac {k_2}{1-(\frac{1}{k_1})^2}<\alpha\wedge \overrightarrow{{\bf e_1}}\wedge \overrightarrow{{\bf e_3}}, \overrightarrow{\bf e_2}>= \frac {k_2}{1-(\frac{1}{k_1})^2} = \frac {k_1^2k_2}{\kappa^2}. \end{array}$$ The only $4$-dimensional spherical non-trivial ccr-curves which are also intrinsic generalized helices of $S^3$ are helices, i.e., curves with all curvatures constant. As it is proven in [@Ba], a curve in $S^3$ is an intrinsic helix if and only if $\tau = 0$ or there exists a constant $b$ such that $\tau = b\kappa\pm 1$. The case $\tau = 0$ implies that $k_1k_2 = 0$ and we get a non-regular curve. In the other case, if the curve is also a ccr-curve (with $k_2 = c k_1$), then $$\frac {ck_1^3}{\kappa^2} = b \kappa \pm 1.$$ Equivalently $$(\frac {ck_1^3}{k_1^2-1}\mp 1)^2 = b (k_1^2-1).$$ That is, the function $k_1$ is the solution of a polynomial equation with constant coefficients; and, therefore, the function $k_1$ is constant, and so the other two curvatures $k_2$ and $k_3$ are also constant. The same happens with $\kappa$ and $\tau$. We are then in the presence of a helix according to the designation in [@Ba], or a geodesic in a flat torus in ${\mathbb R}^4$ according to [@Ro]. [99]{} M. Barros, [*General helices and a theorem of Lancret*]{}, Proceedings of the Am. Math. Soc., vol. 125, 1503-1509 (1997). do Carmo, M. P., [*Differential Geometry of curves and surfaces*]{}, Prentice-Hall Inc., (1976). A. Ferrández, A. Giménez, P. Lucas, [*Null generalized helices in Lorentz-Minkowski spaces*]{}, J. of Physics, A: Math. and general, vol 35, 8243-8251 (2002). H. A. Hayden, [*On a generalized helix in a Riemannian $n$-space*]{}, Proc. London Math. Soc., vol 32, 37-45 (1931). W. Klingenberg, [*A course in Differential Geometry*]{}, Springer-Verlag (1978). S. Rodrigues Costa, [*On closed twisted curves*]{}, Proceedings of the Am. Math. Soc., vol. 109, 205-214 (1990). M.C. Romero-Fuster, E. Sanabria-Codesal, [*Generalized helices, twistings and flattenings of curves in $n$-space*]{}, Matemática Contemporânea, vol. 17, 267-280 (1999). Struick, D. J.; [*Lectures on Classical Differential Geometry*]{}, Dover, New-York, (1988). [^1]: This work was partially supported by a Spanish MCyT grant BFM2002-00770.

--- abstract: 'We complete the classification of the finite special linear groups ${\mathrm{SL}}_n(q)$ which are $(2,3)$-generated, i.e., which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups ${\mathrm{PSL}}_n(q)$ which are $(2,3)$-generated.' address: 'Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Musei 41, 25121 Brescia, Italy' author: - Marco Antonio Pellegrini title: 'The $(2,3)$-generation of the special linear groups over finite fields' --- Introduction ============ It is a well known fact that every finite simple group can be generated by a pair of suitable elements: for alternating groups this is a classical result of Miller [@Mi], for groups of Lie type it is due to Steinberg [@St] and for sporadic groups it was proved by Aschbacher and Guralnick [@AG]. A more difficult problem is to find for a finite nonabelian (quasi)simple group $G$ the minimum prime $r$, if it exists, such that $G$ is $(2, r)$-generated, i.e. such that $G$ can be generated by two elements of respective orders $2$ and $r$. We denote such minimum prime $r$ by $\varpi(G)$ (setting $\varpi(G)=\infty$ if $G$ is not $(2,r)$-generated for any prime $r$). Since groups generated by two involutions are dihedral, we must have $\varpi(G)\geq 3$. Miller himself proved that $\varpi({\mathrm{Alt}(n)})=3$ if $n=5$ or $n\geq 9$, while it is easy to verify that $\varpi({\mathrm{Alt}(n)})=5$ if $n=6,7,8$. The special linear groups were firstly considered in [@T87], where Tamburini showed that $\varpi({\mathrm{SL}}_n(q)) = 3$ for all $n\ge 25$ and all prime power $q$. Woldar [@Wo] proved that all simple sporadic groups are $(2,3)$-generated, except for ${\mathrm{M}}_{11}$, ${\mathrm{M}}_{22}$, ${\mathrm{M}}_{23}$ and ${\mathrm{McL}}$, for which $\varpi(G)=5$. As proved by Lübeck and Malle [@LM], all simple exceptional groups are $(2,3)$-generated with the only exception of the Suzuki groups ${\mathrm{Sz}}(2^{2m+1})$, for which Suzuki himself [@Sz] proved that $\varpi({\mathrm{Sz}}(2^{2m+1}))=5$. Hence, we are left to consider the finite simple classical groups. A key result for such groups is due to Liebeck and Shalev, who proved in [@LS] that, apart from the infinite families ${\mathrm{PSp}}_4(2^m)$ and ${\mathrm{PSp}}_4(3^m)$, all finite simple classical groups are $(2,3)$-generated with a finite number of exceptions. So, the problem of finding the exact value of $\varpi(G)$ reduces to classifying the exceptions to the Liebeck and Shalev’ theorem. However, their result relies on probabilistic methods and does not provide any estimates on the number or the distribution of such exceptions. We remark that King proved in [@K] that $\varpi(G)\neq \infty$ for all finite simple classical groups $G$, but in general the problem of computing the exact value of $\varpi(G)$ is still wide open (see [@Isc] for a recent survey on this topic). In this paper we consider the projective special linear groups ${\mathrm{PSL}}_n(q)$. Many authors, such as Di Martino, Macbeath Tabakov, Tamburini and Vavilov, already dealt with the problem of the $(2,3)$-generation of ${\mathrm{SL}}_n(q)$. Summarizing their results we have the following list of $(2,3)$-generated groups: - ${\mathrm{PSL}}_2(q)$ if $q\neq 9$ (see [@Mac]); - ${\mathrm{SL}}_3(q)$ if $q\neq 4$ (see [@PT35]); - ${\mathrm{SL}}_4(q)$ if $q\neq 2$ (see [@PTV]); - ${\mathrm{SL}}_n(q)$ if $5\leq n\leq 11$ (see [@PT35; @T6; @T7; @GG; @GGT]); - ${\mathrm{SL}}_n(q)$ if $n\geq 13$ (see [@T]); - ${\mathrm{SL}}_n(q)$ if $n\geq 5$ and $q\neq 9$ is odd (see [@DV1; @DV]). We observe that the $(2,3)$-generation of ${\mathrm{SL}}_n(q)$ clearly implies the $(2,3)$-generation of ${\mathrm{PSL}}_n(q)$. Here, using a constructive approach as in many of the above papers and in particular the permutational method illustrated in [@T], we solve the last remaining case, i.e. we prove the $(2,3)$-generation of ${\mathrm{SL}}_{12}(q)$, obtaining the following classification. \[main\] The groups ${\mathrm{PSL}}_2(q)$ are $(2,3)$-generated for any prime power $q$, except when $q=9$. The groups ${\mathrm{SL}}_n(q)$ are $(2,3)$-generated for any prime power $q$ and any integer $n\geq 3$, except when $(n,q)\in \{(3,4), (4,2)\}$. Observe that $\varpi(G)=5$ if $G\in \{{\mathrm{PSL}}_2(9)\cong {\mathrm{Alt}(6)}$, ${\mathrm{SL}}_3(4)$, ${\mathrm{PSL}}_3(4)$, ${\mathrm{PSL}}_4(2)\cong {\mathrm{Alt}(8)}\}$. Clearly ${\mathrm{SL}}_2(q)$ cannot be $(2,r)$-generated when $q$ is odd, as the unique involution is the central one. Regarding the $(2,3)$-generation of the other finite classical groups, we recall that only partial results are available, mainly concerning small or high dimensions, see [@P67; @unit; @PT35; @PTV; @TW; @TWG; @TV]. Finally, we recall that the infinite groups ${\mathrm{PSL}}_n(\Z)$ are $(2,3)$-generated if and only if either $n=2$ or $n\geq 5$, and that the groups ${\mathrm{SL}}_n(\Z)$ are $(2,3)$-generated if and only if $n\geq 5$ (see [@T; @V1; @V2; @V3]). The $(2,3)$-generation of ${\mathrm{SL}}_{12}(q)$ ================================================= Let $q=p^a$, where $p$ is a prime and let $\F_q$ be the field of $q$ elements. Let $V$ be a $12$-dimensional $\F_q$-space, that we identify with the row vectors of $\F_q^{12}$. Let $\mathcal{C}=\{e_1,e_2\ldots,e_{12}\}$ be the canonical basis of $V$. For any element $\sigma \in {\mathrm{Alt}(\mathcal{C})}$, we write $g=\sigma$ to denote the permutation matrix $g\in {\mathrm{SL}}_{12}(q)$ corresponding to $\sigma$ with respect to $\mathcal{C}$. This allows us to consider ${\mathrm{Alt}(\mathcal{C})}$ as a subgroup of ${\mathrm{SL}}_{12}(q)$. Now, let $$\label{eqy} y=(e_1,e_2,e_3)(e_4,e_5,e_6)(e_7,e_8,e_9)(e_{10},e_{11},e_{12})$$ and let $x$ be the matrix, written with respect to $\mathcal{C}$, such that: - $x$ swaps $e_1$ and $e_8$; - $e_2x=-e_2$ and $e_5 x =e_5$; - $x$ swaps $e_{3i}$ and $e_{3i+1}$ for all $1\leq i\leq 3$; - $x$ acts on $\langle e_{11},e_{12}\rangle$ as the matrix $\begin{pmatrix} 1 & 0 \\ t & -1\end{pmatrix}$ with $t\in \F_q$. Clearly $x$ and $y$ have orders, respectively, $2$ and $3$, and $$\label{h} H=\langle x,y\rangle$$ is a subgroup of ${\mathrm{SL}}_{12}(q)$. First of all we prove the following. \[alt\] If $p\neq 5$, then the group $H$ contains ${\mathrm{Alt}(\mathcal{C})}$. Let $c=[x,y]=x^{-1} y^{-1} x y$ e define $\gamma$ according to the following rule: - $\gamma=c^{12}$, if $p=2$; - $\gamma=c^{12p}$, if $p\equiv 1 \pmod{10}$; - $\gamma=c^{24p}$, if $p\equiv 3 \pmod{10}$; - $\gamma=c^{6p}$, if $p\equiv 7 \pmod{10}$; - $\gamma=c^{18p}$, if $p\equiv 9 \pmod{10}$. It is easy to see that $e_1\gamma=-e_3$, $e_3\gamma=e_5$, $e_5\gamma=e_4$, $e_4\gamma=-e_8$ and $e_8\gamma=e_1$. Furthermore, $e_i\gamma=e_i$ for all $i\in \{2,6,7,9,10,11,12\}$. Also taking $\delta=\gamma^y$, we define $$\eta_1 = (\gamma^{4}\delta^{3}\gamma^2\delta^2)^2,\quad \eta_2 = (\gamma^4\delta^3\gamma^2\delta^2 \gamma^2\delta^2)^2,\quad \eta_3 = (\delta\gamma^2\delta\gamma^2\delta\gamma^3\delta^4\gamma^2)^2.$$ Since $$\eta_1=(e_2,e_5)(e_4,e_8), \quad \eta_2=(e_1,e_6)(e_4,e_9),\quad \eta_3=(e_1,e_3)(e_2,e_8)(e_4,e_9)(e_5,e_6),$$ we obtain that $\langle \eta_1,\eta_2,\eta_3\rangle ={\mathrm{Alt}(\Delta)}$, where $\Delta=\{e_1,e_2,e_3, e_4,e_5,e_6,e_8,e_9\}\subset\mathcal{C}$. It follows that $\langle \gamma, \delta \rangle $ contains the subgroup ${\mathrm{Alt}(\Delta)}$ and in particular the element $g=(e_1,e_4,e_9)$. Since $g^x=(e_3,e_{10},e_8)$, we conclude that $H$ contains the subgroup $\langle{\mathrm{Alt}(\Delta)}, g^x, y\rangle ={\mathrm{Alt}(\mathcal{C})}$. The next key ingredient is the following result, which is a particular case of [@T Lemma 4.1]. As usual, $E_{i,j}$ denotes the elementary matrix having $1$ at position $(i,j)$ and $0$ elsewhere. \[5\] Let $t\neq 0,2$ be such that $\F_q=\F_p(t)$. Then, the normal closure $N$ of the involution $w=I_5-2E_{5,5}+tE_{5,4}$ under ${\mathrm{Alt}(5)}$ is $\langle {\mathrm{SL}}_5(q), {\mathrm{diag}({-1,1,1,1,1})}\rangle$. We can now prove the following proposition that, combined with the known results on the $(2,3)$-generation of ${\mathrm{SL}}_n(q)$ described in the Introduction, immediately gives Theorem \[main\]. \[propNo5\] For all primes $p\neq 5$ and all integers $a\geq 1$, the groups ${\mathrm{SL}}_{12}(p^a)$ are $(2,3)$-generated. Set $q=p^a$. Let $H=\langle x,y\rangle $ be as in , where the element $t\in \F_q$ in $x$ is chosen in such a way that $t\neq 0,2$ and $\F_p(t)=\F_q$. As already observed, $H\leq {\mathrm{SL}}_{12}(q)$. So, we have to prove that ${\mathrm{SL}}_{12}(q)\leq H$. First, consider the element $g=(e_1,e_8)(e_9,e_{10})\in {\mathrm{Alt}(\mathcal{C})}$. Then $w=gx$ acts on $\langle e_8,\ldots,e_{12} \rangle$ as the involution $I_5-2E_{5,5}+tE_{5,4}$. By Lemma \[5\], we get that ${\mathrm{SL}}_5(q)$ is contained in $K=\langle w, {\mathrm{Alt}( {\{e_8,\ldots,e_{12}\}} )}\rangle$. It follows that $T=\langle K', {\mathrm{Alt}(\mathcal{C})}\rangle$ is ${\mathrm{SL}}_{12}(q)$. Since, by Lemma \[alt\], ${\mathrm{Alt}(\mathcal{C})}$ is a subgroup of $H$ we have $T\leq H$, whence $H={\mathrm{SL}}_{12}(q)$. For sake of completeness, using the permutational method we now prove the $(2,3)$-generation of ${\mathrm{SL}}_{12}(5^a)$ for all $a\geq 1$.\ Let $\tilde y=y$ be as in and let $\tilde x$ be the matrix, written with respect to $\mathcal{C}$, such that: - $e_1\tilde x=-e_1$, $e_5 \tilde x =e_5$ and $e_8 \tilde x =e_8$; - $\tilde x$ swaps $e_{3i}$ and $e_{3i+1}$ for $i=2,3$; - $\tilde x$ acts on $\langle e_{2},e_{3},e_4\rangle$ as the involution $x_3=\begin{pmatrix} 3 & 3 & 2 \\ 2 & 3 & 1\\ 3 & 1 & 3\end{pmatrix}$; - $\tilde x$ acts on $\langle e_{11},e_{12}\rangle$ as the matrix $\begin{pmatrix} 1 & 0 \\ t & -1\end{pmatrix}$ with $t\in \F_q$. Also in this case, $\tilde x$ and $\tilde y$ have orders, respectively, $2$ and $3$, and $$\label{h5} \widetilde H=\langle \tilde x,\tilde y\rangle$$ is a subgroup of ${\mathrm{SL}}_{12}(q)$. \[alt5\] The group $\widetilde H$ contains ${\mathrm{Alt}(\mathcal{C})}$. Let $\tilde c=[\tilde x,\tilde y]$ and define $\tilde \gamma=\tilde c^{12}$ and $\tilde \delta=\tilde \gamma^{y^2}$. We firstly observe that both $\tilde \gamma$ and $\tilde \delta$ fix the decomposition $V=\langle e_1,\ldots,e_8\rangle \oplus \langle e_9\rangle\oplus \ldots\oplus \langle e_{12}\rangle$. Since $\tilde \gamma\tilde\delta^2$ and $\tilde\gamma\tilde\delta\tilde\gamma^3\tilde\delta^3$ have orders, respectively, $313$ and $19531$, we obtain that $K=\langle \tilde\gamma,\tilde\delta\rangle$ coincides with the subgroup $\left\{\left(\begin{array}{c|c} A & 0 \\ \hline 0 & I_{4} \end{array} \right): A \in {\mathrm{SL}}_8(5)\right\}\cong {\mathrm{SL}}_8(5)$ (use, for instance, [@LPS]). In particular, $K$ contains the elements $g_1={\mathrm{diag}({1,x_3,I_{8}})}$, $g_2=(e_1,e_2,e_3,e_4,e_5,e_6,e_7)$ and $g_3=(e_6,e_7,e_8)$. Now, as $g_3^{\tilde yg_1\tilde x}=(e_4,e_8,e_{10}) $, we obtain that $\widetilde H$ contains the subgroup $\langle g_2,g_3^{\tilde yg_1\tilde x},\tilde y\rangle={\mathrm{Alt}(\mathcal{C})}$. For all integers $a\geq 1$, the groups ${\mathrm{SL}}_{12}(5^a)$ are $(2,3)$-generated. It suffices to repeat the proof of Proposition \[propNo5\] using $\tilde x,\tilde y,\widetilde H$ instead of $x,y,H$, respectively, and defining $w=g\tilde x$, where $g=(e_6,e_7)(e_9,e_{10})$. [50]{} M. Aschbacher R. Guralnick, Some applications of the first cohomology group, *J. Algebra* **90** (1984), 446–460. L. Di Martino N. Vavilov, $(2,3)$-generation of ${\mathrm{SL}}(n,q)$. I. Cases $n=5,6,7$, *Comm. Algebra* **22** (1994), 1321–1347. L. Di Martino N. Vavilov, $(2,3)$-generation of ${\mathrm{SL}}(n,q)$. II. Cases $n\geq 8$. *Comm. Algebra* **24** (1996), 487–515. E. Gencheva Ts. Genchev, $(2,3)$-generation of the special linear groups of dimension $8$, *Mathematics and Education in Mathematics, 2015*, Proceedings of the Forty Fourth Spring Conference of the Union of Bulgarian Mathematicians, SOK “Kamchia”, April 2–6 (2015), 167–173. E. Gencheva, Ts. Genchev K. Tabakov, $(2,3)$-generation of the special linear groups of dimensions $9$, $10$ and $11$, `arxiv.org/pdf/1412.8631v5.pdf`. C.S.H. King, Generation of finite simple groups by an involution and an element of prime order, preprint. M. Liebeck, C.E. Praeger J. Saxl, Transitive Subgroups of Primitive Permutation Groups, *J. Algebra* **234** (2000), 291–361. M.W. Liebeck A. Shalev, Classical groups, probabilistic methods, and the $(2,3)$-generation problem, *Ann. Math. (2)* **144** (1996), 77–125. F. Lübeck and G. Malle, $(2,3)$-generation of exceptional groups, *J. London Math. Soc. (2)* **59** (1999), 109–122. A.M. Macbeath, Generators for the linear fractional groups, *Proc. Simp. Pure Math.* **12** (1969), 14–32. G.A. Miller, On the groups generated by two operators, *Bull. AMS* **7** (1901), 424–426. M.A. Pellegrini, The $(2,3)$-generation of the classical simple groups of dimensions $6$ and $7$, *Bull. Aust. Math. Soc.* **93** (2016), 61–72. M.A. Pellegrini, M. Prandelli M.C. Tamburini Bellani, The $(2,3)$-generation of the special unitary groups of dimension $6$, *J. Algebra Appl.* **15** (2016) 1650171, 12 pp. M.A. Pellegrini M.C. Tamburini, Finite simple groups of low rank: Hurwitz generation and $(2,3)$-generation, *Int. J. Group Theory* **4** (2015), 13–19. M.A. Pellegrini M.C. Tamburini Bellani, The simple classical groups of dimension less than $6$ which are $(2,3)$-generated, *J. Algebra Appl.* **14** (2015), 1550148, 15 pp. M.A. Pellegrini, M.C. Tamburini Bellani M.A. Vsemirnov, Uniform $(2,k)$-generation of the $4$-dimensional classical groups, *J. Algebra*, **369** (2012), 322–350. R. Steinberg, Generators for simple groups, *Canad. J. Math* **14** (1962), 277–283. M. Suzuki, On a class of doubly transitive groups, *Ann. of Math. (2)* **75** (1962), 105–145. K. Tabakov, $(2,3)$-generation of the groups ${\mathrm{PSL}}_7(q)$, *Proceedings of the Forty Second Spring Conference of the Union of Bulgarian Mathematicians*, Borovetz, April 2–6 (2013), 260–264. K. Tabakov, K. Tchakerian, $(2,3)$-generation of the groups ${\mathrm{PSL}}_6(q)$, *Serdica Math J.* **37** (2011), 365–370. M.C. Tamburini, Generation of certain simple groups by elements of small order, *Istit. Lombardo Accad. Sci. Lett. Rend. A* **121** (1987), 21–27. M.C. Tamburini, The $(2,3)$-generation of matrix groups over the integers, *Ischia group theory 2008*, 258–264, World Sci. Publ., Hackensack, NJ, 2009. M.C. Tamburini J.S. Wilson, On the $(2,3)$-generation of some classical groups. II, *J. Algebra* **176** (1995), 667–680. M.C. Tamburini, J.S. Wilson N. Gavioli, On the (2,3)-generation of some classical groups. I, *J. Algebra* **168** (1994), 353–370. M.C. Tamburini Bellani M. Vsemirnov, Hurwitz generation of ${\mathrm{PSp}}_6(q)$, *Comm. Algebra* **43** (2015), 4159–4169. M.A. Vsemirnov, Is the group ${\mathrm{SL}}(6,Z)$ $(2,3)$-generated?, *Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)* **330** (2006), *Vopr. Teor. Predst. Algebr. i Grupp.* **13**, 101–130, 272; English translation in *J. Math. Sci. (N. Y.)* **140** (2007), 660–675. M.A. Vsemirnov, On the $(2,3)$-generation of matrix groups over the ring of integers, *Algebra i Analiz* **19** (2007), 22–58; English translation in *St. Petersburg Math. J.* **19** (2008), 883–910. M.A. Vsemirnov, On the $(2,3)$-generation of small rank matrix groups over integers, *Quaderni del Seminario Matematico di Brescia* **30** (2008), 1–15. A.J. Woldar, On Hurwitz generation and genus actions of sporadic groups, *Illinois Math. J. (3)* **33** (1989), 416–437.

--- abstract: 'The hybrid relay selection (HRS) scheme, which adaptively chooses amplify-and-forward (AF) and decode-and-forward (DF) protocols, is very effective to achieve robust performance in wireless networks. This paper analyzes the frame error rate (FER) of the HRS scheme in general cooperative wireless networks without and with utilizing error control coding at the source node. We first develop an improved signal-to-noise ratio (SNR) threshold-based FER approximation model. Then, we derive an analytical average FER expression as well as an asymptotic expression at high SNR for the HRS scheme and generalize to other relaying schemes. Simulation results are in excellent agreement with the theoretical analysis, which validates the derived FER expressions.' author: - 'Tianxi Liu, Lingyang Song,  Yonghui Li,  Qiang Huo and Bingli Jiao,  [^1] [^2]' bibliography: - 'IEEEabrv.bib' - 'HRS.bib' title: Performance Analysis of Hybrid Relay Selection in Cooperative Wireless Systems --- Hybrid relay selection, frame error rate, SNR threshold-based approximation, cooperative communications. Introduction ============ ooperative relaying has been shown to be an effective technique to improve the system performance in wireless networks by allowing users to cooperate with each other in their transmissions. Two of the most typical cooperative schemes are amplify-and-forward (AF) and decode-and-forward (DF). The performance of AF protocol is mainly limited by the noise amplified at the relay during the forwarding process, especially at low SNR. The performance of DF will be degraded when the relay fails to decode the received signals correctly and the process of decoding and re-encoding will cause serious error propagation. To overcome these limitations, various cooperative schemes have been reported, such as signal-to-noise-ratio (SNR) threshold based selective DF [@Laneman2004; @Onat2008; @Onat2008a; @Onat2007], cooperative-maximum-ratio-combining based DF [@Wang2007; @Wang2005], decode-amplify-forward (DAF) [@Bao2007; @Bao2005DAF], link adaptive relaying DF [@Vien2009; @Wang2008; @wang2006smart] and log-likelihood-ratio (LLR) threshold based selective DF [@kwon2010LLRDF; @van2006llr], etc. Recently, the hybrid relaying protocol (HRP) has received a lot of attention. It adaptively combines the merits of both DF and AF by forwarding *clean* packets in DF if decodes correctly and forwarding *soft* represented packets in AF if decodes incorrectly) [@Can2006DAF-OFDM; @Eslamifar2009; @Souryal2006; @Li2007; @Yu2005; @Bao2007]. Its performance can be further improved by incorporating relay selection in HRP, which is referred to as hybrid relay selection (HRS) [@Can2006DAF-OFDM; @Eslamifar2009; @Hasan2009; @Li2009; @Li2007; @Li2008; @Song2009; @Souryal2006; @Yang2009DAF; @Yao2009; @Yu2005]. In the HRS scheme, for each transmission, all relays are divided into two groups, a DF group and an AF group. A relay is included either into the DF group if it decodes correctly or into the AF group if it decodes incorrectly. The destination node selects a single optimal relay node with maximum destination SNR from either the DF group or the AF group to forward the packet. Although the principle of the HRS scheme is simple, the calculation of the analytical FER is non-trivial even for single-relay cooperative networks. Therefore, most earlier work have been focusing on the approximate average FER analysis. Nonetheless, still very limited results have been reported in the literature so far. In [@Souryal2006], Souryal analyzed the FER of single-relay networks using hybrid forward scheme in block Rayleigh fading channel. In [@Li2007; @Li2008], Li proposed the hybrid relay selection scheme by combining hybrid forwarding and relay selection. In [@Song2009], Song extended the HRS scheme with differential modulation. Only very loose upper bounds were provided for the FER of the HRS scheme. In [@Huo2010P1], the authors mainly analyze the FER of the all-participate scheme without relay selection, where all relays participate in forwarding signals from the source. In this paper, we derive the analytical FER of the HRS scheme in general cooperative wireless networks with and without applying convolutional coding at the source node. The contribution of this paper is threefold: 1) we develop an improved SNR threshold-based FER approximation model, which is simple and accurate for general diversity systems, and obtain the SNR threshold in an analytical expression; 2) we derive an analytical approximate average FER expression, and its simplified asymptotic FER expression at high SNRs for the HRS scheme; 3) we generalized the derived theoretical FER expressions to other relaying schemes, e.g. the AF-RS scheme and the PDF-RS scheme. The rest of the paper is organized as follows: We describe the system model and the HRS scheme in Section \[sec.system.model\]. In Section \[sec.FER.approx.model\], we develop an improved SNR-threshold based FER approximation model based on cumulative distribution functions (CDF). In Section \[sec:FER.HRS\], we derive the analytical approximate and asymptotic FER expression at high SNRs of the HRS scheme. In Section \[sec.simulation\], we present some numerical simulation results. And in Section \[sec.conclusion\], we draw the main conclusions. ***Notation***: Boldface lower-case letters denote vectors. $\mathcal{Z}_n$ represents the $n$-dimensional binary space $\left\{0,1\right\}^{n}$. For a random variable $X$, $Pr(\cdot)$ denotes its probability, $f_X(\cdot)$ denotes its probability density function (PDF), $F_X(\cdot)$ is its CDF, and $\mathbb{E}[X]$ represents its expectation. $X\sim \mathcal{CN}(0, \Omega)$ is a circular symmetric complex Gaussian variable with a zero mean and variance $\Omega$. $Q(x)$ denotes the $Q$-function, i.e. $Q(x)=\frac{1}{\sqrt{2\pi}}\int_x^{\infty}e^{-t^2/2}\,\mathrm{d}t$. ![Diagram of the cooperative wireless system with the HRS scheme.[]{data-label="fig:SystemDiagram"}](SystemDiagram){width="80.00000%"} System Model and The HRS Scheme {#sec.system.model} =============================== In this paper, we consider a general $2$-hop relay network, as shown in Fig. \[fig:SystemDiagram\], consisting of one source node, $n$ relay nodes and one destination node. The link between any two nodes is modeled as a block Rayleigh fading channel with additive white Gaussian noise (AWGN), where the fading coefficients of the channels are fixed within one frame and vary independently from one frame to another. $h_0$, $h_{1,i}$, and $h_{2,i}$ are the fading coefficients of the channels from the source node to the destination node, from the source node to the $i$-th relay node, and from the $i$-th relay node to the destination node, respectively. We assume $h_0\sim \mathcal{CN}(0,\Omega_0)$, $h_{1i} \sim\mathcal{CN}(0,\Omega_{1i})$, and $h_{2,i} \sim\mathcal{CN}(0,\Omega_{2i})$. Similarly, $n_0$, $n_{1i}$ and $n_{2i}$ are the corresponding additive Gaussian noises. We assume $n_0 \sim \mathcal{CN}(0,N_0)$, $n_{1i} \sim \mathcal{CN}(0,N_0)$, and $n_{2i} \sim \mathcal{CN}(0,N_0)$. Without loss of generality, we consider that all the nodes transmit with the same power $\mathcal{E}$. In addition, all channel state information (CSI) needed for decoding is available at the relay nodes and the destination node. We assume that each terminal in the network is equipped with a single antenna working in the half-duplex mode. We consider an orthogonal transmission scheme in which only one terminal is allowed to transmit at each time slot. Therefore, one frame transmission in the HRS scheme consists of two phases: The First Phase --------------- The source node broadcasts the transmit signals with length $L$, denoted by $\textbf{s}$, to both the destination node and all of the relay nodes. The received signals at the destination node and the $i$-th relay node, denoted by $\textbf{y}_0$ and $\textbf{y}_{1i}$, are $ \textbf{y}_0 = \sqrt{\mathcal{E}}h_0 \textbf{s} + \textbf{n}_0, $ and $ \textbf{y}_{1i} = \sqrt{\mathcal{E}} h_{1i} \textbf{s} +\textbf{n}_{1i}, $ respectively. Then the corresponding instantaneous SNRs can be calculated as $$\label{eq:SNR_S2D} \gamma_0 = \frac{\mathbb{E}(\lvert \sqrt{\mathcal{E}} h_0 \textbf{s} \rvert ^2)}{\mathbb{E}(\lvert \textbf{n}_0 \rvert ^2)}= \frac{\mathcal{E}}{N_0} |h_0|^2 = \bar{\gamma} |h_0|^2 ,$$ and $$\label{eq:SNR_S2Ri} \gamma_{1i} = \frac{ \mathcal{E} }{N_0}|h_{1i}|^2 = \bar{\gamma} |h_{1i}|^2,$$ respectively, where $\mathbb{E}(\lvert \mathbf{s}\rvert ^2)=L$, $\mathbb{E}(\lvert \mathbf{n}\rvert ^2)=LN_0$, and $\bar{\gamma} $ is defined as $$\bar{\gamma} \triangleq\frac{\mathcal{E}}{N_0}.$$ The Second Phase ---------------- In the second phase, every relay node first decodes it’s receive signals. Then it is assigned to either the AF group $\mathcal{G}_{AF}$ or the DF group $\mathcal{G}_{DF}$ according to its CRC checking result. If the CRC checking result is correct, the relay node is included into $\mathcal{G}_{DF}$, otherwise, it is included into $\mathcal{G}_{AF}$. As a result, the instantaneous destination SNR of the link through the $i$-th relay node, denoted by $\gamma_i$, can be expressed as $$\label{eq:SNR_i_cases} \begin{split} \gamma_i &= \begin{cases} \frac{\gamma_{1i}\gamma_{2i}}{\gamma_{1i}+\gamma_{2i}+1}, & \text{if $i \in \mathcal{G}_{AF}$ }, \\ \gamma_{2i}, & \text{if $ i \in \mathcal{G}_{DF}$ }, \end{cases} \\ %&= \left(\frac{\gamma_{1i}\gamma_{2i}}{\gamma_{1i}+\gamma_{2i}+1} \right)^{z_i} %\left(\gamma_{2i} \right)^{1-z_i}, \end{split}$$ where $\gamma_{2i}$ represents the instantaneous SNR of the link between the $i$-th relay node and the destination node, which is given by $$\gamma_{2i} = \frac{ \mathcal{E} }{N_0} |h_{2i}|^2 = \bar{\gamma} |h_{2i}|^2.$$ Then, the destination node selects the relay node with maximum SNR $\gamma_m$ to transmit at the second phase $$\label{eq:Selection} \gamma_m = \max_{1 \leq i \leq n} \{ \gamma_i \}.$$ The selected relay node $m$ transmits through AF if $m \in \mathcal{G_{AF}}$ and using DF if $m \in \mathcal{G_{DF}}$. Finally, the destination combines the signals received at both phases by maximal ratio combining (MRC) to decode. After MRC, the effective SNR of the received signal, denoted by $\gamma_{HRS}$, is calculated as $$\label{eq:gamma_vz} %\gamma_{HRS} = \gamma_{0} + \max_{1\leq i \leq n} \left\{ \left(\frac{\gamma_{1i}\gamma_{2i}}{\gamma_{1i}+\gamma_{2i}+1} \right)^{z_i} % \left(\gamma_{2i} \right)^{1-z_i} \right\} . \gamma_{HRS} = \gamma_{0} + \gamma_m.$$ The selection process in the HRS scheme is exactly the same as in the conventional AF or DF selection schemes and does not add any extra complexity in system implementation. The only requirement for the HRS scheme is that each relay needs send one bit indicator to inform the destination if it uses AF or DF protocol. Destination then calculates the overall received SNR from each relay accordingly and selects the best relay with largest SNR. For simplicity, we define $\lambda_{0}$, $\lambda_{1i}$ and $\lambda_{2i}$ as: $$\label{eq.lambda.define} \lambda_{0} = \frac{1}{\bar{\gamma} \Omega_{0}}, \lambda_{1i} = \frac{1}{\bar{\gamma} \Omega_{1i}}, \lambda_{2i} = \frac{1}{\bar{\gamma} \Omega_{2i}}.$$ An Improved SNR Threshold-Based FER Approximation Model {#sec.FER.approx.model} ======================================================= In this section, we develop an improved SNR threshold-based FER approximation model for general diversity systems. We first describe the SNR threshold-based FER approximation model in Subsection \[subsec.FER.model.introduction\], in Subsection \[subsec.previous.SNR\], some earlier approaches are presented, and then we propose an improved criterion to calculate the SNR threshold in Subsection \[subsec.improved.SNR\]. Introduction of the SNR threshold-based FER approximation model {#subsec.FER.model.introduction} --------------------------------------------------------------- The average FER over a block fading channel, denoted by $\bar{P}_f$, can be computed by integrating the instantaneous FER over AWGN channel, represented by $P_f^G(\gamma)$, over the fading distribution [@Proakis1995] $$\label{eq:P_f_B} \bar{P}_{f}(\bar{\gamma})= \int_{0}^{\infty} P_f^G(\gamma) f_{\gamma}(\gamma,\bar{\gamma}) \text{d} \gamma,$$ where $\gamma$ and $\bar{\gamma}$ denote the instantaneous and average SNR, and $f_{\gamma}(\cdot)$ denotes the PDF of $\gamma$. Although (\[eq:P\_f\_B\]) is an exact expression for $\bar{P}_f $, its closed-form expression is difficult to evaluate. By assuming the instantaneous FER is $1$ when instantaneous SNR $\gamma$ is below a threshold $\gamma_t$, otherwise it is $0$: $$P_f^G(\gamma|\gamma \leq \gamma_t) \approx 1 \; \text{and} \; P_f^G(\gamma|\gamma > \gamma_t) \approx 0 ,$$ the average FER can be rewritten as $$\begin{split} \label{eq:ThresholdModelFER} \bar{P}_f &= \int_{0}^{\gamma_t} P_f^G(\gamma) f_{\gamma}(\gamma,\bar{\gamma}) \text{d} \gamma + \int_{\gamma_t}^{\infty} P_f^G(\gamma) f_{\gamma}(\gamma,\bar{\gamma}) \text{d} \gamma \\ &\approx \int_{0}^{\gamma_t} f_{\gamma}(\gamma,\bar{\gamma}) \text{d} \gamma = F_{\gamma}(\gamma_t,\bar{\gamma}), \end{split}$$ where $f_{\gamma}(\cdot)$ and $F_{\gamma}(\cdot)$ are the PDF and CDF of $\gamma$, respectively. Note that, according to the approximation model, the analytical FER can be calculated as an outage probability, and thus, the accuracy is mainly determined by the SNR threshold selection. Some Existing SNR Threshold Approaches {#subsec.previous.SNR} -------------------------------------- In [@ElGamal2001], El Gamal and Hammons demonstrated the SNR threshold-based FER approximation model for iteratively decoded systems employing turbo codes. And the optimal SNR threshold has been proved to coincide with the convergence threshold of the iterative turbo decoder. Recently, in [@Chatzigeorgiou2008; @Chatzigeorgiou2009], Chatzigeorgiou extended this model to non-iterative coded and uncoded systems. To get the optimal SNR threshold, proper error criterion should be used. In [@Chatzigeorgiou2008; @Chatzigeorgiou2009], the *minimum absolute error sum criterion* is adopted to minimize the sum of absolute error $$\label{eq:AbsoluteCriterion} \gamma_{t} = \text{arg} \; \min\left\{ \int_{0}^{\infty} \left| \bar{P}_{f}(\bar{\gamma}) - F_{\gamma}(\gamma_t,\bar{\gamma})\right| \text{d} \bar{\gamma} \right\} \;,$$ where $\bar{\gamma}$ is the average SNR, and the SNR threshold can be calculated as $$\label{eq.SNR.threshod.old} \gamma_t = \left(\int_{0}^{\infty} \frac{1-P_f^G(\gamma)}{\gamma^2} \text{d} \gamma \right)^{-1}.$$ However, the model on the basis of the *minimum absolute error sum criterion* might not be sufficiently accurate since it does not consider the fact that FER decreases more quickly at high SNR region in high diversity order systems. Hence, it can be improved. Model improvement and the SNR threshold result {#subsec.improved.SNR} ---------------------------------------------- In this subsection, we propose an improved SNR threshold-based FER approximation model for the general diversity systems. The improvements are twofold. Firstly, taking into account the fact that the FER decreases quickly when SNR increases, *minimum absolute relative error sum criterion*, which minimizing the sum of absolute *relative* error, can be adopted $$\label{eq:RelativeCriterion} \gamma_{t} = \text{arg} \; \min\left\{ \int_{0}^{\infty} \left| \frac{ \bar{P}_{f}(\bar{\gamma}) - F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right| \text{d} \bar{\gamma} \right\} \; .$$ As Eq. (\[eq:RelativeCriterion\]) is difficult to solve, we instead use a suboptimal error criterion. Reasonably, suppose $\min\left\{ \int_{0}^{\infty} \left| \frac{ \bar{P}_{f}(\bar{\gamma}) - F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right| \text{d} \bar{\gamma} \right\} < \infty$ and notice that the integration in Eq. (\[eq:RelativeCriterion\]) is from $0$ to $\infty$, and then, the absolute relative error should approach zero when $\bar{\gamma} \rightarrow \infty$: $$\begin{split} \label{eq:ZECriterion} \gamma_{t} &= \text{arg} \lim_{\bar{\gamma} \rightarrow \infty} \left\{ \left| \frac{ \bar{P}_{f}(\bar{\gamma}) - F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right| =0 \right\} \\ & \approx \text{arg} \lim_{\bar{\gamma} \rightarrow \infty} \left\{\bar{P}_{f}(\bar{\gamma}) -F_{\gamma}(\gamma_t,\bar{\gamma}) =0 \right\}. \end{split}$$ Otherwise, for a sufficiently big value $T$ ($0<T<\infty$) and a small enough value $\delta$ ($0< \delta < \infty$), the absolute relative error can be greater than $\delta$, i.e. $\left| \frac{ \bar{P}_{f}(\bar{\gamma}) - F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right| >\delta, \text{when } \bar{\gamma}>T$. Hence, the absolute relative error sum can’t be minimized as it approaches infinity: $\int_{0}^{\infty} \left| \frac{ \bar{P}_{f}(\bar{\gamma}) - F_{\gamma}(\gamma_t,\bar{\gamma})}{\bar{P}_{f}(\bar{\gamma})}\right| \text{d} \bar{\gamma} > \int_{T}^{\infty} \delta \text{d} \bar{\gamma} = \infty$. Secondly, the SNR threshold should include the factor of the diversity order of general diversity systems. For example, in a Rayleigh fading channel, the CDF of SNR is $F_{\gamma}(\gamma, \bar{\gamma}) = 1- e^{-\gamma / \bar{\gamma}}$, then $\lim_{\bar{\gamma} \rightarrow \infty }F_{\gamma}(\gamma, \bar{\gamma}) < {\bar{\gamma}}^{-1} \gamma $ and in a Nakagami-$m$ fading channel, the CDF of SNR is given by: $F_{\gamma}(\gamma, \bar{\gamma}) = \frac{\Gamma(m,m\gamma/\bar{\gamma})}{\Gamma(m)}$, where $\Gamma(m)$ is the Gamma function, and $\Gamma(m,x)$ is the lower part incomplete Gamma function given by: $\Gamma(m,x)=\int_{0}^x t^{m-1}e^{-t}\text{d}t$, then using the fact that $\lim_{t\rightarrow 0}e^{-t}<1$, we can get $\lim_{\bar{\gamma} \rightarrow \infty }F_{\gamma}(\gamma, \bar{\gamma}) < \Gamma(m)^{-1}\int_{0}^{\bar{\gamma}} (\frac{\gamma}{\bar{\gamma}})^{m-1}\text{d}t =\frac{1}{m \Gamma(m) (\bar{\gamma})^m } \gamma^m $. Suppose for some diversity system with a diversity order of $d$, the CDF of SNR can be approximated in a form [@Chatzigeorgiou2009; @Zheng2003; @Rodrigues2008Turbo]: $$\label{eq:CDFLimit} \lim_{\bar{\gamma} \rightarrow \infty }F_{\gamma}(\gamma, \bar{\gamma}) \approx G(\bar{\gamma}) \gamma ^d,$$ where $G(\bar{\gamma})$ is a constant related to $\bar{\gamma}$. Then, combining Eq. (\[eq:P\_f\_B\]), Eq. (\[eq:ZECriterion\]) and Eq. (\[eq:CDFLimit\]), we can get the analytical SNR threshold for our improved model as $$\label{eq:SNR_threshold_continue} \begin{split} \gamma_{t,d} &\approx \text{arg} \lim_{\bar{\gamma} \rightarrow \infty} \left\{ \int_{0}^{\infty} f_{\gamma}(\gamma, \bar{\gamma}) P_f^G(\gamma) \text{d} \gamma - G(\bar{\gamma}) \gamma_t ^d= 0\right\} \\ &=\text{arg} \lim_{\bar{\gamma} \rightarrow \infty} \left\{ \int_{0}^{\infty} G(\bar{\gamma}) d \gamma ^{d-1} P_f^G(\gamma) \text{d} \gamma - G(\bar{\gamma}) \gamma_t ^d = 0\right\} \\ &= \text{arg} \lim_{\bar{\gamma} \rightarrow \infty} \left\{ {\gamma_t}^d = d \int_{0}^{\infty} \gamma^{d-1} P_f^G(\gamma) \text{d} \gamma \right\} \\ &\approx \left( d \int_{0}^{\infty} \gamma^{d-1} P_f^G(\gamma) \text{d} \gamma \right)^{1/d}. \end{split}$$ In uncoded systems, $ P_f^G(\gamma) $ can be given in closed-form. For example, for linear modulation, $ P_f^G(\gamma) $ can be approximated as [@Chatzigeorgiou2008] $$\label{eq:P_f_G_uncode} P_f^G(\gamma) \approx 1 - \left(1 - Q\left(\sqrt{c \gamma}\right)\right)^L,$$ where $c$ is modulation constant ($c=2$ for binary-phase-shift-keying (BPSK)), $Q(\cdot)$ is the $Q$ function, $L$ is the frame length. Substituting Eq. (\[eq:P\_f\_G\_uncode\]) into Eq. (\[eq:SNR\_threshold\_continue\]), we can get SNR threshold $\gamma_{t,d}$ for uncoded systems: $$\label{eq:SNR_threshold_uncoded} \gamma_{t,d} \approx \left( d \int_{0}^{\infty} \gamma^{d-1} \left(1 - \left(1 - Q\left(\sqrt{c \gamma}\right)\right)^L \right) \text{d} \gamma \right)^{1/d}.$$ In coded systems, we can calculate Eq. (\[eq:SNR\_threshold\_continue\]) using numerical methods. We can first get $P_f^G(\gamma) $, i.e. the instantaneous FER for the scheme over AWGN channel, using Monte Carlo methods. When the SNR values $\gamma_i^{\prime}$, $i=1,2,\cdots,N$, are equally spaced with $\Delta \gamma$ and ordered, the following equivalent expression for discrete SNR values can be obtained $$\label{eq:SNR_threshold_quanti} \gamma_{t,d} \approx \left( d \sum_{i=1}^{N}\gamma_i^{\prime d-1} P_f^G(\gamma_i^{\prime}) \Delta \gamma\right)^{1/d} .$$ Note that when $c=2$ (BPSK), $L=1$, and $d=1$, we can obtain $\gamma_{t,1}=\frac{1}{4}$ using Eq. (\[eq:SNR\_threshold\_uncoded\]) [@GammaBetaErf], and $\bar{P}_f = 1-e^{-\frac{1}{4\bar{\gamma}}} \approx \frac{1}{4\bar{\gamma}}$, which is equivalent to the well known average BER of BPSK over Rayleigh fading channel at high SNR: $\bar{P}_b=\frac{1}{2}(1-\frac{\bar{\gamma}}{1+\bar{\gamma}}) \approx \frac{1}{4\bar{\gamma}}$ [@Proakis1995]. If we consider a multiple-input multiple-output (MIMO) channel having $N_T$ inputs and $N_R$ outputs. The transmitter uses space-time block coding [@alamouti1998simple], while the receiver coherently combines the $N=N_T N_R$ independent fading paths. If $\gamma$ now corresponds to the instantaneous SNR at the output of the combiner, its probability distribution is given by [@Proakis1995; @Chatzigeorgiou2009]: $$f_{\bar{\gamma}}(\gamma) = \frac{\gamma^{N-1}e^{-\gamma/(\bar{\gamma}/N_T)}}{(\bar{\gamma}/N_T)^N (N-1)!},$$ where $\bar{\gamma}$ is the average SNR per receive antenna. Using the fact that $e^{-x} < 1$ ($x>0$), the CDF of $\gamma$ can be approximated as: $$\begin{split} F_{\bar{\gamma}}(\gamma) &\approx \int_0^{\gamma} \frac{t^{N-1}}{(\bar{\gamma}/N_T)^N (N-1)!} \text{d} t\\ &=\frac{1}{(\bar{\gamma}/N_T)^N N!} \gamma^{N}\\ & =G(\bar{\gamma}) \gamma^{N}, \end{split}$$ where $G(\bar{\gamma}) =\frac{1}{(\bar{\gamma}/N_T)^N N!}$. So, we can obtain the SNR threshold $\gamma_{t,N}$ using above method, and the approximated FER of the system for MIMO quasistatic fading channels can then be approximated as [@Proakis1995; @Chatzigeorgiou2009] $$\begin{split} \bar{P}_f &\approx F_{\bar{\gamma}}(\gamma_{t,N}) \\ &\approx 1- e^{-\gamma_{t,N}N_T/\bar{\gamma}}\sum_{k=0}^{N-1}\frac{(\gamma_{t,N} N_T /\bar{\gamma})^k}{k!} \\ &< \frac{1}{(\bar{\gamma}/N_T)^N N!} \gamma_{t,N}^{N}. \end{split}$$ So far, we have developed an improved SNR threshold-based FER approximation model for general diversity systems. In this model, FER is approximated as an outage probability as Eq. (\[eq:ThresholdModelFER\]), the SNR threshold is given by Eq. (\[eq:SNR\_threshold\_continue\]) and can be calculated using Eq. (\[eq:SNR\_threshold\_uncoded\]) for uncoded systems and using Eq. (\[eq:SNR\_threshold\_quanti\]) for coded systems. FER Analysis of the HRS Scheme {#sec:FER.HRS} ============================== In this section, we perform the FER analysis of the HRS scheme in cooperative systems. Analytical FER Analysis of the HRS Scheme {#subsec:FER_HRS} ----------------------------------------- According to the proposed FER approximation model, a frame error only occurs if SNR is below the SNR threshold $\gamma_{t,d}$. Then, the average FER of the HRS scheme, denoted by $\bar{P}_f$, can be expressed as $$\label{eq:Pf} \begin{split} \bar{P}_f &= Pr({\gamma_{HRS}} < \gamma_{t,d}) , \end{split}$$ where $\gamma_{HRS}$ is the SNR of the HRS scheme and $d$ is the diversity order of the cooperative system with the HRS scheme. Hence, to get the FER for the HRS scheme, we merely need to derive the CDF of $\gamma_{HRS}$. Since the channels are Rayleigh block fading, the instantaneous SNR $\gamma_{1i}$ is exponentially distributed. Then, the CDF of $\gamma_{1i}$ is given by $$\label{eq:CDF_SNR_{1i}} %F_{\gamma_{0}}(\gamma) = 1 - e^{-\lambda_0 \gamma } \text{ and } F_{\gamma_{1i}}(\gamma_{t,1}) = 1 - e^{-\lambda_{1i} \gamma_{t,1} },$$ where $\lambda_{1i}=1/\gamma_{1i}$ is defined in Eq. (\[eq.lambda.define\]). For simplicity, we introduce a vector variable $\mathbf{z}$, where $$\mathbf{z}=\left[z_1, z_2, \cdots, z_n\right], z_i = \begin{cases} 0, & \text{if $\gamma_{1i} \geq \gamma_{t,1}$}, \\ 1, & \text{if $\gamma_{1i}< \gamma_{t,1}$}. \end{cases}$$ In the HRS scheme, the $i$-th relay node is assigned to either $\mathcal{G}_{DF}$ or $\mathcal{G}_{AF}$ according to its SNR $\gamma_{1i}$. If $\gamma_{1i} \geq \gamma_{t,1}$ ($z_i = 0$) then $i \in \mathcal{G}_{DF}$, and if $\gamma_{1i} < \gamma_{t,1}$ ($z_i = 1$) then $i \in \mathcal{G}_{AF}$. By using the fact that for any value $x$, $x^0=1$, then the probability of $z_i$, denoted by $Pr(z_i)$, can be written as $$\begin{split} Pr(z_i) &= [Pr(z_i=0)]^{1-z_i}[ Pr(z_i=1) ]^{z_i}\\ &= [Pr(\gamma_{1i} \geq \gamma_{t,1})]^{1-z_i}[ Pr(\gamma_{1i} < \gamma_{t,1}) ]^{z_i}\\ &=[ F_{\gamma_{1i}}(\gamma_{t,1}) ]^{z_i} [ 1- F_{\gamma_{1i}}(\gamma_{t,1}) ]^{1 - z_i}. \end{split}$$ Then, the CDF of $\gamma_{HRS}$ is derived in Appendix \[Proof:CDF\_SNR\_Z\] as $$\label{eq:CDF_SNR_Z} F_{\gamma_{HRS}}(\gamma_{t,d}) \approx \sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b}\neq0} \mathcal{C}_1 \frac{\mathcal{C}_2(1-e^{- \lambda_{0} \gamma_{t,d}})-\lambda_0(1-e ^{- \mathcal{C}_2 \gamma_{t,d}}) }{\lambda_0-\mathcal{C}_2},$$ where $\mathbf{b}=\{b_i, i=1,\cdots,n\} \in \mathcal{Z}_n, b_i \in \{0,1\}$, $\mathcal{C}_1 = (-1)^{\sum_{i=1}^n b_i} $ and $ \mathcal{C}_2 = \sum_{i=1}^n b_i (\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i})$. Substituting Eq. (\[eq:CDF\_SNR\_Z\]) into Eq. (\[eq:Pf\]), we can obtain the FER for the HRS scheme $$\begin{split} \label{eq:FER_HRS} \bar{P}_{f} \approx \sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b}\neq0} \mathcal{C}_1 \frac{\mathcal{C}_2(1-e^{- \lambda_{0} \gamma_{t,d}})-\lambda_0(1-e ^{- \mathcal{C}_2 \gamma_{t,d}}) }{\lambda_0-\mathcal{C}_2}. \end{split}$$ FER Simplification at High SNR {#subsec:FER_Simplify} ------------------------------ In this subsection, we derive the simplified FER for the HRS scheme at high SNR. At high SNR, $\bar{\gamma_0}$, $\bar{\gamma_{1i}}$ and $\bar{\gamma_{2i}}$ are large, then according to Eq. (\[eq.lambda.define\]): $\lambda_0=1/\bar{\gamma_0}$, $\lambda_{1i}=1/\bar{\gamma_{1i}}$ and $\lambda_{2i}=1/\bar{\gamma_{2i}}$ are small. Using the approximation that $ 1- \exp(- x) \approx x, \text{when $x \rightarrow 0$}, $ we can get the simplified CDF of $\gamma_{HRS}$ at high SNR is given by $$\label{eq:CDF_SNR_Z_approx} F_{\gamma_{HRS}} (\gamma_{t,d}) \approx \frac{\lambda_0 \gamma_{t,d}^{n+1}}{n+1} \prod_{i=1}^n (\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}).$$ The proof of Eq. (\[eq:CDF\_SNR\_Z\_approx\]) is given at appendix \[Proof:CDF\_SNR\_Z\_approx\]. Substituting Eq. (\[eq:CDF\_SNR\_Z\_approx\]) into Eq. (\[eq:Pf\]), we can obtain the simplified FER of the HRS scheme at high SNR $$\begin{split} \label{eq:FER_HRS.simplify} \bar{P}_{f} &\approx %\sum_{\textbf{z} \in \mathcal{Z}_n} %\prod_{i=1}^{n} \left(\lambda_{1i} \gamma_{t,1} \right)^{z_i} % \frac{\lambda_0 (\gamma_{t,d})^{n+1}}{n+1} \prod_{i=1}^n %{\gamma_{t,d}}^{-z_i} %\left(\lambda_{2i}\right)^{1-z_i} \\ %&=\frac{(\gamma_{t,d})^{n+1}}{n+1} \lambda_0 \sum_{\textbf{z} \in \mathcal{Z}_n} % \prod_{i=1}^n %(\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i})^{z_i} %\left(\lambda_{2i}\right)^{1-z_i} \\ %& = \frac{(\gamma_{t,d})^{n+1}}{n+1} \lambda_0 \prod_{i=1}^n \lambda_{2i} \left(1+ \frac{\gamma_{t,1}}{\gamma_{t,d}} \frac{\lambda_{1i}}{ \lambda_{2i}}\right), \end{split}$$ where $\lambda_0 \prod_{i=1}^n \left(\lambda_{1i} + \lambda_{2i}\right) \propto (\bar{\gamma})^{-(n+1)}$, which indicates that the diversity order of $n+1$ can be achieved, $d=n+1$, $\gamma_{t,d}=\gamma_{t,n+1}$ and see also Eq. (\[eq.binomial\]) for the last step. The simplified FER is more intuitive and simple compared to the FER expression in Eq. (\[eq:FER\_HRS\]). Note that both Eq. (\[eq:FER\_HRS\]) and Eq. (\[eq:FER\_HRS.simplify\]) are analytical approximate average FER expressions for the HRS scheme in cooperative networks. The calculation methods of parameters $\gamma_{t,1}$ and $\gamma_{t,d}$ are presented in Section \[sec.FER.approx.model\]. Extending to other Relaying Schemes {#subsec.AF.PDF} ----------------------------------- The FER results for the HRS scheme in Eq. (\[eq:FER\_HRS\]) and Eq. (\[eq:FER\_HRS.simplify\]) can be easily extended to the cooperative systems using the AF with relay selection (AF-RS) scheme [@Zhao2006] and the perfect DF with relay selection (PDF-RS) scheme [@Beres2006]. In the AF-RS scheme, a single relay with the maximum source-relay-destination SNR will be selected to forward signals from the source to the destination using AF [@Zhao2006]. In the PDF-RS scheme, it assumes that all relay nodes can correctly decode the received signals and a single relay with the maximum relay-destination SNR will be selected to forward signals from the source to the destination. For simplicity, we only present the simplified results at high SNR [@Beres2006]. For PDF-RS, the transmissions from the source to any relay are always correct, it means that $\gamma_{1i}=\infty$ and $z_i=0$. Thus, the average FER of PDF-RS, denoted by $\bar{P}_f^{PDF-RS}$, is given by $$\label{eq.FER.PDF-RS} \bar{P}_f^{PDF-RS}\approx \frac{(\gamma_{t,d})^{n+1}}{n+1} \lambda_0 \prod_{i=1}^n \lambda_{2i}.$$ For AF-RS, the SNR of the $i$-th Rayleigh relay link (from the source to the $i$-th Rayleigh relay and to the destination) can be approximated as “combined” Rayleigh fading with parameter of $\lambda_{i}^{AF}\approx\lambda_{1i}+\lambda_{2i}$ at high SNR [^3]. Using this approximation, AF-RS can be viewed as a special case of PDF-RS with $\lambda_{2i}^{'}=\lambda_{i}^{AF}$. The average FER of AF-RS, denoted by $\bar{P}_f^{AF-RS}$, is given by $$\label{eq.FER.AF-RS} \bar{P}_f^{AF-RS}\approx \frac{(\gamma_{t,d})^{n+1}}{n+1} \lambda_0 \prod_{i=1}^n \lambda_{2i} \left(1+ \frac{\lambda_{1i}}{ \lambda_{2i}}\right).$$ Defining the gain of the HRS scheme over the AF-RS scheme as the ratio of FER, we can get $$\label{eq.gain.HRS.AF-RS} G_{\frac{HRS}{AF-RS}} = \frac{\bar{P}_f^{AF-RS}}{\bar{P}_{f} } \approx \prod_{i=1}^n \frac{ 1+ \frac{\lambda_{1i}}{ \lambda_{2i}}}{1+ \frac{\gamma_{t,1}}{\gamma_{t,d}} \frac{\lambda_{1i}}{ \lambda_{2i}} } \text{(dB)}.$$ It indicates that the gain improves when $\frac{\bar{\gamma}_{2i}}{ \bar{\gamma}_{1i}}$ increases, and when the number of relay nodes increases, the gain increases quickly. Simulation results {#sec.simulation} ================== ![FER comparison of proposed model and the model of [@Chatzigeorgiou2008] for case 0: general MIMO channels with $N_T=1$, $N=N_R=1,2,4$, uncoded.[]{data-label="fig.case.0"}](FER_MIMO_Comparison_Result_Without_Outaget_Uncoded){width="80.00000%"} ![Average FER of the HRS scheme for case 1: $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$, uncoded.[]{data-label="fig.case.1"}](UnCoded_L100_case1){width="80.00000%"} ![Theoretical and simulated FER of the HRS scheme for case 4 and 5: $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$, coded.[]{data-label="fig.case.4.5"}](Coded_1n2_case_4_5){width="80.00000%"} ![Theoretical and simulated FER of the HRS scheme for case 6 and 7: $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$, coded.[]{data-label="fig.case.6.7"}](Coded_2n3_case_6_7){width="80.00000%"} ![Average FER of the HRS, AF-RS and PDF-RS schemes for case 4 with $n=4$: $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$, coded.[]{data-label="fig.case.4.AF.PDF.N4"}](Coded_1n2_case_4_AF_PDF_HRS_N4){width="80.00000%"} In this section, we compare the proposed approximate FER expressions with the exact FER obtained by Monte-Carlo simulations. Unless specifically mentioned, the simulations are performed for a BPSK modulation and a frame size of $100$ or $200$ symbols over block Rayleigh fading channels. We consider two basic cases: (1) the *uncoded* case, where none channel code is used; and (2) the *coded* case, where a systematic convolutional code with a code rate of $1/2$ and the generator matrix of $(5,7)_8$ or a code rate of $2/3$ and the generator matrix of $(23,35,0; 0,5,13)_8$ is used. It is practically infeasible to verify the accuracy of analytical results for all the possible scenarios, as a prohibitively large number of combinations can be generated by varying relay number $n$ and average SNR parameters $\Omega$. Therefore, the scenarios in Table \[tb.scenarios\] are chosen such that a range of diverse cases are covered. For instance, cases 1, 2, and 3 are uncoded and case 4, 5, 6 and 7 are coded. Cases 1, 4, 5, and 6 are symmetric situations, while cases 2 and 3 are dissimilar situations. In all cases, 1, 2 and 4 relay nodes are considered. We also choose case 0 for comparing our proposed FER approximation model to the results of [@Chatzigeorgiou2008]. And in all cases, $\gamma_{t,d}=\gamma_{t,n+1}$. The SNR threshold values are obtained using the methods developed in Section \[sec.FER.approx.model\], and are shown in Table \[tb.snr.threshold\]. -------- -------- ------- ---------- ------ ------------------------ --------------------------------------------------- System Frame uncoded/ Code Number of \*[SNRs]{} Type Size coded Rate Nodes case 0 MIMO 100 uncoded - N$_T$=1, N=N$_R$=1,2,4 $\Omega=1$ case 1 HRS 100 uncoded - n=1, 2, 4 $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$ case 2 HRS 100 uncoded - n=1, 2, 4 $\Omega_0=1$, $\Omega_{1i}=16$, $\Omega_{2i}=1$ case 3 HRS 100 uncoded - n= 1, 2, 4 $\Omega_0=1$, $\Omega_{1i}=1/16$, $\Omega_{2i}=1$ case 4 HRS 100 coded 1/2 n= 1, 2, 4 $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$ case 5 HRS 200 coded 1/2 n= 1, 2, 4 $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$ case 6 HRS 100 coded 2/3 n= 1, 2, 4 $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$ case 7 HRS 200 coded 2/3 n= 1, 2, 4 $\Omega_0=\Omega_{1i}=\Omega_{2i}=1$ -------- -------- ------- ---------- ------ ------------------------ --------------------------------------------------- System Type uncoded/coded Number of Nodes Diversity Order SNR Threshold ------------------ ------------- ---------------- -------------------- ----------------- --------------- \*[case 0]{} \*[MIMO]{} \*[uncoded ]{} N$_T$=1, N=N$_R$=1 1 5.10 dB N$_T$=1, N=N$_R$=2 2 5.36 dB N$_T$=1, N=N$_R$=4 4 5.89 dB \*[case 1,2,3]{} \*[HRS]{} \*[uncoded ]{} n=1 2 5.36 dB n=2 3 5.62 dB n=4 5 6.16 dB \*[case 4]{} \*[HRS]{} \*[coded ]{} n=1 2 -1.48 dB n=2 3 -1.26 dB n=4 5 -0.81 dB \*[case 5]{} \*[HRS]{} \*[coded ]{} n=1 2 -0.85 dB n=2 3 -0.68 dB n=4 5 -0.33 dB \*[case 6]{} \*[HRS]{} \*[coded ]{} n=1 2 0.06 dB n=2 3 0.20 dB n=4 5 0.47 dB \*[case 7]{} \*[HRS]{} \*[coded ]{} n=1 2 0.60 dB n=2 3 0.70 dB n=4 5 0.90 dB Fig. \[fig.case.0\] shows the analytical FER curves by using the proposed model and the model of [@Chatzigeorgiou2008] for case 0. For case 0, the SNR threshold was found to be $4.6$ dB based on Eq. (\[eq.SNR.threshod.old\]) (the model of [@Chatzigeorgiou2008]), and was found to be $5.10$ dB, $5.36$ dB and $5.89$ dB for $N=1, 2, 4$, respectively, by our proposed model based on Eq. (\[eq:P\_f\_G\_uncode\]). As shown in the figure, our proposed model converge with the simulated FER quickly as the SNR increases while the relative error using [@Chatzigeorgiou2008] cannot be ignored even at high SNR. The FER results based on the model in [@Chatzigeorgiou2008] become less accurate when the diversity order increaeses, but our proposed model is still accurate. While not shown here, similar trends can be observed for other scenarios. Fig. \[fig.case.1\] shows the proposed approximate FER results and simulated results of the HRS scheme for case 1. From the figure we can see that the analytical theoretical curves obtained by Eq. (\[eq:FER\_HRS\]) converge well with the simulated results, and the simplified results computed by Eq. (\[eq:FER\_HRS.simplify\]) converge to the simulated results at high SNRs. Figs \[fig.case.4.5\] and \[fig.case.6.7\] show the different encoders and other block lengths for coded scheme and confirm the good performance of the proposed model for different parameters. Case 2 and case 3 are also verified through simulation but omitted here for brevity. Fig. \[fig.case.4.AF.PDF.N4\] shows our proposed simplified FER of the AF-RS and PDF-RS for case 4 with 4 relay nodes. From the figures we can see that the simplified results converge to the simulated results at high SNRs for both AF-RS and PDF-RS. And the HRS scheme has considerable performance gain over AF-RS. Conclusions {#sec.conclusion} =========== In this paper, we have analyzed the average FER of the HRS scheme in general cooperative wireless networks with and without applying channel coding at transmitting nodes. We proposed an improved SNR threshold-based FER approximation model. We then apply this model to HRS system and derived the analytical approximate average FER expression and the simplified asymptotic FER expression at high SNRs for the HRS scheme. Simulation results match well with the theoretical analysis, which validates our derived FER expressions. The PDF and CDF of $\gamma_{HRS}$: The Proof of Eq. (\[eq:CDF\_SNR\_Z\]) {#Proof:CDF_SNR_Z} ======================================================================== According to Eq. (\[eq:gamma\_vz\]), $\gamma_{HRS} = \gamma_{0} + \gamma_m$. In the following, we first derive the PDF of $\gamma_0$ and $\gamma_m$ and then get the PDF of $\gamma_{HRS} $ as the convolution of the PDF of $\gamma_0$ and $\gamma_m$. As $\gamma_0$, $\gamma_{1i}$ and $\gamma_{2i}$ are exponentially distributed, their PDF and CDF are given as $$f_{\gamma_k}(\gamma) = \lambda_{k} e^{- \lambda_{k} \gamma }, k \in \{0,1i,2i\},$$ and $$F_{\gamma_k}(\gamma) = 1 - e^{- \lambda_{k} \gamma }, k \in \{0,1i,2i\},$$ respectively. According to Eq. (\[eq:SNR\_i\_cases\]) and Eq. (\[eq:Selection\]), $\gamma_m = \max_{1 \leq i \leq n} \{ \gamma_i \}$ and $$\gamma_i = \begin{cases} \frac{\gamma_{1i}\gamma_{2i}}{\gamma_{1i}+\gamma_{2i}+1}, & \text{if $z_i=1$, }\\ \gamma_{2i}, & \text{if $z_i=0$. } \end{cases}$$ To get the distribution of $\gamma_m$, we first derive the distribution of $\gamma_i$. If $z_i=0$, $\gamma_i=\gamma_{2i}$. Then, the conditional CDF of $\gamma_i$ given $z_i=0$ is $$\label{eq:CDF_gamma_i_zi0} F_{\gamma_i}(\gamma|z_i=0) = Pr(\gamma_{2i}<\gamma) = 1-e^{-\lambda_{2i} \gamma}.$$ If $z_i=1$, using the approximation $\frac{x y}{x+y+1} \approx min\{x,y\}$ [@Anghel2004], then the conditional CDF of $\gamma_{i}$ given $z_i=1$ is [@David2003]. $$\begin{split} \label{eq.CDF.gamma_i_z1} F_{\gamma_i|_{z_i=1}}(\gamma) &= Pr(min\{\gamma_{1i}|_{z_i=1},\gamma_{2i}\} < \gamma) \\ &=1-(1-F_{\gamma_{1i}|_{z_i=1}}(\gamma))(1-F_{\gamma_{2i}}(\gamma)). \end{split}$$ The conditional PDF and CDF of $\gamma_{1i}|_{z_i=1}$ are $$\begin{split} f_{\gamma_{1i}|_{z_i=1}}(\gamma) &= \frac{f_{\gamma_{1i}}(\gamma) Pr(\gamma_{1i}=\gamma, z_i=1)}{Pr(z_i=1)}\\ &= \frac{\lambda_{1i} e^{-\lambda_{1i} \gamma} P_{f,\gamma_{1i}}^G(\gamma)}{P_{f,\gamma_{1i}}}\\ &\approx \begin{cases} \frac{\lambda_{1i} e^{-\lambda_{1i} \gamma} } {1-e^{-\lambda_{1i} \gamma_{t,1}}}, & \text{if } \gamma < \gamma_{t,1}, \\ 0,& \text{if } \gamma \geq \gamma_{t,1}, \end{cases} \end{split}$$ $$\label{eq.CDF.gamma_1i_z1} F_{\gamma_{1i}|_{z_i=1}}(\gamma) = \begin{cases} \frac{ 1-e^{-\lambda_{1i} \gamma}} {1-e^{-\lambda_{1i} \gamma_{t,1}}}, & \text{if } \gamma < \gamma_{t,1}, \\ 1,& \text{if } \gamma \geq \gamma_{t,1}, \end{cases}$$ where $P_{f,\gamma_{1i}}^G(\gamma)$ is the FER of $\gamma_{1i}$ at AWGN channel and $P_{f,\gamma_{1i}}^G(\gamma) \approx 1$ if $\gamma < \gamma_{t,1}$ and $P_{f,\gamma_{1i}}^G(\gamma) \approx 0$ if $\gamma \geq \gamma_{t,1}$. Combining Eq. (\[eq.CDF.gamma\_i\_z1\]) and Eq. (\[eq.CDF.gamma\_1i\_z1\]), the corresponding conditional CDF of $\gamma_i$ given $z_i=1$ can be written as $$\begin{split} \label{eq:CDF_gamma_i_zi1} F_{\gamma_i|z_i=1}(\gamma) & \approx \begin{cases} 1-\frac{ e^{-\lambda_{1i} \gamma}-e^{-\lambda_{1i} \gamma_{t,1}}} {1-e^{-\lambda_{1i} \gamma_{t,1}}} e^{-\lambda_{2i} \gamma}, & \text{if } \gamma < \gamma_{t,1}, \\ 1,& \text{if } \gamma \geq \gamma_{t,1}, \end{cases} \\ %& \approx %\begin{cases} %1-e^{-(\frac{1}{\gamma_{t,d}} + \lambda_{1i} + \lambda_{2i}) \gamma}, & \text{if } \gamma < \gamma_{t,d}, \\ %1,& \text{if } \gamma \geq \gamma_{t,d}, %\end{cases} \end{split}$$ where at the second step, we extend the range of $\gamma$ from $\gamma_{t,1}$ to $\gamma_{t,d}$ using an exponential function to approximate $F_{\gamma_i|z_i=1}(\gamma)$. The simulation results in Section \[sec.simulation\] indicate that this approximation is accurate. Combining Eq. (\[eq:CDF\_gamma\_i\_zi0\]) and Eq. (\[eq:CDF\_gamma\_i\_zi1\]) and noting that $Pr(z_i=1)=1-e^{-\lambda_{1i}\gamma_{t,1}}$ and $Pr(z_i=0)=e^{-\lambda_{1i}\gamma_{t,1}}$, the CDF of $\gamma_i$ can be rewritten as $$\begin{split} \label{eq:CDF_gamma_i} F_{\gamma_i}(\gamma) &= Pr(z_i=1)F_{\gamma_i}(\gamma|z_i=1) + Pr(z_i=0)F_{\gamma_i}(\gamma|z_i=0)\\ &\approx \begin{cases} 1-e^{-(\lambda_{1i} + \lambda_{2i}) \gamma}, & \text{if } \gamma < \gamma_{t,1}, \\ 1-e^{-(\lambda_{1i}\gamma_{t,1} + \lambda_{2i}\gamma) },& \text{if } \gamma \geq \gamma_{t,1}, \end{cases}\\ &\approx \begin{cases} 1-e^{-(\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}) \gamma}, & \text{if } \gamma < \gamma_{t,d}, \\ 1-e^{-(\lambda_{1i}\gamma_{t,1} + \lambda_{2i}\gamma) },& \text{if } \gamma \geq \gamma_{t,d}, \end{cases} \end{split}$$ Noting that $\gamma_{t,1}\lessapprox \gamma_{t,d}$, we can use a exponential function to approximate the CDF of $\gamma_i$ when $\gamma < \gamma_{t,d}$. In order to keep the value $F_{\gamma_i}(\gamma_{t,d})$ unchanged, we have the last step. Then the CDF of $\gamma_m$ (when $\gamma < \gamma_{t,d}$) is given by [@David2003] $$\label{eq:CDF_SNR_Max_1} \begin{split} F_{\gamma_m}(\gamma) &= Pr(\max_{1\leq i \leq n}\{\gamma_i\} < \gamma)\\ %= Pr(\{\gamma_1,\cdots,\gamma_i,\cdots,\gamma_n \} < \gamma) %= \prod_{i=1}^n Pr(\gamma_i<\gamma) &\approx \prod_{i=1}^n \left(1 -e ^{-(\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}) \gamma}\right)\\ &= {\sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1e ^{- \mathcal{C}_2 \gamma} }, \end{split}$$ with the help of that $$\label{eq.binomial} \prod_{i=1}^n (1 + x_i) =\prod_{i=1}^n {x_i}^{0} + x_1^1\prod_{i=2}^n {x_i}^{0} + \cdots + \prod_{i=1}^n {x_i}^{1} = \sum_{\mathbf{b} \in \mathcal{Z}_n} \prod_{i=1}^n {x_i}^{b_i},$$ where $\mathbf{b}=\{b_i, i=1,\cdots,n\} \in \mathcal{Z}_n, b_i \in \{0,1\}$, $\mathcal{C}_1 = (-1)^{\sum_{i=1}^n b_i} $ and $ \mathcal{C}_2 = \sum_{i=1}^n b_i (\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}) $ . The PDF of $\gamma_m$ is given by $$f_{\gamma_m}(\gamma) \approx -{\sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b} \neq 0} \mathcal{C}_1 \mathcal{C}_2 e ^{- \mathcal{C}_2 \gamma} }.$$ As $\gamma_{HRS}=\gamma_0+\gamma_m$, the PDF of $\gamma_{HRS}$ when $\gamma \leq \gamma_{t,d}$ can be expressed as $$\begin{split} \label{eq:PDF_gamma_z} f_{\gamma_{HRS}}(\gamma) &= \int_0^{\gamma} f_{\gamma_0}(\gamma-t) f_{\gamma_m}(t) \text{d} t \\ &=- \int_0^{\gamma} \lambda_0 e^{- \lambda_{0} (\gamma-t)}{\sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1 \mathcal{C}_2 e ^{- \mathcal{C}_2 t} } \text{d} t \\ &\approx - \lambda_0 e^{- \lambda_{0} \gamma} {\sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b}\neq0} \mathcal{C}_1 \mathcal{C}_2 \int_0^{\gamma} e ^{- (\mathcal{C}_2 - \lambda_0)t} } \text{d} t \\ &= - \lambda_0 e^{- \lambda_{0} \gamma}\sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b}\neq0} \mathcal{C}_1 \mathcal{C}_2 \frac{1-e ^{- (\mathcal{C}_2 - \lambda_0)\gamma} }{\mathcal{C}_2 - \lambda_0} \\ &= \sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b}\neq0} \mathcal{C}_1 \lambda_0 \mathcal{C}_2 \frac{e^{- \lambda_{0} \gamma}-e ^{- \mathcal{C}_2\gamma} }{ \lambda_0 -\mathcal{C}_2}. \end{split}$$ Finally, the CDF of $\gamma_{HRS}$ is given by $$\begin{split} \label{eq:CDF_gamma_z} F_{\gamma_{HRS}}(\gamma_{t,d}) &= \int_0^{\gamma_{t,d}} f_{\gamma_{HRS}}(t)\text{d}t \\ &\approx \int_0^{\gamma_{t,d}} \sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b}\neq0} \mathcal{C}_1 \lambda_0 \mathcal{C}_2 \frac{e^{- \lambda_{0} t}-e ^{- \mathcal{C}_2t} }{ \lambda_0-\mathcal{C}_2} \text{d}t \\ &= \sum_{\mathbf{b} \in \mathcal{Z}_n, \mathbf{b}\neq0} \mathcal{C}_1 \frac{\mathcal{C}_2(1-e^{- \lambda_{0} \gamma_{t,d}})-\lambda_0(1-e ^{- \mathcal{C}_2 \gamma_{t,d}}) }{\lambda_0-\mathcal{C}_2}. \end{split}$$ Simplified CDF of $\gamma_{HRS}$ at high SNR: The Proof of Eq. (\[eq:CDF\_SNR\_Z\_approx\]) {#Proof:CDF_SNR_Z_approx} =========================================================================================== Using Eq. (\[eq:PDF\_gamma\_z\]) and the approximation that $e^{-x} \approx 1- x$, the PDF of $\gamma_{HRS}$ at high SNR can be approximated as $$\begin{split} \label{eq:PDF_gamma_z_a0} f_{\gamma_{HRS}}(\gamma) &\approx \sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1 \lambda_0 \mathcal{C}_2 \frac{(1- \lambda_{0} \gamma)-(1- \mathcal{C}_2\gamma) }{ \lambda_0 -\mathcal{C}_2}\\ &=- \lambda_0 \sum_{\mathbf{b} \in \mathcal{Z}_n} { \mathcal{C}_1 \mathcal{C}_2 \gamma}. \end{split}$$ Applying the approximation that $e^{-x} \approx 1- x$ to Eq. (\[eq:CDF\_gamma\_i\]) and Eq. (\[eq:CDF\_SNR\_Max\_1\]), we can get $$\label{eq:CDF_gamma_m_1} F_{\gamma_m}(\gamma) \approx \gamma^n \prod_{i=1}^n (\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}),$$ and $$\label{eq:CDF_gamma_m_2} F_{\gamma_m}(\gamma) \approx \sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1 (1-\mathcal{C}_2\gamma) = \sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1 -\sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1 \mathcal{C}_2 \gamma.$$ As $F_{\gamma_m}$ is CDF, $F_{\gamma_m}(0) = 0$, we can get $$\label{eq:CDF_gamma_m_0} F_{\gamma_m}(0) = \sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1 = 0.$$ Combining Eq. ([\[eq:CDF\_gamma\_m\_1\]]{}), Eq. (\[eq:CDF\_gamma\_m\_2\]), and Eq. (\[eq:CDF\_gamma\_m\_0\]), we can get $$\label{eq:CDF_gamma_m_3} -\sum_{\mathbf{b} \in \mathcal{Z}_n} \mathcal{C}_1 \mathcal{C}_2 \gamma \approx \gamma^n \prod_{i=1}^n (\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}).$$ Substituting Eq. (\[eq:CDF\_gamma\_m\_3\]) into Eq. (\[eq:PDF\_gamma\_z\_a0\]), we can get the approximated PDF of $\gamma_{HRS}$ as $$f_{\gamma_{HRS}}(\gamma) \approx \lambda_0 \gamma^n \prod_{i=1}^n (\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}),$$ And the CDF of $\gamma_{HRS}$ can be expressed $$\begin{split} \label{eq:CDF_gamma_z_a} F_{\gamma_{HRS}}(\gamma_{t,d}) = \int_0^{\gamma_{t,d}} f_{\gamma_{HRS}}(t) \text{d} t \approx \frac{\lambda_0 \gamma_{t,d}^{n+1}}{n+1} \prod_{i=1}^n (\frac{\gamma_{t,1}}{\gamma_{t,d}}\lambda_{1i} + \lambda_{2i}). \end{split}$$ [^1]: Tianxi Liu, Lingyang Song, Qiang Huo and Bingli Jiao are with the State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Engineering and Computer Science, Peking University, Beijing, China, 100871. e-mail: ; ; ; . [^2]: Yonghui Li is with the University of Sydney, Sydney, NSW 2006, Australia e-mail: . [^3]: This is a lower bound for AF [@Anghel2004; @LiuTianxi2010].

--- abstract: 'For an electron in the plane subjected to a perpendicular constant magnetic field and a homogeneous Gau[ss]{}ian random potential with a Gau[ss]{}ian covariance function we approximate the averaged density of states restricted to the lowest Landau level. To this end, we extrapolate the first 9 coefficients of the underlying continued fraction consistently with the coefficients’ high-order asymptotics. We thus achieve the first reliable extension of Wegner’s exact result \[Z. Phys. B [**51**]{}, 279 (1983)\] for the delta-correlated case to the physically more relevant case of a non-zero correlation length.' address: - ' ${}^1$Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstr. 7, D-91058 Erlangen, Germany' - ' ${}^2$Institut für Theoretische Physik, Universität Göttingen, Bunsenstr. 9, D-37073 Göttingen, Germany' author: - 'Markus Böhm,${}^1$ Kurt Broderix,${}^2$ and Hajo Leschke${}^1$' title: ' Lowest Landau level broadened by a Gau[ss]{}ian random potential with an arbitrary correlation length: An efficient continued-fraction approach ' --- Nearly ideal two-dimensional electronic structures have attracted great attention for more than a decade not only because of their varied and important applications, but also because of the discovery of the quantum Hall effect[@review]. For a microscopic understanding of the occurring phenomena it is essential to know the spectral properties of electrons confined to two dimensions under the influence of a perpendicular constant magnetic field taking into account the presence of disorder. A commonly studied minimal model is that of non-interacting electrons which is characterized by the one-electron Hamiltonian given by the Schrödinger operator $$\label{H} \hat{H} := \hat{K} + \hat{V}$$ $$\hat{K} := \frac{1}{2m}\left( \frac{\hbar}{\text{i}} \frac{\partial}{\partial x_1} - \frac{\text{e}B}{2}x_2 \right)^2 + \frac{1}{2m}\left( \frac{\hbar}{\text{i}} \frac{\partial}{\partial x_2} + \frac{\text{e}B}{2}x_1 \right)^2 {}.$$ Here $x:=(x_1,x_2)$ are Cartesian coordinates of the Euclidean plane ${\Bbb R}^2$, $\hbar$ is Planck’s constant, e is the elementary charge, $m$ is the (effective) mass of the (spinless) electron, and $B>0$ the strength of a perpendicular magnetic field. The static random potential $V$ is added to mimic the interaction with quenched disorder. Its probability distribution is assumed to be Gau[ss]{}ian with zero mean and Gau[ss]{}ian covariance, that is $$\label{covariance} \overline{V(x)}=0 , \quad \overline{V(x)V(x')} = \sigma^2 \exp\!\left[-\frac{(x-x')^2}{2\lambda^2}\right] {}.$$ The overbar denotes averaging with respect to the probability distribution. Finally, $\sigma>0$ is the strength and $\lambda>0$ the correlation length of the fluctuations of the potential. The spectral resolution of the unperturbed Hamiltonian $\hat{K}$ reads $\hat{K}=\sum_{n=0}^\infty \varepsilon_n \hat{E}_n$, where the eigenvalue $\varepsilon_n := (2n+1) \frac{\hbar \text{e} B}{2 m}$ is called the $n$-th Landau level and $\hat{E}_n$ denotes the corresponding eigenprojector. With increasing $B$ the magnetic length $l:=\sqrt{\hbar/(\text{e}B)}$ decreases, while the degeneracy $\langle x | \hat{E}_n | x \rangle = (2\pi l^2)^{-1}$ (per area) of each Landau level and the distance between successive levels increase. Hence, for a fixed concentration of electrons, sufficiently high fields, and low temperatures it is reasonable to simplify the model by restricting its Hamiltonian (\[H\]) to the (still infinite dimensional) eigenspace $\hat{E}_0L^2({\Bbb R}^2)$ of the lowest Landau level. This has been done, for example, in the important work of Wegner [@wegner], where the averaged restricted density of states (per area) $$\label{dos} \varrho(\varepsilon) := \overline{ \langle y|\hat{E}_0 \delta(\varepsilon - \hat{E}_0 \hat{H} \hat{E}_0) \hat{E}_0 |y \rangle }$$ has been calculated exactly in the delta-correlated limit $\lambda \downarrow 0$, $\sigma \to \infty$, $\lambda^2 \sigma^2 = \mbox{const}$. The purpose of the present Letter is twofold. One goal is to present what we think is an accurate extension of Wegner’s result to arbitrary values of $\sigma$ and $\lambda$. Clearly, this is of physical interest because in the high-field limit the actual correlation length is no longer small in comparison with the magnetic length. The other goal is to demonstrate thereby that the power of the well-known continued-fraction approach to spectral densities of non-trivial quantum problems [@cookbook; @czycholl] can considerably be enhanced in cases, where one has an [*a-priori*]{} information about their asymptotic high-frequency decay. To get rid of physical dimensions we write $\varrho$ in the standardized form $$\label{dimlessdos} \varrho(\varepsilon) =: \frac{\sqrt{1+(l/\lambda)^2}}{2\pi l^2 \sigma} \; W\!\left( \frac{\lambda^2}{l^2}, \frac{(\varepsilon-\varepsilon_0)}{\sigma}\sqrt{1+(l/\lambda)^2} \right) {}.$$ In this way we have introduced a one-parameter family of even probability densities on the real line ${\Bbb R}$ with normalized second moment $$W(a,u) = W(a,-u) \ge 0$$ $$\label{normalization} \int_{\Bbb R} \! \text{d} u \, W(a,u) = 1 = \int_{\Bbb R} \! \text{d} u \, W(a,u) \, u^2 {}.$$ Here we have used Eq. (7) of [@jphysa] to normalize the second moment. In this notation Wegner’s result for the delta-correlated limit reads [@wegner; @brezin] $$\label{wegnerdos} W(0,u) = %\frac{2}{\pi^{3/2}} \frac{ 2 \pi^{-3/2} \exp(u^2) }{1+ \left[ %\frac{2}{\sqrt{\pi}} 2 \pi^{-1/2} \int_0^u \text{d}\xi \, \exp(\xi^2) \right]^2} {}.$$ For the other extreme of the spatial extent of correlations, namely the constantly correlated case $\lambda = \infty$, one simply has [@zphys] the Gau[ss]{}ian $$\label{constantdos} W(\infty,u) = (2\pi)^{-1/2}\exp\!\left(-u^2/2\right) {}.$$ For intermediate values of $a$ no exact expression for $W(a,u)$ is known. What is exactly known, however, is the fact that $W(a,u)$ falls off for sufficiently large $|u|$ like a Gau[ss]{}ian. More precisely, by Eqs. (17) and (15) of [@jphysa] (see also [@apel]) $$\label{decay} \lim_{u \to \pm \infty} \frac{1}{u^2} \ln( W(a,u)) = -\frac{a+2}{2a+2} {}.$$ In the sequel we will design approximations to the Stieltjes transform $$R(a,z) := \int_{\Bbb R} \! \text{d}u \, \frac{W(a,u)}{z - \text{i}u} , \quad \mbox{Re} z > 0 ,$$ of the standardized density of states $W$ which in turn will yield approximations to $W$ by means of the inversion formula $$\label{inversion} W(a,u) = \frac{1}{\pi} \lim_{v \downarrow 0} \mbox{Re}\left[ R(a,v+\text{i}u) \right] {}.$$ According to Stieltjes’ classical theory, see for example [@perron; @wall], $R$ can be expanded into a Jacobi-type continued fraction $$R(a,z) = \mathop{\text{K}}_{j=1}^\infty (\frac{r_j(a)}{z}) , \quad r_j(a) \ge 0 {}.$$ Here we are using the notation $$\mathop{\text{K}}_{j=1}^\infty (\frac{\Delta_j}{z}) := \frac{\displaystyle 1}{\displaystyle z + \frac{\displaystyle \Delta_1}{\displaystyle z + \frac{\displaystyle \Delta_2}{\displaystyle z + \ddots } } }$$ for the continued fraction with coefficients $\Delta_1, \Delta_2, \ldots$ and variable $z$. To derive the continued-fraction coefficients $\{ r_j(0) \}$ and $\{ r_j(\infty) \}$ corresponding to (\[wegnerdos\]) and (\[constantdos\]), respectively, we use the identity $$\label{terminator} \mathop{\text{K}}_{j=1}^\infty (\frac{\beta + \gamma j}{z}) = \frac{ D_{-(\beta/\gamma)-1} ( \gamma^{-1/2} z ) }{ \gamma^{1/2} D_{-\beta/\gamma} ( \gamma^{-1/2} z )} =: T(\beta,\gamma,z) ,$$ valid if $\gamma>0$, $\beta+\gamma>0$, and $\mbox{Re}z>0$. Here $D_\nu$ denotes Whittaker’s parabolic cylinder function with index $\nu$, see Section 8.1 of [@magnus]. The identity follows by iteration from the observation that the rhs of (\[terminator\]) obeys $$\label{recurrence} T(\beta,\gamma,z) = [z+(\beta+\gamma)T(\beta+\gamma,\gamma,z)]^{-1}$$ which itself is a consequence of the first of the four recurrence relations for the $D_\nu$’s in Section 8.1.3 of [@magnus]. In fact, Eq. (\[terminator\]) is equivalent to Eq. (14) in §50 of [@perron]. With the help of (\[inversion\]) one now checks that $$r_j(0) = \frac{1}{2} + \frac{1}{2} j , \quad r_j(\infty) = j ,$$ for all $j \ge 1$. According to [@growth-rate] the asymptotic behavior (\[decay\]) implies the following asymptotic linear growth for the continued-fraction coefficients $$\label{asympty} \lim_{j\to\infty} \frac{r_j(a)}{j} = \frac{a+1}{a+2} {}.$$ It is natural to match this linear growth to the first $J < \infty$ coefficients $r_1(a)$, …, $r_J(a)$ to construct the announced approximations $R^{(J)}(a,z)$ to $R(a,z)$ by means of $$R^{(J)}(a,z) := \mathop{\text{K}}_{j=1}^\infty (\frac{r^{(J)}_j(a)}{z}),$$ where $$\label{match} r_j^{(J)}(a) := \left\{ \begin{array}{ll} r_j(a) & \quad \mbox{for} \quad j \le J \\ r_J(a) + \frac{a+1}{a+2} (j-J) & \quad \mbox{for} \quad j > J \end{array} \right. {}.$$ Not surprisingly, the approximation $R^{(J)}$ to $R$ results in an approximation $W^{(J)}$ to $W$ with the same Gaussian fall-off (\[decay\]) in the tails. This can be seen as follows. By virtue of (\[terminator\]) the approximation $R^{(J)}$ can be expressed as a terminating continued fraction $$R^{(J)}(a,z) = \frac{\displaystyle 1}{\displaystyle z + \frac{\displaystyle r_1(a)}{\displaystyle z +_{\ddots\displaystyle + \frac{\displaystyle r_{J-1}(a)}{\displaystyle z + r_J(a)T(r_J(a),{\textstyle\frac{a+1}{a+2}},z) } } } }.$$ In view of (\[inversion\]) it is therefore sufficient to show $$\label{t-prop-re} \lim_{u \to \pm\infty} \frac{1}{u^2} \ln(\mbox{Re}[T(\beta,\gamma,iu)]) = -\frac{1}{2\gamma}$$ and $$\label{t-prop-im} \lim_{u \to\pm\infty} \mbox{Im}[T(\beta,\gamma,iu)] = 0,$$ because these asymptotic properties are conserved under the (repeated) substitution $$T(\beta,\gamma,iu) \mapsto [iu + \alpha T(\beta,\gamma,iu)]^{-1}$$ for any $\alpha>0$. The validity of (\[t-prop-re\]) und (\[t-prop-im\]) can be deduced from an asymptotic evaluation [@bender] of the following Riccati differential equation $$\gamma \frac{d}{du} T(\beta,\gamma,iu) = i\beta \left[T(\beta,\gamma,iu)\right]^2 - u T(\beta,\gamma,iu) - i$$ which itself follows from the aforementioned recurrence relations. To summarize, Eq. (\[terminator\]) adds a two-parameter family of terminators for affine-linear extrapolations to the toolkit of [@cookbook]. In order to compute the first $J$ continued-fraction coefficients $r_1(a)$, …, $r_J(a)$, we employ the well-known one-to-one correspondence [@perron; @wall; @cookbook] to the first $J$ even moments $M_2(a)$, …, $M_{2J}(a)$ of the standardized density of states. According to (\[dos\]) and (\[dimlessdos\]) one has for the latter $$\begin{aligned} \label{momdef} M_{2j}(\lambda^2/l^2) := \int_{\Bbb R} \! \text{d}u\, W(\lambda^2/l^2,u)u^{2j} \nonumber \\ = 2\pi l^2 \left(\frac{1+l^2/\lambda^2}{\sigma^2}\right)^j \overline{ \langle y|(\hat{E}_0 \hat{V} \hat{E}_0)^{2j}|y \rangle }\end{aligned}$$ if $j \ge 1$. We now substitute $\hat{V}=\int_{{\Bbb R}^2}\text{d}^2x \, V(x) | x \rangle \langle x |$ into the rhs of (\[momdef\]) and use the standard reduction formula $$\overline{\prod_{k=1}^{2j}V(x(k))} = \sum_{k=1}^{(2j-1)!!} \prod_{s=1}^j \overline{V(x({P_k(2s-1)}))V(x({P_k(2s)}))}$$ for the average of a product of $2j$ jointly Gau[ss]{}ian random variables with zero mean. Here $x(1),\ldots,x(2j) \in {\Bbb R}^2$ are $2j$ points of the plane and $P_k$ denotes the $k$-th of the $(2j-1)!!$ permutations of the first $2j$ natural numbers which lead to different sets $\left\{ \{ P_k(1),P_k(2) \} , \ldots ,\{ P_k(2j-1),P_k(2j) \} \right\}$ of $j$ pairs $\{ P_k(2s-1),P_k(2s) \}$. Since the covariance function (\[covariance\]) of the random potential $V$ and the position representation $$\langle x|\hat{E}_0|x' \rangle = \frac{1}{2\pi l^2} \exp\!\left[ \frac{\text{i}}{2 l^2}(x_1x_2'-x_2x_1') - \frac{1}{4l^2}(x-x')^2 \right]$$ of the eigenprojector corresponding to the lowest Landau level are both Gau[ss]{}ian, one ends up with a sum over $4j$-dimensional Gau[ss]{}ian integrals which can be performed to yield $$\label{momsum} M_{2j}(a) = \Big(1+\frac{1}{a}\Big)^j \: \sum_{k=1}^{(2j-1)!!} \frac{1}{\det\left(A_k(a)\right)} , \quad a>0.$$ Here the $2j\times 2j$-matrix $A_k(a)$ is defined through its entries in terms of the Kronecker delta $$\left(A_k(a)\right)_{\mu,\nu} := \Big(1+\frac{1}{a}\Big) \delta_{\mu,\nu} - \delta_{\mu+1,\nu} - \frac{1}{a}\sum_{s=1}^j \left( \delta_{\mu,P_k(2s-1)} \delta_{\nu,P_k(2s)}+ \delta_{\nu,P_k(2s-1)} \delta_{\mu,P_k(2s)} \right) {}.$$ We have computed the sum (\[momsum\]) for $j=1, 2, \ldots, 6$ using the [Axiom]{} symbolic system [@axiom] for general $a$. The results for $M_{2j}(a)$, $j=1,2,3,4$, are given by the following rational functions $$\begin{array}{c} M_{2}(b-1) = 1 \\[1ex] M_{4}(b-1) = (3 b^2+2) / (b^2+1) \\[1ex] M_{6}(b-1) = ({{{{15} {b^6}}+{{75} {b^4}}+{{102} {b^2}}+{30}}) / ({{b^6}+{6 {b^4}}+{{11} {b^2}}+6} }) \\[1ex] M_{8}(b-1) = ( {{105} {b^{24}}} +{{2835} {b^{22}}} +{{32830} {b^{20}}} +{{213697} {b^{18}}} +{{861130} {b^{16}}} +{{2231807} {b^{14}}}\\ +{{3750338} {b^{12}}} +{{4038543} {b^{10}}} +{{2717117} {b^8}} +{{1105510} {b^6}} +{{261216} {b^4}} +{{32792} {b^2}} +{1680} )/ \\ ( {b^{24}} +{{29} {b^{22}}} +{{365} {b^{20}}} +{{2620} {b^{18}}} +{{11854} {b^{16}}} +{{35276} {b^{14}}} +{{69974} {b^{12}}} +{{91906} {b^{10}}} \\ +{{78025} {b^8}} +{{41015} {b^6}} +{{12461} {b^4}} +{{1954} {b^2}} +{120} ). \end{array}$$ Since the expressions for $M_{10}(a)$ and $M_{12}(a)$ are extremely lengthy, we only give, as an example, their values for $a=1$: $$\begin{array}{c} M_{10}(1)= \frac{158\,659\,605\,940\,126\,452\,841}% {294\,310\,802\,651\,335\,470} \\[1ex] M_{12}(1)= \frac{388\,336\,271\,072\,847\,928\,549\,926\,597\,% 113\,071\,401\,088\,677\,997\,478\,405\,727\,555\,223\,031}% {82\,363\,680\,790\,265\,452\,914\,044\,225\,729\,941\,% 466\,953\,642\,191\,484\,142\,275\,602\,750}. \end{array}$$ The moments $M_{14}(a)$ and $M_{16}(a)$ have been computed exactly as a reduced fraction for $a=\frac{1}{4},\frac{1}{2},1,2,4$ only, by employing a C-program. For the same set of values for $a$ we also have computed $M_{18}(a)$, but only have been able to gather up the $17!! \approx 3.4 \cdot 10^7$ terms occurring in (\[momsum\]) to a sum of approximately $10^4$ reduced fractions, the value of which has been computed accurately by floating-point arithmetics. For lack of space we only show the first 35 digits for the example $a=1$: $$\begin{aligned} M_{14}(1)=47657.946072630475536554559639613945\ldots \\ M_{16}(1)=545841.43592501744224295351679205436\ldots \\ M_{18}(1)=6980770.3795705620571551127542699202\ldots\end{aligned}$$ From Fig. \[fig:delta\] it can be seen that the resulting continued-fraction coefficients $r_1(a),\ldots,r_9(a)$ approach quite rapidly their asymptotic behavior given in (\[asympty\]). For example, the relative extrapolation error $|r_9^{(8)}(a) - r_9(a)|/r_9(a)$ varies between $0.2\%$ for $a=4$ and $0.02\%$ for $a=\frac{1}{4}$. In Fig. \[fig:convergence\] we demonstrate the convergence of the resulting approximations $W^{(J)}(1,u)$ to $W(1,u)$ for increasing $J=1,2,\ldots,9$. The differences of two successive approximations, $W^{(J)}(a,u)-W^{(J-1)}(a,u)$, seem to form a nearly alternating sequence with geometric decrease for fixed $u$. Both observations, taken together, suggest rapid pointwise convergence of the sequence $\{W^{(J)}\}$. Therefore, we may safely conclude that $W^{(9)}$ constitutes a reliable approximation to $W$. Figure \[fig:dos\] shows a plot of the corresponding approximation $\varrho^{(9)}$ to the averaged density of states $\varrho$ for different $\lambda$. Note that, by construction, $\varrho^{(9)}$ is exact in the limiting cases of a delta-correlated and a constantly correlated random potential; more importantly, $\varepsilon\mapsto 2\pi l^2\varrho^{(9)}(\varepsilon+\varepsilon_0)$ is an even probability density with the same first $18$ moments and the same Gau[ss]{}ian fall-off in the tails as the true density for general values of the correlation length $\lambda$. For review-type literature see T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. [**54**]{}, 437 (1982); I. V. Kukushkin, S. V. Meshkov, and V. B. Timofeev, Usp. Fiz. Nauk. [**155**]{}, 219 (1988) \[Sov. Phys. Usp. [**31**]{}, 511 (1988)\]; G. Landwehr (ed), [*High Magnetic Fields in Semiconductor Physics I, II, III*]{} (Springer, Berlin, 1987, 1989, 1992). M. Jan[ß]{}en, O. Viehweger, U. Fastenrath, and J. Hajdu, [*Introduction to the Theory of the Integer Quantum Hall Effect*]{} (VCH, Weinheim, 1994) V. S. Viswanath and G. Müller, [*The Recursion Method — Application to Many-Body Dynamics*]{}, Lecture Notes in Physics, Vol m23 (Springer, Berlin, 1994), and references therein. Continued fractions were applied successfully to a lattice version of (\[H\]) by G. Czycholl and W. Ponischowski, Z. Phys. B [**73**]{}, 343 (1988). K. Broderix, N. Heldt, and H. Leschke, J. Phys. A [**24**]{}, L825 (1991). F. Wegner, Z. Phys. B [**51**]{}, 279 (1983). see also E. Brézin, D. J. Gross, and C. Itzykson, Nucl. Phys. B [**235**]{}, 24 (1984); A. Klein and J. F. Perez, Nucl. Phys. B [**251**]{}, 199 (1985). see, for example, K. Broderix, H. Heldt, and H. Leschke, Z. Phys. B [**68**]{}, 19 (1987). W. Apel, J. Phys. C [**20**]{}, L577 (1987). O. Perron, [*Die Lehre von den Kettenbrüchen, Vols. I, II*]{} (Teubner, Stuttgart, 1977). H. S. Wall, [*Continued Fractions*]{} (Van Nostrand, New York, 1948) W. Magnus, F. Oberhettinger, and R. P. Soni, [*Formulas and Theorems for the Special Functions of Mathematical Physics*]{}, 3rd enlarged ed. (Springer, Berlin, 1966). D. S. Lubinsky, H. N. Mhaskar, and E. B. Saff, Constr. Approx. [**4**]{}, 65 (1988), and references therein. see, for example, C. M. Bender and S. A. Orszag, [*Advanced Mathematical Methods for Scientists and Engineers*]{}, (McGraw-Hill, Auckland, 1978) R. D. Jenks and R. S. Sutor, [AXIOM: The Scientific Computation System]{} (Springer, Berlin, 1992). -33mm -28mm -12mm -25pt -15mm -33mm -28mm

--- author: - | N. Severijns$^{1}$, O. Zimmer$^{2}$, H.-F. Wirth$^{2}$ and D. Rich$% ^{2}$\ $^{1}$Katholieke Universiteit Leuven, B-3001 Leuven, Belgium\ $^{2}$Technische Universität München, 85747 Garching, Germany title: 'Comment to Observation of the neutron radiative decay by R.U. Khafizov et al., published in JETP Letters 83 (2006) 5 (Pis’ma v ZhETF 83 (2006) 7)' --- Herewith we formally deny any responsibility for the content of the commented paper. The manuscript was submitted for publication without informing at least four of the other authors, viz. N. Severijns, O. Zimmer, H.-F. Wirth and D. Rich. This violates our rights as collaborators. The analysis presented and the manuscript itself have not been discussed and have also not been approved by the entire collaboration prior to submission. Besides this formal incorrectness, we also criticise the content of the paper (low quality and premature). Not only the interpretation, but already the presentation of data is not comprehensive and does not fulfill minimum scientific standards. These views are shared by J. Byrne who is collaborating on the project as well, but was not mentioned on the paper. The coincidence spectrum shown in fig. 5 contains a forest of peaks. Only the two major peaks are explained. The highest peak is called zero or false coincidences with reference to a publication by another group, where this peak is not discussed at all. To our understanding it might be due to a physics event as for example detection by the proton detector of an electron bremsstrahlung photon created in the electron detector, coincident with detection of an electron by the electron detector (as seems to be suggested in the paper), or cross-talk of electronics of the corresponding detectors, or some other reason. No attempt for explanation is made for the smaller peaks in Fig. 5. Figure 6 seems to show a spectrum of triple coincidences between three different detectors. Without that it is mentioned explicitly in the figure caption, the explanations given seem to indicate that the horizontal axis represents the time between an event in the electron detector and an event in the proton detector. It was said in the text that the feature in channel 120 in Fig. 6 contains coincidences between decay electrons and protons, along with an event in one of the gamma detectors. However, no width of the coincidence window is stated. Further, since exactly this type of events was announced to be used for determination of the branching ratio of radiative neutron decay, the statement that the analysis of the branching ratio was based on the leftmost peak with maximum in channel 102 remains completely incomprehensible. Also, no attempt for explanation is made for the broad bump with maximum at about channel 165 in Fig. 6. Besides these deficiencies of the presented treatment it is obvious that the detector setup allows for many types of events which were not discussed. Without a careful analysis of backgrounds one just cannot extract any quantitative information about neutron decay, and in particular not seriously extract a branching ratio of a hitherto unuobserved effect. We therefore do not endorse the claim of the first author to have observed radiative neutron decay.

--- abstract: 'Using an effective Hamiltonian including the Zeeman and internal interactions, we describe the quantum theory of magnetization dynamics when the spin system evolves non-adiabatically and out of equilibrium. The Lewis-Riesenfeld dynamical invariant method is employed along with the Liouville-von Neumann equation for the density matrix. We derive a dynamical equation for magnetization defined with respect to the density operator with a general form of damping that involves the non-equilibrium contribution in addition to the Landau-Lifshitz-Gilbert equation. Two special cases of the radiation-spin interaction and the spin-spin exchange interaction are considered. For the radiation-spin interaction, the damping term is shown to be of the Gilbert type, while in the spin-spin exchange interaction case, the results depend on a coupled chain of correlation functions.' author: - 'F. M. Saradzhev' - 'F. C. Khanna' - Sang Pyo Kim - 'M. de Montigny' title: 'General form of magnetization damping: Magnetization dynamics of a spin system evolving nonadiabatically and out of equilibrium' --- Introduction ============ Magnetization dynamics in nanomagnets and thin films is rich in content, including such phenomena as giant magnetoresistance [@bai], spin-current-induced magnetization reversal [@slon],[@baza] and adiabatic spin pumping [@tser]. The study of the magnetization dynamics is motivated by theoretical interest in a deep and fundamental understanding of the physics of magnetic systems on a short time scale and out of equilibrium. The results are important in gaining an understanding of the technological applications of magnetic systems to areas such as high-density memory and data storage devices. Recent results, in real time, provide the behavior of spins in particular set-ups of magnetic fields and measuring the spin flips as a function of time [@myers],[@free]. The Landau-Lifshitz-Gilbert (LLG) equation [@dau],[@gil] provides a plausible phenomenological model for many experimental results. Recently, the LLG equation and the Gilbert damping term have been derived from an effective Hamiltonian including the radiation-spin interaction (RSI) [@ho]. It has been assumed there that the spin system maintains quasiadiabatic evolution. However, a magnetic system whose Hamiltonian $\hat{\cal H}(t)$ evolves nonadiabatically, i.e. in a non-equilibrium state, deviates far from quasi-equilibrium, and its density operator satisfies the quantum Liouville - von Neumann (LvN) equation $$i \hbar \frac{{\partial}\hat{\rho}}{{\partial}t} + [ \hat{\rho}, \hat{\cal H} {]}_{-}=0. \label{llg}$$ In the present work, we aim to derive a magnetization equation including a damping term for such a nonequilibrium magnetic system. To find its nonadiabatic quantum states, we employ the Lewis-Riesenfeld (LR) dynamical invariant method [@lewis]. This method originally designed for the nonequilibrium evolution of time-dependent quantum systems has been successfully applied to a variety of problems, including the nonadiabatic generalization of the Berry phase for the spin dynamics, nonequilibrium fermion systems etc [@miz],[@san],[@sengupta]. In [@kim], the time-dependent invariants have been used for constructing the density operator for nonequilibrium systems. Following [@kim], we construct the density operator and then use it to determine the time evolution of magnetization. The damping term represents magnetization relaxation processes due to the dissipation of magnetic energy. Various kinds of relaxation processes are usually melded together into a single damping term. Relativistic relaxation processes result in the Gilbert damping term with one damping parameter, while for the case of both exchange and relativistic relaxation the damping term is a tensor with several damping parameters [@bar], [@saf]. The relaxation processes are specified by interactions of spins with each other and with other constituents of the magnetic system. A derivation of the damping term from first principles should therefore start with a microscopic description of the interactions. Even though such microscopic derivation of damping has been performed for some relaxation processes (for instance, [@kam]), a full version of derivation for the Gilbert damping term has not yet been given, and in particular for a system in non-equilibrium. Herein, we consider a general spin system without specifying the interaction Hamiltonian and related relaxation processes. We start with a system of spins precessing in the effective magnetic field $\vec{\bf H}_{eff}$ neglecting for a moment mutual interactions. Then at a fixed time later interactions in the system are switched on and influence the original precessional motion. The interactions are assumed to be time-dependent, and the spin system evolves nonadiabatically out of equilibrium trying to relax to a new equilibrium magnetization. We perform a transformation, which is analogous to the one used in the transition to the interaction picture, to connect the density and magnetic moment operators before and after the time, when a new nonequilibrium dynamics starts, and to find an explicit expression for the interaction contribution to the magnetization equation. Our paper is organized as follows. In Sec. II, the magnetization equation for the system of spins with a general form of interactions is derived. In Sec. III, the case of the radiation-spin interaction is considered, which is shown to produce the Gilbert damping term even for systems that are not in equilibrium. The contribution of the non-dissipative part of the radiation field to the magnetization equation and magnetization algebra is discussed. Sec. IV focuses on a special type of spin-spin interactions. We conclude with discussions in Sec. V. Magnetization equation ====================== Let us consider a quantum spin system defined by $$\hat{\cal H} = \hat{\cal H}_0 + {\lambda} \hat{\cal H}_{I}, \label{1}$$ where $\hat{\cal H}_0$ is the Zeeman Hamiltonian describing the interaction of spins with an effective magnetic field $$\hat{\cal H}_0 = - {\gamma} \sum_{i} \hat{S}_{i} \cdot \vec{\bf H}_{eff}(t), \label{2}$$ ${\gamma}$ being the gyromagnetic ratio and $\hat{S}_i$ being the spin operator of the ith atom, while the Hamiltonian $\hat{\cal H}_{I}$ describes the internal interactions between the constituents of the spin system, including, for instance, the exchange and dipolar interactions between the atomic spins, as well as higher-order spin-spin interactions. In Eq.(\[2\]), the effective magnetic field $\vec{\bf H}_{eff}$ is given by the energy variational with magnetization , $\vec{\bf H}_{eff}=- {\delta}E(M)/{\delta}\vec{\bf M}$, where $E(M)$ is the free energy of the magnetic system. This field includes the exchange field, the anisotropy field, and the demagnetizing field, as well as the external field, $\vec{\bf H}_{ext}$. The interaction terms included in $\hat{\cal H}_{I}$ are in general time-dependent, being switched on adiabatically or instantly at a fixed time $t_0$. The parameter ${\lambda}$ in (\[1\]) can be chosen small in order to take into account the higher order effects perturbatively. We introduce next the magnetic moment operator $$\hat{\cal M} \equiv - \frac{{\delta}\hat{\cal H}}{{\delta}\vec{\bf H}_{ext}}, \label{3}$$ which is the response of the spin system to the external field. The magnetization is defined as an ensemble average of the response $$\vec{\bf M}=\langle \hat{\cal M} \rangle \equiv \frac{1}{V} {\bf \rm Tr}\{ \hat{\rho} \hat{\cal M} \}, \label{4}$$ where $\hat{\rho}$ is the density operator satisfying the LvN equation (\[llg\]) and $V$ is the volume of the system. The explicit form of the density operator will be shown below. For systems in equilibrium, the Hamiltonian itself satisfies the LvN equation and the density operator is expressed in terms of the Hamiltonian. For nonequilibrium systems, whose Hamiltonians are explicitly time dependent, the density operator is constructed by making use of the time-dependent adiabatic invariants. Zeeman precession ----------------- Let us first derive the magnetization equation for the Zeeman Hamiltonian. If the interactions, $\hat{\cal H}_{I}$, are switched off, the magnetic moment operator and the magnetization are $$\hat{\cal M}_0 = - \frac{{\delta}\hat{\cal H}_0}{{\delta}\vec{\bf H}_{ext}} = {\gamma} \sum_{i} \hat{S}_i \label{6}$$ and $$\vec{\bf M}_0 = {\langle \hat{\cal M}_0 \rangle}_{0} = \frac{\gamma}{V} \sum_{i} {\bf \rm Tr}\{ \hat{\rho}_{0} \hat{S}_{i} \}, \label{7}$$ respectively, the operators $\hat{\cal M}_{0}^{a}$, $a=1,2,3$, fulfilling the $SU(2)$ algebra $$\Big[ \hat{\cal M}_{0}^{a} , \hat{\cal M}_{0}^{b} {\Big]}_{-} = i{\hbar}{\gamma} {\varepsilon}^{abc} \hat{\cal M}_{0}^{c}, \label{8}$$ where the summation over repeated indices is assumed. In Eq.(\[7\]), the subscript “0” in the symbol $\langle ... {\rangle}_{0}$ indicates using of the density operator $\hat{\rho}_0$, which satisfies the equation $$i{\hbar} \frac{{\partial}\hat{\rho}_0}{{\partial}t} + [ \hat{\rho}_0 , \hat{\cal H}_0 {]}_{-}=0. \label{9}$$ Let $\hat{\cal I}_0(t)$ be a non-trivial Hermitian operator, which is a dynamical invariant. That is, $\hat{\cal I}_0(t)$ satisfies the LvN equation $$\frac{d\hat{\cal I}_0}{dt} \equiv \frac{{\partial}\hat{\cal I}_0}{{\partial}t} + \frac{1}{i{\hbar}} [ \hat{\cal I}_0 , \hat{\cal H}_0 {]}_{-}=0.$$ As shown in [@lewis], the eigenstates of $\hat{\cal I}_0(t)$ can be used for evaluating the exact quantum states that are solutions of the Schrödinger equation. The linearity of the LvN equation allows us to state that any analytic functional of $\hat{\cal I}_0(t)$ satisfies the LvN equation provided that $\hat{\cal I}_0(t)$ satisfies the same equation. In particular, we can use $\hat{\cal I}_0(t)$ to define the density operator $\hat{\rho}_0$ as [@kim] $$\hat{\rho}_0 (t) = \frac{1}{\cal Z}_0 e^{ - \beta \hat{\cal I}_0 (t)}, \quad {\cal Z}_0 = {\bf Tr} \{ e^{ - \beta \hat{\cal I}_0 (t)} \}.$$ Here $\beta$ is a free parameter and will be identified with the inverse temperature for the equilibrium system. The LvN equation for $\hat{\cal I}_0$ is formally solved by $$\hat{\cal I}_0(t) = \hat{U}(t,t_0) \hat{\cal I}_0(t_0) \hat{U}(t_0,t), \label{10}$$ where $$\hat{U}(t_0,t) \equiv T\exp \Big\{ \frac{i}{\hbar} \int_{t_0}^{t} d{\tau} \hat{\cal H}_0({\tau}) \Big\} \label{11} ,$$ and $T$ denotes the time-ordering operator. As the Zeeman Hamiltonian is linear in spin operators, we can take $\hat{\cal I}_0(t)$ of the same form [@khanna] $$\hat{\cal I}_0 (t) = \sum_{i} \hat{S}_i \cdot \vec{\bf R}_0 (t), \label{inv op0}$$ where $\vec{\bf R}_0$ is a vector parameter to be determined by a dynamical equation. Then, Eq.(\[9\]) becomes $$\sum_{i} \hat{S}_i \cdot \Biggl( \frac{d \vec{\bf R}_0}{dt} - {\gamma} \vec{\bf R}_0 \times \vec{\bf H}_{eff} \Biggr) = 0$$ and we obtain an equation for the vector parameter given by $$\frac{d \vec{\bf R}_0}{dt} = - |\gamma| \vec{\bf R}_0 \times \vec{\bf H}_{eff}. \label{rr}$$ This equation explicitly describes the vector precessing with respect to the field $\vec{\bf H}_{eff}$. Thus, without loss of generality, we can identify $\vec{\bf R}_0$ with the magnetization vector $\vec{\bf M}_0$ (up to a dimensional constant factor) and Eq.(\[rr\]) with the equation of motion of magnetization for the Zeeman Hamiltonian $\hat{\cal H}_0$. An alternative method to obtain the magnetization equation is just to differentiate both sides of Eq.(\[7\]) with respect to time and use Eq.(\[9\]). In this way, we arrive at Eq.(\[rr\]) with $\vec{\bf R}_0 = \vec{\bf M}_0$. Interactions and damping ------------------------ In the case, when the interactions are present, the density operator for the full Hamiltonian may be written as $$\hat{\rho} (t) = \frac{1}{\cal Z} e^{ - \beta \hat{\cal I} (t)}, \quad {\cal Z} = {\bf Tr} \{ e^{ - \beta \hat{\cal I} (t)} \}, \label{den op}$$ where $\hat{\cal I}$ is the invariant operator for the system. As the interaction Hamiltonian is in general non-linear in spin operators, $\hat{\cal I}(t)$ cannot be taken in the form given by Eq.(\[inv op0\]). Moreover, the form of $\hat{\cal I}(t)$ can not be determined without specifying the interactions. To derive the magnetization equation in this case, we proceed as follows. We perform, on $\hat{\rho}(t)$, the transformation defined by the operator (\[11\]), $$\hat{\rho} \to \hat{\rho}_{int} \equiv \hat{U}(t_0,t) \hat{\rho}(t) \hat{U}(t,t_0), \label{13}$$ removing the Zeeman interaction. For systems with $\hat{\cal H}_0$ constant in time, the operator $\hat{U}(t_0,t)= \exp\{ (i/{\hbar}) \hat{\cal H}_0 (t-t_0) \}$ leads to the interaction picture, which proves to be very useful for all forms of interactions since it distinguishes among the interaction times. For our system with both $\hat{\cal H}_0$ and $\hat{\cal H}_{I}$ dependent on time, the operator (\[11\]) plays the same role, removing the unperturbed part of the Hamiltonian from the LvN equation. Substituting Eq.(\[13\]) into (\[llg\]), yields $$i{\hbar} \frac{{\partial}\hat{\rho}_{int}}{{\partial}t} = {\lambda} \Big[ \hat{\cal H}_{int}, \hat{\rho}_{int} {\Big]}_{-}, \label{14}$$ where $$\hat{\cal H}_{int}(t) \equiv \hat{U}(t_0,t) \hat{\cal H}_{I}(t) \hat{U}(t,t_0). \label{15}$$ The magnetic moment operator and the magnetization become $$\hat{\cal M}=\hat{\cal M}_0 + \hat{\cal M}_{I}, \label{16}$$ where $$\hat{\cal M}_{I} \equiv - {\lambda} \frac{{\delta} \hat{\cal H}_{I}}{{\delta} \vec{\bf H}_{ext}}, \label{17}$$ and $$\vec{\bf M} = \frac{1}{V} {\bf \rm Tr} \{ \hat{\rho}_{int} ( \hat{\cal M}_{0,int} + \hat{\cal M}_{I,int} ) \}, \label{18}$$ $\hat{\cal M}_{0,int}$ and $\hat{\cal M}_{I,int}$ being related with $\hat{\cal M}_0$ and $\hat{\cal M}_{I}$, respectively, in the same way as $\hat{\cal H}_{int}$ is related with $\hat{\cal H}_{I}$. The operators $\hat{\cal M}_{0,{int}}$, $\hat{\cal M}_{I,{int}}$ are generally used to calculate the magnetic susceptibility [@white]. Let us show now how these operators determine the time evolution of magnetization. The evolution in time of $\hat{\cal M}_{0,{int}}$ is given by the equation $$\frac{{\partial}\hat{\cal M}_{0,{int}}}{{\partial}t} = \frac{i}{\hbar} \hat{U}(t_0,t) \Big[ \hat{\cal H}_0, \hat{\cal M}_0 {\Big]}_{-} \hat{U}(t,t_0)$$ $$= {\gamma} \hat{\cal M}_{0,{int}} \times \vec{\bf H}_{eff}, \label{19}$$ which is analogous to Eq.(\[rr\]). It describes the magnetization precessional motion with respect to $\vec{\bf H}_{eff}$. The equation for $\hat{\cal M}_{I,{int}}$, $$\frac{{\partial}\hat{\cal M}_{I,int}}{{\partial}t} = \frac{i}{\hbar} \hat{U}(t_0,t) \Big[ \hat{\cal H}_0, \hat{\cal M}_I {\Big]}_{-} \hat{U}(t,t_0)$$ $$+ \hat{U}(t_0,t) \frac{{\partial}\hat{\cal M}_{I}}{{\partial}t} \hat{U}(t,t_0)$$ describes more complex magnetization dynamics governed by the interaction Hamiltonian $\hat{\cal H}_{I}$. In addition, $\hat{\cal M}_{I}$ depends on time explicitly. However, this dynamics includes the precessional motion as well, sine the interactions induced magnetization is a part of the precessing total magnetization $\vec{\bf M}$. Introducing $$\vec{\bf D}_{I} \equiv \frac{i}{\hbar} \Big[ \hat{\cal H}_0 , \hat{\cal M}_{I} {\Big]}_{-} - {\gamma} \hat{\cal M}_{I} \times \vec{\bf H}_{eff} \label{20}$$ to represent deviations from the purely precessional motion, we bring the equation for $\hat{\cal M}_{I,{int}}$ into the following form $$\frac{{\partial}\hat{\cal M}_{I,{int}}}{{\partial}t} = {\gamma} \hat{\cal M}_{I,{int}} \times \vec{\bf H}_{eff}$$ $$+ \hat{U}(t_0,t) \Big( \frac{{\partial}\hat{\cal M}_{I}}{{\partial}t} + \vec{\bf D}_{I} \Big) \hat{U}(t,t_0). \label{21}$$ Taking the time-derivative of $\vec{\bf M}$ given by Eq.(\[18\]) and using Eq.(\[14\]), we obtain $$\frac{d\vec{\bf M}}{dt} = \frac{1}{V} {\bf \rm Tr} \Big\{ \hat{\rho}_{int} \Big( \frac{{\partial}\hat{\cal M}_{0,int}}{{\partial}t} + \frac{{\partial}\hat{\cal M}_{I,int}}{{\partial}t} \Big)$$ $$+ \frac{\lambda}{i{\hbar}} \Big[ \hat{\cal M}_{0,int} + \hat{\cal M}_{I,int} , \hat{\cal H}_{int} {\Big]}_{-} \Big\}. \label{inter}$$ Substituting next Eqs.(\[19\]) and (\[21\]) into Eq.(\[inter\]), finally yields $$\frac{d\vec{\bf M}}{dt} = - |{\gamma}| \vec{\bf M} \times \vec{\bf H}_{eff} + \vec{\bf D}, \label{22}$$ where $$\vec{\bf D} \equiv {\lambda} \langle \frac{1}{i{\hbar}} \Big[ \hat{\cal M}, \hat{\cal H}_{I} {\Big]}_{-} \rangle + \langle \frac{{\partial}\hat{\cal M}_{I}}{{\partial}t} + \vec{\bf D}_{I} \rangle. \label{23}$$ Therefore, Eq.(\[22\]) is the magnetization equation for the system specified by (\[1\]). This equation is general since it is derived without specifying $\hat{\cal H}_{I}$. The $\vec{\bf D}$-term contains all effects that the interactions, $\hat{\cal H}_{I}$, can have on the magnetization precession, so that Eq.(\[22\]) is complete. The contribution of $\hat{\cal H}_{I}$ to the $\vec{\bf D}$-term in the magnetization equation can be divided into two parts. One is proportional to $\langle [ \hat{\cal M} , \hat{\cal H}_{I} {]}_{-} \rangle$ and is related to the change in the density matrix when the interactions of $\hat{\cal H}_{I}$ are switched on. The second part $\langle \frac{{\partial} \hat{\cal M}_{I}}{{\partial}t} + \vec{\bf D}_{I} \rangle$ originates from the change in the magnetization itself. Which part of $\vec{\bf D}$ is dominating depends on the nature of the interactions. Radiation-spin interaction ========================== One of the important issues in the study of the magnetization dynamics is the relaxation phenomena. The magnetization relaxation mechanism can be introduced by various interactions such as spin-orbit coupling and two-magnon scattering. In this section, we calculate the $\vec{\bf D}$-term for the magnetization relaxation process, which is induced by the RSI. Dissipative radiation field --------------------------- The RSI approach is an effective field method, in which contributions to the magnetization relaxation are effectively represented by the radiation-spin interaction [@ho]. In this method, the damping imposed on the precessing magnetization originates from the magnetization precessional motion itself. It is assumed that a radiation field is induced by the precession and that this field acts back on the magnetization producing a dissipative torque. The Hamiltonian for the RSI is $${\lambda} \hat{\cal H}_{I}= - {\gamma} \sum_{i} \hat{S}_{i} \cdot \vec{\bf H}_{r}^{d}, \label{24}$$ where $$\vec{\bf H}_{r}^{d} \equiv {\lambda} ( \vec{\bf M} \times \vec{\bf H}_{eff} - {\alpha} M^2 \vec{\bf H}_{eff}) \label{25}$$ is the dissipative part of the effective radiation field, i.e. the part responsible for the dissipative torque, and $M$ is the magnitude of magnetization. The parameter ${\alpha}$ will be specified below, while ${\lambda}$ in Eq.(\[1\]) is now the radiation parameter. The RSI contribution to the magnetic moment is $$\hat{\cal M}_{I,d} = - {\lambda} ({\alpha} M^2 \hat{\cal M}_{0} - \hat{\cal M}_{0} \times \vec{\bf M} ), \label{26}$$ and the total magnetization becomes $$\vec{\bf M} = (1-{\lambda}{\alpha}M^2) \langle \hat{\cal M}_{0} \rangle + {\lambda} \langle \hat{\cal M}_{0} \rangle \times \vec{\bf M}. \label{27}$$ It turns out that the vectors $\vec{\bf M}$ and $\langle \hat{\cal M}_0 \rangle$ are parallel. Indeed, taking the vector product of both sides of Eq.(\[27\]) with $\langle \hat{\cal M}_{0} \rangle$, yields the equation $$\langle \hat{\cal M}_{0} \rangle \times \vec{\bf M} = {\lambda} \langle \hat{\cal M}_{0} \rangle \times \langle \hat{\cal M}_{0} \rangle \times \vec{\bf M}$$ $$= {\lambda} {\langle \hat{\cal M}_{0} \rangle}^2 \Big[ (1-{\lambda} {\alpha}M^2) \langle \hat{\cal M}_{0} \rangle - \vec{\bf M} \Big], \label{28}$$ which is valid only if $\vec{\bf M}$ is parallel to $\langle \hat{\cal M}_{0} \rangle$, so that $$\langle \hat{\cal M}_{0} \rangle = \frac{\vec{\bf M}}{1-{\lambda}{\alpha}M^2}. \label{29}$$ A straightforward calculation shows that $$\langle \vec{D}_{I} \rangle = - {\lambda} \langle \frac{1}{i{\hbar}} \Big[ \hat{\cal M}, \hat{\cal H}_{I} {\Big]}_{-} \rangle$$ $$= - \frac{{\gamma}{\lambda}}{1-{\lambda}{\alpha}M^2} \vec{\bf M} \times \vec{\bf M} \times \vec{\bf H}_{eff}, \label{30}$$ so that the contributions of $\vec{\bf D}_{I}$ and $[ \hat{\cal M}, \hat{H}_{I} {]}_{-}$ to Eq.(\[23\]) cancel each other. Calculating next the time derivative of $\hat{\cal M}_{I,d}$ and using Eq.(\[29\]), we obtain $$\vec{\bf D} = {\alpha} \vec{\bf M} \times \frac{d\vec{\bf M}}{dt}, \label{31}$$ provided the following relation between the parameters $\alpha$ and $\lambda$ holds $${\alpha} = \frac{\lambda}{1-{\lambda}{\alpha}M^2}. \label{32}$$ Therefore, Eq.(\[22\]) takes the form of the LLG equation, ${\alpha}$ becoming a damping parameter. The relation given by Eq.(\[32\]) reflects the origin of the radiation field. Both the radiation and damping parameters depend on the magnetization. The dimensionless parameters independent of $M$ are ${\lambda}_0 \equiv {\lambda} M$ and ${\alpha}_0 \equiv {\alpha} M$ with the relation $${\lambda}_0 = \frac{{\alpha}_0}{1+{\alpha}_0^2}.$$ The equivalent form of the LLG equation, which is more suitable for calculations, is $$\frac{d\vec{\bf M}}{dt} = - |{\gamma}| \vec{\bf M} \times \Big[ (1-{\lambda}{\alpha}M^2) \vec{\bf H}_{eff} + {\lambda}\vec{\bf M} \times \vec{\bf H}_{eff} \Big]. \label{34}$$ Let us assume that $\vec{\bf H}_{eff}$ is a uniform static field in the $z$-direction, i.e. $\vec{\bf H}_{eff}=(0,0,H_z)$. Then the magnetization equation becomes, in component form, $$\begin{aligned} \frac{d}{dt} M_p^2 & = & -2{\lambda}{\omega}_0 M_z M_p^2,\\ \frac{d}{dt} M_z & = & {\lambda}{\omega}_0 M_p, \label{35-36}\end{aligned}$$ where ${\omega}_0 \equiv |{\gamma}| H_z$ is the frequency of magnetization precession and $M_p^2 \equiv M_x^2+M_y^2=M^2-M_z^2$. These equations are solved exactly by $$M_z=M \cdot \frac{{\tanh}[{\lambda}_0 {\omega}_0 (t-t_0)] +d}{1+d \cdot {\tanh}[{\lambda}_0 {\omega}_0 (t-t_0)]} \label{37}$$ and $$M_p= \frac{M\sqrt{1-d^2}}{{\cosh}[{\lambda}_0 {\omega}_0(t-t_0)]}$$ $$\cdot \frac{1}{1+d \cdot {\tanh}[{\lambda}_0 {\omega}_0 (t-t_0)]}, \label{38}$$ where $d$ stands for the initial condition $$d \equiv \frac{M_z(t=t_0)}{M}.$$ In the limit $t \to \infty$, $M_z$ tends to $M$ and $M_p$ tends to zero, so that during the relaxation process the magnetization vector tends to be parallel to the effective magnetic field, and the relaxation characteristic time ${\tau}$ is $1/({\lambda}_0 {\omega}_0)$. General radiation field ----------------------- The RSI approach can be directly used for studying the effect of the real radiation-spin interaction in the magnetization relaxation process. In that case, one has to distinguish the real radiation field contribution to the damping parameter from the effective one. It is therefore important to consider possible generalizations of the ansatz given by Eq.(\[25\]) to get a more detailed picture of the RSI. The dissipative radiation field in Eq.(\[25\]) is a composite of two fields, which are parallel and perpendicular to $\vec{\bf H}_{eff}$, respectively. The field parallel to $\vec{\bf H}_{eff}$ changes only the frequency of the magnetization precessional motion, while the torque $\vec{\bf H}_{eff} \times \vec{\bf M}$ introduces a damping effect as well. The choice of the field $\vec{\bf H}_r^d$ made in Eq.(\[25\]) is not a general one. Let us modify it by applying additional fields and see how the parameters in the magnetization equation are changed. If we apply an additional field in the direction of $\vec{\bf H}_{eff}$ and modify $\vec{\bf H}_r^d$ as $$\vec{\bf H}_r^d \to \vec{\bf H}_r^d - \overline{\lambda}_0 \vec{\bf H}_{eff}, \label{39}$$ where $\overline{\lambda}_0$ is an arbitrary dimensionless parameter, then this results in a change of the magnitude of the effective magnetic field and therefore in the change of the frequency of the magnetization precession as $${\omega}_0 \to {\omega} \equiv {\omega}_0 (1-\overline{\lambda}_0 ). \label{40}$$ For $\overline{\lambda}_0 >1$, the magnetization vector turns upside down and the precession continues in an opposite direction. The magnetization damping is not affected by the additional field. Rescaling of $\vec{\bf H}_{eff}$ in $\vec{\bf H}_r^d$ leads to a change of the radiation parameter as well, $${\lambda}_0 \to \frac{1}{1-\overline{\lambda}_0} {\lambda}_0, \label{lam}$$ so that the relaxation characteristic time remains the same. If we apply an additional torque, $$\vec{\bf H}_r^d \to \vec{\bf H}_r^d + \overline{\lambda}_0 \vec{\bf M} \times \vec{\bf H}_{eff}, \label{41}$$ then the parameters in the ansatz (\[25\]) change as follows $$\begin{aligned} {\lambda}_0 & \to & {\lambda}_0 + \overline{\lambda}_0,\\ {\alpha}_0 & \to & \frac{{\lambda}_0}{{\lambda}_0 + \overline{\lambda}_0} {\alpha}_0, \label{42-43}\end{aligned}$$ and the relaxation characteristic time becomes $${\tau} \to \overline{\tau} \equiv {\tau} {\Big( 1 + \frac{\overline{\lambda}_0}{{\lambda}_0} \Big)}^{-1}, \label{44}$$ decreasing for $\overline{\lambda}_0 >0$ and increasing for $-{\lambda}_0 < \overline{\lambda}_0 < 0$. If the additional torque is stronger than the original one and in the opposite direction, i.e. $\overline{\lambda}_0 < -{\lambda}_0$, then the magnetization vector turns over again. For $\overline{\lambda}_0 =-{\lambda}_0$, the additional and original torques cancel each other, and there is no damping. In Fig.\[fig1\] and Fig.\[fig2\] we plot the solutions for $M_z$ and $M_p$ given by Eq.(\[37\]) and Eq.(\[38\]), respectively, in the presence of an additional torque (nonzero $a$’s) and without it ($a=0$), where $a \equiv \overline{\lambda}_{0}/{\lambda}_{0}$. We observe a change in the relaxation time depending on the sign of $a$. For $a>0$, the additional torque is in the same direction as the original one, while for $a<0$ their directions are opposite. A drastic change occurs for $a<-1$, when $M_z$ tends to $(-M)$ indicating the overturning of $\vec{\bf M}$. The magnetization vector first becomes perpendicular to $\vec{\bf H}_{eff}$, when $M_p/M$ reaches its maximum value, and then it turns upside down. The radiation field $\vec{\bf H}_{r}$ can contain a non-dissipative part as well, i.e. $\vec{\bf H}_{r} = \vec{\bf H}_{r}^{d} + \vec{\bf H}_{r}^{n}$. Let us assume that $\vec{\bf H}_r^n$ is parallel to $\vec{\bf M}$ and we take it of the form $$\vec{\bf H}_r^n = {\kappa} {\lambda} \frac{\vec{\bf M}}{M} (\vec{\bf M} \cdot \vec{\bf H}_{eff} ), \label{45}$$ where ${\kappa}$ is an arbitrary parameter. The field given by Eq.(\[45\]) does not change the magnetization precessional motion. However, the magnetic moment operator is given as $$\hat{\cal M} \to \hat{\cal M} + \hat{\cal M}_{I,n} \equiv \hat{\cal M} + {\kappa} {\lambda} \frac{\vec{\bf M}}{M} (\hat{\cal M}_0 \cdot \vec{\bf M}). \label{46}$$ The relation between the vectors $\langle \hat{\cal M}_0 \rangle$ and $\vec{\bf M}$ becomes $$\langle \hat{\cal M}_0 \rangle = \frac{\vec{\bf M}}{1 - {\lambda}({\alpha}M+{\kappa})M}. \label{47}$$ Proceeding in the same way as before, when the non-dissipative part of the radiation field was omitted, to calculate the total radiation field contribution to the ${\vec{\bf D}}$-term, we obtain the same Gilbert-type structure of the damping term with the same damping parameter, i.e. the part of the radiation field that is parallel to $\vec{\bf M}$ does not contribute to the magnetization equation. Magnetization algebra --------------------- The magnetization algebra, i.e. the algebra of magnetic moment operators, gives a further insight into the RSI. Let us study how the non-dissipative part of the radiation field contributes to this algebra. Although we can omit the non-dissipative radiation field in the procedure obtaining the equation of motion for magnetization, it is important to take this field into account when we construct the magnetization algebra. Indeed, the operators $\hat{\cal M}^{a}=\hat{\cal M}^{a}_0 + \hat{\cal M}^{a}_{I,d}$ without the non-dissipative radiation field contribution fulfil the algebra $$\Big[ \hat{\cal M}^{a} , \hat{\cal M}^{b} {\Big]}_{-} = i{\hbar}{\gamma} {\varepsilon}^{abc} \{ (1-{\lambda}{\alpha} M^2 ) \cdot \hat{\cal M}^{c}$$ $$+ \frac{1}{\kappa} {\lambda}M \cdot \hat{\cal M}_{I,n}^{c} \}, \label{48}$$ which, however, contains $\hat{\cal M}_{I,n}$ on its right-hand side and therefore the algebra is not closed. Let us define, as in Eq.(\[46\]), the total magnetic moment operator $\hat{\cal M}_{tot} \equiv \hat{\cal M} + \hat{\cal M}_{I,n}$. Then the magnetization algebra becomes $$\Big[ \hat{\cal M}_{tot}^{a} , \hat{\cal M}_{tot}^{b} {\Big]}_{-} = i{\hbar}{\gamma} {\varepsilon}^{abc} \{ (1-{\lambda}{\alpha} M^2 + {\kappa} {\lambda} M ) \cdot \hat{\cal M}^{c}$$ $$+ ( \frac{1}{\kappa} {\lambda}M - 1 + {\lambda} {\alpha} M^2 ) \cdot \hat{\cal M}_{I,n}^{c} \}. \label{49}$$ Fixing next the parameter ${\kappa}$ by taking it as a solution of the equation $$\frac{1}{\kappa} - {\kappa} = \frac{2}{{\alpha}_0}, \label{50}$$ that is $${\kappa}_{\pm} = - \frac{1}{{\alpha}_0} (1 \mp \sqrt{1+{\alpha}_0^2}),$$ brings the algebra into the closed, standard form for $SU(2)$ algebra $$\Big[ \hat{\cal M}_{tot}^{a} , \hat{\cal M}_{tot}^{b} {\Big]}_{-} = i{\hbar}{\gamma}_{tot} {\varepsilon}^{abc} \hat{\cal M}_{tot}^{c}, \label{51}$$ where $${\gamma}_{tot} \equiv \pm {\gamma} \frac{1}{\sqrt{1+{\alpha}_0^2}}. \label{par}$$ For $0<{\alpha}_0<1$, ${\kappa}_{+}>0$ and ${\kappa}_{-}<0$, the sign $(+)$ in Eq.(\[par\]) corresponding to the case ${\kappa}={\kappa}_{+}$ and sign $(-)$ to ${\kappa}={\kappa}_{-}$. Therefore, the RSI preserves the form of the magnetization algebra (c.f. Eq.(\[8\])), by renormalizing the gyromagnetic ratio, only if the interaction of spins with the non-dissipative part of the radiation field is included in a proper way. If the direction of $\vec{\bf H}_r^n$ is the same as that of $\vec{\bf M}$ and ${\kappa}={\kappa}_{+}$, the gyromagnetic ratio remains negative, while its absolute value decreases. If the direction of $\vec{\bf H}_r^n$ is opposite to that of $\vec{\bf M}$ and ${\kappa}={\kappa}_{-}$, then the gyromagnetic ratio changes its sign and becomes positive. Spin-spin interactions ====================== The spin-spin interactions among the spins in the system introduce many body effects, which can be treated perturbatively in the weak coupling regime. In this case the $\vec{\bf D}$-term can be expanded in powers of $\lambda$. To demonstrate this, we consider the spin-spin interactions of a specific type. The interaction between spins is usually an exchange interaction of the form $$-2J \sum_{i,j} \hat{S}_i \hat{S}_j = - \frac{2J}{{\gamma}^2} \hat{\cal M}_0^2, \label{52}$$ the coupling constant $J$ being called the exchange integral. We generalize the ansatz given by Eq.(\[52\]) by assuming that the exchange integral depends on the magnetization and introduce the spin-spin interactions as follows $${\lambda}\hat{\cal H}_{I} = \sum_{i,j} J^{ab}(M) \hat{S}_i^a \hat{S}_j^b, \label{53}$$ where $J^{ab}={\lambda}M^aM^b$. Since $\hat{\cal H}_{I}$ does not depend explicitly on the external field, its contribution to the magnetic moment operator vanishes, $\hat{\cal M}_{I}=0$. The non-vanishing commutator $[\hat{\cal M}_0, \hat{\cal H}_{I} {]}_{-}$ in Eq.(\[23\]) is the only contribution of the spin-spin interaction to the magnetization equation, resulting in $$\vec{\bf D} = \frac{\lambda}{\gamma} \vec{\bf M} \times \vec{\bf{\Omega}}, \label{54}$$ where $${\Omega}^a \equiv \langle \Big[ \hat{\cal M}_0^a, \hat{\cal M}_0^b {\Big]}_{+} \rangle M^b, \label{55}$$ and $$\Big[ \hat{\cal M}_0^a , \hat{\cal M}_0^b {\Big]}_{+} \equiv \hat{\cal M}_0^a \hat{\cal M}_0^b + \hat{\cal M}_0^b \hat{\cal M}_0^a. \label{56}$$ The correlation function $G^{ab} \equiv \langle \Big[ \hat{\cal M}_0^a , \hat{\cal M}_0^b {\Big]}_{+} \rangle$ is the sum of spin correlation functions, $$G^{ab} = 2{\gamma}^2 \sum_{i} \sum_{j \neq i} \langle \hat{S}_i^a \hat{S}_j^b \rangle, \label{57}$$ excluding the self-interaction of spins. For the standard ansatz given in Eq.(\[52\]), $\vec{\bf D}=0$ and the magnetization equation does not change. If the spin-spin interactions are turned on at $t=t_0$, so that $\hat{\rho}(t_0)=\hat{\rho}_0(t_0)$, then, integrating both sides of Eq.(\[14\]), we find $$\hat{\rho}_{\lambda}(t) = \hat{\rho}_0(t_0) + \frac{\lambda}{i{\hbar}} \int_{t_0}^{t} d{\tau} \Big[ \hat{\cal H}_{\lambda}(\tau), \hat{\rho}_{\lambda}(\tau) {\Big]}_{-}. \label{58}$$ Substituting Eq.(\[58\]) into the definition of $G^{ab}$, yields the equation $$G^{ab}(t) = G_0^{ab}(t_0)$$ $$+ \frac{1}{\gamma} \int_{t_0}^t d{\tau} J^{cd}({\tau}) \Big( {\varepsilon}^{ace} G^{ebd}({\tau}) + {\varepsilon}^{bce} G^{aed}({\tau}) \Big), \label{59}$$ where $$G_0^{ab} \equiv \langle \Big[ \hat{\cal M}_0^a , \hat{\cal M}_0^b {\Big]}_{+} {\rangle}_{0}, \label{60}$$ which relates $G^{ab}$ to the third order correlation function, i.e. the correlation function of the product of three magnetic moment operators, $$G^{abc} \equiv \langle \Big[ \Big[ \hat{\cal M}_0^a , \hat{\cal M}_0^b {\Big]}_{+} , \hat{\cal M}_0^c {\Big]}_{+} \rangle. \label{61}$$ The correlation function $G^{abc}$, in turn, is related to the fourth order correlation function and etc., and we have therefore an infinite number of coupled equations for spin correlation functions. For any practical calculation this infinite hierarchy has to be truncated. That then defines the approximation scheme which may be considered on the basis of the physical requirements for the system. The approximation scheme will depend on the physical properties such as density and on the strength of the interactions. If the Hamiltonian $\hat{\cal H}_{I}$ is a small perturbation to the original $\hat{\cal H}_0$, we can solve Eq.(\[58\]) perturbatively. In the lowest, zeroth order in $\lambda$, we replace $\hat{\rho}_{\lambda}(t)$ by $\hat{\rho}_0(t_0)$, so that $G^{ab} \approx G_0^{ab}(t_0)$. We choose the initial value for $G^{ab}$ as $$\sum_{i} \sum_{j \neq i} \langle \hat{S}_i^a \hat{S}_j^b {\rangle}_0 = I^{ab} \label{62}$$ with $I^{xx}=I^{yy}=0$, $I^{zz}=I$ and $I^{ab}=0$ for $a \neq b$. We also define again the $z$-direction as the direction of the effective magnetic field that is chosen uniform and static. Then the $\vec{\bf D}$-term becomes, in component form, $$\begin{aligned} D_x & = & 2{\lambda} {\gamma} I M_y M_z,\\ D_y & = & -2{\lambda} {\gamma} I M_x M_z, \\ D_z & = & 0. \end{aligned}$$ producing two effects in the magnetization equation: the magnetization is now precessing with respect to $(H_z - 2{\lambda}I M_z)$, its z-component remaining constant in time, $(d/{dt})M_z=0$, and the frequency of the precession is $$\overline{\omega}_{0} \equiv {\omega}_0 \Big( 1- {\lambda} \frac{2IM_z}{H_z} \Big). \label{64}$$ Therefore, in the lowest order of perturbations, when the $\vec{\bf D}$ is linear in ${\lambda}$, it shifts the direction and the frequency of the precessional motion without introducing damping effects. To find a role of the higher powers of ${\lambda}$ in $\vec{\bf D}$ and to determine how they affect the magnetization equation, a truncation of the chain of spin correlation equations is needed. This would require a consistent perturbation approach to the hierarchy of the coupled equations for the correlation functions. Discussions =========== We have derived a general form of magnetization equation for a system of spins precessing in an effective magnetic field without specifying the internal interactions. It can be applied in the study of magnetization dynamics of any type, including nonequilibrium and nonlinear effects, provided the interaction of individual spins with each other and with other degrees of freedom of the system is specified. For the interactions related to the relaxation processes, this equation provides a general form of magnetization damping. This paper uses the dynamical invariant method introduced by Lewis and Riesenfeld [@lewis]. It extends its applicability to magnetic systems that are in a non-equilibrium state i.e. the various components in the defining Hamiltonian are time-dependent. This is in contrast to the earlier attempts to sove a problem for quasi-adiabatic evolution. The $\vec{\bf D}$-term in the magnetization equation has been obtained without using any approximation scheme. It is exact, accumulates all effects of the internal interactions on the magnetization precessional motion and can be a starting point for practical calculations. We have evaluated the $\vec{\bf D}$-term in two special cases, the RSI and the spin-spin interactions. For the RSI, which is linear in spin operators, it takes the form of the Gilbert damping term, the damping and radiation parameters being interrelated. For the spin-spin interactions, it is determined by the spin correlation functions, which fulfil an infinite chain of equations. A further analysis of the $\vec{\bf D}$-term requires an approximation scheme to truncate the chain in a consistent approach to higher order calculations. In our work, we have considered a specific type of the spin-spin interactions, which do not contribute to the magnetization algebra. However, if spin-spin interactions depend explicitly on the external field, the form of the algebra can change. In this case, the total magnetic moment operator becomes nonlinear in $\hat{\cal M}_0$, and this results in the magnetization algebra with an infinite chain of commutation relations. The chain has to be truncated in a way consistent with the truncation of the chain of equations for the spin correlation functions in the same approximation scheme. We wish to thank the Natural Sciences and Engineering Research Council of Canada for financial support. The work of S.P.K is funded by the Korea Research Foundation under Grant No. KRF-2003-041-C20053. [99]{} M. N. Baibich, J. M. Broto, A. Fert, F.  Nguyen Van Dau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. [**61**]{}, 2472 (1988); P. M. Levy, Solid State Phys. [**47**]{}, 367 (1994); M. A. M.  Gijs and G. E. W.  Bauer, Adv. Phys. [**46**]{}, 285 (1997); G. A.  Prinz, Science [**282**]{}, 1660 (1998). J. C.  Slonczewski, J. Magn. Magn. Mater. [**159**]{}, L1 (1996); [**195**]{}, L261 (1999); L.  Berger, Phys. Rev. [**B 54**]{}, 9353 (1996); J. Appl. Phys. [**89**]{}, 5521 (2001); Ya. B.  Bazaliy, B. J.  Jones, and S. C.  Zhang, Phys. Rev. [**B 57**]{}, R3213 (1998); [**B 69**]{}, 094421 (2004); J. Appl. Phys. [**89**]{}, 6793 (2001); X.  Waintal, E. B. Myers, P. W.  Brouwer, and D. C. Ralph, Phys. Rev. [**B 62**]{}, 12317 (2000); J. Z.  Sun, Phys. Rev. [**B 62**]{}, 570 (2000); A.  Brataas, Yu. V.  Nazarov, and G. E. W.  Bauer, Phys. Rev. Lett. [**84**]{}, 2481 (2000); C. Heide, P. E.  Zilberman, and R. J.  Elliott, Phys. Rev. [**B 63**]{}, 064424 (2001); S. Zhang, P. M.  Levy, and A. Fert, Phys. Rev. Lett. [**88**]{}, 236601 (2002); M. D.  Stiles and A. Zangwill, Phys. Rev. [**B 66**]{}, 014407 (2002); J. Appl. Phys. [**91**]{}, 6812 (2002); M. D.  Stiles, J.  Xiao, and A.  Zangwill, cond-mat/0309289. Y.  Tserkovnyak, A.  Brataas, and G. E. W.  Bauer, Phys. Rev. Lett. [**88**]{}, 117601 (2002); Phys. Rev. [**B 66**]{}, 224403 (2002); [**B 67**]{}, 140404 (2003); A.  Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. [**B 66**]{}, 060404(R) (2002); B.  Heinrich, Y. Tserkovnyak, G.  Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett. [**90**]{}, 187601 (2003); E.  Šimánek and B.  Heinrich, Phys. Rev. [**B 67**]{}, 144418 (2003). M.  Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. [**80**]{}, 4281 (1998); [**81**]{}, 493(E) (1998); Nature (London) [**406**]{}, 46 (2000); J. A.  Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D.  C. Ralph, Phys. Rev. Lett. [**84**]{}, 3149 (2000); E. B.  Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A.  Buhrman, Science [**285**]{}, 867 (1999); Phys. Rev. Lett. [**89**]{}, 196801 (2002); F. J.  Albert, J. A. Katine, R. A.  Buhrman, D. C. Ralph, Appl. Phys. Lett. [**77**]{}, 3809 (2000); Phys. Rev. Lett. [**89**]{}, 226802 (2002); J.  Grollier, V. Cros, A. Hamzic, J. M. George, H. Jaffrès, A.  Fert, G. Faini, J. Ben Youssef, and H. Legall, Appl. Phys. Lett. [ **78**]{}, 3663 (2001); Phys. Rev. [**B 67**]{}, 174402 (2003); J. E.  Wegrowe, D.  Kelly, T. Truong, Ph.  Guittienne, and J. -Ph.  Ansermet, Europhys. Lett. [**56**]{}, 748 (2001); J. Appl. Phys. [**91**]{}, 6806 (2002); J. Z.  Sun, D. J. Monsma, T. S.  Kuan, M. J.  Rooks, D. W.  Abraham, B.  Oezyilmaz, A. D.  Kent, R. H.  Koch, J. Appl. Phys. [**93**]{}, 6859 (2003); B.  Oezyilmaz, A. D.  Kent, D.  Monsma, J. Z.  Sun, M. J.  Rooks, and R. H.  Koch, Phys. Rev. Lett. [**91**]{}, 067203 (2003); S.  Urazhdin, N.   O.  Birge, W. P.  Pratt, Jr., and J.  Bass, Phys. Rev. Lett. [**91**]{}, 146803 (2003); A.  Barman, V. V.  Kruglyak, R. J.  Hicken, J. M.  Rowe, A. Kundrotaite, J.  Scott, and M.  Rahman, Phys. Rev. [**B 69**]{}, 174426 (2004); M.  Wu, B. A.  Kalinikos, P.  Krivosik, and C. E.  Patton, J. Appl. Phys. [**99**]{}, 013901 (2006); S. -K.  Lee, E. L.  Hahn, and J.  Clarke, Phys. Rev. Lett. [**96**]{}, 257601 (2006). W. Y.  Lee, A. Samad, T. A.  Moore, J. A. C.  Bland, B. C. Choi, Phys. Rev. [**B 61**]{}, 6811 (2000); B. C.  Choi, G. E.  Ballentine, M.  Belov, and M. R.  Freeman, Phys. Rev. [**B 64**]{}, 144418 (2001); Phys. Rev. Lett. [**86**]{}, 728 (2001); [**95**]{}, 237211 (2005); X.  Zhu, Z.  Liu, V.  Metlushko, P.  Grütter, M. R.  Freeman, Phys. Rev. [**B 71**]{}, 180408 (2005); M. H.  Park, Y. K.  Hong, B. C.  Choi, M. J.  Donahue, H.  Han, and S. H.  Gee, Phys. Rev. [**B 73**]{}, 094424 (2006); Q. F.  Xiao, J.  Rudge, B. C.  Choi, Y. K.  Hong, and G.  Donohoe, Phys. Rev. [**B 73**]{}, 104425 (2006); Z.  Liu, F.  Giesen, X.  Zhu, R. D.  Sydora, and M. R.  Freeman, cond-mat/0606235. L. D.  Landau and E. M.  Lifshitz, Phys. Z. Sowjetunion [**8**]{}, 153 (1935); L. D.  Landau, E. M.  Lifshitz, and L. P.  Pitaevski, [*Statistical Physics, Part 2*]{} (Pergamon, Oxford, 1980), 3rd ed. T. L.  Gilbert, [*Armor Research Foundation Rep. No. 11*]{} (Chicago, IL, 1955); Phys. Rev. [**100**]{}, 1243 (1955). J.  Ho, F. C.  Khanna, and B. C.  Choi, Phys. Rev. Lett. [**92**]{}, 097601 (2004). H. R.  Lewis  Jr. and W. B.  Riesenfeld, J. Math. Phys. [**10**]{}, 1458 (1969). S. S.  Mizrahi, Phys. Lett. [**A 138**]{}, 465 (1989); X. -C.  Gao, J. -B.  Xu, and T. -Z.  Qian, Phys. Lett. [**A 152**]{}, 449 (1991). S. P.  Kim, A. E.  Santana, and F. C.  Khanna, Phys. Lett. [**A 272**]{}, 46 (2000); J. K.  Kim and S. P.  Kim, J. Phys. [**A 32**]{}, 2711 (1999); S. P.  Kim and D. N.  Page, Phys. Rev. [**A 64**]{}, 012104 (2001); S. P.  Kim, J.  Korean Phys.  Soc.  [**43**]{}, 11 (2003); H. K.  Kim and S. P.  Kim, J. Korean Phys. Soc. [**48**]{}, 119 (2006). S. P.  Kim, S.  Sengupta, and F. C.  Khanna, Phys. Rev. [**D 64**]{}, 105026 (2001); S. P.  Kim and C. H.  Lee, Phys. Rev. [**D 65**]{} 045013 (2002); G.  Flores-Hidalgo and R.  O.  Ramos, Physica [**A 326**]{}, 159 (2003). S. P.  Kim and C. H.  Lee, Phys. Rev. [**D 62**]{}, 125020 (2000); S. P.  Kim, J. Korean Phys. Soc. [**41**]{}, 643 (2002). V. G.  Baryakhtar, B. A.  Ivanov, A. L.  Sukstanskii and E. Yu.  Melikhov, Phys. Rev. [**B 56**]{}, 619 (1997) and references therein. V. L.  Safonov, J. Magn. Magn. Mater. [**195**]{}, 526 (1999); J. Appl. Phys. [**85**]{}, 4370 (1999); V. L.  Safonov and H. N.  Bertram, J. Appl. Phys. [**87**]{}, 5681 (2000); X. Wang, H. N.  Bertram, and V. L.  Safonov, J. Appl. Phys. [**91**]{}, 6920 (2002); [**92**]{}, 2064 (2002). V. Kambersky, Canadian J. Phys. [**48**]{}, 2906 (1970); V. L.  Safonov and H. N.  Bertram, Phys. Rev. [**B 61**]{}, R14893 (2000). J. Ho, F. C.  Khanna, and B. C. Choi, Phys. Rev. [**B 70**]{}, 172402 (2004). R. M.  White, [*Quantum Theory of Magnetism*]{} (Springer-Verlag, Berlin, Heidelberg, New York, 1983), 2nd ed.

--- abstract: 'We consider families of special cycles, as introduced by Kudla, on Shimura varieties attached to anisotropic quadratic spaces over totally real fields. By augmenting these cycles with Green currents, we obtain classes in the arithmetic Chow groups of the canonical models of these Shimura varieties (viewed as arithmetic varieties over their reflex fields). The main result of this paper asserts that generating series built from these cycles can be identified with the Fourier expansions of non-holomorphic Hilbert-Jacobi modular forms. This result provides evidence for an arithmetic analogue of Kudla’s conjecture relating these cycles to Siegel modular forms.' author: - Siddarth Sankaran bibliography: - 'refs.bib' title: Arithmetic special cycles and Jacobi forms --- Introduction ============ The main result of this paper is a modularity result for certain generating series of “special" cycles that live in the arithmetic Chow groups of Shimura varieties of orthogonal type. We begin by introducing the main players. Let $F$ be a totally real extension of $\mathbb Q$ with $d = [F:{\mathbb Q}]$, and let $\sigma_1, \dots , \sigma_d$ denote the archimedean places of $F$. Suppose $V$ is a quadratic space over $F$ that is of signature $((p,2), (p+2,0), (p+2,0), \cdots , (p+2,0))$ with $p>0$. In other words, we assume that $V \otimes_{F, \sigma_1} {\mathbb R}$ is a real quadratic space of signature $(p,2)$ and that $V$ is positive definite at all other real places. *We assume throughout that $V$ is anisotropic over F.* Note that the signature condition guarantees that $V$ is anisotropic whenever $d > 1$. Let $H = \mathrm{Res}_{F/{\mathbb Q}} \mathrm{GSpin}(V)$. The corresponding Hermitian symmetric space ${\mathbb D}$ has two connected components; fix one component ${\mathbb D}^+$ and let $H^+({\mathbb R})$ denote its stabilizer in $H({\mathbb R})$. For a neat compact open subgroup $K_f \subset H({\mathbb A}_f)$, let $\Gamma := H^+({\mathbb R}) \cap H({\mathbb Q}) \cap K_f$, and consider the quotient $$X({\mathbb C}) \ := \ \Gamma \big\backslash {\mathbb D}^+.$$ This space is a (connected) Shimura variety; in particular, it admits a canonical model $X$ over a number field $E \subset {\mathbb C}$ depending on $K_f$, see [@KudlaAlgCycles] for details. Moreover, as $V$ is anisotropic, $X$ is a projective variety. Fix a $\Gamma$-invariant lattice $L \subset V$ such that the restriction of the bilinear form $\langle \cdot, \cdot, \rangle$ to $L$ is valued in ${\mathcal O}_F$, and consider the dual lattice $$L' = \{ {\mathbf x}\in L \ | \ \langle {\mathbf x}, L \rangle \subset \partial_F^{-1} \}$$ where $\partial_F^{-1}$ is the inverse different. For an integer $n$ with $1 \leq n \leq p$, let $S(V({\mathbb A}_f)^n)$ denote the Schwartz space of compactly supported, locally constant functions on $V({\mathbb A}_f)^n$, and consider the subspace $$\label{eqn:intro S(L) defn} S(L^n) \ := \ \{ \varphi \in S(V({\mathbb A}_f)^n)^{\Gamma} \ | \ \mathrm{supp}(\varphi) \subset (\widehat{L'})^n \text{ and } \varphi({\mathbf x}+ l) = \varphi({\mathbf x}) \text{ for all } l \in L^n \}.$$ For every $T \in \operatorname{Sym}_n(F)$ and $\Gamma$-invariant Schwartz function $\varphi \in S(L^n)$, there is an $E$-rational “special" cycle $$Z(T,\varphi)$$ of codimension $n$ on $X$, defined originally by Kudla [@KudlaAlgCycles]; this construction is reviewed in below. It was conjectured by Kudla that these cycles are closely connected to automorphic forms; more precisely, he conjectured that upon passing to the Chow group of $X$, the generating series formed by the classes of these special cycles can be identified with the Fourier expansions of Hilbert-Siegel modular forms. When $F = \mathbb Q$, the codimension one case of this conjecture follows from results of Borcherds [@BorcherdsGKZ], and the conjecture for higher codimension was established by Zhang and Bruinier-Raum [@BruinierRaum; @ZhangThesis]; when $F \neq {\mathbb Q}$, conditional proofs have been given by Yuan-Zhang-Zhang [@YuanZhangZhang] and Kudla [@KudlaTotallyReal]. More recently, attention has shifted to the arithmetic analogues of this result, where one replaces the Chow groups with an “arithmetic" counterpart, attached to a model ${\mathcal X}$ of $X$ defined over a subring of the reflex field of $E$; these arithmetic Chow groups were introduced by Gillet-Soulé [@GilletSouleIHES] and subsequently generalized by Burgos-Kramer-Kühn [@BurgosKramerKuhn]. Roughly speaking, in this framework cycles are represented by pairs $({\mathcal Z}, g_{{\mathcal Z}})$, where ${\mathcal Z}$ is a cycle on ${\mathcal X}$, and $g_{{\mathcal Z}}$ is a *Green object*, a purely differential-geometric datum that encodes cohomological information about the archimedean fibres of ${\mathcal Z}$. In this paper, we consider the case where the model ${\mathcal X}$ is taken to be $X$ itself. In order to promote the special cycles to the arithmetic setting, we need to choose the Green objects: for this, we employ the results of [@GarciaSankaran], where a family $\{ { \mathfrak{g}}(T,\varphi; {\mathbf v})\}$ of Green forms was constructed. Note that these forms depend on an additional parameter ${\mathbf v}\in \operatorname{Sym}_{n}(F \otimes_{{\mathbb Q}} {\mathbb R})_{\gg 0}$, which should be regarded as the imaginary part of a variable in the Hilbert-Siegel upper half space. We thereby obtain classes $$\widehat Z(T,{\mathbf v}) \in {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X) \otimes_{{\mathbb C}} S(L^n)^{\vee} ,$$ where ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X) $ is the Gillet-Soulé arithmetic Chow group attached to $X$, by the formula $$\widehat Z(T,{\mathbf v})(\varphi) \ =\ \big(Z(T,\varphi), \, { \mathfrak{g}}(T, \varphi; {\mathbf v}) \big) \ \in \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X).$$ For reasons that will emerge in the course of the proof our main theorem, we will also need to consider a larger arithmetic Chow group ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X, {{\mathcal D}_{\mathrm{cur}}})$, constructed by Burgos-Kramer-Kühn [@BurgosKramerKuhn]. This group appears as an example of their general cohomological approach to the theory of Gillet-Soulé. There is a natural injective map ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X) \hookrightarrow {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}})$; abusing notation, we identify the special cycle $\widehat Z(T,{\mathbf v})$ with its image under this map. \[thm:Intro main theorem\] (i) Suppose $1 < n \leq p$. Fix $T_2 \in \operatorname{Sym}_{n-1}(F)$, and define the formal generating series $$\label{eqn:main thm n} {\widehat{\mathrm{FJ}}}_{T_2}({\bm{\tau}}) \ = \ \sum_{T = {\left( \begin{smallmatrix} * & * \\ * & T_2 \end{smallmatrix} \right)}} \widehat Z(T,{\mathbf v}) \, q^T$$ where ${\bm{\tau}}\in {\mathbb H}_n^d$ lies in the Hilbert-Siegel upper half space of genus $n$, and ${\mathbf v}= \mathrm{Im}(\tau)$. Then ${\widehat{\mathrm{FJ}}}_{T_2}({\bm{\tau}})$ is the $q$-expansion of a (non-holomorphic) Hilbert-Jacobi modular form of weight $p/2 + 1$ and index $T_2$, taking values in ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}}) \otimes S(L^n)^{\vee}$ via the Weil representation. (ii) When $n=1$, the generating series $$\label{eqn:main thm genus 1} \widehat \phi_1({\bm{\tau}}) \ = \ \sum_{t \in F} \widehat Z(t,{\mathbf v}) q^t$$ is the $q$-expansion of a (non-holomorphic) Hilbert modular form of weight $p/2+1$, valued in ${\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X) \otimes S(L)^{\vee}$. Some clarification is warranted in the interpretation of this theorem. The issue is that there is no apparent topology on the arithmetic Chow groups for which the series and can be said to converge in a reasonable sense; in a similar vein, the Green forms ${ \mathfrak{g}}(T,{\mathbf v})$ vary smoothly in the parameter ${\mathbf v}$, but there does not appear to be a natural way in which the family of classes $\widehat Z(T,{\mathbf v})$ can be said to vary smoothly. What is being asserted in the theorem is the existence of: (i) finitely many classes $\widehat Z_1, \dots \widehat Z_r \in \widehat{\mathrm{CH}}{}^n(X, \mathcal D_{\mathrm{cur}})$ (or in ${\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X)$ when $n=1$), (ii) finitely many Jacobi modular forms (in the usual sense) $f_1, \dots, f_r$, (iii) and a Jacobi form $g({\bm{\tau}})$ valued in the space of currents on $X$ that is locally uniformly bounded in ${\bm{\tau}}$, such that the $T$’th coefficient of the Jacobi form $ \sum_i f_i({\bm{\tau}}) \widehat Z_i + a( g({\bm{\tau}})) $ coincides with $\widehat Z(T,{\mathbf v})$. Here $a(g({\bm{\tau}})) \in \widehat{\mathrm{CH}}{}^n(X, \mathcal D_{\mathrm{cur}})$ is an “archimedean class" associated to the current $g({\bm{\tau}})$. A more precise account may be found in . To prove the theorem, we first prove the $n=1$ case, using a modularity result due to Bruinier [@Bruinier-totally-real] that involves a different set of Green functions; the theorem in this case follows from a comparison between his Green functions and ours. For $n>1$, we exhibit a decomposition $$\label{eqn:intro Zhat decomp} \widehat Z(T,{\mathbf v}) = \widehat A(T,{\mathbf v}) + \widehat B(T,{\mathbf v})$$ in ${\widehat{\mathrm{CH}}{}^{n}}(X,{{\mathcal D}_{\mathrm{cur}}}) \otimes S(L^n)^{\vee}$; this decomposition is based on a mild generalization of the star product formula [@GarciaSankaran Theorem 4.10]. The main theorem then follows from the modularity of the series $$\widehat \phi_A({\bm{\tau}}) := \sum_{T = {\left( \begin{smallmatrix} * & * \\ * & T_2 \end{smallmatrix} \right)}} \widehat A(T,{\mathbf v}) \, q^T \qquad \text{and} \qquad \widehat \phi_B({\bm{\tau}}) \ = \ \sum_{T = {\left( \begin{smallmatrix} * & * \\ * & T_2 \end{smallmatrix} \right)}} \widehat B(T,{\mathbf v}) \, q^T ,$$ which are proved in and respectively. The classes $\widehat A(T,{\mathbf v})$ are expressed as linear combinations of pushforwards of special cycles along sub-Shimura varieties of $X$, weighted by the Fourier coefficients of classical theta series; the modularity of $\widehat \phi_A({\bm{\tau}})$ follows from this description and the $n=1$ case. The classes $\widehat B(T,{\mathbf v})$ are purely archimedean, and the modularity of $\widehat \phi_B({\bm{\tau}})$ follows from an explicit computation involving the Kudla-Millson Schwartz form, [@KudlaMillsonIHES]. This result provides evidence for the arithmetic version of Kudla’s conjecture, namely that the generating series $$\widehat \phi_n({\bm{\tau}}) \ = \ \sum_{T \in \operatorname{Sym}_n(F)} \, \widehat Z(T,{\mathbf v}) \, q^T$$ is a Hilbert-Siegel modular form; indeed, the series ${\widehat{\mathrm{FJ}}}_{T_2}({\bm{\tau}})$ is a formal Fourier-Jacobi coefficient of $\widehat\phi_n({\bm{\tau}})$. Unfortunately, there does not seem to be an obvious path by which one can infer the more general result from the results in this paper; the decomposition $\widehat Z(T,{\mathbf v})$ depends on the lower-right matrix $T_2$, and it is not clear how to compare the decompositions for various $T_2$. Acknowledgements {#acknowledgements .unnumbered} ---------------- The impetus for this paper emerged from discussions during an AIM SQuaRE workshop; I’d like to thank the participants – Jan Bruinier, Stephan Ehlen, Stephen Kudla and Tonghai Yang – for the stimulating discussion and insightful remarks, and AIM for the hospitality. I’d also like to thank Craig Cowan for a helpful discussion on the theory of currents. This work was partially supported by an NSERC Discovery grant. Preliminaries {#sec:Prelims} ============= Notation -------- - Throughout, we fix a totally real field $F$ with $[F:{\mathbb Q}]=d$. Let $\sigma_1 , \dots, \sigma_d$ denote the real embeddings. Using these embeddings, we identify $F\otimes_{{\mathbb Q}} {\mathbb R}$ with $\mathbb R^d$, and denote by $\sigma_i(\mathbf t)$ the $i$’th component of $\mathbf t \in F \otimes_{{\mathbb Q}}{\mathbb R}$ under this identification. - For any matrix $A $, we denote the transpose by $A'$. - If $A \in {\mathrm{Mat}}_{n}(F \otimes_{{\mathbb Q}}{\mathbb R})$, we write $$e(A) \ := \ \prod_{i=1}^d \, \exp\big( 2 \pi i\, \mathrm{tr} \left( \sigma_i(A) \right)\big)$$ - If $(V,Q)$ is a quadratic space over $F$, let $\langle {\mathbf x}, {\mathbf y}\rangle$ denote the corresponding bilinear form. Here we take the convention $Q({\mathbf x}) = \langle {\mathbf x}, {\mathbf x}\rangle$. If ${\mathbf x}\in V$ and ${\mathbf y}= ({\mathbf y}_1, \dots , {\mathbf y}_{n}) \in V^n$ , we set $\langle {\mathbf x}, {\mathbf y}\rangle = ( \langle {\mathbf x}, {\mathbf y}_1 \rangle , \dots, \langle {\mathbf x}, {\mathbf y}_{n} \rangle ) \in {\mathrm{Mat}}_{1,n}(F)$. - For $i = 1, \dots d$, we set $V_i = V \otimes_{F,\sigma_i} {\mathbb R}$. - Let $${\mathbb H}_n^d = \{ {\bm{\tau}}= {\mathbf u}+ i{\mathbf v}\in \operatorname{Sym}_n(F \otimes_{{\mathbb Q}} {\mathbb R}) \ | \ {\mathbf v}\gg 0 \}$$ denote the Hilbert-Siegel upper half-space of genus $n$ attached to $F$. Via the fixed embeddings $\sigma_1, \dots, \sigma_d$, we may identify $\operatorname{Sym}_n(F \otimes {\mathbb R}) \simeq \operatorname{Sym}_n({\mathbb R})^d$; we let $\sigma_i({\bm{\tau}}) = \sigma_i({\mathbf u}) + i \sigma_i({\mathbf v})$ denote the corresponding component, so that, in particular, $\sigma_i({\mathbf v}) \in \operatorname{Sym}_n({\mathbb R})_{>0}$ for $i=1,\dots, d$. If ${\bm{\tau}}\in {\mathbb H}_n^d$ and $T \in \operatorname{Sym}_n(F)$, we write $$q^T = e ({\bm{\tau}}T).$$ Arithmetic Chow groups ---------------------- In this section, we recall the theory of arithmetic Chow groups ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X)$, as conceived by Gillet and Soulé [@GilletSouleIHES]; note here and throughout this paper, we work with complex coefficients. Recall that $X$ is defined over a number field $E$ endowed with a fixed complex embedding $\sigma: E \to {\mathbb C}$. We view $X$ as an arithmetic variety over the “arithmetic ring" $(E, \sigma, \text{complex conjugation})$ in the terminology of [@GilletSouleIHES §3.1.1]. An *arithmetic cycle* is a pair $(Z,g)$, where $Z$ is a formal ${\mathbb C}$-linear combination of codimension $n$ subvarieties of $X$, and $g$ is a *Green current* for $Z$; more precisely, $g$ is a current of degree $(p-1,p-1)$ on $X({\mathbb C})$ such Green’s equation $$\label{eqn:ChowHat Green} {\mathrm{dd^c}}g \ + \ \delta_{Z({\mathbb C})} = \omega$$ holds, where the right hand side is the current defined by integration[^1] against some smooth form $\omega$. Given a codimension $n-1$ subvariety $Y$ and a rational function $f \in k(Y)^{\times}$ on $Y$, let $$\widehat{\mathrm{div}} (f) := (\mathrm{div}(f), - \log | f |^2 \, \delta_Y)$$ denote the corresponding principal arithmetic divisor. The arithmetic Chow group ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X)$ is quotient of the space of arithmetic cycles by the subspace spanned by (a) the principal arithmetic divisors and (b) classes of the form $(0, \eta)$ with $\eta \in \mathrm{im}(\partial) + \mathrm{im}(\overline \partial)$. For more details, see [@GilletSouleIHES; @SouleBook] In their paper [@BurgosKramerKuhn], Burgos, Kramer and Kühn give an abstract reformulation and generalization of this theory: their main results describe the construction of an arithmetic Chow group ${\widehat{\mathrm{CH}}{}^{*}}(X, {\mathcal C})$ attached to a “Gillet complex" ${\mathcal C}$. One of the examples they describe is the group attached to the complex of currents ${\mathcal D}_{\mathrm{cur}}$; we will content ourselves with the superficial description of this group given below, which will suffice for our purposes, and the reader is invited to consult [@BurgosKramerKuhn §6.2] for a thorough treatment. Unwinding the formal definitions in [@BurgosKramerKuhn], one finds that classes in ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X, {{\mathcal D}_{\mathrm{cur}}})$ are represented by tuples $(Z, [T, g])$, with $Z$ as before, but now $T$ and $g$ are currents of degree $(n,n)$ and $(n-1, n-1)$ respectively such that[^2] $${\mathrm{dd^c}}g + \delta_{Z({\mathbb C})} = T + {\mathrm{dd^c}}(\eta)$$ for some current $\eta$ with support contained in $Z({\mathbb C})$; we can view this as a relaxation of the condition that the right hand side of is smooth. A nice consequence of this description is that any codimension $n$ cycle $Z$ on $X$ gives rise to a *canonical class* (see [@BurgosKramerKuhn Definition 6.37]) $$\widehat Z^{{\mathrm{can}}} \ := \ \left( Z, [\delta_Z, 0] \right).$$ By [@BurgosKramerKuhn Theorem 6.35], the natural map $${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X) \to {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}}), \qquad (Z,g) \mapsto (Z, [\omega, g])$$ is injective. Moreover while ${\widehat{\mathrm{CH}}{}^{*}}_{{\mathbb C}}(X, {{\mathcal D}_{\mathrm{cur}}})$ is not a ring in general, it is a module over ${\widehat{\mathrm{CH}}{}^{*}}_{{\mathbb C}}(X)$. As a special case of this product, let $(Z,g) \in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X)$ be an arithmetic divisor, where $g$ is a Green function with logarithmic singularities along the divisor $Z$. Suppose $\widehat Y^{{\mathrm{can}}} \in {\widehat{\mathrm{CH}}{}^{m}}(X, {{\mathcal D}_{\mathrm{cur}}})$ is the canonical class attached to a cycle $Y$ that intersects $Z$ properly; then by inspecting the proofs of [@BurgosKramerKuhn Theorem 6.23, Proposition 6.32] we find $$(Z,g) \cdot \widehat Y^{{\mathrm{can}}} \ = \ \left(Z \cdot Y, [\omega \wedge \delta_{Y({\mathbb C})}, g \wedge \delta_{Y({\mathbb C})}] \right) \in {\widehat{\mathrm{CH}}{}^{m+1}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}}).$$ \[remark:Chow vanishing\] One consequence of our setup is the vanishing of certain “archimedean rational" classes in ${\widehat{\mathrm{CH}}{}^{n}}(X)$ and ${\widehat{\mathrm{CH}}{}^{n}}(X, {{\mathcal D}_{\mathrm{cur}}})$. More precisely, if $Y$ is a codimension $n-1$ subvariety, then $$(0, \delta_{Y({\mathbb C})}) = 0 \ \in \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X).$$ To see this, let $c \in {\mathbb Q}$ be any rational number such that $c \neq 0$ or $1$, and view $c$ as a rational function on $Y$; its divisor is trivial, and so $$0 = \widehat{\mathrm{div}}(c) \ = \ (0, - \log|c|^2 \delta_{Y({\mathbb C})}) \ = \ - \log|c|^2 \cdot (0, \delta_{Y({\mathbb C})}).$$ and hence $(0, \delta_{Y({\mathbb C})}) = 0$. As a special case, we have $(0,1) = 0 \in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X)$. Special cycles {#sec:Prelim cycles} -------------- Here we review Kudla’s construction of the family $\{Z(T)\}$ of special cycles on $X$, [@KudlaAlgCycles]. First, recall that the symmetric space ${\mathbb D}$ has a concrete realization $$\label{eqn:symmetric space proj model} {\mathbb D}= \{ z \in {\mathbb P}^1(V\otimes_{\sigma_1, F} {\mathbb C}) \ | \ \langle z, z \rangle = 0, \langle z , \overline z \rangle < 0 \}$$ where $\langle \cdot, \cdot\rangle$ is the ${\mathbb C}$-bilinear extension of the bilinear form on $V$; the two connected components of ${\mathbb D}$ are interchanged by conjugation. Given a collection of vectors ${\mathbf x}= ({\mathbf x}_1, \dots, {\mathbf x}_n) \in V^n$, let $${\mathbb D}^+_{{\mathbf x}} \ := \ \{ z \in {\mathbb D}^+ \ | \ z \perp \sigma_1({\mathbf x}_i) \ \text{ for } i = 1, \dots, n\}.$$ where, abusing notation, we denote by $\sigma_1 \colon V \to V_1 = V \otimes_{F, \sigma_1} {\mathbb R}$ the map induced by inclusion in the first factor. Let $\Gamma_{{\mathbf x}}$ denote the pointwise stabilizer of ${\mathbf x}$ in $\Gamma$; then the inclusion ${\mathbb D}^+_{{\mathbf x}} \subset {\mathbb D}^+$ induces a map $$\Gamma_{{\mathbf x}} \big\backslash {\mathbb D}^+_{{\mathbf x}} \to \Gamma \big\backslash {\mathbb D}^+ = X,$$ which defines a complex algebraic cycle that we denote $Z({\mathbf x})$. If the span of $\{ {\mathbf x}_1, \dots, {\mathbf x}_r \}$ is not totally positive definite, then ${\mathbb D}^+_{{\mathbf x}}= \emptyset$ and $Z({\mathbf x}) = 0$; otherwise, the codimension of $Z({\mathbf x})$ is the dimension of this span. Now suppose $T \in \operatorname{Sym}_n(F)$ and $\varphi \in S(L^n)$, and set $$Z(T, \varphi)^{\natural} \ := \ \sum_{\substack{ {\mathbf x}\in \Omega(T) \\ \text{mod } \Gamma}} \varphi({\mathbf x}) \cdot Z({\mathbf x}),$$ where $$\Omega(T) \ := \ \{ {\mathbf x}= ({\mathbf x}_1, \dots, {\mathbf x}_n)\in V^n \ | \ \langle {\mathbf x}_i, {\mathbf x}_j \rangle = T_{ij} \} .$$ This cycle is rational over $E$. If $Z(T,\varphi)^{\natural} \neq 0$, then $T$ is necessarily totally positive semidefinite, and in this case $Z(T,\varphi)^{\natural}$ has codimension equal to the rank of $T$. Finally, we define a $S(L^n)^{\vee}$-valued cycle $Z(T)^{\natural}$ by the rule $$Z(T)^{\natural} \colon \varphi \mapsto Z(T, \varphi)^{\natural}.$$ for $\varphi \in S(L^n)$. The cotautological bundle ------------------------- Let ${\mathcal E}\to X $ denote the tautological bundle: over the complex points $X({\mathbb C}) = \Gamma \backslash {\mathbb D}^+$, the fibre ${\mathcal E}_z$ at a point $z \in {\mathbb D}^+$ is simply the line corresponding to $z$ in the model . There is a natural Hermitian metric $\| \cdot \|^2_{{\mathcal E}}$ on ${\mathcal E}({\mathbb C})$, defined at a point $z \in {\mathbb D}^+$ by the formula $ \| v_z \|^2_{{\mathcal E},z} \ = \ - \langle v_z, v_z \rangle$ for $ v_z \in z$. Consider the arithmetic class $$\widehat \omega = - \widehat c_1({\mathcal E}, \| \cdot \|_{{\mathcal E}}) \ \in \ {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X);$$ concretely, $\widehat\omega = - (\mathrm{div} s, - \log \| s\|_{{\mathcal E}}^2 ) $, where $s$ is any meromorphic section of ${\mathcal E}$. Finally, for future use, we set $$\Omega := - c_1({\mathcal E}, \| \cdot \|_{{\mathcal E}}) \in A^{1,1}(X({\mathbb C}))$$ where $-\Omega = c_1({\mathcal E}, \| \cdot \|_{{\mathcal E}}) $ is the first Chern form attached to $({\mathcal E}, \|\cdot \|_{{\mathcal E}})$; here the Chern form is normalized as in [@SouleBook §4.2]. Note that $-\Omega$ is a Kähler form, cf. [@GarciaSankaran §2.2]. Elsewhere in the literature, one often finds a different normalization (i.e. an overall multiplicative constant) for the metric $\| \cdot \|_{{\mathcal E}}$ that is better suited to certain arithmetic applications; for example, see [@KRYbook §3.3]. In our setting, however, implies that rescaling the metric does not change the Chern class in ${\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X)$. Green forms and arithmetic cycles {#sec:prelim arith cycles} --------------------------------- In this section, we sketch the construction of a family of Green forms for the special cycles, following [@GarciaSankaran]. We begin by recalling that for any tuple $x = (x_1, \dots x_n) \in V_1^n = (V \otimes_{\sigma_1, F} {\mathbb R})^n$, Kudla and Millson (see [@KudlaMillsonIHES]) have defined a Schwartz form $\varphi_{{\mathrm{KM}}}(x)$, which is valued in the space of closed $(n,n)$ forms on ${\mathbb D}^+$, and is of exponential decay in $x$. Let $T(x) \in \operatorname{Sym}_{n}({\mathbb R})$ denote the matrix of inner products, i.e. $T(x)_{ij} = \langle x_i, x_j \rangle$, and consider the normalized form $$\label{eqn:PhiKM normalized} \varphi^o_{{\mathrm{KM}}}(x) \ := \ \varphi_{{\mathrm{KM}}}(x) \, e^{2 \pi \mathrm{tr} T(x) } .$$ In [@GarciaSankaran §2.2], another form $\nu^o(x)$, valued in the space of smooth $(n-1,n-1)$ forms on ${\mathbb D}^+$ is defined (this form is denoted by $\nu^o(x)_{[2n-2]}$ there). It satisfies the relation $$\label{eqn:nu transgression} {\mathrm{dd^c}}\nu^o( \sqrt{u} x) \ = \ - u \frac{\partial}{\partial u} \varphi_{{\mathrm{KM}}}^o(\sqrt{u} x), \qquad u \in {\mathbb R}_{>0}.$$ For a complex parameter $\rho \gg 0$, let $${ \mathfrak{g}}^o(x;\rho) \ := \ \int_1^{\infty} \nu^o(\sqrt u x) \frac{du}{u^{1+\rho}};$$ then ${ \mathfrak{g}}^o(x, \rho)$ defines a smooth form for $Re(\rho) \gg 0$. The corresponding current admits a meromorphic continuation to a neighbourhood of $\rho=0$ and we set $${ \mathfrak{g}}^o(x) := \mathop{CT}_{\rho=0} \, { \mathfrak{g}}^o(x;\rho).$$ Note that, for example, $$\label{eqn:green current zero} { \mathfrak{g}}^o(0) \ = \ \nu^o(0) \, \mathop{CT}_{\rho = 0} \int_1^{\infty} \frac{du}{u^{1+\rho}} = 0.$$ In general, the current ${ \mathfrak{g}}^o(x)$ satisfies the equation $$\label{eqn:Green equation for g} {\mathrm{dd^c}}{ \mathfrak{g}}^o(x) + \delta_{{\mathbb D}^+_{{\mathbf x}}} \wedge \Omega^{n - r(x)} = \varphi^o_{{\mathrm{KM}}}(x)$$ where $r(x) = \dim\mathrm{span}(x) = \dim\mathrm{span}(x_1, \dots, x_n)$; for details regarding all these facts, see [@GarciaSankaran §2.6]. Now suppose $T \in \operatorname{Sym}_n(F)$. Following [@GarciaSankaran §4], we define an $S(L^n)^{\vee}$-valued current ${ \mathfrak{g}}^o(T,{\mathbf v})$, depending on a parameter ${\mathbf v}\in \operatorname{Sym}_n(F\otimes_{{\mathbb Q}} {\mathbb R})_{\gg 0}$, as follows: let $v = \sigma_1({\mathbf v})$ and choose any matrix $a \in {\mathrm{GL}}_n({\mathbb R})$ such that $v = a a'$. Then ${ \mathfrak{g}}^o(T,{\mathbf v})$ is defined by the formula $${ \mathfrak{g}}^o(T, {\mathbf v})(\varphi) \ := \ \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \ { \mathfrak{g}}^o \left( \sigma_1({\mathbf x})a \right),$$ where $ \sigma_1({\mathbf x}) \in V_1^n$; by [@GarciaSankaran Proposition 2.12], this is independent of the choice of $a \in {\mathrm{GL}}_n({\mathbb R})$. Note that ${ \mathfrak{g}}^o(T, {\mathbf v})$ is a $\Gamma$-invariant current on ${\mathbb D}^+$ and hence descends to $X({\mathbb C}) = \Gamma \backslash {\mathbb D}^+$. Next, consider the $S(L^n)^{\vee}$-valued differential form $\omega(T,{\mathbf v})$, defined by the formula $$\label{eqn:KM theta q coeff} \omega(T,{\mathbf v})(\varphi) \ := \ \sum_{{\mathbf x}\in \Omega(T)} \varphi(x) \, \varphi_{{\mathrm{KM}}}^o( \sigma_1({\mathbf x}) a), \qquad \sigma_1({\mathbf v}) = aa',$$ and which is a $q$-coefficient of the *Kudla-Millson theta series* $$\label{eqn:Kudla Millson theta} \Theta_{{\mathrm{KM}}}({\bm{\tau}}) \ = \ \sum_{T \in \operatorname{Sym}_n(F)} \, \omega(T,{\mathbf v}) \, q^T, \qquad$$ where $ \tau \in {\mathbb H}^d_n$, and $ {\mathbf v}= \mathrm{Im}({\bm{\tau}})$. We then have the equation of currents $$\label{eqn:global green eqn} {\mathrm{dd^c}}{ \mathfrak{g}}^o(T,{\mathbf v}) \ + \ \delta_{Z(T)({\mathbb C})} \wedge \Omega^{n - \mathrm{rank}T} \ = \ \omega(T,{\mathbf v})$$ on $X$, see [@GarciaSankaran Proposition 4.4] In particular, if $T$ is non-degenerate, then ${\mathrm{rank}}(T) = n$ and ${ \mathfrak{g}}^o(T,{\mathbf v})$ is a Green current for the cycle $Z(T)^{\natural}$; in this case, we obtain an arithmetic special cycle $$\widehat Z(T, {\mathbf v}) := (Z(T)^{\natural}, { \mathfrak{g}}^o(T,{\mathbf v}) ) \ \in \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X) \otimes_{{\mathbb C}} S(L^n)^{\vee}.$$ Now suppose $T \in \operatorname{Sym}_n(F)$ is arbitrary, let $r = {\mathrm{rank}}(T)$, and fix $\varphi \in S(L^n)$. We may choose a pair $(Z_0, g_0)$ representing the class $\widehat\omega^{n-r} \in {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X)$, such that $Z_0$ intersects $Z(T, \varphi)^{\natural}$ properly and $g_0$ has logarithmic type [@SouleBook §II.2]. We then define $$\widehat Z(T, {\mathbf v}, \varphi) := \left(Z(T, \varphi)^{\natural} \cdot Z_0, \ { \mathfrak{g}}^o(T,{\mathbf v}, \varphi) + g_0 \wedge \delta_{Z(T,\varphi)({\mathbb C})} \right) \ \in \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X).$$ The reader may consult [@GarciaSankaran §5.4] for more detail on this construction, including the fact that it is independent of the choice of $(Z_0, g_0)$. Finally, we define a class $\widehat Z(T,{\mathbf v}) \in {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X) \otimes S(L^n)^{\vee}$ by the rule $$\widehat Z(T,{\mathbf v})(\varphi) = \widehat Z(T, {\mathbf v}, \varphi).$$ In [@GarciaSankaran], the Green current ${ \mathfrak{g}}^o(T,{\mathbf v})$ is augmented by an additional term, depending on $\log(\det {\mathbf v})$, when $T$ is degenerate see [@GarciaSankaran Definition 4.5]. This term was essential in establishing the archimedean arithmetic Siegel-Weil formula in the degenerate case; however, in the setting of the present paper, implies that this additional term vanishes upon passing to ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X)$, and can be omitted from the discussion without consequence. In particular, according to our definitions, we have $$\widehat Z(0_n, {\mathbf v})(\varphi) = \varphi(0) \cdot \widehat\omega^{n}.$$ Hilbert-Jacobi modular forms {#sec:modularity def} ---------------------------- In this section, we briefly review the basic definitions of vector-valued (Hilbert) Jacobi modular forms, mainly to fix notions. For convenience, we work in “classical" coordinates and only with parallel scalar weight. Throughout, we fix an integer $n \geq 1.$ We begin by briefly recalling the theory of metaplectic groups and the Weil representation; a convenient summary for the facts mentioned here, in a form useful to us, is [@JiangSoudryGenericityII §2]. For a place $v \leq \infty$, let $\operatorname{\widetilde{Sp}}_n(F_v)$ denote the metaplectic group, a two-fold cover of $\operatorname{Sp}_n(F_v)$; as a set, $\operatorname{\widetilde{Sp}}_n(F_v) = \operatorname{Sp}_n(F_v) \times \{\pm 1\}$. When $F_v = {\mathbb R}$, the group $\operatorname{\widetilde{Sp}}_n({\mathbb R})$ is isomorphic to the group of pairs $(g, \phi)$, where $g = {\left( \begin{smallmatrix}A & B \\ C & D \end{smallmatrix} \right)} \in \operatorname{Sp}_n({\mathbb R})$ and $\phi \colon {\mathbb H}_n \to {\mathbb C}$ is a function such that $\phi(\tau)^2 = \det (C \tau + D)$; in this model, multiplication is given by $$(g, \phi(\tau)) \cdot (g', \phi'(\tau)) = (gg', \phi(g' \tau) \phi'(\tau) ).$$ At a non-dyadic finite place, there exists a canonical embedding $\operatorname{Sp}_n({\mathcal O}_v) \to \operatorname{\widetilde{Sp}}_n(F_v)$. Consider the restricted product $\prod'_{v \leq \infty} \operatorname{\widetilde{Sp}}_n(F_v)$ with respect to these embeddings; the global double cover $\operatorname{\widetilde{Sp}}_{n,{\mathbb A}}$ of $\operatorname{Sp}_n({\mathbb A})$ is the quotient $\prod'_{v \leq \infty} \operatorname{\widetilde{Sp}}_n(F_v)/ I$ of this restricted direct product by the subgroup $$I := \{ (1, \epsilon_v)_{v \leq \infty} \, | \, \prod_v \epsilon_v = 1, \, \epsilon_v = 1 \text{ for almost all } v \}.$$ Moreover, there is a splitting $$\label{eqn:metaplectic canonical splitting} \iota_F \colon \operatorname{Sp}_n(F) \hookrightarrow \operatorname{\widetilde{Sp}}_{n,{\mathbb A}}, \qquad \gamma \mapsto \prod_v (\gamma, 1)_v \cdot I.$$ Let $\widetilde \Gamma'$ denote the full inverse image of $\operatorname{Sp}_{n}({\mathcal O}_F)$ under the covering map $\prod_{v | \infty} \operatorname{\widetilde{Sp}}_n(F_v) \to \operatorname{Sp}_n(F \otimes_{{\mathbb Q}} {\mathbb R})$. We obtain an action $\rho$ of $\widetilde \Gamma'$ on the space $ S(V({\mathbb A}_f)^n)$ as follows. Let $\omega $ denote the[^3] Weil representation of $\operatorname{\widetilde{Sp}}_{n,{\mathbb A}}$ on $S(V({\mathbb A})^n)$. Given $\widetilde \gamma \in \widetilde \Gamma'$, choose $\widetilde \gamma_f \in \prod'_{v < \infty} \operatorname{\widetilde{Sp}}_n(F_v)$ such that $\widetilde \gamma \widetilde \gamma_f \in \mathrm{im}(\iota_F)$ and set $$\rho(\widetilde \gamma) \ := \ \omega(\widetilde \gamma_f).$$ Recall that we had fixed a lattice $L \subset V$. The subspace $S(L^n) \subset S(V({\mathbb A}_f)^n)$, as defined in , is stable under the action of $\widetilde{\Gamma}'$; when we wish to emphasize this lattice, we denote the corresponding action by $\rho_L$. For a half-integer $\kappa \in \frac12 \mathbb Z$, we define a (parallel, scalar) weight $\kappa$ slash operator, for the group $\widetilde \Gamma'$ acting on the space of functions $f \colon {\mathbb H}^d_n \to S(L^n)^{\vee}$, by the formula $$\label{eqn:slash operators} f|_{\kappa}[\widetilde \gamma] ({\bm{\tau}}) \ = \ \prod_{v|\infty} \phi_v(\sigma_v({\bm{\tau}}))^{- 2 \kappa} \rho_L^{\vee}(\widetilde \gamma^{-1})\cdot f( g {\bm{\tau}}), \qquad \widetilde \gamma = (g_v, \phi_v(\tau))_{v | \infty}$$ where $g = (g_v)_v$. If $n >1$, consider the Jacobi group $G^J = G^J_{n,n-1} \subset \operatorname{Sp}_{n}$; for any ring $R$, its $R$-points are given by $$G^J(R) := \left\{ g = \left( \begin{array}{cc|cc} a & 0 & b & a\mu - b \lambda \\ \lambda^t & 1_{n-1} & \mu^t & 0 \\ \hline c & 0 & d & c\mu - d \lambda \\ 0&0&0&1_{n-1} \end{array} \right) \ | \ \begin{pmatrix} a&b\\c&d\end{pmatrix} \in {\mathrm{SL}}_2(R) , \ \mu, \tau \in M_{1,n-1}(R)\right\}.$$ Define $\widetilde \Gamma^J \subset \widetilde \Gamma'$ to be the inverse image of $G^J({\mathcal O}_F)$ in $\widetilde G'_{{\mathbb R}}$. \[def:Jacobi form\] Suppose $$f \colon {\mathbb H}_n^d \to S(L^n)^{\vee}$$ is a smooth function. Given $T_2 \in \operatorname{Sym}_{n-1}(F)$, we say that $f({\bm{\tau}})$ transforms like a Jacobi modular form of genus $n$, weight $\kappa$ and index $T_2$ if the following conditions hold. (a) For all ${\mathbf u}_2 \in \operatorname{Sym}_{n-1}(F_{{\mathbb R}})$, $$f \left( {\bm{\tau}}+ {\left( \begin{smallmatrix} 0 & \\ & {\mathbf u}_2 \end{smallmatrix} \right)} \right) \ = \ e( T_2 {\mathbf u}_2) f({\bm{\tau}}).$$ (b) For all $\widetilde \gamma \in \widetilde \Gamma^J$, $$f|_{\kappa}[\widetilde \gamma] ({\bm{\tau}}) \ = \ f({\bm{\tau}}).$$ Let $A_{\kappa,T_2}(\rho_L^{\vee})$ denote the space of $S(L^n)^{\vee}$-valued smooth functions that transform like a Jacobi modular form of weight $\kappa$ and index $T_2$. 1. If desired, one can impose further analytic properties of $f$ (holomorphic, real analytic, etc.). 2. If $n = 1$, then we simply say that a function $f \colon {\mathbb H}^d_1 \to S(L)^{\vee}$ transforms like a (Hilbert) modular form of weight $\kappa$ if it satisfies $f|_{\kappa}[ \tilde \gamma]({\bm{\tau}}) = f({\bm{\tau}})$ as usual. 3. An $S(L^n)^{\vee}$-valued Jacobi modular form $f$, in the above sense, has a Fourier expansion of the form $$\label{eqn:Fourier expansion of Jacobi form} f ({\bm{\tau}}) \ := \ \sum_{T = {\left( \begin{smallmatrix} * & * \\ * & T_2 \end{smallmatrix} \right)}} c_{ f}(T, {\mathbf v}) \,q^T$$ where the coefficients $c_f(T,{\mathbf v})$ are smooth functions $ c_f(T,{\mathbf v}) \colon \operatorname{Sym}_n(F\otimes_{{\mathbb Q}}{{\mathbb R}})_{\gg 0} \to S(V^n)^{\vee}$. The dependence on ${\mathbf v}$ arises from the natural expectation that the Fourier-Jacobi coefficients of non-holomorphic Siegel modular forms should be Jacobi forms. We now clarify what it should mean for generating series with coefficients in arithmetic Chow groups, such as those appearing in , to be modular. First, let $D^{n-1}(X)$ denote the space of currents on $X({\mathbb C})$ of complex bidegree $(n-1,n-1)$, and note that there is a map $$a \colon D^{n-1}(X) \ \to \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}}), \qquad a(g) = ( 0 , \, [{\mathrm{dd^c}}g, g]).$$ \[def:smooth currents\] Define the space $A_{\kappa, T_2}( \rho^{\vee}; D^{n-1}(X))$ of “Jacobi forms valued in $S(L^n)^{\vee} \otimes_{{\mathbb C}} D^{n-1}(X)$" as the space of functions $$\xi \colon {\mathbb H}^d_n \to D^{n-1}(X)\otimes_{{\mathbb C}} S(L^n)^{\vee}$$ such that the following two conditions hold. (a) For every smooth form $\alpha$ on $X$, the function $\xi({\bm{\tau}})(\alpha)$ is an element of $A_{\kappa, T_2}( \rho_L^{\vee})$, and in particular, is smooth in the variable ${\bm{\tau}}$. (b) Fix an integer $k\geq 0$ and let $\| \cdot \|_k$ be an algebra seminorm, on the space of smooth differential forms on $X$, such that given a sequence $\{ \alpha_i \} $, we have $\| \alpha_i \|_k \to 0$ if and only if $\alpha_i$, together with all partial derivatives of order $\leq k$, tends uniformly to zero. We then require that for every compact subset $ C\subset {\mathbb H}^d_n$, there exists a constant $c_{k,V}$ such that $$|\xi({\bm{\tau}})(\alpha)| \leq c_{k,C} \| \alpha\|_k$$ for all ${\bm{\tau}}\in C$ and all smooth forms $\alpha$. The second condition ensures that any such function admits a Fourier expansion as in whose coefficients are continuous in the sense of distributions, i.e. they are again $S(L^n)^{\vee}$-valued currents. \[definition of modularity\] Given a collection of classes $\widehat Y(T,{\mathbf v}) \in {\widehat{\mathrm{CH}}{}^{n}}(X,{{\mathcal D}_{\mathrm{cur}}}) \otimes_{{\mathbb C}}S(L^n)^{\vee}$, consider the formal generating series $$\widehat \Phi_{T_2}({\bm{\tau}}) \ := \ \sum_{\substack{T = {\left( \begin{smallmatrix} * & * \\ * & T_2 \end{smallmatrix} \right)}}} \widehat Y(T, {\mathbf v}) \, q^T.$$ Roughly speaking, we say that $\widehat\Phi_{T_2}({\bm{\tau}})$ is modular (of weight $\kappa$ and index $T_2$) if there is an element $$\label{eqn:modularity def spaces} \widehat \phi({\bm{\tau}}) \in A_{\kappa, T_2}(\rho_L^{\vee})\otimes_{{\mathbb C}} {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}}) \ + \ a\left( A_{\kappa, T_2}(\rho_L^{\vee} ; D^{n-1}(X)) \right)$$ whose Fourier expansion coincides with $\widehat \Phi_{T_2}({\bm{\tau}})$. More precisely, we define the modularity of $\widehat\Phi_{T_2}({\bm{\tau}})$ to mean that there are finitely many classes $$\widehat Z_1, \dots, \widehat Z_r \in {\widehat{\mathrm{CH}}{}^{n}}(X,{{\mathcal D}_{\mathrm{cur}}})$$ and Jacobi forms $$f_1, \dots f_r \in A_{\kappa,T_2}(\rho^{\vee}), \qquad g \in A_{\kappa,T_2}(\rho^{\vee}; D^{n-1}(X))$$ such that $$\begin{aligned} \label{eqn:def of modularity cycle decomp} \widehat Y(T, {\mathbf v}) =& \sum_i c_{f_i}(T, {\mathbf v}) \, \widehat Z_i + a \left( c_{g}(T, {\mathbf v}) \right) \ \in \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}}) \otimes_{{\mathbb C}} S(L^n)^{\vee} \end{aligned}$$ for all $T = {\left( \begin{smallmatrix}* & * \\ * & T_2 \end{smallmatrix} \right)}$ . \[remark:modularity Chow\] 1. If $\widehat Z_1, \dots, \widehat Z_r \in {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X)$ and $g({\bm{\tau}})$ takes values in the space of (currents represented by) smooth differential forms on $X$, then we say that $\widehat \Phi_{T_2}({\bm{\tau}})$ is valued in ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X) \otimes S(L^n)^{\vee}$; indeed, in this case, the right hand side of lands in this latter group. 2. As before, one may also impose additional analytic conditions on the forms $f_i, g$ appearing above if desired. 3. Elsewhere in the literature (e.g.  [@BorcherdsGKZ; @Bruinier-totally-real; @ZhangThesis]), one finds a notion of modularity that amounts to omitting the second term in ; this notion is well-adapted to the case that the generating series of interest are holomorphic, i.e. the coefficients are independent of the imaginary part of ${\bm{\tau}}$. In contrast, the generating series that figure in our main theorem depend on these parameters in an essential way. Indeed, the Green forms ${ \mathfrak{g}}^o(T,{\mathbf v})$ vary smoothly in ${\mathbf v}$; however, to the best of the author’s knowledge, there is no natural topology on ${\widehat{\mathrm{CH}}{}^{n}}(X)$, or ${\widehat{\mathrm{CH}}{}^{n}}(X, {{\mathcal D}_{\mathrm{cur}}})$, for which the corresponding family $\widehat Z(T,{\mathbf v})$ varies smoothly in $v$. As we will see in the course of the proof of the main theorem, the additional term in will allow us enough flexibility to reflect the non-holomorphic behaviour of the generating series. Similar considerations appear in [@EhlenSankaran] in the codimension one case. The genus one case ================== In this section, we prove the main theorem in the case $n =1$; later on, this will be a key step in the proof for general $n$. The proof of this theorem amounts to a comparison with a generating series of special divisors equipped with a different family of Green functions, defined by Bruinier. A similar comparison appears in [@EhlenSankaran] for unitary groups over imaginary quadratic fields; in the case at hand, however, the compactness of $X$ allows us to apply spectral theory and simplify the argument considerably. Suppose $t \gg 0$. In [@Bruinier-totally-real], Bruinier constructs an $S(L)^{\vee}$-valued Green function $\Phi(t)$ for the divisor $Z(t) = Z(t)^{\natural}$. To be a bit more precise about this, recall the Kudla-Millson theta function $\Theta_{{\mathrm{KM}}}({\bm{\tau}})$ from . As a function of ${\bm{\tau}}$, the theta function $\Theta_{{\mathrm{KM}}}$ is non-holomorphic and transforms as a Hilbert modular form of parallel weight $\kappa = p/2 + 1$. It is moreover of moderate growth, [@Bruinier-totally-real Prop. 3.4] and hence can be paired, via the Petersson pairing, with cusp forms. Let $\Lambda_{{\mathrm{KM}}}({\bm{\tau}}) \in S_{\kappa}(\rho_L)$ denote the cuspidal projection, defined by the property $$\langle \Theta_{{\mathrm{KM}}}, g \rangle^{\mathrm{Pet}} = \langle \Lambda_{{\mathrm{KM}}}, g\rangle^{\mathrm{Pet}}$$ for all cusp forms $g \in S_{\kappa}(\rho)$. Writing the Fourier expansion $$\Lambda_{{\mathrm{KM}}}({\bm{\tau}}) = \sum_t c_{\Lambda}(t) \ q^t, \qquad \Lambda_{{\mathrm{KM}},t} \in A^{1,1}(X) \otimes S(L)^{\vee}$$ it follows from [@Bruinier-totally-real Corollary 5.16, Theorem 6.4] that $\Phi(t)$ satisfies the equation $$\label{eqn:Bruinier Green equation} {\mathrm{dd^c}}[\Phi(t)] \ + \ \delta_{Z(t)} \ = \ \left[ c_{\Lambda}(t) + B(t) \cdot \Omega \right]$$ of currents on $X$, where $$B(t) \ := \ - \frac{ \deg(Z(t))}{\mathrm{vol}(X, (-\Omega)^p) } \ \in \ S(L)^{\vee}.$$ recall here that $(-\Omega)^p$ induces a volume form on $X$. Finally, define classes $ \widehat Z_{\mathrm{Br}}(t) \in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X) \otimes S(L)^{\vee}$ as follows: $$\widehat Z_{\mathrm{Br}}(t) \ = \ \begin{cases} \left( Z(t), \Phi(t) \right), & \text{if } t \gg 0 \\ \widehat{ \omega} \otimes {\mathrm{ev}}_0, & \text{if } t = 0 . \\ 0, & \text{otherwise,} \end{cases}$$ where ${\mathrm{ev}}_0 \in S(L)^{\vee}$ is the functional $\varphi \mapsto \varphi(0)$. We then have the generating series $$\widehat\phi_{\mathrm{Br}}(\tau) = \sum_t \widehat Z_{\mathrm{Br}}(t) \, q^t.$$ \[Bruinier modularity gen fibre\] The generating series $\widehat\phi_{\mathrm{Br}}(\tau)$ is a (holomorphic) Hilbert modular form of parallel weight $\kappa = p/2+1$. More precisely, there are finitely many classes $\widehat Z_1, \dots \widehat Z_r \in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X)$ and holomorphic Hilbert modular forms $f_1, \dots, f_r$ such that $\widehat Z_{\mathrm{Br}}(t) = \sum c_{f_i}(t) \, \widehat Z_i$ for all $t\in F$. The proof follows the same argument as [@Bruinier-totally-real Theorem 7.1], whose main steps we recall here. Bruinier defines a space $M^!_{k}(\rho_L)$ of weakly holomorphic forms [@Bruinier-totally-real §4] of a certain“dual" weight $k $; each $f \in M^!_{k}(\rho_L)$ is defined by a finite collection of vectors $c_f(m) \in S(L)^{\vee}$ indexed by $m \in F$. Applying Bruinier’s criterion for the modularity of a generating series, cf.  [@Bruinier-totally-real (7.1)], we need to show that $$\sum_m c_f(m) \widehat Z_{\mathrm{Br}}(m) = 0 \in {\widehat{\mathrm{CH}}{}^{1}}(X)$$ for all $f \in M_k^!(\rho_L)$. Let $c_0 = c_f(0)(0)$, and assume $c_0 \in \mathbb Z$. By [@Bruinier-totally-real Theorem 6.8], after replacing $f$ by a sufficiently large integer multiple, there exists an analytic meromorphic section $\Psi^{an} $ of $(\omega^{an})^{-c_0}$ such that $$\mathrm{div} \, \Psi^{an} = \sum_{m \neq 0} c_f(m) \cdot Z(m)^{an}.$$ and with $$\mathrm - \log \| \Psi^{an} \|^2 = \sum_{m \neq 0} c_f(m) \cdot \Phi(m).$$ Recall that $X$ is projective; by GAGA and the fact that the $Z(m)$’s are defined over $E$, there is an $E$-rational section $\psi$ of $\omega^{-c_0}$ and a constant $C \in {\mathbb C}$ such that $$\mathrm{div} (\psi) = \sum_{m\neq 0} c_f(m) Z(m), \qquad - \log \| \psi^{an} \|^2 = - \log \| \Psi^{an} \| ^2 + C.$$ Thus $$-c_0 \cdot \widehat \omega \ = \ \widehat{\mathrm{div}}( \psi) = \sum_{m \neq 0} c_f(m) \cdot \widehat Z_{\mathrm{Br}}(m) \ + \ (0, C) \ \in \ {\widehat{\mathrm{CH}}{}^{1}}(X).$$ However, as in , the class $(0, C) = 0$, and thus we find $$\sum_m c_f(m) \widehat Z_{\mathrm{Br}}(m) \ =\ c_0 \cdot \widehat \omega + \sum_{m \neq 0 } c_f(m) \widehat Z_{\mathrm{Br}}(m) \ = \ 0$$ as required. Now we consider the difference $$\widehat\phi_1({\bm{\tau}}) - \widehat \phi_{\mathrm{Br}}({\bm{\tau}}) = \sum_t (0, { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t)) \, q^t$$ whose terms are classes represented by purely archimedean cycles. Comparing the Green equations and , we have that for $t \neq 0$ and any smooth form $\eta$, $${\mathrm{dd^c}}[{ \mathfrak{g}}^o(t,v) - \Phi(t)] (\eta) = \int_X \left( { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t) \right) \, {\mathrm{dd^c}}\eta = \int_X \left( \omega(t,{\mathbf v}) - c_{\Lambda}(t) - B(t) \Omega \right) \wedge \eta$$ where $\omega(t,{\mathbf v})$ is the $t$’th $q$-coefficient of $\Theta_{{\mathrm{KM}}}({\bm{\tau}})$; in particular, elliptic regularity implies that the difference ${ \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t)$ is smooth on $X({\mathbb C})$. \[genus one diff modularity\] There exists a smooth $S(L)^{\vee}$-valued function $s({\bm{\tau}},z) $ on ${\mathbb H}_1^d \times X({\mathbb C})$ such that the following holds. (i) For each fixed $z \in X({\mathbb C})$, the function $s({\bm{\tau}},z)$ transforms like a Hilbert modular form in ${\bm{\tau}}$. (ii) Let $$s({\bm{\tau}},z) = \sum_t c_s(t,{\mathbf v},z) \ q^t$$ denote its $q$-expansion in ${\bm{\tau}}$; then for each $t$, we have $$(0, { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t) ) \ = \ (0, c_s(t, {\mathbf v},z)) \ \in \ {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X) \otimes_{{\mathbb C}}S(L)^{\vee}$$ Combining this theorem with , we obtain: \[genus one modularity\] The generating series $\widehat \phi_1({\bm{\tau}}) $ is modular, valued in ${\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X) \otimes S(L)^{\vee}$, in the sense of (i). Recall that the $(1,1)$ form $- \Omega$ is a Kähler form on $X$. Let $- \Delta_X$ denote the corresponding Laplacian; the eigenvalues of $- \Delta_X$ are non-negative, discrete in ${\mathbb R}_{\geq 0}$, and each eigenspace is finite dimensional. Write $\Delta_X = 2 (\partial \partial^* + \partial^* \partial)$ and let $L \colon \eta \mapsto -\eta \wedge (-\Omega)$ denote the Lefschetz operator. From the Kähler identities $[L, \partial] = [L,\overline \partial] = [L, \Delta_S] = 0$ and $[L, \partial^*] = i \overline \partial$, an easy induction argument shows that $$\partial^* \circ L^k \ =\ L^k \circ \partial^* \ - \ i k \, \overline \partial \circ L^{k-1}$$ for $k \geq 1$. Thus, for a smooth function $\phi$ on $X$, we have $$\begin{aligned} \Delta_X (\phi) \cdot (-\Omega)^p = \Delta_X \circ L^p(\phi) &= 2 \partial \partial ^* \circ L^p (\phi) \\ &= \ 2 \ \partial \circ \left( L^p \circ \partial^* - i p \overline \partial \circ L^{p-1} \right)(\phi ) \\ &= \ - 2 i p \ \partial \overline \partial \left( \phi \wedge (-\Omega)^{p-1} \right) \\ &= \ - 4 \pi p \ {\mathrm{dd^c}}\left( \phi \wedge (-\Omega)^{p-1} \right); \end{aligned}$$ note here that $p = \dim_{{\mathbb C}} (X).$ Consider the Hodge pairing $$\langle f,g \rangle_{L^2} = \int_X f \, \overline g \, (-\Omega)^p = (-1)^p\int_X f \, \overline g \, \Omega^p.$$ If $\lambda >0$ and $\phi_{\lambda}$ is a Laplace eigenfunction, we have that for any $t\neq0$, $$\begin{aligned} \langle { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t), \phi_{\lambda}\rangle_{L^2} &= \lambda^{-1} \langle { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t), - \Delta_X \phi_{\lambda} \rangle_{L^2} \\ &= (-1)^p \lambda^{-1} \int_X ( { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t)) \cdot (\overline{- \Delta_X \phi_{\lambda}}) \cdot \Omega^p \\ &= (-1)^{p+1} \frac{ 4 \pi p }{\lambda} \int_X \left( { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t) \right) \cdot {\mathrm{dd^c}}\left( \overline \phi_{\lambda} \, \Omega^{p-1} \right) \\ &= (-1)^{p+1 } \frac{ 4 \pi p }{\lambda} \int_X \left( \omega(t,{\mathbf v}) - c_{\Lambda}(t) - B(t) \Omega \right) \wedge \overline \phi_{\lambda} \Omega^{p-1} \end{aligned}$$ Note that $\int_X \overline \phi_{\lambda} \Omega^p = \langle 1, \phi_{\lambda} \rangle_{L^2} = 0$, as $\lambda >0$ and so $\phi_{\lambda}$ is orthogonal to constants; thus the term involving $B(m) \Omega$ vanishes, and so $$\langle { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t), \phi_{\lambda}\rangle_{L^2} = (-1)^{p+1 }\frac{ 4 \pi p }{\lambda} \int_X \left( \omega(t,{\mathbf v}) - c_{\Lambda}(t) \right) \wedge \overline \phi_{\lambda} \Omega^{p-1}$$ for all $t \neq 0$. This equality also holds for $t=0$, as both sides of this equation vanish. Indeed, for the left hand side we have ${ \mathfrak{g}}^o(0,{\mathbf v}) = 0$, cf. , and $\Phi(0) = 0$ by definition; on the right hand side, $c_{\Lambda}(0) =0 $ as $\Lambda_{{\mathrm{KM}}}({\bm{\tau}})$ is cuspidal, and the constant term of the Kudla-Millson theta function is given by $$\omega(0, {\mathbf v}) = \Omega \otimes {\mathrm{ev}}_0.$$ Now define $$h({\bm{\tau}},z) = (L^*)^{p-1} \circ \ast \left( \Theta_{{\mathrm{KM}}}({\bm{\tau}}) - \Lambda_{{\mathrm{KM}}}({\bm{\tau}}) \right)$$ where $\ast$ is the Hodge star operator, and $L^*$ is the adjoint of the Lefschetz map $L$. Then $h({\bm{\tau}},z)$ is smooth, and transforms like a modular form in ${\bm{\tau}}$, since both $\Theta_{{\mathrm{KM}}}({\bm{\tau}})$ and $\Lambda_{{\mathrm{KM}}}({\bm{\tau}})$ do; writing its Fourier expansion $$h({\bm{\tau}},z) \ =\ \sum_t c_h(t, {\mathbf v},z) \, q^t,$$ we have $$\langle c_h(t, {\mathbf v},z), \phi \rangle_{L^2} = (-1)^{p-1} \int_X \left( \omega(t,{\mathbf v}) -c_{ \Lambda}(t) \right) \wedge \overline{\phi } \, \Omega^{p-1}$$ for any smooth function $\phi$. Note that for any integer $N$ and $L^2$ normalized eigenfunction $\phi_{\lambda}$ with $\lambda \neq 0$, $$\label{eqn:laplace eigenfunction} \langle h , \phi_{\lambda} \rangle_{L^2} = \lambda^{-N} \langle - \Delta^N_X (h), \varphi_{\lambda} \rangle \leq \lambda^{-N} \| - \Delta^N_X(h) \|^2_{L^2}.$$ Choose an orthonormal basis $\{ \phi_{\lambda}\}$ of $L^2(X)$ consisting of eigenfunctions, and consider the sum $$\label{eqn:s(tau,z) def} s(\tau,z) = 4 \pi p \sum_{\lambda >0} \lambda^{-1} \langle h, \phi_{\lambda} \rangle_{L^2} \, \phi_{\lambda};$$ by Weyl’s law, there are positive constants $C_1$ and $C_2$ such that $$\# \{ \lambda \ | \ \lambda < x \} \sim x^{C_1}$$ and $\| \phi_{\lambda}\|_{L^{\infty}} = O(\lambda^{C_2})$. Thus taking $N$ sufficiently large in , we conclude that the sum converges uniformly, and hence defines a smooth function in $({\bm{\tau}},z)$. Writing its Fourier expanison as $$s({\bm{\tau}},z) = \sum_t c_s(t,{\mathbf v},z) q^t,$$ we have $$\langle c_s(t,{\mathbf v},z) , \phi_{\lambda} \rangle_{L^2} = \langle { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t), \phi_{\lambda} \rangle_{L^2}$$ for any eigenfunction $\phi_{\lambda}$ with $\lambda \neq 0$. Thus $ c_s(t,{\mathbf v},z) $ and $ g^o(t,{\mathbf v}) - \Phi(t)$ differ by a function that is constant in $z$; as $(0,1) = 0 \in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X)$, we have $$(0, { \mathfrak{g}}^o(t,{\mathbf v}) - \Phi(t)) = (0, c_s(t,{\mathbf v},z)) \in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X) \otimes_{{\mathbb C}} S(L)^{\vee},$$ which concludes the proof of the theorem. Decomposing Green currents ========================== We now suppose $n > 1$ and fix $T_2 \in \operatorname{Sym}_{n-1}(F)$. The aim of this section is to establish a decomposition $ \widehat Z(T, {\mathbf v}) = \widehat A(T,{\mathbf v}) + \widehat B(T,{\mathbf v})$, where $T = {\left( \begin{smallmatrix}*&*\\ *& T_2 \end{smallmatrix} \right)}$. Our first step is to decompose Green forms in a useful way; the result can be seen as an extension of the star product formula [@GarciaSankaran Theorem 4.10] to the degenerate case. Let $x = (x_1, \dots, x_n) \in (V_1 )^n =( V\otimes_{F, \sigma_1} {\mathbb R})^n$ and set $y = (x_2, \dots, x_n) \in V_1^{n-1}$ . By [@GarciaSankaran Proposition 2.6.(a)], we may decompose $$\label{eqn: go decomposition} { \mathfrak{g}}^o(x, \rho) = \int_1^{\infty} \nu^o(\sqrt t \, x_1) \wedge \varphi^o_{{\mathrm{KM}}}(\sqrt t \, y) \ \frac{dt}{t^{1+\rho}} \ + \ \int_1^{\infty} \varphi_{{\mathrm{KM}}}^o(\sqrt t \, x_1) \wedge \nu^o(\sqrt t \, y) \ \frac{dt}{t^{1+\rho}}$$ for $Re(\rho) \gg 0$. By the transgression formula , we may rewrite the second term in as $$\begin{aligned} \int_1^{\infty} & \varphi_{{\mathrm{KM}}}^o (\sqrt t \, x_1) \wedge \nu^o(\sqrt t \, y) \ \frac{dt}{t^{1+\rho}} \\ &= \int_1^{\infty} \left( \int_1^t \, \frac{\partial}{\partial u} \varphi_{{\mathrm{KM}}}^o(\sqrt u \, x_1) \, du\right) \wedge \nu^o(\sqrt t \, y) \ \frac{dt}{t^{1+\rho}} + \varphi_{{\mathrm{KM}}}^o(x_1) \wedge \int_1^{\infty} \nu^o(\sqrt t \, y) \frac{dt}{t^{1+\rho}} \\ &= \int_1^{\infty} \left( \int_1^t \, - {\mathrm{dd^c}}\nu^o(\sqrt u \, x_1) \, \frac{du}{u}\right) \wedge \nu^o(\sqrt t \, y) \ \frac{dt}{t^{1+\rho}} + \varphi_{{\mathrm{KM}}}^o(x_1) \wedge { \mathfrak{g}}^o(y,\rho) . \label{eqn: g0 second piece} \end{aligned}$$ For $t>1$, define smooth forms $$\alpha_t(x_1,y) := \int_1^{t} \overline{\partial} \nu^o(\sqrt u \, x_1) \, \frac{du}{u} \wedge \nu^o(\sqrt{t} \, y)$$ and $$\beta_t(x_1,y) := \int_1^t \nu^o(\sqrt{u} \, x_1) \, \frac{du}{u} \wedge \partial \nu^o(\sqrt{t} y)$$ so that $$\begin{aligned} \eqref{eqn: g0 second piece} = \frac{i}{2 \pi } \int_1^{\infty} \partial & \alpha_t(x_1,y) + \overline{\partial} \beta_t(x_1,y) \, \frac{dt}{t^{1+\rho}} \ - \ \int_1^{\infty} \left[ \int_1^{t} \nu^o(\sqrt{u} \, x_1) \, \frac{du}{u} \right] \wedge {\mathrm{dd^c}}\nu^o(\sqrt t \, y) \, \frac{dt}{t^{1+ \rho}} \notag \\ &+ \varphi_{{\mathrm{KM}}}^o(x_1) \wedge { \mathfrak{g}}^o(y, \rho). \end{aligned}$$ Finally, we consider the second integral above; as $Re(\rho)$ is large, we may interchange the order of integration and obtain $$\begin{aligned} \int_1^{\infty} & \left(\int_1^{t} \nu^o(\sqrt{u} \, x_1) \, \frac{du}{u} \right) \wedge {\mathrm{dd^c}}\nu^o(\sqrt t \, y) \, \frac{dt}{t^{1+ \rho}} \\ & = \int_1^{\infty} \nu^o(\sqrt u x_1) \wedge \left( \int_u^{\infty} {\mathrm{dd^c}}\nu^o(\sqrt t y) \frac{dt}{t^{1+\rho}} \right) \frac{du}{u} \\ &= \int_1^{\infty} \nu^o(\sqrt u x_1) \wedge \left( \int_u^{\infty} - \frac{\partial}{\partial t} \varphi_{{\mathrm{KM}}}^o(\sqrt t y) \frac{dt}{t^{\rho}} \right) \frac{du}{u} \\ &= \int_1^{\infty} \nu^o(\sqrt u \, x_1) \wedge \varphi_{{\mathrm{KM}}}^o(\sqrt{u} \, y) \, \frac{du}{u^{1+ \rho}} - \rho \int_1^{\infty} \nu^o(\sqrt u\, x_1) \wedge \left( \int_u^{\infty} \varphi_{{\mathrm{KM}}}^o(\sqrt t \, y) \frac{dt}{t^{1+\rho}} \right) \, \frac{du}{u} \end{aligned}$$ Note that the first term here coincides with the first term in . Combining these computations, it follows that $$\begin{aligned} { \mathfrak{g}}^o(x,\rho) &= \varphi^o_{{\mathrm{KM}}}(x_1) \wedge { \mathfrak{g}}^o(y,\rho) + \frac{i}{2 \pi } \int_1^{\infty} \partial \alpha_t(x_1,y) + \overline{\partial} \beta_t(x_1,y) \, \frac{dt}{t^{1+\rho}} \notag \\ & \qquad \qquad + \rho \int_1^{\infty} \nu^o(\sqrt u\, x_1) \wedge \left( \int_u^{\infty} \varphi_{{\mathrm{KM}}}^o(\sqrt t \, y) \frac{dt}{t^{1+\rho}} \right) \, \frac{du}{u}. \end{aligned}$$ This identity holds for arbitrary $x =(x_1,y) \in V_1^n$ and $Re(\rho) \gg 0$, and is an identity of smooth differential forms on ${\mathbb D}$. To continue, we view the above line as an identity of currents, and consider meromorphic continuation.[^4] Note that (as currents) $$\begin{aligned} \rho \int_1^{\infty} & \nu^o(\sqrt u\, x_1) \wedge \left( \int_u^{\infty} \varphi_{{\mathrm{KM}}}^o(\sqrt t \, y) \frac{dt}{t^{1+\rho}} \right) \, \frac{du}{u} \\ &= \rho \int_1^{\infty} \nu^o(\sqrt u \, x_1) \wedge \int_u^{\infty} \left( \varphi_{{\mathrm{KM}}}^o(\sqrt t \, y) - \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} \right) \frac{dt}{t^{1+\rho}} \frac{du}{u} \notag \\ & \qquad + \int_1^{\infty} \nu^o(\sqrt u x_1) \wedge \delta_{{\mathbb D}^+_y } \wedge \Omega^{n-1-r(y)} \ \frac{du}{u^{1+\rho}} \ \end{aligned}$$ where $r(y) = \dim \, \mathrm{span} (y)$. The first term vanishes at $\rho = 0$; indeed, the double integral in the first term is holomorphic at $\rho = 0$, as can easily seen by by Bismut’s asymptotic [@BismutInv90 Theorem 3.2] $$\varphi_{{\mathrm{KM}}}^o(\sqrt{t} y) - \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} \ =\ O(t^{-1/2})$$ as $t \to \infty$. Next, let $$\alpha(x_1,y; \rho) \ := \ \int_1^{\infty} \, \alpha_t(x_1,y) \frac{dt}{t^{1+ \rho}}, \qquad \qquad \beta(x_1,y; \rho) \ := \ \int_1^{\infty} \, \beta_t(x,y) \frac{dt}{t^{1+ \rho}}.$$ A straightforward modification of the proof of [@GarciaSankaran Proposition 2.12.(iii)] can be used to show that $\alpha(x_1, y; \rho)$ and $\beta(x_1, y; \rho)$ have meromorphic extensions, as currents, to a neighbourhood of $\rho = 0$. We denote the constant terms in the Laurent expansion at $\rho = 0$ by $\alpha(x_1, y)$ and $\beta(x_1, y)$ respectively. Thus, as currents on ${\mathbb D}$, we have $$\begin{aligned} { \mathfrak{g}}^o(x_1,y) &= \varphi_{{\mathrm{KM}}}^o(x_1) \wedge { \mathfrak{g}}^o(y) + d \alpha(x_1,y) + d^c \beta(x_1,y) \\ & \qquad \qquad + CT_{\rho = 0} \int_1^{\infty} \nu^o(\sqrt u x_1) \wedge \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} \ \frac{du}{u^{1+\rho}} \label{eqn:go xy decomp} \end{aligned}$$ for all $x_1 \in V_1$ and $y \in (V_1)^{n-1}$. As a final observation, note that if $x_1 \in \mathrm{span}(y)$, then $\nu^o(\sqrt u x_1) \wedge \delta_{{\mathbb D}^+_y} =\delta_{{\mathbb D}^+_y}$, see [@GarciaSankaran Lemma 2.4]. Thus $$\begin{aligned} \gamma(x_1, y) &:= CT_{\rho = 0} \int_1^{\infty} \nu^o(\sqrt u x_1) \wedge \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} \ \frac{du}{u^{1+\rho}} \\ & = \begin{cases} { \mathfrak{g}}^o(x_1) \wedge \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} , & \text{if } x_1 \notin \mathrm{span}(y) \\ 0 , & \text{if } x_1 \in \mathrm{span}(y). \end{cases} \end{aligned}$$ In the case that the components of $x = (x_1, y) = (x_1, \dots , x_n)$ are linearly independent, we recover the star product formula from [@GarciaSankaran Theorem 2.16]. Now we discuss a decomposition of the global Green current ${ \mathfrak{g}}^o(T, {\mathbf v})$, for ${\mathbf v}\in \operatorname{Sym}_{n}(F_{{\mathbb R}})_{\gg 0}$. Write $$v := \sigma_1({\mathbf v}) = \begin{pmatrix} v_1 & v_{12} \\ v_{12}' & v_2 \end{pmatrix}$$ with $v_1 \in {\mathbb R}_{>0}$ and $v_{12} \in M_{1,n-1}({\mathbb R})$; recall that $\sigma_1 \colon F \to {\mathbb R}$ is the distinguished real embedding. Set $$v_2^* := v_2 - v'_{12}v_{12} / v_1 \in \operatorname{Sym}_{n-1}({\mathbb R})_{>0},$$ and fix a matrix $a_2^* \in {\mathrm{GL}}_{n-1}({\mathbb R})$ such that $v_2^* = a_2^* \cdot (a_2^*)'$. \[gT decomp\] Let $T \in \operatorname{Sym}_n(F)$ and ${\mathbf v}\in \operatorname{Sym}_n(F_{{\mathbb R}})_{\gg 0}$ as above, and define $S(L)^{\vee}$-valued currents ${ \mathfrak{a}}(T,{\mathbf v})$ and ${ \mathfrak{b}}(T,{\mathbf v})$ on $X$ by the formulas $$\label{eqn:lie a def} { \mathfrak{a}}(T,{\mathbf v})(\varphi) := \sum_{{\mathbf x}\in \Omega(T) } \varphi({\mathbf x}) \gamma(\sqrt{v_1} x_1, y),$$ where we have written $\sigma_1({\mathbf x}) = (x_1, y) \in V_1 \oplus (V_1)^{n-1}$, and $${ \mathfrak{b}}(T,{\mathbf v})(\varphi) = \sum_{{\mathbf x}\in \Omega(T) } \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}} \left(\sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } \right) \wedge { \mathfrak{g}}^o(y a_2^*).$$ Then $${ \mathfrak{g}}^o(T,{\mathbf v}) (\varphi) \equiv { \mathfrak{a}}(T,{\mathbf v})(\varphi) + { \mathfrak{b}}(T,{\mathbf v})(\varphi) \pmod{\mathrm{im} \,\partial \, + \, \mathrm{im} \, \overline{\partial}}.$$ First, the fact that the sums defining ${ \mathfrak{a}}(T,{\mathbf v})$ and ${ \mathfrak{b}}(T,{\mathbf v})$ converge to currents on $X$ follows from the same argument as [@GarciaSankaran Proposition 4.3]. Now recall that $${ \mathfrak{g}}^o(T,{\mathbf v}) (\varphi) = \sum_{{\mathbf x}\in \Omega(T) } \varphi({\mathbf x}) \, { \mathfrak{g}}^o( xa)$$ where $x = \sigma_1({\mathbf x})$, and $a \in {\mathrm{GL}}_n({\mathbb R})$ is any matrix satisfying $v = aa'$. Note that $$v = \begin{pmatrix} v_1 & v_{12} \\ v_{12}' & v_2 \end{pmatrix} = \theta \begin{pmatrix} v_1 & \\ & v_2 ^* \end{pmatrix} \theta', \qquad \text{ where } \theta = \begin{pmatrix} 1 & \\ v_{12}' / v_1 & 1_{n-1} \end{pmatrix} .$$ Thus, we may take $$\label{eqn:a theta} a = \theta \cdot {\left( \begin{smallmatrix} \sqrt{v_1} & \\ & a_2^* \end{smallmatrix} \right)} ,$$ and so, applying , we find $$\begin{aligned} { \mathfrak{g}}^o(T,{\mathbf v})(\varphi) &= \sum_{{\mathbf x}\in \Omega(T)} \, \varphi({\mathbf x}) \, { \mathfrak{g}}^o\left( (x_1, y) \theta {\left( \begin{smallmatrix} \sqrt{v_1} & \\ & a_2^* \end{smallmatrix} \right)} \right) \qquad \qquad x = (x_1, y)\\ &= \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \, { \mathfrak{g}}^o \left( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } , \, y a_2^* \right) \\ &= \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \Big( \varphi^o_{{\mathrm{KM}}} \left( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } \right) \wedge { \mathfrak{g}}^o(y a_2^*) + \partial \alpha( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } ,y a_2^*) \notag \\ & \qquad \qquad\qquad\qquad\qquad \ + \overline \partial \beta( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } , y a_2^*) + \gamma( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } ,y a_2^*) \Big). \label{eqn:go(T) decomp big sum} \end{aligned}$$ Again, an argument as in [@GarciaSankaran Proposition 4.3] shows that the sums $$\eta_1 := \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \, \alpha( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } ,y a_2^*)$$ and $$\eta_2 := \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \, \beta( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } ,y a_2^*)$$ converge to $\Gamma$-invariant currents on ${\mathbb D}$, and hence define currents on $X$. Moreover, it follows easily from the definitions that $$\gamma( \sqrt{v_1} x_1 + \frac{y \cdot v_{12}'}{\sqrt{ v_1} } ,y a_2^*) = \gamma(\sqrt{v_1} x_1, y).$$ Thus, we find $${ \mathfrak{g}}^o(T,{\mathbf v})(\varphi)= { \mathfrak{a}}(T,{\mathbf v})(\varphi)+ { \mathfrak{b}}(T,{\mathbf v})(\varphi) + \partial \eta_1 + \overline{\partial } \eta_2,$$ as required. Next, we define an $S(L^n)^{\vee}$-valued current $\psi(T,{\mathbf v})$ as follows. For ${\mathbf x}\in \Omega(T)$, write $\sigma_1({\mathbf x}) = x = (x_1, y) \in V_1 \oplus V_1^{n-1}$ as above; then $$\begin{aligned} \label{eqn:psi current def} \psi(T,{\mathbf v})(\varphi) \ &:= \ \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}}( \sqrt{v_1} x_1) \wedge \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} \end{aligned}$$ defines a $\Gamma$-equivariant current on $\mathbb D^+$, and hence descends to a current (also denoted $\psi(T,{\mathbf v})$) on $X({\mathbb C})$. \[a and b ddc eqn\] (i) Let $\omega(T,{\mathbf v})$ be the $T$th coefficient of the Kudla-Millson theta function, as in ; then $${\mathrm{dd^c}}{ \mathfrak{b}}(T,{\mathbf v}) \ = \ \omega(T,{\mathbf v}) - \psi(T,{\mathbf v}).$$ (ii) We have $${\mathrm{dd^c}}\, { \mathfrak{a}}(T,{\mathbf v}) + \delta_{Z(T)({\mathbb C})}\wedge \Omega^{n - r(T)} \ = \ \psi(T,{\mathbf v}),$$ where $r(T) = \mathrm{rank}(T)$. With $v = \sigma_1({\mathbf v})$ and taking $a = \theta \cdot {\left( \begin{smallmatrix} \sqrt{v_1} & \\ & a_2^* \end{smallmatrix} \right)}$ as , we have $$\begin{aligned} \omega(T,{\mathbf v})(\varphi) \ &= \ \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}}(xa) \\ &= \ \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}} \left( \sqrt{v_1} x_1 + \frac{y \, v_{12}'}{\sqrt{ v_1} } , \ y a_2^* \right) \\ &= \sum_{{\mathbf x}\in \Omega(T)} \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}} \left( \sqrt{v_1} x_1 + \frac{y v_{12}'}{\sqrt{ v_1} } \right) \wedge \varphi^o_{{\mathrm{KM}}} \left( y a_2^* \right)\end{aligned}$$ for $\varphi \in S(L^n)$, where the last line follows from [@KudlaMillsonIHES Theorem 5.2(i)]. Therefore, $$\begin{aligned} {\mathrm{dd^c}}{ \mathfrak{b}}(T, {\mathbf v})(\varphi) &= \sum_{{\mathbf x}\in \Omega(T) } \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}} \left(\sqrt{v_1} x_1 + \frac{y \, v_{12}'}{\sqrt{ v_1} } \right) \wedge {\mathrm{dd^c}}{ \mathfrak{g}}^o(y a_2^*) \\ &= \sum_{{\mathbf x}\in \Omega(T) } \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}} \left(\sqrt{v_1} x_1 + \frac{y \, v_{12}'}{\sqrt{ v_1} } \right) \wedge \Big\{ - \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} + \varphi^o_{{\mathrm{KM}}}(ya_2^*) \Big\} \\ &= - \sum_{{\mathbf x}\in \Omega(T) } \varphi({\mathbf x}) \, \varphi^o_{{\mathrm{KM}}} \left(\sqrt{v_1} x_1 + \frac{y \, v_{12}'}{\sqrt{ v_1} } \right) \wedge \delta_{{\mathbb D}^+_y} \wedge \Omega^{n-1-r(y)} + \omega(T,{\mathbf v})(\varphi).\end{aligned}$$ For $v \in V_1 $, the restriction $\varphi^o_{{\mathrm{KM}}}(v) \wedge \delta_{{\mathbb D}^+_y}$ depends only on the orthogonal projection of $v$ onto $\mathrm{span}(y)^{\perp}$; see, for example, [@GarciaSankaran Lemma 2.4]. In particular, $$\varphi^o_{{\mathrm{KM}}} \left(\sqrt{v_1} x_1 + \frac{y \, v_{12}'}{\sqrt{ v_1} } \right) \wedge \delta_{{\mathbb D}^+_y} = \varphi^o_{{\mathrm{KM}}} \left(\sqrt{v_1} x_1 \right) \wedge \delta_{{\mathbb D}^+_y} .$$ The first part of the lemma follows upon applying the definition of $\gamma(T,{\mathbf v})$ in . The second part then follows from the first, together with and . We finally arrived at the promised decomposition of $\widehat Z(T,{\mathbf v})$. Recall that in defining the cycle $\widehat Z(T,{\mathbf v})$ in , we fixed a representative $(Z_0, g_0)$ for $\widehat \omega^{n- r(T)}$ such that $Z_0$ intersects $Z(T)$ properly. By the previous proposition, $${\mathrm{dd^c}}\left( { \mathfrak{a}}(T,{\mathbf v}) + g_0 \wedge \delta_{Z(T)({\mathbb C}) } \right) + \delta_{Z(T)\cap Z_0 ({\mathbb C})} \ = \ \psi(T,{\mathbf v});$$ we then obtain classes in ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X, {{\mathcal D}_{\mathrm{cur}}})\otimes_{{\mathbb C}}S(L^n)$ by setting $$\widehat A(T,{\mathbf v}) \ := \ \left( Z(T) \cdot Z_0, \, [ \psi(T,{\mathbf v}), \, { \mathfrak{a}}(T,{\mathbf v}) + g_0 \wedge \delta_{Z(T)({\mathbb C})} ] \right)$$ and $$\widehat B(T,{\mathbf v}) \ := \ \left(0, \, [ \omega(T,{\mathbf v}) - \psi(T,{\mathbf v}) , \, { \mathfrak{b}}(T,{\mathbf v})] \right),$$ so that $$\widehat Z(T,{\mathbf v}) = \widehat A(T,{\mathbf v}) + \widehat B(T,{\mathbf v}) \ \in \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X,{{\mathcal D}_{\mathrm{cur}}}) \otimes_{{\mathbb C}} S(L)^{\vee}$$ \[remark: A vanishes if not semi definite\] Suppose $T = {\left( \begin{smallmatrix}* & * \\ * &T_2 \end{smallmatrix} \right)}$ as above; if $T_2$ is not totally positive semidefinite, then ${\mathbb D}^+_y = \emptyset$ for any ${\mathbf y}\in \Omega(T_2)$, and hence $\widehat A(T,{\mathbf v}) = 0$. Modularity I ============ In this section, we establish the modularity of the generating series $$\widehat \phi_B ({\bm{\tau}}) = \sum_{T = {\left( \begin{smallmatrix}* & * \\ * & T_2 \end{smallmatrix} \right)}} \, \widehat B(T,{\mathbf v}) \, q^T.$$ Note that $$\widehat B(T,{\mathbf v}) = (0, [{\mathrm{dd^c}}{ \mathfrak{b}}(T,{\mathbf v}), { \mathfrak{b}}(T,{\mathbf v})]) = a({ \mathfrak{b}}(T,v));$$ thus in light of , it suffices to establish the following theorem. \[thm:B series\] Fix $T_2 \in \operatorname{Sym}_{n-1}(F)$, and consider the generating series $$\label{eqn:B's gen series} \xi({\bm{\tau}}) = \sum_{T = {\left( \begin{smallmatrix}* & * \\ * & T_2 \end{smallmatrix} \right)}} { \mathfrak{b}}(T,{\mathbf v}) \, q^T ,$$ Then $ \xi({\bm{\tau}})$ is an element of $A_{\kappa, T_2}(\rho_L^{\vee}; D^*(X))$, see . We begin by showing the convergence of the series . By definition, $$\sum_{T = {\left( \begin{smallmatrix}T_1 &T_{12}\\ T_{12}' & T_2 \end{smallmatrix} \right)}} \, { \mathfrak{b}}(T, {\mathbf v})(\varphi) \, q^T = \sum_{T} \sum_{({\mathbf x}_1, {\mathbf y}) \in \Omega(T)} \varphi({\mathbf x}_1,{\mathbf y}) \, \varphi_{{\mathrm{KM}}}^o \left( \sqrt{v_1} x_1+ \frac{y \cdot v'_{12} }{ \sqrt{v_1}} \right)\wedge { \mathfrak{g}}^o(ya_2^*) \ q^T \label{eqn:expansion of b sum}$$ where $ x_1 = \sigma_1({\mathbf x}_1)$ and $y = \sigma_1({\mathbf y})$, and here we are working with $\Gamma$-equivariant currents on ${\mathbb D}^+$. For $v \in V_1$, consider the normalized Kudla-Millson form $$\varphi_{{\mathrm{KM}}}(v) \ := \ e^{- 2 \pi \langle v, v \rangle} \, \varphi^o_{{\mathrm{KM}}}(v)$$ which is a Schwartz form on $V_1$, valued in closed forms on ${\mathbb D}^+$; more precisely, for any integer $k$ and relatively compact open subset $U \subset {\mathbb D}^+$, there exists a positive definite quadratic form $Q_U$ on $V_1$ such that $$\label{eqn:varphi estimate} \| \varphi_{{\mathrm{KM}}}(v) \|_{k,\overline U} \ll e^{ - Q_U(v)}$$ where $\| \cdot \|_{k,\overline U}$ is an algebra seminorm measuring uniform convergence of all derivatives of order $\leq k$ on the space of smooth forms supported on $\overline U$, and the implied constant depends on $k$ and $\overline U$. Similarly, for $y \in V_1^{n-1}$, write $${ \mathfrak{g}} (y) = e^{-2 \pi \sum \langle y_i, y_i \rangle } { \mathfrak{g}}^o(y);$$ if ${\mathbb D}^+_y \cap \overline U = \emptyset$, then ${ \mathfrak{g}}(y)$ is smooth on $U$, and the form $Q_U$ may be chosen so that $$\label{eqn:g(y) estimate} \| { \mathfrak{g}}(y) \|_{k,\overline U} \ \ll \ e^{- \sum_{i=1}^{n-1} Q_U(y_i)} , \qquad y = (y_1, \dots, y_{n-1}) \in V_1^{n-1},$$ see [@GarciaSankaran §2.1.5]. Finally, for the remaining real embeddings $\sigma_2, \dots \sigma_d$, let $\varphi_{\infty_i} \in S(V_i^n)$ denote the standard Gaussian on the positive definite space $V_i = V \otimes_{F, \sigma_i} {\mathbb R}$, defined by $\varphi_{\infty_i}(x_1, \dots, x_n) = e^{ - 2 \pi \sum \langle x_i, x_i \rangle}$. Then a brief calculation gives $$\xi({\bm{\tau}})(\varphi) = \sum_{T} \sum_{({\mathbf x}_1, {\mathbf y}) \in \Omega(T)} \varphi({\mathbf x}_1,{\mathbf y}) \varphi_{{\mathrm{KM}}} \left( \sqrt{v_1} x_1+ \frac{y \cdot v'_{12} }{ \sqrt{v_1}} \right)\wedge { \mathfrak{g}}(ya_2^*) \cdot \prod_{i=2}^d \varphi_{\infty_i}( \sigma_i ({\mathbf x}_1, {\mathbf y}) a_i) \ e( T \mathbf u )$$ where we have chosen matrices $a_i \in GL_n({\mathbb R})$ for $i = 2, \dots, d$, such that $\sigma_i({\mathbf v}) = a_i \cdot a_i'$. Let $$S_1 := \{ {\mathbf y}\in (L')^{n-1} \ | \langle {\mathbf y}, {\mathbf y}\rangle = T_{2} \text{ and } {\mathbb D}^+_{y} \cap \overline U \neq \emptyset \}$$ which is a finite set, and let $$S_2 := \{ {\mathbf y}\in (L')^{n-1} \ | \langle {\mathbf y}, {\mathbf y}\rangle = T_{2} \text{ and } {\mathbb D}^+_{y} \cap \overline U = \emptyset \}.$$ Using the estimates and , and standard arguments for convergence of theta series, it follows that the sum $$\label{eqn:xi_0 sum} \sum_{T = {\left( \begin{smallmatrix}* & * \\ * & T_2 \end{smallmatrix} \right)}} \sum_{\substack{({\mathbf x}_1, {\mathbf y}) \in \Omega(T) \\ {\mathbf y}\in S_2}}\varphi({\mathbf x}_1,{\mathbf y}) \varphi_{{\mathrm{KM}}} \left( \sqrt{v_1} x_1+ \frac{y \cdot v'_{12} }{ \sqrt{v_1}} \right)\wedge { \mathfrak{g}}(ya_2^*) \prod_{i=2}^d \varphi_{\infty_i}( \sigma_i ({\mathbf x}_1, {\mathbf y}) a_i) \ e( T \mathbf u )$$ converges absolutely to a smooth form on ${\mathbb H}^d_n \times U$. The (finitely many) remaining terms, corresponding to ${\mathbf y}\in S_1$, can be written as $$\sum_{{\mathbf y}\in S_1} f_{{\mathbf y}}({\bm{\tau}})(\varphi)\wedge { \mathfrak{g}}(y a_2^*)$$ where, for any ${\mathbf y}\in V^{n-1}$ and $\varphi \in S(L^n)$, we set $$f_{{\mathbf y}}({\bm{\tau}}) (\varphi) \ = \ \sum_{ {\mathbf x}_1 \in V} \varphi({\mathbf x}_1, {\mathbf y}) \varphi_{{\mathrm{KM}}} \left( \sqrt{v_1} x_1+ \frac{y \cdot v'_{12} }{ \sqrt{v_1}} \right) \prod_{i=2}^d \varphi_{\infty_i}( \sigma_i ({\mathbf x}_1, {\mathbf y})a_i) \ e( T({\mathbf x}_1, {\mathbf y})\mathbf u ),$$ where $T({\mathbf x}_1, {\mathbf y}) = {\left( \begin{smallmatrix} \langle {\mathbf x}_1, {\mathbf x}_1 \rangle & \langle {\mathbf x}_1, {\mathbf y}\rangle \\ \langle {\mathbf x}_1, {\mathbf y}\rangle' & \langle {\mathbf y}, {\mathbf y}\rangle \end{smallmatrix} \right)} $. Again, the estimate shows that the series defining $f_{{\mathbf y}}({\bm{\tau}})$ converges absolutely to a smooth form on ${\mathbb H}_d^n \times {\mathbb D}^+$. Moreover, for a fixed $y \in V_1^{n-1}$ and any compactly supported test form $\alpha$ on ${\mathbb D}^+$, the value of the current ${ \mathfrak{g}}^o(y a_2^*)[\alpha]$ varies smoothly in the entries of $a_2^*$ (this fact follows easily from the discussion in [@GarciaSankaran §2.1.4]). Taken together, the above considerations imply that the series $\xi({\bm{\tau}})(\varphi)$ converges absolutely to a $\Gamma$-invariant current on ${\mathbb D}^+$, and therefore descends to a current on $X$ that satisfies part (b) of as ${\bm{\tau}}$ varies. In addition, this discussion shows that given any test form $\alpha$, the value of the current $\xi({\bm{\tau}})[\alpha]$ is smooth in ${\bm{\tau}}$. It remains to show that $\xi({\bm{\tau}})$ transforms like a Jacobi modular form, i.e. is invariant under the slash operators . Recall that the form $\varphi_{{\mathrm{KM}}}$ is of weight $p/2 +1$; more precisely, let $\widetilde U(1) \subset \operatorname{\widetilde{Sp}}_1({\mathbb R})$ denote the inverse image of $U(1)$, which admits a genuine character $\chi$ whose square is the identity on $U(1)$. Then $\omega(\widetilde k) \varphi_{{\mathrm{KM}}} = (\chi(\widetilde k))^{p+2} \varphi_{{\mathrm{KM}}}$ for all $\widetilde k \in \widetilde U(1)$, where $\omega$ is the Weil representation attached to $V_1$, cf. [@KudlaMillsonIHES Theorem 5.2]. To show that $\xi({\bm{\tau}})$ transforms like a Jacobi form, note that (by Vaserstein’s theorem), every element of $\widetilde \Gamma^J$ can be written as a product of the following elements. (i) For each $i = 1, \dots, d$, let $$\label{eqn:Jacobi generator epsilon} \widetilde \epsilon(i) = (\widetilde \epsilon (i))_v \in \prod_{v| \infty} \operatorname{\widetilde{Sp}}_n(F_v)$$ be the element whose $v$’th component is $({\mathrm{Id}}, 1)$ if $v \neq \sigma_i$, and $({\mathrm{Id}}, -1)$ if $v= \sigma_i$. (ii) For $\mu, \lambda \in \mathrm{M}_{1, n-1}({\mathcal O}_F)$, let $$\label{eqn:Jacobi generator gamma_lambda mu} \gamma_{\lambda, \mu} =\left( \begin{array}{cc|cc} 1 & 0 & 0 & \mu \\ \lambda' & 1_{n-1} & \mu' & 0 \\ \hline 0 & 0 & 1 & - \lambda \\ 0&0&0&1_{n-1} \end{array} \right) \in G^J({\mathcal O}_F).$$ Let $\iota_F(\gamma_{\lambda,\mu}) \in \operatorname{\widetilde{Sp}}_{n,{\mathbb A}}$ denote its image under the splitting ; we choose $\widetilde \gamma_{\lambda, \mu} \in \widetilde \Gamma^J$ to be the archimedean part of a representative $\iota_F(\gamma_{\lambda, \mu}) = \widetilde \gamma_{\lambda,\mu} \cdot \widetilde \gamma_f$. (iii) For $r \in {\mathcal O}_F$, let $$\label{eqn:Jacobi generator gamma_r} \gamma_r = \left( \begin{array}{cc|cc} 1 & 0 & r & 0 \\ 0 & 1_{n-1} & 0 & 0 \\ \hline 0 & 0 & 1 & 0 \\ 0&0&0&1_{n-1} \end{array} \right) \in G^J({\mathcal O}_F),$$ and choose an element $\widetilde \gamma_r$ as the archimedean part of a representative of $\iota_F(\gamma_r)$, as before. (iv) Finally, let $$\label{eqn:Jacobi generator S} S = \left( \begin{array}{cc|cc} 0 & 0 & -1 & 0 \\ 0 & 1_{n-1} & 0 & 0 \\ \hline 1 & 0 & 0 & 0 \\ 0&0&0&1_{n-1} \end{array} \right)$$ and take $\widetilde S \in \widetilde \Gamma^J$ to be the archimedean part of a representative of $\iota_F(S)$. Now, rearranging the absolutely convergent sum , we may write $$\xi({\bm{\tau}}) \ = \ \sum_{{\mathbf y}\in \Omega(T_2)} f_{{\mathbf y}}({\bm{\tau}}) \wedge { \mathfrak{g}}(y a_2^*).$$ Using the aforementioned generators, a direct computation shows that ${\mathbf v}_2^* = {\mathbf v}_2 - {\mathbf v}'_{12} {\mathbf v}_{12} / {\mathbf v}_1$, viewed as a function on ${\mathbb H}^d_n$, is invariant under the action of $\widetilde \Gamma^J$; it therefore suffices to show that for a fixed ${\mathbf y}$, the $S(L^n)^{\vee}$-valued function $f_{{\mathbf y}}({\bm{\tau}})$ transforms like a Jacobi form. It is a straightforward verification to check that $ f_{{\mathbf y}}({\bm{\tau}})$ is invariant under the action of $\widetilde \epsilon(i)$, $\tilde \gamma_{\lambda, \mu}$, and $\tilde \gamma_r$. For example, the element $\widetilde \gamma_{\lambda, \mu}^{-1}$ acts on $S(L^n)$ by the formula $$\rho(\widetilde \gamma_{\lambda, \mu}^{-1}) (\varphi) \left( {\mathbf x}_1, {\mathbf y}\right) \ = \ e \big( 2 \langle {\mathbf x}_1,{\mathbf y}\rangle \mu' - \langle {\mathbf y}\lambda' , {\mathbf y}' \rangle \mu'\big) \, \varphi( {\mathbf x}_1 -{\mathbf y}\lambda', \, {\mathbf y})$$ and $\gamma_{\lambda, \mu} $ acts on $ {\mathbb H}_n^d$ by the formula $$\gamma_{\lambda, \mu} \cdot {\bm{\tau}}= \begin{pmatrix} {\bm{\tau}}_1 & {\bm{\tau}}_{12} + {\bm{\tau}}_1 \lambda + \mu \\ {\bm{\tau}}_{12}' + {\bm{\tau}}_1 \lambda' + \mu' & {\bm{\tau}}_2 + \left( \lambda' \cdot {\bm{\tau}}_{12} + {\bm{\tau}}_{12}' \cdot \lambda \right) + \mu' \cdot \lambda \end{pmatrix}$$ where ${\bm{\tau}}= {\left( \begin{smallmatrix}{\bm{\tau}}_1 & {\bm{\tau}}_{12} \\ {\bm{\tau}}_{12}' & {\bm{\tau}}_2 \end{smallmatrix} \right)}$. Moreover, writing $\widetilde \gamma_{\lambda, \mu} = (\gamma_{\lambda, \mu}, (\phi_v))_v$ as in , we have $\prod \phi_v(\tau) = 1$. For ${\mathbf x}_1 \in V$ and ${\mathbf y}\in V^{n-1}$, a direct computation gives $$\mathrm{tr} \Big( T({\mathbf x}_1, {\mathbf y}) \cdot \mathrm{Re}(\gamma_{\lambda, \mu} \cdot {\bm{\tau}})\Big) = \mathrm{tr} \Big( T({\mathbf x}_1 + {\mathbf y}\lambda', {\mathbf y}) {\mathbf u}\Big) + 2 \langle {\mathbf x}_1, {\mathbf y}\rangle \mu' + \langle {\mathbf y}\lambda', {\mathbf y}\rangle\mu';$$ therefore, applying the above identity and the change of variables ${\mathbf x}_1 \mapsto {\mathbf x}_1 - {\mathbf y}\cdot \lambda'$, we find $$\begin{aligned} f_{{\mathbf y}}(\gamma_{\lambda, \mu}\cdot {\bm{\tau}})(\varphi) &= \sum_{{\mathbf x}_1 \in V} \varphi({\mathbf x}_1, {\mathbf y}) \varphi_{{\mathrm{KM}}} \left( \sqrt{v_1}( x_1 + y \cdot \lambda')+ \frac{y \cdot v'_{12} }{ \sqrt{v_1}} \right) \notag \\ & \qquad \qquad \times \left\{\prod_{i=2}^d \varphi_{\infty_i} \left( \sigma_i ({\mathbf x}_1, {\mathbf y}){\left( \begin{smallmatrix}1 & \\ \lambda' & 1 \end{smallmatrix} \right)} a_i \right) \right\} \, e \Big(T({\mathbf x}_1, {\mathbf y}) \mathrm{Re}(\gamma_{\lambda, \mu} {\bm{\tau}}) \Big) \\ &= \sum_{{\mathbf x}_1 \in V} \left\{ \varphi({\mathbf x}_1 - {\mathbf y}{\bm{\lambda}}') \, e(2 \langle {\mathbf x}_1, {\mathbf y}\rangle \mu ' - \langle {\mathbf y}\lambda ', {\mathbf y}\rangle \mu') \right\} \varphi_{{\mathrm{KM}}} \left( \sqrt{v_1} x_1+ \frac{y \cdot v'_{12} }{ \sqrt{v_1}} \right) \notag \\ & \qquad \qquad \times \prod_{i=2}^d \varphi_{\infty_i}( \sigma_i ({\mathbf x}_1, {\mathbf y})a_i) \ e( T({\mathbf x}_1, {\mathbf y})\mathbf u ) \\ &= f_{{\mathbf y}}({\bm{\tau}}) \left( \rho( \widetilde \gamma_{\lambda, \mu}) \varphi \right) \end{aligned}$$ as required. As for $\widetilde S$, recall that $\iota_F(S)$ acts on $S(V({\mathbb A})^n)$ by the partial Fourier transform in the first variable; the desired invariance follows from Poisson summation on ${\mathbf x}_1$ and straightforward identities for the behaviour of the Fourier transform under translations and dilations. Modularity II ============= In this section, we prove the modularity of the generating series $\widehat \phi_A({\bm{\tau}})$. By , we only need to consider totally positive semidefinite matrices $T_2$; assume that this is case throughout this section. We begin by fixing an element ${\mathbf y}= ({\mathbf y}_1, \dots, {\mathbf y}_{n-1}) \in \Omega(T_2)$, and setting $y = \sigma_1({\mathbf y})$. Let $$U_{{\mathbf y}} = \mathrm{span}({\mathbf y}_1, \dots, {\mathbf y}_{n-1}) \subset V,$$ so that $U_{{\mathbf y}}$ is totally positive definite. Let $$\Lambda_{{\mathbf y}} := U_{{\mathbf y}} \cap L, \qquad \text{and} \qquad \Lambda_{{\mathbf y}}^{\perp} := U_{{\mathbf y}}^{\perp} \cap L$$ and set $$\Lambda := \Lambda_{{\mathbf y}} \oplus \Lambda_{{\mathbf y}}^{\perp} \subset L,$$ so that $$\Lambda \subset L \subset L' \subset \Lambda'.$$ In light of the definition , we have a natural inclusion $S(L^n) \to S(\Lambda^n)$, and the composition $$\label{eqn:lattice embedding} S(L^n) \to S(\Lambda^n) \stackrel{\sim}{\to} S(\Lambda_{{\mathbf y}}^n) \otimes S((\Lambda_{{\mathbf y}}^{\perp})^{n}).$$ is equivariant for the action of $\widetilde \Gamma^J$, via $\rho_L$ on the left hand side, and via $\rho_{\Lambda_{{\mathbf y}}} \otimes \rho_{\Lambda_{{\mathbf y}}^{\perp}}$ on the right; this latter fact can be deduced from explicit formulas for the Weil representation, cf. [@KudlaCastle Proposition II.4.3]. Note that $U_{{\mathbf y}}^{\perp}$ is a quadratic space of signature $((p',2 ), (p'+2, 0 ), \dots (p'+2, 0)) $ with $p' = p- \mathrm{rank}(T_2)$, so the constructions in apply equally well in this case. In particular, let $X_{{\mathbf y}}({\mathbb C}) = \Gamma_{{\mathbf y}} \big\backslash {\mathbb D}^+_{y}$. Then for $m \in F$ and ${\mathbf v}_1 \in (F \otimes_{{\mathbb R}} {{\mathbb R}})_{\gg0}$, we have a special divisor $$\label{eqn:genus one Zy def} \widehat Z_{U_{{\mathbf y}}^{\perp}}(m, {\mathbf v}_1) \in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X_{{\mathbf y}}) \otimes S(\Lambda_{{\mathbf y}}^{\perp})^{\vee},$$ where we introduce the subscript $U_{{\mathbf y}}^{\perp}$ in the notation to emphasize the underlying quadratic space being considered. Let $$\pi_{{\mathbf y}} \colon X_{{\mathbf y}} \to X$$ denote the natural embedding, whose image is the cycle $Z({\mathbf y})$ of codimension ${\mathrm{rank}}(T_2)$, and define a class $$\widehat Z_{{\mathbf y}}(m, {\mathbf v}_1) \in {\widehat{\mathrm{CH}}{}^{{{ \mathrm{rk}}(T_2)+1}}}(X, {{\mathcal D}_{\mathrm{cur}}}) \otimes_{{\mathbb C}} S((\Lambda_{{\mathbf y}}^{\perp})^n)$$ as follows: suppose $\varphi \in S((\Lambda_{{\mathbf y}}^{\perp})^n) $ is of the form $\varphi_1 \otimes \varphi_2$ with $\varphi_1 \in S(\Lambda_{{\mathbf y}}^{\perp}) $ and $\varphi_2 \in S((\Lambda_{{\mathbf y}}^{\perp})^{n-1})$. Then, using the pushforward $\pi_{{\mathbf y},*}$, set $$\widehat Z_{{\mathbf y}}(m, {\mathbf v}_1)(\varphi_1 \otimes \varphi_2) := \varphi_2(0) \cdot \pi_{{\mathbf y},*} \left( \widehat Z_{U_{{\mathbf y}}^{\perp}}(m, {\mathbf v}_1,\varphi_1)\right) \ \in \ {\widehat{\mathrm{CH}}{}^{{ \mathrm{rk}}(T_2) + 1}}(X, {{\mathcal D}_{\mathrm{cur}}}),$$ and extend this definition to arbitrary $\varphi$ by linearity. Observe that the pushforward is an element of ${\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X, {{\mathcal D}_{\mathrm{cur}}})$; the existence of pushforward maps along arbitrary proper morphisms, which are not available in general for the Gillet-Soulé Chow groups, are an essential feature of the extended version, [@BurgosKramerKuhn §6.2]. Finally, for ${\bm{\tau}}= {\left( \begin{smallmatrix} {\bm{\tau}}_1 & {\bm{\tau}}_{12} \\ {\bm{\tau}}_{12}' & {\bm{\tau}}_2 \end{smallmatrix} \right)} \in {\mathbb H}_d^n$, we define the generating series $$\widehat \phi_{{\mathbf y}}({\bm{\tau}}_1) \ := \ \sum_{m \in F} \widehat Z_{{\mathbf y}}(m, {\mathbf v}_1) \, q_1^m$$ where ${\bm{\tau}}_1 \in {\mathbb H}_1^d$ with ${\mathbf v}_1 = \mathrm{Im}({\bm{\tau}}_1)$, and $q_1^m = e(m {\bm{\tau}}_1)$. There is also a classical theta function attached to the totally positive definite space $U_{{\mathbf y}}$, defined as follows: let $\varphi \in S(\Lambda_{{\mathbf y}}^n)$ and suppose $\varphi = \varphi_1 \otimes \varphi_2$ with $\varphi_1 \in S(\Lambda_{{\mathbf y}})$ and $\varphi_2 \in S(\Lambda_{{\mathbf y}}^{n-1})$. Then we set $$\theta_{{\mathbf y}} ({\bm{\tau}})(\varphi_1 \otimes \varphi_2) \ := \ \varphi_2({\mathbf y}) \sum_{\lambda \in U_{{\mathbf y}}} \varphi_1(\lambda) \ e\left( \langle \lambda, \lambda \rangle {\bm{\tau}}_1 + 2 \langle \lambda, {\mathbf y}\rangle {\bm{\tau}}_{12}' \right) \, e(T_2\cdot {\bm{\tau}}_2),$$ and again, extend to all $\varphi \in S(\Lambda_{{\mathbf y}}^n)$ by linearity. It is well-known that $\theta_{{\mathbf y}}({\bm{\tau}})$ is a holomorphic Jacobi modular form of weight $\dim U_{{\mathbf y}} / 2 = { \mathrm{rk}}(T_2)/2$ and index $T_2$, see e.g. [@EichlerZagier §II.7]. The Fourier expansion of $ \theta_{{\mathbf y}} ({\bm{\tau}})(\varphi)$ can be written, for $\varphi = \varphi_1 \otimes \varphi_2$ as above, as $$\theta_{{\mathbf y}}({\bm{\tau}})(\varphi_1\otimes \varphi_2) \ =\ \varphi_2({\mathbf y})\sum_{T = {\left( \begin{smallmatrix}* & * \\ * & T_2 \end{smallmatrix} \right)}} r_{{\mathbf y}}(T, \varphi_1) \ q^T,$$ where $r_{{\mathbf y}}(T) \in S(\Lambda_{{\mathbf y}})^{\vee}$ is given by the formula $$r_{{\mathbf y}} \left( {\left( \begin{smallmatrix}T_1 & T_{12} \\ T_{12}' & T_2 \end{smallmatrix} \right)}, \varphi_1 \right) = \sum_{\substack{\lambda \in U_{{\mathbf y}} \\ \langle \lambda, \lambda \rangle = T_1 \\ \langle \lambda, {\mathbf y}\rangle = T_{12}} } \, \varphi_1 (\lambda)$$ Finally, note that given $T$ as above, we must have either ${\mathrm{rank}}(T) = {\mathrm{rank}}(T_2) + 1$, or ${\mathrm{rank}}(T) = {\mathrm{rank}}(T_2)$. \[r vanishing non-degenerate\] Suppose ${\mathrm{rank}}(T) = {\mathrm{rank}}(T_2) + 1$. Then for any ${\mathbf y}\in \Omega(T_2)$, we have $r_{{\mathbf y}}(T) = 0.$ Suppose $r_{{\mathbf y}}(T) \neq 0$; then there exists a tuple $(\lambda, {\mathbf y}) \in \Omega(T)$ with $ \mathrm{span}(\lambda, {\mathbf y}) = \mathrm{span} ({\mathbf y})$, which contradicts the assumption on ${\mathrm{rank}}(T)$. \[PhiA Prop\] As formal generating series, we have $$\label{eqn:PhiA Prop main eqn} \widehat{ \phi}_A({\bm{\tau}}) = \sum_{T = {\left( \begin{smallmatrix} * & * \\ * & T_2 \end{smallmatrix} \right)}} \widehat A(T, {\mathbf v}) \, q^T = \sum_{\substack{ {\mathbf y}\in \Omega(T_2) \\ \mod{\Gamma}} } \widehat{ \phi}_{{\mathbf y}}({\bm{\tau}}) \cdot \widehat{ \omega}^{n - r(T_2) - 1} \otimes \theta_{{\mathbf y}}({\bm{\tau}}) ,$$ where $$\widehat{ \phi}_{{\mathbf y}}({\bm{\tau}}) \cdot \widehat{ \omega}^{n - r(T_2) - 1} \ := \ \sum_{m \in F} \widehat Z_{{\mathbf y}}(m, {\mathbf v}_1) \cdot \widehat \omega^{n-r(T_2) - 1} \, q_1^m;$$ here, we view the right hand side of as valued in $S(L^n)^{\vee}$ by dualizing . By linearity, it suffices to evaluate both sides of the desired relation at a Schwartz function $\varphi \in S(L^n)$ of the form $\varphi = \varphi_1 \otimes \varphi_2$ for $\varphi_1 \in S(L) $ and $\varphi_2 \in S(L^{n-1})$. Then we may write $$\begin{aligned} Z(T)(\varphi_1 \otimes \varphi_2) \ &= \ \sum_{\substack{{\mathbf x}\in \Omega(T) \\ \text{mod } \Gamma}} (\varphi_1 \otimes \varphi_2) ({\mathbf x}) \, Z({\mathbf x}) \\ &= \sum_{\substack{{\mathbf y}\in \Omega(T_2) \\ \text{mod } \Gamma}} \varphi_2({\mathbf y}) \sum_{\substack{ {\mathbf x}_1 \in \Omega(T_1) \\ \langle {\mathbf x}_1, {\mathbf y}\rangle = T_{12} \\ \text{mod } \Gamma_{{\mathbf y}} }} \varphi_1({\mathbf x}_1) \Gamma_{({\mathbf x}_1, {\mathbf y})} \big\backslash {\mathbb D}_{({\mathbf x}_1, {\mathbf y})}. \end{aligned}$$ We may further assume that $$\varphi_1 = \varphi_1' \otimes \varphi_1'' \in S(U_{{\mathbf y}}) \otimes S(U_{{\mathbf y}}^{\perp}) \qquad \text{ and } \qquad \varphi_2 = \varphi_2' \otimes \varphi_2'' \in S(\Lambda_{{\mathbf y}}^{n-1}) \otimes S((\Lambda_{{\mathbf y}}^{\perp})^{n-1});$$ in this case, $\varphi_2({\mathbf y}) = \varphi_2'({\mathbf y}) \varphi_2''(0)$. For a vector ${\mathbf x}_1 \in V$ as above, write its orthogonal decomposition as $${\mathbf x}_1 = {\mathbf x}_1' \ + \ {\mathbf x}_1'' \ \in \ U_{{\mathbf y}} \oplus U_{{\mathbf y}}^{\perp},$$ and note that ${\mathbb D}^+_{(x_1, y)} = {\mathbb D}^+_{(x_1'', y)}$, where $x_1 = \sigma_1({\mathbf x}_1)$, etc., and $\Gamma_{({\mathbf x}_1, {\mathbf y})} = \Gamma_{({\mathbf x}_1'', {\mathbf y})}$. Thus, decomposing the sum on ${\mathbf x}_1$ as above and writing $T ={\left( \begin{smallmatrix}T_1 & T_{12} \\ T_{12}' & T_2 \end{smallmatrix} \right)}$, we have $$\begin{aligned} \notag Z(T)& (\varphi_1\otimes \varphi_2) \\ &= \sum_{\substack{ {\mathbf y}\in \Omega(T_2) \\ \text{mod } {\Gamma}}} \varphi_2({\mathbf y}) \, \sum_{m \in F} \left( \sum_{\substack{ {\mathbf x}_1'' \in U_{{\mathbf y}}^{\perp} \\ \langle {\mathbf x}_1'', {\mathbf x}_1'' \rangle = m \\ \text{mod } \Gamma_{{\mathbf y}} }} \varphi_1''({\mathbf x}_1'') \Gamma_{({\mathbf x}_1'', {\mathbf y})} \big\backslash {\mathbb D}_{({\mathbf x}_1'', {\mathbf y})}. \right) \cdot \left( \sum_{\substack{ {\mathbf x}_1' \in U_{{\mathbf y}} \\ \langle {\mathbf x}_1' , {\mathbf x}_1' \rangle = T_1 - m \\ \langle {\mathbf x}_1', {\mathbf y}\rangle = T_{12} } } \varphi'_1({\mathbf x}_1') \right) \notag \\ &= \sum_{\substack{ {\mathbf y}\in \Omega(T_2) \\ \text{mod } {\Gamma}}} \varphi_2({\mathbf y}) \, \sum_{m \in F} \left( \sum_{\substack{ {\mathbf x}_1'' \in U_{{\mathbf y}}^{\perp} \\ \langle {\mathbf x}_1'', {\mathbf x}_1'' \rangle = m \\ \text{mod } \Gamma_{{\mathbf y}} }} \varphi_1''({\mathbf x}_1'') \Gamma_{({\mathbf x}_1'', {\mathbf y})} \big\backslash {\mathbb D}_{({\mathbf x}_1'', {\mathbf y})}. \right) \cdot r_{{\mathbf y}}\left( {\left( \begin{smallmatrix}T_1 - m & T_{12} \\ T_{12'} & T_2 \end{smallmatrix} \right)}, \varphi_1' \right). \end{aligned}$$ which we may rewrite as $$\begin{aligned} \label{eqn:Z(T) cycle relation} Z(T)\left( \varphi_1 \otimes \varphi_2 \right) &= \sum_{ \substack{ {\mathbf y}\in \Omega(T_2) \\ \text{mod } \Gamma} } \varphi_2''(0)\varphi_2'({\mathbf y}) \sum_{m } \pi_{{\mathbf y}, *} \left( Z_{U_{{\mathbf y}}^{\perp}} (m) (\varphi_1'') \right) \cdot r\left( {\left( \begin{smallmatrix}T_1 - m & T_{12} \\ T_{12'} & T_2 \end{smallmatrix} \right)}, \varphi_1' \right) \\ &= \sum_{ \substack{ {\mathbf y}\in \Omega(T_2) \\ \text{mod } \Gamma} } \sum_{m } Z_{{\mathbf y}} (m) (\varphi_1'' \otimes \varphi_2'') \cdot \left\{ \varphi_2'({\mathbf y}) \, r\left( {\left( \begin{smallmatrix}T_1 - m & T_{12} \\ T_{12'} & T_2 \end{smallmatrix} \right)}, \varphi_1' \right) \right\} \end{aligned}$$ where in the second line, $Z_{{\mathbf y}}(m)$ denotes the $S((\Lambda_{{\mathbf y}}^{\perp})^n)^{\vee}$-valued cycle $$Z_{{\mathbf y}}(m) \colon \varphi'' \mapsto \varphi_2''(0) \, \pi_{{\mathbf y},*} Z_{U_{{\mathbf y}}^{\perp} } (m,\varphi_1'').$$ Now suppose that ${ \mathrm{rk}}(T) = { \mathrm{rk}}(T_2)+1$. Then, by , the term $m=0$ does not contribute to , and so all the terms $Z_{U_{{\mathbf y}}^{\perp}}(m)$ that do contribute are divisors. To incorporate Green currents in the discussion, note that, at the level of arithmetic Chow groups, the pushforward is given by the formula $$\begin{aligned} \widehat Z_{{\mathbf y}}(m, {\mathbf v}_1)(\varphi'') &= \varphi_2''(0) \ \pi_{{\mathbf y},*} \widehat Z_{U^{\perp}_{{\mathbf y}} } (m, {\mathbf v}_1, \varphi_1'') \\ &= \left( \pi_{{\mathbf y}, *} Z_{U^{\perp}_{{\mathbf y}}}(m,\varphi_1''), \ \left[ \omega_{U_{{\mathbf y}}^{\perp}}(m, {\mathbf v}_1,\varphi_1'')\wedge \delta_{Z({\mathbf y})}, { \mathfrak{g}}^o_{U_{{\mathbf y}}^{\perp}}(m,{\mathbf v}_1,\varphi_1'') \wedge \delta_{Z({\mathbf y})} \right] \right), \end{aligned}$$ where, as before, we use the subscript $U_{{\mathbf y}}^{\perp}$ to denote objects defined with respect to that space. This may be rewritten as $$\begin{aligned} \widehat Z_{{\mathbf y}}(m, {\mathbf v}_1)(\varphi'') &= \widehat Z_{{\mathbf y}}(m)^{{\mathrm{can}}}(\varphi'') \notag \\ & \qquad + \varphi''_2(0)\left(0, \left[ \omega_{U^{\perp}_{{\mathbf y}}}(m, {\mathbf v}_1, \varphi''_1) \wedge \delta_{Z({\mathbf y})} - \delta_{Z_{{\mathbf y}}(m) } , \, { \mathfrak{g}}^o_{U_{{\mathbf y}}^{\perp}}(m,{\mathbf v}_1,\varphi''_1) \wedge \delta_{Z({\mathbf y})} \right] \right) \label{eqn:Z(y) pushforward} \end{aligned}$$ where $\widehat Z_{{\mathbf y}}(m)^{{\mathrm{can}}} = (Z_{{\mathbf y}}(m), [ \delta_{Z_{{\mathbf y}}(m) }, 0])$ is the canonical class associated to $Z_{{\mathbf y}}(m)$. Thus, $$\widehat Z_{{\mathbf y}}(m,{\mathbf v}_1) \cdot \widehat \omega ^{n-{ \mathrm{rk}}(T)} = \widehat Z_{{\mathbf y}}(m)^{{\mathrm{can}}} \cdot \widehat \omega^{n-{ \mathrm{rk}}(T)} + \left( 0, \, [\beta_{{\mathbf y}}(m, {\mathbf v}_1) , \alpha_{{\mathbf y}}(m, {\mathbf v}_1)]\right)$$ where $\alpha_{{\mathbf y}}(m,{\mathbf v}_1)$ and $\beta_{{\mathbf y}}(m,{\mathbf v}_1)$ are $S((\Lambda_{{\mathbf y}}^{\perp})^n)^{\vee}$-valued currents defined by $$\alpha_{{\mathbf y}}(m, {\mathbf v}_1)(\varphi'') \ = \ \varphi''_2(0) \ { \mathfrak{g}}^o_{U_{{\mathbf y}}^{\perp}}(m, {\mathbf v}_1,\varphi''_1) \wedge \delta_{Z({\mathbf y}) } \wedge \Omega^{n-{ \mathrm{rk}}(T)}$$ and $$\beta_{{\mathbf y}}(m, {\mathbf v}_1)(\varphi'') \ = \ \varphi''_2(0) \ \omega_{U^{\perp}_{{\mathbf y}} }(m,{\mathbf v}_1,\varphi_1'') \wedge \delta_{Z({\mathbf y})} \wedge \Omega^{n-{ \mathrm{rk}}(T)} - \delta_{Z_{{\mathbf y}}(m)(\varphi'') } \wedge \Omega^{n-{ \mathrm{rk}}(T)}$$ where $\varphi'' = \varphi''_1 \otimes \varphi''_2$ as before. Turning to the class $\widehat A(T,{\mathbf v})$, it can be readily verified that $$\widehat A(T,{\mathbf v}) = \widehat{Z(T)}{}^{{\mathrm{can}}} \cdot \widehat{ \omega}^{n-{ \mathrm{rk}}(T)} \ + \ \left( 0, [ \psi(T,{\mathbf v}) - \delta_{Z(T)} \wedge \Omega^{n-{ \mathrm{rk}}(T)}, { \mathfrak{a}}(T,{\mathbf v})] \right)$$ where the currents ${ \mathfrak{a}}(T,{\mathbf v})$ and $\psi(T,{\mathbf v})$ are defined in and , respectively. Now, by the same argument as in , and under the assumption ${\mathrm{rank}}(T) = {\mathrm{rank}}(T_2)+1$, we have (as a $\Gamma$-invariant current on ${\mathbb D}$) $$\begin{aligned} { \mathfrak{a}}(T,{\mathbf v})(\varphi_1 \otimes \varphi_2) \ &= \ \sum_{{\mathbf y}\in \Omega(T_2)} \varphi_2({\mathbf y}) \sum_{\substack{ {\mathbf x}_1 \in \Omega(T_1) \\ \langle {\mathbf x}_1, {\mathbf y}\rangle = T_{12} }} \varphi_1({\mathbf x}_1) { \mathfrak{g}}^o(\sqrt{v_1} x_1 ) \wedge \delta_{{\mathbb D}^+_{y}} \wedge \Omega^{n-r(T)} \\ &= \sum_{{\mathbf y}\in \Omega(T_2)} \varphi_2({\mathbf y}) \cdot \sum_{m \in F} \left( \sum_{\substack{ {\mathbf x}_1'' \in U_{{\mathbf y}}^{\perp} \\ \langle {\mathbf x}_1'', {\mathbf x}_1'' \rangle = m }} \varphi_1''({\mathbf x}_1'') \, { \mathfrak{g}}^o(\sqrt{v_1} x_1'') \wedge \delta_{{\mathbb D}^+_{y}} \wedge \Omega^{n-r(T)} \right) \\ & \qquad \qquad \qquad \qquad \times r\left( {\left( \begin{smallmatrix}T_1 - m & T_{12} \\ T_{12'} & T_2 \end{smallmatrix} \right)} , \varphi_1' \right) , \end{aligned}$$ where we use the fact that ${ \mathfrak{g}}^o(\sqrt{v_1} x_1) \wedge \delta_{{\mathbb D}^+_y}$ only depends on the orthogonal projection $x_1''$ of $x_1$ onto $U_{y}^{\perp} = \sigma_1(U_{{\mathbf y}}^{\perp})$. Thus, as $S(L^n)^{\vee}$-valued currents on $X$, we obtain the identity $${ \mathfrak{a}}(T,{\mathbf v})(\varphi_1 \otimes \varphi_2) = \sum_{{\mathbf y}\text{ mod } \Gamma} \sum_{m \in F} \alpha_{{\mathbf y}}(m,{\mathbf v}_1)(\varphi_1''\otimes \varphi_2'') \cdot \left\{ \varphi_2'({\mathbf y}) \, r_{{\mathbf y}}\left( {\left( \begin{smallmatrix}T_1 - m & T_{12} \\ T_{12'} & T_2 \end{smallmatrix} \right)} , \varphi_1'\right) \right\}$$ with $\varphi_i =\varphi_i' \otimes \varphi_i ''$ as above. A similar argument gives $$\psi(T,{\mathbf v})(\varphi) - \delta_{Z(T)(\varphi)} \wedge \Omega^{n-{ \mathrm{rk}}(T)} = \sum_{{\mathbf y}\text{ mod } \Gamma} \sum_{m \in F} \beta_{{\mathbf y}}(m, {\mathbf v}_1)(\varphi_1'' \otimes \varphi_2'') \cdot \left\{\varphi_2'({\mathbf y}) \, r_{{\mathbf y}} \left( {\left( \begin{smallmatrix}T_1 - m & T_{12} \\ T_{12'} & T_2 \end{smallmatrix} \right)}, \varphi_1' \right) \right\},$$ and so in total, we have $$\label{eqn:final AHat eqn} \widehat A(T,{\mathbf v})(\varphi_1\otimes \varphi_2) = \sum_{{\mathbf y}\text{ mod } \Gamma} \sum_m \widehat Z_{{\mathbf y}}(m, {\mathbf v}_1) (\varphi_1'' \otimes \varphi_2'')\cdot \widehat{\omega}^{n - { \mathrm{rk}}(T_2) - 1} \cdot \left\{ \varphi_2'({\mathbf y}) \, r_{{\mathbf y}}\left( {\left( \begin{smallmatrix}T_1 - m & T_{12} \\ T_{12'} & T_2 \end{smallmatrix} \right)} \right) (\varphi_1')\right\}$$ whenever ${\mathrm{rank}}(T) = {\mathrm{rank}}(T_2) + 1$. Now suppose ${\mathrm{rank}}(T) = {\mathrm{rank}}(T_2)$. Then for any tuple $({\mathbf x}_1, {\mathbf y}) \in \Omega(T)$, we must have ${\mathbf x}_1 \in U_{{\mathbf y}}$, and in particular, the only terms contributing to the right hand side of are those with $m=0$. On the other hand, we have $${ \mathfrak{a}}(T,{\mathbf v}) = 0, \qquad \psi(T,{\mathbf v}) = \delta_{Z(T)} \wedge \Omega^{n-{ \mathrm{rk}}(T)},$$ and hence $$\widehat A(T,{\mathbf v}) = \widehat{Z(T)}{}^{{\mathrm{can}}} \cdot \widehat \omega^{n-{ \mathrm{rk}}(T)};$$ with these observations, it follows easily from unwinding definitions that continues to hold in this case. Finally, the statement in the proposition follows by observing that the $T$’th $q$ coefficient on the right hand side of is precisely the right hand side of . \[thm:A series\] The series $\widehat \phi_A({\bm{\tau}})$ is a Jacobi modular form of weight $\kappa := (p+2)/2$ and index $T_2$, in the sense of . Fix ${\mathbf y}\in \Omega(T_2)$. By , applied to the space $U_{{\mathbf y}}^{\perp}$, there exist finitely many $\widehat z_{{\mathbf y},1}, \dots, \widehat z_{{\mathbf y},r}\in {\widehat{\mathrm{CH}}{}^{1}}_{{\mathbb C}}(X_{{\mathbf y}})$, finitely many (elliptic) forms $f_{{\mathbf y},1}, \dots, f_{{\mathbf y}, r} \in A_{\kappa}(\rho^{\vee}_{\Lambda^{\perp}_{{\mathbf y}}})$ and an element $g_{{\mathbf y}} \in A_{\kappa}(\rho^{\vee}_{\Lambda_{{\mathbf y}}^{\perp}}; D^*(X))$ such that the identity $$\sum_{m \in F} \widehat Z_{U_{{\mathbf y}}^{\perp}}(m, {\mathbf v}_1) q^m = \sum_{i = 1}^r f_{{\mathbf y}, i}({\bm{\tau}}_1) \widehat z_{{\mathbf y},i}\ + \ a( g_{{\mathbf y}} ({\bm{\tau}}_1))$$ holds at the level of $q$-coefficients; here ${\bm{\tau}}_1 \in {\mathbb H}_1^d$ and ${\mathbf v}_1 = \mathrm{Im}({\bm{\tau}}_1)$. Moreover, from the proof of , we see that $g_{{\mathbf y}}(\tau)$ is smooth on $X$. Therefore, applying and unwinding definitions, we obtain the identity $$\widehat \phi_{A}({\bm{\tau}}) \ = \ \sum_{\substack{ {\mathbf y}\in \Omega(T_2) \\ \text{mod } {\Gamma} }} \ \sum_{i=1}^r \left( F_{{\mathbf y}, i} ({\bm{\tau}}) \otimes \theta_{{\mathbf y}}({\bm{\tau}}) \right) \widehat Z_{{\mathbf y}, i} \ + \ a\left( \left( G_{{\mathbf y}}({\bm{\tau}}) \otimes \theta_{{\mathbf y}}({\bm{\tau}}) \right) \wedge \delta_{Z({\mathbf y})} \wedge \Omega^{n-{\mathrm{rank}}(T_2) - 1} \right)$$ of formal generating series, where $$\widehat Z_{{\mathbf y}, i} \ := \ \pi_{{\mathbf y}, *} \left( \widehat z_{{\mathbf y},i} \right) \cdot \widehat \omega^{n - { \mathrm{rk}}(T_2) - 1} \ \in \ {\widehat{\mathrm{CH}}{}^{n}}_{{\mathbb C}}(X, {{\mathcal D}_{\mathrm{cur}}}),$$ and we promote the elliptic forms $f_{{\mathbf y}, i}$ and $g_{{\mathbf y}}$ to $S((\Lambda_{{\mathbf y}}^{\perp})^n)^{\vee}$-valued functions by setting $$F_{{\mathbf y}, i}({\bm{\tau}})(\varphi) \ := \ \varphi_2(0) \cdot f_{{\mathbf y}, i}({\bm{\tau}}_1)(\varphi_1) , \qquad G_{{\mathbf y}}({\bm{\tau}}) = \varphi_2(0) \cdot g_{{\mathbf y}}({\bm{\tau}}_1)(\varphi_1)$$ for $\varphi = \varphi_1 \otimes \varphi_2 \in S(\Lambda_{{\mathbf y}}^{\perp}) \otimes S((\Lambda_{{\mathbf y}}^{\perp})^{n-1}) $ and ${\bm{\tau}}= {\left( \begin{smallmatrix}{\bm{\tau}}_1 & {\bm{\tau}}_{12} \\ {\bm{\tau}}_{12}' & {\bm{\tau}}_2 \end{smallmatrix} \right)}$. It remains to show that $F_{{\mathbf y},i}({\bm{\tau}})\otimes \theta_{{\mathbf y}}({\bm{\tau}})$ and $G_{{\mathbf y}}(\tau)\otimes \theta_{{\mathbf y}}({\bm{\tau}})$ are invariant under the slash operators for elements of $\widetilde \Gamma^J$; this can be verified directly using the generators – , the modularity in genus one of $f_{{\mathbf y}, i}$ and $g_{{\mathbf y}}$, and explicit formulas for the Weil representation (as in e.g. [@KudlaCastle]). [^1]: Here and throughout this paper, we will abuse notation and write $\omega$ both for the form and the current it defines. [^2]: The reader is cautioned that in [@BurgosKramerKuhn], the authors normalize delta currents and currents defined via integration by powers of $2 \pi i$, resulting in formulas that look slightly different from those presented here; because we are working with ${\mathbb C}$-coefficients, the formulations are equivalent. [^3]: Here we take the Weil representation for the standard additive character $\psi_F \colon {\mathbb A}_F / F \to {\mathbb C}$, which we suppress from the notation. [^4]: More precisely, we mean that for every smooth form $\alpha$, the function $[{ \mathfrak{g}}^o(x,\rho)](\alpha) = \int_X { \mathfrak{g}}^o(x, \rho) \wedge \alpha$ admits a meromorphic continuation in $\rho$, such that the Laurent coefficients are continuous in $\alpha$ in the sense of currents.

--- abstract: 'We apply an analysis of time-dependent spin-polarized current in a semiconductor channel at room temperature to establish how the magnetization configuration and dynamics of three ferromagnetic terminals, two of them biased and third connected to a capacitor, affect the currents and voltages. In a steady state, the voltage on the capacitor is related to spin accumulation in the channel. When the magnetization of one of the terminals is rotated, a transient current is triggered. This effect can be used for electrical detection of magnetization reversal dynamics of an electrode or for dynamical readout of the alignment of two magnetic contacts.' author: - 'Ł. Cywi[ń]{}ski' - 'H. Dery' - 'L. J. Sham' title: 'Electric readout of magnetization dynamics in a ferromagnet-semiconductor system' --- Integration of non-volatile magnetic memory into semiconductor electronics is one of the goals of spintronics. Current magnetic random-access memory (MRAM) implementations are based on all-metallic systems,[@Tehrani_IEEE00] maintaining a physical separation between the memory and the logic parts in computer architectures. Several theoretical proposals of spin transistors involving semiconductors have been put forth, [@DattaDas_APL90; @Schliemann_PRL03; @Ciuti_APL02; @Fabian_PRB04; @Flatte_APL03] but the experimental verification is yet to be made. Recently, we have proposed a system consisting of three ferromagnetic metal contacts on top of a planar semiconductor channel, [@Dery_MCT_PRB06] in which the flexibility provided by a third terminal allows to directly express the magnitude of spin accumulation in the semiconductor channel by electrical means. This system relies on currently available parameters of metal/semiconductor spin injecting structures involving Fe and GaAs . [@Hanbicki_APL02; @Hanbicki_APL03; @Adelmann_PRB05; @Crooker_Science05] In this Letter, we explore the possibility of a [*dynamical*]{} read-out scheme in a three-terminal system in Fig. 1 based on a time-dependent analysis of the [*lateral*]{} spin diffusion under the ferromagnetic contact. The lateral scale of the planar structure is set by the spin diffusion length $L_{sc}$, about 1 $\mu$m in GaAs at room temperature, the distance within which the electron spin polarization is preserved. The ferromagnetic terminals have collinear magnetizations. Bias is applied between the left (L) and the middle (M) contacts. The right contact (R) is connected to a capacitor $C$ which blocks the current in steady state. The voltage on the capacitor depends on the alignment of L and M magnetizations as well as on the spin selective properties of the R terminal. This is known as the non-local spin-valve effect [@Johnson_Science93; @Jedema_Nature01; @Ji_APL06]. In the following, we fix the M terminal magnetization, which can be realized by using an antiferromegnetic capping layer. [@Nogues_JMMM99] To change the magnetizations of L and R terminals the planar structure is augmented by a set of current-carrying lines known from MRAM devices. [@Tehrani_IEEE00] We discuss two possible modes of operation for this system. In the first mode, the magnetization of L contact is perturbed, and its dynamics is driving a current in the R contact. This leads to the possibility of an all-electrical measurement of magnetization reversal. In the second mode, the L magnetization represents a bit of memory, and the rotation of the R contact triggers a transient current, the magnitude of which is related to the relative alignment of L and M magnetizations. {height="8cm"} The spin accumulation in the channel and its connection to the alignment of the contacts is crucial for understanding of the system’s operation. The current passing between L and M electrodes is spin polarized due to the spin selectivity of thin Schottky barriers. [@Hanbicki_APL02; @Hanbicki_APL03] The amount of spin accumulation (proportional to the spin splitting of electrochemical potential $\Delta \xi$ defined below) in the semiconductor depends on whether the L and M magnetizations are parallel (P) or anti-parallel (AP). In the P case, the excess injected spin population is easily drained from the channel, while in the AP case opposite spins are more efficiently injected and extracted, leading to much larger spin accumulation. Using the notations in Fig. 1 the effective length of the active channel covered by the L and M terminals is $l$$\approx$$d$$+$$w_L$$+$$w_M$. Beneath the R contact, we then have from Ref.  approximately $\Delta \xi_{AP}/\Delta \xi_{P}$$\approx$$(2L_{sc}/l)^2$. In the steady state, when no current is flowing through R electrode, its electrochemical potential $\mu_{R}$ (having spin splitting negligible compared to the splittings in the semiconductor) depends on the spin accumulation beneath the contact, and on the direction of R magnetization relative to the reference direction of M magnet. The boundary condition [@Yu_Flatte_long_PRB02; @Dery_lateral_PRB06] connecting the electrochemical potential $\xi_{s}$ (spin $s$$=$$\pm$) with the spin current $j_{s}$ at the interface is $ej_{s}$$=$$G_{s}(\mu_{R}-\xi_{s})$, where $G_{s}$ is the barrier conductance for spin $s$. Using this, the requirement of zero net current gives $$\mu_{R} = \xi + \frac{G^{R}_{+}-G^{R}_{-}}{G^{R}_{+}+G^{R}_{-}} \, \frac{\Delta \xi}{2} \,\, , \label{eq:steady}$$ where $\xi$ is the mean potential beneath R contact and $\Delta \xi$ is its spin splitting. When the L magnetization is rotated, $\Delta \xi$ changes, and when R magnet is switched, $G^{R}_{+}$ and $G^{R}_{-}$ trade places. In both cases, perturbation of one of the magnets leads to transient currents charging the capacitor. The values of steady-state $\mu_R$ for different contact alignments are illustrated in Fig. 2 (dashed lines) with respect to the spin-resolved potentials in the channel beneath the R contact (solid lines). The perturbation of either R or L magnet changes $\mu_{R}$ between these values. If the RC time constant of the entire circuit is shorter than a time-scale of magnetization dynamics, it is possible to trace out the magnetization dynamics by electrical means. If $\Delta \xi$ is unchanged (when only R is rotated), then the signal is expected to differ strongly in magnitude depending of the L/M alignment, due to the ratio of spin splittings mentioned above. This leads to a dynamical readout of the L magnetization direction by rotating the R magnet. {height="6cm"} In contrast to previous treatments of time-dependent spin diffusion [@Rashba_APL02; @Zhang_PRB02], we treat the transport in a planar semiconductor by a method [@Dery_lateral_PRB06], in which the effect of traversing under finite width of the metal contacts and barrier capacitance are taken into account. The electrochemical potential is defined [@Yu_Flatte_long_PRB02] for spin $s$$=$$\pm$ in a non-degenerate semiconductor as $\mu_{s}$$=$$2k_{B}Tn_{s}/n_{0}$$-$$e\phi$, where $n_{s}$ is the non-equilibrium part of the spin density, $n_{0}$ is the free electron concentration, and $\phi$ is the electrostatic potential. The spin selectivity of the barrier is described by the finesse $F$$=$$(G_{+}-G_{-})/G$ with $G$$=$$G_{+}+G_{-}$. We introduce two dimensionless parameters to quantify, respectively, the total conductance of the barrier and its spin selectivity: $\alpha$$=$$2L_{sc}^{2}G/(\sigma h)$ and $\beta$$=$$\alpha F$, where $h$ is the thickness of the conducting channel (see Fig. 1) and $\sigma$ is the conductivity of the semiconductor. The $y$-average of $\mu_{s}$ over the thickness of the channel, denoted by $\xi_{s}$ is used in the transport equations. [@Dery_lateral_PRB06] In terms of the splitting $\Delta \xi $$=$$ \xi_{+}-\xi_{-}$ and the mean $\xi $$=$$ (\xi_{+}+\xi_{-})/2$ we have the spin diffusion equation $$\frac{\partial \Delta \xi}{\partial t} = D \frac{ \partial^{2} \Delta \xi}{\partial x^{2}} + \frac{\beta_{i}(t)}{\tau_{s}}(\mu_{i}-\xi) -\frac{\alpha_{i}}{2\tau_{s}} \Delta \xi - \frac{\Delta \xi}{\tau_{s}}, \label{eq:Dxi}$$ where $\mu_i$ is the electrochemical potential in the $i^{th}$ ferromagnet and $\tau_{s}$ is the semiconductor spin relaxation time. To complete the equations for $\xi$ and $\Delta \xi$, we use the excellent approximation of quasi-neutrality in the channel at all times ($n_{+}+n_{-}$$=$$0$). In steady state, the quasi-neutrality condition follows from the smallness of the ratio of Fermi screening length to spin diffusion length. [@Hershfield_PRB97] In the time-dependent case, deviations form neutrality are screened out on the scale of the dielectric relaxation time $\tau_{d}=\epsilon\epsilon_{0}/\sigma$, which is $\sim$100 fs for the semiconductor channel in our case [@Smith_Semiconductors]. For the dynamics on longer time-scales (at least tens of picoseconds), we can assume that at every time-step the quasi-neutrality is preserved. Consequently, $\xi$ is proportional to $\phi$ and it satisfies the Laplace equation with von Neumann boundary conditions related to currents at the boundaries of the channel, which in the time-dependent case include also a displacement current connected with charging of the barrier capacitance $C_{B}$. The equation for $\xi$ in the channel is then: $$\frac{ \partial^{2} \xi}{\partial x^{2}} = -\frac{\alpha_{i}}{2L_{sc}^{2}}(\mu_{i}-\xi) +\frac{\beta_{i}(t)}{4L_{sc}^2} \Delta \xi - \frac{c_{B}}{\sigma h}\frac{\partial}{\partial t} (\mu_{i}-\xi) \,. \label{eq:xi}$$ where $c_{B}$ is the barrier capacitance per unit area, and the right hand side of Eq. (\[eq:xi\]) is non-zero only under the contacts. The barrier conductances $G_{s}$ refer to the two spin directions $s$$=$$\pm$ along the quantization axis parallel to the M magnetization. During the magnetization dynamics, we employ the barrier finesse $F(t)$ value proportional to the projection of the magnetization on the quantization axis, while keeping total $G$ constant. Thus, we neglect the effects of “mixing conductance”, [@Brataas] which are expected to be small for tunneling barriers. The magnetization dynamics of $i^{th}$ contact translates into time-dependence of $\beta_{i}$, driving the spin diffusion in Eq. (\[eq:Dxi\]) and electric potential in the channel in Eq. (\[eq:xi\]). From $\xi_{s}$ we calculate the current $I_{R}$ flowing into the right contact and charging the capacitor $C$, and consequently the electrochemical potential of the R terminal $\mu_{R}$$=$$-eV_{R}$ changes according to $dV_{R}/dt$$=$$I_{R}/C$. For the electrical tracing of L magnetization dynamics, both M and R magnets should be pinned in the same direction. In the case of the dynamical readout of L/M alignment, we need to write separately the memory bit (direction of L magnet) and read by rotating the R magnet. A proper choice of different coercivities of two magnets and magnetic field pulses should allow for separate addressing. The half-selection (unintentional perturbation of magnetization) of L when rotating R should be diminished, in order not to mix the signal from the L dynamics with the readout of L alignment. {height="8cm"} For the calculations, we use the parameters of GaAs at room temperature: $\tau_{s}$$=$$80$ ps, doping $n$$=$$10^{16}$cm$^{-3}$, mobility $\nu$$=$5000 cm$^{2}$/Vs with corresponding diffusion constant $D$$=$$\nu kT/e$. The dimensions of the system are given in Fig. 1. Beneath the barriers we assume a heavily doped profile [@Hanbicki_APL02] so that the Schottky barriers are thin ($<$$10$ nm), enabling spin injection [@Rashba_PRB00]. We employ the experimentally verified [@Hanbicki_APL03] spin selectivity $G_{+}/G_{-}$$=$$2$ and take the barrier conductance to be $G$$=$$10^{4}\,$$\Omega^{-1}$cm$^{-2}$. For such barriers of 1 $\mu$m$^{2}$ area and 10 nm thickness, $R_{B}$$=$$10$ k$\Omega$ and $C_{B}$$=$$10$ fF . The external capacitance is taken as $C$$=$$40$ fF, and the resulting RC time is about 1 ns. The applied voltage $V_L$ is $0.1$ V, and the ratio of forward to reverse biased $G$ is set to 2. In Fig. 3a we present the calculated $I_{R}$ induced by reversal of the L magnet from AP to P alignment relative to M. In Fig. 3b the transient $I_{R}$ for the rotation of R occurring in 3 ns is shown. While the average current is zero, the average power of the current pulse is much higher for the L/M$=$AP than for P. Two signals of such clearly different magnitude can be easily distinguished, provided that the stronger signal is above the noise level (dominated by Johnson noise in our system). In Fig. 3b the power of AP pulse is slightly above the noise power in 0.3 GHz bandwidth. In summary, we have proposed a metal-semiconductor system in which the dynamics of one of magnets can be sensed electrically. This opens up a possibility for electrical detection of magnetization switching dynamics in buried structures, inaccessible to magneto-optical techniques. We have also discussed a possibility for dynamical read-out of magnetization direction of one of the terminals, which can be used for magnetic memory purposes. Our ideas are supported by calculations of time-dependent spin diffusion, taking into account realistic geometry of the structure. Further developments of including a scheme of all-magnetic logic gate will be presented in upcoming publications. We thank Parin Dalal for useful discussions. This work is supported by NSF DMR-0325599. [99]{} S. Tehrani, B. Engel, J. M. Slaughter, E. Chen, M. DeHerrera, M. Durlam, P. Naji, R. Whig, J. Janesky, J. Calder, IEEE Trans. Magn. [**36**]{}, 2752 (2000). S. Datta and B. Das, Appl. Phys. Lett **56**, 665 (1990). J. Schliemann, J. C. Egues and D. Loss, Phys. Rev. Lett. [**90**]{}, 146801 (2003). C. Ciuti, J. P. McGuire, and L. J. Sham, Appl. Phys. Lett **81**, 4781 (2002). J. Fabian and I. [Z]{}uti[ć]{}, Phys. Rev. B **69**, 115314 (2004). M. E. Flatt[é]{}, Z. G. Yu, E. Johnston-Halperin and D. D. Awschalom, Appl. Phys. Lett **82**, 4740 (2003). H. Dery, Ł. Cywi[ń]{}ski, and L. J. Sham, Phys. Rev. B **73** 161307(R) (2006). A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou, and A. Petrou, Appl. Phys. Lett. **80**, 1240 (2002). A. T. Hanbicki, O. M. J. van’ t Erve, R. Magno, G. Kioseoglou, C. H. Li, B. T. Jonker, G. Itskos, R. Mallory, M. Yasar, and A. Petrou, Appl. Phys. Lett. **82**, 4092 (2003). C. Adelmann, X. Lou, J. Strand, C. J. Palmstrom, and P. A. Crowell, Phys. Rev. B **71**, 121301(R) (2005). S. A. Crooker, M. Furis, X. Lou, C. Adelmann, D. L. Smith, C. J. Palmstrom, and P. A. Crowell, Science **309**, 2191 (2005). M. Johnson, Science **260**, 320 (1993). F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature **410**, 345 (2001). Y. Ji, A. Hoffman, J.E. Pearson, and S.D. Bader, Appl. Phys. Lett. **88**, 052509 (2006). J. Nogu[é]{}s and I.K. Schuller, J. Magn. Magn. Mat. **192**, 203 (1999). E.I. Rashba, Appl. Phys. Lett. **80**, 2329 (2002). S. Zhang and P.M. Levy, Phys. Rev. B **65**, 052409 (2002). H. Dery, [Ł]{}. Cywi[ń]{}ski, and L. J. Sham, Phys. Rev. B [**73**]{} 041306(R) (2006). Z. G. Yu and M. E. Flatt[é]{}, Phys. Rev. B **66**, 235302 (2002). S. Hershfield and H. L. Zhao, Phys. Rev. B **56**, 3296 (1997). R. A. Smith, [*Semiconductors*]{} (Cambridge University Press, Cambridge, England, 1978). A. Brataas. Y. V. Nazarov and G. E. W. Bauer, Phys. Rev. Lett. [**84**]{}, 2481 (2000); Eur. Phys. J. B [**22**]{}, 99 (2001). E. I. Rashba, Phys. Rev. B **62**, R16267 (2000).

--- abstract: 'We present the results of an investigation of the dredge-up and mixing during the merger of two white dwarfs with different chemical compositions by conducting hydrodynamic simulations of binary mergers for three representative mass ratios. In all the simulations, the total mass of the two white dwarfs is $\lesssim1.0~{\rm M_\odot}$. Mergers involving a CO and a He white dwarf have been suggested as a possible formation channel for R Coronae Borealis type stars, and we are interested in testing if such mergers lead to conditions and outcomes in agreement with observations. Even if the conditions during the merger and subsequent nucleosynthesis favor the production of $^{18}{\mathrm O}$, the merger must avoid dredging up large amounts of $^{16}{\mathrm O}$, or else it will be difficult to produce sufficient $^{18}{\mathrm O}$ to explain the oxygen ratio observed to be of order unity. We performed a total of 9 simulations using two different grid-based hydrodynamics codes using fixed and adaptive meshes, and one smooth particle hydrodynamics (SPH) code. We find that in most of the simulations, $>10^{-2}~{\rm M_\odot}$ of $^{16}{\mathrm O}$ is indeed dredged up during the merger. However, in SPH simulations where the accretor is a hybrid He/CO white dwarf with a $\sim 0.1~{\rm M_\odot}$ layer of helium on top, we find that no $^{16}{\mathrm O}$ is being dredged up, while in the $q=0.8$ simulation $<10^{-4}~{\rm M_\odot}$ of $^{16}{\mathrm O}$ has been brought up, making a WD binary consisting of a hybrid CO/He WD and a companion He WD an excellent candidate for the progenitor of RCB stars.' author: - 'Jan. E. Staff' - Brandon Wiggins - Dominic Marcello - 'Patrick M. Motl' - 'Wesley Even and Chris L. Fryer' - Cody Raskin - 'Geoffrey C. Clayton and Juhan Frank' title: 'The role of dredge-up in double white dwarf mergers' --- Introduction ============ R Coronae Borealis (RCB) stars are hydrogen deficient, supergiant stars, with distinguishing chemical abundance patterns and photometric properties. They are carbon rich, observed to consist mainly of helium with $1\%$ carbon by mass [@clayton96; @clayton12]. They produce clumps of dust that may obscure our view [@okeefe39], making them fade at irregular intervals by up to 8 magnitudes over a short timescale (weeks), and then gradually re-brighten on a timescale of months to years. Cool RCB stars have not been observed in binary systems, making it difficult to measure the mass. Based on stellar pulsation models, the mass is assumed to be of the order $1~{\rm M_\odot}$ [@saio08; @han98]. The oxygen isotopic ratio $^{16}\mathrm{O}$ to $^{18}\mathrm{O}$ has been found to be of the order unity in RCB stars [measured to be between 0.3 and 20: @clayton07; @garcia09; @garcia10]. In the solar neighborhood, this ratio is found to be $\sim500$ [@scott06], which is also a typical value in the Galactic interstellar medium [@wilson94]. What causes this unusually low oxygen ratio in RCB stars? In a single star, partial He burning can produce $^{18}\mathrm{O}$, but if the process continues $^{18}\mathrm{O}$ will be turned into $^{22}\mathrm{Ne}$ [@clayton05]. Likewise, proton capture processes: $^{17}{\mathrm O}({\rm p, \gamma})^{18}{\mathrm F}({\rm \beta^+})^{18}{\mathrm O}$ can also produce $^{18}{\mathrm O}$, but this process also continues ($^{18}{\mathrm O}({\rm p, \gamma})^{19}{\mathrm F}$). Two scenarios for the progenitors of RCB stars are often discussed; the final helium shell flash and the double degenerate white dwarf (WD) merger [@webbink84; @renzini90]. A variation of the double degenerate merger model is that the merger is between two He WDs. However, @zhang12 found that less than $1\%$ of RCB stars may be formed this way. No overproduction of $^{18}\mathrm{O}$ is expected in the final-flash scenario but in a WD merger, partial helium burning may take place leading to enhanced $^{18}\mathrm{O}$. In the double degenerate merger scenario, two situations are possible: a cold merger where no nucleosynthesis occurs, or a hot merger where additional nucleosynthesis takes place. Using non-LTE model atmospheres, @pandey11 argued that a cold merger (ie. no additional nucleosynthesis) could explain the observed oxygen abundances. Based on one dimensional stellar evolution models, @jeffery11 found that both cold and hot mergers could result in the oxygen surface abundances that can explain the observations. In @staff12 we presented the results of several grid-based hydrodynamics simulations of the merger of a He-WD (the donor) and a CO-WD (the accretor) with varying mass ratio but constant total mass of $\sim0.9~{\rm M_\odot}$. We found that for mass ratio $q>0.7$ ($q={\rm M_{donor}}/{\rm M_{accretor}}$), the temperature does not get sufficiently high to allow much nucleosynthesis to happen. However, for $q\lesssim0.7$, we found that a hot “shell of fire” (SOF) formed, with temperatures up to and above $2\times10^8~{\rm K}$, with lower mass ratio leading to higher temperatures. The densities in the SOF are around $10^5~{\rm g~cm^{-3}}$. Under these conditions, helium can start burning. It is therefore in the SOF that $^{18}{\mathrm O}$ can form, which we found is needed in order to get the unusual oxygen ratios seen in RCB stars. @zhang14 investigated the post-merger evolution of CO+He WD binaries, and found that to explain the $^{12}\mathrm{C}$ abundances in RCB stars, the accretor mass must be around $0.55~{\rm M_\odot}$, while the He WD must have a mass $>0.3~{\rm M_\odot}$. They also found that $^{18}\mathrm{O}$ can be produced and can survive, to account for the observed oxygen ratio. @staff12 found that under very special conditions, it was possible to achieve an oxygen ratio of $\sim4$ (comparable to observed oxygen ratios), however these conditions may be difficult to achieve in nature (they found this in their $q=0.7$ simulation, the same as the non-AMR $q=0.7$ simulation presented here, by assuming that density, temperature, etc. remains unchanged in the SOF for $\sim 100$ years). @longland11, using SPH simulations of WD mergers, also found it difficult to achieve the unusually low oxygen ratio found in RCB stars. Using an idealized post-merger configuration based on the hydrodynamic simulations of WD mergers in @staff12, @menon13 were able to reproduce the observed abundance ratios in RCB stars using a one-dimensional stellar evolution code. In @menon13 a four-zone model was assumed, consisting of the cold core of the merged object, the SOF, a thin buffer zone between them, and the relatively cold envelope. They assumed that most of the dredged up material from the accretor ended up in the buffer zone. Therefore they could find lower $^{16}\mathrm{O}$ to $^{18}\mathrm{O}$ ratios than that of @staff12, in better agreement with observations. The issue is not that $^{18}\mathrm{O}$ is not produced, but rather that much $^{16}\mathrm{O}$ is being dredged up from the accretor (the oxygen in a CO WD is mainly $^{16}\mathrm{O}$) into the SOF where the $^{18}\mathrm{O}$ can form. Hence, in order to get an oxygen ratio of order unity, as much $^{18}\mathrm{O}$ must be produced as $^{16}\mathrm{O}$ is being dredged up. Therefore we are interested in particular in how much $^{16}{\mathrm O}$ is being dredged up from the accretor to the SOF. As the $^{18}{\mathrm O}$ must be formed in the SOF, it must be brought from the SOF up into the atmosphere of the star where it is observed. However, it is implausible that only $^{18}{\mathrm O}$ is brought up from the SOF, and not $^{16}{\mathrm O}$ also present at the same place. Hence the oxygen ratio in the SOF will to some extent reflect the observed oxygen ratio[^1]. In this paper, we present the results of simulations of the merger of two WDs. We have simulated the merger of three different mass ratios: $q=0.5$, $q=0.7$, and $q=0.8$ with total mass $M_{\rm tot}<1~{\rm M_\odot}$, using three different simulation codes. Many binary WD systems are known, and also systems that will merge within a Hubble time. @kilic11 lists 12 such binary systems where at least one component is a He WD with $M<0.25~{\rm M_\odot}$. He WD must form through common envelope interactions (with an envelope ejection), and hence must be in short period binaries, as the main sequence lifetime for low mass stars that would lead to He WDs are much longer than a Hubble time. Many He WDs are in binary systems with another WD. SDSS J092345.60+302805.0 [@brown10] has one WD with mass of $0.23~{\rm M_\odot}$ and the most likely mass of the other (with an inclination of $60^\circ$) is $0.44~{\rm M_\odot}$, which will merge in $\sim 130\times10^6~{\rm yrs}$. This is very similar to our simulated system with $q=0.5$ in this paper. J1436+5010 [@mullally09] has one component with mass $M=0.24~{\rm M_\odot}$, and the other has a mass $>0.46~{\rm M_\odot}$ (but for $i=60^\circ$ the other component has mass $M=0.60~{\rm M_\odot}$) that will merge in $<100\times10^6~{\rm yrs}$, so this system may also be similar to our $q=0.5$ simulated system. @hermes12 reported on SDSS J065133.338+284423.37, a binary WD system with masses of $0.26~{\rm M_\odot}$ and $0.50~{\rm M_\odot}$ and an orbital period of 12.75 minutes, which is also close to the masses in our $q=0.5$ simulation. @nelemans05 observed five close WD binary systems. One of these, WD1013-010, has one component with mass $0.44~{\rm M_\odot}$, while the component has a mass $>0.38~{\rm M_\odot}$ with an orbital period of $0.44~{\rm days}$. These masses are not too dissimilar from our simulated $q=0.8$ system. Another binary WD system, SDSS J104336.275+055149.90, reported by @brown17, has WD with masses of $0.30~{\rm M_\odot}$ and $0.52~{\rm M_\odot}$, and is expected to merge in $20\times10^6~{\rm yrs}$. The masses in this system is not that dissimilar from our simulated $q=0.7$ system. In this paper we want to investigate the dredge-up of $^{16}\mathrm{O}$ from the core of the accretor into the SOF in more detail. We will investigate how much $^{16}{\mathrm O}$ is at densities below $\rho<10^{5.2}~{\rm g~cm^{-3}}$, and at $\rho<10^{5}~{\rm g~cm^{-3}}$ for each of the simulations in this paper. These fiducial values are chosen to represent approximately the transition between the core and the envelope of the accretor. The numerical tools used in this paper are described in section \[methodsection\]. We will compare the results of three different hydrodynamics codes, one fixed-grid-based code (See §2.1), one grid-based code with Adaptive Mesh Refinement (AMR) (See §2.2), and one SPH code (See §2.3) to test if the numerical method affects the results. We investigate three different mass ratios, $q=0.5$ with $M_{\rm tot}=0.7~{\rm M_\odot}$, $q=0.7$ with $M_{\rm tot}=0.9~{\rm M_\odot}$ [similar to one of the simulations in @staff12], and $q=0.8$ with $M_{\rm tot}=0.9~{\rm M_\odot}$. The accretor in the $q=0.5$ and $q=0.8$ are “hybrid” WDs, that is they have a CO core and a thick layer of $0.1~{\rm M_\odot}$ He on top. Perhaps this layer of He can prevent dredge-up of $^{16}\mathrm{O}$ from the core? Our results are presented in section \[resultssection\], we have a discussion of our results and of other recent work on similar topics in section \[discussionsection\], and finally we conclude in section \[conclusionsection\]. Methods {#methodsection} ======= Using three different codes, we investigate and compare three different sets of initial conditions: $q=0.7$ and $M_{\rm tot}=0.9~{\rm M_\odot}$, $q=0.5$ and $M_{\rm tot}=0.71~{\rm M_\odot}$ where the accretor is a hybrid WD of which $0.13~{\rm M_\odot}$ is $^4{\rm He}$, and $q=0.8$ and $M_{\rm tot}=0.9~{\rm M_\odot}$ where the accretor is a slightly more massive hybrid WD of mass $0.5~{\rm M_\odot}$, of which $0.13~{\rm M_\odot}$ is $^4{\rm He}$. We briefly describe the codes below, two of which are grid-based, one using AMR and one not, and the last code is a smooth particle hydrodynamics (SPH) code. We are in particular interested in how much $^{16}{\mathrm O}$ is being dredged up from the accretor into the SOF or further out in the star. In the simulations that formed a SOF [we found those with $q\lesssim 0.7$ formed a SOF in @staff12], the SOF sits on top of the core. The SOF exists at densities of $\sim10^4-10^5~{\rm g~cm^{-3}}$. For the purpose of estimating how much $^{16}{\mathrm O}$ has been dredged-up to the SOF or outside of it, experience shows the core boundary to be located at densities between $10^5~{\rm g~cm^{-3}}$ and $10^{5.2}~{\rm g~cm^{-3}}$. Furthermore, we also require the core to be at a temperature $T<10^8~{\rm K}$. Everything else is therefore in the SOF or further outside, and we estimate how much $^{16}{\mathrm O}$ exists there. This method likely underestimates the size of the core, and therefore overestimates the amount of $^{16}{\mathrm O}$ that has been dredged up from the core. To get a better handle on this, we use two different density limits to define the core. More elaborate and refined methods may be used to identify the core, however this will not significantly change the amount of $^{16}{\mathrm O}$ that we find has been dredged up from the core. In the grid-based simulations, we calculate the mass of $^{16}{\mathrm O}$ below the density limits by multiplying the mass of the gas in a cell with the mass fraction associated with the CO and dividing by 2 (since we assume half the CO mass to be $^{16}{\mathrm O}$), then summing over all cells with density below the density limits. In the SPH simulations we likewise sum the $^{16}{\mathrm O}$ mass in all SPH particles below the density limits to find the mass of $^{16}{\mathrm O}$ below the density limits. In the SPH code, we keep track of the amount of $^{16}{\mathrm O}$ in each SPH particle. In the grid-based codes, we advect two different mass fractions. One of these is assigned to the CO, the other to the He. We will assume that 50% (by mass) of the CO mass fraction is $^{16}{\mathrm O}$. The equation of state is that of a zero-temperature Fermi gas of electrons, and an ideal gas of ions. Initially, the WDs in all the simulations are assumed to have zero temperature. Heating can occur through shocks or adiabatic compression. The temperature is calculated as in @staff12: $$T=\frac{E_{\rm gas}}{\rho c_v}, \label{eq:T}$$ where $E_{\rm gas}$ is the gas internal energy and $c_v$ is the specific heat capacity at constant volume [@segretain97] given by: $$c_v=\frac{(<Z>+1)k_B}{<A>m_H(\gamma-1)}=1.24\times10^8{\rm ergs~g^{-1}~K^{-1}}= ~\frac{(<Z>+1)}{<A>}=6.2\times10^7 {\rm ergs~g^{-1}~K^{-1}} \label{eq:cv}$$ $k_B$ is Bolzmann’s constant and $m_H$ is the mass of the hydrogen atom, and we have assumed that $(<Z>+1)/<A>=0.5$, with $<Z>$ and $<A>$ being the average charge and mass for a fully ionized gas. This is approximately correct for a CO mixture, but it is an overestimate when He is present. Fixed grid simulations ---------------------- The fixed grid hydrodynamics code used in this work is the same as that used in @staff12, and an earlier version of the code was described in @motl07 and @dsouza06. In fact, the $q=0.7$ simulation is the same as in @staff12, and we include it to compare the codes with our previous work. This hydrodynamics code uses a cylindrical grid, with equal spacing between the grid cells in the radial and vertical directions. The resolution is $(r,z,\phi)=(226,146,256)$ cells for most of the simulations. In addition, we ran the $q=0.5$ hybrid simulation at a higher resolution of $(r,z,\phi)=(354,226,512)$ cells to test if the resolution plays a role in the results. We found good agreement between the two resolutions (see section \[q05subsection\] for details), and in order to conserve computer resources we therefore used the lower resolution for the other simulations. The physical size of the grid is: $r: 0-6.1\times10^9~{\rm cm}$, $z: -2.0\times10^9~{\rm cm}\text{ to }2.0\times10^9~{\rm cm}$. The simulations are full 3D, so the $\phi$ direction covers $2\pi$. The initial setup for the non-AMR simulations were also made in the same way as in @staff12, using a self consistent field code [@even09]. Two synchronously rotating WDs are constructed, so that the donor almost fills its Roche Lobe. We then artificially remove orbital angular momentum from the system at a rate of $1\%$ per orbit for a couple of orbits to force the stars into contact faster. The exact duration of the artificial angular momentum removal does not appear to affect the results [@motl16]. In the grid based simulations, we found that much CO material was found outside of the core, even at early times before the merger. Speculating that this might be a numerical artifact, we decided to “reset” the mass fractions in the hybrid accretor in the $q=0.5$ non-AMR grid-based simulation shortly before the merger, to enforce that all the CO material is in the core. We thereby ignored all CO material that had so far been dredged up, ie. we assumed all material outside of the core of the accretor was helium shortly before the merger. AMR code -------- The LSU code, Octo-tiger, a 3-D, finite-volume adaptive mesh refinement (AMR) hydrodynamics code with Newtonian gravity, is a successor to previous LSU hydrodynamics codes [@lindblom01; @ott05; @dsouza06; @motl07; @kadam16; @motl16]. Octo-tiger decomposes the spatial domain into a variable depth octree structure, with each octree node containing a single Cartesian 12 X 12 X 12 subgrid. The hydrodynamic variables are evolved using the central scheme of @kurganov00, while the gravitational field is computed using the fast multipole method (FMM) presented by @dehnen00. The AMR simulations presented here use 8 levels of refinement. These simulations were also presented in @montiel15 in order to study mass loss from WD mergers. ### Initial Conditions To generate our initial models for the AMR simulations, we used a method similar to the self consistent field (SCF) technique described by @even09. The SCF method solves the hydrostatic balance equation in the presence of gravity, $$h + \Psi = \Psi_0,$$ where $\Psi_0$ is a constant unique to each star. The isentropic enthalpy, $h$, is defined as $$\label{hbal} h\left[\rho\right] = \int_0^{P=P\left[\rho\right]} \frac{dP'}{\rho'}.$$ For the zero temperature WD equation of state, $$h\left[\rho\right] = \frac{8 A}{B}\sqrt{ \left(\frac{\rho}{B}\right)^\frac{2}{3} + 1} .$$ @even09 require the choosing of two boundary points for the donor star, each on the line of centers between the stars and on opposite sides of the donor. For our initial model, instead of fixing the boundary point closest to the accretor in space, we define it to be the L1 Lagrange point. This sets $\Psi_0$ for the accretor to $\Psi_{L1} + h\left[0\right]$, where $\Psi_{L1}$ is the effective potential at the L1 point. This ensures the donor Roche lobe is filled. The donor WD is taken to be 100 % helium. For hybrid accretor models, the core is taken to contain an evenly distributed mixture of equal parts of carbon and oxygen, while the envelope is 100 % helium. The non-hybrid accretor contains equal parts carbon and oxygen throughout. Because the discretization used for the initial conditions and the discretization that results from writing the time invariant version of the semi-discrete evolution equations are not exactly the same, the initial model is not in exact equilibrium when it begins evolving. As a result, the outer edges of each star diffuse slightly at the very beginning of the simulation. In the case of the donor, this causes Roche lobe overflow, leading to mass transfer. SPH code -------- Smoothed particle hydrodynamic (SPH) merger calculations were carried out in SNSPH [@fryer06]. The code contains a highly scalable hashed oct-tree data structure [@warren95] to support efficient neighbor finding and is coupled with a multipole expansion to calculate the gravitational potential and was run with a traditional SPH scheme with the cold WD EOS and standard ($\alpha = 2, \beta = 1$) artificial viscosity parameters. Our SPH simulations contain the same initial data as the grid-based calculations, being generated from the self-consistent-field (SCF) code. Initial particle distributions were prepared with the method in @diehl15 which converts cylindrical grid-based data into a particle representation and iteratively re-arranges particles to recover a distribution reminiscent of Weighted Voronoi Tessellation (WVT) initial conditions. Setups for each pair of stars contained $\sim 20$M particles of nearly equal mass[^2]. Each star in the WD binary system was subsequently allowed to relax individually for a short period of time prior to the merger calculation to prevent stellar oscillations and premature WD heating. Merger calculations were run on 256 cores and required approximately 3 weeks ($\sim100,000$ CPU hours) to run to completion. White dwarf binaries were allowed to orbit about a half-dozen times before the stars were driven into contact by removing a small amount ($1\%$) of the system’s angular momentum during each orbit. Once mass transfer was initiated, angular momentum was no longer extracted from the system. Calculations were run out to $\sim 5$ orbital periods post-merger. Formation of hybrid WDs ----------------------- The scenario forming a hybrid WD in a short period binary with a He WD is complex and therefore it is valuable to review it in detail here to show that such objects can form, and that they do so in binaries. @rappaport09 outlined how this may happen, and here we just briefly summarize the scenario they discussed for Regulus: A binary system consists of two main sequence stars in a binary with an orbital period of $\sim40$ hours and with masses $2.1~{\rm M_\odot}$ for the primary and $1.74~{\rm M_\odot}$ for the companion, that eventually will become the hybrid WD. The more massive primary evolves off the main sequence first, transferring its envelope mass to the companion, which also causes the period to expand to $\sim40~{\rm days}$, which corresponds to a separation of $\sim76~{\rm R_\odot}$. This way, a $\sim3.4~{\rm M_\odot}$ star is orbited by a $\sim0.3~{\rm M_\odot}$ He WD. It is crucial that the separation is of this size, since that will cause the $3.4~{\rm M_\odot}$ star to overflow its Roche lobe when it is near the tip of the red giant branch. At this point it has a He core of roughly $0.48~{\rm M_\odot}$. The mass transfer leads to a common envelope, and if this interaction ejects the envelope before the helium core merges with the He WD, a short period binary results consisting of the He WD and the helium core. The helium core is sufficiently massive that it will ignite, and @rappaport09 found that the period should be larger than about 80 minutes or else the helium star would overflow its Roche lobe while still burning helium in the center. Once core helium burning ceases, the star will cool and contract to form a hybrid He/CO WD. Emission of gravitational waves will bring the WDs into contact, with the bigger, less massive He WD as the donor. Other masses than those discussed in @rappaport09 can likely also lead to a binary consisting of a hybrid He/CO WD and a He WD, with the required periods adjusted accordingly. However, since some fine-tuning is needed in order to get the correct separation following both the initial mass transfer that created the He WD, and following the common envelope interaction that exposed the He core, it is likely that this is a rare process. The formation rate of RCB stars is not known. Assuming that RCB stars are formed by the merger of a CO and an He WD, @karakas15 found an RCB birthrate of $1.8\times10^{-3}~{\rm yr}^{-1}$. @brown16 found a similar merger rate for CO+He WDs. Depending on the RCB lifetime this can mean that there are a few hundred RCB stars in the galaxy, in agreement with estimates in @lawson90, but less than the several thousands estimated in @han98. In any case, considerable uncertainty surrounds the number of RCB stars in the galaxy, their lifetime, and hence their birthrate. It is therefore not unthinkable that a rare process like the merger of a hybrid He/CO WD with a CO WD could be the formation channel of RCB stars. Results {#resultssection} ======= Non-hybrid accretor, $M_{\rm tot}\approx 0.9 M_\odot$, $q=0.7$ -------------------------------------------------------------- ![The results of the $q=0.7$ non-AMR grid simulation [this simulation was also presented in @staff12] showing equatorial slices in the left column and slices perpendicular to this in the right column. The top row shows logarithm of density, the middle row shows temperature, and the bottom row shows the accretor mass fraction. The blob is clearly visible to the right of the core in the equatorial plots. The perpendicular slices in the right column stretch from the center of the grid to $3.6\times10^9~{\rm cm}$, from $-1.7\times10^9$ to $1.7\times10^9~{\rm cm}$ in the vertical direction, and are made through the blob.[]{data-label="jan07"}](f1.pdf){width="85.00000%"} ![The results of the $q=0.7$ non-hybrid SPH simulation, showing equatorial slices in the left column and slices perpendicular to this through the middle of the core in the right column. The top row shows logarithm of density, the middle row shows temperature, and the bottom row shows the $^{16}{\mathrm O}$ mass fraction. A weak blob-like feature is visible to the right of the merged core in the equatorial plots, and the vertical slices are made through this.[]{data-label="brandon07"}](f2.pdf){width="\textwidth"} ![The results of the $q=0.7$ AMR grid simulation. [*Upper panel:* ]{} Logarithm of density, [*middle panel:* ]{} temperature, [*lower panel:* ]{}CO mass fraction. The left image in each panel shows the equatorial slice, while the right image shows a slice perpendicular to it, through the middle of the grid, taken through the blob, which is visible to the left of the merged core.[]{data-label="dominic07"}](f3.pdf){width="90.00000%"} The results from the grid-based, non-AMR simulations of the $q=0.7$, $M_{\rm tot}=0.9~{\rm M_\odot}$ setup from @staff12 are shown again here in an equatorial slice and a slice perpendicular to the orbital plane in Fig. \[jan07\] [the time of the snapshot is slightly different from that in @staff12]. We find that there is $0.09 M_\odot$ of accretor material at densities $\rho<10^{5.2}~{\rm g~cm^{-3}}$ following the merger, or $0.07~{\rm M_\odot}$ of accretor material at densities $\rho<10^{5}~{\rm g~cm^{-3}}$. If we assume half of this to be $^{16}{\mathrm O}$, we get that there is $0.035-0.045~{\rm M_\odot}$ of $^{16}{\mathrm O}$ outside of the merged core immediately following the merger. The amount of $^{16}{\mathrm O}$ below these densities grows following the merger and we find $0.055-0.07~{\rm M_\odot}$ of $^{16}{\mathrm O}$ below these densities at the end of the simulation. We find temperatures up to $1.5\times10^8~{\rm K}$ in the SOF. The maximum density in the merged core is $\sim10^6~{\rm g~cm^{-3}}$. Figure \[brandon07\] shows the density, temperature, and $^{16}{\mathrm O}$ mass fraction of the $q=0.7$ non-hybrid SPH simulation. Very little of the $^{16}\mathrm{O}$ that is dredged up from the accretor ends up in the equatorial plane. Most of the helium from the donor is found in the equatorial plane. We find $\sim0.01~{\rm M_\odot}$ of $^{16}{\mathrm O}$ at densities $\rho<10^{5}~{\rm g~cm^{-3}}$. Likewise, we find $\sim0.02~{\rm M_\odot}$ of $^{16}{\mathrm O}$ at densities $\rho<10^{5.2}~{\rm g~cm^{-3}}$. We find temperatures above $2.5\times10^8~{\rm K}$ in the SOF in this SPH simulation, and densities $3\times10^6~{\rm g~cm^{-3}}$ in the merged core. In the $q=0.7$ AMR simulations (Fig. \[dominic07\]) we find about $0.055~{\rm M_\odot}$ ($1.1\times10^{32}~{\rm g}$) of CO material at densities $\rho<10^5~{\rm g~cm^{-3}}$, and about $0.075~{\rm M_\odot}$ ($1.5\times10^{32}~{\rm g}$) of CO material at densities $\rho<10^{5.2} ~{\rm g~cm^{-3}}$ following the merger. Again assuming half of this is $^{16}{\mathrm O}$, we get that roughly $0.03-0.04~{\rm M_\odot}$ of $^{16}{\mathrm O}$ is outside of the core following the merger. This is quite similar to what we found in the non-AMR simulations. It is about a factor 2 more than what we found in the SPH simulations. We find that the temperatures barely reach $1\times10^8~{\rm K}$ in the SOF, and densities in the merged core reach $2\times10^6~{\rm g~cm^{-3}}$. In Fig. \[q07amrcofrac\] we show the mass of CO material at densities below $\rho<10^5~{\rm g~cm^{-3}}$ and $\rho<10^{5.2}~{\rm g~cm^{-3}}$ as a function of time in all of the simulations. As this is not a hybrid simulation, even initially there is CO material at these relatively low densities in the grid-based simulations, but not in the SPH simulation, showing that the density structure is slightly different between the simulations. We note that when we transform the SPH data to a grid, we also find similar amount of CO material at these lower densities. Interestingly, during the merger the CO core gets squeezed sufficiently that the amount of CO material below these densities drops in the AMR simulation. We do not see this squeezing in the non-AMR simulations or the SPH simulations. However, the merger leads to dredge up, and following the merger there is $50-80\%$ more CO material at lower densities than initially. ![The mass of oxygen at densities below $\rho<10^5~{\rm g~cm^{-3}}$ (left panel) and $\rho<10^{5.2}~{\rm g~cm^{-3}}$ (right panel) as a function of time in the $q=0.7$ non-AMR simulation (green curve), AMR simulation (red curve), SPH simulation (blue curve), and SPH simulation mapped to a grid (purple curve).[]{data-label="q07amrcofrac"}](f4.pdf){width="50.00000%"} Hybrid accretor, $M_{\rm tot}=0.71 M_\odot$, $q=0.5$ {#q05subsection} ---------------------------------------------------- ![High resolution non-AMR grid-based hybrid simulation with $q=0.5$, showing equatorial slices in the left column and slices perpendicular to this in the right column. The top row shows logarithm of density, the middle row shows temperature, and the bottom row shows the accretor mass fraction. The blob is clearly visible to the left of the core in the equatorial plots. The perpendicular slices in the right column stretches from the center of the grid to $5\times10^9~{\rm cm}$ in the radial direction, and from $-2\times10^9$ to $2\times10^9~{\rm cm}$ in the vertical direction, and it is made through the blob.[]{data-label="jan05hybrid"}](f5.pdf){width="85.00000%"} In the high resolution non-AMR grid simulation, we find that $\sim0.04 M_\odot$ accretor material is at $\rho<10^{5.2}~{\rm g/cm^3}$ ($\sim 0.02 M_\odot$ at $\rho<10^5~{\rm %0.024 Msun actually g/cm^3}$). If we assume that the accretor consists of equal amounts of $^{16}\mathrm{O}$ and $^{12}\mathrm{C}$, then half of this is $^{16}\mathrm{O}$. We also find that this amount keeps increasing with time past the merger event, indicating that there is some artificial diffusion of the mass fractions in our non-AMR grid-based simulations. In the merged core, we find densities of $\sim1.7\times10^6~{\rm g/cm^3}$, and temperatures in the SOF reaches $1.6\times10^8~{\rm K}$ in the high resolution simulation. For comparison, the amount of accretor mass below the threshold densities are $0.05~{\rm M_\odot}$ for $\rho<10^{5.2}~{\rm g/cm^3}$ and $\sim 0.03 M_\odot$ at $\rho<10^5~{\rm g/cm^3}$ for the lower resolution simulation. In the lower resolution simulations we also found densities up to $\sim1.5\times10^6~{\rm g/cm^3}$ and sustained temperatures in the SOF up to $\sim1.8\times10^8~{\rm K}$. Using the amount of accretor material at densities below a certain threshold to compare resolutions is difficult, since the amount of accretor material keeps increasing with time following the merger (see Fig. \[q07amrcofrac\]). Since the merger is a drawn out process, that alone can not be used as a measure of time. The accretor masses quoted here are therefore found 1.5 orbits after the last frame that showed a density maximum for the donor. Using the temperature to compare simulations with different resolutions is also inaccurate, since the temperature can fluctuate quite a bit. The temperatures quoted are therefore temperatures that we found could be sustained for some time in the SOF, but the exact value is somewhat subjective. The central density is resolution dependent, since a higher resolution better resolves also the central region, where the density increases towards the center. Nevertheless, with all this in mind we judge that there is reasonably good agreement between the simulations with different resolution. The accretor material outside of the core is predominantly located in or around the equatorial plane (see Fig. \[jan05hybrid\]). It encapsulates the blob (see below), and very little accretor material is being pushed vertically in this simulation. The majority of the dredge-up has happened in or around the equatorial plane. As in all the other grid-based non-AMR simulations that we have performed, we find a donor-material rich, cold blob forming outside of the high density core (Fig. \[jan05hybrid\]). This blob appears to be some of the last of the donor material to fall onto the accretor, but it manages to stay together and not diffuse out. Over time, however, its size does decrease, perhaps in part because of the artificial diffusion of the accretor and donor mass fractions. ![Log density (top panel), temperature (middle panel), and CO mass fraction (bottom panel) for the $q=0.5$ hybrid simulation with the AMR grid code, taken in the equatorial slice (left frames), and perpendicular to it (right frames). The blob is visible to the right and above the merged core. The perpendicular slices are made through the lower part of the blob at $y=0~{\rm cm}$.[]{data-label="dominic05hybridcomp"}](f6.pdf){width="95.00000%"} In Fig. \[dominic05hybridcomp\] we show the results of the AMR simulation for the hybrid simulation with $q=0.5$ and total mass of $0.758~{\rm M_\odot}$. Again, as in the other grid-based simulations, we find a cold, higher density, donor rich blob outside of the merged core (sitting at the upper left of the core in Fig. \[dominic05hybridcomp\]). In fact, the AMR simulation looks very similar to the non-AMR simulation in many ways. There are “fingers” of donor material “attacking” the core before and during the mergers. However, contrary to the non-AMR simulation, this does not lead to much contamination of the CO core, which maintain a high CO fraction of about $0.9$ after the merger. We find densities up to $2\times10^6~{\rm g/cm^3}$. We again find a SOF surrounding the core, with temperatures up to $2\times10^8~{\rm K}$. As in the non-AMR simulations, the SOF contains a near equal mix of accretor and donor material. In the AMR grid simulation, we find that the amount of accretor material at densities $\rho<10^5~{\rm g/cm^3}$ increases from $0.02~{\rm M_\odot}$ immediately before the merger to $0.06~{\rm M_\odot}$ at the end of the simulation, while at densities below $\rho<10^{5.2}~{\rm g/cm^3}$ it climbs from about $0.04~{\rm M_\odot}$ immediately before the merger to about $0.07~{\rm M_\odot}$ at the end of the simulation. If we again assume that the accretor material consists of equal amounts of $^{16}\mathrm{O}$ and $^{12}\mathrm{C}$, then half of this is $^{16}\mathrm{O}$. We note that this number is quite close to the number found in the non-AMR simulation (remembering that in the non-AMR simulation the CO mass outside of these densities had been artificially set to zero shortly before the merger), which gives us some confidence in this result. Of interest is that as soon as mass transfer starts prior to the merger, the CO mass at low densities rapidly grows and reaches a plateau-value of about $0.05~{\rm M_\odot}$ for $\rho<10^5~{\rm g/cm^3}$ and $0.09~{\rm M_\odot}$ for $\rho<10^{5.2}~{\rm g/cm^3}$. This is already much too high to explain the oxygen ratios in RCBs. ![SPH simulations with hybrid accretor and $q=0.5$ mass ratio, showing equatorial slices in the left column and slices perpendicular to this through the middle of the core in the right column. The top row shows logarithm of density, the middle row shows temperature, and the bottom row shows the $^{16}{\mathrm O}$ mass fraction. The merged object is very axisymmetric, and no blob is visible. The vertical slices are made at $y=0~{\rm cm}$.[]{data-label="brandonq05hybrid"}](f7.pdf){width="\textwidth"} Figure \[brandonq05hybrid\] shows the logarithm of the density, the temperature, and the $^{16}{\mathrm O}$ mass fraction, in the equatorial plane and perpendicular to it for the SPH simulation with $q=0.5$ and a hybrid WD accretor. We find no $^{16}{\mathrm O}$ at densities below $\rho<10^{5.2}~{\rm g/cm^3}$. We find densities up to $3\times10^6~{\rm g/cm^3}$ in the merged core. An SOF did form, with maximum temperature of $\sim2.0\times10^8~{\rm K}$. There is no blob formed in this simulation, and indeed we see that the resulting object is quite symmetric. Hybrid accretor, $M_{\rm tot}\approx 0.9 M_\odot$, $q=0.8$ ---------------------------------------------------------- ![Non-AMR grid-based high-resolution hybrid simulation with $q=0.8$, showing equatorial slices in the left column and slices perpendicular to this in the right column. The top row shows logarithm of density, the middle row shows temperature, and the bottom row shows the accretor mass fraction. The blob is visible to the right of the core in the equatorial plots. The perpendicular slices in the right column stretch from the center of the grid to $3.8\times10^9~{\rm cm}$, and are made through the blob.[]{data-label="jan08hybrid"}](f8.pdf){width="85.00000%"} In the non-AMR grid-based simulations, we find densities up to $3.2\times10^6~{\rm g~cm^{-3}}$. We show the logarithm of the density, the temperature, and the accretor fraction in the equatorial slice and in a slice perpendicular to it in Fig. \[jan08hybrid\]. In @staff12 we found that high-q simulations do not have a SOF as this is destroyed in the merger of the two cores, and a similar thing occurs in the grid-based non-AMR $q=0.8$ simulation we present here. The hottest regions at high densities in this simulation are a few spots deep inside the high-density merged core that reach about $1.25\times10^8~{\rm K}$. Surrounding the core there is, however, a hot shell with temperatures up to $10^8~{\rm K}$, although in the equatorial plane it is noticeably cooler. As a result of the violent core merger, much $^{16}{\mathrm O}$ is dredged-up. We find that $0.06 M_\odot$ of accretor material is at $\rho<10^5~{\rm g/cm^3}$ ($0.087 M_\odot$ of accretor material at $\rho<10^{5.2}~{\rm g/cm^3}$). Again assuming that half of this is $^{16}{\mathrm O}$, then about $0.03M_\odot$ of $^{16}{\mathrm O}$ is outside of $\rho<10^5~{\rm g/cm^3}$. ![Log density (top panel), temperature (middle panel), and CO mass fraction (bottom panel) for the q=0.8 hybrid simulation with the AMR grid code, taken in the equatorial slice (left frames), and perpendicular to it (right frames). The vertical slices are made through the blob, which is visible below the merged core in the equatorial slices.[]{data-label="dominic08"}](f9.pdf){width="95.00000%"} In Fig. \[dominic08\] we show the logarithm of density, accretor mass fraction, and temperature, in the equatorial slice and a slice perpendicular to it for the grid-based AMR simulations for $q=0.8$ and with a total mass of $1.01~{\rm M_\odot}$. The results compare well with the non-AMR grid-based simulations. We find densities up to $5\times10^6~{\rm g~cm^{-3}}$, and temperatures up to $8\times10^7~{\rm K}$, which is found in a “hot spot” in the merged core (as in the non-AMR simulation). The merged core contains much donor material as a result of a core merger, and the accretor rich part of the core is highly distorted and not spherical. There is no clear SOF, although there is a region surrounding the merged core with a slightly higher temperature ($T\sim5\times10^7~{\rm K}$), at a density of $10^4 - 10^5~{\rm g~cm^{-3}}$ in the equatorial plane. As in the non-AMR simulation, the equatorial plane is noticeably cooler. Also as in the non-AMR grid-based simulation, we find a cold, donor rich blob outside of the merged core. Even before the merger, much accretor mass is at densities below the density thresholds. Following the merger, we find $0.08~{\rm M_\odot}$ of CO mass at densities below $10^{-5}~{\rm g~cm^{-3}}$, and $0.12~{\rm M_\odot}$ at densities below $10^{-5.2}~{\rm g~cm^{-3}}$, corresponding to $0.04-0.06~{\rm M_\odot}$ of $^{16}{\mathrm O}$ in the SOF or outside. This is slightly more than what we found in the non-AMR grid-based simulations. The SPH simulation is very different. There was no core-merger, and the accretor core remains very much like the original accretor core, with the donor material mostly smeared out around it. Contrary to the $q=0.5$ hybrid SPH simulation, we do find that some $^{16}{\mathrm O}$ has been dredged up in the $q=0.8$ hybrid SPH simulation. We find that $5\times10^{-5} M_\odot$ of $^{16}{\mathrm O}$ is at $\rho<10^{5.2}~{\rm g/cm^3}$ ($2\times10^{-5} M_\odot$ of $^{16}{\mathrm O}$ at $\rho<10^5~{\rm g/cm^3}$). With an SPH particle mass of $\sim4.5\times10^{-8}~{\rm M_\odot}$, this means that $\sim10^3$ CO SPH particles have been dredged up from the CO core of the hybrid accretor. The logarithm of density, temperature, and $^{16}{\mathrm O}$ mass fraction is shown in Fig. \[brandon08hybrid\]. A clear SOF formed around the core also in this simulation, reaching temperatures above $2\times10^8~{\rm K}$, much higher than the highest temperatures found in the grid-based simulations. As in the grid-based simulations, the equatorial plane is noticeably cooler than the rest of the SOF. The highest density found in this SPH simulation is $2\times10^6~{\rm g~cm^{-3}}$. Of interest is also that a blob does form, and is clearly visible in Fig. \[brandon08hybrid\] to the right of the merged core in that figure. It has the same features as in the grid-based simulations, that it is colder and has higher density than the surroundings. Because so little $^{16}{\mathrm O}$ has been dredged up, no clear feature can be seen in the $^{16}{\mathrm O}$ mass fraction plots. This blob persists to the end of the simulation, but it gets gradually smeared out over time, as in the grid-based simulations. ![SPH based hybrid simulation with $q=0.8$, total mass of $0.9 M_\odot$, showing equatorial slices in the left column and slices perpendicular to this through the middle of the core. The top row shows logarithm of density, the middle row shows temperature, and the bottom row shows the $^{16}{\mathrm O}$ mass fraction. A blob is visible to the right of the merged core in the equatorial slices, and the vertical slices are taken through this blob.[]{data-label="brandon08hybrid"}](f10.pdf){width="\textwidth"} Discussion {#discussionsection} ========== In this paper we are interested in estimating the amount of dredged-up CO material from the accretor in mergers between a normal CO WD or a hybrid CO/He WD and a He WD. In @staff12 we presented five hydrodynamics simulations of the merger of a CO WD with a He WD with a range of mass ratios between $q=0.5$ and $q=0.99$ and a total mass of $M_{\rm tot}=0.9 M_\odot$. We found that in all cases much $^{16}\mathrm{O}$ was dredged up, making it difficult to produce the observed oxygen ratio of order unity in RCB stars during the dynamic merger phase. In that paper, we also speculated that the amount of dredged-up $^{16}\mathrm{O}$ could be limited if the accretor is a “hybrid” CO/He WD [@rappaport09; @iben85], ie. a WD that has a CO core with a thick helium layer on top of it. We have performed two simulations with such a hybrid CO/He WD accretor, one with a $0.48 M_\odot$ accretor and a $0.24 M_\odot$ He WD donor (ie. a mass ratio $q=0.5$), and one with accretor mass of $0.5~{\rm M_\odot}$ and donor mass of $0.4~{\rm M_\odot}$ (mass ratio of 0.8). We find that in both of these simulations, thanks to the thick outer layer of He on the accretor, less $^{16}\mathrm{O}$ is being dredged up than in simulations with a pure CO WD accretor. In the $q=0.5$ SPH simulation we even find that absolutely no $^{16}\mathrm{O}$ is found at lower densities following the merger. ![Richardson number (color) and density contours in the equatorial plane for the $q=0.5$ SPH simulation (left panel) and the $q=0.8$ SPH simulation (right panel). Black color indicates $R_i \geq 0.5$. The snapshots are taken shortly before the final disruption of the donor. The dashed line indicates the boundary of the CO core in the hybrid accretor, which has a radius of $5.7\times10^8~{\rm cm}$ in the $q=0.5$ simulation and a radius of $6.3\times10^8~{\rm cm}$ in the $q=0.8$ simulation. The radius of the CO core remains reasonably constant in both simulations. The axes are in centimeters.[]{data-label="richardsonnumber"}](f11a.pdf "fig:"){width="45.00000%"} ![Richardson number (color) and density contours in the equatorial plane for the $q=0.5$ SPH simulation (left panel) and the $q=0.8$ SPH simulation (right panel). Black color indicates $R_i \geq 0.5$. The snapshots are taken shortly before the final disruption of the donor. The dashed line indicates the boundary of the CO core in the hybrid accretor, which has a radius of $5.7\times10^8~{\rm cm}$ in the $q=0.5$ simulation and a radius of $6.3\times10^8~{\rm cm}$ in the $q=0.8$ simulation. The radius of the CO core remains reasonably constant in both simulations. The axes are in centimeters.[]{data-label="richardsonnumber"}](f11b.pdf "fig:"){width="45.00000%"} We have calculated the Richardson number (see the appendix) in the hybrid SPH simulations, and found that indeed the low Richardson numbers are only in the helium layer (see Fig. \[richardsonnumber\]). A low Richardson numbers indicates that there is sufficient free kinetic energy in the velocity sheer to overturn the fluid. Hence, only helium is being dredged up from the hybrid accretor, to mix in with helium from the donor. Though in the $q=0.8$ simulation, we see that there are low Richardson numbers near the CO core boundary around where the stream impacts the accretor, and it is not unthinkable that at another time these high Richardson numbers could be found slightly deeper, explaining how a small amount of $^{16}\mathrm{O}$ is eventually found at low densities in this simulation. However, it turns out that in the grid-based hybrid simulations, even before the merger, much CO material was found outside the CO core of the accretor despite the overlying helium layer. Within roughly one orbit, several percent of a solar mass of accretor material is found at densities below the threshold densities. It may be that there is artificial (numerical) diffusion of CO accretor material out into the SOF, and therefore we decided to reset the mass fractions to artificially ensure that all the CO material was in the core (as discussed in section 2.1). Even with this numerical trick, we find that much CO material is still being brought to lower densities during the merger. With the hybrid accretor and $q=0.5$ in the grid-based simulations, we find that $0.01-0.03~{\rm M_\odot}$ of $^{16}{\mathrm O}$ finds its way out of the high density core and to the SOF. We did not do this resetting of mass fractions in the $q=0.8$ simulation, in part explaining why we find more $^{16}{\mathrm O}$ at lower densities in that simulation. However, due to the violent core merger occurring in this case[^3], it is not surprising that we find that much material from the core is being brought out to lower densities. The $q=0.8$ SPH simulation did not result in such a violent core merger, and instead the merger behaved more like the lower q simulations, where the donor material more gently puts itself on top of the accretor. Furthermore, secondary effects (for instance squeezing of the accretor due to the impact of the accretion stream) also do not lead to dredge up of CO material from deep in the core of the accretor in the SPH simulations. The SPH simulations do not suffer from the artificial diffusion of the mass fractions that we suspect may cause the large dredge-up effect in the grid based codes, since the mass fractions are inherent to each SPH particle. The SPH simulations may, however, underestimate the amount of dredge up, since in these simulations the smallest mass unit that can be dredged up is one SPH particle. One SPH particle is slightly less than $10^{26}~{\rm g}$, depending on the simulation. We do emphasize that dredge up is possible also in the SPH simulations, as observed in the $q=0.7$ and $q=0.8$ SPH simulations. An important difference between SPH and grid-based simulations is the treatment of viscosity in the two types of codes, and this may well affect the amount of mass being dredged up. The SPH code only applies viscosity in converging flows, while the grid-based codes applies viscosity over all discontinuities and local extrema. This would mean, for example, that the SPH code does not apply viscosity at contact discontinuities, while the grid based codes do. Another aspect that affects the amount of $^{16}{\mathrm O}$ being dredged up is our assumption that the CO WD contains 50% C and 50% O, uniformly distributed throughout the star (or the CO core of the hybrid stars). In fact, CO WDs likely contains more $^{16}{\mathrm O}$ than $^{12}{\mathrm C}$, but it will not be uniformly distributed in the star. The internal $^{16}{\mathrm O}$ profile depends on the rate of the $^{12}{\mathrm C}(\alpha,\gamma)^{16}{\mathrm O}$ reaction [@salaris97]. This leads to oxygen rich cores surrounded by a carbon rich layer [see a model of a $0.58~{\rm M_\odot}$ CO WD in @staff12]. By using that model, instead of assuming 50% C and 50% O uniformly distributed, we found that the amount of $^{16}{\mathrm O}$ in the SOF was reduced a factor of $\sim2$ in the $q=0.7$ simulation (the difference was smaller in the other models that we used). Hence the amount of $^{16}{\mathrm O}$ that we find (which is listed in Table \[jansimrestable\]) might be overestimated by up to a factor of $\sim2$. The resulting density distribution looks similar between the non-AMR and AMR grid-based simulations, but the grid-based simulations do not show a separation of donor and accretor material in the same way as we found in the $q=0.7$ SPH simulation. In all the grid simulations (see e.g. Fig. \[jan07\]) we find the blob [discussed in @staff12], which is donor rich. In some of the SPH simulations a blob-like structure is visible for a short time after the merger, but this disappears over time and the core appears quite axisymmetric at the end of the simulation. This feature is visible around “3-4 o’clock” from the merged core in the equatorial slice in Fig. \[brandon07\], as a colder and slightly denser structure. It is also visible in the vertical density plot, as the $10^4-10^5~{\rm g~cm^{-3}}$ contour level is elongated to the right. Since very little accretor material ends up in the equatorial plane in this SPH simulation, this blob-like structure is not associated with a gas that is poorer in accretor material than the surroundings, as we see in the grid-based simulations. Whether this is the reason for the disappearance of the blob in the SPH simulations remains unclear, but it is supported by the fact that we find no such blob in the $q=0.5$ SPH simulation, where we also find no dredge up of $^{16}{\mathrm O}$ from the core. We find that the maximum density in the core of the merged object slightly higher in the non-AMR simulations than in the AMR simulations, and even higher in the SPH simulations. Even initially, the maximum density is slightly higher in the SPH simulations. This is likely resolution dependent, since SPH simulations have higher resolution at higher density, and therefore can resolve the core better. The temperature in the SOF is also higher in the non-AMR simulations, and even higher in the SPH simulations. In the SOF, the equatorial plane is noticeably cooler than the rest of the SOF, a feature seen in all the simulations forming a SOF. Another noticeable difference is that the core in the AMR simulations appears to remain more pure CO compared with the non-AMR simulation. This is likely due to the diffusion of the mass fractions mentioned in @staff12, which may be limited with the higher resolution in the AMR simulations. The merged core in the SPH simulations remains very CO pure. We have talked about much $^{16}\mathrm{O}$ being dredged up, and one may reasonably wonder how much is too much dredge-up of $^{16}\mathrm{O}$ in the context of RCB stars? Based on @zhang14 we can find the amount of $^{18}\mathrm{O}$ that may be synthesized. Since the oxygen ratio should be of order unity, no more than that amount of $^{16}\mathrm{O}$ should be dredged up, as the $^{16}\mathrm{O}$ is not being destroyed [@staff12], and this assumes that no significant amount of $^{16}\mathrm{O}$ is being produced. @zhang14 found that in most of their models the surface oxygen mass fraction is 0.005 to 0.008 (see their tables 2 and 3). In most of their models they found the $^{16}\mathrm{O}$ to $^{18}\mathrm{O}$ ratio to be of order unity (comparable to observations), so roughly half of this is $^{16}\mathrm{O}$. The envelope mass is roughly equivalent to the mass of the He WD donor, or $\sim0.3~{\rm M_\odot}$. Hence the $^{16}\mathrm{O}$ and $^{18}\mathrm{O}$ mass in the envelope is $\sim10^{-3}~{\rm M_\odot}$, and this is therefore roughly how much $^{18}\mathrm{O}$ @zhang14 found could be synthesized. In our grid-based simulations we find more than ten times as much $^{16}\mathrm{O}$ being dredged up in the $q=0.5$ simulation (the “best case”), and even more in the higher q simulations (see Table \[jansimrestable\]). In the hybrid SPH simulations, however, no or very little $^{16}\mathrm{O}$ is dredged up, and these could therefore be excellent candidates for producing the oxygen ratio seen in RCB stars. @dan14 ran 225 SPH simulations of WD mergers with varying masses, total masses, mass ratios, composition, etc. Relevant to this paper they find that not much mass is lost in the merger event, up to only $3.4\times10^{-2}M_\odot$, where increasing mass ratio leads to less mass ejected. They also investigate the degree of mixing of donor and accretor material, and find that for $q\lesssim0.45$ there is hardly any mixing, while most mixing occurs for near equal mass ratios. We have not done $q<0.5$, but this result is qualitatively in agreement with what we found in @staff12, that for $q\sim1$ the merged core consist of a mix of donor and accretor material, while for $q=0.5$ the core after merger consists mainly of accretor material. @dan14 also find that for $M_{\rm tot}\lesssim1M_\odot$ (which is the mass range that we focus on), nuclear burning is negligible on the time scale that they simulate, in agreement with our SPH simulations. This is likely because they, like us, have not included hydrogen in their simulations[^4], as we found in @staff12 that it can react very rapidly, releasing much energy. In that paper, we also found that while helium burning can occur, it is on a timescale that is long compared to the dynamical timescale that we simulate. @zhu13 performed a parameter study of the merger of unsynchronized CO WDs. They found many results similar to @dan14, although the fact that the stars in @zhu13 are not tidally locked results in differences regarding the amount of mixing compared with the simulations in @dan14 who, like us, simulates WDs that are initially synchronously rotating. The nucleosynthesis that can form $^{18}{\mathrm O}$ will occur in the SOF (or possibly in hot-spots in the core; see below), which is located just outside the core at a radius of roughly $10^9~{\rm cm}$. If these mergers are to produce RCB stars, they will have to swell up to giant sizes ($\sim10^{13}~{\rm cm}$). The $^{18}{\mathrm O}$ and other elements produced in the SOF will then have to be brought up to the surface. As $^{18}{\mathrm O}$ is produced in the SOF, it may become buoyant and therefore be transported out of the SOF to regions of the star where nucleosynthesis cannot occur. If many elements have already been dredged-up to areas outside of the SOF, this can be mixed in with the newly formed elements in the SOF. Of relevance for the oxygen ratio, it is therefore not only the oxygen ratio in the SOF that is important, as the oxygen ratio might be further diluted if much $^{16}{\mathrm O}$ is present outside the SOF. The strength of the mixing and depth at which elements are brought up from (i.e. from the core, from the SOF,...) following the merger is therefore of great relevance. @menon13 found that both the magnitude of the mixing and the depth must decrease over time in order to get the observed elemental abundances found in RCBs. The depth of the mixing relates to the question of how to define the core in the simulations. In this work we have implemented two density thresholds ($\rho=10^5~{\rm g~cm^{-3}}$ and $\rho=10^{5.2}~{\rm g~cm^{-3}}$) and found the amount of $^{16}{\mathrm O}$ below these density thresholds. We have then assumed that all this $^{16}{\mathrm O}$ is being brought into the envelope and affects the oxygen ratio, as the $^{18}{\mathrm O}$ would also have to be produced in the same region. While the nucleosynthesis processes will likely proceed faster in hotter and denser environments (in the SOF), that does not necessarily result in more $^{18}{\mathrm O}$ being formed there, as it can also be destroyed. $^{16}{\mathrm O}$ is also produced in this region, but as we found in @staff12 it is for the most part not being destroyed. A more detailed study will need to be performed in order to understand exactly where in the SOF the $^{18}{\mathrm O}$ forms and how deep the mixing can go in order to obtain the observed oxygen ratios. In this paper we focused on investigating how much $^{16}{\mathrm O}$ is being brought up during the dynamical merger to the region where it can affect the observed oxygen ratio, and we chose these density thresholds to get some feel for the importance of the depth. We note that these density thresholds for the core are not inconsistent with the core boundaries found in @paxton13 for a somewhat more massive WD than what we use here. In @staff12 we did not consider the high-q simulations when discussing possible nucleosynthesis and the resulting element abundances, as these simulations did not form a SOF. However, hot spots may still develop in the cores of these simulations as in the grid-based $q=0.8$ simulation. @clayton07 discussed the importance of a small amount of hydrogen to the nucleosynthesis. Can the resulting energy release further heat up these hot spots, allowing for helium to react? In these high-q simulations, the cores merged causing much dredge-up of $^{16}{\mathrm O}$, and therefore it seems difficult to form sufficient $^{18}{\mathrm O}$ to achieve the oxygen ratios required for RCB stars. What will such high-q merged object then look like? In order to begin answering this question, properly resolving the hydrogen in the outer part of the He WD will be necessary, as well as treating the nuclear reactions, including hydrogen, in the hydrodynamics simulations. In the simulations presented in this paper, we have ignored the effects of magnetic fields. Magnetic fields, however, may suppress Kelvin-Helmholz instabilities from growing (for instance on the surface between the SOF and the core), thereby possibly reducing the amount of dredge-up of $^{16}{\mathrm O}$ from the core. This may make it easier to achieve the oxygen ratios. Summary {#conclusionsection} ======= ------------ ------------- --------------------------------- ------------------------------- ------------------ ----------------- --------------------- simulation total mass $^{16}{\mathrm O}$ at $^{16}{\mathrm O}$ at max(T) in $\overline{T}$ $\overline{\rho}$ type and $\rho<10^{5.2}~{\rm g~cm^{-3}}$ $\rho<10^{5}~{\rm g~cm^{-3}}$ SOF SOF SOF mass ratio $[M_\odot]$ $[M_\odot]$ $[M_\odot]$ $[{\rm K}]$ $[{\rm K}]$ $[{\rm g~cm^{-3}}]$ $q=0.7$: non AMR $0.9$ $0.045$ $0.035$ $1.5\times10^8$ $1.1\times10^8$ $4.3\times10^4$ AMR $0.9$ $0.04$ $0.03$ $7.0\times10^7$ $9.3\times10^7$ $3.3\times10^5$ SPH $0.9$ $0.02$ $0.01$ $2.5\times10^8$ $1.9\times10^8$ $1.1\times10^5$ $q=0.5$: non AMR $0.71$ $0.02$ $0.012$ $2.1\times10^8$ $1.6\times10^8$ $2.3\times10^4$ AMR $0.758$ $<0.035$ $<0.03$ $1.5\times10^8$ $1.2\times10^8$ $5.5\times10^4$ SPH $0.71$ $0.0$ $0.0$ $2.5\times10^8$ $2.0\times10^8$ $4.6\times10^4$ $q=0.8$: non AMR $0.9$ $0.044$ $0.03$ $1.25\times10^8$ $1.1\times10^8$ $8.5\times10^4$ (hot spot) AMR $1.01$ $0.06$ $0.04$ $8.1\times10^7$ $8.0\times10^7$ $6.4\times10^5$ (hot spot) SPH (SOF) $0.9$ $5\times10^{-5}$ $2\times10^{-5}$ $2.0\times10^8$ $1.8\times10^8$ $1.0\times10^5$ ------------ ------------- --------------------------------- ------------------------------- ------------------ ----------------- --------------------- : Summary of the simulations showing the mass ratio and type, and the main results showing amount of $^{16}{\mathrm O}$ dredged up to lower densities, the maximum temperature found in the SOF during and after the merger (or in hot spots in the merged core in the $q=0.8$ grid-based simulations), the mean temperature in the SOF (or hot spots), and estimates of the mean density in the SOF (or hot spot). \[jansimrestable\] We have run three different WD merger simulations, using three different codes: a non-AMR grid-based hydrodynamics code on a cylindrical grid, an AMR grid-based hydrodynamics code on a Cartesian grid, and a smooth particle hydrodynamics code. We summarize the simulations and the main results in Table \[jansimrestable\]. Between the two grid-based codes we find very good agreement, and they also agree reasonably well with the SPH code. However, in the SPH simulations we find much less $^{16}{\mathrm O}$ (accretor material) at lower densities than in the grid-based codes. Also, we found that in the $q=0.8$ simulation, the two cores merged in the grid-based simulations and no SOF was formed. Instead, “hot spots” formed in and around the merged core. In the SPH simulation, we find no He in the merged core indicating that the cores did not merge, and some kind of a SOF does form although it is much cooler in the equatorial plane than elsewhere. The temperature in the $q=0.8$ simulations does not get very high in either of the grid-based simulations in high density regions, and little nucleosynthesis can occur. In the SPH simulation, however, temperatures of $\sim2\times10^8~{\rm K}$ is found in an SOF above and below the equatorial plane, making nucleosynthesis possible. The $q=0.5$ simulation with a hybrid accretor may be the most interesting in the context of formation of RCB stars, as temperatures of the order of $1.5-2\times10^8~{\rm K}$ is found in the SOF, at densities of the order $10^5~{\rm g~cm^{-3}}$. This is sufficiently hot and dense that nucleosynthesis processes including helium burning can occur. We also found in the SPH simulation that no accretor material is dredged up to lower densities, allowing for oxygen ratios of order unity if equal amounts of $^{18}{\mathrm O}$ is produced in the SOF. Not much nucleosynthesis is expected in the $q=0.8$ SPH simulation, and therefore it is likely not favorable for producing the high oxygen ratio even though not much $^{16}{\mathrm O}$ is being dredged up. However, if the SOF becomes hotter with time in the high density regions, nucleosynthesis may occur, which could lead to the production of $^{18}{\mathrm O}$ and consequently a high oxygen ratio. In the grid-based simulations, we find that much $^{16}{\mathrm O}$ is being dredged up from the accretor during the merger event, even if the accretor is a hybrid CO/He WD with a thick $>0.1~{\rm M_\odot}$ layer of He on top. In fact, we found that right from the begining of the simulation the amount of $^{16}{\mathrm O}$ at lower densities grows rapidly, indicating that this is a numerical effect. In the non-AMR $q=0.5$ simulation, we tried to artificially “reset” the hybrid accretor shortly before the merger event to ensure that no accretor material was at lower densities then. Despite this, we still found much $^{16}{\mathrm O}$ at densities where nucleosynthesis could occur, making it very difficult to reach the observed oxygen ratios. It seems clear that the grid-based codes overestimate the amount of dredge-up, due to the artificial diffusion of the mass fractions. The SPH code, run in very high resolution ($\sim20\times10^6$ particles) may more accurately track the amount of dredge-up, and our conclusion is that a hybrid accretor with a thick outer layer of helium can prevent dredge-up of oxygen from the core. A merger between a He WD and a hybrid CO/He WD with a mass ratio ($q\lesssim0.8$; in which the cores do not merge and an SOF forms) is therefore a good candidate for the progenitor system of RCB stars. This would, however, indicate that the mass of RCB stars should be significantly lower than $1~{\rm M_\odot}$, since the hybrid CO/He WDs have a maximum mass of $<0.5~{\rm M_\odot}$. We would like to thank J. E. Tohline for helpful discussions, and the anonymous referee for many helpful comments. Some of the work was done by J.E.S. while he was a post doc. at Macquarie University and at the University of Florida, and he acknowledges support from the Australian Research Council Discovery Project (DP12013337) program. This work has been supported, in part, by grant NNX10AC72G from NASA’s ATP program, and in part by NASA grant NNX15AP95A. We wish to acknowledge the support from the National Science Foundation through CREATIV grant AST-1240655. The computations were carried out in part using the computational resources of the Louisiana Optical Network Initiative (LONI), and the XSEDE machine Kraken through grant TG-AST090104 and TG-AST110034. LANL simulations were carried out on Turquoise network platform Wolf under Institutional Computing (IC) allocations. One simulation was carried out on UVI’s Bucc cluster. Measurement of the Richardson Number ==================================== The Richardson number [$Ri$; @drazin04] is given by: $$Ri = - \frac{ \nabla \Phi_{eff} \nabla \rho}{\rho \left( \nabla v \right)^{2} }.$$ Where $Ri<1/4$, the flow can over-turn so these are regions where the Kelvin-Helmholtz instability can mix the fluid. We have calculated the Richardson number in the SPH simulations. The SPH data had been put on a grid for this calculation. The code measures the center of mass of each component, the accretor has center of mass coordinates in the equatorial plane given by $x_{2}, y_{2}$. The velocity field is transformed to one where the accretor’s center of mass is at rest and we then use an effective potential that corresponds to this angular frequency $$\Phi_{eff} = \Phi - \frac{1}{2} \Omega^{2} \left( \vec{\mathbf{r}} - \vec{\mathbf{r}}_{2} \right)^{2}$$ The directions “perpendicular” and “parallel” to the flow locally are constructed from the gradient of the effective potential as a normal direction from $$\hat{\mathbf{n}} = \frac{ \vec{\mathbf{\nabla}} \Phi_{eff}}{ \left| \vec{\mathbf{\nabla}} \Phi_{eff} \right| }$$ and a tangent direction parallel to the equipotentials such that $$\hat{\mathbf{t}} \cdot \hat{\mathbf{n}} = 0$$ and both $\hat{\mathbf{n}}$ and $\hat{\mathbf{t}}$ must lie in the equatorial plane due to symmetry about the equatorial plane. The cylindrical coordinate system centered on the origin of the grid will have unit vectors given by $\hat{\mathbf{e}}_{r}$ and $\hat{\mathbf{e}}_{\phi}$ in the equatorial plane. $$\hat{\mathbf{e}}_{r_{x}} = \cos{\phi}$$ $$\hat{\mathbf{e}}_{r_{y}} = \sin{\phi}$$ $$\hat{\mathbf{e}}_{\phi_{x}} = - \sin{\phi}$$ $$\hat{\mathbf{e}}_{\phi_{y}} = \cos{\phi}$$ The gradients that go in the calculation of the Richardson number are then computed from the gradient in the code’s coordinate system projected with the local directions normal and tangential to equipotential curves in the equatorial plane as $$\nabla \Phi_{eff} = \hat{\mathbf{n}} \cdot \vec{\mathbf{\nabla}} \Phi_{eff} = \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{r_{x}} + \hat{\mathbf{n}}_{y} \cdot \hat{\mathbf{e}}_{r_{y}} \right) \frac{ \partial \Phi_{eff}}{ \partial r} + \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{\phi_{x}} + \hat{\mathbf{n}}_{y} \cdot \hat{\mathbf{e}}_{\phi_{y}} \right) \frac{ \partial \Phi_{eff}}{r \partial \phi}$$ $$\nabla \rho = \hat{\mathbf{n}} \cdot \vec{\mathbf{\nabla}} \rho = \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{r_{x}} + \hat{\mathbf{n}}_{y} \cdot \hat{\mathbf{e}}_{r_{y}} \right) \frac{ \partial \rho}{ \partial r} + \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{\phi_{x}} + \hat{\mathbf{n}}_{y} \cdot \hat{\mathbf{e}}_{\phi_{y}} \right) \frac{ \partial \rho}{r \partial \phi}$$ $$\begin{aligned} \nabla v = \hat{\mathbf{t}} \cdot \left( \hat{\mathbf{n}} \cdot \vec{\mathbf{\nabla}} \vec{\mathbf{v}} \right) & = & \left( \hat{\mathbf{e}}_{r_{x}} \cdot \hat{\mathbf{t}}_{x} + \hat{\mathbf{e}}_{r_{y}} \cdot \hat{\mathbf{t}}_{y} \right) \left( \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{r_{x}} + \hat{\mathbf{n}}_{y} \cdot \hat{\mathbf{e}}_{r_{y}} \right) \frac{\partial v_{r}}{\partial r} + \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{\phi_{x}} + \hat{\mathbf{n}}_{y} \cdot + \hat{\mathbf{e}}_{\phi_{y}} \right) \frac{\partial v_{r}}{ r \partial \phi} \right)\\ &+ & \left( \hat{\mathbf{e}}_{\phi_{x}} \cdot \hat{\mathbf{t}}_{x} + \hat{\mathbf{e}}_{\phi_{y}} \cdot \hat{\mathbf{t}}_{y} \right) \left( \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{r_{x}} + \hat{\mathbf{n}}_{y} \cdot \hat{\mathbf{e}}_{r_{y}} \right) \frac{\partial v_{\phi}}{\partial r} + \left( \hat{\mathbf{n}}_{x} \cdot \hat{\mathbf{e}}_{\phi_{x}} + \hat{\mathbf{n}}_{y} \cdot \hat{\mathbf{e}}_{\phi_{y}} \right) \frac{\partial v_{\phi}}{r \partial \phi} \right).\end{aligned}$$ The derivatives in the previous expressions were computed numerically from 4th ordered centered finite differences. Brown, W. R., Kilic, M., Prieto, C. A., & Kenyon, S. J. 2010, ApJ, 723, 1072 Brown, W. R., Kilic, M., Kenyon, S. J., & Gianninas, A. 2016, ApJ, 824, 46 Brown, W. R., Kilic, M, Kosakowski, A., & Gianninas, A. 2017, ApJ, 847, 10 Clayton, G. C. 1996, PASP, 108, 225 Clayton, G. C., Herwig, F., Geballe, T. R., Asplund, M., Tenenbaum, E. D., Engelbracht, C. W., & Gordon, K. D. 2005, ApJL, 623, 141 Clayton, G. C., Geballe, T. R., Herwig, F., Fryer, C. L., Asplund, M. 2007, ApJ, 662, 1220 Clayton, G. C. 2012, JAAVSO, 40, 539 D’Souza, M. C. R., Motl, P. M., Tohline, J. E., & Frank, J. 2006, ApJ, 643, 381 Dan, M., Rosswog, S., Br[ü]{}ggen, M., & Podsiadlowski, P. 2014, MNRAS, 438, 14 Dehnen, W. 2000, ApJL, 536, 39 Diehl, S., Rockefeller, G., Fryer, C., Rockefeller, D., & Statler, T. S. 2015, PASA, 32, 48 Drazin, P. G. & Reid, W. H. 2004, Hydrodynamic Stability, 626 Even, W. & Tohline, J. E., 2009, ApJS, 184, 248 Fryer, C., Rockefeller, G., Warren, M. 2006, ApJ, 643, 292 Garc[í]{}a-Hernandez, D. A., Hinkle, K. H., Lambert, D. L., & Eriksson, K., 2009, ApJ, 696, 1733 Garc[í]{}a-Hernandez, D. A. and Lambert, D. L. and Kameswara Rao, N. and Hinkle, K. H. and Eriksson, K. 2010 , ApJ, 714, 144 Han, Z. 1998, MNRAS, 296, 1019 Hermes, J. J., et al. 2012, ApJL, 757, 21 Iben, I. Jr. & Tutukov, A. V., 1985, ApJS, 58, 661 Jeffery, C. S., Karakas, A. I., & Saio, H. 2011, MNRAS, 414, 3599 Kadam, K., Motl, P. M., Frank, J., Clayton, G. C., & Marcello, D. C. 2016, MNRAS, 462, 2237 Karakas, A. I., Ruiter, A. J., & Hampel, M. 2015, ApJ, 809, 184 Kilic, M., Brown, W. R., Prieto, C. A., Ag[ü]{}eros, M. A., Heinke, C., & Kenyon, S. J. 2011, ApJ, 727, 3 Kurganov, A. & Tadmor, E. 2000, JCoPH, 160, 241 Lawson, W. A., Cottrell, P. L., Kilmartin, P. M., & Gilmore, A. C. 1990, MNRAS, 247, 91 Lindblom, L., Tohline, J. E., & Vallisneri, M. 2001, Physical Review Letters, 86, 1152 Longland, R., Lor[é]{}n-Aguilar, P., Jos[é]{}, J., Garc[í]{}a-Berro, E., Althaus, L. G., & Isern, J. 2011, ApJL, 737, 34 Menon, A., Herwig, F., Denissenkov, P. A., Clayton, G. C., Staff, J., Pignatari, M., & Paxton, B. 2013, ApJ, 772, 59 Montiel, E. J., Clayton, G. C., Marcello, D. C., & Lockman, F. J. 2015, AJ, 150, 14 Motl, P. M., Frank, J., Tohline, J. E., & D’Souza, M. C. R. 2007, ApJ, 670, 1314 Motl, P. M., et al. 2017, ApJS, 229, 27 Mullally, F., Badenes, C., Thompson, S. E., & Lupton, R. 2009, ApJL, 707, 51 Nelemans, G. et al. 2005, A&A, 440, 1087 O’Keefe, J. A., 1939, ApJ, 90, 294 Ott, C. D., Ou, S., Tohline, J. E., & Burrows, A. 2005, ApJ, 625, L119 Pandey, G. & Lambert, D. L. 2011, ApJ, 727, 122 Paxton, B., et al. 2013, ApJS, 208, 4 Rappaport, S., Podsiadlowski, Ph., & Horev, I., 2009, ApJ, 698, 666 Renzini, A., 1990, ASPC, 11, 549 Saio, H. 2008, ASPC, 391, 69 Salaris, M., Dom[í]{}nguez, I., Garc[í]{}a-Berro, E., Hernanz, M., Isern, J., & Mochkovitch, R. 1997, ApJ, 486, 413 Scott, P. C., Asplund, M., Grevesse, N., & Sauval, A. J., 2006, A&A, 456, 675 Segretain, L., Chabrier, G., & Mochkovitch, R. 1997, ApJ, 481, 355 Staff, J. E., et al., 2012, ApJ, 757, 76 Warren M. & J. Salmon. 1995, Comp. Phys. Comm., 87, 266 Webbink, R. F., 1984, ApJ, 277, 355 Wilson, T. L. & Rood, R. 1994, ARA&A, 32, 191 Zhang, X. & Jeffery, C. S., 2012 MNRAS, 426, L81 Zhang, X., Jeffery, C. S., Chen, X., & Han, Z. 2014, MNRAS, 445, 660 Zhu, C., Chang, P., van Kerkwijk, M. H., & Wadsley, J. 2013, ApJ, 767, 164 [^1]: The observed $^{16}{\mathrm O}$ to $^{18}{\mathrm O}$ ratio may be higher than in the SOF, since there may be some $^{16}{\mathrm O}$ present in the atmosphere prior to the production of $^{18}{\mathrm O}$ in the SOF, that can further dilute the ratio. [^2]: for a simulation with 20 million particles and $M_{\rm tot}=0.7~{\rm M_\odot}$, each particle has a mass of $\approx7\times10^{25}~{\rm g}=3.5\times10^{-8}~{\rm M_\odot}$. [^3]: By a “less violent merger”, we mean a merger where the donor material “gently” ends up on top of the accretor, leaving the accretor relatively unchanged except from the outermost part. A “violent merger” then, is one where the accretor core is changed significantly. [^4]: The He WDs have a low mass layer of hydrogen on top of the WD.

--- bibliography: - 'ref.bib' --- 0.6cm DESY 15-103\ FTPI-MINN-15/33\ IPMU15-0098 1.1cm [ **Probing Bino-Wino Coannihilation at the LHC** ]{} 1.2cm Natsumi Nagata${}^{1}$, Hidetoshi Otono${}^{2}$, and Satoshi Shirai${}^{3}$ 0.5cm [*$^1$ William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,\ University of Minnesota, Minneapolis, MN 55455, USA,\ and Kavli Institute for the Physics and Mathematics of the Universe (WPI),\ The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa 277-8583, Japan\ \[5pt\] $^2$ Research Center for Advanced Particle Physics, Kyushu University, Fukuoka 812-8581, Japan\ $^3$ [Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, Germany]{}* ]{} 1.0cm Introduction {#sec:intro} ============ Split supersymmetry (SUSY) [@Wells:2003tf; @*Wells:2004di; @ArkaniHamed:2004fb; @*Giudice:2004tc; @*ArkaniHamed:2004yi; @*ArkaniHamed:2005yv] is an interesting candidate for physics beyond the Standard Model (SM). This class of SUSY models have a mass spectrum in which scalar particles except the SM-like Higgs boson have masses much larger than the electroweak (EW) scale, while gauginos, probably Higgsino as well, lie not far from the EW scale. With such heavy scalars, the split SUSY can overcome disadvantages of the weak-scale SUSY, such as SUSY flavor/CP problems [@Gabbiani:1996hi] and too rapid proton decay in the minimal SUSY grand unified theory (GUT) [@Goto:1998qg; @Murayama:2001ur]. The light fermionic SUSY partners enable the model to take over advantages of the weak-scale SUSY: the lightest SUSY particle (LSP) as a dark matter (DM) candidate and gauge coupling unification. Thus, although the origin of the EW scale is not easily elucidated in this model, its phenomenological aspects are quite appealing. After the discovery of the SM-like Higgs boson [@Aad:2012tfa; @*Chatrchyan:2012ufa] with a mass of around 125 GeV [@Aad:2015zhl], this framework becomes increasingly convincing. Especially, a spectrum with mild splitting  [@Hall:2011jd; @Hall:2012zp; @Ibe:2011aa; @*Ibe:2012hu; @Arvanitaki:2012ps; @ArkaniHamed:2012gw; @Evans:2013lpa], where the scalar mass scale falls into from several tens of TeV to a PeV and the gaugino mass scale is ${\cal O}(1)$ TeV, is getting more popular—this mass spectrum is often called spread or mini-split SUSY, as the 125 GeV Higgs mass is a sweet spot of this spectrum [@Okada:1990vk; @*Okada:1990gg; @Ellis:1990nz; @Haber:1990aw; @Ellis:1991zd]. Such models can also improve gauge coupling unification [@Hisano:2013cqa], and accommodate a simple GUT without suffering from the rapid proton decay problem [@Hisano:2013exa; @Nagata:2013sba; @Evans:2015bxa]. Moreover, reasonable assumptions on the multiverse lead to this spectrum [@Nomura:2014asa]. The realization of such a mass spectrum is rather easy. We may simply assume some charge on the SUSY breaking field $X$. Then, the symmetry associated with the charge forbids Lagrangian terms $[XW^{\alpha} W_{\alpha}/M_*]_{\theta^2}$, where $W^\alpha$ is a gauge field strength superfield and $M_*$ is a cut-off scale, *e.g.*, the Planck scale. Since these terms reduce to the gaugino mass terms after the SUSY breaking, absence of these terms implies suppression of gaugino masses. On the other hand, terms like $[XX^{\dagger} \Phi \Phi^{\dagger} / M_*^2]_{\theta^4}$, where $\Phi$ is a matter chiral superfield, are generally allowed and lead to soft scalar masses of $\widetilde{m} = {\cal O}(F_X/M_*)$, with $F_X$ the $F$-term of the SUSY breaking field $X$. The Higgsino mass $\mu$ is usually expected to be of the same order of sfermion masses: $|\mu| \sim \widetilde{m}$, though there are several models that predict a smaller value for $|\mu|$. Gaugino masses may come from the anomaly mediation mechanism [@Randall:1998uk; @Giudice:1998xp] as well as threshold effects of the Higgs fields [@Pierce:1996zz] or extra matter fields [@Pomarol:1999ie; @*Nelson:2002sa; @*Hsieh:2006ig; @*Gupta:2012gu; @*Nakayama:2013uta; @*Harigaya:2013asa; @*Evans:2014xpa]. In these cases, the gaugino masses are suppressed by a loop factor compared to the scalar mass scale $\widetilde{m}$, and thus the spread/mini-split spectrum can be realized. In addition, this hierarchical mass spectrum makes the mixing among gauginos and Higgsino negligible, and thus they can be regarded as almost pure states. If the contribution from the anomaly mediation is dominant, the LSP is a pure wino. This wino LSP has various interesting features as a DM candidate. Since winos are charged under the EW interactions, their self-annihilation cross sections are rather large, which allows a wino with a mass up to 3 TeV to be consistent with the current observed DM density [@Hisano:2006nn]. Such a large annihilation cross section also makes the indirect detection of wino DM through cosmic ray signals quite promising [@Hisano:2003ec; @*Hisano:2004ds; @Cohen:2013ama; @Fan:2013faa; @Ibe:2015tma; @Hamaguchi:2015wga]. Direct detection of wino DM has also been intensively studied [@Hisano:2010fy; @*Hisano:2010ct; @*Hisano:2011cs; @*Hisano:2012wm; @*Hisano:2015rsa]. In addition, a pure wino offers a unique collider signal. In the spread/mini-split SUSY scenario, the mass splitting between the neutral and charged winos is predicted to be fairly small. This renders the charged wino live long, with a decay length of $c\tau_{\widetilde{W}^0}={\cal O}(1)$ cm. At collider, a produced charged wino leaves a charged track of this length, which is very useful for the discovery and measurement of the wino LSP [@Ibe:2006de; @*Asai:2008sk; @*Asai:2008im]. On the other hand, when other contributions like threshold effects are comparable to that of the anomaly mediation, it is questionable whether the wino LSP is the case or not. Actually, it turns out that an $M_*$ which is slightly smaller than the reduced Planck scale easily leads to the bino LSP [@Hall:2012zp; @Nomura:2014asa]. The presence of extra matters may also favor the bino LSP, since a wino tends to receive larger quantum corrections from the extra matters due to its larger gauge coupling compared to a bino. The bino LSP is, however, often disfavored on the basis of cosmology; on the assumption of the $R$-parity conservation and conventional cosmological history, the bino LSP case usually suffers from the overproduction of DM because of its small self-annihilation cross section. For the bino abundance to be consistent with the current observation, we need to rely on some exceptional situation [@Griest:1990kh]: gaugino coannihilation or Higgs funnel. If the Higgsino mass is larger than ${\cal O}(10)$ TeV, the Higgs funnel cannot work effectively since the bino-Higgs coupling is highly suppressed. In this case, gaugino coannihilation is the only possibility. We previously studied the coannihilation of the bino LSP with a gluino [@Nagata:2015hha]. In this work, we focus on the bino-wino coannihilation [@Baer:2005jq; @ArkaniHamed:2006mb; @Ibe:2013pua; @Harigaya:2014dwa]. The bino DM is quite sterile compared to the wino DM, since both the self-annihilation cross section and the direct detection rate are suppressed by heavy masses of sfermions and Higgsino. Thus, probing this spectrum with the DM experiments is extremely challenging. Instead, the bino-wino coannihilation scenario has a specific mass spectrum; wino should have a mass fairly close to the bino mass in order to make coannihilation effective and to assure that the relic abundance of the bino DM is less than or equal to the observed DM density. Previous works have revealed that the mass difference $\Delta M$ should be $\lesssim {\cal O}(10)$ GeV [@Baer:2005jq; @ArkaniHamed:2006mb; @Ibe:2013pua; @Harigaya:2014dwa] to satisfy the condition. Such a small mass difference makes it possible to probe the scenario in collider experiments, since a degenerate mass spectrum often gives rise to a long-lived particle, which offers a distinct signature. Indeed, we find that a neutral wino can actually be long-lived in our setup and thus be a nice target to probe the scenario. In this work, we study in detail the decay of a neutral wino in the bino-wino coannihilation scenario and show that it typically has a decay length larger than ${\cal O}(1)$ mm. Then, we consider the detectability of the neutral wino decay at the LHC. It is found that searches for a displaced vertex (DV) can actually be a powerful probe for a neutral wino with such a long decay length, and thus provide a promising way of testing the bino-wino coannihilation scenario. This paper is organized as follows. In the next section, we formulate an effective theory for bino and wino to study the wino decay in the spread/mini-split spectrum. Then, in Sec. \[sec:lhcsearch\], we discuss the current constraints and future prospects on the searches for the decay signature of a long-lived neutral wino. Finally, Sec. \[sec:conclusion\] is devoted to conclusion and discussion. Wino decay ========== In this section, we discuss the decay properties of a neutral wino in the case where the neutral wino is highly degenerate with the bino LSP in mass and Higgsino and sfermions are much heavier than these EW gauginos. To adequately deal with this hierarchical setup, in Sec. \[sec:effectivethe\], we first construct a low-energy effective theory for the EW gauginos by integrating out these heavy particles. Then, in Sec. \[sec:decayl\], we discuss the decay properties of the neutral wino. We also show expected values for its decay length taking into account the thermal relic abundance of the bino LSP. Before going into the detailed discussion, let us first summarize the results of this section. In Fig. \[fig:spectrum\], we show the mass spectrum for the EW gauginos and the suppression factors in their decay rates. We assume the bino-wino mass difference, $\Delta M$, to be ${\cal O}(10)$ GeV, which leads to successful DM coannihilation [@Baer:2005jq; @ArkaniHamed:2006mb; @Ibe:2013pua; @Harigaya:2014dwa]. The decay rate of wino into the bino LSP is suppressed by the heavy Higgsino mass. Charged wino can decay into bino promptly for $|\mu|< {\cal O}(10)$ PeV, since the decay occurs via a dimension-five operator. On the other hand, the neutral wino decay is not so rapid. If $\Delta M$ is less than the $Z$ boson mass, neutral wino decays into bino only through the virtual $Z$ boson or Higgs boson $h$ exchange, or via the two-body decay process with emitting a photon at loop level. As we see below, the $Z$ boson mediated decay and the two-body photon-emitting processes are suppressed by a factor of $|\mu|^{-4}$. Regarding the Higgs boson mediated decay, on the other hand, its decay rate is only suppressed by a factor of $|\mu|^{-2}$, though the small couplings between the Higgs boson and the SM fermions prevent neutral wino from decaying rapidly. As a result, for $|\mu|\gtrsim 10$ TeV, the decay length $c\tau_{\widetilde{W}^0}$ of neutral wino gets macroscopic: $c\tau_{\widetilde{W}^0} \gtrsim {\cal O}(1)$ mm. Effective theory for wino decay {#sec:effectivethe} ------------------------------- Here, we discuss the wino decay based on the effective field theoretical approach. To begin with, we introduce the full theory containing Higgsino with renormalizable interactions. Then, we obtain a relevant effective field theory by integrating out the heavy Higgsino. The Higgsino contributions are described by higher-dimensional operators in the effective theory, which causes the wino decay into the bino LSP. ### Full theory above Higgsino scale {#full-theory-above-higgsino-scale .unnumbered} First, let us consider the full theory. In the mini-spit/spread SUSY, it is reasonable to assume that the sfermion mass scale $\widetilde{m}$ is similar to or greater than the Higgsino mass $\mu$: $\widetilde{m} \gtrsim |\mu|$. In this case, the decay of wino is dominantly controlled by the gaugino-Higgsino-Higgs couplings, rather than the interactions with sfermions and heavy Higgs bosons, and thus we can safely neglect their contributions in the following discussion. The gaugino-Higgsino-Higgs interactions are given by $$\begin{aligned} {\cal L}_{\text{int}} =& -\frac{1}{\sqrt{2}} \{g_{1u}^{} H^{\dagger} \widetilde{H}_u+ g_{1d}^{} \epsilon^{\alpha\beta} (H)_\alpha(\widetilde {H}_d)_\beta \} \widetilde{B} \nonumber \\ &- {\sqrt{2}}\{ g_{2u}^{} H^{\dagger}T^A \tilde H_u -g_{2d}^{} \epsilon^{\alpha\beta}(H)_\alpha (T^A \widetilde{H}_d)_\beta \}\widetilde{W}^A +\text{h.c.}~, \label{eq:gauginohiggsinocoup}\end{aligned}$$ where $\widetilde{H}_{u,d}$, $\widetilde{B}$, and $\widetilde{W}^A$ ($A=1,2,3$) denote the Higgsino, bino, and wino fields, respectively; $H$ is the SM Higgs field; $T^A$ are the SU(2)$_L$ generators. In this paper, we mainly use the two-component notation for fermion fields unless otherwise noted. At the leading order, the above coupling constants are given by $$\begin{aligned} g_{1u}^{} &= g^\prime \sin\beta, ~~~~~~g_{1d}^{} = g^\prime \cos\beta~, \nonumber\\ g_{2u}^{} &= g \sin\beta, ~~~~~~~g_{2d}^{} = g \cos\beta~, \label{eq:gauginocoupmatch}\end{aligned}$$ at the SUSY breaking scale $\widetilde{m}$. Here, $g^\prime$ and $g$ are the U(1)$_Y$ and SU(2)$_L$ gauge coupling constants, respectively, and $\tan\beta \equiv \langle H_u^0\rangle /\langle H_d^0\rangle$. The gaugino and Higgsino mass terms are defined by $$\begin{aligned} {\cal L}_{\text{mass}}= -\frac{M_1}{2}\widetilde{B}\widetilde{B} -\frac{M_2}{2}\widetilde{W}^A \widetilde{W}^A -\mu ~\epsilon^{\alpha\beta}(\widetilde{H}_u)_\alpha (\widetilde{H}_d)_\beta +\text{h.c.}~,\end{aligned}$$ with $\epsilon^{\alpha\beta}$ the antisymmetric tensor. In the following subsection, we construct an effective filed theory for the gauginos by integrating out the Higgsinos $\widetilde{H}_u$ and $\widetilde{H}_d$, which are supposed to be much heavier than the gauginos. ### Effective theory below Higgsino scale {#effective-theory-below-higgsino-scale .unnumbered} Next, we formulate an effective theory which describes the wino decay into the bino LSP. The decay is caused by effective interactions expressed by higher-dimensional operators, which are induced when we integrate out heavier particles than gauginos—Higgsinos and scalar particles whose masses are ${\cal O}(10$–$10^3)$ TeV. Let us write down relevant operators up to dimension six. For the dimension-five operators, we have $$\begin{aligned} {\cal O}_1^{(5)} &= \widetilde{B}\widetilde{W}^A H^\dagger T^A H ~, \\ {\cal O}_2^{(5)} &= \widetilde{B}\sigma^{\mu\nu} \widetilde{W}^A W^A_{\mu\nu} ~, \\ {\cal Q}_1 &= \frac{1}{2}{\widetilde{B}}\widetilde{B} |H|^2 ~, \\ {\cal Q}_2 &= \frac{1}{2}{\widetilde{W}^A}\widetilde{W}^A |H|^2 ~,\end{aligned}$$ where $W^A_{\mu\nu}$ is the SU(2)$_L$ gauge field strength tensor; $\sigma_{\mu\nu}\equiv \frac{i}{2}(\sigma_\mu \overline{\sigma}_\nu -\sigma_\nu \overline{\sigma}_\mu)$, where $\sigma^\mu =(\sigma^0, \sigma^i)$ and $\overline{\sigma}^\mu =(\sigma^0, -\sigma^i)$ with $\sigma^i$ ($i=1,2,3$) the Pauli matrices. The first two operators contain a bino and a wino, and thus directly contribute to the wino decay into the bino LSP. The latter two are, on the other hand, only relevant to the mass matrix for the neutral bino and wino; these operators reduce to the mass terms for them after the EW symmetry breaking. As for dimension-six, we have $$\begin{aligned} {\cal O}^{(6)} &= {\widetilde{B}}^\dagger \overline{\sigma}^\mu \widetilde{W}^A H^\dagger T^A i\overleftrightarrow{D}_\mu H ~, \end{aligned}$$ where $D_\mu$ is the covariant derivative and $A\overleftrightarrow{D}_\mu B \equiv AD_\mu B - (D_\mu B) A$ with $A$ and $B$ arbitrary fields. This operator also contributes to the wino decay, though its effect is further suppressed by a heavy mass scale. Then, the effective interactions are given as follows: $$\Delta {\cal L}_{\text{int}} = \sum_{i=1,2} C_i^{(5)} {\cal O}^{(5)}_i + \sum_{i=1,2}\widetilde{C}_i {\cal Q}_i + C^{(6)} {\cal O}^{(6)} + \text{h.c.}$$ The Wilson coefficients of these operators are determined below. By evaluating the tree-level Higgsino exchange diagrams (Fig. \[fig:higgsino\_tree\]), we readily obtain $$\begin{aligned} C^{(5)}_1 &= \frac{1}{\mu}(g_{1u}^{}g_{2d}^{}+g_{1d}^{} g_{2u}^{}) +\frac{1}{2|\mu|^2}[(g_{1u}^* g_{2u}^{}+g_{1d}^*g_{2d}^{})M_1+ (g_{1u}^{} g_{2u}^*+g_{1d}^{}g_{2d}^*)M_2] ~, \label{eq:c51}\\ C^{(6)} &= -\frac{1}{2|\mu|^2}(g_{1u}^*g_{2u}^{}-g_{1d}^* g_{2d}^{}) ~,\\ \widetilde{C}_1 &= \frac{g_{1u}^{}g_{1d}^{}}{\mu} +\frac{M_1}{2|\mu|^2} (|g_{1u}^{}|^2 + |g_{1d}^{}|^2) ~, \\ \widetilde{C}_2 &= \frac{g_{2u}^{}g_{2d}^{}}{\mu} +\frac{M_2}{2|\mu|^2} (|g_{2u}^{}|^2 + |g_{2d}^{}|^2) ~. \end{aligned}$$ Here we have kept effective operators up to dimension six, and used equations of motions for external gaugino fields to eliminate redundant operators. The operator ${\cal O}^{(5)}_2$ is not induced at tree level. However, since this operator gives rise to the two-body decay process $\widetilde{W}^0 \to \widetilde{B} + \gamma$, it could be important even though it is induced at loop level [@Haber:1988px; @Ambrosanio:1996gz; @Diaz:2009zh; @Han:2014xoa], especially when the wino mass is close to the bino mass. Thus, only for this operator, we also consider the one-loop contribution. The one-loop Higgsino-Higgs loop diagram (Fig. \[fig:higgsino\_loop\]) yields $$\begin{aligned} C_2^{(5)} =& +\frac{g}{2(4\pi)^2 \mu} (g_{1u}^{}g_{2d}^{}-g_{1d}^{}g_{2u}^{}) \nonumber \\ &-\frac{g}{8(4\pi)^2}\left[ (g_{1u}^*g_{2u}^{}-g_{1d}^*g_{2d}^{})\frac{M_1}{|\mu|^2} -(g_{1u}^{}g_{2u}^*-g_{1d}^{}g_{2d}^*)\frac{M_2}{|\mu|^2} \right] ~,\end{aligned}$$ where again we have kept terms up to ${\cal O}(|\mu|^{-2})$. Note that the first term vanishes if we use the tree-level relation Eq. . We also find that the heavy Higgs contribution of ${\cal O}(\mu^{-1})$ vanishes in a similar manner. Thus, although the operator ${\cal O}^{(5)}_2$ is dimension five, its Wilson coefficient is suppressed by $|\mu|^{-2}$ and thus subdominant compared to the contribution of ${\cal O}^{(5)}_1$. Moreover, the terms in the second line could also cancel with each other to great extent if $M_1\simeq M_2$. This results in a further suppression of this contribution. Besides, quark-squark loop processes can generate the operator ${\cal O}^{(5)}_2$ at one-loop level. Their contribution is suppressed by a factor of $m_{\tilde{q}}^{-2}$ on top of a loop factor, with $m_{\widetilde{q}}$ the mass of the squark running in the loop, and thus again subdominant. ### EW broken phase {#ew-broken-phase .unnumbered} After the Higgs field acquires a vacuum expectation value (VEV), the operators ${\cal O}^{(5)}_1$, ${\cal Q}_1$, and ${\cal Q}_2$ reduce to the mass terms for bino and wino. The mass matrix for the neutral sector is given by $${\cal L}_{\text{mass}} =-\frac{1}{2} (\widetilde{B}~\widetilde{W}^0) {\cal M} \begin{pmatrix} \widetilde{B} \\ \widetilde{W}^0 \end{pmatrix} ~,$$ with $${\cal M}= \begin{pmatrix} M_1-\frac{v^2}{2}\widetilde{C}_1 & \frac{v^2}{4}C_1^{(5)} \\ \frac{v^2}{4}C_1^{(5)} & M_2 -\frac{v^2}{2}\widetilde{C}_2 \end{pmatrix} \equiv \begin{pmatrix} {\cal M}_{11} & {\cal M}_{12} \\ {\cal M}_{12} & {\cal M}_{22} \end{pmatrix} ~,$$ where $v\simeq 246$ GeV is the Higgs VEV. This mass matrix can be diagonalized with a $2\times 2$ unitary matrix $U$, which we parametrize by $$\begin{aligned} U= \begin{pmatrix} e^{i\alpha}& 0 \\ 0& e^{i\beta} \end{pmatrix} \begin{pmatrix} \cos\theta & e^{-i\phi}\sin\theta \\ -e^{i\phi} \sin\theta & \cos\theta \end{pmatrix} ~.\end{aligned}$$ Then, we find that the matrix $M$ is diagonalized as [@Takagi; @Choi:2006fz] $$U^*{\cal M}U^\dagger = \begin{pmatrix} m_1&0\\ 0 & m_2 \end{pmatrix} ~,$$ where $m_1$ and $m_2$ are real and non-negative, whose values are given by $$m_{1,2}^2 = \frac{1}{2}[|{\cal M}_{11}|^2+|{\cal M}_{22}|^2+2|{\cal M}_{12}|^2\mp \sqrt{(|{\cal M}_{11}|^2-|{\cal M}_{22}|^2)^2 +4|{\cal M}_{11}^*{\cal M}_{12}+{\cal M}_{22}{\cal M}_{12}^*|^2}]~.$$ The mixing angle $\theta$ in the unitary matrix $U$ is given by $$\tan\theta =\frac{|{\cal M}_{11}|^2-|{\cal M}_{22}|^2+\sqrt{(|{\cal M}_{11}|^2-|{\cal M}_{22}|^2)^2 +4|{\cal M}_{11}^*{\cal M}_{12}+{\cal M}_{22}{\cal M}_{12}^*|^2}} {2|{\cal M}_{11}^*{\cal M}_{12}+{\cal M}_{22}{\cal M}_{12}^*|}~,$$ while its phase factors are $$\begin{aligned} e^{i\phi}&=\frac{{\cal M}_{11}^*{\cal M}_{12}+{\cal M}_{22}{\cal M}_{12}^*} {|{\cal M}_{11}^*{\cal M}_{12}+{\cal M}_{22}{\cal M}_{12}^*|}~, \\ \alpha &= \frac{1}{2}\arg\bigl({\cal M}_{11}-{\cal M}_{12}e^{-i\phi} \tan\theta\bigr)~,\\ \beta &= \frac{1}{2}\arg\bigl({\cal M}_{22}+{\cal M}_{12}e^{i\phi}\tan\theta\bigr)~. \end{aligned}$$ In this calculation, we have implicitly assumed that $|{\cal M}_{11}|\leq |{\cal M}_{22}|$. In terms of the gaugino masses and the Wilson coefficients, these parameters are approximately given as $$\begin{aligned} m_1^2 &\simeq |M_1|^2 -v^2 \text{Re}(M_1\widetilde{C}^*_1) ~, \nonumber \\ m_2^2 &\simeq |M_2|^2 -v^2 \text{Re}(M_2\widetilde{C}^*_2) ~, \nonumber \\ \tan\theta &\simeq \frac{v^2}{4}\frac{|M_1^* C_1^{(5)}+M_2C_1^{(5)*}|} {|M_2|^2-|M_1|^2} ~, \nonumber \\ \phi &\simeq \text{arg}(M_1^*C_1^{(5)}+M_2 C_1^{(5)*}) ~, \nonumber \\ \alpha &\simeq \frac{1}{2}\text{arg}(M_1) -\frac{v^2}{4}\text{Im}\left(\frac{\widetilde{C}_1}{M_1}\right) ~, \nonumber \\ \beta &\simeq \frac{1}{2}\text{arg}(M_2) -\frac{v^2}{4}\text{Im}\left(\frac{\widetilde{C}_2}{M_2}\right) ~. \label{eq:paramapp}\end{aligned}$$ If $|M_2|-|M_1|$ is ${\cal O}(10)$ GeV, which is motivated by the bino-wino coannihilation scenario as shown in Refs. [@Baer:2005jq; @ArkaniHamed:2006mb; @Ibe:2013pua; @Harigaya:2014dwa], then the above approximations are valid when $|\mu|\gtrsim {\cal O}(10)$ TeV, with which the bino-wino mixing angle is sufficiently small: $\tan\theta \ll 1$. The mass eigenstates are related to the weak eigenstates through the unitary matrix $U$ by $$\begin{pmatrix} \widetilde{\chi}^0_1 \\ \widetilde{\chi}^0_2 \end{pmatrix} = U \begin{pmatrix} \widetilde{B} \\ \widetilde{W}^0 \end{pmatrix} ~.$$ In this paper, we assume the bino-like state $\widetilde{\chi}^0_1$ is slightly lighter than the neutral wino-like state $\widetilde{\chi}^0_2$: $m_1 \lesssim m_2$ with the mass difference $\Delta M \equiv m_2-m_1$ being ${\cal O}(10)$ GeV. Next, we consider chargino $\widetilde{\chi}^+$, which is related to the weak eigenstate by $$\widetilde{\chi}^+ = \frac{e^{i\gamma}}{\sqrt{2}} (\widetilde{W}^1-i\widetilde{W}^2)~.$$ Its mass eigenvalue and phase factor $\gamma$ are given by $$\begin{aligned} m_{\widetilde{\chi}^+} =\left\vert M_2 -\frac{v^2}{2}\widetilde{C}_2\right\vert ~, ~~~~~~ \gamma \simeq \beta ~. \label{eq:chargmass}\end{aligned}$$ On top of that, EW loop corrections make the chargino heavier than the neutral wino by a small amount; this contribution to the mass splitting is evaluated as $\Delta M_{\text{EW}} \simeq 160$ MeV at two-loop level [@Yamada:2009ve; @Ibe:2012sx]. Note that up to dimension six the higher-dimensional operators do not generate the mass difference between the neutral and charged winos, as we have seen from Eq.  and Eq. . For this reason, although the EW correction $\Delta M_{\text{EW}}$ is quite small, it turns out to be the dominant contribution to the mass difference as long as $|\mu| \gtrsim 10$ TeV. Now we summarize the interactions relevant to the decay of $\widetilde{\chi}^0_2$ and $\widetilde{\chi}^+$. First, we consider the chargino decay. In this scenario, a chargino $\widetilde{\chi}^+$ mainly decays into $\widetilde{\chi}^0_1$ because of the degeneracy between $\widetilde{\chi}^0_2$ and $\widetilde{\chi}^+$. This decay is caused by the tree-level gauge interactions through the bino-wino mixing. In the mass eigenbasis, the gauge interactions are written as $$\begin{aligned} {\cal L}_{\widetilde{\chi}^0 \widetilde{\chi}^+ W} = -g\sin\theta \overline{\widetilde{\chi}^0_1}{{\ooalign{\hfil/\hfil\crcr$W$}}}^- \left[e^{-i(\phi -\alpha +\beta)}P_L+e^{i(\phi -\alpha +\beta)}P_R \right] \widetilde{\chi}^+ -g\cos\theta \overline{\widetilde{\chi}^0_2}{{\ooalign{\hfil/\hfil\crcr$W$}}}^- \widetilde{\chi}^+ +\text{h.c.}~, \label{eq:gaugeint}\end{aligned}$$ where we have used four-component notation. Notice that these interactions are invariant under the charge conjugation. We use this property below. Using the tree-level relation (\[eq:gauginocoupmatch\]), we can obtain an approximate expression for the $\widetilde{\chi}^0_1$-$\widetilde{\chi}^{\pm}$-$W^{\mp}$ coupling as $$- g\sin\theta \simeq -g \sin 2\beta \frac{m_W^2\tan\theta_W}{\mu \Delta M} ~, \label{eq:approxchichiw}$$ where $\theta_W$ is the weak-mixing angle and $m_W$ is the $W$-boson mass. We take the gaugino masses to be real in the derivation. Second, we discuss the decay of heavier neutralino $\widetilde{\chi}^0_2$ into the lightest neutralino $\widetilde{\chi}^0_1$. This decay process occurs via the (off-shell) Higgs emission induced by the dimension-five operator ${\cal O}^{(5)}_1$. The relevant interaction is given by $${\cal L}_{\widetilde{\chi}^0_1\widetilde{\chi}^0_2 h } =-\frac{v}{2}e^{-i(\alpha + \beta)} \cos 2\theta C_1^{(5)} h{\widetilde{\chi}^0_1} \widetilde{\chi}^0_2 +\text{h.c.} \label{eq:muhiggschi1chi2}$$ Since ${\cal O}^{(5)}_1$ is of dimension five, this interaction is only suppressed by $|\mu|^{-1}$. Similarly to Eq. , we can approximate the coupling by $-g\tan\theta_W \sin(2\beta) m_W/\mu$. Notice that large $\tan\beta$ suppresses this coupling. In this case, the contributions suppressed by $|\mu|^{-2}$ originating from the second term in Eq.  may dominate this contribution. The interactions of $\widetilde{\chi}^0_1$ and $\widetilde{\chi}^0_2$ with a photon and a $Z$ boson are, on the other hand, relatively small. At renormalizable level, there is no such a interaction since $Q=Y=T_3=0$ for both of these particles. As discussed above, the contribution of the dimension-five operator ${\cal O}^{(5)}_2$ is suppressed by $|\mu|^{-2}$ besides the one loop factor. The dimension-six operator ${\cal O}^{(6)}$ also gives rise to such interactions at tree level, $${\cal L}_{\widetilde{\chi}^0_1\widetilde{\chi}^0_2 Z} = \frac{g_Zv^2}{4}e^{i(\alpha-\beta)}C^{(6)} \widetilde{\chi}^{0\dagger}_1 \overline{\sigma}^\mu \widetilde{\chi}^0_2 Z_\mu +\text{h.c.}~, \label{eq:chichiz}$$ where $g_Z \equiv \sqrt{g^{\prime 2}+g^2}$. This interaction is again suppressed by $\cos(2\beta)|\mu|^{-2}$. Consequently, compared to the Higgs interaction in Eq. , the interactions with a photon and a $Z$ boson are fairly small if we take $|\mu|$ to be sufficiently large. An important caveat here is that these interactions could be important if $\tan \beta$ is large, since the Higgs coupling in Eq.  is highly suppressed in this case as mentioned above. Although the mini-split type models favor small $\tan \beta$ to explain the 125 GeV Higgs mass, a moderate size of $\tan \beta$ may be allowed if stop masses are rather light. In such cases, the terms suppressed by $|\mu|^{-2}$ can also be significant. Decay length in bino-wino coannihilation {#sec:decayl} ---------------------------------------- Taking the above discussion into account, we now estimate the decay length and branching ratios of the neutral wino-like state $\widetilde{\chi}^0_2$. As we have seen above, when $|\mu|$ is large enough, the Higgs interaction dominates the others. This interaction gives rise to three-body decay processes shown in Fig. \[fig:h\] when the mass difference between $\widetilde{\chi}^0_1$ and $\widetilde{\chi}^0_2$, $\Delta M$, is as small as ${\cal O}(10)$ GeV. In this case, the dominant decay channel is $\widetilde{\chi}^0_2 \to \widetilde{\chi}^0_1 b \bar{b}$, whose decay amplitude is suppressed by the bottom-quark Yukawa coupling in addition to the three-body phase space factor and $|\mu|^{-1}$. Moreover, the decay rate is kinematically suppressed due to the small mass difference $\Delta M$. For these reasons, the resultant decay rate is considerably small and $\widetilde{\chi}^0_2$ has a sizable decay length, as we see below. Notice that, as we mentioned above, tree-level sfermion exchange contributions to the three-body decay processes are generically much smaller than the Higgs exchange contribution. The former contributions can be expressed by dimension-six effective operators whose coefficients are $\sim gg^\prime /\widetilde{m}^2$. On the other hand, after integrating out the Higgs boson, the diagram in Fig. \[fig:h\] yields a dimension-six operator with its coefficient being $\sim gg^\prime (\sin 2\beta) m_b/(\mu m_h^2)$. Therefore, the Higgs exchange contribution dominates sfermion ones as long as $|\mu| \ll {\cal O}(10^3)~\text{TeV}\times\sin 2\beta \cdot (\widetilde{m}/10^2~\text{TeV})^2$. At tree-level, we also have the virtual $Z$ exchange contribution induced by the interaction , which is illustrated in Fig. \[fig:z\]. As discussed above, the interaction is suppressed by a factor of $|\mu|^{-2}$, and thus their contribution is subdominant. Notice that ordinary strategies on the searches for charginos and neutralinos at the LHC rely on the leptonic decay channel $\widetilde{\chi}^0_2 \to \widetilde{\chi}^0_1 \ell^+ \ell^-$ induced by this contribution [@Aad:2014vma; @Khachatryan:2014qwa]. Since the decay branch into the channel is extremely suppressed in our scenario, we need an alternative way to probe the bino-wino coannihilation region at the LHC. This is the subject of the next section. The two-body decay process $\widetilde{\chi}^0_2 \to \widetilde{\chi}^0_1 \gamma$ shown in Fig. \[fig:gamma\] is induced by the dipole-type operator ${\cal O}^{(5)}_2$, which is generated at one-loop level. As discussed above, this contribution is also suppressed by $|\mu|^{-2}$. As a result, it turns out that the three-body decay processes in Fig. \[fig:h\] dominate this two-body process in most of parameter space we are interested in. One may think that the gauge interactions also induces the process in Fig. \[fig:gamma\] via the chargino-$W$ boson loop diagram. We find, however, that this contribution vanishes. To see the reason, notice that the amplitude of the diagram represented in Fig. \[fig:gamma\] is odd under the charge conjugation $C$, as $\widetilde{\chi}^0_1$ and $\widetilde{\chi}^0_2$ are Majorana fields and a photon is $C$-odd. On the other hand, as noted above, the gauge interactions in Eq.  preserve the $C$ symmetry. The electromagnetic interaction is also invariant under the charge conjugation. Then, it follows that any amplitude induced by these interactions should be $C$-even, hence their contribution to the process in Fig. \[fig:gamma\] vanishes. After all, the $\widetilde{\chi}^0_2 \to \widetilde{\chi}^0_1 \gamma$ decay is sub-dominant as long as $|M_1|, |M_2|, \ll |\mu|$, and therefore we focus on the tree-level processes in Fig. \[fig:treewb\] in the following analysis. However, there are several possibilities in which the above conclusion should be altered. Firstly, if the Higgsino mass is rather small, the contribution of the dipole operator ${\cal O}^{(5)}_2$ can be significant. Indeed, such a situation may be realized in the framework of spread SUSY, as discussed in Refs. [@Hall:2011jd; @Evans:2014pxa; @Nagata:2014wma]. If the decay branch into the $\widetilde{\chi}^0_2 \to \widetilde{\chi}^0_1 \gamma$ channel is sizable, the collider signature discussed in the subsequent section would be modified. Secondly, as already noted above, the Higgs exchange contribution decreases if $\tan\beta$ is large. Such a situation may occur if the SUSY breaking scale is as low as ${\cal O}(10)$ TeV. Again, in this case, the decay branching ratios may change considerably. Thirdly, if the mass difference $\Delta M$ is smaller than ${\cal O}(10)$ GeV, then the $\widetilde{\chi}^0_2 \to \widetilde{\chi}^0_1 b \bar{b}$ decay mode is highly suppressed, and other decay modes can be significant, such as one-loop diagrams shown in Fig. \[fig:box\]. In addition, the two-body decay process containing a bottomonium induced by the virtual Higgs exchange may also be important in such a situation. Since these possibilities are only relevant to rather specific parameter region in our setup, we do not consider them in this paper, and discuss these contributions on another occasion. In summary, the mass spectrum for EW gauginos and the suppression factor in their decay rates are illustrated in Fig. \[fig:spectrum\]. We consider a specific collider signature for this mass spectrum in the following section. Now we evaluate the decay length $c\tau_{\widetilde{W}^0}$ of the neutral wino-like state $\widetilde{\chi}^0_2$. In Fig. \[fig:dl1\], we plot it in the $M_{\widetilde{B}}$–$\Delta M$ plane. Here, we set $\mu =+100$ TeV ($+25$ TeV) and $\tan\beta = 1$ (30) in the black solid (red dashed) lines. The gaugino masses are taken to be real and positive in this figure. The blue dashed line corresponds to the parameter region where the thermal relic abundance of the bino LSP agrees to the observed DM density $\Omega_{\text{DM}} h^2 = 0.12$. As can be seen from this figure, a neutral wino has a sizable decay length in a wide range of parameter region motivated by the bino-wino coannihilation scenario. This observation is a key ingredient for the search strategy discussed in the next section. In Fig. \[fig:dl2\], on the other hand, the decay length $c \tau_{\widetilde{W}^0}$ is plotted in the $\mu$–$\tan\beta$ plane. Here, we set $M_1 = +400$ GeV and $M_2=+430$ GeV. For reference, we also show the 1$\sigma$ (2$\sigma$) preferred region for the 125 GeV Higgs mass in the dark (light) red shaded area, where the gluino mass is set to be 1.5 TeV and the scalar mass scale is taken to be equal to the Higgsino mass, $\widetilde{m} = \mu$. We find that the decay length grows as $\tan \beta$ is taken to be large. This is because the Higgs interaction is suppressed in this case. Anyway, a neutral wino has a decay length of $\gtrsim 1$ cm over the parameter region in this figure, which could be observed at the LHC as we see in the next section. Before closing this section, let us comment on the bino-wino coannihilation. One may wonder the chemical equilibrium between the wino and bino, which is essential for the successful coannihilation, can be kept even if $\mu$ is significantly large. Unlike the bino-gluino coannihilation case [@Nagata:2015hha; @Ellis:2015vaa], the bino and wino can interact with each other through the dimension-five operators. Thus, the interchange of $\widetilde B \leftrightarrow \widetilde W$ at the freeze-out temperature is rapid enough for the chemical equilibrium between them to be maintained, even though $\mu$ is at PeV scale. LHC search {#sec:lhcsearch} ========== We have seen in the previous section that a neutral wino in our scenario has a sizable decay length. In this section, we discuss the prospects of the searches for such a long-lived wino at the LHC. A decay process of a neutral wino with $c \tau_{\widetilde{W}^0} \gtrsim 1$ mm can be observed as a DV plus missing transverse energy event. The ATLAS collaboration has reported the result of searches for such events by using the 20.3 fb$^{-1}$ data set collected at the LHC with a center-of-mass energy of $\sqrt{s}=8$ TeV [@Aad:2015rba]. In Sec. \[sec:atlasstudy\], we begin with reviewing this ATLAS result for the DV search. It turns out, however, that the existing result cannot be directly applied to the bino-wino coannihilation scenario, since the ATLAS study mainly focus on high-mass DVs. Thus, we need to extend their analysis so that a smaller mass DV can also be probed. We describe our setup in Sec. \[sec:setup\]. Finally, in Sec. \[sec:prospect\], we show the prospects for the long-lived wino search in the cases of both direct and gluino mediated productions. Previous LHC study {#sec:atlasstudy} ------------------ The LHC finished the operation at center of mass energy of 8 TeV in 2012. The ATLAS and the CMS experiments explored many types of neutral long-lived SUSY particles so far, which are listed as follows [@Aad:2015rba; @TheATLAScollaboration:2013yia; @Aad:2012zx; @*Aad:2011zb; @CMS:2014wda; @Aad:2014gfa; @CMS:2014hka; @Khachatryan:2014mea; @Aad:2015asa; @Aad:2015uaa]: - Long-lived gluino in the mini-split SUSY which decays into two quarks and a neutralino. - Long-lived neutralino in the $R$-parity violation scenario which decays into leptons and quarks. - Long-lived neutralino in the gauge mediation scenario which decays into a photon or $Z$ boson, and a gravitino. In these searches, the decay products from the long-lived particles have large impact parameters, which are distinct from background events that originate from the primary vertex. Our search strategy proposed in this paper mainly considers signals including two quarks from a DV, and thus the long-lived gluino search in the mini-split SUSY performed by the ATLAS experiment [@Aad:2015rba] would be a good reference. The ATLAS study [@Aad:2015rba] employs several triggers for long-lived particles. Among them, the trigger utilizing missing transverse energy ($E^{\rm{miss}}_{\rm{T}}$) has the best sensitivity for a long-lived gluino; $E^{\rm{miss}}_{\rm{T}} > 80$ GeV is required for the trigger, $E^{\rm{miss}}_{\rm{T}} > 100$ GeV by offline filters, and eventually $E^{\rm{miss}}_{\rm{T}} > 180$ GeV at the offline selection. Reconstruction of DVs is CPU-intensive due to many tracks with high impact parameters. In order to reduce the computation time, only the events with two “trackless” jets of $P_{\rm{T}} > 50$ GeV are processed. The “trackless” jets are defined such that the scalar sum of the transverse momenta of the tracks in the jet should be less than 5 GeV with the standard track reconstruction. Such jets could be accompanied by decays of long-lived particles which occur away from the primary vertex. At the end of the selections, a DV with more than four tracks whose invariant mass is more than 10 GeV is treated as a signal event. Here, each track is supposed to have the mass of the charged pion $m_{\pi^{\pm}}$. After the selections, the reconstruction efficiency of at least a DV from a pair of produced gluinos with a decay length of $c\tau_{\widetilde{g}}={\cal O}(1)~\rm{mm}$ is almost zero, which increases up to more than 50% when $c\tau_{\widetilde{g}}={\cal O}(10)~\rm{mm}$. Here, a gluino mass of 1400 GeV is assumed in the estimation. Beyond the point, the reconstruction efficiency gradually decreases and reaches zero for $c\tau_{\widetilde{g}} \gtrsim{\cal O}(1)~\rm{m}$. In the bino-wino coannihilation scenario, a favored value of $\Delta M$ is $\lesssim 30$ GeV, as can be seen from Fig. \[fig:dl1\]. In this case, jets from a decay of a neutral wino are too soft to satisfy the $P_\text{T}$ condition of the trackless jets. To probe the scenario with the DV searches, therefore, we need to relax the above requirements to some extent, which we discuss in what follows. Signal simulation setup {#sec:setup} ----------------------- Now we adjust the ATLAS DV search method such that it has sensitivity to long-lived neutral winos, and discuss its prospects at the 14 TeV LHC running. At the LHC, there are two channels to produce a neutral wino. One is its direct production and the other is gluino mediated production in which a produced gluino decays into a neutral wino. In both cases, the $E^{\rm{miss}}_{\rm{T}}$ trigger is the most efficient among the DV search triggers, just like the long-lived gluino search. In this study, we adopt $E^{\rm{miss}}_{\rm{T}}>100$ GeV and 200 GeV for the 8 TeV and 14 TeV LHC cases, respectively. As for the DV detection criteria, we assume the same setup as the ATLAS study: - The number $N_{\rm tr}$ of charged tracks forming a DV should be grater than four, where each track should have a transverse momentum of $P_{\rm T}>1$ GeV. - The invariant mass of the sum of the momenta of the tracks, $m_{\rm DV}$, should be greater than 10 GeV, with the mass of each track being assumed to be $m_{\pi^{\pm}}$. Notice that we have dropped the $P_{\text{T}}$ condition for the trackless jets adopted in the case of the long-lived gluino search. To see expected size of $m_{\rm DV}$, in Fig. \[mdv\], we show the distributions of $m_{\rm DV}$. Here we consider the wino direct production case, with setting the wino and bino masses to be $M_{\widetilde{W}}=400$ GeV and $M_{\widetilde{B}} = 370$ GeV, respectively. A produced neutral wino is assumed to decay into a bino via the Higgs exchange process. As can be seen, $m_{\rm DV}$ is expected to be $\sim 10$ GeV, if $\Delta M = 30$ GeV, and a sizable number of events provide $m_{\rm DV}>10$ GeV. We also need the DV detection efficiency to estimate the number of signals. We estimate it from the result of the ATLAS gluino search [@Aad:2015rba]. Figure 19 of Ref. [@Aad:2015rba] shows a lower-bound on the gluino mass as a function of the gluino decay length $c\tau_{\widetilde g}$, with the LSP mass of 100 GeV being assumed. In this case, the DV+$E^{\rm{miss}}_{\rm{T}}$ search provides the strongest constraints in most of the parameter region. The mass difference between the gluino and the LSP is typically greater than 1 TeV in this plot. Since the transverse momenta of jets and the LSP produced by the gluino decay are large, acceptance rates for the trigger and the event filter are expected to be almost 100%. Moreover, the acceptance rate of the truth-level events for the requirements of the DV criteria and the trackless jets is also expected to be almost 100%. As a consequence, we can estimate the decay-length dependent detection efficiency of DVs, $\epsilon_{\widetilde{g}}$, as $$\begin{aligned} \epsilon_{\widetilde{g}} (c\tau) = \frac{\sigma_{\rm obs} }{\sigma({ p p \to \widetilde g \widetilde g})}~, \end{aligned}$$ where $\sigma_{\rm obs} ~(= 0.15~{\rm fb})$ is observed upper-bound on the cross section of the DV+$E^{\rm{miss}}_{\rm{T}}$ events [@Aad:2015rba]. In this case, for each long-lived gluino DV event, the number of DVs is two. Therefore, the reconstruction efficiency of a single DV, $\epsilon_{\text{DV}}$, is about a half of $\epsilon_{\widetilde{g}}$. For $c\tau_{\widetilde g} \gtrsim 1$ m, the DV reconstruction efficiency would be inversely proportional to decay length, since gluinos must decay within the tracker system. Although Fig. 19 of Ref. [@Aad:2015rba] only shows the case of $1<c\tau_{\widetilde g} <1000$ mm, we extrapolate the efficiency up to $c\tau_{\widetilde g} > 1$ m taking into account a suppression factor of $\propto (c\tau_{\widetilde g})^{-1}$. In Fig. \[eff\], we show the estimated efficiency $\epsilon_{\text{DV}}$ as a function of the decay length of the neutral wino. This figure shows that the DV reconstruction efficiency is sizable for $1~\text{cm}\lesssim c\tau_{\widetilde{W}^0} \lesssim 1$ m, which is maximized when $c\tau_{\widetilde{W}^0} \sim 10$ cm. This estimation assumes the velocity distribution of the long-lived winos to be the same as that of long-lived gluinos in Ref. [@Aad:2015rba]. Although these two distributions are potentially different, we do not expect that this difference alters our results drastically. The total acceptance rate is the product of the efficiency and the rate of passing the above $E^{\rm{miss}}_{\rm{T}}$ triggers and the DV criteria. To estimate the acceptance rate, we use the program packages [Madgraph5]{} [@Alwall:2011uj], [Pythia6]{} [@Sjostrand:2006za], and [Delphes3]{} [@deFavereau:2013fsa], while for the cross sections of the SUSY particles we use [Prospino2]{} [@Beenakker:1996ed]. Note that the estimation of the prospects discussed below possibly suffers from large uncertainties. At first, the DV detection efficiency depends on the momentum distributions of the particles which generate DVs. The small mass difference $\Delta M$ reduces the number of tracks and makes their momenta lower, which may impair the DV reconstruction efficiency. On the other hand, the current DV criteria is designed for the high-mass DVs, and not optimized for the low-mass DVs like the present wino case. A small modification in the DV criteria can drastically change the acceptance rate as seen in Fig. \[mdv\] and can potentially improve the detection efficiency. Moreover, although the present DV reconstruction efficiency is obtained for DVs associated with light-quark jets, we assume the neutral wino mainly decays into $b$ jets in the following analysis. With the present ATLAS analysis, the reconstruction efficiency of a DV accompanied with $b$ jets is expected to be worse than those with light quarks. However, the neutral wino decaying into two $b$ quarks provides characteristic signatures compared to light-quark decays; namely, the $b$ quarks cause additional DVs originating from a DV given by a decay of a long-lived neutral wino. This specific signature may give further optimization of the search of the wino DV and improve the reconstruction efficiency in the future LHC experiments. For more precise estimation, we need full detector simulation and this is out of the scope of this paper. We use the above simplified setup to see the prospects qualitatively in what follows. LHC Prospects {#sec:prospect} ------------- First, let us discuss the case where winos are directly produced. The crucial difference between the ATLAS gluino search and the present direct wino search is the sizes of the missing transverse momentum and the invariant mass of the tracks from DVs, due to the small mass difference between the wino and bino. Both factors reduce the signal acceptance of the wino processes. The missing energy mainly comes from the back reaction of the initial state radiations in the wino production. For $M_{\widetilde{W}} = 400$ GeV and $\Delta M = 30$ GeV, the acceptance rates for the missing energy ($E^{\rm{miss}}_{\rm{T}}>100$ GeV for the 8 TeV running and 200 GeV for 14 TeV) and DV are about 3% and 1%, respectively. Here, we assume the neutral wino decays into a pair of bottom quarks and a bino via the Higgs boson exchange process. In Fig. \[direct-wino\], we show the prospects for the long-lived wino search at the LHC. Here we assume zero background and require three signal events. The mass difference $\Delta M$ is set to be 30 GeV. The red and blue solid lines show the prospects for 8 TeV and 14 TeV LHC run, respectively. Note that this estimation is based on our simplified method described in the previous subsection. The DV reconstruct efficiency in the real detector could be different, since the DV masses in the current case are much smaller than those expected for the ATLAS model points [@Aad:2015rba]. Moreover, $b$ jets from DVs may worsen the efficiency. On the other hand, the current ATLAS analysis is not optimized for the low-mass DV and $b$ jets, and thus future development on search techniques for such DVs may improve the efficiency. These possibilities result in large uncertainties in the present estimation. Here, we estimate the uncertainties by scaling the acceptance rate by factors of three and one third, and show them as the bands in the figure. For reference, we also show the decay length of the neutral wino for $\mu=25$, 100, 500 TeV with $\tan\beta=2$ in the black solid, dashed, and dotted lines, respectively, with $\Delta M$ taken so that it realizes the correct DM abundance. From this figure, we find that using the DV searches we may probe a wino with a mass of 400 GeV (800 GeV) at the 8 TeV (14 TeV) LHC if its decay length is ${\cal O}(10)$ cm. This should be contrasted with other wino searches which do not use DVs. Without DVs, the discovery of a wino having $\Delta M ={\cal O}(10)$ GeV is extremely difficult; for such a neutral wino decaying into a bino via the Higgs boson exchange, the soft leptons or mono-jet plus missing energy searches at the LHC can provide essentially no constraints on this mass spectrum. Next, let us discuss the case in which winos are produced from decays of gluinos. If gaugino masses are determined by only the anomaly mediation effects and the threshold effects of the Higgsino loop, the gluino mass is about 2–4 times larger than the wino mass in the case of the bino-wino coannihilation. If the gluino mass is not so large, wino productions through gluino decays are also active. Unlike the direct wino production case, the acceptance rate of the missing energy trigger is close to 100% as long as the gluino-wino mass difference is sufficiently large. The rate for passing the above DV criteria is around 5%. The gluino decay mode strongly depends on the squark mass sector. Here we assume that left-handed squarks dominantly contribute the gluino decay, and the branching fraction of a gluino decaying into a wino, $\widetilde{g} \to \widetilde{W}^0$, is about 30%. In Fig. \[gluino\], we show the prospects for the wino search via this channel. Here, we take $M_{\widetilde g}=2 M_{\widetilde W}$, and $\Delta M = 30$ GeV. Again, the black solid, dashed, and dotted lines show contours corresponding to $\mu = 25$, 100, and 500 TeV, respectively, with $\tan\beta=2$ and $\Delta M$ taken so that it gives the correct DM abundance. We find that in this case the 14 TeV LHC reach for the gluino mass is as high as 2 TeV when $c\tau_{\widetilde{W}^0} \sim 10$ cm. For the gluino-mediated case, a sizable mass difference between gluino and wino leads to rather high jet activity and large missing energy. Therefore, usual jets plus missing energy search (without DVs) can probe a wide range of gluino masses. For $M_{\widetilde{g}}=2 M_{\widetilde{W}}$, a gluino with a mass of 1 TeV (2 TeV) can be probed with conventional jets plus missing energy search at the LHC8 (14). The gluino search with DVs discussed here may not drastically improve the gluino discovery range, since the DV reconstruction efficiency is quite low. Optimization of the DV criteria may increase this efficiency, though. Even if gluinos are first discovered in the ordinary jets plus missing energy searches, however, it is still important to search for the gluino-mediated wino production events using DVs as well; it is difficult for the former searches to distinguish the bino-wino coannihilation scenario from merely the wino LSP case, while this is possible for the DV searches. In addition, the DV search strategy may allow us to extract the bino-wino mass difference through energy measurements of decay product, which is quite important to test the bino-wino coannihilation scenario. Conclusion and discussion {#sec:conclusion} ========================= In this paper, we study the neutral wino decay in the bino-wino coannihilation scenario and discuss its collider signature. We find that the neutral wino has a considerably long lifetime in the mini-split spectrum, and is detectable at the LHC by means of the DV searches. To assess the prospects for the detectability, we study the direct and gluino-mediated productions of the neutral wino at the 8 and 14 TeV LHC running. It turns out that winos (gluinos) with a mass of 800 GeV (2 TeV) can be probed at the next stage of the LHC in the former (latter) production case. The search for the directly produced winos with DVs is very powerful, compared to the conventional searches. For the gluino, this is not so efficient and comparable to the conventional jets plus missing energy searches. However, note that this conclusion can be altered since our estimation of the detection rate has potentially large uncertainty. A more realistic detector simulation is needed to obtain robust prospects. In this paper, we focus on the parameter region where the tree-level contributions are dominant. However, as discussed above, higher loop processes may significantly contribute to the wino decay in certain parameter region: a moderate $\mu$ case $|\mu|\lesssim {\cal O}(10)$ TeV, a high-degeneracy mass case $\Delta M\sim {\cal O}(1)$ GeV, and so on. In these cases, the decay branches of wino are altered, accordingly we may need to modify our collider search strategy. Detailed study with a more realistic detector simulation and more accurate evaluation of the wino decay will be done elsewhere. If the Higgsino mass is rather small, then the DV search strategy discussed in this paper does not work as the neutral wino decay length is too short, $c\tau_{\widetilde{W}^0}<1$ mm. However, it is possible to probe such a region with low-energy precision experiments like the DM direct detection experiments and the measurements of electric dipole moments [@Nagata:2014wma; @Hisano:2014kua; @Nagata:2014aoa]. The interplay between the DV search and these experiments will also be discussed on another occasion. Lastly let us comment on the prospects of future lepton colliders. The high-energy lepton collider is very powerful tool to probe directly or indirectly the wino sector [@Gunion:2001fu; @Harigaya:2015yaa]. However it is non-trivial whether the lepton collider can observe the wino DV signal, since the neutral wino production cross section is quite small. Still, neutral winos may be produced through a loop process or multi-particle production channel. In contrast, they can be considerably produced both directly and indirectly at the LHC. In this sense, the LHC experiment is quite suitable for the DV wino search. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Kazuki Sakurai for informing us of his related work in Ref. [@Rolbiecki:2015gsa]. The work of N.N. is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.

--- abstract: 'For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $({\overline}{X},{\overline}{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on ${\overline}{X}$, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated to the complete conformal metric $g={\overline}{g}/\rho^2$ where $\rho$ is a smooth function on ${\overline}{X}$ which equals the distance to the boundary near ${\partial}{\overline}{X}$. We show that ${\frac{1}{2}}({\rm Id}+{\widetilde}{S}(0))$ is the orthogonal Calderón projector, where ${\widetilde}{S}({\lambda})$ is the holomorphic family in $\{\Re({\lambda})\geq 0\}$ of normalized scattering operators constructed in [@GMP], which are classical pseudo-differential of order $2{\lambda}$. Finally we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.' address: - | DMA, U.M.R. 8553 CNRS\ Ecole Normale Supérieure,\ 45 rue d’Ulm\ F 75230 Paris cedex 05\ France - | Institutul de Matematică al Academiei Române\ P.O. Box 1-764\ RO-014700 Bucharest, Romania - | School of Mathematics\ Korea Institute for Advanced Study\ 207-43\ Hoegiro 87\ Dongdaemun-gu\ Seoul 130-722\ Republic of Korea author: - Colin Guillarmou - Sergiu Moroianu - Jinsung Park title: Bergman and Calderón projectors for Dirac operators --- Introduction ============ Let $({\overline}{X},{\overline}{g})$ be a compact spin Riemannian manifold with boundary, and denote by $(M,h)$ its boundary with the induced spin structure and Riemannian metric. Let also $D_{\bar{g}}$ denote the associated Dirac operator acting on the spinor bundle $\Sigma$ over ${{\overline}{X}}$. The purpose of this paper is to clarify some aspects of the interaction between the space of smooth spinors of $D_{\bar{g}}$ on ${{\overline}{X}}$ which are harmonic in the interior, and the space of their restrictions to the boundary. More precisely, we will examine the orthogonal projectors on these spaces in $L^2$ sense, the operator of extension from the boundary to a harmonic spinor, and its adjoint. Before stating our results in the general case, let us review the situation for the unit disc where one can give explicit constructions for these objects. Example: the unit disc {#example-the-unit-disc .unnumbered} ---------------------- Keeping the notation $({{\overline}{X}},\bar{g})$ for the closed unit disc in $\mathbb{C}$ equipped with the Euclidean metric and $M=S^1$, let $$\label{chcc} {\mathcal{H}}:=\{\phi\in C^\infty({{\overline}{X}});D_{\bar{g}}\phi=0\}, \qquad {\mathcal{H}}_{\partial}:=\{\phi|_{M};\phi\in{\mathcal{H}}\}$$ where for the moment $D_{\bar{g}}={\overline{\partial}}$ is the Cauchy-Riemann operator. The functions $z^k,k\geq 0$ clearly are dense in ${\mathcal{H}}$ with respect to the $L^2$ norm. Their restrictions to the boundary $e^{ikt}$, $k\geq 0$, span the space of those smooth functions whose Fourier coefficients corresponding to negative frequencies vanish. The orthogonal projection $P_{{\mathcal{H}}_{{\partial}}}$ onto the $L^2$-closure of ${\mathcal{H}}_{\partial}$ is easily seen to be pseudodifferential; if $A={-}id/dt$ is the self-adjoint Dirac operator on $M$, then $P_{{\mathcal{H}}_{{\partial}}}$ is the Atiyah-Patodi-Singer projection on the [non-negative]{} part of the spectrum of $A$, whose kernel is given by $(2\pi(1-z\bar{w}))^{-1}$ with respect to the measure $dt$ where $w=e^{it}$. Let $K:C^\infty(M)\to C^\infty({{\overline}{X}})$ be the operator which to $\phi_{|M}\in {\mathcal{H}}_{\partial}$ associates $\phi$, extended by $0$ on the orthogonal complement of ${\mathcal{H}}_{\partial}$. Then $K$ has a smooth kernel on ${X}\times M$ where $X=\mathrm{int}({\overline}{X})$ given by $$\label{form1} K(z,w)= \frac{1}{2\pi(1-z{\overline}{w})}$$ with respect to the standard measure on the circle, where $w=e^{it}$. This kernel extends to ${{\overline}{X}}\times M$ with a singularity at the boundary diagonal $\{(z,w);z=w\}$. If we set $$\begin{aligned} z=(1-x)e^{i(t+y)},&&\rho:=\sqrt{x^2+y^2}\end{aligned}$$ we see that the leading term of the singularity is $\rho^{-1}$, moreover $K(z,w)$ admits a power series expansion near $\rho=0$. The coefficients live on the “polar coordinates”, or blow-up space which will play an essential role in the rest of this paper. The adjoint of $K$, denoted by $K^*$, has a smooth kernel on $M\times X$ with respect to the standard measure $\frac{1}{2 i}dz\wedge d{\overline}{z}$, given formally by . This has the same type of singularity as $K$ near $\{z=w\}$. The kernel of $K^*K$ on $M\times M$ is given by $$-\frac{1}{4\pi}\frac{\log(1-z{\overline}{w})}{z{\overline}{w}},$$ which is the kernel of a classical pseudo-differential operator of order $-1$ (actually given by ${\frac{1}{2}}P_{{\mathcal{H}}_{{\partial}}}(|D_t|+1)^{-1}$). The remaining composition $KK^*$ has a smooth kernel on $X\times X$ given by $(2\pi (1-z{\overline}{w}))^{-1}$ with respect to the Euclidean measure in $w$. This kernel extends to ${{\overline}{X}}\times {{\overline}{X}}$ with the same type of singularity as in the case of $K$ and $K^*$, only that now the singular locus is of codimension $3$, and there are two, instead of one, extra boundary hyperfaces. To finish our example, consider the projector on (the closure of) ${\mathcal{H}}$. Its kernel with respect to $\frac{1}{2i}dw\wedge d{\overline}{w}$ is $$\frac{1}{\pi(1-z{\overline}{w})^2}$$ which is of the same nature as the kernel of $KK^*$ but with a higher order singularity. Harmonic spinors on manifolds with boundary {#harmonic-spinors-on-manifolds-with-boundary .unnumbered} ------------------------------------------- One can extend the above example to higher complex dimensions. One direction would be to study holomorphic functions smooth up to the boundary, however in this paper we will consider another generalization. Let thus ${\overline{X}}$ be a compact domain in ${\mathbb{C}}^n$, and $D_{\bar{g}}={\overline{\partial}}+{\overline{\partial}}^*$ acting on $\Lambda^{0,*}X$. A form is called *harmonic* if it belongs to the nullspace of $D_{\bar{g}}$. Then the above analysis of the operators $K,K^*$ and of the projection on the space of harmonic forms can be carried out, describing the singularities of the kernels involved. In fact, even more generally, we will consider the Dirac operator $D_{\bar{g}}$ acting on the spinor bundle $\Sigma$ over a compact spin manifold ${\overline}{X}$ with boundary $M$. We assume that the metric ${\overline}{g}$ on ${{\overline}{X}}$ is smooth at the boundary but not necessarily of product type (which would mean that the gradient of the distance function $\rho$ to the boundary were Killing near the boundary). We then denote by ${\mathcal}{H}(D_{\bar{g}})$ and ${\mathcal{H}}_{\partial}(D_{\bar{g}})$ the space of smooth harmonic spinors and the Cauchy data space of $D_{\bar{g}}$ respectively, $$\begin{aligned} {\mathcal}{H}(D_{\bar{g}}):=\{\phi\in C^{\infty}({\overline}{X};\Sigma); D_{\bar{g}}\phi=0\},&& {\mathcal}{H}_{\partial}(D_{\bar{g}}):=\{\phi|_{M}; \phi\in {\mathcal}{H}(D_{\bar{g}})\}\end{aligned}$$ and let ${\overline}{{\mathcal}{H}}(D_{\bar{g}})$ and ${\overline}{{\mathcal}{H}}_{{\partial}}(D_{\bar{g}})$ be their respective $L^2$ closures. When the dependence on $D_{\bar{g}}$ is clear, we may omit $D_{\bar{g}}$ in the notations ${\mathcal{H}}(D_{\bar{g}})$, ${\mathcal{H}}_{{\partial}}(D_{\bar{g}})$ for simplicity. We denote by $P_{{\overline}{{\mathcal}{H}}}$ and $P_{{\overline}{{\mathcal}{H}}_{\partial}}$ the respective orthogonal projectors on ${\overline}{{\mathcal}{H}}$ (that we call the *Bergman projector*) and ${\overline}{{\mathcal}{H}}_{\partial}$ (the *Calderón projector*) for the $L^2$ inner product induced by ${\overline}{g}$ and ${\overline}{g}|_{M}$. Let $K:L^2(M,\Sigma)\to L^2({\overline}{X},\Sigma)$ be the *Poisson operator*, i.e., the extension map which sends ${\overline}{{\mathcal}{H}}_{\partial}$ to ${\overline}{{\mathcal}{H}}$, that is, $D_{\bar{g}}K\psi=0$ and $K\psi|_{{\partial}{\overline}{X}}=\psi$ for all $\psi\in {\mathcal}{H}_{{\partial}}$, and denote by $K^*:L^2({{\overline}{X}},\Sigma)\to L^2(M,\Sigma)$ its adjoint. The main results in this paper concern the structure of the Schwartz kernels of these operators, which also gives new proofs for some known results. Let us remark that the construction of the orthogonal projector $P_{{\overline}{{\mathcal}{H}}_{\partial}}$ called here *Calderón projector*, and its applications, have been a central subject in the global analysis of manifolds with boundary since the works of Calderón [@Calderon] and Seeley [@Seeley66], [@Seeley69]. The Calderón projector of Dirac-type operators turned out to play a fundamental role in geometric problems related to analytic-spectral invariants. This was first observed by Bojarski in the linear conjugation problem of the index of a Dirac type operator [@Bor]. Following Bojarski, Booss and Wojciechowski extensively studied the geometric aspect of the Calderón projector [@BoW]. The Calderón projector also appears in the gluing formulae of the analytic-spectral invariants studied in [@Nic], [@SW], [@KL], [@LP] since the use of the Calderón projector provides us with more refined proofs of these formulae in more general settings. We also refer to [@Epstein1], [@Epstein2] for an application of the Calderón projector of the ${\rm Spin}\sb {\mathbb C}$ Dirac operator, and a recent paper of Booss-Lesch-Zhu [@BLZ] for other generalizations of the work in [@BoW]. Extensions of the Calderón projector for non-smooth boundaries were studied recently in [@AmNis; @loya]. Polyhomogeneity {#polyhomogeneity .unnumbered} --------------- Before we state the main results of this paper, let us fix a couple of notations and definitions. If $W$ and $Y$ are smooth compact manifolds (with or without boundary) such that the corner of highest codimension of $W{\times}Y$ is diffeomorphic to a product $M{\times}M$ where $M$ is a closed manifold, we will denote, following Mazzeo-Melrose [@MM], by $W{\times}_0Y$ the smooth compact manifold with corners obtained by blowing-up the diagonal $\Delta$ of $M{\times}M$ in $W{\times}Y$, i.e., the manifold obtained by replacing the submanifold $\Delta$ by its interior pointing normal bundle in $W{\times}Y$ and endowed with the smallest smooth structure containing the lift of smooth functions on $W{\times}Y$ and polar coordinates around $\Delta$. The bundle replacing the diagonal creates a new boundary hypersurface which we call the *front face* and we denote by ${\textrm{ff}}$. A smooth boundary defining function of ${\textrm{ff}}$ in $W{\times}_0Y$ can be locally taken to be the lift of $d(\cdot,\Delta)$, the Riemannian distance to the submanifold $\Delta$. On a smooth compact manifold with corners $W$, we say that a function (or distribution) has an *integral polyhomogeneous expansion* at the boundary hypersurface $H$ if it has an asymptotic expansion at $H$ of the form $$\label{intphgexp} \sum_{j=-J}^\infty\sum_{\ell=0}^{\alpha(j)} q_{j,\ell}\, \rho_{H}^{j}(\log\rho_H)^{\ell}$$ for some $J\in{\mathbb{N}}_0{:=\{0\}\cup{\mathbb{N}}}$, a non-decreasing function $\alpha:{\mathbb{Z}}\to {\mathbb{N}}_0$, and some smooth functions $q_{j,\ell}$ on ${\rm int}(H)$, where $\rho_H$ denotes any smooth boundary defining function of $H$ in $W$. \[th1\] Let $({\overline}{X},{\overline}{g})$ be a smooth compact spin Riemannian manifold with boundary $M$. Let $K$ be the Poisson operator for $D_{\bar{g}}$ and let $K^*$ be its adjoint. Then the following hold true: 1. The Schwartz kernels of $K$, $K^*$ and $KK^*$ are smooth on the blown-up spaces ${\overline}{X}{\times}_0 M$, $M{\times}_0{\overline}{X}$, respectively ${\overline}{X}{\times}_0{\overline}{X}$ with respect to the volume densities induced by ${\overline}{g}$. 2. [The operator $K^*K$ is a classical pseudo-differential operator of order $-1$ on $M$ which maps $L^2(M,\Sigma)$ to $H^1(M,\Sigma)$, and there exists a pseudo-differential operator of order $1$ on $M$ denoted by $(K^*K)^{-1}$ such that the Calderón projector $P_{{\overline}{{\mathcal}{H}}_{{\partial}}}$ is given by $(K^*K)^{-1}K^*K$. In particular, $P_{{\overline}{{\mathcal}{H}}_{{\partial}}}$ is classical pseudo-differential of order $0$.]{} 3. [The Bergman orthogonal projection $P_{{\overline}{{\mathcal}{H}}}$ from $L^2({\overline}{X},\Sigma)$ to ${\overline}{{\mathcal}{H}}$ is given by $K(K^*K)^{-1}K^*$ and its Schwartz kernel on ${\overline}{X}{\times}_0{\overline}{X}$ is smooth except at the front face ${\rm ff}$ where it has integral polyhomogeneous expansion as in with $\alpha\leq 3$.]{} Note that an alternate description of these kernels in terms of oscillatory integrals is given in Appendix \[appB\]. Our method of proof is to go through an explicit construction of all these operators which does not seem to be written down in the literature in this generality for the Dirac operator, although certainly some particular aspects are well known (especially those involving the Calderón projector $P_{{\overline}{{\mathcal}{H}}_{\partial}}$, see [@BoW]). We use the fundamental property that the Dirac operator is conformally covariant to transform the problem into a problem on a complete non-compact manifold $(X,g)$ conformal to $({\overline}{X},{\overline}{g})$ obtained by simply considering $X:={\rm int}({\overline}{X})$ and $g:={\overline}{g}/\rho^2$ where $\rho$ is a smooth boundary defining function of the boundary $M={\partial}{\overline}{X}$ which is equal to the distance to the boundary (for the metric ${\overline}{g}$) near ${\partial}{\overline}{X}$. This kind of idea is not really new since this is also in spirit used for instance to study pseudoconvex domains by considering a complete Kähler metric in the interior of the domain (see Donnelly-Fefferman [@Donnelly-Fefferman], Fefferman [@Fe], Cheng-Yau [@Cheng-Yau], Epstein-Melrose [@Ep-Mel]), and obviously this connection is transparent for the disc in ${\mathbb{C}}$ via the Poisson kernel and the relations with the hyperbolic plane. One of the merits of this method, for instance, is that we do not need to go through the invertible double of [@BoW] to construct the Calderón projector and thus we do not need the product structure of the metric near the boundary. We finally remark that the bound $\alpha\leq 3$ in (3) of the Theorem is almost certainly not optimal, we expect instead $\alpha\leq 1$ to be true. Conformally covariant operators {#conformally-covariant-operators .unnumbered} ------------------------------- We also obtain, building on our previous work [@GMP], \[th2\] There exists a holomorphic family in $\{{\lambda}\in {\mathbb{C}}; \Re({\lambda})\geq 0\}$ of elliptic pseudo-differential operators ${\widetilde}{S}({\lambda})$ on $M={\partial}{\overline}{X}$ of complex order $2{\lambda}$, invertible except at a discrete set of ${\lambda}$’s and with principal symbol $i{\mathrm{cl}}(\nu){\mathrm{cl}}(\xi)|\xi|^{2{\lambda}-1}$ where $\nu$ is the inner unit normal vector field to $M$ with respect to $\bar{g}$, such that 1. [${\frac{1}{2}}({\rm Id}+{\widetilde}{S}(0))$ is the Calderón projector $P_{{\overline}{{\mathcal}{H}}_{\partial}}$;]{} 2. [For $k\in{\mathbb{N}}_0$, $L_k:=-{\mathrm{cl}}(\nu){\widetilde}{S}(1/2+k)$ is a conformally covariant differential operator whose leading term is $D_M^{1+2k}$ where $D_M$ denotes the Dirac operator on $M$, and $L_0=D_M$.]{} By using the existence of ambient (or Poincaré-Einstein) metric of Fefferman-Graham [@FGR; @FGR2], this leads to the construction of natural conformally covariant powers of Dirac operators in degree $2k+1$ on any spin Riemannian manifolds $(M,h)$ of dimension $n$, for all $k\in {\mathbb{N}}_0$ if $n$ is odd and for $k\leq n/2$ if $n$ is even. We explicitly compute $L_1$. \[corL1\] Let $(M,h)$ be a Riemannian manifold of dimension $n\geq 3$ with a fixed spin structure, and denote by ${\mathrm{scal}},{\mathrm{Ric}}$ and $D$ the scalar curvature, the Ricci curvature, and respectively the Dirac operator with respect to $h$. Then the operator $L_1$ defined by $$L_1:=D^3 -\frac{{\mathrm{cl}}(d({\mathrm{scal}}))}{2(n-1)}-\frac{2\,{\mathrm{cl}}\circ{\mathrm{Ric}}\circ\nabla}{n-2} +\frac{{\mathrm{scal}}}{(n-1)(n-2)}D$$ is a natural conformally covariant differential operator: $$\hat{L}_1= e^{-\frac{n+3}{2}\omega}L_1e^{\frac{n-3}{2}}$$ if $\hat{L}_1$ is defined in terms of the conformal metric $\hat{h}=e^{2\omega}h$. Cobordism invariance of the index and local Wodzicki-Guillemin residue for the Calderón projector {#cobordism-invariance-of-the-index-and-local-wodzicki-guillemin-residue-for-the-calderón-projector .unnumbered} ------------------------------------------------------------------------------------------------- As a consequence of Theorem \[th2\] and the analysis of [@GMP], we deduce the following Let $({\overline}{X},{\overline}{g})$ be a smooth compact spin Riemannian manifold with boundary $M$. 1. The Schwartz kernel of the Calderón projector $P_{{\overline}{{\mathcal}{H}}_{\partial}}$ associated to the Dirac operator has an asymptotic expansion in polar coordinates around the diagonal without log terms. In particular, the Wodzicki-Guillemin local residue density of $P_{{\overline}{{\mathcal}{H}}_{\partial}}$ vanishes. 2. When the dimension of $M$ is even, the spinor bundle $\Sigma$ splits in a direct sum $\Sigma_+\oplus\Sigma_-$. If $D^+_M$ denotes $D_M|_{\Sigma_+}:\Sigma_+\to \Sigma_-$, then the index ${\rm Ind}(D^+_M)$ is $0$. As far as we know, the first part of the corollary is new. It is known since Wodzicki [@Wod] that the *global* residue trace of a pseudo-differential projector of order $0$ vanishes, however the local residue density does not vanish for general projectors (e.g. see [@Gil]). What is true is that the APS spectral projector has also vanishing local residue, a fact which is equivalent to the conformal invariance of the eta invariant. For metrics of product type near the boundary the Calderón and APS projectors coincide up to smoothing operators; thus our result was known for such metrics. The second statement is the well-known cobordism invariance of the index for the Dirac operator; there exist several proofs of this fact for more general Dirac type operators (see for instance [@AS; @Moro; @Lesch; @Nico; @Brav]) but we found it worthwhile to point out that this fact can be obtained as a easy consequence of the invertibility of the scattering operator. In fact, a proof of cobordism invariance using scattering theory for cylindrical metrics has been found recently by Müller-Strohmaier [@MuSt], however their approach does not seem to have implications about the Caldéron or Bergman projectors. More general operators {#more-general-operators .unnumbered} ---------------------- Our approach does not seem to work for more general Dirac type operators. However it applies essentially without modifications to twisted spin Dirac operators, with twisting bundle and connection smooth on ${\overline}{X}$. For simplicity of notation, we restrict ourselves to the untwisted case. Acknowledgements {#acknowledgements .unnumbered} ---------------- This project was started while the first two authors were visiting KIAS Seoul, it was continued while C.G. was visiting IAS Princeton, and finished while S.M. was visiting ENS Paris; we thank these institutions for their support. We also thank Andrei Moroianu for checking (with an independent method) the formula for $L_1$ in Corollary \[corL1\]. C.G. was supported by the grant NSF-0635607 at IAS. S.M. was supported by the grant PN-II-ID-PCE 1188 265/2009 and by a CNRS grant at ENS. Dirac operator on asymptotically hyperbolic manifold {#AH} ==================================================== We start by recalling the results of [@GMP] that we need for our purpose. Let $(X,g)$ be an $(n+1)$-dimensional smooth complete non-compact spin manifold which is the interior of a smooth compact manifold with boundary ${\overline}{X}$. We shall say that it is *asymptotically hyperbolic* if the metric $g$ has the following properties: there exists a smooth boundary defining function $x$ of ${\partial}{\overline}{X}$ such that $x^2g$ is a smooth metric on ${\overline}{X}$ and $|dx|_{x^2g}=1$ at ${\partial}{\overline}{X}$. It is shown in [@GRL; @JSB] that for such metrics, there is a diffeomorphism $\psi:[0,{\epsilon})_t{\times}{\partial}{\overline}{X}\to U\subset{\overline}{X}$ such that $$\label{psig} \psi^*g=\frac{dt^2+h(t)}{t^2}$$ where ${\epsilon}>0$ is small, $U$ is an open neighborhood of ${\partial}{\overline}{X}$ in ${\overline}{X}$ and $h(t)$ is a smooth one-parameter family of metrics on ${\partial}{\overline}{X}$. The function $\psi_*(t)$ will be called *geodesic boundary defining function* of ${\partial}{\overline}{X}$ and the metric $g$ will be said *even to order $2k+1$* if ${\partial}^{2j+1}_th(0)=0$ for all $j<k$; such a property does not depend on $\psi$, as it is shown in [@GuiDMJ]. The *conformal infinity* of ${\overline}{X}$ is the conformal class on ${\partial}{\overline}{X}$ given by $$[h_0]:=\{(x^2g)|_{T{\partial}{\overline}{X}}\ ; \ x\textrm{ is a boundary defining function of }{\overline}{X}\}.$$ On ${\overline}{X}$ there exists a natural smooth bundle ${^0T}{\overline}{X}$ whose space of smooth sections is canonically identified with the Lie algebra ${\mathcal}{V}_0$ of smooth vector fields which vanish at the boundary ${\partial}{\overline}{X}$, its dual ${^0T}^*{\overline}{X}$ is also a smooth bundle over ${\overline}{X}$ and $g$ is a smooth metric on ${^0T}{\overline}{X}$. Consider the ${\rm SO}(n+1)$-principal bundle ${^0_o}F({\overline}{X})\to {\overline}{X}$ over ${\overline}{X}$ of orthonormal frames in ${^0T}{\overline}{X}$ with respect to $g$. Since ${\overline}{X}$ is spin, there is a ${\rm Spin}(n+1)$-principal bundle ${^0_s}F({\overline}{X})\to {\overline}{X}$ which double covers ${^0_o}F({\overline}{X})$ and is compatible with it in the usual sense. The $0$-Spinor bundle $^0\Sigma({\overline}{X})$ can then be defined as a bundle associated to the ${\rm Spin}(n+1)$-principal bundle ${^0_s}F({\overline}{X})$, with the fiber at $p\in{\overline}{X}$ $$^0\Sigma_p({\overline}{X})=({^0_s}F_p{\times}S(n+1))/\tau$$ where $\tau:{\rm Spin}(n+1)\to {\rm Hom}(S(n+1))$ is the standard spin representation on $S(n+1)\simeq {\mathbb{C}}^{2^{[(n+1)/2]}}$. If $x$ is any geodesic boundary defining function, the unit vector field $x{\partial}_x:=\nabla^{g}\log(x)$ is a smooth section of ${^0T}{\overline}{X}$. The Clifford multiplication ${\rm cl}(x{\partial}_x)$ restricts to the boundary to a map denoted by ${\rm cl}(\nu)$, independent of the choice of $x$, satisfying ${\rm cl}(\nu)^2=-{\rm Id}$ which splits the space of 0-spinors on the boundary into $\pm i$ eigenspaces $$\begin{aligned} {^0\Sigma_\pm}:=\ker ({\rm cl}(\nu)\mp i),&& {^0\Sigma}|_{M}={^0\Sigma}_+\oplus {^0\Sigma}_-\end{aligned}$$ The Dirac operator $D_g$ associated to $g$ acts in $L^2(X,{^0\Sigma})$ and is self-adjoint since the metric $g$ is complete. Let us denote by $\dot{C}^\infty({\overline}{X},{^0\Sigma})$ the set of smooth spinors on ${\overline}{X}$ which vanish to infinite order at ${\partial}{\overline}{X}$. We proved the following result in [@GMP Prop 3.2]: \[resolvent\] The spectrum of $D_g$ is absolutely continuous and given by the whole real line $\sigma(D_g)={\mathbb{R}}$. Moreover the $L^2$ bounded resolvent $R_\pm({\lambda}):=(D_g\pm i{\lambda})^{-1}$ extends from $\{\Re({\lambda})>0\}$ meromorphically in ${\lambda}\in{\mathbb{C}}\setminus {-{\mathbb{N}}/2}$ as a family of operators mapping $\dot{C}^\infty({\overline}{X},{^0\Sigma})$ to $x^{{\frac{n}{2}}+{\lambda}}C^\infty({\overline}{X},{^0\Sigma})$, and it is analytic in $\{\Re({\lambda})\geq 0\}$. Finally, we have $[x^{-{\frac{n}{2}}-{\lambda}}R_\pm({\lambda})\sigma]|_{{\partial}{\overline}{X}}\in C^{\infty}({\partial}{\overline}{X},{^0\Sigma}_\mp)$ for all $\sigma\in\dot{C}^\infty({\overline}{X},{^0\Sigma})$. Using this result, in [@GMP] we were able to solve the following boundary value problem \[poisson\] Let ${\lambda}\in U:=\{z\in{\mathbb{C}};\Re(z)\geq 0, z\notin {\mathbb{N}}/2\}$. For all $\psi\in C^{\infty}({\partial}{\overline}{X},{^0\Sigma}_\pm)$ there is a unique $\sigma_\pm({\lambda})\in C^{\infty}(X,{^0\Sigma})$ such that there exist $\sigma_\pm^+({\lambda}),\sigma_{\pm}^-({\lambda})\in C^{\infty}({\overline}{X},{^0\Sigma})$ satisfying $\sigma_\pm({\lambda})=x^{{\frac{n}{2}}-{\lambda}}\sigma_\pm^-({\lambda})+x^{{\frac{n}{2}}+{\lambda}}\sigma_\pm^+({\lambda})$ and $$\begin{aligned} \label{eq-sigma} (D_g\pm i{\lambda})\sigma_\pm({\lambda})=0, && \sigma^-_\pm({\lambda})|_{{\partial}{\overline}{X}}=\psi.\end{aligned}$$ Moreover $\sigma_\pm^+({\lambda}),\sigma_\pm^-({\lambda})$ are analytic in ${\lambda}\in U$ and one has $\sigma_\pm^+({\lambda})|_{{\partial}{\overline}{X}}\in C^\infty({\partial}{\overline}{X},{^0\Sigma}_\mp)$. The solution $\sigma_{\pm}({\lambda})$ of Proposition \[poisson\] is constructed in Lemma 4.4 of [@GMP] as a sum $$\label{constsigma} \sigma_\pm({\lambda})=\sigma_{\infty,\pm}({\lambda})-R_\pm({\lambda})(D_g\pm i{\lambda})\sigma_{\infty,\pm}({\lambda})$$ where $\sigma_{\infty,\pm}({\lambda})\in x^{{\frac{n}{2}}-{\lambda}}C^{\infty}({\overline}{X},{^0\Sigma})$ satisfies $$\begin{aligned} \label{sigmainfty} (D_g\pm i{\lambda})\sigma_{\infty,\pm}({\lambda})\in \dot{C}^\infty({\overline}{X},{^0\Sigma}), && [x^{-{\frac{n}{2}}+{\lambda}}\sigma_{\infty,\pm}({\lambda})]|_{{\partial}{\overline}{X}}=\psi\end{aligned}$$ with the additional property that it is analytic in $\{\Re({\lambda})\geq 0, {\lambda}\notin {\mathbb{N}}/2\}$. Since $R_\pm({\lambda})$ are analytic in $\{\Re({\lambda})\geq 0\}$, this shows that $\sigma_\pm({\lambda})$ is analytic in the same domain, and we have $D_g\sigma_{\pm}(0)=0$. Since this will be useful below, we recall briefly the construction of the approximate solution $\sigma_{\infty,\pm}({\lambda})$ near the boundary from Lemma 4.4 in [@GMP]. The principle is to write the Dirac operator near ${\partial}{\overline}{X}$ in the product decomposition $[0,{\epsilon})_x{\times}{\partial}{\overline}{X}$ $$\label{diracAH} D_g=x^{{\frac{n}{2}}}({\rm cl}(x{\partial}_x)x{\partial}_x+xD_{h_0})x^{-{\frac{n}{2}}}+ xP$$ where $D_{h_0}$ is the Dirac operator on the boundary for the metric $h_0$ and $P$ is a first order differential operator with smooth coefficients which in local coordinates $(x,y)$ near the boundary can be written $$P=P_0(x,y)x{\partial}_x +\sum_{j=1}^nP_j(x,y)x{\partial}_{y_i}$$ for some smooth sections $P_j$ of ${^0\Sigma}\otimes {^0\Sigma}^*$. Consequently, one has for any $\psi_\pm\in C^{\infty}({\partial}{\overline}{X},{^0\Sigma}_{\pm})$ and $k\in{\mathbb{N}}_0$ the indicial equation $$\begin{aligned} \label{indicialeq} (D_g\pm i{\lambda})&x^{{\frac{n}{2}}-{\lambda}+k}(\psi_++\psi_-)\\ =&ix^{{\frac{n}{2}}-{\lambda}+k}\Big((k-{\lambda}\pm{\lambda})\psi_++({\lambda}-k\pm{\lambda})\psi_-\Big)+ x^{{\frac{n}{2}}-{\lambda}+k+1}F_{\lambda}^k\notag\end{aligned}$$ where $F^k_{\lambda}\in C^{\infty}({\overline}{X},{^0\Sigma})$ is holomorphic near ${\lambda}=0$. From this, using formal series and Borel lemma, it is easy to see that one can construct near ${\lambda}=0$ a spinor $\sigma_{\infty,\pm}({\lambda})\in x^{{\frac{n}{2}}-{\lambda}}C^{\infty}({\overline}{X},{^0\Sigma})$, holomorphic near ${\lambda}=0$, solving whose formal Taylor series is determined locally and uniquely by $\psi_\pm$. Let $\sigma_{\pm}({\lambda})$ be the spinor of Proposition \[poisson\] (thus depending on $\psi$), we can then define linear Poisson operators and scattering operators $$\begin{aligned} E_\pm({\lambda}):C^{\infty}({\partial}{\overline}{X},{^0\Sigma}_\pm)\to &C^{\infty}(X,{^0\Sigma}), & \psi \mapsto& \sigma_{\pm}({\lambda}), \\ \ \ \, S_\pm ({\lambda}): C^{\infty}({\partial}{\overline}{X},{^0\Sigma}_\pm)\to &C^{\infty}({\partial}{\overline}{X},{^0\Sigma}_\mp), & \psi \mapsto & \sigma^+_{\pm}({\lambda})|_{{\partial}{\overline}{X}}\end{aligned}$$ which are holomorphic in $\{\Re({\lambda})\geq 0, {\lambda}\notin {\mathbb{N}}/2\}$. We extend the definition of $E_\pm({\lambda})$ to the whole bundle ${^0\Sigma}$ by setting that it acts by $0$ on ${^0\Sigma}_\mp$. Then from Proposition 4.6 of [@GMP], the Schwartz kernel $E_\pm({\lambda};m,y')\in C^{\infty}(X{\times}{\partial}{\overline}{X}; {^0\Sigma}\otimes{^0\Sigma}^*)$ of $E_\pm({\lambda})$ is given by $$\label{kernelE} E_\pm({\lambda};m,y')=[R_\pm({\lambda};m,x',y'){x'}^{-{\frac{n}{2}}-{\lambda}}]|_{x'=0}{\rm cl}(\nu)$$ where $R_\pm({\lambda};m,m')$ is the Schwartz kernel of $R_\pm({\lambda})$. We can also define $$\begin{aligned} \label{defelasla} \, E({\lambda}):C^{\infty}({\partial}{\overline}{X},{^0\Sigma})\to &C^{\infty}(X,{^0\Sigma}),& \psi_++\psi_-\mapsto & E_+({\lambda})\psi_++E_-({\lambda})\psi_-, \\ S({\lambda}):C^{\infty}({\partial}{\overline}{X},{^0\Sigma})\to &C^{\infty}({\partial}{\overline}{X},{^0\Sigma})\,& \psi_++\psi_-\mapsto & S_+({\lambda})\psi_++S_-({\lambda})\psi_- .\nonumber\end{aligned}$$ The main features of $S({\lambda})$, also proved in Section 4.3 of [@GMP], are gathered in \[propofS\] For $\Re({\lambda})\geq 0$ and ${\lambda}\notin {\mathbb{N}}/2$, the operator $S({\lambda})$ depends on the choice of the boundary defining function $x$ but changes under the law $$\begin{aligned} \label{change} \hat{S}({\lambda})=e^{-({\frac{n}{2}}+{\lambda})\omega_0}S({\lambda})e^{({\frac{n}{2}}-{\lambda})\omega_0}, && \omega_0:=\omega|_{x=0}\end{aligned}$$ if $\hat{S}({\lambda})$ is the scattering operator defined using the boundary defining function $\hat{x}=e^{\omega}x$ for some $\omega\in C^{\infty}({\overline}{X})$. Moreover $S({\lambda})\in \Psi^{2{\lambda}}({\partial}{\overline}{X},{^0\Sigma})$ is a classical pseudodifferential operator of order $2{\lambda}$, and its principal symbol is given by $$\sigma_{\rm pr}(S({\lambda}))(\xi)=i2^{-2{\lambda}}\frac{\Gamma(1/2-{\lambda})}{\Gamma(1/2+{\lambda})}{\rm cl }(\nu){\rm cl}(\xi)|\xi|^{2{\lambda}-1}_{h_0}$$ where $h_0=(x^2g)|_{T{\partial}{\overline}{X}}$. If ${\lambda}\in i{\mathbb{R}}$, $S({\lambda})$ extends as a unitary operator on $L^2({\partial}{\overline}{X},{^0\Sigma})$, its inverse is given by $S(-{\lambda})$ and extends meromorphically in $\{\Re({\lambda})\geq 0,{\lambda}\notin {\mathbb{N}}/2\}$ as a family of classical pseudo-differential operators in $\Psi^{-2{\lambda}}({\partial}{\overline}{X},{^0\Sigma})$. Finally $S({\lambda})$ is self-adjoint for ${\lambda}\in (0,\infty)$. The conformal change law and the invertibility are easy consequences of the definition of $S({\lambda})$ and the uniqueness of the solution $\sigma_{\pm}({\lambda})$ in Proposition \[poisson\], the pseudodifferential properties and the meromorphic extension are more delicate and studied in Section 4.3 of [@GMP]. In particular, by letting ${\lambda}\to 0$ in , we deduce easily the following \[p:harmonic\] Let $\psi\in C^\infty({\partial}{\overline}{X},{^0\Sigma})$, then $\sigma:=E(0)\psi$ is a harmonic spinor for $D$, which lives in $x^{\frac{n}{2}}C^\infty({\overline}{X},{^0\Sigma})$ and has the following behavior at the boundary $$\sigma= x^{{\frac{n}{2}}}({\rm Id}+S(0))\psi +O(x^{{\frac{n}{2}}+1}).$$ Remark from Proposition \[propofS\] that $S(0)^*=S(0)^{-1}=S(0)$ and so the operator $$\label{defc} {\mathcal}{C}:={\frac{1}{2}}({\rm Id}+S(0))$$ is an orthogonal projector on a subspace of $L^2({\partial}{\overline}{X},{^0\Sigma})$ for the measure ${\rm dv}_{h_0}$ where $h_0=(x^{2}g)|_{T{\partial}{\overline}{X}}$. Notice from that, under a change of boundary defining function $\hat{x}=e^{\omega}x$, the operator ${\mathcal}{C}$ changes according to conjugation $\hat{{\mathcal}{C}}=e^{-{\frac{n}{2}}\omega_0}{\mathcal}{C} e^{{\frac{n}{2}}\omega_0}$. Now we want to prove that the range of $E(0)$ acting on $C^{\infty}({\partial}{\overline}{X},{^0\Sigma})$ is exactly the set of harmonic spinors in $x^{{\frac{n}{2}}}C^{\infty}({\overline}{X},{^0\Sigma})$. \[span\] Let $\phi\in x^{{\frac{n}{2}}}C^{\infty}({\overline}{X},{^0\Sigma})$ such that $D_g\phi=0$ and let $\psi:=(x^{-\frac n2}\phi)|_{{\partial}{\overline}{X}}$. Then we have $E(0)\psi=2\phi$. First let us write $\psi=\psi_++\psi_-$ with $\psi_\pm\in{^0\Sigma}_\pm$. Then we construct the approximate solution $\sigma_{\infty,+}({\lambda})$ of associated to $\psi_+$. Let us set $\phi_+({\lambda}):=\sigma_{\infty,+}({\lambda})$ and $\phi_-({\lambda}):=\phi-\phi_+({\lambda})$. One has $(x^{-{\frac{n}{2}}}\phi_-({0}))|_{x=0}=\psi_-\in{^0\Sigma}_-$ and $D_g\phi_-(0)=-D_g\phi_+(0)$. As in the proof of Proposition \[poisson\], we have $$\sigma_{+}({\lambda})=\phi_+({\lambda})-R_+({\lambda})(D_g+ i{\lambda})\phi_+({\lambda})=E_+({\lambda})\psi_+.$$ and in particular, since all the terms in the composition on the right hand side are holomorphic near ${\lambda}=0$, we obtain that $$E_+(0)\psi_+=\phi_+(0)-R_+(0)D_g\phi_+(0)=\phi_+(0)+R_+(0)D_g\phi_-(0).$$ Now we use Green’s formula on a region $\{x\leq {\epsilon}\}$ for ${\epsilon}>0$ small and by letting ${\epsilon}\to 0$ we deduce easily from that $$R_+(0)D_g\phi_-(0)=\phi_-(0)-E_+(0)\psi_-=\phi_-(0).$$ Consequently, we have proved that $E_+(0)\psi_+=\phi_+(0)+\phi_-(0)=\phi$. A similar reasoning shows that $E_-(0)\psi_-=\phi$ and this achieves the proof. As a corollary we deduce that $S(0)\psi=\psi$ for $\psi$ as in Proposition \[span\], so \[projector\] The following identity holds for ${\mathcal}{C}={\frac{1}{2}}({\rm Id}+S(0))$ $$\{(x^{-{\frac{n}{2}}}\sigma)|_{{\partial}{\overline}{X}}; \sigma\in x^{{\frac{n}{2}}}C^{\infty}({\overline}{X},{^0\Sigma}), D_g\sigma=0\}= \{{\mathcal}{C}\psi;\psi\in C^{\infty}({\partial}{\overline}{X},{^0\Sigma})\}.$$ Dirac operator on compact manifolds with boundary ================================================= Calderón projector and scattering operator at $0$ ------------------------------------------------- Now we let $D_{\bar{g}}$ be the Dirac operator on a smooth compact spin manifold with boundary $({\overline}{X},{\overline}{g})$, and we denote by $\Sigma$ the spinor bundle. We recall that the *Cauchy data space* of $D_{\bar{g}}$ is given by $${\mathcal}{H}_{\partial}:=\{\phi|_{{\partial}{\overline}{X}}, \phi\in C^{\infty}({\overline}{X},\Sigma), D_{\bar{g}}\phi=0\}$$ i.e., it is the space of boundary values of smooth harmonic spinors on ${\overline}{X}$ for $D_{\bar{g}}$. The orthogonal *Calderón projector* $P_{{\overline}{{\mathcal{H}}}_{\partial}}$ is a projector acting on $L^2({\partial}{\overline}{X},\Sigma)$ and whose range is the $L^2$-closure ${\overline}{{\mathcal}{H}}_{\partial}$. Booss and Wojciechowski [@BoW] studied Fredholm properties of boundary value problems for Dirac type operators on manifolds with boundary, they found that if $P$ is a pseudo-differential projector on the boundary, the operator $D_P^+:{\rm Dom}(D_P^+)\to C^{\infty}({\overline}{X},{\Sigma}^+)$ with domain $${\rm Dom}(D_P^+):=\{\phi\in C^{\infty}({\overline}{X},\Sigma^+); P(\phi|_{{\partial}{\overline}{X}})=0\}$$ is Fredholm if and only if $P\circ P_{{\overline}{{\mathcal{H}}}_{\partial}}:{\mathcal}{H}_{\partial}\to {\rm ran}(P)$ is Fredholm, and their indices agree. One of the main problems in this setting is to construct Calderón projectors, there exist methods by Wojciechowski [@BoW] which use the invertible double construction, but a special product structure near the boundary has to be assumed. Our purpose is to construct the Calderón projector in a general setting for the Dirac operator using its conformal covariance and the scattering theory of Dirac operators on asymptotically hyperbolic manifolds developed in [@GMP]. Let $x$ be the distance to the boundary, which is smooth near ${\partial}{\overline}{X}$, and modify it on a compact set of $X$ so that it becomes smooth on ${\overline}{X}$, we still denote it by $x$. Define a metric $g$ conformal to ${\overline}{g}$ by $$g:=x^{-2}{\overline}{g},$$ this is a complete metric on the interior $X$ which is asymptotically hyperbolic. The associated Dirac operator $D$ is related to $D_{\bar{g}}$ by the conformal law change $$D_g=x^{{\frac{n}{2}}+1}D_{\bar{g}}x^{-{\frac{n}{2}}}.$$ Notice that this formula appears with a wrong exponent in several places in the literature, e.g. [@Hitchin Prop. 1.3], [@LawMik Thm. II.5.24]. Let ${^0\Sigma}$ be the rescaled spin bundle defined in Section \[AH\], then there is a canonical identification between $\Sigma$ and ${^0\Sigma}$. We deduce that the Cauchy data space may also be given by $${\mathcal}{H}_{\partial}=\{(x^{-{\frac{n}{2}}}\sigma)|_{{\partial}{\overline}{X}}; \sigma\in x^{{\frac{n}{2}}}C^{\infty}({\overline}{X},{^0\Sigma}), D_g\sigma=0\}.$$ Combining this and Theorem \[projector\], we obtain \[proj2\] The $L^2$-closure of the Cauchy data space ${\overline}{{{\mathcal}{H}}}_{\partial}$ is given by the range of ${\mathcal}{C}={\frac{1}{2}}({\rm Id}+S(0))$ on $L^2({\partial}{\overline}{X},{^0\Sigma})$, in particular, $P_{{\overline}{{\mathcal}{H}}_{\partial}}={\mathcal}{C}$. Remark that no assumption is needed on the geometry of $({\overline}{X},{\overline}{g})$ (this was needed for instance for the double construction in [@BoW]). Another consequence of our construction is that $S(0)$ anti-commutes with the endomorphism ${\rm cl}(\nu)$ of Section \[AH\] and thus The operator ${\mathcal}{C}$ satisfies $-\rm{cl}(\nu)\, {\mathcal}{C}\, \rm{cl}(\nu)=\mathrm{Id}-{\mathcal}{C}$, in other words, the $L^2$-closure of the Cauchy data space ${\overline}{{\mathcal}{H}}_{\partial}$ is a Lagrangian subspace in $L^2(\partial {\overline}{X}, {^0\Sigma})$ with respect to the symplectic structure $(v,w):=\langle {\rm cl}(\nu) v, w \rangle_{h_0}$ for $v,w\in L^2(\partial {\overline}{X}, {^0\Sigma})$ where $h_0=({\overline}{g})|_{T{\partial}{\overline}{X}}$. The equality $-\rm{cl}(\nu)\, {\mathcal}{C}\, \rm{cl}(\nu)=\mathrm{Id}-{\mathcal}{C}$ follows easily from $\rm{cl}(\nu) S(0) = -S(0) \rm{cl}(\nu)$ since $$-\frac12 \rm{cl}(\nu) (\mathrm{Id}+ S(0))\rm{cl}(\nu) =\frac12 (\mathrm{Id}-\rm{cl}(\nu) S(0)\rm{cl}(\nu)) =\frac12 (\mathrm{Id} -S(0)).$$ This immediately implies that ${\overline}{{\mathcal}{H}}_{\partial}$ and ${\overline}{{\mathcal}{H}}^\perp$ are both isotropic subspaces in $L^2(\partial {\overline}{X}, ^0\Sigma)$, which completes the proof. Calderón projector and the operator $K$ {#caldproj} --------------------------------------- By Propositions \[p:harmonic\] and \[span\], the extension map $K:C^\infty({\partial}{\overline}{X},{^0\Sigma})\to C^{\infty}({\overline}{X},{^0\Sigma})$ from spinors on $M$ to harmonic spinors on ${\overline}{X}$ is given by $$K\psi={\frac{1}{2}}x^{-{\frac{n}{2}}}E(0)\psi$$ where $E(0)$ is the operator defined in for the Dirac operator $D$ associated to $g={\overline}{g}/x^2$. The adjoint $E(0)^*$ of $E(0)$ with respect to ${\rm dv}_g$ is a map from $\dot{C}^\infty({\overline}{X},{^0\Sigma})$ to $C^\infty({\partial}{\overline}{X},{^0\Sigma})$ such that $$\int_X {\langle}E(0)\varphi,\psi{\rangle}_g{\rm dv}_g=\int_{{\partial}{\overline}{X}}{\langle}\varphi,E(0)^*\psi{\rangle}_{h_0}{\rm dv}_{h_0}$$ for all $\psi\in\dot{C}^\infty({\overline}{X},{^0\Sigma})$ and $\varphi\in C^{\infty}({\partial}{\overline}{X},{^0\Sigma})$. Here $h_0$ denotes the metric over ${\partial}{\overline}{X}$ given by the restriction of ${\overline}{g}$ to the bundle $T{\partial}{\overline}{X}$. Similarly the adjoint of $K$ with respect to the metric ${\rm{dv}}_{{\overline}{g}}$ satisfies $$\int_X {\langle}K\varphi,\psi{\rangle}_g{\rm dv}_{{{\overline}{g}}}=\int_{{\partial}{\overline}{X}}{\langle}\varphi,K^*\psi{\rangle}_{h_0}{\rm dv}_{h_0}$$ and since ${\rm dv}_{{g}}=x^{-(n +1)}{\rm dv}_{{\overline}{g}}$, we obtain $$K^*={\frac{1}{2}}E(0)^*x^{{\frac{n}{2}}+1}$$ where the adjoint for $E(0)$ is with respect to $g$ while the adjoint for $K$ is with respect to $\bar{g}$. The Schwartz kernels of $E({\lambda}),E^*({\lambda})$ and $R_\pm({\lambda})$ are studied in [@GMP]. They are shown to be polyhomogeneous conormal on a blown-up space. Let us now describe them, by referring the reader to the Appendix for what concerns blown-up manifolds and polyhomogeneous conormal distributions. The first space is the stretched product (see for instance [@MM; @MaCPDE] where it was first introduced) $$\begin{aligned} {\overline}{X}{\times}_0{\overline}{X}=[{\overline}{X}{\times}{\overline}{X};\Delta_{{\partial}}],&& \Delta_{\partial}:=\{(m,m)\in{\partial}{\overline}{X}{\times}{\partial}{\overline}{X}\}\end{aligned}$$ obtained by blowing-up the diagonal $\Delta_{\partial}$ in the corner, the blow-down map is denoted by $\beta:{\overline}{X}{\times}_0{\overline}{X}\to {\overline}{X}{\times}{\overline}{X}$. This is a smooth manifold with corners which has $3$ boundary hypersurfaces: the front face ${\textrm{ff}}$ obtained from blowing-up $\Delta_{\partial}$, and the right and left boundaries ${\textrm{rb}}$ and ${\textrm{lb}}$ which respectively project down to ${\overline}{X}{\times}{\partial}{\overline}{X}$ and ${\partial}{\overline}{X}{\times}{\overline}{X}$ under $\beta$. One can similarly define the blow-ups $$\begin{aligned} \label{stretched} {\overline}{X}{\times}_0{\partial}{\overline}{X}:=[{\overline}{X}{\times}{\partial}{\overline}{X};\Delta_{\partial}] , && {\partial}{\overline}{X}{\times}_0{\overline}{X}:=[{\partial}{\overline}{X}{\times}{\overline}{X};\Delta_{\partial}]\end{aligned}$$ which are manifolds with $1$ corner of codimension $2$ and $2$ boundary hypersurfaces: the front face ${\textrm{ff}}$ obtained from the blow-up and the left boundary ${\textrm{lb}}$ which projects to ${\partial}{\overline}{X}{\times}{\partial}{\overline}{X}$ for ${\overline}{X}{\times}_0{\partial}{\overline}{X}$, respectively the front face ${\textrm{ff}}$ and right boundary ${\textrm{rb}}$ for ${\partial}{\overline}{X}{\times}_0{\overline}{X}$. We call $\beta_l,\beta_r$ the blow-down maps of and we let $\rho_{{\textrm{ff}}},\rho_{{\textrm{lb}}}$ and $\rho_{{\textrm{rb}}}$ be boundary defining functions of these hypersurfaces in each case. Notice that the two spaces in are canonically diffeomorphic to the submanifolds $\{\rho_{{\textrm{rb}}}=0\}\subset{\overline}{X}{\times}_0{\overline}{X}$ and $\{\rho_{{\textrm{lb}}}=0\}\subset{\overline}{X}{\times}_0{\overline}{X}$. Like in Section 3.2 in [@GMP], the bundle ${^0\Sigma}\boxtimes{^0\Sigma}^*$ lifts smoothly to these 3 blown-up manifolds through $\beta, \beta_l$ and $\beta_r$, we will use the notation $$\begin{aligned} {\mathcal}{E}:=\beta^*({^0\Sigma}\boxtimes{^0\Sigma}^*),&& {\mathcal}{E}_j:=\beta_j^*({^0\Sigma}\boxtimes{^0\Sigma}^*) \textrm{ for }j=l,r\end{aligned}$$ for these bundles. The interior diagonal in $X{\times}X$ lifts to a submanifold $\Delta_{\iota}$ in ${\overline}{X}{\times}_0{\overline}{X}$ which intersects the boundary only at the front face (and does so transversally). Then it follows from [@GMP Prop 3.2] that the resolvent $R_\pm({\lambda})$ has a Schwartz kernel $R_\pm({\lambda};m,m')\in C^{-\infty}({\overline}{X}{\times}{\overline}{X};{\mathcal}{E})$ which lifts to ${\overline}{X}{\times}_0{\overline}{X}$ to a polyhomogeneous conormal distribution on ${\overline}{X}{\times}_0{\overline}{X}\setminus \Delta_\iota$ $$\label{e:R(0)} \beta^*R_\pm({\lambda})\in (\rho_{{\textrm{rb}}}\rho_{{\textrm{lb}}})^{{\lambda}+{\frac{n}{2}}}C^{\infty}({\overline}{X}{\times}_0{\overline}{X}\setminus \Delta_\iota;{\mathcal}{E}).$$ Combined with Theorem \[proj2\], this structure result on $R_\pm({\lambda})$ implies \[vanishing residue\] The Schwartz kernel of the Calderón projector $P_{{\overline}{{\mathcal}{H}}_{\partial}}$ associated to the Dirac operator has an asymptotic expansion in polar coordinates around the diagonal without log terms. In particular, the Wodzicki-Guillemin local residue density of $P_{{\overline}{{\mathcal}{H}}_{\partial}}$ vanishes. Using Theorem \[proj2\], it suffices to show that $S(0)$ has this property. From [@GMP eq (4.10), Sec. 3], the kernel of $S({\lambda})$ is given outside the diagonal by $$S({\lambda};y,y')=i[(xx')^{-{\lambda}-{\frac{n}{2}}}R_+({\lambda};x,y,x',y')|_{x=x'=0}-(xx')^{-{\lambda}-{\frac{n}{2}}}R_-({\lambda};x,y,x',y')|_{x=x'=0}]$$ Since a boundary defining function $x'$ of ${\overline}{X}{\times}{\partial}{\overline}{X}$ in ${\overline}{X}{\times}{\overline}{X}$ lifts to $\beta^*x'=\rho_{{\textrm{rb}}}\rho_{{\textrm{ff}}}F$ for some $F>0$ smooth on ${\overline}{X}{\times}_0{\overline}{X}$ (and similarly $\beta^*x=\rho_{{\textrm{lb}}}\rho_{{\textrm{ff}}}F$ for some smooth $F>0$), one can use to obtain $$\beta^*((xx')^{-{\lambda}-{\frac{n}{2}}}R_\pm({\lambda})\in \rho_{{\textrm{ff}}}^{-2{\lambda}-n}C^\infty({\overline}{X}{\times}_0{\overline}{X}; {\mathcal}{E}).$$ Restricting to $x=x'=0$, $y\not=y'$ corresponds to restricting to the corner ${\textrm{lb}}\cap {\textrm{rb}}$ which is canonically diffeomorphic to $M{\times}_0 M=[M{\times}M; \Delta_{\partial}]$ and thus the pull-back $\beta_{\partial}^* S({\lambda})$ of the kernel of $S({\lambda})$ has an expansion in polar coordinates at $\Delta_{\partial}$ with no log terms after setting ${\lambda}=0$. From , we deduce that the kernel $E({\lambda};m,y')$ of $E({\lambda})$ lifts to $$\beta_l^*E({\lambda})\in \rho_{\textrm{lb}}^{{\lambda}+{\frac{n}{2}}}\rho_{{\textrm{ff}}}^{-{\lambda}-{\frac{n}{2}}}C^{\infty}({\overline}{X}{\times}_0{\partial}{\overline}{X};{\mathcal}{E}_l)$$ where we used the identification between $\{\rho_{{\textrm{rb}}}=0\}\subset {\overline}{X}{\times}_0{\overline}{X}$ and ${\overline}{X}{\times}_0{\partial}{\overline}{X}$. Here, obviously, this is the kernel of the operator acting from $L^2(M,{^0\Sigma};{\rm dv}_{h_0})$ to $L^2(X,{^0\Sigma};{\rm dv}_{g})$. We have a similar description $$\beta_r^*E^*(\lambda)\in\rho_{\textrm{rb}}^{{\lambda}+{\frac{n}{2}}}\rho_{{\textrm{ff}}}^{-{\lambda}-{\frac{n}{2}}}C^{\infty}({\partial}{\overline}{X}{\times}_0{\overline}{X};{\mathcal}{E}_r).$$ So we deduce that the Schwartz kernel $K^*(y,x',y')\in C^{\infty}({\partial}{\overline}{X}{\times}{\overline}{X};{^0\Sigma}\boxtimes{^0\Sigma}^*)$ of $K^*$ with respect to the density $|{\rm dv}_{h_0}\otimes{\rm dv}_{{\overline}{g}}|=x^{n+1}|{\rm dv}_{h_0}\otimes{\rm dv}_{g}|$ lifts through $\beta_r$ to $$\label{kernelK*} \beta_{r}^*K^*={\frac{1}{2}}\beta_r^*(x'^{-{\frac{n}{2}}}E^*(0))\in \rho_{{\textrm{ff}}}^{-n}C^{\infty}({\partial}{\overline}{X}{\times}_0{\overline}{X};{\mathcal}{E}_r).$$ Similarly, for $K$ we have $$\label{e:k1} \beta_{l}^*K\in \rho_{{\textrm{ff}}}^{-n}C^{\infty}({\overline}{X}{\times}_0{\partial}{\overline}{X};{\mathcal}{E}_l).$$ When it is clear, we may omit $^0\Sigma$ in the notations $L^2({\overline}{X},^0\Sigma, {\rm dv}_g)$, $L^2({\partial}X, ^0\Sigma, {\rm dv}_{h_0})$ for simplicity. Now we have \[Kbounded\] The operator $K$ is bounded from $L^2({\partial}{\overline}{X},{\rm dv}_{h_0})$ to $L^2({\overline}{X},{\rm dv}_{{\overline}{g}})$, and so is its adjoint $K^*$ from $L^2({\overline}{X},{\rm dv}_{{\overline}{g}})$ to $L^2({\partial}{\overline}{X},{\rm dv}_{h_0})$. The range of $K^*$ acting on $L^2({\overline}{X},{\rm dv}_{{\overline}{g}})$ is contained in ${\overline}{{\mathcal}{H}}_{\partial}$ and the kernel of $K$ contains ${\overline}{{\mathcal}{H}}_{\partial}^\perp$. It is shown in Lemma 4.7 of [@GMP] the following identity $$R_+(0)-R_-(0)=-\frac{i}{2}(E_+(0)E_+(0)^*+ E_-(0)E_-(0)^*)=-\frac{i}{2}E(0)E(0)^*$$ as operators from $\dot{C}^\infty({\overline}{X},{^0\Sigma})$ to $x^{{\frac{n}{2}}}C^{\infty}({\overline}{X},{^0\Sigma})$, so in particular this implies that $$KK^*={\frac{1}{2}}ix^{-{\frac{n}{2}}}(R_+(0)-R_-(0))x^{{\frac{n}{2}}+1}$$ as operators. Using the isometry $\psi\to x^{-(n+1)/2}{\psi}$ from $L^2(X,{\rm dv}_g)$ to $L^2(X,{\rm dv}_{{\overline}{g}})$, we see that the operator $KK^*$ is bounded on $L^2({\overline}{X},{\rm dvol}_{{\overline}{g}})$ if and only if $x^{\frac{1}{2}}(R_+(0)-R_-(0))x^{{\frac{1}{2}}}$ is bounded on $L^2(X,{\rm dv}_g)$. Now by , the Schwartz kernel of $x^{{\frac{1}{2}}}R_\pm(0)x'^{\frac{1}{2}}$ lifts on the blown-up space ${\overline}{X}{\times}_0{\overline}{X}$ as a conormal function $$\beta^*(x^{\frac12}R_\pm(0)x'^{\frac12}) \in \rho_{{\textrm{lb}}}^{\frac{n+1}{2}}\rho_{{\textrm{rb}}}^{\frac{n+1}{2}}\rho_{{\textrm{ff}}}\,C^{\infty}({\overline}{X}{\times}_0{\overline}{X};{\mathcal}{E})$$ since $(xx')^{\frac{1}{2}}$ lifts to ${\overline}{X}{\times}_0{\overline}{X}$ to $(\rho_{{\textrm{rb}}}\rho_{{\textrm{lb}}})^{\frac{1}{2}}\rho_{\textrm{ff}}F$ for some $F>0$ smooth on ${\overline}{X}{\times}_0{\overline}{X}$. We may then use Theorem 3.25 of Mazzeo [@MaCPDE] to conclude that it is bounded on $L^2(X,{\rm dv}_{g})$, and it is even compact according to Proposition 3.29 of [@MaCPDE]. As a conclusion, $K^*$ is bounded from $L^2(X,{\rm dv}_{{\overline}{g}})$ to $L^2({\partial}{\overline}{X},{\rm dv}_{h_0})$ and $K$ is bounded on the dual spaces. The fact that the range of $K^*$ is contained in ${\overline}{{\mathcal}{H}}_{\partial}$ comes directly from a density argument and the fact that for all $\psi\in\dot{C}^\infty({\overline}{X};{^0\Sigma})$, $K^*\psi={-{\frac{1}{2}}i[x^{-{\frac{n}{2}}}(R_+(0)-R_-(0))(x^{{\frac{n}{2}}+1}\psi)]|_{{\partial}{\overline}{X}}}$, and $x^{-{\frac{n}{2}}}(R_+(0)-R_-(0))(x^{{\frac{n}{2}}+1}\psi)$ is a smooth harmonic spinor of $D_{\bar{g}}$ on ${\overline}{X}$. The operator $K^*K$ acts on $L^2({\partial}{\overline}{X},{\rm dv}_{h_0})$ as a compact operator, we actually obtain \[pseudo-1\] The operator $K^*K$ is a classical pseudo-differential operator of order $-1$ on ${\partial}{\overline}{X}$ and its principal symbol is given by $$\sigma_{\rm pr}(K^*K)(y;\mu)=\frac{1}{4} |\mu|^{-1}_{h_0}\Big({\rm Id}+i{\rm cl}(\nu){\rm cl}\Big(\frac{\mu}{|\mu|_{h_0}}\Big)\Big)$$ According to and Lemma \[rel0calculus1\], the operator $K={\frac{1}{2}}x^{-\frac{n}{2}}E(0)$ is a log-free classical pseudodifferential operator in the class $I^{-1}_{\rm lf}({\overline}{X}{\times}M;{\mathcal}{E})$ in the terminology of Subsection \[interiortoboundary\], while $K^*$ is in the class $I^{-1}_{\rm lf}(M{\times}{\overline}{X}\;{\mathcal}{E})$. We can therefore apply Proposition \[compositionKL\] to deduce that $K^*K\in \Psi^{-1}(M;{\mathcal}{E})$ is a classical pseudo-differential operator of order $-1$ on $M$. Moreover, from Proposition \[compositionKL\], the principal symbol is given by $$\sigma_{K^*K}(y,\mu)=(2\pi)^{-2}\int_{0}^\infty \hat{\sigma}_{K^*}(y;-x,\mu).\hat{\sigma}_K(y;x,\mu)dx$$ where hat denotes Fourier transform in the variable $\xi$ and $\sigma_{K^*}(y,\xi,\mu),\sigma_{K}(y;\xi,\mu)$ are the principal symbols of $K^*,K$. We have to compute for $|\mu|$ large the integral above. We know from [@GMP] that the leading asymptotic in polar coordinates around $\Delta_{\partial}$ (or equivalently the normal operator at the front face) of $K={\frac{1}{2}}x^{-n/2}E(0)$ at the submanifold $\Delta_{\partial}$ is given in local coordinates by $$K(x,y,y+z)\sim {\frac{1}{2}}\pi^{-\frac{n+1}{2}}\Gamma\left(\frac{n+1}{2}\right)\rho^{-n-1}(x+{\mathrm{cl}}(\nu){\mathrm{cl}}(z))$$ where $\rho:=(x^2+|z|^2)^{1/2}$ is the defining function for the front face of ${\overline}{X}{\times}_0 M$. To obtain the symbol, we need to compute the inverse Fourier transform in $(x,z)$ variables of the homogeneous distribution $\rho^{-n-1}(x+{\mathrm{cl}}(\nu){\mathrm{cl}}(z))$. To do this, we use the analytic family of $L^1$ tempered distributions $\omega(\lambda)=\rho^{-n-1+\lambda}$ for $\Re(\lambda)>0$. We have $${\mathcal{F}}_{(x,z)\to (\xi,\mu)}(\omega(\lambda))= (2\pi)^{\frac{n+1}{2}} 2^{\lambda-\frac{n+1}{2}} \frac{\Gamma\left(\frac{\lambda}{2}\right)} {\Gamma\left(\frac{n+1-\lambda}{2}\right)} R^{-\lambda}$$ for $R:=|(\xi,\mu)|$. This allows us to compute ${\mathcal{F}}(x\omega(\lambda))$ and ${\mathcal{F}}(z_j \omega(\lambda))$, which turn out to be regular at $\lambda=0$. Thus by setting $\lambda=0$ we get after a short computation $$\sigma_K(y;\xi,\mu)=i(\xi^2+|\mu|^2)^{-1}(\xi+{\mathrm{cl}}(\nu){\mathrm{cl}}(\mu)).$$ This gives $\sigma_{K^*}(y,\xi',\mu)=-i((\xi')^2+|\mu|^2)^{-1}(\xi'-{\mathrm{cl}}(\nu){\mathrm{cl}}(\mu))$. Use the fact that the Fourier transform of the Heaviside function is $\pi\delta-\frac{i}{\xi}$. Then $$4\sigma_{K^*K}(y;\mu)=\pi^{-1} \int_{{\mathbb{R}}} R^{-2}d\xi -\pi^{-2}i \int_{{\mathbb{R}}^2}(RR')^{-2}(\xi\xi'+|\mu|^2 +(\xi'-\xi){\mathrm{cl}}(\nu){\mathrm{cl}}(\mu))\frac{d\xi d\xi'}{\xi-\xi'}$$ in the sense of principal value for $(\xi-\xi')^{-1}$. The first term gives $|\mu|^{-1}$. In the second term, by symmetry in $\xi,\xi'$, only the term $\pi^{-2}i{\mathrm{cl}}(\nu){\mathrm{cl}}(\mu)\int_{{\mathbb{R}}^2}(RR')^{-2}d\xi d\xi'$ contributes, and it gives $i|\mu|^{-2}{\mathrm{cl}}(\nu){\mathrm{cl}}(\mu)$. This ends the proof. In fact, we could also compute the principal symbol using the push-forward approach but the computation is slightly more technical. We deduce easily from the two last lemmas \[KstarKinv\] There exists a pseudo-differential operator of order $1$ on ${\partial}{\overline}{X}$, denoted $(K^*K)^{-1}$ such that $(K^*K)^{-1}K^*K={\mathcal}{C}$. Using Lemmas \[Kbounded\], \[pseudo-1\], we deduce that if $D_{h_0}$ is the Dirac operator on the boundary ${\partial}{\overline}{X}$ equipped with the metric $h_0={\overline}{g}|_{T{\partial}{\overline}{X}}$, then $A:= K^*K+\frac14({\rm Id}-{\mathcal}{C})({\rm Id}+D^2_{h_0})^{-{\frac{1}{2}}}({\rm Id}-{\mathcal}{C})$ is a classical pseudo-differential operator of order $-1$, and by Lemma \[pseudo-1\] its principal symbol on the cosphere bundle equals ${\rm Id}$. Moreover, it is straightforward that $\ker A=0$ since $K$ is injective on ${\overline}{{\mathcal}H}_{{\partial}}$. This implies that $A$ is elliptic and has a classical pseudo-differential inverse $B$ which is of order $1$. Let us define $(K^*K)^{-1}:=B{\mathcal}{C}$, which is classical pseudo-differential of order $1$, then one has $(K^*K)^{-1}K^*K=(K^*K)^{-1}A={\mathcal}{C}$. The orthogonal projector on harmonic spinors on ${\overline}{X}$ ---------------------------------------------------------------- We will construct and analyze the projector on the $L^2({\overline}{X},{\rm dv}_{{\overline}{g}})$-closure ${\overline}{{\mathcal}{H}}(D_{\bar{g}})$ of $${\mathcal}{H}(D_{\bar{g}}):=\{\psi\in C^{\infty}({\overline}{X};{^0\Sigma}); D_{\bar{g}}\psi=0\}.$$ For this, let us now define the operator $$P:=K(K^*K)^{-1}K^*$$ which maps continuously $\dot{C}^\infty({\overline}{X};{^0\Sigma})$ to $C^{\infty}({\overline}{X};{^0\Sigma})$. Since $K$ is bounded on $L^2({\partial}{\overline}{X},{\rm dv}_{h_0})$, Lemma \[Kbounded\] and Corollary \[KstarKinv\] imply easily the following \[PK=K\] The operator $P$ satisfies $PK=K{\mathcal}{C}=K$ on $L^2({\partial}{\overline}{X}, {\rm dv}_{h_0})$. We want to show that $P$ extends to a bounded operator on $L^2({\overline}{X},{\rm dv}_{{\overline}{g}})$ and study the structure of its Schwartz kernel. We first use the following composition result which is a consequence of Melrose’s push-forward theorem [@Me]. The definition of polyhomogeneous functions and index sets is recalled in Appendix \[appA\]. \[structP\] The operator $P:=K(K^*K)^{-1}K^*$ has a Schwartz kernel in $C^{-\infty}({\overline}{X}{\times}{\overline}{X};({^0\Sigma}\boxtimes{^0\Sigma}^*)\otimes \Omega^{\frac{1}{2}})$ on ${\overline}{X}{\times}{\overline}{X}$ which lifts to ${\overline}{X}{\times}_0{\overline}{X}$ through $\beta$ to $k_P\beta^*(|{\rm dv}_{{\overline}{g}}\otimes {\rm dv}_{{\overline}{g}}|^{\frac{1}{2}})$ with $$\begin{aligned} k_P\in {\mathcal}{A}_{\rm phg}^{J_{\rm ff},J_{\rm rb},J_{\rm lb}}({\overline}{X}{\times}_0{\overline}{X};{\mathcal}{E}), && J_{\rm ff}=-(n+1)\cup(-2,1)\cup(0,3),&& J_{\rm rb}=J_{\rm lb}=0\end{aligned}$$ where $|{\rm dv}_{{\overline}{g}}\otimes{\rm dv}_{{\overline}{g}}|$ is the Riemannian density trivializing $\Omega({\overline}{X}{\times}{\overline}{X})$ induced by ${\overline}{g}$. We start by composing $A\circ B$ where $A:=(K^*K)^{-1}({\rm Id}+D_{h_0}^2)^{-1}$ and $B:=({\rm Id}+D_{h_0}^2)K^*$. From Corollary \[KstarKinv\], we know that $A$ is a classical pseudo-differential operator on $M$ of order $-1$, so its kernel lifts to $M{\times}_0 M$ as a polyhomogeneous conormal kernel and its index set (as a $b$-half-density) $E$ is of the form $-{\frac{n}{2}}+1+{\mathbb{N}}_0\cup ({\frac{n}{2}}+{\mathbb{N}}_0,1)$. Now, since the lift of vector fields on $M$ by the b-fibration $M{\times}_0{\overline}{X}\to M{\times}{\overline}{X}\to M$ is smooth, tangent to the right boundary in $M{\times}_0{\overline}{X}$ and transverse to the front face ${\textrm{ff}}$, we deduce that applying ${\rm Id}+D_{h_0}^2$ to $K^*$ reduces its order at ${\textrm{ff}}$ by $2$ and leaves the index set at ${\textrm{rb}}$ invariant, so $({\rm Id}+D_{h_0}^2)K^*$ has a kernel which lifts on $M{\times}_0{\overline}{X}$ to an element in ${\mathcal}{A}_{\rm phg}^{F_{{\textrm{ff}}},F_{{\textrm{rb}}}}(M{\times}_0{\overline}{X};{\mathcal}{E}_r\otimes \Omega_b^{\frac{1}{2}})$ with $$\begin{aligned} F_{{\textrm{ff}}}=-{\frac{n}{2}}-\frac{3}{2}, && F_{{\textrm{rb}}}={\frac{1}{2}}.\end{aligned}$$ So using Lemma \[composition\], we deduce that $A\circ B$ has a kernel which lifts to $M{\times}_0{\overline}{X}$ as an element in $$\begin{aligned} {\mathcal}{A}_{\rm phg}^{H_{{\textrm{ff}}},H_{{\textrm{rb}}}}(M{\times}_0{\overline}{X};{\mathcal}{E}_r\otimes \Omega_b^{\frac{1}{2}}),&& H_{{\textrm{ff}}}\subset (-{\frac{n}{2}}-\frac{1}{2}) \cup ({\frac{n}{2}}-\frac{3}{2},1)\cup ({\frac{n}{2}}+{\frac{1}{2}},2),&& H_{{\textrm{rb}}}=-\frac{3}{2}\, {\overline{\cup}}\, {\frac{1}{2}}\end{aligned}$$ The index set $H_{\textrm{rb}}$ must in fact be ${\frac{1}{2}}$ since the dual of this composition maps $C^\infty(M;{^0\Sigma})$ into $C^\infty({\overline}{X};{^0\Sigma})$ (with respect to the density $|{\rm dv}_{\bar{g}}|^{\frac{1}{2}}$). Now the operator $K$ has a kernel lifted to ${\overline}{X}{\times}_0M$ which is in $\rho_{{\textrm{ff}}}^{-{\frac{n}{2}}+{\frac{1}{2}}}\rho_{{\textrm{lb}}}^{{\frac{1}{2}}}C^{\infty}({\overline}{X}{\times}_0M;{\mathcal}{E}_l\otimes \Omega_b^{\frac{1}{2}})$ thus using Lemma \[composition\] (and the same argument as above to show that the index set is ${\frac{1}{2}}$ at ${\textrm{lb}},{\textrm{rb}}$), we deduce that the lift $k_P$ of the Schwartz kernel of $P$ is polyhomogeneous conormal on ${\overline}{X}{\times}_0{\overline}{X}$, and the index set of $k_P$ satisfies (as a b-half-density) $$\begin{aligned} \label{defJ} J_{{\textrm{ff}}} =-{\frac{n}{2}}\cup ({\frac{n}{2}}-1,1)\cup ({\frac{n}{2}}+1,3),&& J_{{\textrm{lb}}}= J_{{\textrm{rb}}}={\frac{1}{2}}.\end{aligned}$$ Now this completes the proof since the lift of the half-density $|{\rm dv}_{{\overline}{g}}\otimes {\rm dv}_{{\overline}{g}}|^{\frac{1}{2}}$ is of the form $\rho_{{\textrm{ff}}}^{{\frac{n}{2}}+1}\rho_{{\textrm{lb}}}^{\frac{1}{2}}\rho_{{\textrm{rb}}}^{\frac{1}{2}}\mu_b^{\frac{1}{2}}$ where $\mu_b$ is a non vanishing smooth section of $\Omega_b$. The operator $P=K(K^*K)^{-1}K^*$ is bounded on $L^2({\overline}{X},{\rm dv}_{{\overline}{g}})$ and is the orthogonal projector on the $L^2$-closure of the set of smooth harmonic spinors for $D_{\bar{g}}$ on ${\overline}{X}$, that is, $P=P_{{\overline}{{\mathcal}{H}}}$. Let $P':=x^{\frac{n+1}{2}}Px^{-\frac{n+1}{2}}$ acting on $\dot{C}^\infty({\overline}{X};{^0\Sigma})$, then it suffices to prove that $P'$ extends to a bounded operator on $L^2(X,{\rm dv}_{g})$. But, in terms of half-densities, the half density $|{\rm dv}_g\otimes {\rm dv}_g|^{\frac{1}{2}}$ is given by $(xx')^{-\frac{n+1}{2}}|{\rm dv}_{{\overline}{g}}\otimes {\rm dv}_{{\overline}{g}}|^{\frac{1}{2}}$ and Theorem \[structP\] shows that the Schwartz kernel of $P'$ lifts on ${\overline}{X}{\times}_0{\overline}{X}$ to a half-density $k_{P'}\beta^*(|{\rm dv}_g\otimes {\rm dv}_g|^{\frac{1}{2}})$ where $$\begin{aligned} k_{P'}\in {\mathcal}{A}_{\rm phg}^{J'_{\rm ff},J'_{\rm rb},J'_{\rm lb}}({\overline}{X}{\times}_0{\overline}{X};{\mathcal}{E}) && J'_{\rm ff}\geq 0, && J'_{\rm rb}=J'_{\rm lb}=\frac{n+1}{2}.\end{aligned}$$ It is proved in Proposition 3.20 of Mazzeo [@MaCPDE] that such operators are bounded on $L^2({\overline}{X},{\rm dv}_g)$. To conclude, we know from Corollary \[PK=K\] that $P$ is the identity on the range of $K$ acting on $C^{\infty}({\partial}{\overline}{X};{^0\Sigma})$, which coincides with the space of smooth harmonic spinors for $D_{\bar{g}}$ on ${\overline}{X}$, and we also know that $P$ vanishes on $\ker (K^*)={\overline}{{\rm Im}(K)}^{\perp}$, so this achieves the proof. Conformally covariant powers of Dirac operators and cobordism invariance of the index ===================================================================================== In this section, we define some conformally covariant differential operators with leading part given by a power of the Dirac operator. The method is the same as in Graham-Zworski [@GRZ], using our construction of the scattering operator in Section \[AH\]. Since this is very similar to the case of functions dealt with in [@GRZ], we do not give much details. Let $(X,g)$ be an asymptotically hyperbolic manifold with a metric $g$, and let $x$ be a geodesic boundary defining function of ${\partial}{\overline}{X}$ so that the metric has a product decomposition of the form $g=(dx^2+h(x))/x^2$ near ${\partial}{\overline}{X}$ as in . \[finitemero\] Let $C({\lambda}):=2^{-2{\lambda}}\Gamma(1/2-{\lambda})/\Gamma(1/2+{\lambda})$. If the metric $g$ is even to infinite order, the operator ${\widetilde}{S}({\lambda}):=S({\lambda})/C({\lambda})$ is finite meromorphic in ${\mathbb{C}}$, and it is holomorphic in $\{\Re({\lambda})\geq 0\}$. Moreover for $k\in{\mathbb{N}}_0$, the operator $L_k:={\widetilde}{S}(1/2+k)$ is a conformally covariant self-adjoint differential operator on ${\partial}{\overline}{X}$ with leading part ${\rm cl}(\nu) D_{h_0}^{1+2k}$, and it depends only on the tensors ${\partial}_x^{2j}h(0)$ in a natural way for $j\leq k$. For $k=0$, one has ${\widetilde}{S}(1/2)={\rm cl}(\nu)D_{h_0}$. The first statement is proved in Corollary 4.11 of [@GMP]. The last statement about ${\widetilde}{S}(1/2+k)$ is a consequence of the construction of $\sigma_{\pm}({\lambda})$ in Proposition \[poisson\], by copying mutatis mutandis the proof of Theorem 1 of Graham-Zworski [@GRZ]. Indeed, by construction, the term $\sigma_{\infty,\pm}$ satisfying has a Taylor expansion at $x=0$ of the form $$\sigma_{\infty,\pm}({\lambda})=\psi+\sum_{j=1}^{k}x^{j}(p_{j,{\lambda}}\psi)+O(x^{k+1})$$ for all $k\in{\mathbb{N}}$ where $p_{j,{\lambda}}$ are differential operators acting on $C^{\infty}({\partial}{\overline}{X},{^0\Sigma})$ such that $\frac{p_{j,{\lambda}}}{\Gamma(1/2-{\lambda})}$ are holomorphic in $\{\Re({\lambda})\geq 0\}$ and depend in a natural way only on the tensors $({\partial}_x^{\ell}h(0))_{\ell\leq j}$. Following Proposition 3.5 and Proposition 3.6 in [@GRZ], the operator $\textrm{Res}_{{\lambda}=1/2+k}S({\lambda})$ is also equal to $-{\rm Res}_{{\lambda}=1/2+k}(p_{2k+1,{\lambda}})$. The computation of ${\widetilde}{S}(1/2)$ is then rather straightforward by checking that $$p_{1,{\lambda}}=-\frac{{\rm cl}(\nu)D_{h_0}}{2{\lambda}-1}$$ using the indicial equation and the decomposition . A first corollary of Lemma \[finitemero\] is the cobordism invariance of the index of the Dirac operator. Let $D_{h_0}$ be the Dirac operator on a $2k$-dimensional closed spin manifold $(M,h_0)$ which is the oriented boundary of a compact manifold with boundary $({\overline}{X},{\overline}{g})$. Let $D_{h_0}^+$ be the restriction of $D_{h_0}$ to the sub-bundle of positive spinors $\Sigma^+:=\ker(\omega-1)$, where $\omega$ is the Clifford multiplication by the volume element when $k$ is even, respectively $\omega=i{\mathrm{cl}}(\mathrm{vol}_{h_0})$ for $k$ odd. Then ${\rm Ind}(D_{h_0}^+)=0$. By topological reasons, we may assume that ${\overline}{X}$ is also spin and that the spin structure on $M$ is induced from that on ${\overline}{X}$. Using the isomorphism between the usual spin bundle $\Sigma(X)$ and the 0-spin bundle ${^0\Sigma}(X)$ in Section \[AH\], we see that $D_{h_0}$ can be considered as acting in the restriction of the 0-spin bundle ${^0\Sigma}$ to $M$. Since the odd-dimensional spin representation is chosen such that ${\mathrm{cl}}(\nu)=i\omega$, the $\pm i$ eigenspaces of ${\rm cl}(\nu)$ on ${^0\Sigma}(X)|_{M}$ correspond to the splitting in positive, respectively negative spinors defined by $\omega$ on $\Sigma(M)$. We have seen that ${\widetilde}{S}(1/2)={\rm cl}(\nu)D_{h_0}$. Then by the homotopy invariance of the index, it suffices to use the fact that ${\widetilde}{S}({\lambda})$ is invertible for all ${\lambda}$ except in a discrete set of ${\mathbb{C}}$, which follows from Lemma \[finitemero\] and Proposition \[propofS\]. We refer for instance to [@AS; @Moro; @Lesch; @Nico; @Brav] for other proofs of the cobordism invariance of the index of $D^+$. Now, let us consider $M$ a compact manifold equipped with a conformal class $[h_0]$. A $(n+1)$-dimensional *Poincaré-Einstein* manifold $(X,g)$ associated to $(M,[h_0])$ is an asymptotically hyperbolic manifold with conformal infinity $(M,[h_0])$ and such that the following extra condition holds near the boundary $M={\partial}{\overline}{X}$ $$\begin{aligned} {\rm Ric}(g)=-ng+O(x^{N-2}),&& N= \begin{cases} \infty & \textrm{ if }n+1\textrm{ is even},\\ n & \textrm{ if }n+1{\textrm{ is odd.}} \end{cases}\end{aligned}$$ Notice that by considering the disjoint union $M_2:=M\sqcup M$ instead of $M$, one sees that either $M$ or $M_2$ can be realized as the boundary of a compact manifold with boundary ${\overline}{X}$. Fefferman and Graham [@FGR; @FGR2] proved that for any $(M,[h_0])$ which is the boundary of a compact manifold ${\overline}{X}$, there exist Poincaré-Einstein manifolds associated to $(M,[h_0])$. Moreover writing $g=(dx^2+h(x))/x^2$ for a geodesic boundary defining function $x$, the Taylor expansion of the metric $h(x)$ at $M=\{x=0\}$ is uniquely locally (and in a natural way) determined by $h_0=h(0)$ and the covariant derivatives of the curvature tensor of $h_0$, but not on the Poincaré-Einstein metrics associated to $(M,[h_0])$. If $M$ is spin, we can always construct a Poincaré-Einstein $(X:=[0,1]{\times}M,g)$ associated to $M_2$ with a spin structure induced naturally by that of $M$. \[corcovdif\] If $(X,g)$ is a spin Poincaré-Einstein manifold associated to a spin conformal manifold $(M,[h_0])$, then for $k\leq N/2$ the operators $L_k=:-{\mathrm{cl}}(\nu){\widetilde}{S}(1/2+k)$ acting on $C^{\infty}(M,{^0\Sigma})$ are self-adjoint natural (with respect to $h_0$), conformally covariant differential operators of the form $L_k=D_{h_0}^{2k+1}+\textrm{ lower order terms}$. Hence we can then always define the operators $L_k$ on $M_2=M\sqcup M$ and thus, since the construction is local and natural with respect to $h_0$, this defines naturally $L_k$ on any $M$. As above, when $(M,[h_0])$ is a boundary, the index of the restriction $L^\pm_k$ to ${^0\Sigma}_\pm=\ker(\omega\mp 1)$ (when $n$ is even) is always $0$. In general, the index of $L_k^\pm$ is the index of $L^\pm_0$, which equals the $\hat{A}$-genus of $M$ by the Atiyah-Singer index theorem [@AS]. The Dirac operator ------------------ For $k=0$, the operator $L_0$, which is essentially the pole of the scattering matrix at $\lambda=1/2$, is just the Dirac operator $D_{h_0}$ on $(M,h_0)$ when the dimension of $M$ is even, respectively two copies of $D_{h_0}$ when $\dim(M)$ is odd. A conformally covariant operator of order $3$ --------------------------------------------- For $k=1$ in Corollary \[corcovdif\] we get a conformally covariant operator of order $3$ on any spin manifold of dimension $n\geq 3$, with the same principal symbol as $D_{h_0}^3$. Let $(M,h_0)$ be a Riemannian spin manifold of dimension $n\geq 3$. Then the differential operator of order $3$ acting on spinors $$L_1:=D_{h_0}^3 -\frac{2{\mathrm{cl}}\circ{\mathrm{Ric}}_{h_0}\circ\nabla^{h_0}}{n-2} +\frac{{\mathrm{scal}}_{h_0}}{(n-1)(n-2)}D_{h_0}-\frac{{\mathrm{cl}}(d({\mathrm{scal}}_{h_0}))}{2(n-1)}$$ is conformally covariant with respect to $h_0$ in the following sense: if $\omega \in C^\infty(M)$ and $\hat{h}_0=e^{2\omega}h_0$, then $$\hat{L}_1=e^{-\frac{n+3}{2}\omega}L_1e^{\frac{n-3}{2}\omega}$$ where $\hat{L}_1$ is defined as above but using the metric $\hat{h}_0$ instead of $h_0$. The existence of the operator $L_1$ with the above covariance property is already established, we are now going to compute it explicitly. The asymptotic expansion of the Poincaré-Einstein metric $g=x^{-2}(dx^2+h_x)$ at the boundary is given in [@FGR2] by $$\begin{aligned} {\overline}{g}=x^2 g=dx^2+h_0-x^2P+O(x^4),&& P=\tfrac{1}{n-2}\left({\mathrm{Ric}}_{h_0}-\tfrac{{\mathrm{scal}}_{h_0}}{2(n-1)}\right).\end{aligned}$$ We trivialize the spinor bundle on $({\overline}{X},{\overline}{g})$ from the boundary using parallel transport along the gradient vector field $X:={\partial_{x}}$. Let us write the limited Taylor series of $D_{\bar{g}}$ in this trivialization: $$\label{bard} D_{\bar{g}}={\mathrm{cl}}(\nu){\partial_{x}}+D_0+xD_1+x^2D_2+O(x^3).$$ Use the conformal change formula $$\label{ccf} D_g=x^{\frac{n+2}{2}}D_{\bar{g}}x^{-\frac{n}{2}}$$ valid in dimension $n+1$. The idea from [@GRZ] is to use the formal computation giving the residue of the scattering operator at $\lambda=\frac32$ in terms of the $x^{{\frac{n}{2}}+3}\log(x)$ coefficient in the asymptotic expansion of formal solution to $(D_g-\frac{3}{2}i)\omega=0$ (the same method has been used in [@AuGu] for forms): there is a unique solution $\omega$ modulo $O(x^{{\frac{n}{2}}+3})$ of $(D_g-\frac{3}{2}i)\omega=O(x^{{\frac{n}{2}}+3})$ of the form $$\label{ansatz} \omega=x^{\frac{n}{2}}\left(x^{-\frac{3}{2}}\omega^-_0+\sum_{j=1}^2 x^{j-\frac{3}{2}} \omega^\pm_j + x^{\frac{3}{2}} \log x \cdot \nu^+\right)+O(x^{{\frac{n}{2}}+3})$$ and $\nu^+= C_k {\rm Res}_{{\lambda}=3/2}S({\lambda})\omega_0^-= C'_k {\mathrm{cl}}(\nu)L_1(\omega_0^-)$ for some non-zero constants $C_k,C'_k$. Since we know the principal term of $L_1$ is $D_{h_0}^3$, we can renormalize later and the constant $C'_k$ is irrelevant in the computation. Recall that spinors in the $\pm i$ eigenspaces of ${\mathrm{cl}}(\nu)$ are denoted with a $\pm$ symbol. The conformally covariant operator of order $3$ from Corollary \[corcovdif\] is given on by $$\label{forml1} L_1= D_0^3 +2{\mathrm{cl}}(\nu)(D_1D_0+D_0D_1)-4D_2.$$ From , and we derive by a straightforward computation the identity on negative spinors. The same formula is obtained when we start with $\omega_0^+$, so the lemma is proved. The operators $D_1, D_2$ are given by $$\begin{aligned} D_1=&\ -\frac{{\mathrm{scal}}_{h_0}{\mathrm{cl}}(\nu)}{4(n-1)},\\ -4D_2=&\ -2{\mathrm{cl}}\circ P\circ\nabla =-\tfrac{2}{n-2}\sum_{i,j=1}^n {\mathrm{cl}}_i {\mathrm{Ric}}_{h_0;ij}\nabla_j +\frac{2{\mathrm{scal}}_{h_0} D_0}{2(n-1)(n-2)}.\end{aligned}$$ We write $\langle U,V\rangle$ for the scalar product with respect to the ${\overline}{g}$ metric, and $\nabla$ for the Riemannian connection. Notice that for $U,V$ vectors tangent to the $\{x=x_0\}$ slices, and for $A$ defined by the identity $P(U,V)=h_0(AU,V)$, we have $$\begin{aligned} \langle U,V\rangle=h_0(U-x^2 AU,V)+O(x^4).\end{aligned}$$ Let $U,V$ be local vector fields on $M$. We first extend them to be constant in the $x$ direction with respect to the product structure $(0,\epsilon)_x\times M$. Then $$\langle\nabla_X U,V\rangle=-xh_0(AU,V)+O(x^3)$$ which implies that the vector field $$\tilde{U}:=U+\frac{x^2}{2} AU$$ is parallel with respect to $X$ modulo $O(x^3)$. Let $(U_j)_{1\leq j\leq n}$ be a local orthonormal frame on $M$. Then $(X,{\tilde{U}}_1,\ldots,{\tilde{U}}_n)$ is an orthonormal frame on $(0,{\epsilon}){\times}M$ up to order $O(x^4)$ and parallel with respect to $X$ to order $O(x^3)$. To compute the Dirac operator of ${\overline}{g}$, we use the trivialization of the spinor bundle “from the boundary” given by the Gram-Schmidt orthonormalisation of this frame with respect to $\bar{g}$, which introduces an extra error term of order $O(x^4)$ (therefore harmless). Notice that $$[{\tilde{U}},\tilde{V}]=\widetilde{[U,V]} -\frac{x^2}{2}\left(A[U,V]-[U,AV]-[AU,V]\right).$$ Then we compute from the Koszul formula $$\begin{aligned} \nabla_X{\tilde{U}}_j=O(x^3),&&\nabla_X X=0,&& \nabla_{{\tilde{U}}_j}X=-xA{\tilde{U}}_j+O(x^3),\end{aligned}$$ $$\begin{split} 2\langle\nabla_{{\tilde{U}}_j}{\tilde{U}}_i,{\tilde{U}}_k\rangle=2h_0(\nabla^{h_0}_{U_j}U_i,U_k) -\frac{x^2}{2}&\left\{h_0(A[U_j,U_i]-[U_j,AU_i]-[AU_j,U_i],U_k) \right.\\ &+h_0(A[U_k,U_j]-[U_k,AU_j]-[AU_k,U_j],U_i)\\ &\left. +h_0(A[U_k,U_i]-[U_k,AU_i]-[AU_k,U_i],U_j) \right\}+O(x^3). \end{split}$$ We continue the computation at a point $p$ assuming that the frame $U_j$ is radially parallel from $p$, in particular at $p$ we have $(\nabla^{h_0}_{U_j}U_i)(p)=0$, $[U_j,U_i](p)=0$ and $U_j(p)=\partial_j$ i.e., at $p$ the vector fields $U_j$ are just the coordinate vectors of the geodesic normal coordinates. Then the coefficient of $\frac{x^2}{2}$ in $2\langle\nabla_{{\tilde{U}}_j}{\tilde{U}}_i,{\tilde{U}}_k\rangle$ simplifies a lot, and we get at $p$ $$2\langle\nabla_{{\tilde{U}}_j}{\tilde{U}}_i,{\tilde{U}}_k\rangle=2h_0(\nabla^{h_0}_{U_j}U_i,U_k) -x^2(\partial_iA_{kj}-\partial_k A_{ij})+O(x^3).$$ From the local formula for the Dirac operator [@BGV Eq 3.13] we obtain $$\begin{aligned} D_{\bar{g}}=& {\mathrm{cl}}(X){\partial_{x}}+{\mathrm{cl}}_j(U_j+\frac{x^2}{2}AU_j) -\frac{1}{2} \sum_{j,k=1}^n xA_{jk}{\mathrm{cl}}_j{\mathrm{cl}}(X){\mathrm{cl}}_k +\frac{1}{2} \sum_{i<k}h_0(\nabla^{h_0}_{U_j}U_i,U_k){\mathrm{cl}}_j{\mathrm{cl}}_i{\mathrm{cl}}_k\\ &-\frac{x^2}{4} \sum_{j=1}^n\sum_{i<k}(\partial_iA_{kj}-\partial_k A_{ij}){\mathrm{cl}}_j{\mathrm{cl}}_i{\mathrm{cl}}_k+ O(x^3).\end{aligned}$$ It follows that $D_0$ is just the Dirac operator for $h_0$. For $D_1$, we could additionally assume that at $p$, the vectors $U_j$ are eigenvectors of $A$, thus $D_1=\frac{1}{2}{\mathrm{tr}}_{h_0}(A){\mathrm{cl}}(X)$ which in view of the definition of $P$ (recall that $A$ is the transformation corresponding to $P$ with respect to $h_0$) implies after a short computation the first formula of the lemma. We also get $$D_2=\frac{1}{2} {\mathrm{cl}}_j AU_j-\frac{1}{4} \sum_{j=1}^n\sum_{i<k}(\partial_iA_{kj}-\partial_k A_{ij}){\mathrm{cl}}_j{\mathrm{cl}}_i{\mathrm{cl}}_k,$$ but in the first term the action of $U_j$ at $p$ clearly coincides with the covariant derivative (the frame is parallel at $p$) so we get the advertised formula. As for the second term, it turns out to vanish miraculously because of the coefficients inside $P$. Indeed, due to the Clifford commutations we first check that the sum where $j,i,k$ are all distinct vanishes. The remaining sum is given at $p$ by $$\sum_{i,k} {\mathrm{cl}}_k(\partial_k A_{ii}-\partial_i A_{ik})$$ which in invariant terms reads $${\mathrm{cl}}(d({\mathrm{tr}}_{h_0}(A)))+{\mathrm{cl}}(\delta^\nabla(A))$$ where $\delta^\nabla$ is the formal adjoint of the symmetrized covariant derivative with respect to $h_0$. It is known that $$\delta^\nabla{\mathrm{Ric}}_{h_0}+\frac{d({\mathrm{scal}}_{h_0})}{2}=0,$$ and from $$\begin{aligned} {\mathrm{tr}}_{h_0}({\mathrm{Ric}}_{h_0})={\mathrm{scal}}_{h_0}, && {\mathrm{tr}}_{h_0}(A)=\frac{{\mathrm{scal}}_{h_0}}{2(n-1)}, && \delta^\nabla({\mathrm{scal}}_{h_0}\cdot I)=-d({\mathrm{scal}}_{h_0})\end{aligned}$$ we get the result. This lemma ends the proof of the theorem by using . Polyhomogeneous conormal distributions, densities, blow-ups and index sets {#appA} =========================================================================== On a compact manifold with corners ${\overline}{X}$, consider the set of boundary hypersurfaces $(H_{j})_{j=1}^m$ which are codimension $1$ submanifolds with corners. Let $\rho_1, \dots, \rho_m$ be some boundary defining functions of these hypersurfaces. An index set ${\mathcal}{E}=({\mathcal}{E}_1,\dots, {\mathcal}{E}_m)$ is a subset of $({\mathbb{C}}{\times}{\mathbb{N}}_0)^m$ such that for each $M \in {\mathbb{R}}$ the number of points $(\beta, j) \in {\mathcal}{E}_{j}$ with $\Re(\beta) \leq M$ is finite, if $(\beta, k) \in {\mathcal}{E}_{j}$ then $(\beta + 1, k)\in {\mathcal}{E}_{j} $, and if $k > 0$ then also $(\beta, k-1) \in {\mathcal}{E}_{j}$. We define the set $$\dot{C}^\infty({\overline}{X}):=\{f\in C^\infty({\overline}{X}); f\textrm{ vanishes to all orders on each } H_{j}\}.$$ Its dual $C^{-\infty}({\overline}{X})$ is called the set of *extendible distributions* (the duality pairing is taken with respect to a fixed smooth $1$-density on ${\overline}{X}$). Conormal distributions on manifolds with corners were defined and analyzed by Melrose [@Me; @APS], we refer the reader to these works for more details, but we give here some definitions. We say that an extendible distribution $f$ on a manifold with corners $X$ with boundary hypersurfaces $(H_1,\dots,H_m)$ is *polyhomogeneous conormal* (phg for short) at the boundary, with index set ${\mathcal}{E}=({\mathcal}{E}_1,\dots,{\mathcal}{E}_m)$, if it is smooth in the interior $X$, conormal (i.e., if it remains in a fixed weighted $L^2$ space under repeated application of vector fields tangent to the boundary of ${\overline}{X}$) and if for each $s \in {\mathbb{R}}$ we have $$\left( \prod_{j=1}^m \prod_{\substack{(z, p) \in {\mathcal}{E}_j\\ \text{ s.t. } \Re (z) \leq s}} (V_j - z) \right) f = O\big( (\prod_{j=1}^m \rho_j )^s \big)$$ where $V_j$ is a smooth vector field on ${\overline}{X}$ that takes the form $V_j = \rho_j \partial_{\rho_j} + O(\rho_j^2)$ near $H_j$. This implies that $f$ has an asymptotic expansion in powers and logarithms near each boundary hypersurface. In particular, near the interior of $H_j$, we have $$f = \sum_{\substack{{(z,p)} \in {\mathcal}{E}_j\\ \text{ s.t. } \Re (z) \leq s}} a_{(z,p)} \rho_j^z (\log \rho_j)^p +O(\rho_j^s)$$ for every $s \in {\mathbb{R}}$, where $a_{(z,p)}$ is smooth in the interior of $H_j$, and $a_{(z,p)}$ is itself polyhomogeneous on $H_j$. The set of polyhomogeneous conormal distributions with index set ${\mathcal}{E}$ on ${\overline}{X}$ with values in a smooth bundle $F\to {\overline}{X}$ will be denoted by $${\mathcal}{A}^{{\mathcal}{E}}_{\rm phg}({\overline}{X};F).$$ Recall the operations of addition and extended union of two index sets $E_1$ and $E_2$, denoted by $E_1 + E_2$ and $E_1 {\overline{\cup}}E_2$ respectively: $$\begin{split} &E_1 + E_2 = \{ (\beta_1 + \beta_2, j_1 + j_2) \mid (\beta_1, j_1) \in E_1 \text{ and } (\beta_2 , j_2) \in E_2 \} \\ &E_1\, {\overline{\cup}}\, E_2 = E_1 \cup E_2 \cup \{ (\beta, j) \mid \exists (\beta, j_1) \in E_1, (\beta, j_2) \in E_2 \text{ with } j = j_1 + j_2 + 1 \}. \end{split}$$ In what follows, we shall write $q$ for the index set $\{ (q + n, 0) \mid n = 0, 1, 2, \dots \}$ for any $q \in {\mathbb{R}}$. For any index set $E$ and $q \in {\mathbb{R}}$, we write $E \geq q$ if $\Re(\beta) \geq q$ for all $(\beta, j) \in E$ and if $(\beta, j) \in E$ and $\Re(\beta) = q$ implies $j = 0$. Finally we say that $E$ is integral if $(\beta, j) \in E$ implies that $\beta \in \mathbb{Z}$. On ${\overline}{X}$, the most natural densities are the $b$-densities introduced by Melrose [@Me; @APS]. The bundle $\Omega_b({\overline}{X})$ of $b$-densities is defined to be $\rho^{-1}\Omega({\overline}{X})$ where $\rho=\prod_{j}\rho_j$ is a total boundary defining function and $\Omega({\overline}{X})$ is simply the usual smooth bundle of densities on ${\overline}{X}$. In particular a smooth section of the $b$-densities bundle restricts canonically on each $H_j$ to a smooth $b$-density on $H_j$. The bundle of $b$-half-densities is simply $\rho^{-{\frac{1}{2}}}\Omega^{\frac{1}{2}}({\overline}{X})$. A natural class of submanifolds, called *p-submanifolds*, of manifolds with corners is defined in Definition 1.7.4 in [@Melbook]. If $Y$ is a closed $p$-submanifold of ${\overline}{X}$, one can define the blow-up $[{\overline}{X};Y]$ of ${\overline}{X}$ around $Y$, this is a smooth manifold with corners where $Y$ is replaced by its inward pointing spherical normal bundle $S^+NY$ and a smooth structure is attached using polar coordinates around $Y$. The new boundary hypersurface is diffeomorphic to $S^+NY$ and is called *front face* of $[{{\overline}{X}};Y]$, there is a canonical smooth blow-down map $\beta:[{{\overline}{X}};Y]\to {{\overline}{X}}$ which is the identity outside the front face and the projection $S^+NY\to Y$ on the front face. See section 5.3 of [@Melbook] for details. The pull-back $\beta^*$ maps continuously $\dot{C}^\infty({\overline}{X})$ to $\dot{C}^{\infty}([{\overline}{X};Y])$ and it is a one-to-one correspondence, giving by duality the same statement for extendible distributions. Compositions of kernels conormal to the boundary diagonal {#appB} ========================================================= In this section, we introduce a symbolic way to describe conormal distributions associated to the diagonal $\Delta_{\partial}$ inside the corner of ${\overline}{X}{\times}{\overline}{X}$, ${\overline}{X}{\times}{\partial}{\overline}{X}$, or ${\partial}{\overline}{X}{\times}{\overline}{X}$. In particular, we compare the class of operators introduced by Mazzeo-Melrose (the $0$-calculus) to a natural class of pseudo-differential operators we define by using oscillatory integrals. We will prove composition results using both the push-forward Theorem of Melrose [@Me] and some classical symbolic calculus. We shall use the notations from the previous sections. Operators on ${\overline}{X}$ ----------------------------- We say that an operator $K:\dot{C}^\infty({\overline}{X})\to C^{-\infty}({\overline}{X})$ is in the class $I^s({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ if its Schwartz kernel $K(m,m')\in C^{-\infty}({\overline}{X}{\times}{\overline}{X})$ is the sum of a smooth function $K_\infty\in C^\infty({\overline}{X}{\times}{\overline}{X})$ and a singular kernel $K_s$ supported near $\Delta_{\partial}$, which can be written in local coordinates $(x,y,x',y')$ near a point $(0,y_0,0,y_0)\in \Delta_{\partial}$ under the form (here $x$ is a boundary defining function on ${\overline}{X}$ and $y$ some local coordinates on ${\partial}{\overline}{X}$ near $y_0$, and prime denotes the right variable version of them) $$\label{koscill} K_s(x,y,x',y')=\frac{1}{(2\pi)^{n+2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{{\mathbb{R}}^n} e^{-ix\xi-ix'\xi'-i(y-y')\mu}a(x,y,x',y';\xi,\xi',\mu)d\mu d\xi d\xi'$$ where $a$ is a smooth classical symbol of order $s\in {\mathbb{R}}$ in the sense that it satisfies for all multi-indices $\alpha,\alpha',\beta$ $$|{\partial}_{m}^\alpha{\partial}_{m'}^{\alpha'}{\partial}^\beta_{\zeta}a(m,m';\zeta)|\leq C_{\alpha,\alpha',\beta}(1+|\zeta|^2)^{s-|\beta|}$$ where $m=(x,y)\in {\mathbb{R}}^+{\times}{\mathbb{R}}^n$ and $\zeta:=(\xi,\xi',\mu)\in {\mathbb{R}}{\times}{\mathbb{R}}{\times}{\mathbb{R}}^n$. The integral in makes sense as an oscillatory integral: we integrate by parts a sufficient number $N$ of times in $\zeta$ to get $\Delta_\zeta^N a(m,m';\zeta)$ uniformly $L^1$ in $\zeta$; of course we pick up a singularity of the form $(x^2+{x'}^2+|y-y'|^2)^{-N}$ by this process but the outcome still makes sense as an element in the dual of $\dot{C}^{\infty}({\overline}{X}{\times}{\overline}{X})$. If ${\widetilde}{X}$ is an open manifold extending ${\overline}{X}$, such a kernel can be extended to a kernel ${\widetilde}{K}$ on the manifold ${\widetilde}{X}{\times}{\widetilde}{X}$ so that ${\widetilde}{K}$ is classically conormal to the embedded closed submanifold $\Delta_{\partial}$. Therefore our kernels (which are extendible distributions on ${\overline}{X}{\times}{\overline}{X}$) can freely be considered as restriction of distributional kernels acting on a subset of functions of ${\widetilde}{X}{\times}{\widetilde}{X}$, i.e. the set $\dot{C}^\infty({\overline}{X}{\times}{\overline}{X})$ which corresponds to smooth functions with compact support included in ${\overline}{X}{\times}{\overline}{X}$. Standard arguments of pseudodifferential operator theory show that we can require that $K_s$ in charts is, up to a smooth kernel, of the form $$K_s(x,y,x',y')=\frac{1}{(2\pi)^{n+2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{{\mathbb{R}}^n} e^{-ix\xi-ix'\xi'-i(y-y')\mu}a(y;\xi,\xi',\mu)d\mu d\xi d\xi'.$$ Indeed, it suffices to apply a Taylor expansion of $a(x,y,x',y';\zeta)$ at $\Delta_{\partial}=\{x=x'=y-y'=0\}$ and use integration by parts to show that the difference obtained by quantizing these symbols and the symbols of the form $a(y,\zeta)$ is given by smooth kernels. We say that the symbol $a$ is *classical of order $s$* if it has an asymptotic expansion as $\zeta:=(\xi,\xi',\mu)\to \infty $ $$\label{expansion} a(y;\zeta)\sim \sum_{j=0}^\infty a_{s-j}(y;\zeta)$$ where $a_j$ are homogeneous functions of degree $s-j$ in $\zeta$. It is clear from their definition that operators in $I^s({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ have smooth kernels on $({\overline}{X}{\times}{\overline}{X})\setminus \Delta_{\partial}$. Let us consider the diagonal singularity of $K$ when its symbol is classical. \[phgexp\] An operator $K_s\in I^{s-n-2}({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ has a kernel which is the sum of a smooth kernel together with a kernel which is smooth outside $\Delta_{\partial}$ and has an expansion at $\Delta_{\partial}$ in local coordinates $(x,y,x',y')$ of the form $$\label{Kxy} K_s(x,y,x',y')\sim \begin{cases} R^{-s}\sum_{j=0}^\infty R^jK^j(y,\omega) & \textrm{ if }s\notin {\mathbb{Z}},\\ R^{-s}\sum_{j=0}^\infty R^jK^j(y,\omega) +\log(R)\sum_{j=0}^\infty R^jK^{j,1}(y,\omega) & \textrm{ if }s\in {\mathbb{N}}_0,\\ R^{-s}(\sum_{j=0}^\infty R^jK^j(y,\omega) +\log(R)\sum_{j=0}^\infty R^jK^{j,1}(y,\omega)) & \textrm{ if }s\in -{\mathbb{N}}, \end{cases}$$ where $R:=(x^2+{x'}^2+|y-y'|^2)^{\frac{1}{2}}$, $(x,x',y-y'):=R\omega$ and $K^j,K^{j,1}$ are smooth. Assume $K$ has a classical symbol $a$ like in . First, we obviously have that for any $N\in{\mathbb{N}}$, $K\in C^{N}({\overline}{X}{\times}{\overline}{X})$ if $s<-N$. Let us write $t=s-n-2$, then we remark that for all $y$, the homogeneous function $a_{t-j}(y,.)$ has a unique homogeneous extension as a homogeneous distribution on ${\mathbb{R}}^{n+2}$ of order $t-j$ if $s\notin j-{\mathbb{N}}_0$ (see [@Ho Th 3.2.3]), and its Fourier transform is homogeneous of order $-s+j$. Clearly, $K(x,y,x',y')$ can be written as the Fourier transform in the distribution sense in $\zeta$ of $A_N+B_N$ where for $N\in{\mathbb{N}}$ $$\begin{aligned} A_N(y,\zeta):=\sum_{j=0}^N a_{t-j}(y;\zeta), && B_N(y,\zeta):=a(y,\zeta)-A_N(y,\zeta).\end{aligned}$$ Now $|\zeta|^{-s+N} B_N(y,\zeta)$ is in $ L^1(d\zeta)$ in $|\zeta|>1$ thus ${\mathcal}{F}_{\zeta\to Z}((1-\chi(\zeta))B_N(y,\zeta))$ is in $C^{[N-s]}$ with respect to all variables if $\chi\in C_0^\infty({\mathbb{R}}^{n+2})$ equals $1$ near $0$, while the Fourier transform ${\mathcal}{F}(\chi B_N)$ and ${\mathcal}{F}(\chi A_N)$ have the same regularity and are smooth since the convolution of ${\mathcal}{F}(\chi)$ with a homogeneous function is smooth. This implies the expansion of $K$ at the diagonal when $t\notin {\mathbb{Z}}$. For the case $t\in {\mathbb{Z}}$, this is similar but a bit more complicated. We shall be brief and refer to Beals-Greiner [@BeGr Chap 3.15] for more details (this is done for the Heisenberg calculus there but their proof obviously contains the classical case). Let us denote $\delta_{\lambda}$ the action of dilation by ${\lambda}\in{\mathbb{R}}^+$ on the space ${\mathcal}{S}'$ of tempered distributions on ${\mathbb{R}}^{n+2}$, then any homogeneous function $f_k$ of degree $-n-2-k\in -n-2-{\mathbb{N}}_0$ on ${\mathbb{R}}^{n+2}$ can be extended to a distribution ${\widetilde}{f}_k\in{\mathcal}{S}'$ satisfying $$\label{deltala} \delta_{\lambda}({\widetilde}{f}_k)= {\lambda}^{-n-2-k} {\widetilde}{f}_k + {\lambda}^{-n-2-k}\log ({\lambda}) P_k$$ for some $P_k\in{\mathcal}{S}'$ of order $k$ supported at $0$. This element $P_k$ is zero if and only if $f_k$ can be extended as a homogeneous distribution on ${\mathbb{R}}^{n+2}$, or equivalently $$\begin{aligned} \label{homogene} \int_{S^{n+1}}f_k(\omega)\omega^\alpha d\omega=0,&& \forall \alpha\in {\mathbb{N}}_0^{n+2}\textrm{ with } |\alpha|=k.\end{aligned}$$ According to Proposition 15.30 of [@BeGr], the distribution ${\widetilde}{f}_k$ has its Fourier transform which can be written outside $0$ as $${\mathcal}{F}({\widetilde}{f}_k)(Z)=L_k(Z)+M_k(Z)\log |Z|$$ where $L_k$ is a homogeneous function of degree $k$ on ${\mathbb{R}}^{n+2}\setminus\{0\}$ and $M_k$ a homogeneous polynomial of degree $k$. Thus reasoning as above when $t\notin {\mathbb{Z}}$, this concludes the proof. It can be noted from � that in the expansion at $\Delta_{\partial}$ in , one has $K^{j,1}=0$ for all $j=0,\dots, k$ for some $k\in{\mathbb{N}}$ if the symbols satisfy the condition $$\begin{aligned} \label{condition} \int_{S^{n+1}}a_{-n-2-j}(y,\omega)\omega^\alpha d\omega=0,&& \forall \alpha\in {\mathbb{N}}_0^{n+2}\textrm{ with } |\alpha|=j\end{aligned}$$ for all $ j=0,\dots,k$ and all $y\in M$. Using the expression of the symbol expansion after a change of coordinates, it is straightforward to check that this condition is invariant with respect to the choice of coordinates. A consequence of this Lemma (or another way to state it) is that if $K\in I^{s-n-2}({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ is classical, then its kernel lifts to a conormal polyhomogeneous distribution on the manifold with corners ${\overline}{X}{\times}_0{\overline}{X}$ obtained by blowing-up $\Delta_{\partial}$ inside ${\overline}{X}{\times}{\overline}{X}$ and $$\label{betaK} \beta^*K\in C^\infty({\overline}{X}{\times}_0{\overline}{X})+ \begin{cases} \rho_{{\textrm{ff}}}^{-s}C^\infty({\overline}{X}{\times}_0{\overline}{X}) & \textrm{ if }s\notin {\mathbb{Z}},\\ \rho_{{\textrm{ff}}}^{-s}C^\infty({\overline}{X}{\times}_0{\overline}{X})+\log(\rho_{{\textrm{ff}}})C^\infty({\overline}{X}{\times}_0{\overline}{X}) & \textrm{ if }s\in {\mathbb{N}}_0,\\ \rho_{{\textrm{ff}}}^{-s}(C^\infty({\overline}{X}{\times}_0{\overline}{X})+\log(\rho_{{\textrm{ff}}})C^\infty({\overline}{X}{\times}_0{\overline}{X})) & \textrm{ if }s\in -{\mathbb{N}}. \end{cases}$$ Therefore $I^s({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ is a subclass of the full $0$-calculus of Mazzeo-Melrose [@MM], in particular with no interior diagonal singularity. Let us make this more precise: \[rel0calculus\] Let $\ell\in-{\mathbb{N}}$, then a classical operator $K\in I^{\ell}({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ with a local symbol expansion has a kernel which lifts to $\beta^*K\in \rho_{\rm ff}^{-\ell-n-2}C^\infty({\overline}{X}{\times}_0{\overline}{X})+C^\infty({\overline}{X}{\times}_0{\overline}{X})$ if the symbol satisfies the condition for all $j\in{\mathbb{N}}_0$. Conversely, if $K\in C^{-\infty}({\overline}{X}{\times}{\overline}{X})$ is a distribution which lifts to $\beta^*K$ in $\rho_{\rm ff}^{-\ell-n-2}C^\infty({\overline}{X}{\times}_0{\overline}{X})+C^\infty({\overline}{X}{\times}_0{\overline}{X})$, then it is the kernel of a classical operator in $I^{-n-2}({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ with a symbol satisfying for all $j\in{\mathbb{N}}_0$. Let us start with the converse: we can extend smoothly the kernel $\beta^*K$ to the blown-up space $[{\widetilde}{X}{\times}{\widetilde}{X}, \Delta_{\partial}]$ where ${\widetilde}{X}$ is an open manifold extending smoothly ${\overline}{X}$. Then the extended function has an expansion to all order in polar coordinates $(R,\omega)$ at $\{R=0\}$ (i.e., around $\Delta_{\partial}$) where $R=(x^2+{x'}^2+|y-y'|^2)^{\frac{1}{2}}$ and $R\omega=(x,x',y-y')$ $$\begin{aligned} K(x,y,x',y')- \sum_{j=0}^k R^{-\ell -n-2+j}K^j(y,\omega) \in C^{k}({\overline}{X}{\times}{\overline}{X}),&& \forall k\in{\mathbb{N}}\end{aligned}$$ for some smooth $K^j$, in particular using Fourier transform in $Z=(x,x',y-y')$ one finds that for all $k\in{\mathbb{N}}$, there exists a classical symbol $a^{k}(y,\zeta)$ $$K(x,y,x',y')-\frac{1}{(2\pi)^{n+2}}\int e^{ix\xi+ix'\xi'+i(y-y')\mu} a^k(y;\xi,\xi',\mu)d\xi d\xi' d\mu \in C^k({\overline}{X}{\times}{\overline}{X})$$ with $a^k$ being equal to $\sum_{j=0}^ka^k_j(y;\zeta)$ when $|\zeta|>1$ for some homogeneous functions $a^k_j$ of degree $\ell-j$. Moreover, the $a^k_j$ can be extended as homogeneous distribution on ${\mathbb{R}}^{n+2}$ since they are given by Fourier transforms of the homogeneous distributions $K^j(y,Z)$ in the variable $Z$. Using that a homogeneous function on ${\mathbb{R}}^{n+2}\setminus \{0\}$ which extends as a homogeneous distribution on ${\mathbb{R}}^{n+2}$ has no $\log {\lambda}$ terms in , or equivalently satisfies , this ends one way. To prove the first statement, it suffices to consider the kernel in local coordinates and locally $\beta^*K$ has the structure with no $\log(\rho_{\textrm{ff}})$ if the local symbol satisfies . Notice that having locally the structure $\rho_{\textrm{ff}}^{-s}C^\infty({\overline}{X}{\times}{\overline}{X})$ for a function is a property which is independent of the choice of coordinates. But from what we just proved above, this implies that in any choice of coordinates the local symbol satisfies . We shall call the subclass of operators in Lemma \[rel0calculus\] the class of *log-free classical operators* of order $\ell\in-{\mathbb{N}}$, and denote it $I_{\rm lf}^\ell({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$. For (log-free if $s\in-{\mathbb{N}}$) classical operators in $I^s({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$, there is also a notion of *principal symbol* which is defined as a homogeneous section of degree $s$ of the conormal bundle $N^*\Delta_{\partial}$: if $a$ has an expansion $a(y,\zeta)\sim \sum_{j=0}^\infty a_{s-j}(y,\zeta)$ as $\zeta\to \infty$ with $a_{s-j}$ homogeneous of degree $s-j$ in $\zeta$, then the principal symbol is given by $\sigma_{\rm pr}(K)=a_s$. The principal symbol is actually not invariantly defined if one considers $K$ as an extendible distribution on ${\overline}{X}{\times}{\overline}{X}$ : if $a(y,\zeta)$ and $a'(y,\zeta)$ are two classical symbols for the kernel $K$, then if $Z=(x,x',z)$ $${\mathcal}{F}_{\zeta \to Z} (a_s(y,\zeta)-a_s'(y,\zeta))=0 \textrm{ when }x>0 \textrm{ and } x'>0,$$ thus it is defined only up to this equivalence relation. To make the correspondence with the 0-calculus of Mazzeo-Melrose [@MM], we recall that the normal operator of an operator $K\in C^\infty({\overline}{X}{\times}_0{\overline}{X})$ is given by the restriction to the front face: if $y\in \Delta_{\partial}$, $N_y(K):= K|_{{\textrm{ff}}_y}$ where ${\textrm{ff}}_y$ is the fiber at $y$ of the unit interior pointing spherical normal bundle $S^+N\Delta_{\partial}$ of $\Delta_{\partial}$ inside ${\overline}{X}{\times}{\overline}{X}$, then we remark that the normal operator at $y\in\Delta_{\partial}$ of an admissible operator $K\in I_{\rm lf}^{-n-2}({\overline}{X}{\times}{\overline}{X};\Delta_{\partial})$ is given by the homogeneous function of degree $0$ on ${\mathbb{R}}^+{\times}{\mathbb{R}}^+{\times}{\mathbb{R}}^n\simeq {\textrm{ff}}_y{\times}{\mathbb{R}}_+$ $$N_y(K)(Z)={\mathcal}{F}_{\zeta\to Z}(\sigma_{\rm pr}(K)(y,\zeta)).$$ Operators from ${\overline}{X}$ to ${\partial}{\overline}{X}$ and conversely {#interiortoboundary} ---------------------------------------------------------------------------- We define operators in $I^s({\overline}{X}{\times}{\partial}{\overline}{X},\Delta_{\partial})$ and $I^s({\partial}{\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ by saying that their respective distributional kernels are the sum of a smooth kernel on ${\overline}{X}{\times}{\partial}{\overline}{X}$ (resp. ${\partial}{\overline}{X}{\times}{\overline}{X}$) and of a singular kernel $K_s\in C^{-\infty}({\overline}{X}{\times}{\partial}{\overline}{X})$ (resp. $L_s\in C^{-\infty}({\partial}{\overline}{X}{\times}{\overline}{X})$) supported near $\Delta_{\partial}$ of the form (in local coordinates) $$\label{operatorsKL} \begin{split} &K_s(x,y,y')=\frac{1}{(2\pi)^{n+1}}\int e^{-ix\xi+i(y-y')\mu}a(y';\xi, \mu)d\xi d\mu ,\\ & L_s(y,x',y')=\frac{1}{(2\pi)^{n+1}}\int e^{ix'\xi'+i(y-y')\mu}b(y;\xi', \mu)d\xi' d\mu \end{split}$$ with $a$ and $b$ some smooth symbols $$\begin{aligned} |{\partial}_y^\alpha {\partial}_{\zeta}^\beta a(y,\zeta)|\leq C_{\alpha,\beta}{\langle}\zeta{\rangle}^{s-|\beta|}, && |{\partial}_y^\alpha {\partial}_{\zeta}^\beta b(y,\zeta)|\leq C_{\alpha,\beta}{\langle}\zeta{\rangle}^{s-|\beta|}\end{aligned}$$ for all $\alpha,\beta$. We shall say they are *classical* if their symbols have an expansion in homogeneous functions at $\zeta\to \infty$, just like above for operators on ${\overline}{X}$. It is easy to see that such operators map respectively $\dot{C}^\infty({\overline}{X})$ to $C^\infty({\partial}{\overline}{X})$ and $C^\infty({\partial}{\overline}{X})$ to $C^{-\infty}({\overline}{X})\cap C^\infty(X)$. Using the exact same arguments as for operators on ${\overline}{X}$, we have the following \[rel0calculus1\] Let $\ell\in-{\mathbb{N}}$, then a classical operator $K\in I^{\ell}({\overline}{X}{\times}{\partial}{\overline}{X},\Delta_{\partial})$ with a local symbol expansion $a(y,\zeta)\sim \sum_{j=0}^\infty a_{-n-1-j}(y,\zeta)$ has a kernel which lifts to $\beta_1^*K\in \rho_{\rm ff}^{-\ell-n-1}C^\infty({\overline}{X}{\times}_0{\partial}{\overline}{X})+C^\infty({\overline}{X}{\times}_0{\partial}{\overline}{X})$ if $$\begin{aligned} \label{condition2} \int_{S^{n}}a_{-n-1-j}(y,\omega)\omega^\alpha d\omega=0,&& \forall \alpha\in {\mathbb{N}}_0^{n+1}\textrm{ with } |\alpha|=j\end{aligned}$$ for all $j\in{\mathbb{N}}_0$. Conversely, if $K\in C^{-\infty}({\overline}{X}{\times}{\overline}{\partial}{X})$ is a distribution which lifts to $\beta_1^*K$ in $\rho_{\rm ff}^{-\ell-n-1}C^\infty({\overline}{X}{\times}_0{\partial}{\overline}{X})+C^\infty({\overline}{X}{\times}_0{\partial}{\overline}{X})$, then it is the kernel of a classical operator in $I^{\ell}({\overline}{X}{\times}{\partial}{\overline}{X},\Delta_{\partial})$ with a symbol satisfying for all $j\in{\mathbb{N}}_0$. The symmetric statement holds for operators in $I^{\ell}({\partial}{\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$. We shall also call the operators of Lemma \[rel0calculus1\] *log-free classical operators* and denote this class by $I^\ell_{\rm lf}({\overline}{X}{\times}{\partial}{\overline}{X},\Delta_{\partial})$ and $I^\ell_{\rm lf}({\partial}{\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$. Notice that, since the restriction of a function in $C^\infty({\overline}{X}{\times}_0{\overline}{X})$ to the right boundary gives a function in $C^\infty({\overline}{X}{\times}_0{\partial}{\overline}{X})$, we deduce that an operator $I^{-n-2}({\overline}{X}{\times}{\overline}{X},\Delta_{\partial})$ satisfying condition induces naturally (by restriction to the boundary on the right variable) an operator in $I^{-n-1}({\overline}{X}{\times}{\partial}{\overline}{X},\Delta_{\partial})$ satisfying . This can also be seen by considering the oscillatory integrals restricted to $x'=0$ but it is more complicated to prove. Compositions ------------ We start with a result on the composition of operators mapping from $\bar{X}$ to $M$ with operators mapping $M$ to $M$ or $M$ to $\bar{X}$. This is will be done using the push-forward theorem of Melrose [@Me Th. 5] \[composition\] Let $A:C^\infty(M;{^0\Sigma}\otimes \Omega^{\frac{1}{2}})\to C^\infty(M;{^0\Sigma}\otimes \Omega^{\frac{1}{2}})$ be a pseudo-differential operator of negative order with lifted kernel in ${\mathcal}{A}_{\rm phg}^{{E_{\rm ff}}}(M{\times}_0 M; {\mathcal}{E}\otimes \Omega_b^{\frac{1}{2}})$. Let $B: \dot{C}^{\infty}({\overline}{X};{^0\Sigma}\otimes\Omega_b^{\frac{1}{2}}) \to C^\infty(M;{^0\Sigma}\otimes\Omega^{\frac{1}{2}})$ be an operator with lifted kernel in ${\mathcal}{A}_{\rm phg}^{F_{\rm ff},F_{\rm rb}}(M{\times}_0{\overline}{X};{\mathcal}{E}_r\otimes \Omega_b^{\frac{1}{2}})$ and let $C:C^{\infty}(M;{^0\Sigma}\otimes\Omega^{\frac{1}{2}})\to C^{-\infty}({\overline}{X};{^0\Sigma}\otimes \Omega_b^{\frac{1}{2}})$ be an operator with lifted kernel on ${\mathcal}{A}_{\rm phg}^{G_{\rm ff},G_{\rm lb}}({\overline}{X}{\times}_0{M}; {\mathcal}{E}_l\otimes \Omega_b^{\frac{1}{2}})$. Then the Schwartz kernels of $A\circ B$ and $C\circ B$ lift to polyhomogeneous conormal kernels $$\begin{aligned} k_{A\circ B}\in {\mathcal}{A}_{\rm phg}^{H_{\rm ff},H_{\rm rb}}(M{\times}_0{\overline}{X}; {\mathcal}{E}_r\otimes \Omega_b^{\frac{1}{2}}),&& k_{C\circ B}\in {\mathcal}{A}_{\rm phg}^{I_{\rm ff},I_{\rm lb},I_{\rm rb}}({\overline}{X}{\times}_0{\overline}{X}; {\mathcal}{E}\otimes \Omega_b^{\frac{1}{2}})\end{aligned}$$ and the index sets satisfy $$\begin{aligned} &H_{\rm ff}=(E_{\rm ff}+ F_{\rm ff}+{\frac{n}{2}})\, {\overline{\cup}}\,(F_{\rm rb}+{\frac{n}{2}}),& H_{\rm rb}=F_{\rm rb}\,{\overline{\cup}}\,(F_{\rm ff}+{\frac{n}{2}})&& \\ &I_{\rm ff}= (F_{\rm ff}+G_{\rm ff}+{\frac{n}{2}})\,{\overline{\cup}}\, (F_{\rm rb}+G_{\rm lb}+{\frac{n}{2}}),& I_{\rm lb}= G_{\rm lb}\,{\overline{\cup}}\,(G_{\rm ff}+{\frac{n}{2}}), && I_{\rm rb}=F_{\rm rb}\,{\overline{\cup}}\,(F_{\rm ff}+{\frac{n}{2}}).\end{aligned}$$ The proof is an application of Melrose push-forward theorem. Let us discuss first the composition $A\circ B$. We denote by $\Delta$ both the diagonal in $M{\times}M$ and the submanifold $\{(m,m')\in M{\times}{\overline}{X};m=m'\}$, by $(\pi_j)_{j=l,c,r}$ the canonical projections of $M{\times}M{\times}{\overline}{X}$ obtained by projecting-off the $j$ factor (here $l,c,r$ mean left, center, right), and let $$\begin{aligned} \Delta_3:=\{(m,m',m'')\in M{\times}M{\times}{\overline}{X}; m=m'=m''\}, && \Delta_{2,j}=\pi_j^{-1}(\Delta) \textrm{ for }j=l,c,r.\end{aligned}$$ The triple space $M{\times}_0M{\times}_0{\overline}{X}$ is the iterated blow-up $$\label{triplespace} M{\times}_0M{\times}_0{\overline}{X}:=[M{\times}M{\times}{\overline}{X};\Delta_3,\Delta_{2,l},\Delta_{2,c},\Delta_{2,r}].$$ The submanifolds to blow-up are $p$-submanifolds, moreover $\Delta_3$ is contained in each $\Delta_{2,j}$ and the lifts of $\Delta_{2,j}$ to the blow-up $[M{\times}M{\times}{\overline}{X};\Delta_3]$ are disjoint. Consequently (see for instance [@GuHa Lemma 6.2]) the order of blow-ups can be commuted and the canonical projections $\pi_j$ lift to maps $$\beta_l:M{\times}_0 M{\times}_0{\overline}{X}\to M{\times}_0{\overline}{X},\, \beta_c:M{\times}_0 M{\times}_0{\overline}{X}\to M{\times}_0{\overline}{X}, \, \beta_r:M{\times}_0 M{\times}_0{\overline}{X}\to M{\times}_0 M$$ which are $b$-fibrations. The manifold $M{\times}_0M{\times}_0{\overline}{X}$ has $5$ boundary hypersurfaces, the front face ${\textrm{ff}}'$ obtained by blowing up $\Delta_3$, the faces ${\rm lf},{\rm cf}, {\rm rf}$ obtained from the respective blow-up of $\Delta_{2,l},\Delta_{2,c},\Delta_{2,r}$ and finally the face ${\textrm{rb}}'$ obtained from the lift of the original face $M{\times}M{\times}M\subset M{\times}M{\times}{\overline}{X}$. We denote by $\rho_{f}$ a smooth boundary defining function of the face $f\in\{{\textrm{ff}}',{\rm rf},{\rm cf},{\rm lf},{\textrm{rb}}'\}$. If $k_A$ and $k_B$ are the lifted kernel of $A$ and $B$ to respectively $M{\times}_0M$ and $M{\times}_0{\overline}{X}$ then it is possible to write the composition as a push-forward $$k_{A\circ B}.\mu ={\beta_c}_*\Big(\beta_r^*k_A.\beta_l^*k_B.\beta_c^*\mu\Big)$$ if $\mu\in C^{\infty}(M{\times}_0{\overline}{X};{\mathcal}{E}_r\otimes \Omega_b^{\frac{1}{2}})$. An easy computation shows that a smooth b-density $\omega$ on $M{\times}M{\times}{\overline}{X}$ lifts through $\beta$ to an element $$\beta^*\omega\in \rho_{{\textrm{ff}}'}^{2n}(\rho_{\rm lf}\rho_{\rm rf}\rho_{\rm cf})^n C^\infty(M{\times}_0M{\times}_0{\overline}{X}; \Omega_b)$$ so by considering the lifts through $\beta_{l},\beta_c,\beta_r$ of boundary defining functions in $M{\times}_0{\overline}{X}$, $M{\times}_0{\overline}{X}$ and ${\overline}{X}{\times}_0{\overline}{X}$ respectively we deduce that there is some index set $K=(K_{{\textrm{ff}}'},K_{{\textrm{rb}}'},K_{\rm lf},K_{\rm rf},K_{\rm cf})$ such that $$\begin{aligned} \lefteqn{\beta_r^*k_A.\beta_l^*k_B.\beta_c^*\mu\in {\mathcal}{A}_{\rm phg}^{K}(M{\times}_0M{\times}_0{\overline}{X};\Omega_b),}\\ &K_{{\textrm{ff}}'}=E_{{\textrm{ff}}}+ F_{{\textrm{ff}}}+{\frac{n}{2}},&& K_{{\textrm{rb}}'}=F_{{\textrm{rb}}},&& K_{\rm lf}=F_{{\textrm{ff}}}+{\frac{n}{2}},&& K_{\rm rf}=E_{{\textrm{ff}}}+{\frac{n}{2}}, && K_{\rm cf}=F_{{\textrm{rb}}}+{\frac{n}{2}}.\end{aligned}$$ Then from the push-forward theorem of Melrose [@Me Th. 5], we obtain that $$\begin{aligned} \lefteqn{(\beta_c)_*(\beta_r^*k_A.\beta_l^*k_B.\beta_c^*\mu)\in {\mathcal}{A}_{\rm phg}^{H_{{\textrm{ff}}},H_{{\textrm{rb}}}}(M{\times}_0{\overline}{X},\Omega_b),}\\ &H_{{\textrm{ff}}}=(E_{{\textrm{ff}}}+ F_{{\textrm{ff}}}+{\frac{n}{2}})\, {\overline{\cup}}\,(F_{{\textrm{rb}}}+{\frac{n}{2}}), && H_{{\textrm{rb}}}=F_{{\textrm{rb}}}\,{\overline{\cup}}\,(F_{{\textrm{ff}}}+{\frac{n}{2}})&&\end{aligned}$$ and this shows the first composition result for $A\circ B$. Remark that to apply [@Me Th.5], we need the index of $K_{\rm rf}>0$, i.e., $E_{{\textrm{ff}}}+n/2>0$, but this is automatically satisfied with our assumption that $A$ is a pseudodifferential operator of negative order on $M$. The second composition result is very similar, except that there are more boundary faces to consider. One defines $\Delta_3:=\{(m,m',m'')\in {\overline}{X}{\times}M{\times}{\overline}{X}; m=m'=m''\}$ and let $$\Delta_{2,j}=\{(m_l,m_c,m_r)\in{\overline}{X}{\times}M{\times}{\overline}{X}; m_i=m_k \textrm{ if }j\notin\{i,k\}\}$$ similarly as before. The triple space is defined like , it has now $6$ boundary faces which we denote as in the case above but with the additional face, denoted ${\textrm{lb}}'$, obtained from the lift of the original boundary $M{\times}M{\times}{\overline}{X}$. The same arguments as above show that the canonical projections from ${\overline}{X}{\times}_0 M{\times}_0{\overline}{X}$ obtained by projecting-off one factor lift to b-fibrations $\beta_r,\beta_l,\beta_c$ from the triple space to ${\overline}{X}{\times}_0M$, $M{\times}_0{\overline}{X}$ and ${\overline}{X}{\times}_0{\overline}{X}$. Like for the case above, one has to push-forward a distribution $\beta_r^*k_C.\beta_l^*k_B.\beta_c^*\mu$, and a computation gives that there is an index set $L=(L_{{\textrm{ff}}'},L_{{\textrm{rb}}'},L_{{\textrm{lb}}'},L_{\rm lf},L_{\rm rf},L_{\rm cf})$ such that $$\begin{aligned} \lefteqn{\beta_r^*k_C.\beta_l^*k_B.\beta_c^*\mu\in {\mathcal}{A}_{\rm phg}^{L}({\overline}{X}{\times}_0M{\times}_0{\overline}{X};\Omega_b),}\\ &L_{{\textrm{ff}}'}=F_{{\textrm{ff}}}+ G_{{\textrm{ff}}}+{\frac{n}{2}},&& L_{{\textrm{rb}}'}=F_{{\textrm{rb}}},&& L_{{\textrm{lb}}'}=G_{{\textrm{lb}}},\\ &L_{\rm lf}=F_{{\textrm{ff}}}+{\frac{n}{2}},&& L_{\rm rf}=G_{{\textrm{ff}}}+{\frac{n}{2}}, && L_{\rm cf}=F_{{\textrm{rb}}}+G_{{\textrm{rb}}}+{\frac{n}{2}}\end{aligned}$$ and by pushing forward through $\beta_c$ using Melrose [@Me Th. 5], we deduce that the result is polyhomogeneous conormal on ${\overline}{X}{\times}_0{\overline}{X}$ with the desired index set. In order to analyze the composition $K^*K$ in Subsection \[caldproj\], we use the symbolic approach since it is a slightly more precise (in terms of log terms at the diagonal) than the push-forward Theorem in this case, and a bit easier to compute the principal symbol of the composition. We are led to study the composition between classical operators $K$ and $L$ where $K:C^\infty({\overline}{X}) \to C^\infty({\partial}{\overline}{X})$ is an operator in $I^{-1}({\overline}{X}{\times}{\partial}{\overline}{X})$ and $L:C^\infty({\partial}{\overline}{X}) \to C^\infty({\overline}{X})$ is in $I^{-1}({\partial}{\overline}{X}{\times}{\overline}{X})$. We show \[compositionKL\] Let $K\in I^{-1}({\overline}{X}{\times}{\partial}{\overline}{X})$ and $L\in I^{-1}({\partial}{\overline}{X}{\times}{\overline}{X})$ with principal symbols $\sigma_K(y;\xi,\mu)$ and $\sigma_L(y;\xi,\mu)$. The composition $L\circ K$ is a classical pseudodifferential operator on ${\partial}{\overline}{X}$ in the class $L\circ K\in \Psi^{-1}({\partial}{\overline}{X})$. Moreover the principal symbol of $LK$ is given by $$\label{calculsymbpr} \sigma_{\rm pr}(L\circ K)(y;\mu)=(2\pi)^{-2}\int_{0}^\infty\hat{\sigma}_{L}(y;-x,\mu). \hat{\sigma}_{K}(y;x,\mu) dx.$$ where $\hat\sigma$ denotes the Fourier transform of $\sigma$ in the variable $\xi$. Since the composition with smoothing operators is easier, we essentially need to understand the composition of singular kernels like . Writing the kernel of $K$ and $L$ as a sum of elements $K_j,L_j$ of the form , we are reduced to analyze in a chart $U$ $$L_jK_j f(y)= \frac{1}{(2\pi)^{2n+2}}\int e^{ix'(\xi'-\xi)+iy'(\mu'-\mu)+iy\mu-iy''\mu'}b(y;\xi', \mu)\chi(x',y') a(y'';\xi, \mu') f(y'')dy''d\Omega$$ where $d\Omega:=dy'dx'd\xi d\xi'd\mu d\mu'$, $\chi\in C_0^\infty(U)$ and $a,b$ are compactly supported in $U$ in the $y$ and $y''$ coordinates. If $U$ intersects the boundary ${\partial}{\overline}{X}$, then $\chi$ is supported in $x'\geq 0$. The kernel of the composition $L_jK_j$ in the chart $U$ is then $$\begin{split} F(y,y'')=&\, \frac{1}{(2\pi)^{2n+2}}\int e^{ix'(\xi'-\xi)+iy'(\mu'-\mu)+iy\mu-iy''\mu'}b(y;\xi', \mu)\chi(x',y') a(y'';\xi, \mu') d\Omega \\ =&\, \frac{1}{(2\pi)^{n}}\int e^{i\mu(y-y'')} c(y,y'';\mu)d\mu \end{split}$$ where $$c(y,y'';\mu):=\frac{1}{(2\pi)^{n+2}}\int e^{-iy''.\mu'}b(y;\xi',\mu)a(y''; \xi, \mu-\mu')\hat{\chi}(\xi-\xi',\mu') d\mu'd\xi d\xi'.$$ We want to prove that $c(y,y'';\mu)$ is a symbol of order $-1$ with an expansion in homogeneous terms in $\mu$ as $\mu\to \infty$. We shall only consider the case where $U\cap {\partial}{\overline}{X}\not=\emptyset$ since the other case is simpler. First, remark that in $U$ the function $\chi$ can be taken of the form $\chi(x,y)=\varphi(x)\psi(y)$ with $\psi\in C_0^\infty({\mathbb{R}}^n)$ and $\varphi\in C_0^\infty([0,1))$ equal to $1$ in $[0,1/2]$, therefore $\hat{\chi}(\xi,\mu)=\hat{\varphi}(\xi)\hat{\psi}(\mu)$ with $\hat{\psi}$ Schwartz and by integration by parts one also has $$\hat{\varphi}(\xi)= \frac{1}{i\xi}( 1+ \hat{\varphi'}(\xi)).$$ with $\hat{\varphi'}$ Schwartz. We first claim that $|{\partial}_y^\alpha{\partial}^\beta_{y''}{\partial}_\mu^\gamma c(y,y'';\mu)|\leq C {\langle}\mu{\rangle}^{-1-|\gamma|}$ uniformly in $y,y''$: indeed using the properties of $\hat{\chi}$ and the symbolic assumptions on $a,b$, we have that for any $N\gg |\beta|$, there is a constant $C>0$ such that $$\begin{aligned} \lefteqn{|{\partial}_y^\alpha{\partial}^\beta_{y''}{\partial}_\mu^\gamma c(y,y'';\mu)| }\\ &&\leq C\int \Big(\frac{1}{1+|\xi|'+|\mu|}\Big)^{1+k}\Big(\frac{1}{1+|\xi|+|\mu|}\Big)^{1+j}{\langle}\xi-\xi'{\rangle}^{-1}{\langle}\mu'{\rangle}^{-N+|\beta|} d\mu'd\xi d\xi'\end{aligned}$$ where $j+k=|\gamma|$. Using polar coordinates $i\xi+\xi'=re^{i\theta}$ in ${\mathbb{C}}\simeq {\mathbb{R}}^2$, the integral above is bounded by $$C\int \Big(\frac{1}{1+r|\cos(\theta)|+|\mu|}\Big)^{1+k}\Big(\frac{1}{1+r|\sin\theta|+|\mu|}\Big)^{1+j}\frac{1}{1+r|\cos\theta-\sin\theta|}r drd\theta$$ which, by a change of variable $r\to r|\mu|$ and splitting the $\theta$ integral in different regions, is easily shown to be bounded by $C{\langle}\mu {\rangle}^{-1-|\beta|}$. To prove that $LK$ it is a classical operator of order $-1$ (with an expansion in homogeneous terms), we can modify slightly the usual proof of composition of pseudo-differential operators, like in Theorem 3.4 of [@GrSj]. Let $\theta\in C_0^\infty({\mathbb{R}})$ be an even function equal to $1$ near $0$. We write for $\mu={\lambda}\omega$ with $\omega\in S^{n-1}$ $$\begin{split} F(y,y'')=&\, \frac{1}{(2\pi)^{2n+2}}\int e^{ix'(\xi'-\xi)-i(\mu'-\mu)(y''-y')}\chi(x',y')b(y;\xi',\mu)a(y'';\xi,\mu')dy'dx'd\xi d\mu d\mu' d\xi'\\ =&\, \frac{1}{(2\pi)^{n}}\int e^{i(y-y'')\mu}c(y,y'';\mu) d\mu \end{split}$$ with $$\begin{split} \lefteqn{c(y,y';\mu)}\\ &= \frac{{\lambda}^{n+1}}{(2\pi)^{n+2}}\int e^{-i{\lambda}x'\zeta-i{\lambda}\sigma. s}\chi(x',y''-s)b(y;\xi',{\lambda}\omega)a(y'';\xi'+{\lambda}\zeta,{\lambda}(\omega+\sigma)) d\Omega d\xi' \\ &= \frac{{\lambda}^{n+1}}{(2\pi)^{n+2}}\int e^{-i{\lambda}x'\zeta-i{\lambda}\sigma. s}\varphi(x')\theta(\zeta) \psi(y''-s)b(y;\xi',{\lambda}\omega)a(y'';\xi'+{\lambda}\zeta,{\lambda}(\omega+\sigma))d\Omega d\xi' \\ & \quad + \frac{{\lambda}^{n+1}}{(2\pi)^{n+2}}\int e^{-i{\lambda}x'\zeta-i{\lambda}\sigma. s}\varphi(x')(1-\theta)(\zeta) \psi(y''-s)\\ &\qquad \qquad \qquad \quad \times b(y;\xi',{\lambda}\omega)a(y'';\xi'+{\lambda}\zeta,{\lambda}(\omega+\sigma))d\Omega d\xi' \\ & =:c_1(y,y'';\mu)+c_2(y,y'';\mu) \end{split}$$ where $\Omega=(\sigma,s,\zeta,x')$. Let us denote the phase by $\Phi:= x'\zeta+\sigma.s$. The last integral can be dealt with by integrating by parts in $x'$: $$\label{c_2} \begin{split} &{c_2(y,y'';\mu)}\\ &=\frac{{\lambda}^{n}}{i(2\pi)^{n+2}}\int e^{-i{\lambda}\Phi}\varphi'(x')\frac{(1-\theta)(\zeta)}{\zeta} \psi(y''-s)b(y;\xi',{\lambda}\omega)a(y'';\xi'+{\lambda}\zeta,{\lambda}(\omega+\sigma))d\Omega d\xi'\\ & + \frac{{\lambda}^{n}}{i(2\pi)^{n+2}}\int e^{-i{\lambda}\sigma.s}\frac{(1-\theta)(\zeta)}{\zeta} \psi(y''-s)b(y;\xi',{\lambda}\omega)a(y'';\xi'+{\lambda}\zeta,{\lambda}(\omega+\sigma))d\sigma dsd\zeta d\xi'. \end{split}$$ We can extend $\varphi'={\partial}_x\varphi$ by $0$ on $(-\infty,0]$ to obtain a $C_0^\infty({\mathbb{R}})$ function which vanishes near $0$. Since $\varphi'$ now vanishes near $0$, one easily proves that the first integral in is a $O({\lambda}^{-N})$ for all $N$, uniformly in $y,y''$ by using integrations by parts $N$ times in $x'$ and ${\partial}_{x'}(e^{-i{\lambda}x'\zeta})=-i{\lambda}\zeta e^{-i{\lambda}x'\zeta}$. Now for the second integral in , we use stationary phase in $(\sigma,s)$, one has for any $N\in {\mathbb{N}}$ $$\label{statphase} \begin{split} \lefteqn{\int e^{-i{\lambda}\sigma. s}\psi(y''-s)\frac{(1-\theta)(\zeta)}{\zeta}b(y;\xi',{\lambda}\omega)a(y'';\xi'+{\lambda}\zeta,{\lambda}(\omega+\sigma))d\sigma ds}\\ &= (2\pi)^n\frac{(1-\theta)(\zeta)}{\zeta}(\sum_{|\alpha|\leq N} \frac{i^{|\alpha|}}{\alpha!}{\partial}^\alpha\psi(y'')b(y;\xi',\mu){\partial}^\alpha_\mu a(y'';\xi'+{\lambda}\zeta,\mu) +S_N(y,y'';\xi',\zeta,\mu)) \end{split}$$ with $|S_N(y,y'';\xi',\zeta,\mu)|\leq C {\langle}(\xi',\mu){\rangle}^{-1} {\langle}(\xi'+|\mu|\zeta,\mu){\rangle}^{-1-N}$. Now, both $a$ and $b$ can be written under the form $a=a_N+a_h$ and $b=b_h+b_N$ where $a_N(y;\xi,\mu),b_N(y;\xi,\mu)$ are bounded in norm by $C{\langle}(\xi,\mu){\rangle}^{-N}$ and $a_h(y;\xi,\mu),b_h(y,\xi,\mu)$ are finite sums of homogeneous functions $a_h^{-j},b_h^{-j}$ of order $-j$ in $|(\xi,\mu)|>1$ for $j=1,\dots N-1$. Replacing $a,b$ in by their decomposition $a_N+a_h$ and $b_N+b_h$ we get that $c(y,y'',\mu)$ is the sum of a term bounded uniformly by $C{\langle}\mu{\rangle}^{-N+2}$ and some terms of the form $${\lambda}\int \frac{1-\theta(\zeta)}{\zeta}b_h^{-j}(y;\xi',\mu){\partial}^\alpha \psi(y''){\partial}_\mu^\alpha a_h^{-k}(y'';\xi'+{\lambda}\zeta,\mu)d\zeta d\xi'.$$ The integral is well defined and is easily seen (by changing variable $\xi'\to {\lambda}\xi'$) to be homogeneous of order $-k-j-|\alpha|+1$ for ${\lambda}=|\mu|>1$ . This shows that $c_2(y,y'';\mu)$ has an expansion in homogeneous terms. It remains to deal with $c_1$. We first apply stationary phase in the $(\sigma,s)$ variables and we get $$\begin{aligned} c_1(y,y'';\mu)= &\frac{{\lambda}}{(2\pi)^2}\sum_{|\alpha|\leq N} \frac{i^{|\alpha|}}{\alpha!}{\partial}^\alpha \psi(y'')\int e^{-i{\lambda}x'\zeta}b(y,\xi',\mu) \varphi(x')\theta(\zeta){\partial}^\alpha_\mu a(y'';\xi'+{\lambda}\zeta,\mu)dx'd\xi' d\zeta \\ &+\int \varphi(x')S_N'(y,y'';\xi',\zeta,\mu)d\xi' d\zeta dx'\end{aligned}$$ for some $S_N'$ which will contribute $O({\lambda}^{-N-2})$ like for $c_2$ above. Decomposing $a(y,\xi,\mu)$ and $b(y,\xi,\mu)$ as above in homogeneous terms outside a compact set in $(\xi,\mu)$, it is easy to see that up to a $O({\lambda}^{-N})$ term, we can reduce the analysis of $c_1(y,y'';\mu)$ to the case where $a,b$ are replaced by terms $a_h^{-j},b_h^{-k}$ homogeneous of orders $-j,-k$ outside compacts. We then have $$\label{homoexp} \begin{split} \lefteqn{\int e^{-i{\lambda}x'\zeta}b_h^{-j}(y,\xi',\mu) \varphi(x')\theta(\zeta){\partial}^\alpha_\mu a_h^{-k}(y'';\xi'+{\lambda}\zeta,\mu)dx'd\xi' d\zeta}\\ &={\lambda}^{-j-k-|\alpha|+1}\int e^{-i{\lambda}x'\zeta}b_h^{-j}(y,\xi',\omega) \varphi(x')\theta(\zeta){\partial}^\alpha_\mu a_h^{-k}(y'';\xi'+\zeta,\omega)dx'd\xi' d\zeta \end{split}$$ and we write by Taylor expansion at $\zeta=0$ $$\label{taylorexp} \theta(\zeta){\partial}_\mu^\alpha a_h^{-k}(y'';\xi'+\zeta,\omega)=\theta(\zeta){\partial}_\mu^\alpha a_h^{-k}(y'';\xi',\omega)+ \zeta \theta(\zeta) a'(y'',\xi',\zeta,\omega)$$ for some $a'(y'';\xi',\zeta,\mu)$ smooth in $y''$ and homogeneous of degree $-k-1$ in $|(\xi,\zeta,\mu)|>1$. For the term with $a'$, we have by integration by parts in $x'$ $$\label{a'} \begin{split} \lefteqn{\int \zeta e^{-i{\lambda}x'\zeta}b_h^{-j}(y,\xi',\omega) \varphi(x')\theta(\zeta) {\partial}^\alpha_\mu a'(y'';\xi',\zeta,\omega)dx'd\xi' d\zeta}\\ &=(i{\lambda})^{-1}\int e^{-i{\lambda}x'\zeta}\varphi'(x')b_h^{-j}(y,\xi',\omega)\theta(\zeta) {\partial}^\alpha_\mu a'(y'';\xi',\zeta,\omega)dx'd\xi' d\zeta\\ &\quad +(i{\lambda})^{-1}\int b_h^{-j}(y,\xi',\omega)\theta(\zeta){\partial}^\alpha_\mu a'(y'';\xi',\zeta,\omega)d\xi' d\zeta \end{split}$$ and the first term is $O({\lambda}^{-\infty})$ by non-stationary phase while the second one is homogeneous of order $-1$ in ${\lambda}$ (the integrals in all variables are converging). It remains to deal with the first term in , we notice that $\theta$ is even and so $$\int \varphi(x') e^{-i{\lambda}x'\zeta}\theta(\zeta)dx'd\zeta={\lambda}^{-1}\int \hat{\theta}(x')\varphi(x'/{\lambda})dx' ={\lambda}^{-1}\pi-\int \hat{\theta}(x')(1-\varphi(x'/{\lambda}))dx'.$$ Since $\hat{\theta}$ is Schwartz, the last line clearly has an expansion of the form $\pi{\lambda}^{-1}+O({\lambda}^{-\infty})$ for some constant $C$, and combining with , we deduce that is thus homogeneous of degree ${\lambda}^{-j-k-1}$ modulo $O({\lambda}^{\infty})$. This ends the proof of the fact that $KL$ is a classical pseudo-differential operator on $M$. Now, we compute the principal symbol. According to the discussion above, it is given by $$\begin{aligned} &-i(2\pi)^{-2}\int \frac{1}{\zeta}\Big(\sigma_{L}(y;\xi',\mu)((1-\theta(\zeta))\sigma_{K}(y;\xi'+\zeta,\mu)+ \theta(\zeta)\zeta\sigma_K'(y;\xi',\zeta,\mu)\Big)d\xi' d\zeta\\ &+(2\pi)^{-2}\pi \int \sigma_L(y,\xi',\mu) \sigma_K(y;\xi',\mu)d\xi'\end{aligned}$$ where $\zeta\sigma_K'(y;\xi',\zeta,\mu):=\sigma_K(y;\xi'+\zeta,\mu)-\sigma_K(y;\xi',\mu)$. It is straightforward to see that this is equal to by using the fact that the Fourier transform of the Heaviside function is the distribution $\pi\delta-i/\zeta$. Notice that the integral makes sense since $\sigma_K,\sigma_L$ are $L^2$ in the $\xi'$ variable. [99]{} B. Ammann, V. Nistor, *Weighted Sobolev spaces and regularity for polyhedral domains,* Comput. Methods Appl. Mech. Engrg. **196** (2007), no. 37-40, 3650–3659. M. F. Atiyah, I. M. Singer, *The index of elliptic operators on compact manifolds.*, Bull. Amer. Math. Soc. **69** (1963) 422–433. E. Aubry, C. Guillarmou, *Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity,*, to appear, Journ. Eur. Math. Soc., arXiv:0808.0552. R. Beals, P. Greiner, *Calculus on Heisenberg manifolds*, Annals of Math. Studies **119**, Princeton University Press. N. Berline, E. Getzler, M. Vergne, *Heat kernel and Dirac operators*, 2004 edition of Vol 298 Grundlehren der mathematishen Wissenschaften 1992, Springer Verlag. B. Booss-Bavnbek, M. Lesch, C. Zhu, *The Calderón projection: New definition and applications*, Journal of Geometry and Physics **59** (2009), No. 7, 784-826 . B. Booss-Bavnbek, K. P. Wojciechowski, *Elliptic boundary problems for Dirac operators*, Birkhäuser Boston, Inc., Boston, MA 1993. B. Bojarski, *The abstract linear conjugation problem and Fredholm pairs of subspaces*, In Memoriam I. N. Vekua (Tbilisi Univ. 1979) 45–60. M. Braverman, *New proof of the cobordism invariance of the index*, Proc. Amer. Math. Soc. [**130**]{} (2002) 1095–1101. A. P. Calderón, *Boundary value problems for elliptic equations*, 1963 Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), 303–304. S. Y. Cheng, S. T.  Yau, *On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation*, Comm. Pure Appl. Math. **33** (1980), no. 4, 507–544. H. Donnelly, C.  Fefferman, *$L\sp{2}$-cohomology and index theorem for the Bergman metric*, Ann. of Math.**118** (1983), no. 3, 593–618. C. L. Epstein, *Subelliptic ${\rm Spin}\sb {\mathbb C}$ Dirac operators. I*, Ann. of Math. (2) **166** (2007), no. 1, 183–214. C. L. Epstein, *Subelliptic ${\rm Spin}\sb {\mathbb C}$ Dirac operators. II. Basic estimates*, Ann. of Math. (2) **166** (2007), no. 3, 723–777. C. L. Epstein, R. Melrose, *Shrinking tubes and the d-bar Neumann problem*, preprint available online at http://www.math.upenn.edu/$\sim$cle/papers/index.html. C. Fefferman *The Bergman kernel and biholomorphic mappings of pseudoconvex domains,* Invent. Math. **26** (1974), 1–65. C. Fefferman, C. R. Graham, *Conformal invariants*, SMF Astérisque, hors série (1985), 95–116. C. Fefferman, C. R. Graham, *The ambient metric*, preprint arXiv:0710.0919. P. Gilkey, *The residue of the local eta function at the origin*, Math. Ann. **240** (1979), no. 2, 183–189. C. R. Graham, J. M. Lee, *Einstein metrics with prescribed conformal infinity on the ball*, Adv. Math. **87** (1991), no. 2, 186–225. C. R. Graham, M. Zworski, *Scattering matrix in conformal geometry*, Invent. Math. **152** (2003), 89–118. A. Grigis, J. Sjöstrand, *Microlocal analysis for differential operators*, Lecture note series **196** (1994) London Math. Soc., Cambridge Univ. Press. C. Guillarmou, *Meromorphic properties of the resolvent for asymptotically hyperbolic manifolds*, Duke Math. J. **129** no. 1 (2005), 1–37. C. Guillarmou, A. Hassell, *Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I,* Math. Ann. **341** (2008), no. 4, 859–896. C. Guillarmou, S. Moroianu, J. Park, *Eta invariant, Dirac operator and odd Selberg zeta function on convex co-compact hyperbolic manifolds*, Adv. Math. **225** (2010), no. 5, 2464–2516. N. Hitchin, *Harmonic spinors,* Advances in Math. **14** (1974), 1–55. L. Hörmander, *The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis.* Springer-Verlag, Berlin, 2003. M. Joshi, A. Sá Barreto, *Inverse scattering on asymptotically hyperbolic manifolds*, Acta Math. **184** (2000), 41–86. P. Kirk, M. Lesch, *The $\eta$-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary*, Forum Math. **16** (2004), no. 4, 553–629. B.H. Lawson, M-L. Michelsohn, *Spin geometry,* Princeton Mathematical Series **38**, Princeton University Press, Princeton, NJ, 1989. M. Lesch, *Deficiency indices for symmetric Dirac operators on manifolds with conic singularities*, Topology **32** 611–623. P. Loya, *Geometric BVPs, Hardy spaces, and the Cauchy integral and transform on regions with corners,* J. Differential Equations [**239**]{} (2007), no. 1, 132–195. P. Loya, J. Park, *On the gluing problem for the spectral invariants of Dirac operators*, Adv. Math. **202** (2006), no. 2, 401–450. R. Mazzeo, *Elliptic theory of differential edge operators I*, Comm. Partial Diff. Equations **16** (1991), no. 10, 1615–1664. R. Mazzeo, R. Melrose, *Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature*, J. Funct. Anal. **75** (1987), 260–310. R. B. Melrose, *Calculus of conormal distributions on manifolds with corners*, Int. Math. Res. Not. **3** (1992), 51–61. R. B. Melrose, *The Atiyah-Patodi-Singer index theorem* (AK Peters, Wellesley, 1993). R. B. Melrose, *Differential analysis on manifolds with corners*, book in preparation, available online at www-math.mit.edu/$\sim$rbm/book.html. S. Moroianu, *Cusp geometry and the cobordism invariance of the index*, Adv. Math. **194** (2005), 504–519. W. Müller, A. Strohmaier *Scattering of low energies on manifolds with cylindrical ends and stable systoles*, GAFA, to appear. L. I.  Nicolaescu, *The Maslov index, the spectral flow, and decompositions of manifolds*, Duke Math. J. 80 (1995), no. 2, 485–533. L. I. Nicolaescu, *On the cobordism invariance of the index of Dirac operators*, Proc. AMS **125** (1997), 2797–2801. S. G. Scott, K. P. Wojciechowski, *The $\zeta$-determinant and Quillen determinant for a Dirac operator on a manifold with boundary*, Geom. Funct. Anal. **10** (2000), no. 5, 1202–1236. R. T. Seeley, *Singular integrals and boundary value problems*, Amer. J. Math. **88** (1966), 781–809. R. T. Seeley, *Topics in pseudo-differential operators*, 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) 167–305. M. Wodzicki, *Local invariants of spectral asymmetry,* Invent. Math. **75** (1984), no. 1, 143–177.

--- abstract: 'If a family symmetry exists for the quarks and leptons, the Higgs sector is expected to be enlarged to be able to support the transformation properties of this symmetry. There are however three possible generic ways (at tree level) of hiding this symmetry in the context of the Standard Model with just one Higgs doublet. All three mechanisms have their natural realizations in the unification symmetry $E_6$ and one in $SO(10)$. An interesting example based on $SO(10) \times A_4$ for the neutrino mass matrix is discussed.' --- UCRHEP-T391\ June 2005 [**Hiding the Existence of a Family\ Symmetry in the Standard Model\ **]{} There are three families of quarks and leptons. The pattern of their masses and mixing angles has been under study for a long time. If a family symmetry exists at the Lagrangian level, broken presumably only spontaneously and by explicit soft terms, the Yukawa couplings $f_{ijk} q_i q^c_j \phi_k$ and $f'_{ijk} l_i l^c_j \phi_k$ should have two or more Higgs doublets $\phi_k$. Otherwise $f_{ijk}$ and $f'_{ijk}$ would reduce to $f_{ij}$ and $f'_{ij}$. Since the number of bilinear invariants of any given symmetry is very much limited, this would not result in a realistic description of quark and lepton mass matrices. On the other hand, the well-tested Standard Model (SM) requires only one Higgs doublet (although it remains to be discovered experimentally). One Higgs doublet is also preferred phenomenologically as an explanation of the natural suppression of flavor-changing neutral currents [@gw77]. Thus an important theoretical question is whether a family symmetry can be hidden in the context of the SM and how. The answer is yes and there are three generic mechanisms (at tree level) for achieving it, as shown below. Specific new particles with masses well above the electroweak scale are required, but some of these exist already in well-known unification symmetries such as $E_6$ and $SO(10)$. The idea is very simple. The information concerning the family symmetry is encoded in the Yukawa couplings $f_{ijk}$ of quarks through the various $\phi_k$ Higgs doublets. If only one Higgs doublet is allowed, the same information can be encoded using the dimension-five operator [@fn79] $${\cal L}_Y = {f_{ijk} \over \Lambda} q_i q^c_j \phi \sigma_k + H.c.,$$ where $\sigma_k$ are heavy scalar singlets. This mechanism is widely used in model building but without any discussion of how it may arise from fundamental interactions. Of course, if the new interactions occur near the Planck scale, then they may be very strong and the effective operator of Eq. (1) is nonperturbative in general. However, if the new physics responsible for the family symmetry is at or below the quark-lepton unification scale of about $10^{16}$ GeV, then it is reasonable to ask how it may be realized at tree level. In the following it is shown that there are three generic ways of doing this, and each will be discussed also in the context of the complete underlying new physics involved. The assumption of this operator means that one or more of these generic mechanisms is likely to be correct and is thus an important clue to physics beyond the SM. Even a casual observation of Eq. (1) shows that the four fields involved can be grouped into the product of two pairs in only three ways, in exact analogy to the classic analysis of the scattering of two particles into two particles. The intermediate states must then have well-defined transformation properties under the standard $SU(3)_C \times SU(2)_L \times U(1)_Y$ gauge group. These are depicted in Figures 1 to 3. The heavy quarks $Q_{1,2}$ and $Q^c_{1,2}$ are $SU(2)_L$ singlets (doublets) and $H$ are heavy scalar doublets. (400,225)(0,0) (125,200)(200,150) (200,100)(200,150) (200,100)(200,50) (125,0)(200,50) (275,200)(200,150)3 (275,0)(200,50)3 (115,200)\[\][$q$]{} (115,0)\[\][$q^c$]{} (215,125)\[\][$Q^c_1$]{} (201,100)\[\][$\times$]{} (215,75)\[\][$Q_1$]{} (285,200)\[\][$\phi$]{} (285,0)\[\][$\sigma$]{} Consider first Fig. 1. Under $SU(3)_C \times SU(2)_L \times U(1)_Y$, $q \sim (3,2,1/6)$ and $\phi \sim (1,2,\pm1/2)$, hence $Q^c_1 \sim (3^*,1,-2/3)$ or $(3^*,1,1/3)$ is required. This means $Q_1 \sim (3,1,2/3)$ or $(3,1,-1/3)$, i.e. heavy quark singlets with charges equal to either those of the $u$ quarks or $d$ quarks. This mechanism is thus equivalent to that of the canonical seesaw mechanism for Dirac fermions [@dwr87]. The effective family structure of Eq. (1) is then given by $${f_{ijk} \over \Lambda} = y_{ia} (M^{-1})_{ab} h_{bjk},$$ where $y_{ia}$ are the couplings of $q_i (Q^c_1)_a \phi$, $h_{bjk}$ those of $(Q_1)_b q^c_j \sigma_k$, and $M$ the mass matrix of $Q_1 Q^c_1$. Since the family symmetry applies to all three of these quantities, it is well hidden in the resulting effective operator of Eq. (1) and even more so in the resulting mass matrix $$m_{ij} = {f_{ijk} \over \Lambda} \langle \sigma_k \rangle \langle \phi \rangle.$$ On the positive side, if this particular mechanism is assumed, specific models of family structure may be considered and then compared to the data. Singlet quarks of charge $-1/3$ are contained in the fundamental representation of $E_6$. Hence the $d$ quarks of the SM may owe their family structure wholly or partly [@r00] to such a mechanism. (400,225)(0,0) (125,200)(200,150) (200,100)(200,150) (200,100)(200,50) (125,0)(200,50) (275,200)(200,150)3 (275,0)(200,50)3 (115,200)\[\][$q$]{} (115,0)\[\][$q^c$]{} (215,125)\[\][$Q^c_2$]{} (201,100)\[\][$\times$]{} (215,75)\[\][$Q_2$]{} (285,200)\[\][$\sigma$]{} (285,0)\[\][$\phi$]{} Consider next Fig. 2. Here $Q_2 \sim (3,2,1/6)$ and $Q^c_2 \sim (3^*,2,-1/6)$ are required. Whereas these heavy vector quark doublets are not present in the of $E_6$, the corresponding heavy lepton doublets $L_2 \sim (1,2,-1/2)$ and $L^c_2 \sim (1,2,1/2)$ are, and they have been used for example in a recently proposed model [@hmp05] of late neutrino mass and baryogenesis. Thus the observed lepton family structure may be encoded with this mechanism. (400,125)(0,0) (75,100)(150,50) (75,0)(150,50) (325,100)(250,50)3 (325,0)(250,50)3 (150,50)(250,50)3 (65,100)\[\][$q$]{} (65,0)\[\][$q^c$]{} (200,60)\[\][$H$]{} (335,0)\[\][$\sigma$]{} (335,100)\[\][$\phi$]{} Finally consider Fig. 3. The analog of Eq. (2) is $${f_{ijk} \over \Lambda} = h_{ija} (M^2)^{-1}_{ab} \mu_{bk},$$ where $h_{ija}$ are the couplings of $q_i q^c_j H_a$, $\mu_{bk}$ those of $H^\dagger_b \phi \sigma_k$, and $M^2$ the mass-squared matrix of $H$. This mechanism is realized naturally for example in $SO(10)$ (as well as $E_6$), where $q$ and $q^c$ are $SU(2)_L$ and $SU(2)_R$ doublets respectively, $H$ the heavy scalar bidoublets which carry the family structure, and $\phi$ the SM scalar doublet. It is appplicable to all Dirac fermions, including the $u$ quarks. It differs from the usual realization of quark masses in left-right gauge models where $H$ is a scalar bidoublet at the electroweak scale. Since $q q^c$ couples to $\phi \sigma$ through $H$ in Fig. 3, the full Higgs potential involving all 3 scalar fields should be considered. As a simple example, consider the case where $\sigma$ and $H$ transform in the same way under an extra $U(1)$ symmetry but $\phi$ is trivial, so that $H^\dagger \phi \sigma$ is an allowed term in the Lagrangian but $H^\dagger \phi$ is not. The most general Higgs potential involving $\sigma$, $H$, and $\phi$ is then given by $$\begin{aligned} V &=& m_\sigma^2 \sigma^\dagger \sigma + m_H^2 H^\dagger H + m_\phi^2 \phi^\dagger \phi + {1 \over 2} \lambda_1 (\sigma^\dagger \sigma)^2 + {1 \over 2} \lambda_2 (H^\dagger H)^2 + {1 \over 2} \lambda_3 (\phi^\dagger \phi)^2 \nonumber \\ &+& \lambda_4 (\sigma^\dagger \sigma) (H^\dagger H) + \lambda_5 (\sigma^\dagger \sigma)(\phi^\dagger \phi) + \lambda_6 (H^\dagger H)(\phi^\dagger \phi) + \lambda_7 (H^\dagger \phi) (\phi^\dagger H) \nonumber \\ &+& [\mu H^\dagger \phi \sigma + H.c.]\end{aligned}$$ Let $\mu$ be real, as well as $\langle \sigma \rangle = x$, $\langle H \rangle = u$, and $\langle \phi \rangle = v$. Then the minimization of $V$ results in the 3 conditions: $$\begin{aligned} && x[m_\sigma^2 + \lambda_1 x^2 + \lambda_4 u^2 + \lambda_5 v^2] + \mu u v = 0, \\ && u[m_H^2 + \lambda_2 u^2 + \lambda_4 x^2 + (\lambda_6 + \lambda_7)v^2] + \mu v x = 0, \\ && v[m_\phi^2 + \lambda_3 v^2 + \lambda_5 x^2 + (\lambda_6 + \lambda_7)u^2] + \mu u x = 0.\end{aligned}$$ Since $u,v << x$ is required for electroweak symmetry breaking, $$x^2 \simeq {-m_\sigma^2 \over \lambda_1}$$ is obtained from Eq. (6). Assuming now that $m_H^2 + \lambda_4 x^2 > 0$, Eq. (7) then yields $$u \simeq {-\mu v x \over m_H^2 + \lambda_4 x^2}.$$ Substituting the above into Eq. (8), the following effective condition for $v$ is obtained: $$m_\phi^2 + \lambda_5 x^2 - {\mu^2 x^2 \over m_H^2 + \lambda_4 x^2} + \left[ \lambda_3 + {(\lambda_6 + \lambda_7) \mu^2 x^2 \over (m_H^2 + \lambda_4 x^2)^2} \right] v^2 = 0.$$ Using Eqs. (6) to (8), the mass-squared matrix spanning the neutral real components of $\sigma$, $H$, and $\phi$ is given by $${\cal M}^2_{\sigma,H,\phi} = \pmatrix{2 \lambda_1 x^2 - \mu u v/x & 2 \lambda_4 x u + \mu v & 2 \lambda_5 x v + \mu u \cr 2 \lambda_4 x u + \mu v & 2 \lambda_2 u^2 - \mu v x/u & 2 (\lambda_6 + \lambda_7) u v + \mu x \cr 2 \lambda_5 x v + \mu u & 2 (\lambda_6 + \lambda_7) u v + \mu x & 2 \lambda_3 v^2 - \mu u x/v}.$$ Since $x >> u,v$, two approximate eigenstates are $\sigma$ and $(v H - u \phi)/\sqrt{v^2+u^2}$ with $m^2 \simeq 2 \lambda_1 x^2$ and $ -\mu x(v^2+u^2)/vu$ respectively. Thus all scalar fields are heavy except for the linear combination $(v \phi + u H)/\sqrt{v^2+u^2}$ which is identical to the single Higgs doublet of the SM. If the latter is extended to include supersymmetry, then there will be two Higgs doublets, as in the MSSM (Minimal Supersymmetric Standard Model). In the mechanism of Fig. 3, it is clear that the $q q^c$ mass matrix may also be written as $$m_{ij} = h_{ijk} \langle H_k \rangle,$$ where $\langle H_k \rangle$ is given by the generalization of Eq. (10). The family structure is determined not only by $h_{ijk}$ which may come from an assumed symmetry, but also by $\langle H_k \rangle$ which is hidden in the dynamics of the scalar sector much above the electroweak scale. However, if the family symmetry is global, and broken only spontaneously, then a massless Goldstone boson, the familon, will appear [@w82]. As an application of the mechanism of Fig. 3, consider the non-Abelian discrete symmetry $A_4$, the group of the even permutation of 4 objects which is also the symmetry group of the tetrahedron. It has been discussed [@mr01; @bmv03; @m04; @af05; @m05] as a family symmetry for the understanding of the neutrino mass matrix. Suppose it is combined with $SO(10)$. Then all quarks and leptons are naturally assigned as $({\bf 16}; \underline{3})$ under $SO(10) \times A_4$. \[There are 3 inequivalent irreducible singlet representations of $A_4$, $\underline{1}$, $\underline{1}'$, $\underline{1}''$, and 1 irreducible triplet representation $\underline{3}$.\] This assignment differs from the original one [@mr01; @bmv03] where $q,l \sim \underline{3}$ but $q^c,l^c \sim \underline{1}, \underline{1}', \underline{1}''$, which cannot be embedded into $SO(10)$. The heavy scalar $H$ should then be assigned as $(\overline{\bf 10}; \underline{1}, \underline{1}', \underline{1}'')$, $\sigma$ as $(\overline{\bf 16}; \underline{1}, \underline{1}', \underline{1}'')$, and $\phi$ as $(\overline{\bf 16}; \underline{1})$. For $a_{1,2,3} \sim \underline{3}$ and $b_{1,2,3} \sim \underline{3}$ under $A_4$, $$\begin{aligned} && a_1 b_1 + a_2 b_2 + a_3 b_3 \sim \underline{1}, \\ && a_1 b_1 + \omega^2 a_2 b_2 + \omega a_3 b_3 \sim \underline{1}', \\ && a_1 b_1 + \omega a_2 b_2 + \omega^2 a_3 b_3 \sim \underline{1}'',\end{aligned}$$ where $\omega = e^{2 \pi i/3}$. Hence all quark and lepton Dirac mass matrices are diagonal but with arbitrary eigenvalues. This is actually a rather good approximation in the quark sector, where all mixing angles are known to be small. In fact, if the theory is also supersymmetric, then the explicit breaking of $A_4$ in the soft supersymmetry-breaking terms themselves could be used to generate a realistic quark mixing matrix [@bdm99]. In the lepton sector, the same could be accomplished [@hrsvv04] in the case of the BMV model [@bmv03]. On the other hand, in the context of $SO(10)$, the $\overline{\bf 126}$ representation can be used to obtain Majorana neutrino masses according to the well-known seesaw formula [@seesaw] $${\cal M}_\nu = {\cal M}_L - {\cal M}_D {\cal M}_R^{-1} {\cal M}_D^T,$$ where $${\cal M}_D = \pmatrix{x & 0 & 0 \cr 0 & y & 0 \cr 0 & 0 & z}.$$ As for ${\cal M}_{L,R}$, using the assignment $(\overline{\bf 126}; \underline{3})$, they are both naturally of the form $${\cal M}_L = \pmatrix{0 & d & d \cr d & 0 & d \cr d & d & 0}, ~~~ {\cal M}_R = \pmatrix{0 & D & D \cr D & 0 & D \cr D & D & 0}.$$ Thus $${\cal M}_R^{-1} = {1 \over 2D} \pmatrix{-1 & 1 & 1 \cr 1 & -1 & 1 \cr 1 & 1 & -1},$$ and $${\cal M_D} {\cal M}_R^{-1} {\cal M}_D^T = {1 \over 2D} \pmatrix{-x^2 & xy & xz \cr xy & -y^2 & yz \cr xz & yz & -z^2} = - \pmatrix{a & b & c \cr b & b^2/a & -bc/a \cr c & -bc/a & c^2/a},$$ which has 3 zero $2 \times 2$ subdeterminants as expected [@ma05]. Together with ${\cal M}_L$, this becomes a four-parameter hybrid description [@cfm05] of the neutrino mass matrix. Let $b=c$, then in the basis spanning $\nu_e$, $(\nu_\mu+\nu_\tau)/\sqrt{2}$, and $(-\nu_\mu+\nu_\tau)/\sqrt{2}$, $$\begin{aligned} {\cal M}_\nu = \pmatrix{a & \sqrt{2} (d+b) & 0 \cr \sqrt{2} (d+b) & d & 0 \cr 0 & 0 & -d+2b^2/a},\end{aligned}$$ which is a new and very interesting pattern. It implies that $\theta_{13} = 0$ and $\theta_{23} = \pi/4$ in the neutrino mixing matrix, in agreement with data. Assuming $d,a,b$ to be real, the solar mixing angle is given by $$\tan 2 \theta_{12} = {2 \sqrt{2} (d+b) \over d-a},$$ which yields $\tan^2 \theta_{12} = 0.5$ in the limit $a=b=0$, again in agreement with data and realizing the so-called tri-bimaximal mixing pattern suggested [@hps02] some time ago. Assuming thus that $$a << b << d << b^2/a,$$ the 3 neutrino mass eigenvalues in this case are approximately $-d$, $2d$, and $2b^2/a$, resulting in $$\Delta m^2_{sol} \simeq 3d^2, ~~~ \Delta m^2_{atm} \simeq 4b^4/a^2.$$ As a numerical example, let $$a = 2.6 \times 10^{-5}~{\rm eV}, ~~~ b = -8.5 \times 10^{-4}~{\rm eV}, ~~~ d = 5.5 \times 10^{-3}~{\rm eV},$$ then $$\tan^2 \theta_{12} = 0.45, ~~~ \Delta m^2_{sol} = 7.9 \times 10^{-5} ~{\rm eV}^2, ~~~ \Delta m^2_{atm} = 2.4 \times 10^{-3}~{\rm eV}^2.$$ More details of this model will be presented elsewhere. In conclusion, it has been shown how the Standard Model with one Higgs doublet may hide the existence of a family symmetry for the quarks and leptons. The new heavy particles involved in 3 tree-level realizations of this mechanism have been identified and found to be available in the unification symmetries $E_6$ and $SO(10)$. A new and very interesting model based on $SO(10) \times A_4$ for the neutrino mass matrix is obtained. This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837. [99]{} S. L. Glashow and S. Weinberg, Phys. Rev. [**D15**]{}, 1958 (1977). C. Froggatt and H. B. Nielsen, Nucl. Phys. [**B147**]{}, 277 (1979). A. Davidson and K. C. Wali, Phys. Rev. Lett. [**59**]{}, 393 (1987); S. Rajpoot, Phys. Rev. [**D36**]{}, 1479 (1987). J. L. Rosner, Phys. Rev. [**D61**]{}, 097303 (2000). L. J. Hall, H. Murayama, and G. Perez, hep-ph/0504248. F. Wilczek, Phys. Rev. Lett. [**49**]{}, 1549 (1982). E. Ma and G. Rajasekaran, Phys. Rev. [**D64**]{}, 113012 (2001). K. S. Babu, E. Ma, and J. W. F. Valle, Phys. Lett. [**B552**]{}, 207 (2003). E. Ma, Phys. Rev. [**D70**]{}, 031901(R) (2004). G. Altarelli and F. Feruglio, hep-ph/0504165. E. Ma, hep-ph/0505209. K. S. Babu, B. Dutta, and R. N. Mohapatra, Phys. Rev. [**D60**]{}, 095004 (1999). M. Hirsch, J. C. Romao, S. Skadhauge, J. W. F. Valle, and A. Villanova del MOral, Phys. Rev. [**D69**]{}, 093006 (2004). M. Gell-Mann, P. Ramond, and R. Slansky, in [*Supergravity*]{}, edited by P. van Nieuwenhuizen and D. Z. Freedman (North-Holland, Amsterdam, 1979), p. 315; T. Yanagida, in [*Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe*]{}, edited by O. Sawada and A. Sugamoto (KEK Report No. 79-18, Tsukuba, Japan, 1979), p. 95; R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. [**44**]{}, 912 (1980). E. Ma, hep-ph/0501056. This differs from the proposal of S.-L. Chen, M. Frigerio, and E. Ma, hep-ph/0504181, where ${\cal M}_L$ is proportional to the unit matrix. P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys. Lett. [**B530**]{}, 167 (2002).

--- abstract: 'The observed morphological evolution since $z\sim 0.5$ within galaxy clusters provides evidence for a hierarchical universe. This evolution is driven by dynamical effects that operate within the cluster environment – suppression of star-formation by ram-pressure and viscous stripping of the intra-galactic medium and tidal heating of disks by gravitational encounters.' author: - 'Ben Moore, Vicent Quilis & Richard Bower' title: Dynamical effects on galaxies in clusters --- Evidence for a hierarchical universe? ===================================== A hierarchical universe with a low matter density is the currently favoured model for structure formation. However, observational evidence for evolution is scarce - structures appear old and similar over moderate look back times (Peebles 1999). Galaxy clusters are the most massive virialised objects in the universe and are the latest systems to form, therefore they may reveal evidence for recent evolution. Indeed, perhaps the strongest indication for evolution can be found by examining galaxy morphologies in rich clusters over the past 5 Gyrs ([*i.e.*]{} references within: Sandage 1970 Butcher & Oemler 1978, Dressler 1980, Dressler & Gunn 1983, Couch 1987, Binggeli 1988, Ferguson & Sandage 1991, Binggeli 1991, Thompson & Gregory 1993, Peterson & Caldwell 1993, Lavery & Henry 1994, Barger 1996, Dressler 1997, Balogh 1998, Couch 1998, Poggianti 1999, Bower 1999). At $z\sim 0.5$ we see that clusters of galaxies contain predominantly late type disk galaxies that have undergone a transformation into dwarf ellipticals (dSph) by the present day. There is also compelling evidence that the fraction of S0 galaxies decreases over the same look back time (Dressler 1997), whilst at even earlier epochs, tentative evidence for the hierarchical growth of the cluster elliptical galaxies may be found (van Dokkum 1999). The data suggest that the cluster environment is causing a morphological transformation between galaxy types. Other indicators of environmental influences in clusters include the morphology-density relation (Dressler 1980), signatures of star-bursts followed by a rapid truncation of star-formation in cluster spirals and k+a/a+k galaxies (Poggianti 1999), the absence of low surface brightness disks in clusters (Bothun 1993) and the presence of a large component of diffuse star-light (Bernstein 1993, Freeman - this volume). In this paper we will discuss the role of environment and the proposed mechanisms that are responsible for driving galactic evolution in clusters and creating the “morphological and spectroscopic Butcher-Oemler effects”. Mergers, Winds, Harassment & Stripping ====================================== Several mechanisms have been proposed that can induce a morphological transformation between galaxy classes. Toomre & Toomre’s (1972) pioneering n-body simulations of merging spirals gave rise to elliptical remnants. This process is most effective when the encounter velocity is comparable to the galaxies internal velocity, therefore mergers should be rare within rich virialised systems, a fact pointed out by many authors ([*e.g.*]{} Mamon - this volume), however see van Dokkum (1999). Numerical simulations of clusters that formed naturally in a hierarchical universe show that mergers are indeed rare (Ghigna 1998). Galactic winds from supernovae are frequently cited as important mechanisms for dwarf galaxy evolution (Dekel & Silk 1986) yet observational evidence for this process is not compelling (Martin 1998) and it is unlikely that feedback can reshape the stellar configuration. We therefore focus on the two mechanisms that are most likely to operate extensively in the cluster environment ([*c.f.*]{} Fujita 1999). Impulsive heating via rapid tidal encounters has been demonstrated to be an efficient mechanism at heating disk galaxies in clusters – termed “galaxy harassment” by Moore (1996, 1998), tidal forces and internal cluster dynamics have been studied by many authors, including Richstone & Malmuth 1983, Icke 1985, Merritt 1985, Byrd & Valtonen 1990, Valluri & Jog 1991, Valluri 1993, Gnedin (1999). Gunn & Gott (1978) suggested that the ram-pressure force of the hot intra-cluster medium could remove the atomic hydrogen from disks infalling into clusters. Nulsen (1982) proposed that the viscous and turbulent stripping of gas would be equally as effective. Morphological transformation by gravitational encounters is a process that requires several strong or frequent tidal collisions and acts over a timescale of several $\times 10^9$ years. In contrast, hydrodynamic effects operate on a much shorter timescale, $\sim 10^7$ years. Observational evidence for gas-dynamical interactions in clusters is scarce due to the speed at which the process occurs (Stevens 1999), however the literature does contain some examples ([*i.e.*]{} Warmels 1988, Cayatte 1990, Gavazzi 1995, Kenny & Koopman 1999, and several recent studies in this volume by Vollmer (1999), Ryder (1997) & J. Solanes ). The first 3-dimensional calculation of ram pressure stripping using realistic disk galaxy models was carried out by Abadi (1999). Earlier work had studied the properties of spherical systems in two dimensions ([*i.e.*]{} Balsara 1994). Abadi found that the simple analytic argument proposed by Gunn & Gott (1972) balancing the ram pressure, $\rho_{_{ICM}} v^2$, with the disk’s gravitational restoring force works very well. Simulations at $45^\circ$ and edge on to the motion through the intra-cluster medium suffered slightly less stripping than face on. However, even in the best case scenario for stripping, an $L_*$ disk galaxy moving through the center of the Coma cluster at 3000 km s$^{-1}$, a significant fraction of gas remained (HI extending to a radius $\sim 5$ kpc) that would continue to form stars. =5.0truein [   The time evolution of the diffuse HI component within a spiral galaxy falling into the Coma cluster (Quilis 1999).]{} A more realistic treatment of the stripping process using a Eulerian finite difference code revealed that the turbulent and viscous stripping processes help to remove more gas than just the ram pressure alone (Quilis 1999). Quilis also realised that disks are not smooth homogeneous mediums, but have a complex structure containing many holes and regions that are completely devoid of gas (Brinks & Bajaja 1986). Once these are included, the ICM streams through the holes, preventing “backside infall” and ablates away the gas from their edges, rapidly removing the entire HI content of disks. Thus, these authors find that the stripping process alone, is in general 100% efficient at removing the HI component within a timescale $\lsim 10^7$ years (see Figure 1). Without fresh material to replenish GMC’s, star formation will be rapidly truncated (Elmgreen & Efremov 1997). Continued heating by tidal encounters with massive galaxies increases the disk scale height by factors of 2–3. Spiral features will be suppressed and these galaxies will resemble the S0 galaxies observed in nearby clusters. The efficiency of the stripping process increases rapidly towards the cluster center where $\rho_{_{ICM}}$ and $v$ are maximum, thus we have a natural way of creating the morphology-density relation. Whether or not the transformation of Sa/Sb galaxies to S0’s can reproduce the same scaling as observed remains to be tested. Galaxy harassment will transform dI/Sc/Sd disks into dwarf elliptical (dSph) galaxies over a timescale of several billion years - [*i.e.*]{} several cluster orbital periods. Simulations of this process appear to produce remnant galaxies that closely match the observed cluster dwarf’s (Moore 1996, 1998), although the sample of kinematical data is small. A fundamental prediction of the formation of dE (dSph) galaxies by tidal heating is that the remnant galaxies are embedded in diffuse streams of stars, tidally removed from the progenitor disks (see Figure 2). The surface brightness profiles of these galaxies are expected to be well fitted by an exponential profile over their central regions but a significant excess of stars will be found at 4–5 scale lengths (Moore 1999). The surface brightness of this excess will occur fainter than $27-28M_{_B}$ per arcsec$^{-2}$, an observable effect with 4m or 8m class telescopes. The presence of freely orbiting planetary nebulae in clusters (Freeman, this meeting) provides strong support for the efficiency of this process. Not surprisingly, low surface brightness disks are not found within clusters (Bothun 1993). The shear extent of their stellar distributions and low central potentials make them unstable to tidal forces. Once these galaxies enter the cluster environment, most of there stars are tidally removed and a diffuse spheroidal remnant remains (Moore 1999). Numerical simulations of this process can be used to study the diffuse cluster light component. =3.0truein [   An example of tidal tails from the galaxy harassment process from the cluster CL0054-27 at z=0.56 courtesy of Ian Smail. The image is roughly 150 kpc on a side. Features this prominent are rare at this surface brightness ($\sim 27 M_{_B}$ per arcsec$^{-2}$) but are expected to surround all cluster dE (dSph) galaxies at lower surface brightness levels.]{} Conclusions =========== Two mechanisms are necessary to reproduce the observed morphological evolution of galaxies in clusters. At the faint end of the luminosity function, disk galaxies are easily transformed into dE (dSph) galaxies by galaxy harassment. Numerical simulations of this process predict that all dwarf galaxies in nearby clusters will be embedded within very diffuse streams of stripped stars. Low surface brightness disks with slowly rising rotation curves and shallow central potentials will not survive the fluctuating potentials of rich clusters and will be tidally shredded to form the bulk of the diffuse intra-cluster light. Spirals with bulges are more stable to gravitational encounters. They would retain their stars and gas and continue to form stars to the present day. Most, if not all of the atomic hydrogen of these galaxies must be rapidly removed upon entering the cluster environment in order to explain their star-formation histories. Fluid dynamic simulations of realistic disks moving through a hot ICM demonstrates that a combination of ram-pressure and viscous stripping can remove 100% of the HI from the IGM within $10^7$ years. Continued tidal heating by encounters will complete a transformation to the S0 class. Abadi, M.G., Moore, B., & Bower, R.G. 1999. [*M.N.R.A.S.*]{}, [**308**]{}, 947. Balogh, M.L., Schade, D., Morris, S.L., Yee, H.K.C., Carlberg, R.G. & Ellingson, E. 1998 [*Ap.J.Lett.*]{}, [**504**]{}, 75. Balsara, D., Livio, M., O’dea, C. P. 1994, [*Ap.J.*]{}, [**437**]{}, 83. Barger, A.J., Aragon-Salamanca, A., Ellis, R.S., Couch, W.J. Smail, I., Sharples, R. M. 1996 [*M.N.R.A.S.*]{}, [**279**]{}, 1. Bernstein, G.M., Nichol R.C., Tyson J.A., Ulmer M.P. & Wittman D. 1995, [*A.J.*]{}, [**110**]{}, 1507. Binggeli, B., Sandage, A. & Tammann G.A. 1988, [*Ann. Rev. Astr. Ap.*]{}, [**26**]{}, 509. Binggeli, B. & Cameron, L.M. 1991, [*Astr. Ap.*]{}, [**252**]{}, 27. Butcher, H. and Oemler, A. 1978, [*Ap.J.*]{}, [**219**]{}, 18. Bothun G.D., Schombert J.M., Impey C.D., Sprayberry D. & McGaugh S.S. 1993, [*A.J.*]{}, [**106**]{}, 530. Bower R.G., Kodama T. & Terlevich, A. 1999 [*M.N.R.A.S.*]{}, [**299**]{}, 1193. Brinks, E. & Bajaja, E. 1986, [*A.A.*]{}, [**169**]{}, 14. Byrd, G. & Valtonen, M. 1990, [*Ap.J.*]{}, [**350**]{}, 89. Cayatte, V., Balkowski, C., Van Gorkom, J. H., Kotanyi, C. 1990, [*A.J.*]{}, [**100**]{}, 604. Couch, W. J. & Sharples, R. M. 1987 [*M.N.R.A.S.*]{}, [**229**]{}, 423. Couch, W. J., Barger, A. J., Smail, I., Ellis, R. S., Sharples, R. M. 1998 [*Ap.J.*]{}, [**497**]{}, 188. Dekel, A. & Silk, J. 1986, [*Ap.J.*]{}, [**303**]{}, 39. Dressler, A. 1980 [*Ap.J.*]{}, [**42**]{}, 565. Dressler, A., Gunn, J. E. 1983 [*Ap.J.*]{}, [**270**]{}, 7. Dressler, A., Oemler, A., Couch, W. J., Smail, I., Ellis, R. S., Barger, A., Poggianti, B. M., Sharples, R. M. 1997 [*Ap.J.*]{}, [**490**]{}, 577. Elmergreen, B. G. & Efremov, Y. N. 1997 [*Ap.J.*]{}, [**480**]{}, 235. Ferguson, H.C. & Sandage, A. 1991, [*A.J.*]{}, [**101**]{}, 765. Fujita, Y., & Nagashima, M. 1999 [*Ap.J.*]{}, [**516**]{}, 619. Gavazzi, G., Conursi, A., Carrasco, L., Boseli, A., Kennicutt, R., Scodeggio, M. & Jaffe, W. 1995 [*Astron. Astrophys.*]{}, [**304**]{}, 325. Ghigna, S., Moore, B, Governato, F., Lake, G., Quinn, T. & Stadel, J. 1998 [*M.N.R.A.S.*]{}, [**300**]{}, 146. Gnedin, O.Y. (1999), [*Ap.J.*]{}, submitted. Gunn, J. E. & Gott, J. R. 1972 [*Ap.J.*]{}, [**176**]{}, 1. Icke, V. 1985, [*Astr. Ap.*]{}, [**144**]{}, 115-23. Kenny, J.D.P. & Koopmann, R.A. 1999, [*A.J.*]{}, [**117**]{}, 181. Kormendy, J. 1985, [*Ap.J.*]{}, [**295**]{}, 73. Lavery, R.J. & Henry, J. P. [*Ap.J.*]{}, [**426**]{}, 524. Martin C.L. 1998, [*Ap.J.*]{}, [**506**]{}, 222. Merritt, D. 1985, [*Ap.J.*]{}, [**289**]{}, 18. Moore, B., Katz, N., Lake, G., Dressler, A. & Oemler, A. 1996 [*Nature*]{}, [**379**]{}, 6566, 613. Moore, B., Katz, N. & Lake, G. 1996, [*Ap.J.*]{}, [**457**]{}, 455. Moore, B., Lake, G., Quinn, T. & Stadel, J. 1999 [*M.N.R.A.S.*]{}, [**304**]{}, 465. Nulsen, P.E.J. 1982 [*M.N.R.A.S.*]{}, [**198**]{}, 1007. Peebles, J. 1999, many contributions. Peterson, R. C. & Caldwell, N. 1993, [*A. J.*]{}, [**105**]{}, 1411. Poggianti, B. M., Smail, I., Dressler, A., Ellis, R. S., Couch, W. J., Oemler, A. 1999 [*Ap.J.*]{}, [**518**]{}, 576. Quilis, V., Moore, B. & Bower, R.G. 1999, [*Nature*]{}, submitted. Richstone, D.O. & Malmuth, E.M. 1983, [*Ap.J.*]{}, [**268**]{}, 30. Ryder, S.D., Purcell, G., Davis, D. & Anderson, V. 1997 [*PASP*]{}, [**14**]{}, 1. Sandage, A., Freeman K.C. & Stokes, N.R. 1970 [*Ap.J.*]{}, [**160**]{}, 831. Stevens, I.R., Acreman, D.M. & Ponman, T.J. 2000 [*M.N.R.A.S.*]{} in press, astro-ph-9909066. Thompson, L.A. & Gregory, S.A. 1993, [*A.J.*]{}, [**106**]{}, 2197. Toomre A. & Toomre J. 1972 [*Ap.J.*]{}, [**178**]{}, 623. Valluri, M. 1993, [*Ap.J.*]{}, [**408**]{}, 57. Valluri, M. and Jog, C. J. 1991, [*Ap.J.*]{}, [**374**]{}, 103. van Dokkum, P.G., Franx, M., Fabricant, D., Kelson, D.D., Illingworth, G.D. 1999, [*Ap.J.Lett.*]{}, [**520**]{}, L20. Vollmer, B., Cayatte, V., Boseli, A., Balkowski, C. & Duschl, W.J. 1999 [*Astron. Astrophys.*]{} submitted. Warmels, R. H. 1988, [*A&AS.*]{}, [**72**]{}, 57.

--- abstract: 'Exciton diffusion plays a vital role in the function of many organic semiconducting opto-electronic devices, where an accurate description requires precise control of heterojunctions. This poses a challenging problem because the parameterization of heterojunctions in high-dimensional random space is far beyond the capability of classical simulation tools. Here, we develop a novel method based on deep neural network to extract a function for exciton diffusion length on surface roughness with high accuracy and unprecedented efficiency, yielding an abundance of information over the entire parameter space. Our method provides a new strategy to analyze the impact of interfacial ordering on exciton diffusion and is expected to assist experimental design with tailored opto-electronic functionalities.' author: - Liyao Lyu - Zhiwen Zhang - Jingrun Chen bibliography: - 'achemso-demo.bib' title: Connecting exciton diffusion with surface roughness via deep learning --- {width="9cm" height="3.5cm"} INTRODUCTION ============ Over the past decades, much attention has been paid on organic semiconductors for applications in various opto-electronic devices[@Jason2012; @Su2012; @Martin1999; @Forrest2004911]. These materials include small molecules[@Peumans20033693; @Lin2014280], oligomers[@Sanaur2006; @Murphy2007], and polymers[@Pettersson1999487; @Mikhnenko20126960]. Exciton diffusion is one of the key processes behind the operation of organic opto-electronic devices[@DeOliveiraNeto20123039; @Brabec200550; @Bredas20091691]. From a microscopic perspective, exciton, a bound electron-hole pair, is the elementary excitation in opto-electronic devices such as light emitting diodes and organic solar cells. The exciton diffusion length (EDL) is the characteristic distance that excitons are able to travel during their lifetime[@Lin2014280]. A short diffusion length in organic photovoltaics limits the dissociation of excitons into free charge[@Terao2007; @Menke2013152]. Conversely, a large diffusion length in organic light emitting diodes may limit luminous efficiency if excitons diffuse to non-radiative quenching sites [@Enhancedcarrier]. As quasi-particles with no net charge, excitons are difficult to probe directly by electrical means.[@Mullenbach2017] This is particularly true in organic semiconductors where the exciton binding energy is $\sim$1 electronvolt.[@Tang1986183] Reported techniques to measure EDL include photoluminescence (PL) surface quenching[@Terao2007; @Rim2007; @Markov2005; @Scully2006; @Goh2007; @Shaw20083516; @Wu2005; @Theander200012957; @Lin2014280], time-resolved PL bulk quenching modeled with a Monte Carlo simulation[@Mikhnenko20126960; @Mikhnenko201214196], exciton-exciton annihilation [@Cook200933; @Lewis2006452; @Masri20131445; @Shaw2010155], modeling of solar cell photocurrent spectrum [@Qin20111967; @Stubinger20013632; @Yang20031737; @Halls19963120; @Theander200012957; @Ghosh19785982; @Wagner1993423; @Bulovic199588; @Pettersson1999487; @Peumans20033693; @Rim2008; @Huijser2008], time-resolved microwave conductance [@Fravventura20122367; @Savenije2010; @Kroeze20037696], spectrally resolved PL quenching[@Lunt20101233; @Bergemann2011; @Rand20122987] and Förster resonance energy transfer theory [@Lunt2009; @Lunt20101233; @Mullenbach20133689]. From a theoretical perspective, the minimal modeling error is given by the diffusion equation model[@Chen2016754], which is employed in the current work. To be precise, the device used in PL surface quenching experiment includes two layers of organic materials with thickness ranging from dozens of nanometers to hundreds of nanometers. One layer of material is called donor and the other is called acceptor or quencher according to the difference of their chemical properties. Under the illumination of solar lights, excitons are generated in the donor layer and diffuse in the donor. Due to the exciton-environment interaction, some excitions die out and emit photons which lead to the PL. The donor-acceptor interface serves as the absorbing boundary while other boundaries serve as reflecting boundaries due to the tailored properties. Since the donor-acceptor interface is not exposed to the air/vacuum and the resolution of the surface morphology is limited by the resolution of atomic force microscopy, the interface is subject to an uncertainty. It is found that the fitted EDL is sensitive to the uncertainty in some scenarios. From a numerical perspective, the random interface requires a parametrization in high-dimensional random space, which is prohibitively expensive for any simulation tool. For example, Monte Carlo method overcomes the curse of dimensionality but has very low accuracy [@HONG2014]. Stochastic collection method has high accuracy but is only affordable in low dimensional random space [@SC3]. Asymptotics-based method is efficient but its accuracy relies heavily on the magnitude of randomness [@Chen2019894]. In the current work, we propose a novel method based on deep learning with high accuracy and unprecedented efficiency. Recently, increasing attentions have been paid to apply machine learning (ML) techniques to materials-related problems. For example, the classification of crystal structures of transition metal phosphide via support vector machine [@Oliynyk201717870] leads to the discovery of a novel phase [@Oliynyk20166672]. Likewise, a hybrid probabilistic model based on high-throughput first-principle computation and ML was developed to identify stable novel compositions and their crystal structures [@Hautier20103762]. Physical parameters such as band gap [@Isayev2017; @Xie2018], elastic constants[@Isayev2017; @DeJong2016], and Debye temperature[@Isayev2017] have also been predicted using an array of ML techniques. In another line, deep learning (DL) in computer science has had great success in text classification [@Wang2012EndtoendTR], computer vision[@NIPS2012_4824], natural language processing [@Sarikaya:2014:ADB:2687012.2687014], and other data-driven applications. One significant advantage of DL is its strong ability to approximate a complex function in high dimensions and extract features with high precision using composition of simple nonlinear units. Meanwhile, benefiting from recent advances in parallel graphics processing unit - accelerated computing, huge volumes of data can be put into the DL architecture for training. In this work, we employ DL to extract a complex function of EDL in terms of the random interface parametrized in a high-dimensional space. The fitted function has rich information, which explains a few interesting experimental observations. Compared to classical simulation tools, our approach has the following features: quasi-Monte Carlo sampling [@Russel1998] for data collection and ResNet [@He2015] for training. The size of data in the former step grows only linearly with respect to the dimension of random space, thus our approach overcomes the curse of dimensionality. With the usage of ResNet in the latter step, a complex function can be extracted with high accuracy. Therefore, results provided here are completely out of the capability of classical simulation tools. METHODS ======= Our approach consists of four major components: quasi-Monte Carlo sampling over the high-dimensional random space; diffusion equation model for data generation; ResNet for training to approximate a complex function of EDL; Information extraction for analysis (Figure \[fig:schematic\]). ![Flow chart of the deep learning method for extracting exciton diffusion length over the parameter space. Left: data generation; Middle: data training; Right: data prediction. In the stage of data generation, quasi-Monte Carlo method is used to sample the random space, and the actual exciton diffusion length is generated by solving the diffusion equation model. In the stage of data training, a complex function $\sigma(\theta(\omega_1),\theta(\omega_2))$ is approximated over the entire parameter space. In the stage of data prediction, given the full landscape of $\sigma(\theta(\omega_1),\theta(\omega_2))$ , both qualitative and quantitative information can be extracted.[]{data-label="fig:schematic"}](Figure//schematic.pdf){width="100.00000%"} Model Description ----------------- An exciton that diffuses in the donor layer follows a diffusion-type equation over a 3D random domain $D=\{(x,y,z)\left|h(y,z,\omega_1,\omega_2)<x< d,0<y<L_y,0<z<L_z\right.\}$. Here the donor-acceptor interface $x=h(y,z,\omega_1,\omega_2)$ is parameterized by $$\begin{aligned}\label{eqn:3DInterface} h(y,z,\omega_1,\omega_2)=\hat{h}\sum_{k_1=1}^{K_1} \sum_{k_2=1}^{K_2} k_1^{\beta} k_2^{\beta} \theta_{k_1}(\omega_{1})\theta_{k_2} (\omega_{2})\phi_{k_1}(y)\phi_{k_2}(z), \end{aligned}$$ where $\hat{h}$ is the magnitude of length due to the roughness limited by the resolution of atomic force microscopy, [$\theta_{k_1}(\omega_1),\theta_{k_2}(\omega_2)$]{} are i.i.d. random variables, $\phi_{k_1}(y)=\sin(2k_1\pi \frac{y}{L_y})$, $\phi_{k_2}(z)=\sin(2k_2\pi \frac{z}{L_z})$, and $\beta<0$ controls the decay rate of spatial modes $\phi_{k_1}(y), \phi_{k_2}(z)$. The rougher the interface is, the closer the $\beta$ approaches $0$. Given the surface roughness measured in an experiment, parameters in can be extracted via discrete Fourier transform. The 3D diffusion equation reads as $$\left\{ \begin{aligned}\label{eqn:model} &\sigma^2\triangle u -u + G(x,y,z) =0 & x\in D,\\ &u_x(d,y,z)=0 &0<y<L_y,0<z<L_z,\\ &u(h(y,z,\omega_1,\omega_2),y,z)=0 &0<y<L_y,0<z<L_z,\\ &u(x,y,z)=u(x,y+L_y,z)=u(x,y,z+L_z)& h(y,z,\omega_1,\omega_2)<x<d, \end{aligned}\right.$$ where $\sigma$ is the EDL to be extracted, $u$ is the exciton density, $G$ is the normalized exciton generation function by the transfer matrix method [@burkhard_accounting_2010]. $x=d$ serves as the reflecting boundary and Neumann boundary condition is imposed on the boundary exposed in air. $x=h(y,z,\omega_1,\omega_2)$ serves as the absorbing boundary and homogenous Dirichlet boundary condition is imposed on the donor-acceptor interface. Periodic boundary conditions are imposed on in-plane directions $y$ and $z$. For comparison and completeness, 1D and 2D models are given in the Supporting Information. The PL is computed by $$\begin{aligned}\label{eqn:3Dphotoluminescence} I_{\theta(\omega_1),\theta(\omega_2)}[\sigma,d]=\frac{1}{L_z}\frac{1}{L_y}\int_{0}^{L_z}\int_{0}^{L_y} \int_{h(y,z,\omega_1,\omega_2)}^{d} u(x,,y,z) \mathrm{d}x \mathrm{d}y \mathrm{d}z. \end{aligned}$$ For comparison with the 1D model, we divide the PL in the usual sense by lengths in $x$ and $y$ directions. In the experiment, PL data $\{\hat{I}_i\}_{i=1}^N$ are measured by a series of bilayer devices with different thicknesses $\{d_i\}_{i=1}^N$, where $d_i$ is the thickness of the $i$-th donor layer. The optimal EDL $\sigma$ is expected to reproduce the experimental date $\{d_i,\hat{I}_i\}_{i=1}^N$ in the sense of minimized mean square error (MSE) $$\begin{aligned}\label{eqn:Inverseproblem} \min_{\sigma} J_{\theta(\omega_1),\theta(\omega_2)}(\sigma) =\frac{1}{N} \sum_{i=1}^{N}\left(I_{\theta(\omega_1),\theta(\omega_2)}(\sigma,d_i)-\hat{I}_i\right)^2. \end{aligned}$$ Newton’s method is used to solve for $\sigma$ (see the Supporting Information). The calculated $\sigma$ is defined as $\sigma_{\theta(\omega_1),\theta(\omega_2)}$. Therefore, for different parameters $\theta(\omega_1),\theta(\omega_2)$, we get a data set $\left\{\left(\theta(\omega_1)[j],\theta(\omega_2)[j] ,\sigma_{\theta(\omega_1)[j],\theta(\omega_2)[j]}\right)_{j=1}^M\right\}$ with $M$ the size of data set. ResNet ------ ResNet [@He2015] is used to approximate $\sigma_{\theta(\omega_1),\theta(\omega_2)}$. A ResNet consists of a series of blocks. One block is given in Figure \[fig:schematic\] with two linear transformations, two activation functions, and one short cut. Detailed description of ResNet is included in the Supporting Information. Parameters of the surface roughness ($\theta(\omega_{1})$,$\theta (\omega_{2})$) are fed as input, and the EDL $\sigma$ is extracted as the output function over the entire parameter space. Sigmoid function is chosen as the activation function here. The loss function we use is the MSE between the actual EDL $\sigma_{\theta(\omega_1),\theta(\omega_2)}$ given by the diffusion equation model and the predicted EDL $\sigma(\theta(\omega_1)[j],\theta(\omega_2)[j])$ given by the ResNet $$\label{eqn:MSE} MSE= \frac{1}{M}\sum_{j=1}^{M} \left(\sigma_{\theta(\omega_1)[j],\theta(\omega_2)[j]}-\sigma(\theta(\omega_1)[j],\theta(\omega_2)[j])\right)^2,$$ where $\theta$ represents the parameter set in the ResNet, $j$ is the $j$-th sample, and M is the size of training data set. Define the relative $L^{\infty}$ error of EDL as $$Error = \max_{1\leq j\leq M} \dfrac{\left|\sigma_{\theta(\omega_1)[j],\theta(\omega_2)[j]}-\sigma(\theta(\omega_1)[j],\theta(\omega_2)[j])\right|} {\sigma_{\theta(\omega_1)[j],\theta(\omega_2)[j]}},$$ which will be used to quantify the approximation accuracy of DL. Quasi-Monte Carlo Sampling -------------------------- Compared to uniform sampling and Monte-Carlo sampling, quasi-Monte Carlo sampling provides the best compromise between accuracy and efficiency. It overcomes the curse of dimensionality and has high accuracy [@Russel1998]. For the simulations in our work, at least three orders of magnitude reduction in the size of data set is found for quasi-Monte Carlo sampling without loss of accuracy (see the Supporting Information). Results and Discussion ====================== Accuracy check and training data set ------------------------------------ For the accuracy check in the 3D case, the reference PL data are generated using $5$ realizations with out-of-plane thicknesses $d_i=10,15, 20,25\;$nm and $\sigma=5\;$nm in the absence of randomness. Afterwards, randomness is added with $K_1=K_2=5$, i.e., $\theta(\omega_1)$ and $\theta(\omega_2)$ are arrays with $5$ variables. Quasi-Monte Carlo sampling is used to generate 20000 points with the corresponding EDL obtained by solving - . The first $15000$ data are used as the training set, while the remaining data are used to check the predictability of the trained neural network; see Figure \[fig:actualpredict\]. Relative $L^\infty$ errors of EDL are $0.270\%,0.368\%$ and $0.532\%$ for $\beta=-2, -1, 0$, respectively. It is known that the random field is closer to the white noise when $\beta=0$ and thus is more difficult to be trained. However, uniform generalization errors for three different scenarios are observed, implying the robustness of trained neural network. Moreover, the size of training data set is small in the sense that only linear growth with respect to the dimension of random variables is observed, in contrast to other sampling techniques which either have the curse of dimensionality or low accuracy. Similar performance is observed for the 2D model (see the Supporting Information). Information extraction ---------------------- The trained neural network fits a high-dimensional function for EDL in terms of surface roughness. Rich information can be extracted based on the fitted function. We demonstrate this using three examples. ***Modeling error*** Expectations of EDL in 3D are recorded in Table \[tbl:result\] for $\beta = -2, -1, 0$. PL data are generated using the 1D model with the reference EDL $5\;$nm. When $\beta=-2$, the EDL is close to $5\;$nm, which implies the equivalence between the 3D model and the 1D model. However, when $\beta=0$, the EDL is clearly away from $5\;$nm. We attribute this difference to the modeling error between the 1D model and the 3D model with a surface roughness characterized by with $\beta=0$. So far, the 1D model is largely used in the literature to extract the EDL [@Pettersson1999487; @Lin2014280; @Guideetal:2013; @Chen2016754]. The main assumption underlying the modeling is the high crystalline order of the organic material. When $\beta=-2$, long-range ordering exists in the random interface, which implicitly connects with the crystalline ordering of the material. Therefore, in this case, the 3D model and the 1D model are equivalent. However, when $\beta=0$, only short-range ordering exists. As a consequence, the 3D model and the 1D model are not equivalent any more. Given a surface roughness from the experimental measurement, we can fit a function of form using discrete Fourier transform, from which we can get the decay rate $\beta$ and thus decide whether the 1D model is adequate or not. It is worth mentioning that similar results are observed in 2D using the asymptotics-based approach [@Chen2019894]. $\beta=-2$ $\beta=-1$ $\beta=0$ ------------- ------------- ------------- $4.986\;$nm $4.842\;$nm $4.566\;$nm : Expectations of exciton diffusion length in 3D for different surface roughness. The reference value is $5\;$nm. []{data-label="tbl:result"} ***Landscape exploration*** Contour plots of the fitted EDL on random variables are given in Figures \[fig:lambda2contour\] and \[fig:lambda0contour\] when $\beta=-2$, $\beta=0$ and in Figure \[fig:lambda1contour\] when $\beta=-1$ (see the Supporting Information). In each subfigure, EDL $\sigma$ is plotted as a function of $\theta_{k_1}(\omega_1)$ and $\theta_{k_2}(\omega_2)$, where $k_1,k_2=1,2,3,4,5$ and all the remaining random variables are set to be $0$. A direct comparison between Figure \[fig:lambda2contour\] and Figure \[fig:lambda0contour\] illustrates the directional (anisotropic) dependence of EDL on random variables, due to different decay rates of random variables in the surface roughness. ![Contour plot of exciton diffusion length on random variables in 3D when $\beta=-2$.[]{data-label="fig:lambda2contour"}](Figure//lambda2contour.pdf){width="100.00000%"} ![Contour plot of exciton diffusion length on random variables in 3D when $\beta=0$.[]{data-label="fig:lambda0contour"}](Figure//lambda0contour.pdf){width="100.00000%"} ***Mode dependence*** Figure \[fig:lambda0d1\] provides a detailed demonstration of the dependence of EDL on random variables for $\beta=-2,-1,0$. For illustration, we keep $\theta(\omega_2)=[1;0;0;0;0]$ fixed in the left column and $\theta(\omega_1)=[1;0;0;0;0]$ fixed in the right column. One distinct difference between 3D and 2D is that the maximum EDL is approached in the absence of randomness in 3D, in contrast to the minimum EDL in 2D (see the Supporting Information). The 3D result is reasonable since experimentally larger EDL is observed if the effect of surface roughness is minimized, while the 2D result is also of interest due to the unique dimensional dependence. When $\beta=-2$, the EDL is more sensitive to the lower-order modes (smaller $k$) and is less sensitive to the high-order modes (larger $k$). When $\beta=0$, the trend is completely opposite. This observation provides a detailed connection between surface roughness and EDL, which also sheds light on the experimental design. Given a surface roughness characterized by , we have the value of $\beta$, from which we know which mode is of the most importance. Consequently, targeted experimental techniques can be applied to improve the opto-electronic performance. Conclusion ========== In summary, we have developed a novel method based on quasi-Monte Carlo sampling and ResNet to approximate the exciton diffusion length in terms of surface roughness parametrized by a high-dimensional random field. This method extracts a function for exciton diffusion length over the entire parameter space. Rich information, such as landscape profile and mode dependence, can be extracted with unprecedented details. Useful information regarding the modeling error and the experimental design can be provided, which sheds lights on how to reduce the modeling error and how to design better experiments to improve opto-electronic properties of organics materials. Acknowledgments =============== L. Lyu acknowledges the financial support of Undergraduate Training Program for Innovation and Entrepreneurship, Soochow University (Projection 201810285019Z). Z. Zhang acknowledges the financial support of Hong Kong RGC grants (Projects 27300616, 17300817, and 17300318) and National Natural Science Foundation of China via grant 11601457, Seed Funding Programme for Basic Research (HKU), and Basic Research Programme of The Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20180307151603959). J. Chen acknowledges the financial support by National Natural Science Foundation of China via grants 21602149 and 11971021, and the Science and Technology Development Fund, Macau SAR (File No. 101/2016/A3). Part of the work was done when J. Chen was visiting Department of Mathematics, University of Hong Kong. J. Chen would like to thank its hospitality. Supporting Methods ================== 1D and 2D models ---------------- The 1D model is defined over the domain $$\label{eqn:1DInterface} D_1=\{x:x\in[\theta(\omega), d]\},$$ where the interface is reduced to a random point $x=\theta(\omega)$. The corresponding diffusion equation is $$\left\{ \begin{aligned}\label{eqn:one dimensional model} &\sigma^2u_{xx} -u + G(x) =0 , & \theta(\omega)<x<d,\\ &u_x(d)=0 ,\\ &u(\theta(\omega))=0 , \end{aligned}\right.$$ and the photoluminescence (PL) is $$\begin{aligned}\label{eqn:1Dphotoluminescence} I_{\theta(\omega)}[\sigma,d]= \int_{\theta(\omega)}^{d} u(x) \mathrm{d}x. \end{aligned}$$ The 1D model is commonly used to extract the exciton diffusion length (EDL) due to its simplicity and model accuracy[@Lin2014280; @Chen2016754]. The 2D model is defined over a random domain $$\label{eqn:2DInterface} D_2=\{(x,y)\left|h(y,\omega)<x<d,0<y<L_y\right.\},$$ where the interface is a random line parametrized by $h(y,\omega)=\hat{h} \sum_{k=1}^{K} k^{\beta} \theta_{k}(\omega) \phi_{k}(y)$ with $\phi_{k}(y)=\sin(\frac{2\pi k y}{L_y})$. The corresponding diffusion equation is $$\left\{ \begin{aligned}\label{eqn:2Ddiffusionequation} &\sigma^2\triangle u -u + G(x,y) =0 ,& x\in D_2,\\ &u_x(d)=0, \ \ u(h(y,\omega),y)=0, & 0<y<L_y,\\ &u(x,y)=u(x,y+L_y), & h(y,\omega)<x<d, \end{aligned}\right.$$ and the PL is $$\begin{aligned}\label{eqn:2Dphotoluminescence} I_{\theta(\omega)}[\sigma,d]={\frac{1}{L_y}} \int_{0}^{L_y} \int_{h(y,\omega)}^{d} u(x,y) \mathrm{d}x\mathrm{d}y. \end{aligned}$$ At the formal level, when $L_z \rightarrow 0$, the PL of 3D model defined by reduces to the PL of 2D model defined by , and further they reduce to the PL of 1D model defined by as $L_y\rightarrow 0$. Newton’s method --------------- Given $\sigma^{(0)}$, for $n=1,2,\cdots$, until convergence, Newton’s method for solves $$\begin{aligned}\label{eqn:Newtonmethod} \sigma^{(n)} = \sigma^{(n-1)} -\alpha_n \frac{\frac{\partial}{\partial \sigma} J(\sigma^{(n-1)})}{ \frac{\partial^2}{\partial^2 \sigma}J(\sigma^{(n-1)})} \end{aligned}$$ with $\alpha_n\in(0,1]$ given by line search [@Nocedal1999]. Given one realization of the random interface , by solving the 3D diffusion equation model - , we get one datum $(\theta(\omega_1),\theta(\omega_2),\sigma_{\theta(\omega_1),\theta(\omega_2)})$, where $\theta(\omega_1)$ and $\theta(\omega_2)$ are inputs and $\sigma_{\theta(\omega_1),\theta(\omega_2)}$ is the output. A set of data $\left\{(\theta(\omega_1)[j],\theta(\omega_2)[j],\sigma_{\theta(\omega_1)[j],\theta(\omega_2)[j]})_{j=1}^M\right\}$ will be generated for training and testing. Quasi-Monte Carlo sampling -------------------------- In the sampling stage of data preparation, a large $M$ is needed to ensure that the extracted function of EDL has the desired accuracy. There are two classical choices: uniform sampling and random sampling. For uniform sampling, $M$ grows exponentially fast with respect to $K_1$ and $K_2$. For example, in the 3D case, if $K_1=K_2=5$ and points are uniformly distributed for each random variable, the size of training data set is shown in Table \[tbl:uniformly\]. Number of points in each dimension $2$ $3$ $5$ $9$ ------------------------------------ -------- --------- ----------- -------------- Size of training data set $1024$ $59049$ $9765625$ $3486784401$ : Size of training data set for uniform sampling.[]{data-label="tbl:uniformly"} Figure \[fig:uniformlysampling\] plots the points by uniform sampling when $K_1=K_2=1$ (two random variables). Clearly such a sampling strategy has the curse of dimensionality. On the other hand, if random sampling is used, then we do not have this issue. However, Monte-Carlo method has poor accuracy $\sim O(\frac{1}{\sqrt{M}})$. At least millions of data are needed for training. Meanwhile, for each datum, an inverse problem with the diffusion equation model over a curved domain in 3D has to be solved. These together make the network training prohibitively expensive. Fortunately, the quasi-Monte Carlo sampling has accuracy $\sim O(\frac{1}{M})$ [@Russel1998], which reduces the size of training data set by orders of magnitudes in comparison with Monte-Carlo method. Specifically, we use Sobol sequence to generate points over the (high-dimensional) random space. Figure \[fig:QMCsampling\] plots the points generated by Sobol sequence, which is a deterministic way to generate points with better approximation accuracy. The size of data in the quasi-Monte Carlo method grows merely linearly fast with respect to the number of random variables. For the simulations in our work, at least three orders of magnitude reduction in the size of data set is found for quasi-Monte Carlo sampling strategy. Figures \[fig:bigerror\] and \[fig:smallerror\] show the huge advantage of quasi-Monte-Carlo sampling over uniform sampling. For the same size of training data set, the relative $L^\infty$ error is $30.561\%$ and $0.237\%$, implying more than two orders of magnitude improvement in the prediction accuracy. Softwares --------- The following softwares and libraries are used: Julia, Flux and CuArrays. Julia is a high-level programming language designed for high-performance numerical analysis and computational science[@Bezanson201765]. Flux is a library for machine learning. It comes “batteries-included” with many useful tools built in, but also allows taking the full power of the Julia language. CuArrays provides a fully-functional GPU array, which can give significant speedups over normal arrays without code changes. Detailed description of ResNet {#sec:detailofDNN} ------------------------------ The ResNet network we use is stacked by several blocks with each block containing two linear transformations, two activation functions, and one shortcut connection. The $i$-th block can be expressed as $$\label{equ:ith block} t= f_i(s)= g(W_{i,2}\cdot g(W_{i,1}\cdot s +b_{i,1})+b_{i,2})+s.$$ Here $s,t\in R^{m}$ are input and output of the $i$-th block, and weights $W_{i,j}\in R^{m\times m}, b_{i,1}, b_{i,2}\in R^{m}$. Sigmoid function $$g(x)=\frac{1}{1+\exp(-x)}$$ is chosen as the activation function to balance training complexity and accuracy. The last term in is called the shortcut connection or the residual connection. Advantages of using it are 1) It can solve the notorious problem of vanishing/exploding gradients automatically; 2) Without adding any parameters or computational complexity, the shortcut connection performing as an $Identity$ mapping can resolve the degradation issue (with the network depth increasing, accuracy gets saturated and then degrades rapidly). The fully $n$-layer network can be expressed as $$\label{equ:fully network} f_w(x)= f_n \circ f_{n-1} \cdots \circ f_1(x),$$ where $w$ denotes the set of parameters in the whole network. Note that the input $x$ in the first layer is in $R^{\mathrm{dim}}$ and the output of the whole structure $\sigma(\theta(\omega_1),\theta(\omega_2))$ is in $R^1$. To deal with the problem, we apply two linear transformations on both $x$ before putting it into the ResNet structure and on the output of the ResNet structure. For example, we choose $m=30,n=6$ in the 3D model. Both $\theta(\omega_1)$ and $\theta(\omega_2)$ have $5$ random variables, and thus $\mathrm{dim}=10$. Therefore, we apply two linear transforms: one from a $10$ dimensional vector to a $30$ dimensional vector and the other from a $ 30$ dimensional vector to $1$ dimensional vector before and after the ResNet structure. Parameters in these linear transforms also need to be trained. 2D results ========== First, we focus on the 2D problem with only one realization, i.e., only one $d=10$ and $N=1$. PL data are generated when $\sigma=10$ without any randomness. Accuracy of the trained neural network in terms of size of the training set is recorded in Table \[tbl:error in 2D\]. Size of training data set $9$ $25$ $81$ --------------------------- ----------- ------------- ------------- -- Error ($\beta=-2$) $1.638\%$ $0.04466\%$ $0.00209\%$ Error ($\beta=-1$) $0.101\%$ $0.00795\%$ $0.00234\%$ Error ($\beta=0$) $0.765\%$ $0.105\%$ $0.0180\%$ : Generalization error of the trained neural network model for a random field with different decay rates in 2D.[]{data-label="tbl:error in 2D"} From the results, we can find that a random field with the slower decay rate ($\beta=0$) is more difficult to be trained when uniform sampling is used. Figure \[fig:actualpredict\] shows that this issue can be resolved by quasi-Monte Carlo sampling with moderate size of training set. In the literature, asymptotics-based method has been proposed [@Chen2019894] which only works well for random interfaces with small magnitudes. The proposed method works for random interfaces with large magnitudes. For example, consider $\theta(\omega)$ with $2$ random variables ranging over $[-5,5]$ and $\beta = 2$, the relative $L^{ \infty }$ error is $1.071\%$; see Figure \[fig:bigrandomnesserror\]. For a random field with $10$ random variables and $5$ realizations $d=[10, 15 , 20, 30, 40,50]$, generalization errors of the trained neural network are plotted in Figures \[fig:beta2realisticerror\], \[fig:beta1realisticerror\], \[fig:beta0realisticerror\] for $\beta=-2, -1, 0$, respectively. A detailed dependence of EDL $\sigma$ on random variables is given in Figures \[fig:beta2realistic\], \[fig:beta1realistic\], \[fig:beta0realistic\] for $\beta=-2, -1, 0$, respectively. Contour plot of EDL on random variables in 3D when $\beta=-1$ is given in Figure \[fig:lambda1contour\] for comparison. Supporting Figures ================== ![Uniform sampling for two random variables with $100$ points.[]{data-label="fig:uniformlysampling"}](Figure//uniformly.pdf){width="60.00000%"} ![Quasi-Monte Carlo sampling (Sobol sequence) for two random variables with $100$ points.[]{data-label="fig:QMCsampling"}](Figure//qmc.pdf){width="60.00000%"} ![Generalization error of the trained neural network in 2D when the number of random variables is $10$ and $1024$ uniformly distributed points are used. The relative $L^\infty$ error is $30.561\%$.[]{data-label="fig:bigerror"}](Figure//bigerror.pdf){width="55.00000%"} ![Generalization error of the trained neural network in 2D when the number of random variables is $10$ and $1024$ points generated by Sobol sequence are used. The relative $L^\infty$ error is $0.237\%$.[]{data-label="fig:smallerror"}](Figure//smallerror.pdf){width="55.00000%"} ![Generalization error of the trained neural network for random variables ranging over $[-5,5]$ with $1$ photoluminescence datum in 2D. The relative $L^\infty$ error is $1.071\%$.[]{data-label="fig:bigrandomnesserror"}](Figure//beta2bigerror.pdf){width="55.00000%"} ![Generalization error of the trained neural network for random variables ranging over $[-5,5]$ with $6$ photoluminescence data and $\beta=-2$ in 2D. The relative $L^\infty$ error is $5.784\%$.[]{data-label="fig:beta2realisticerror"}](Figure//2dlambda2ralisticerror.pdf){width="55.00000%"} ![Generalization error of the trained neural network for random variables ranging over $[-5,5]$ with $6$ photoluminescence data and $\beta=-1$ in 2D. The relative $L^\infty$ error is $3.327\%$.[]{data-label="fig:beta1realisticerror"}](Figure//2dlambda1ralisticerror.pdf){width="55.00000%"} ![Generalization error of the trained neural network for random variables ranging over $[-5,5]$ with $6$ photoluminescence data and $\beta=0$ in 2D. The relative $L^\infty$ error is $9.184\%$.[]{data-label="fig:beta0realisticerror"}](Figure//2dlambda0ralisticerror.pdf){width="55.00000%"} ![Dependence of exciton diffusion length on random variables in 2D when $\beta=-2$.[]{data-label="fig:beta2realistic"}](Figure//2dlambda2ralistic.pdf){width="60.00000%"} ![Dependence of exciton diffusion length on random variables in 2D when $\beta=-1$.[]{data-label="fig:beta1realistic"}](Figure//2dlambda1ralistic.pdf){width="60.00000%"} ![Dependence of exciton diffusion length on random variables in 2D when $\beta=0$.[]{data-label="fig:beta0realistic"}](Figure//2dlambda0ralistic.pdf){width="60.00000%"} ![Contour plot of exciton diffusion length on random variables in 3D when $\beta=-1$.[]{data-label="fig:lambda1contour"}](Figure//lambda1contour.pdf){width="70.00000%"}

--- abstract: 'The Duggan-Schwartz theorem [@Duggan1992] is a famous result concerning strategy-proof social choice correspondences, often stated as “A social choice correspondence that can be manipulated by neither an optimist nor a pessimist has a weak dictator”. However, this formulation is actually due to @Taylor2002, and the original theorem, at face value, looks rather different. In this note we show that the two are in fact equivalent.' author: - Egor Ianovski bibliography: - 'references.bib' title: 'Two statements of the Duggan-Schwartz theorem' --- Definitions =========== Let $V$ be a finite set of voters, $A$ a finite set of alternatives. A profile $P$ consists of a linear order over $A$ (also known as a *preference order* or a *ballot*), $P_i$, for every voter $i$. The set of all profiles of voters $V$ over alternatives $A$ is denoted ${\mathcal{P}}(V,A)$. We use $P_{-i}$ to refer to the ballots of all voters except $i$. Hence, $P=P_iP_{-i}$ and $P_i'P_{-i}$ is obtained from profile $P$ by replacing $P_i$ with $P_i'$. A *social choice correspondence* produces a nonempty set of alternatives, $F:{\mathcal{P}}(V,A){\rightarrow}2^A{\backslash}{{ \{\, \emptyset \,\} }}$. \[def:sp\] Let $\emptyset\neq W\subseteq A$. We use $\text{best}(P_i,W)$ to denote the best alternative in $W$ according to $P_i$, $\text{worst}(P_i,W)$ the worst. We extend $\succeq_i$ into two weak orders over $2^A{\backslash}{{ \{\, \emptyset \,\} }}$: 1. $X\succeq_i^O Y$ iff ${\textnormal{best}}(P_i,X)\succeq_i{\textnormal{best}}(P_i,Y)$. 2. $X\succeq_i^P Y$ iff ${\textnormal{worst}}(P_i,X)\succeq_i{\textnormal{worst}}(P_i,Y)$. A social choice correspondence is *strategy-proof for optimists* (SPO) if for all $P_i'$, whenever $F(P_iP_{-i})=W$ and $F(P_i'P_{-i})=W'$, $W\succeq_i^O W'$. A social choice correspondence is *strategy-proof for pessimists* (SPP) if for all $P_i'$, whenever $F(P_iP_{-i})=W$ and $F(P_i'P_{-i})=W'$, $W\succeq_i^P W'$. Given a social choice correspondence $F$, a *weak dictator* is some $i\in V$ such that the first choice of $i$ is always in $F(P)$. Proofs ====== \[thm:Taylor\] Let $F$ be a social choice correspondence that satisfies SPP, SPO and is onto with respect to singletons. That is, for every $a\in A$ there exists a $P$ such that $F(P)={{ \{\, a \,\} }}$. For $|A|\geq 3$, $F$ has a weak dictator. Let $F$ be a social choice correspondence that is onto with respect to singletons. That is, for every $a\in A$ there exists a $P$ such that $F(P)={{ \{\, a \,\} }}$. Let each voter $i$ be equipped with a probability function $p_i:{\mathcal{P}}(V,A)\times A\times 2^A{\rightarrow}[0,1]$ such that $\sum_{x\in X}p_i(P,x,X)=1$ and $p_i(P,a,X)>0$ whenever $a={\textnormal{best}}(P_i,X)$ or $a={\textnormal{worst}}(P_i,X)$. Suppose further that for every $u_i$ consistent with $P_i$ ($u_i(a)>u_i(b)$ whenever $a\succ_i b$), and for every $P_i'$, the following is true: $$\sum_{x\in F(P_iP_{-i})}p_i(P_iP_{-i},x,F(P_iP_{-i}))u_i(x)\geq \sum_{x\in F(P_i'P_{-i})}p_i(P_iP_{-i},x,F(P_i'P_{-i}))u_i(x).$$ For $|A|\geq 3$, $F$ has a weak dictator. The notion of manipulation used by @Duggan1992 is obviously more general than that of @Taylor2002, and one is thus tempted to conclude that the original theorem is weaker than Taylor’s reformulation.[^1] However, this would be erroneous as the theorems, strictly speaking, are incomparable. Taylor’s theorem concerns a social choice correspondence $F$, whereas @Duggan1992’s theorem applies to $F$ *together* with a set of probability functions, $p_i$. It is entirely plausible that one could find two sets of probability functions such that $F$ and $p_1,\dots,p_n$ satisfy the hypotheses of the Duggan-Schwartz theorem while $F$ and $p_1',\dots,p_n'$ do not. However, $F$ is unchanged – it either has a weak dictator, or it does not. To more properly compare the two theorems, then, we need to take an existential projection over the original Duggan-Schwartz theorem. \[thm:DSexist\] Let $F$ be a social choice correspondence that is onto with respect to singletons. That is, for every $a\in A$ there exists a $P$ such that $F(P)={{ \{\, a \,\} }}$. Suppose there exist probability functions $p_i:{\mathcal{P}}(V,A)\times A\times 2^A{\rightarrow}[0,1]$ such that $\sum_{x\in X}p_i(P,x,X)=1$ and $p_i(P,a,X)>0$ whenever $a={\textnormal{best}}(P_i,X)$ or $a={\textnormal{worst}}(P_i,X)$. Suppose further that for every $u_i$ consistent with $P_i$ ($u_i(a)>u_i(b)$ whenever $a\succ_i b$), and for every $P_i'$, the following is true: $$\sum_{x\in F(P_iP_{-i})}p_i(P_iP_{-i},x,F(P_iP_{-i}))u_i(x)\geq \sum_{x\in F(P_i'P_{-i})}p_i(P_iP_{-i},x,F(P_i'P_{-i}))u_i(x).$$ For $|A|\geq 3$, $F$ has a weak dictator. Now we claim the two theorems are equivalent. $F$ satisfies the hypotheses of \[thm:Taylor\] if and only if $F$ satisfies the hypotheses of \[thm:DSexist\]. We will first show that if $F$ is manipulable in the sense of Taylor it is manipulable in the sense of Duggan-Schwartz. Pay heed to the order of the quantifiers in \[thm:DSexist\], as they may appear counter-intuitive: $F$ is strategy-proof if for *some* choice of probability functions, for *every* choice of a utility function, voter $i$ cannot improve his expected utility. Hence, $F$ is manipulable just if for *every* choice of probability functions we can construct *some* utility function giving voter $i$ a profitable deviation. Suppose $i$ can manipulate optimistically from $P_iP_{-i}$ to $P_i'P_{-i}$. That is: $$\begin{aligned} F(P_iP_{-i})=X&,\quad F(P_i'P_{-i})=Y,\\ {\textnormal{best}}(P_i,X)=a&,\quad{\textnormal{best}}(P_i,Y)=b,\\ a&\prec_i b. \end{aligned}$$ Now, let $p_i$ be any probability function in the sense of \[thm:DSexist\]. Note that this means that $p_i(P,a,X)=\epsilon$ and $p_i(P,b,Y)=\delta$ are strictly positive. Let $c\in X$ be the next-best alternative after $a$. Observe that an upper bound on the utility voter $i$ obtains sincerely is $\epsilon u_i(a)+(1-\epsilon)u_i(c)$, whereas the lower bound on the utility voter $i$ obtains from the deviation is $\delta u_i(b)$. All we need to do is pick a $u_i$ that satisfies: $$\delta u_i(b)>\epsilon u_i(a)+(1-\epsilon)u_i(c).$$ It is of course easy to do so as, necessarily, $u_i(b)>u_i(a),u_i(c)$, and $\epsilon,\delta$ are constants. For example, let $u_i(a)=1,u_i(c)=2$ and $u_i(b)=\sfrac{3}{\delta}$. Suppose $i$ can manipulate pessimistically from $P_iP_{-i}$ to $P_i'P_{-i}$. That is: $$\begin{aligned} F(P_iP_{-i})=X&,\quad F(P_i'P_{-i})=Y,\\ {\textnormal{worst}}(P_i,X)=a&,\quad{\textnormal{worst}}(P_i,Y)=b,\\ a&\prec_i b. \end{aligned}$$ As before, let $p_i$ be any probability function in the sense of \[thm:DSexist\]. This means that $p_i(P,a,X)=\epsilon$ and $p_i(P,b,Y)=\delta$ are strictly positive. Let $c\in X$ be the best alternative in the set. Observe that an upper bound on the utility voter $i$ obtains sincerely is $\epsilon u_i(a)+(1-\epsilon)u_i(c)$, whereas the lower bound on the utility voter $i$ obtains from the deviation is $u_i(b)$.[^2] All we need to do is pick a $u_i$ that satisfies: $$u_i(b)>\epsilon u_i(a)+(1-\epsilon)u_i(c).$$ This time it is possible that $u_i(c)>u_i(b)$, however $1-\epsilon$ is strictly smaller than 1. One possibility is $u_i(a)=1$, $u_i(b)=\frac{\sfrac{1}{\epsilon}+\epsilon+1}{1-\epsilon}$, $u_i(c)=\frac{\sfrac{1}{\epsilon}+\epsilon+2}{1-\epsilon}$. This leads to the following inequality, which can be verified algebraically: $$\frac{\sfrac{1}{\epsilon}+\epsilon+1}{1-\epsilon}>2\epsilon+2+\sfrac{1}{\epsilon}.$$ Now suppose that $F$ is manipulable in the sense of Duggan-Schwartz. This means for every choice of $p_i$, there is some choice of $u_i$ such that for some choice of $P_iP_{-i}$ and $P_i'P_{-i}$, $i$’s expected utility is higher in the insincere profile. Pick a $p_i$ that attaches a probability of $\sfrac{1}{2}$ to the best alternative in the set and $\sfrac{1}{2}$ to the worst. In other words, we have the following situation: $$\begin{aligned} F(P_iP_{-i})=X&,\quad F(P_i'P_{-i})=Y,\\ {\textnormal{best}}(P_i,X)=x_1&,\quad{\textnormal{best}}(P_i,Y)=y_1,\\ {\textnormal{worst}}(P_i,X)=x_2&,\quad{\textnormal{worst}}(P_i,Y)=y_2,\\ \frac{u_i(x_1)+u_i(x_2)}{2}&<\frac{u_i(y_1)+u_i(y_2)}{2}.\\ \end{aligned}$$ Clearly, a necessary condition for the above to hold is that either $u_i(y_1)>u_i(x_1)$ or $u_i(y_2)>u_i(x_2)$. That is to say, $F$ is manipulable by either an optimist or a pessimist. [^1]: A wider notion of manipulability implies a more narrow notion of strategy-proofness, and hence the theorem would apply to less functions. [^2]: $b$ is the worst element in $Y$, so the utility of any other element must be at least $u_i(b)$, and $p_i(P,y,Y)$ sums to 1.

--- abstract: '**Device-independent protocols use nonlocality to certify that they are performing properly. This is achieved via Bell experiments on entangled quantum systems, which are kept isolated from one another during the measurements. However, with present-day technology, perfect isolation comes at the price of experimental complexity and extremely low data rates. Here we argue that for device-independent randomness generation – and other device-independent protocols where the devices are in the same lab – we can slightly relax the requirement of perfect isolation, and still retain most of the advantages of the device-independent approach, by allowing a little cross-talk between the devices. This opens up the possibility of using existent experimental systems with high data rates, such as Josephson phase qubits on the same chip, thereby bringing device-independent randomness generation much closer to practical application.**' author: - 'J. Silman' - 'S. Pironio' - 'S. Massar' title: 'Device-Independent Randomness Generation in the Presence of Weak Cross-Talk' --- *Introduction* – The great advantage of device-independent (DI) protocols is their reliance on a small set of tests, which are nevertheless sufficient to certify that they are performing properly. This is achieved by carrying out nonlocality tests on entangled quantum systems. In particular, no assumptions are made regarding the inner workings of the devices, such as the Hilbert space dimension of the underlying quantum systems, etc. [@Mayers; @Acin]. Each device is treated as a ‘black box’ with knobs and registers for selecting and displaying (classical) inputs and outputs. Applications include quantum key-distribution [@Barrett; @Acin; @McKague; @Masanes; @Reichardt; @Vazirani], coin flipping [@Silman], state tomography [@Bardyn; @McKague; @3; @Reichardt], genuine multi-partite entanglement detection [@Bancal], self-testing of quantum computers [@Magniez; @McKague; @2], as well as DI randomness generation (RG) [@Colbeck; @Pironio; @Pironio; @2; @Fehr; @Vazirani; @2]. It is often remarked that DI cryptographic protocols remain secure even if the devices have been provided, or sabotaged, by an adversary. This scenario, while conceptually fascinating, is of little (if any) practical relevance. This is because (i) there are many types of attacks available to a malicious provider – the majority being classical – that eliminating them all is an enormous task; (ii) in any case we assume the existence of honest providers of e.g. the source of randomness, the jamming technology to prevent information leakage from the labs, or the classical devices used to process the data. A scenario where one can trust the latter, but an honest provider for the quantum devices cannot be found, is highly implausible. The actual advantage of DI protocols is that they allow us to monitor the performance of the devices irrespectively of noise, imperfections, lack of knowledge regarding their inner workings, or limited control over them. Indeed, even if the devices were obtained from a trusted provider and thoroughly inspected, many things can still unintentionally go wrong (as demonstrated by the attacks on commercial quantum key-distribution systems [@Zhao; @Xu; @Lydersen], which exploited unintentional design flaws). This problem is particularly acute in the case of DI RG, as it is very difficult even for honest parties to maufacture reliable randomness generators (whether classical or quantum) and monitor them for malfunction. The generation of randomness in a DI manner solves many of the shortcomings of customary RG protocols, since, as mentioned above, the degree of violation of a Bell inequality provides an accurate estimate of the amount of randomness generated irrespectively of experimental imperfections and lack of control. DI RG has so far been proven secure against adversaries with classical side-information about the devices (which is the relevant case when the provider is trusted) for arbitrary Bell inequalities and degrees of violation [@Pironio; @2; @Fehr], and against adversaries with quantum side-information in the case of very high violation of the CHSH inequality [@Vazirani; @2]. Unfortunately, DI RG is experimentally highly challenging. It requires a Bell experiment with the detection loophole closed and with the quantum systems isolated from one another. A proof of principle experiment was reported in [@Pironio] using two ions in separate vacuum traps, but this system operates at an extremely low rate ($\sim 1\,\mathrm{mHz}$), precluding any practical application. Nevertheless, there exist today experiments involving, for example, two Josephson phase qubits on the same chip coupled by a radio frequency resonator [@Ansmann], or two ions in the same trap coupled via their vibrational modes [@Rowe; @Monz], which allow for Bell violating experiments (with the detection loophole closed) at much higher data rates ($\gtrsim 1\,\mathrm{kHz}$). In these experiments the quantum systems are very close to one another. This proximity provides the non-negligible coupling required for high entanglement generation rates. Adapting DI RG to these types of experiments would bring it much closer to real-life application. The problem is that precisely because the systems are close to one another and non-negligibly coupled, they can no longer be considered as completely isolated (see [@Martinis] and [@Haffner] for a discussion of the couplings involved). The aim of the present work is to show how to take this coupling into account by relaxing slightly the assumptions behind the DI approach, while keeping as much as possible all of its advantages. We begin by showing how to derive bounds on the RG rate in a DI setting given a known amount of cross-talk (CT). Next, we present methods for estimating the amount of CT present in an experiment. Our approach is then illustrated on Josephson phase qubits, showing that efficient DI RG is possible using already established technology. We start first, however, by recalling briefly the essential ingredients of DI RG relevant to our analysis. We refer to [@Pironio; @Pironio; @2; @Fehr] for a more detailed presentation. *Bell inequalities and device-independent randomness generation* – A Bell experiment is characterized by the probabilities $\mathcal{P}=\left\{ P_{ab\mid xy}\right\} $ of obtaining the outcomes (or outputs) $a$ and $b$ given the measurement settings (or inputs) $x$ and $y$. A Bell expression $\mathcal{I}\left(\mathcal{P}\right)=\sum_{abxy} c_{abxy} P_{ab|xy}$ is a linear function of these probabilities. For instance, the CHSH inequality has the form $\mathcal{I}\left(\mathcal{P}\right)=\sum_{a,\, b,\, x,\, y\in\left\{ 0,\,1\right\} }\left(-1\right)^{a\oplus b\oplus xy}P_{ab\mid xy} \leq2$. To any Bell expression, one can associate a bound on the randomness of the outputs given the inputs $x$ and $y$ through a function $P_{xy}^{*}\left(I\right)$ such that $\max_{a,\, b}P_{ab\mid xy}\leq P_{xy}^{*}\left(I\right)$ holds for any $\mathcal{P}$ for which $\mathcal{I}\left(\mathcal{P}\right)=I$ [@Pironio; @Pironio; @2; @Fehr]. The function $P_{xy}^{*}\left(I\right)$ should be monotonically decreasing and concave in $I$ (if not we can take its concave hull). Higher values of $P_{xy}^{*}(I)$ imply less randomness, in particular when $\min_{x,\, y}P_{xy}^{*}(I)=1$ the system is fully deterministic. Given knowledge of such a function and the degree of Bell violation $I$ observed in an experiment where the devices are used $n$ times in succession, one can infer a lower bound on the min-entropy of the measurement outcomes. By applying a randomness extractor to the resulting string of outcomes, one then obtains a new private string of random numbers of length $\simeq -n\log_2 P^*_{xy}(I)$ which is arbitrarily close (up to a security parameter) to the uniform distribution. Depending on the assumptions made regarding the devices and the adversary, such a protocol may also require an initial random seed (in which case one talks about DI randomness expansion) that may be polynomially [@Pironio] or exponentially [@Pironio; @2; @Fehr] smaller than the output string. *Device-independent randomness generation with weak cross-talk* – In the security analysis of DI RG protocols the assumption that the two Bell violating devices are isolated from one another only appears in the derivation of the bound $P_{xy}^{*}\left(I\right) \geq \max_{a,\, b}P_{ab\mid xy}$. If we introduce a similar bound $P_{xy}^{*}\left(I,\chi\right)$ that is valid in the presence of a given amount of CT $\chi$ (defined below), then the rest of the reasoning of [@Pironio; @Pironio; @2; @Fehr] will apply without modification. To define such a CT-dependent bound, we write the probabilities observed in a Bell experiment as $P_{ab|xy}=\mathrm{Tr}\left(\rho\Pi_{ab|xy}\right)$, where $\rho\in \mathcal{H}_{A}\otimes \mathcal{H}_B$ and $\{ \Pi_{ab|xy}\} $ is a POVM on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ (i.e. $\Pi_{ab|xy}\succeq0$ and $\sum_{ab}\Pi_{ab|xy}=\mathds{1}$). The novelty with respect to the standard mathematical description of Bell experiments is in allowing the measurement $\Pi_{ab|xy}$ to act collectively on the two systems. We will say that such a collective measurement requires no more than $\chi$ amount of CT if there exists a product POVM $\{ \Pi_{a|x}\otimes\Pi_{b|y}\} $ satisfying $$-\chi \mathds{1}\preceq\Pi_{ab|xy}-\Pi_{a|x}\otimes\Pi_{b|y}\preceq\chi \mathds{1}\label{prox}$$ for all combinations of $a$ and $b$. This condition restricts how far each collective POVM may be from a product of two independent POVMs. In particular, when $\chi=0$ the $\Pi_{ab|xy}$ can be expressed as products, while when $\chi=1$ they are unconstrained. Consider now a fixed value of $\chi$ and a Bell violation $I$. Then the solution of the following program provides the minimal amount of randomness $P^*_{xy}(I,\chi)$ compatible with $I$ and $\chi$: $$\begin{gathered} P_{xy}^{*}(I,\chi) = \max_{a,\, b}\max_{Q}\quad P_{ab\mid xy} \label{bound 1}\\ \mathrm{s.t.} \quad P_{ab\mid xy}=\mathrm{Tr}\left(\rho\Pi_{ab|xy}\right),\quad \mathcal{I}\left(\mathcal{P}\right)=I, \nonumber\\ -\chi \mathds{1}\preceq\Pi_{ab|xy}-\Pi_{a|x}\otimes\Pi_{b|y}\preceq\chi \mathds{1}\nonumber\,,\end{gathered}$$ where the optimization runs over the set $Q=\{\rho,\,\{\Pi_{a|x}\},\,\{\Pi_{b|y}\},\,\{\Pi_{ab|xy}\},\,\mathcal{H}_{A},\,\mathcal{H}_{B}\}$ specifying the state, measurements, and the Hilbert spaces. This formulation is therefore DI in spirit, since the bound is formulated without fixing the dimension of the Hilbert spaces, nor how the measurements are implemented, etc. Upper bounds on the optimization problem Eq. (\[bound 1\]) can be obtained using the techniques of [@Navascues; @Pironio; @3], which relax the problem to a hierarchy of semi-definite programs (SDPs). In particular, the resulting series of bounds is guaranteed to converge to the true solution. Nevertheless, depending on the problem, even the lowest order relaxation may be computationally intractable. We may then obtain a weaker bound in terms of $P_{xy}^{*}\left( I,\,0\right)$ – the solution in the absence of CT. Let $\rho'$, $\{ \Pi_{a | x}' \}$, $\{ \Pi_{b| y}' \}$, and $\{ \Pi_{ab | xy}' \}$ be the state and POVMs corresponding to the solution of Eq. (\[bound 1\]), and let $P'_{ab\mid xy}=\mathrm{Tr}(\rho' \Pi_{a | x}' \otimes \Pi_{b | y}' )$ and $\mathcal{P}'=\{P'_{ab|xy}\}$. From the last constraint in Eq. (\[bound 1\]) we have that $ | P_{ab\mid xy}-P'_{ab \mid xy} | \leq \chi$, and so $\mathcal{I}(\mathcal{P}') \geq \mathcal{I}(\mathcal{P})- \gamma\chi$ where $\gamma=\sum_{a,\,b,\,x,\,y}\left|c_{abxy}\right|$ (in the case of the CHSH inequality for instance $\gamma=16$). Taken together, the last two inequalities imply that $$P_{xy}^{*}\left(I,\,\chi\right)\leq P_{xy}^{*}\left(I-\gamma\chi,\,0\right)+\chi\,.\label{bound 2}$$ Fig. 1 displays upper bounds on $P_{00}^{*}$ obtained from Eq. (\[bound 1\]) and Eq. (\[bound 2\]) in the case of the CHSH inequality. Finally, we note that the last constraint in Eq. (\[bound 1\]) implies that the signaling – the extent to which the output of one device depends on the input of the other – is constrained. Specifically, if to each input $x$ and each input $y$ correspond $N$ outputs, $|P_{a|xy}-P_{a|xy'}|\leq 2N\chi$ for all $a,\,x,\,y,\,y'$, etc. (in the case of zero signaling, one has $P_{a|xy}=P_{a|xy'}$). This allows us to derive a simpler bound on $P^{*}_{xy}$, depending solely on the amount of signaling present, in contrast to the bounds Eqs. (\[bound 1\]) and (\[bound 2\]), which rely on the full structure of quantum mechanics. To this end we define the maximal amount of signaling allowed as $$\label{sig} \delta = \max\Bigl\{\max_{a,\,x,\,y,\,y'} |P_{a|xy}-P_{a|xy'}|,\,\max_{b,\,y,\,x,\,x'} |P_{b|xy}-P_{b|x'y}|\Bigr\}\,.$$ When $\delta=0$, $\mathcal{P}$ resides within the no-signaling polytope [@Barrett; @2], while when $\delta>0$ $\mathcal{P}$ resides within a larger, higher-dimensional polytope. The bound can be obtained by solving the linear program $P^*_{xy}(I,\,\delta)=\max_{ab}P_{ab|xy}$, given that $I(\mathcal{P})=I$, $|P_{a|xy}-P_{a|xy'}|\leq \delta$, and $|P_{b|xy}-P_{b|x'y}|\leq \delta$. In the case of the CHSH inequality, one can show that (see Appendix A) $$P^{*}_{xy}(I,\delta)\leq\frac{3}{2}-\frac{1}{4}I+2\delta\,.\label{bound 3}$$ This bound applies to any post-quantum theory which restricts the amount of signaling (as well as to quantum mechanics). *Estimating the amount of cross-talk* – We have just seen how the introduction of a new security parameter $\chi$, quantifying the amount of CT between the devices, allows us to extend the scope of DI RG to settings with a limited amount of CT. To apply this approach, we therefore need a reliable prior estimate of $\chi$, and means of guaranteeing or verifying that the CT will not exceed this estimate during latter operations of the devices. This obviously requires some modeling of the devices’ inner workings. Indeed, it is impossible to upper-bound the amount of CT from first principles only or from any set of observed data $\mathcal{P}$ alone, since communicating devices can deterministically reproduce any $\mathcal{P}$, and therefore simulate any degree of Bell violation. At first, this may seem an unwelcome departure from the purely DI approach (i.e. $\chi=0$). Nevertheless, our approach has the advantage over fully device-dependent approaches that only a *single* parameter $\chi$ must be device-*dependently* estimated to ensure that the protocol performs properly, and this same parameter is used irrespectively of the underlying physical realization. Morever, even in purely DI protocols the absence of communication cannot be deduced from the observed data alone, and to verify that there is indeed no-communication will necessarily involve putting our trust in certain general assumptions regarding the behavior of the devices, or relying on some trusted external hardware. Seen in this light, our approach is not very different from the standard (DI) one, except that instead of verifying in some trusted way that $\chi=0$, we must verify that $\chi$ is no greater than some finite value. Finally, we note that our approach allows as a safeguard to set $\chi$ to be greater than its expected value – a feature that may be useful even in for purely DI protocols with (allegedly) non-communicating devices. Even though a maximal amount of CT $\chi$ cannot be guaranteed without some modeling of the devices, there are several ways to lower-bound $\chi$ from the observed data $\mathcal{P}$ only. If the devices were not fabricated by an adversary and do not act maliciously, then these lower bounds may provide good estimates of $\chi$. A simple way to lower-bound $\chi$ in a DI manner is via the degree of violation of the no-signaling conditions Eq. (\[sig\]), computed from the observed data $\mathcal{P}$. From Eq. (\[sig\]) it follows that $\chi\geq\delta/2N$. Improved DI bounds are obtainable, however, reflecting the fact that $\delta$ does not capture all of the information contained in $\mathcal{P}$. The minimal amount of CT that is compatible with a given $\mathcal{P}$ is given by the solution of the following optimization problem $$\begin{gathered} \min_{\mathcal{Q}}\quad\chi\label{CT}\\ \mathrm{s.t.}\quad \mathrm{Tr}\left(\rho\Pi_{ab|xy}\right)=P_{ab\mid xy},\nonumber\\ -\chi \mathds{1}\preceq\Pi_{ab|xy}-\Pi_{a|x}\otimes\Pi_{b|y}\preceq\chi \mathds{1},\nonumber \end{gathered}$$ which can be lower-bounded using the techniques of [@Navascues; @Pironio; @3]. It is clear that this bound is optimal, since the optimization runs over all possible states $\rho$ and sets of projectors $\{\Pi_{a|x}\}$, $\{\Pi_{b|y}\}$ and $\{\Pi_{ab|xy}\}$ satisfying the constraints in Eq. (\[bound 1\]). That it constitutes an improvement over the bound provided by Eq. (\[sig\]) is seen by considering the case of post-quantum non-signaling distributions (including those that do not violate Tsirelson’s bound [@Tsirelson]). Such distributions will not give rise to a non-vanishing bound via Eq. (\[sig\]). However, since they cannot be realized quantumly without communication, they will give rise to a non-vanishing bound via Eq. (\[CT\]). See Fig. 2. It is possible of course that the true value of $\chi$ is not revealed by the above lower bounds (for instance, points in QR in Fig. 2 can be reproduced either with or without CT and thus the real value of $\chi$ cannot be unambiguously determined from the observed data $\mathcal{P}$ alone). Nevertheless, one can also adopt a more device-dependent approach to estimating $\chi$. In particular, if the lower bound provided by Eq. (\[CT\]) equals zero, one can vary the state and the measurements. Such a procedure could in principle reveal the presence of any fixed interaction Hamiltonian $H$, since it has been shown that for any such interaction there exists a strategy involving only local operations and classical communication that reveals the presence of the interaction as signaling [@Bennett]. However, we do not know of any systematic way for finding this strategy if $H$ is unknown, nor do we know how to relate in a systematic way the observed signaling to $H$. Finally, by modeling the physical systems, their interaction and the measurement procedure, it is possible to estimate the amount of CT. An example of this last approach are given below. *Candidates for real-life implementation* – A system ideally suited for the implemenation of DI RG will (i) give rise to a sufficiently high Bell violation with the detection loophole kept closed, (ii) exhibit a negligible amount of CT, and (iii) allow for very high data rates. We discuss below an experiment based on Josephson phase qubits, which meets all of these requirements. Another possibility is based on trapped ions, as discussed in Appendix B. In the CHSH experiment of [@Ansmann] two Josephson phase qubits, coupled by a radio frequency strip resonator, are used. The qubits are located on the same chip, separated by $3.1\,\mathrm{mm}$, and are entangled by successively coupling them to the strip resonator. Single qubit rotations are effected by applying microwaves at the resonance frequency of the corresponding qubit. Read-out is effected by letting the excited state tunnel to an auxiliary state macroscopically distinct from both the ground state and the excited state. All operations can be carried out on time scales significantly shorter than $1\,\mu\mathrm{s}$. (For a recent review of Josephson phase qubits experiments see [@Martinis].) The constant coupling between the qubits gives rise to some CT. From the analysis of the experimental set up performed in [@Ansmann] and [@Kofman], it appears that the most significant contribution to the CT occurs during the read-out: The tunneling of one qubit from the excited state to a macroscopically distinct state sometimes forces the other qubit to tunnel when in the ground state. This allows us to estimate the CT at $0.0030$ (see Appendix C). The same value is also obtained by solving the second order relaxation of Eq. (\[CT\]) using the set of observed data found in in the Supplementary Information for [@Ansmann]. For the reported degree of CHSH violation $I = 2.0732$, and the above value of the CT, we find that $P_{00}^{*}\leq 0.983$. To establish robustness we note that for as a low a violation as $I=2.002$ $P_{00}^{*}\leq 0.998$. This shows that useful randomness is extractable from this experiment. *Conclusion* – The analysis of any DI protocol requires that we specify the amount of CT between the devices (irrespectively of whether it is vanishing or finite) – a requirement that cannot be fully verified or implemented in a DI manner. In this work we have shown that one can relax the maxims appearing previous works on DI RG, by allowing for a small amount of CT between the quantum systems. In this way we can keep most of the advantages of the DI approach and at the same time reach data rates of practical interest. Finally, we note that our approach can be generalized to other DI protocols where the devices are in the same lab, such as DI tests of genuine multi-partite entanglement [@Bancal]. We acknowledge support from the European project QCS, project nbr. 255961, from the Inter-University Attraction Poles (Belgian Science Policy) project Photonics@be, from the Brussels Capital Region through a BB2B grant, and from the FRS-FNRS. The Matlab toolboxes YALMIP [@YALMIP] and SeDuMi [@SeDuMi] were used to obtain Fig. 2 and solve Eqs. (\[bound 1\]) and (\[CT\]). Appendix A {#appendix-a .unnumbered} ========== We prove here Eq. (\[bound 3\]). Eq. (\[bound 3\]) can be re-expressed as $4P^{*}_{xy}(I,\,\delta)\geq\left(4-I\right)+\left(2+8\delta\right)$. The first term on the right-hand side is now seen to be a sum of probabilities, all of which appear in the CHSH inequality with a minus sign: $$\begin{gathered} 4-I = \sum_{a,\, b,\, x,\, y}\left[1-(-1)^{a\oplus b \oplus xy} \right]P_{ab\mid xy}\nonumber \\ = 2\sum_{a}\left(P_{a\bar{a}\mid 00}+P_{a\bar{a}\mid 01}+P_{a\bar{a}\mid 10}+P_{aa\mid 11}\right)\,,\end{gathered}$$ where $\bar{a} = a \oplus 1$. Hence, any probability, appearing in the CHSH inequality with a negative sign, is smaller or equal to $2-I/2$. Consider now the following relation $$\begin{gathered} 2+8\delta \geq 2+2\sum_{x,\, y}\left[\left(-1\right)^{y}P_{a\mid xy}+\left(-1\right)^{x}P_{b\mid xy}\right]\nonumber \\ = 2+2\bigl(2 P_{ab\mid 00}+P_{a\bar{b}\mid 00}-P_{a\bar{b}\mid 01}+P_{a\bar{b}\mid 10}-2P_{ab\mid 11} -P_{a\bar{b}\mid 11}+P_{\bar{a}b\mid 00}-P_{\bar{a}b\mid 10}+P_{\bar{a}b\mid 01}-P_{\bar{a}b\mid 11}\bigr)\nonumber \\ \geq 2+2\bigl(2 P_{ab\mid 00}-P_{a\bar{b}\mid 01}-2P_{ab\mid 11}-P_{a\bar{b}\mid 11}-P_{\bar{a}b\mid 10}-P_{\bar{a}b\mid 11}\bigr) \\ \geq 2+2\bigl(2 P_{ab\mid 00}-P_{a\bar{b}\mid 01}-P_{ab\mid 11}-1+P_{\bar{a}\bar{b}\mid 11}-P_{\bar{a}b\mid 10}\bigr)\nonumber \\ \geq 2\bigl(2 P_{ab\mid 00}-P_{a\bar{b}\mid 01}-P_{ab\mid 11}-P_{\bar{a}b\mid 10}\bigr)\,.\nonumber \end{gathered}$$ Setting $b=a$ and summing $4-I$ and $2+8\delta$, we get $$\begin{gathered} 6-I+8\delta \geq 4P_{aa\mid 00}+2\left(P_{a\bar{a}\mid 00}+P_{\bar{a}a\mid 00}+P_{\bar{a}a\mid 01}\right.\nonumber \\ \left. +P_{a\bar{a}\mid 10}+P_{\bar{a}\bar{a}\mid 11}\right)\,,\end{gathered}$$ and so $$P_{aa\mid 00}\leq\frac{3}{2}-\frac{1}{4}I+2\delta \,.$$ Similarly, from $$4 \delta \geq \sum_{x,\, y}\bigl[\left(-1\right)^{x+y+1}P_{a\mid xy} + \left(-1\right)^{x}P_{b\mid xy}\bigr]\,,$$ it follows that $$P_{aa\mid 01}\leq\frac{3}{2}-\frac{1}{4}I+2\delta \,,$$ from $$4 \delta \geq \sum_{x,\, y}\bigl[\left(-1\right)^{y}P_{a\mid xy} + \left(-1\right)^{x+y+1}P_{b\mid xy}\bigr]$$ it follows that $$P_{aa\mid 10}\leq\frac{3}{2}-\frac{1}{4}I+2\delta \,,$$ and from $$4 \delta \geq \sum_{x}\Bigl[\sum_{y}\left(-1\right)^{x+y}P_{a\mid xy} + \left(-1\right)^{x+1}\bigl(P_{b\mid x0}+P_{\bar{b}\mid x1}\bigr)\Bigr]$$ it follows that $$P_{a\bar{a}\mid 11}\leq\frac{3}{2}-\frac{1}{4}I+2\delta \,,$$ and so any probability appearing in the CHSH inequality with a positive sign is smaller or equal to $\frac{3}{2}-\frac{1}{4}I+2\delta $. We therefore have that whenever $I\geq 2$ (the case $I\leq 2$ being trivial) $$P^{*}_{xy}(I,\,\delta)\leq\max\Bigl\{ 2-\frac{1}{2}I,\,\frac{3}{2}-\frac{1}{4}I+2\delta \Bigr\} =\frac{3}{2}-\frac{1}{4}I+2\delta \,$$ for all pairs of inputs $x$ and $y$. Appendix B {#appendix-b .unnumbered} ========== We derive here a rough estimate for the amount of CT present in the experiments such as [@Monz]. Vibrationally coupled ions in the same trap are one of the most advanced quantum information processing systems. The system is initialized by preparing the ions in their vibrational ground state, following which they are entangled via the vibrational coupling. Measurements can be realized on each ion individually: First, to choose the measurement setting, single qubit gate operations are performed by addressing the ions individually with focused light beams. The state of each ion is then measured in the computational basis using fluorescence. The whole process takes $\lesssim1\,\mathrm{ms}$ and the fidelities of the gates and measurements are high ($\sim99\,\%$). (For a recent review of quantum information processing using ion traps see [@Haffner].) In these experiments the ions are typically separated by $\sim5\,\mu\mathrm{m}$, resulting in some CT. It seems that the main contribution to the CT is due to the single qubit rotations performed to select the measurement settings. Indeed, the light beams used to address the ions have a width of $2\,\mu\mathrm{m}$. As a result, if the state of one ion is rotated by $\theta$ on the Bloch sphere, the neighbouring ion will be rotated by $\varepsilon \simeq0.03\theta$ (this is the ratio of Rabi frequencies, as discussed in [@Haffner]). More specifically, with no loss of generality we may assume that the ideal entangled state shared by the parties is such that the corresponding ideal measurements are projectors onto the states $\left|\psi_{abxy}\right\rangle =\left|a_{\varphi_{x}}\right\rangle \otimes|b_{\varphi_{y}}\rangle $, where $\left|a_{\varphi_{x}}\right\rangle$ ($|b_{\varphi_{y}}\rangle$) denotes a state on the Bloch sphere parametrized by $\theta=\pi/2$ and $\varphi=\varphi_{x}=\left(-1\right)^{x}\pi/4$ ($\varphi=\varphi_{y}=\left(-1\right)^{y}\pi/4$). As explained above, due to the CT, the rotation of one ion by $\theta $ induces a rotation of the other by $\varepsilon \simeq 0.03 \theta $. The actual measurements therefore consist of projectors onto $\left|\xi_{abxy}\left(\varepsilon\right)\right\rangle =\left|a_{\varphi_{x}\left(\varepsilon\right)}\right\rangle \otimes\left|b_{\varphi_{y}\left(\varepsilon\right)}\right\rangle $ with $\varphi_{x}\left(\varepsilon\right)=\left[\left(-1\right)^{x}+\left(-1\right)^{y}\varepsilon\right]\pi/4$ and $\varphi_{y}\left(\varepsilon\right)=\left[\left(-1\right)^{y}+\left(-1\right)^{x}\varepsilon\right]\pi/4$. An upper bound on $\chi$ is given by the largest eigenvalue (up to a sign) out of the set of eigenvalues of the 16 matrices $\left|\xi_{abxy}\left(\varepsilon\right)\right\rangle \left\langle \xi_{abxy}\left(\varepsilon\right)\right|-\left|\psi_{abxy}\right\rangle \left\langle \psi_{abxy}\right|$. Since in the CHSH experiment $\varphi_x,\,\varphi_y = \pm \pi /4$, we get that $\chi\lesssim 0.015$. Appendix C {#appendix-c .unnumbered} ========== We derive here the estimate for the amount of CT present in the experiment reported in [@Ansmann]. Denote the ground and excited states by $\left|0\right\rangle $ and $\left|1\right\rangle $, respectively. Then, as explained in the main body of the text, the probability of obtaining anti-correlated outcomes is attenuated. Specifically, let $p_{A}$ ($p_{B}$) be the probability that the tunneling of qubit $A$ ($B$) – i.e. obtaining the outcome $1$ for the measurement of the state of qubit $A$ ($B$) – forces qubit $B$ ($A$) to tunnel when in the ground state. We can model the presence of the CT as follows: $$\begin{gathered} \Pi_{00|xy} = \Pi_{0|x}\otimes\Pi_{0|y}\,,\nonumber\\ \mbox{\ensuremath{\Pi}}_{01|xy} = \left(1-p_{B}\right)\Pi_{0|x}\otimes\Pi_{1|y}\,, \\ \mbox{\ensuremath{\Pi}}_{10|xy} = \left(1-p_{A}\right)\Pi_{1|x}\otimes\Pi_{0|y}\,,\nonumber \\ \mbox{\ensuremath{\Pi}}_{11|xy} = \Pi_{1|x}\otimes\Pi_{1|y}+p_{B}\Pi_{0|x}\otimes\Pi_{1|y}+p_{A}\Pi_{1|x}\otimes\Pi_{0|y}\,,\nonumber \end{gathered}$$ where using the notation of Appendix B $\Pi_{a|x}=\left|a_{\varphi_{x}}\right\rangle \left\langle a_{\varphi_{x}}\right|$, etc. To estimate the amount of CT we need to find the nearest set of product POVMs. For simplicity, we assume them to have the form $$\begin{gathered} M_{0|x}=\left(1-q_{A}\right)\Pi_{0|x}+q_{A}\Pi_{1|x}\,,\nonumber\\ M_{0|y}=\left(1-q_{B}\right)\Pi_{0|y}+q_{B}\Pi_{1|y}\,.\end{gathered}$$ Even if this is not the optimal choice it will still provide an upper bound on $\chi$. Setting $p_A=0.0059$ and $p_B=0.0031$ (the values reported in the Supplementary Information for [@Ansmann]), the largest eigenvalue (up to a sign) out of the set of eigenvalues of the 16 matrices $\Pi_{ab|xy}-M_{a|x}\otimes M_{b|y}$, after minimization with respect to $q_A$ and $q_B $, equals $\simeq0.0030$ and is obtained when $q_A\simeq 0.0001$ and $q_B\simeq 0.0029$. [32]{} D. Mayers and A. Yao, Quantum Inf. Comput. **4**, 273 (2004). J. Barrett, L. Hardy, and A. Kent, Phys. Rev. Lett. **95**, 010503 (2005). A. Acín et al., Phys. Rev. Lett. **98**, 230501 (2007); S. Pironio et al., New J. Phys. **11**, 045021 (2009). M. McKague, New J. Phys. **11**, 103037 (2009). Ll. Masanes, S. Pironio, and A. Acín, Nat. Commun. **2**, 238 (2011). B.W. Reichardt, F. Unger, and U. Vazirani, arXiv:1209.0448 \[quant-ph\]. U. Vazirani, T. Vidick, arXiv:1210.1810 \[quant-ph\]. J. Silman et al., Phys. Rev. Lett. **106**, 220501 (2011). C.-E. Bardyn et al., Phys. Rev. A **80**, 062327 (2009). M. McKague, T.H. Yang, and V. Scarani, arXiv:1203.2976 [\[]{}quant-ph[\]]{}. J.-D. Bancal et al., Phys. Rev. Lett. 106, 250404 (2011). F. Magniez et al., in Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (Springer, 2006), p. 72. M. McKague and M. Mosca, in Proceedings of the 5th Conference on Theory of Quantum Computation, Communication, and Cryptography (Springer, 2011), p. 113. R. Colbeck, PhD dissertation, Univ. Cambridge (2007), arXiv:0911.3814 [\[]{}quant-ph[\]]{}; R. Colbeck and A. Kent, J. Phys. A 44, 095305 (2011). S. Pironio et al., Nature 464, 1021 (2010). S. Pironio and S. Massar, arXiv:1111.6056 [\[]{}quant-ph[\]]{}. S. Fehr, R. Gelles, and C. Schaffner, arXiv:1111.6052 [\[]{}quant-ph[\]]{}. U. Vazirani and T. Vidick, Phil. Trans. R. Soc. A **370**, 3432 (2012). Y. Zhao et al., Phys. Rev. A **78**, 042333 (2008). F. Xu, B. Qi, and H.-K. Lo, New J. Phys. **12**, 113026 (2010). L. Lydersen et al., Nature Photonics **4**, 686 (2010). M. Ansmann et al., Nature **461**, 504 (2009). M.A. Rowe et al., Nature 409, 791 (2001). T. Monz et al., Phys. Rev. Lett. **106**, 130506 (2011). J.M. Martinis, Quantum Inf. Process. **8**, 81 (2009). H. Häffner, C.F. Roosa, and R. Blatt, Phys. Rep. **469**, 155 (2008). J.F. Clauser et al., Phys. Rev. Lett. **23**, 880 (1969). M. Navascués, S. Pironio, and A. Acín, Phys. Rev. Lett. **98**, 010401 (2007); M. Navascués, S. Pironio, and A. Acín, New. J. Phys. **10**, 073013 (2008). S. Pironio, M. Navascués, and A. Acín, SIAM J. Optim. **20**, 2157 (2010). J. Barrett et al., Phys. Rev. A **71**, 022101 (2005). B.S. Cirel’son, Lett. Math. Phys. **4**, 93 (1980). S. Popescu and D. Rohrlich, Found. Phys. **24**, 379 (1994). C.H. Bennett, et al., IEEE Trans. Inf. Theory **49**, 1895 (2003). A.G. Kofman and A.N. Korotkov, Phys. Rev. B **77**, 104502 (2008). J. Löfberg, YALMIP: A Toolbox for Modeling and Optimization in MATLAB. Available at http://users.isy.liu.se/johanl/yalmip. J.F. Sturm and I. Pólik, SeDuMi: a package for conic optimization. Available at http://sedumi.ie.lehigh.edu.

--- abstract: 'A sufficiently connected topology linking the constituent units of a complex system is usually seen as a prerequisite for the emergence of collective phenomena such as synchronization. We present a random network of heterogeneous phase oscillators in which the links mediating the interactions are constantly rearranged with a characteristic timescale and, possibly, an extremely low instantaneous connectivity. We show that, provided strong coupling and fast enough rewiring are considered, the network is able to reach partial synchronization even in the vanishing connectivity limit. We also provide an intuitive analytical argument, based on the comparison between the different characteristic timescales of our system in the low connectivity regime, which is able to predict the transition to synchronization threshold with satisfactory precision. In the formal fast switching limit, finally, we argue that the onset of collective synchronization is captured by the time-averaged connectivity network. Our results may be relevant to qualitatively describe the emergence of consensus in social communities with time-varying interactions and to study the onset of collective behavior in engineered systems of mobile units with limited wireless capabilities.' author: - Marco Faggian - Francesco Ginelli - Fernando Rosas - Zoran Levnajić title: 'Synchronization in time-varying random networks with vanishing connectivity' --- Introduction ============ The emergence of collective phenomena in complex systems is related to the interplay between interaction topology and local dynamics [@VespignaniBook; @newman; @costa; @barabasi]. Stationary connections can lead to coherent dynamical patterns, typically studied in the framework of network theory, with the local dynamics taking places on individual nodes and interactions modelled as links. In the context of complex networks, conditions of minimal connectivity are know for enabling the emergence of collective dynamics [@VespignaniBook; @mason]. A prominent example is [*synchronization*]{} in networks of oscillators [@arkadybook; @Arenas2008; @arkadyandme; @arkadymisha], where the connectivity thresholds for a wide range of different network topologies have been determined in great detail [@Rodrigues2016; @acebron]. However, many complex systems, and particularly social and engineered ones, may not maintain a constant connectivity, but rather yield a topology defined by a time-dependent connectivity matrix $\mathcal{A}_{ij}^t$. Examples range from animal groups [@Ermentrout91; @Ballerini08] and time-dependent plasticity in neural networks [@Markram97; @Maistrenko07] to robot swarms [@Pini2011], human social networks [@Sekara16] and communication networks of moving units [@Hua2009]. Synchronization in time-varying networks received considerable attention in the control and nonlinear dynamics literature [@Hasler04; @Stilwell06; @Amritkar06; @Li08; @Lucas18]. Yet, these efforts almost exclusively concentrated on systems composed of homogeneous units, largely relying on standard linear stability analysis. Introducing quenched disorder, that is, considering systems composed by many heterogeneous oscillators is however more challenging, especially for a finite number of units, as the stability of the [*partially synchronized state*]{} cannot be typically treated by simple linear stability analysis [@Strogatz2000]. In this situation, averaging theorems [@AveragingT] may not be trivially applicable, so that different approaches may be needed. A first step towards the study of time-varying networks of heterogeneous oscillators is provided by Ref. [@So08], which focused on two populations of oscillators switching between two fixed topologies at a given frequency. Interestingly, analysis of this “blinking” network revealed that high-frequency switching may induce synchronization, even when the two individual topologies can only sustain an incoherent phase. While these findings provide a first hint that results for time-varying networks of homogeneous units can be extended to the heterogeneous case, here we wish to take a step further and study a time-varying network of heterogeneous oscillators, where individual nodes interact randomly (and possibly quite seldomly) in both time and oscillator space In particular, here we ask under which conditions macroscopic synchronization may emerge in Erdös-Rény networks with random rewiring and arbitrarily small instantaneous connectivity. We thus consider $N$ heterogeneous agents interacting randomly, with a bidirectional and typically [*sparse*]{} connectivity matrix in a regime of strong coupling. We will see that our system is characterized by three different timescales: interacting agents quickly converge towards a common state on a short [*local syncronization*]{} timescale $\tau_{LS}$, while each agent may randomly rewire all his connections with a typical [*rewiring*]{} timescale $T$. When two connected agents are separated, their internal states diverge, with yet another [*local de-sinchronization*]{} timescale $\tau_{LD}$ which depends on their heterogeneity and it is typically larger than $\tau_{LS}$. One can interpret this setup as a crude model of social interactions, where individuals interact in time with different subsets of their common social network. When interacting, and despite their intrinsic differences, they tend to quickly converge towards a common opinion, but when separated, their differences take over again and their opinion diverges. Model parameters allow to control the separation between these characteristic times, which enables a detailed study of the emergence of synchronization in relation to the interplay between different timescales. In the following, we show that – provided the links are rewired frequently enough – dynamics can permanently achieve a partially synchronized state, even when the instantaneous connectivity is far smaller than what is needed to synchronize stationary networks. In particular, via numerical simulations and approximate analytical arguments of a concrete model, we show that for a sufficiently strong coupling and a sufficiently fast rewiring, our system reaches and maintains a macroscopic (partially) synchronized state, even in the limit of vanishing connectivity: it is the high frequency blinking of links that prevents the system from relaxing into an incoherent state as would happen with stationary topologies. This paper is organized as follows: in Section \[sec2\] we define a precise model for our time-varying network and sketch its synchronization phase diagram through direct numerical simulations. In Section \[sec3\] we focus on the low connectivity regime. Analysing the characteristic timescales of the system and invoking an averaging theorem in the limit $T\to0$, we provide an approximate expression for the synchronization threshold which compares favorably with numerical estimates. In Section \[sec4\] we first discuss higher connectivities, where the instantaneous network topology is characterized by a system spanning giant component, and then argue that the transition to synchronization belongs to the standard Kuramoto class in the entire phase diagram. Conclusions are finally drawn in Section \[sec5\]. Kuramoto model on time-varying networks {#sec2} ======================================= Model definition ---------------- We first introduce our model. Let us consider a network of $N$ Kuramoto oscillators, where the state of the $i$-th node is represented by a phase variable $\varphi_i \in [0,2\pi]$. Each oscillator is characterized by a quenched natural frequency $\omega_i$, drawn from a zero-mean Gaussian distribution with standard deviation $\sigma$. Oscillators interact with each others according to a time-varying adjacency matrix $\mathcal{A}_{ij}^t$, with $m_i^t=\sum_j \mathcal{A}_{ij}^t$ being the instantaneous degree of node $i$. For simplicity, we chose the adjacency matrix to be symmetric and with binary values $\mathcal{A}_{ij}^t=0,1$, leaving other cases for future studies. Hence, the dynamics of the oscillators obey the following equation: $$\dot{\varphi}_i = \omega_i + \frac{\varepsilon}{m_i^t} \sum_{j} \mathcal{A}_{ij}^t(T)\sin(\varphi_j - \varphi_i) \; , \label{maineq}$$ where epsilon quantifies the strength of the coupling [@NOTE1]. Obviously, when no edges at all insist on node $i$ we have simply $\dot{{\varphi}}_i = {\omega}_i$. The dynamics of $\mathcal{A}_{ij}^t(T)$ is determined as follows. At each moment, the adjacency matrix corresponds to a random, or Erdös-Rény (ER) network, defined by the vertex number $N$ and the linking probability $p$ [@VespignaniBook]. The random rewiring of edges is then modelled as a Poissonian process, with each individual node rewiring synchronously all its incident edges with probability rate $1/T$, with $T$ being the typical [*rewiring time*]{}. In the instantaneous rewiring process of vertex $i$, all edges incident on $i$ are first deleted; all the potential links of vertex $i$ are then considered, and new edges $i-j$ are created with probability $p$. It is well known that the topological properties of ER networks are essentially determined by the mean degree connectivity $\langle m \rangle = (N-1) p$, so that in the following we find convenient to define $q=p\,N \approx \langle m \rangle $ and adopt $q \approx \langle m \rangle$ as the relevant connectivity parameter. Note finally that Eq. (\[maineq\]) is invariant under the following rescaling: $$\begin{cases} t'=\alpha t\\ \sigma'=\frac{\sigma}{\alpha}\\ \varepsilon'=\frac{\varepsilon}{\alpha}\\ \end{cases} \label{scaling}$$ (with $\alpha\in\mathbb{R}^+$), provided also the rewiring time is rescaled accordingly, $T'=\alpha T$. Due to this invariance, it is easy to show that the dynamics is actually controlled by the two dimensionless quantities $T/\sigma$ and $\varepsilon/\sigma$ and by the connectivity $q$. One can interpret this setup as a crude model for several natural/social phenomena. Consider for example social interactions, where individuals interact in time with different subsets of their common social network, according to a certain frequency of personal encounters/interactions. When interacting, despite intrinsic differences of their opinions (i.e. different quenched natural frequencies), individuals tend to quickly converge towards a common opinion. However, when separated, their differences take over again, and their opinions slowly diverge. The spectrum of natural frequencies $\omega_i$ can thus represent the range of “unperturbed” opinions of a population, while in Eq. (\[maineq\]) interactions with other persons (nodes) leads to the effective frequencies $\omega^{eff}_i (t) = \dot{\varphi}_i$, representing the actual opinion of the agents. It is well known that the Kuramoto model with stationary network connectivity, either globally connected [@Strogatz2000] or with other sufficiently connected topologies [@Rodrigues2016], displays a synchronized solution for large enough couplings ${\epsilon}$. In this synchronized state, a macroscopic fraction of oscillators share a common effective frequency, reaching macroscopic consensus in our point of view. The degree of synchronization can be evaluated through the standard instantaneous Kuramoto order parameter $$R(t) = \left| \frac{1}{N} \sum_{k=1}^N e^{i{\varphi}_k (t)} \right| \; ,$$ which is finite for synchronized states and tends to zero as $1/\sqrt{N}$ in the absence of macroscopic synchronization. In the following, we will typically consider its average over time and disorder (i.e. different natural frequencies realization), $\Delta = \braket{R}_{t\,,\omega}$, and make use of its different finite size scaling behavior to better estimate the transition between (partial) synchronization and disorder. Direct numerical simulations {#sims} ---------------------------- ![Lin-log plot of the order parameter $R$ as function of time for a network with $N=10^4$ and $q=0.8$, with $\varepsilon=8$ and $\sigma=1$. The three curves correspond to three different values of switching time $T$: green $T=6.28$, red $T=0.63$, and black $T=0.31$.[]{data-label="figure1"}](q08.eps){width="0.9\linewidth"} ![ (a) Stationary values of the order parameter $\Delta$ (color-coded according to the right vertical bar) as function of the rewiring frequency ($T$) and network connectivity ($q$). Simulations have been performed for a network of $N=10^4$ Kuramoto oscillators with $\varepsilon=8$ and $\sigma=1$. Values have been averaged over $\Omega=10$ different realizations. For small values of $q$ the system is strongly dependent on the value of the rewiring time $T$ and the phase diagram shows a clear transition from partial synchronization to disorder as $T$ is increased beyond a critical value $T_c(q)$. At larger $q$ values, the transition approaches a vertical asymptote, roughly located at $q=\bar{q}=1.66(6)$ (dashed black line). For $q>\bar{q}$ the dynamics achieves partial synchronisation regardless of the value of $T$. (b) Zoomed view of panel (a) in the range $q\in [0.5, 1.7]$.[]{data-label="figure2"}](ColorPlot.png){width="0.99\linewidth"} In this work, numerical simulation are performed using a standard 4th order Runge-Kutta integrator of step $dt$. After each Runge-Kutta time-step, each vertex may undergo a rewiring event (as defined above) with Poissonian probability $$r=1-\exp(-dt/T) \;.$$ We use a time-step of at most $dt=10^{-2}$. When investigating fast network dynamics however, we are forced to adopt time-steps smaller than the network rearrangement timescale $T$, that is $dt \approx T/10$. In order to illustrate the behavior of our time-varying network dynamics in a strong coupling regime, ${\epsilon}=8$, $\sigma=1$, we begin presenting numerical simulations of the dynamics (\[maineq\]) for a network of $N=10^4$ elements and a mean connectivity $q=0.8$. As it is shown in Fig.\[figure1\], no synchronization emerges when the rewiring is sufficiently slow ($T \approx 6.3$ in this example). As the rewiring time is lowered past a synchronization threshold, we observe macroscopic synchronization with an increasing order parameter $R(t)$. This shows that sufficiently fast network rewiring can overcome the effects of low network connectivity, inducing partial synchronization on the network. We next want to characterize with more details the parameter space $(q,T)$. We do that by repeating the above computation for a grid (lattice) of different values of $q$ and $T$. For each of these values we calculate $\Delta = \braket{R}_{t\,,\omega}$ by averaging over 10 random realizations of the quenched natural frequencies $\omega_i$ and different random initial phases. Time averages are performed over the stationary part of $R(t)$, after a proper initial transient has been discarded. The results are shown via colorplot in Fig.\[figure2\]a (the lighter the color, the larger the value of $\Delta$). We first observe that for a sufficiently large connectivity, $q>\bar{q}$, the system always reaches macroscopic synchronization, regardless of the rewiring time $T$. Analysis of the averaged order parameter $\Delta$ in the large $T$ limit, as reported in more details in Section \[largeC\], suggests $\bar{q} = 1.66(6)$. Here, we just wish to point out that $\bar{q}$ is clearly larger than $q=1$, the threshold for the emergence of a giant connected component in ER graphs [@giantcomponent]. In this regime, synchronization is indeed to be expected in the strong coupling limit, due to sufficient interactions among the oscillators. For smaller values of $q$, on the other hand, where no large components characterize the instantaneous network topologies, sufficiently fast rewiring is needed to achieve synchronization, at least for $q>0.5$. A transition line $T_c(q)$ separating partial synchrony from incoherence (i.e. the violet border between the dark and the bright zone) in the plane $(q, T)$ can be roughly identified from this colour plot. Indeed, a closer look at the phase diagram, as reported in Fig.\[figure2\]b, suggests that the transition line $T_c(q)$ separating partial synchrony from incoherence in the plane $(q, T)$ is initially characterized by a linear behaviour. For larger connectivity values, on the other hand, $T_c(q)$ grows faster than linear, finally diverging as a vertical asymptote is approached at $q=\bar{q}$. Note however that the transition line is characterized by a non zero intercept at $q=q_0\approx 0.5$ with the $T=0$ axis. Thus, for smaller connectivity values ($q \lessapprox 0.5$), no synchronization is possible for the coupling ${\epsilon}/\sigma = 8$, no matter how fast is the rewiring. In the following section we will proceed to better characterize the transition to synchrony in the low average connectivity region $q<1$ by means of approximate analytical arguments and detailed numerical simulations. Synchronization for low and vanishing connectivity {#sec3} ================================================== Characteristic time scales and the onset of synchronization {#scales} ----------------------------------------------------------- We next seek to understand the physical mechanism leading to synchronisation in the low connectivity region via switching. For this we need to grasp the three characteristic timescales governing information flow and the dynamics of our system. The first time scale is the *local synchronisation* time $\tau_{LS}$, related to the synchronisation of a connected pair of oscillators. The second is the the *local desynchronisation* time $\tau_{LD}$, related to the typical desynchronisation time as the link between two synchronized oscillators is severed. The third one, finally, is the *effective rewiring* time $\tau_{ER}$, describing the typical time needed for an oscillator to establish a new link after a rewiring event. We focus on the limit in which $\tau_{LS}$ is much smaller than both $\tau_{LD}$ and $\tau_{ER}$. In this regime, oscillators couples quickly synchronize when connected by a link, starting to loose their relative synchrony when their mutual link is deleted in a rewiring event. In practice, oscillators tend to loose the information gained when linked with the characteristic timescale $\tau_{LD}$. Two possibilities are then in order for low connectivity. Either a new link is forged by one of these two oscillators with a third node in a time shorter than $\tau_{LD}$, propagating the information it carries from its previous local synchronization to a new node, or no link at all is established before this information is completely lost. We argue that [*global*]{} synchronization will take place when, on average, the information gained by local synchronization events does not get lost but is rather able to propagate through the entire network. This will happen when $\tau_{LD} \lessapprox \tau_{ER}$. On the other hand, when $\tau_{ER} \lessapprox \tau_{LD}$, no information can propagate through the network, and macroscopic synchronization cannot take place. The transition from the desynchronized to the synchronized regimes will thus take place when $$\tau_{LD} \approx \tau_{ER} \,. \label{synch_cond}$$ Note that a similar argument, based on the characteristic timescales of information transfer, has been previously successfully applied to estimate the transition line separating disordered from collective motion in the well known Vicsek model for flocking [@Ginelli2008; @Ginelli2016; @Grygera2018]. We now proceed to estimate the three timescales introduced above. First consider the local synchronization scale $\tau_{LS}$, that is, the time needed by two oscillators $i$ and $j$ sharing a non-directed link to synchronize their effective frequencies. In the low connectivity approximation one can assume for a couple of oscillators $m^t_i=m^t_j=1$ , i.e. that they are only connected one to each other. Hence, from Eq. (\[maineq\]), one immediately gets for their mutual phase difference $\delta \varphi=\varphi_i-\varphi_j$ the dynamics $$\delta\dot{\varphi} = \delta\omega -2 \varepsilon \sin \delta\varphi \; ,$$ where $\delta\omega=\omega_i - \omega_j$ is the difference between their natural frequencies. In the strong coupling regime we are interested into, $\varepsilon \gg \sigma $ and one readily sees that the phase difference converges exponentially fast towards the asymptotic solution $\delta\varphi = \delta\omega /(2 \varepsilon)$ while the two oscillators effective frequencies synchronize with a time scale $\tau_{LS} \approx (2 \varepsilon)^{-1}$. In the following we first assume $\tau_{LS} \ll T$, that is, once a link is established oscillators typically synchronize before being rewired. ![Finite size determination of the transition point $T_c$ for $q=0.8$, ${\epsilon}=8$ and $\sigma=1$. The average order parameter $\Delta(N)$ is evaluated for two different system sizes, respectively $N_1=1000$ (black dots) and $N_2=2000$ (red dots). $T_c$ is estimated as the midpoint between the largest value of $T$ such that the values of $\Delta(N_1)$ and $\Delta(N_2)$ overlap, and the smallest value of $T$ such that the scaling $\Delta(N_1)/ \Delta(N_2) \approx \sqrt{2}$ is satisfied. To facilitate the comparison, the black dashed line marks the value $\Delta(N_1)/\sqrt{2}$. Vertical dashed lines mark the estimated transition point (red) and its confidence interval (green). Error bars report the standard error for the average computed over $\Omega=20$ independent realizations. []{data-label="figure4"}](diffsize.eps){width="0.85\linewidth"} Once the link is removed in a rewiring event, nodes can be left without any link, so that the phase of previously connected and synchronized oscillators will start to drift away one from each other due to their natural frequencies difference $\delta \omega$, loosing any information regarding their previous mutual synchronization when their phase difference approaches $\pi /2$. This allows one to define the typical local desynchronization timescale $\tau_{LD}$ such that $$\tau_D \langle \delta \omega \rangle \approx \frac{\pi}{2}$$ with being the average natural frequency difference. For Gaussian distributed natural frequency one of course has $$\langle \delta \omega \rangle =\sqrt{\int_{-\infty}^{\infty} d\omega_1d\omega_2\, P_\sigma(\omega_1)P_\sigma(\omega_2)(\omega_1-\omega_2)^2}=\sqrt{2}\sigma \;$$ which finally yields $$\tau_{LD} \approx \frac{\pi}{2\sqrt{2}\,\sigma}$$ Before proceeding further, one comment is in order about our estimate of the typical local desynchronization timescale. We have computed it as the time required by a typical pair of oscillators to desynchronize. This is of course different from the average of individual couples desynchronization times $ \langle \pi /(2\delta\omega ) \rangle$, which is dominated by oscillators couples with almost degenerate natural frequencies, $\delta \omega \approx 0$. These latters, however, characterized by a very large local desinchronization time, are far from being representative of the typical behavior of random oscillators couples. ![Critical rewiring time $T_c$ as function of $q$ for $q<1$ for $\sigma=1$ and different values of the coupling constant (increasing along the cyan arrow). Respectively, from left to right: $\varepsilon=32$ (blue circles ), $\varepsilon=16$ (red circles) and $\varepsilon=8$ (black circles). Error bars give the estimated upper and lower boundaries for $T_c(q)$ as discussed in the main text. The dashed straight lines (same color coding) mark the linear prediction of Eq. \[eq\_final\] (see Section \[ave\_net\]). (Inset): The slope $s$ of each ${\epsilon}$ curve, evaluated by linear regression of the main panel data, is compared with the theoretical estimate $s= \pi / (2 \sqrt{2})$ (see Eq. (\[mf\])). Data has been averaged over $\Omega=20$ different realizations and error bars measure one standard error.[]{data-label="figure3"}](Fig3.eps){width="0.95\linewidth"} We finally estimate the effective rewiring timescale $\tau_{ER}$. The probability for an oscillator to be linked to at least one other oscillator is equal to: $$P_\text{link}=1-P_\text{not\ link}\, \label{plink}$$ where $P_\text{not\ link}$ is the probability of not having any link at all, that is $$P_\text{not\ link}=\left(1-\frac{q}{N}\right)^{N-1} \xrightarrow[]{N\to\infty} e^{-q}$$ Substituting back into Eq. (\[plink\]) we get $$P_\text{link} \approx 1-e^{-q} \approx q\;\;\;\mbox{for}\;\; q \ll 1\,,$$ with lowest order corrections of order $q^2$ and $q/N$. We thus evaluate the effective rewiring time in the low connectivity limit as $$\tau_{ER} = \frac{T}{P_\text{link}} \approx \frac{T}{q}\,.$$ Summing up, the synchronization condition (\[synch\_cond\]) yields a linear relation between the rewiring time $T$ and the connectivity $q$, yielding the synchronization line $$T_c (q)\approx \frac{\pi}{2\sqrt{2}\,\sigma}\,q \label{mf}$$ We now compare our predictions with numerical simulations. We determine the synchronization threshold by finite size analysis, comparing the averaged order parameter $\Delta(N)$ for system sizes $N_1=1000$ and $N_2=2000$. In the presence of macroscopic synchronization one expects $\Delta(N_1) \approx \Delta(N_2)$, while in the disordered phase we have $$\frac{\Delta(N_1)}{\Delta(N_2)}=\sqrt{\frac{N_2}{N_1}}=\sqrt{2}$$ An example of our procedure is given in Fig. \[figure4\] for $q=0.8$, where we have estimated $T_c=0.53(6)$. Numerical estimates of the synchronization threshold are reported in Fig. \[figure3\] for $\sigma=1$ and different values of the coupling constant $\varepsilon$. They confirm the linear relation between $T_c$ and $q$ in the low connectivity regime, predicting the actual slope $s=\pi/(2\sqrt{2}\, \sigma )$ within numerical accuracy (see inset). However, it is clear that for finite values of the coupling, it is always possible to find sufficiently small values of $q$ such that synchronization cannot be achieved, no matter how small is $T$. Said differently, the critical line $T_c$ has a non-zero intercept $q_0 ({\epsilon})$ with the $T=0$ axis. Interestingly, the value of $q_0({\epsilon})$ of the intercept decreases towards zero as $\varepsilon$ increases, suggesting that Eq. (\[mf\]) can be fully recovered as ${\epsilon}\to \infty$. This is equivalent to the strong coupling limit under which we have derived Eq. (\[mf\]): By taking the limit ${\epsilon}\to \infty$ first, in fact, we assure that the condition $\tau_{LS} = (2 {\epsilon})^{-1} \ll T$ is verified for any non-zero rewiring time $T$. On the other hand, numerical simulations with a finite coupling constant ${\epsilon}$ show that one can always find a sufficiently low connectivity $q$ such that $T_c(q) \lessapprox \tau_{LS} = (2 {\epsilon})^{-1} $ and our approximation breaks down. In the next section, we will attempt to better understand this regime and the behavior of the intercept $q_0 ({\epsilon})$ by means of averaging considerations. Average network for very fast rewiring {#ave_net} -------------------------------------- We now consider the limit of extremely fast rewiring, where $T$ and $\tau_{ER} \approx T/q$ are much smaller than the local synchronization and desynchronization times. In this regime, one expects the instantaneous order parameter to be approximately constant over a timescales $\tau_{av} \lessapprox \mbox{min}(\tau_{LS},\tau_{LD})$, so that $$R(t) \approx \frac{1}{\tau_{av}} \int_t^{t+\tau_{av}} R(t+t')\, dt' \,.$$ Following the argument of Ref. [@So08], we may invoke a well known result from Ott and Antonsen [@Ott-Antonsen] to argue that the low dimensional dynamics of the Kuramoto order parameter is essentially controlled by the time-averaged interaction matrix $$\left\langle \frac{\mathcal{A}_{ij}^t(T)}{m_i^t} \right\rangle \equiv \frac{1}{\tau_{av}} \int_0^{\tau_{av}}\frac{\mathcal{A}_{ij}^t(T)}{m_i^t} \,dt \,. \label{ave}$$ This result, stating that for $T \to 0$ the dynamics of Eq. (\[maineq\]) is the same as the one of the time-average network with stationary connectivity, can be essentially seen as a form of the averaging theorem [@AveragingT]. While the latter typically involves periodic systems, a recent extension to non-periodic systems has been discussed, for instance, in Ref. [@Duccio]. In the limit $T \to 0$ the average in Eq. (\[ave\]) is computed over arbitrarly many rewiring events and we have $$\frac{A_{ij}}{N} \equiv \lim_{T \to 0} \int_0^{\tau_{av}}\frac{\mathcal{A}_{ij}^t(T)}{m_i^t} \,dt = \sum_k^{N-1} \frac{a_{ij} (k)}{k}\,, \label{avA}$$ where $$a_{ij} (k) = p^k (1 - p)^{N-1-k} \frac{(N-2)!}{(k-1)!(N-k-1)!} \label{avB}$$ is the probability that node $i$ has an active link with node $j$ and exactly $k-1$ other links. Note that the binomial factor $$\binom{N-2}{k-1} \equiv \frac{(N-2)!}{(k-1)!(N-k-1)!}$$ accounts for all the different configurations in which the $k-1$ active link can be chosen out of $N-2$ potential ones after the one between $i$ and $j$ has been activated. By recalling that $p=q/N$, using Eqs. (\[avA\])-(\[avB\]) one can find $$\begin{aligned} \frac{A_{ij}}{N} &=& \sum_k^{N-1} \frac{q^k}{k\,(k-1)!} \frac{1}{N^k} \frac{(N-2)!}{(N-k-1)!} \left(1 - \frac{q}{N}\right)^{N-k-1}\nonumber\\ &=&\left(1 - \frac{q}{N}\right)^{N}\frac{1}{N}\sum_k^{N-1} \frac{q^k}{k!} \,g(N,k) \,,\end{aligned}$$ where $$g(N,k)=\frac{1}{N^{k-1}}\frac{(N-2)!}{(N-k-1)!} \left(1 - \frac{q}{N}\right)^{-(k+1)}\,.$$ In the limit $N \gg 1$ we have $$g(N,k) = 1 + O\left(\frac{1}{N}\right)$$ and therefore, to lowest order in $1/N$, $$\frac{A_{ij}}{N}\approx \frac{e^{-q} }{N}\sum_k^{\infty} \frac{q^k}{k!} \approx \frac{e^{-q}}{N} \left( e^q -1\right) = \frac{1 - e^{-q}}{N}$$ so that the average network is characterized by a globally connected topology. Therefore, under our conjecture, in large networks the fast rewiring ($T\to 0$) dynamics (\[maineq\]) can be replaced by the averaged one $$\frac{\partial {\varphi}_i}{\partial t} = {\omega}_i + \frac{\varepsilon\left(1 - e^{-q}\right)}{N} \sum_{j=1}^N \sin({\varphi}_j - {\varphi}_i) \; , \label{aveq}$$ that is, a globally coupled Kuramoto model with coupling constant $$J={\epsilon}\left(1 - e^{-q}\right) \,,$$ which exhibits macroscopic synchronization for $J>J_c$, with the critical point $J_c$ depending on the natural frequency distribution. For $T\to 0$, therefore, macroscopic synchronization can only be achieved provided ${\epsilon}>J_c$ and for connectivities $q>q_0$ with $$q_0 = \ln \left(\frac{{\epsilon}}{{\epsilon}- J_c}\right) \,. \label{q0}$$ We conclude that, according to Eq. (\[q0\]), in the strong coupling limit ${\epsilon}/\sigma \to \infty$ synchronization can be achieved for arbitrarily small connectivity $q$. Furthermore, we can interpret $q_0$ as the intercept of the transition line $T_c(q_0)$ with the $T=0$ axis. In particular, for a Gaussian distribution of natural frequencies with unit standard deviation we have $J_c = \sqrt{8/\pi}$ [@Strogatz2000], which allows us to to compare Eq. (\[q0\]) with the intercept values obtained by extrapolating the best linear fit for the transition lines of Fig. \[figure3\]. Direct comparison (see Fig. \[figure5\]) shows excellent agreement in the coupling range ${\epsilon}\in [8,32]$ we have probed. ![Numerically estimated transition line intercepts $q_0$ as a function of the coupling constant ${\epsilon}$ (black dots) are compared with the analytical prediction given by Eq. (\[q0\]) (dashed red line). Data as in Fig. \[figure3\]. Error bars represents error in the linear extrapolation process (see main text).[]{data-label="figure5"}](eq1.eps){width="0.95\linewidth"} We can now correct Eq. (\[mf\]) by adding a constant term such that $T_c(q_0) = 0$, thus obtaining $$T_c (q)\approx \frac{\pi}{2\sqrt{2}\,\sigma}\,(q - q_0) \label{eq_final}$$ with $$q_0 = \ln \left(\frac{{\epsilon}}{{\epsilon}- \sqrt{8/\pi}}\right) \,, \label{q00}$$ for Gaussian distributed natural frequencies. This is exactly the linear formula we plotted in Fig. \[figure3\] for unit variance ($\sigma=1$), showing good comparison with the numerical transition values $T_c(q,{\epsilon})$ in the small $T$ regime. As $q$ (and thus $T_c$) grows larger, however, deviations from the linear behavior are clearly visible. Indeed, as $q$ is increased, the critical line $T_c(q)$ bends upwards to meet the vertical asymptote at $\bar{q}$. In this regime, contributions from nodes with more than one link at the time becomes relevant, and the simple arguments leading to the linear relation (\[eq\_final\]) are expected to break down. Behavior at finite connectivity {#sec4} =============================== For completeness, in this section we briefly discuss the synchronization transition at finite connectivity $q$. Behaviour for large connectivity {#largeC} -------------------------------- We have already seen that, in order to synchronize for arbitrarily large rewiring times, the connectivity $q$ should be larger than a threshold $\bar{q} > 1$, so that the typical emergences of connected components of macroscopic size, taking place for $q>1$, is not sufficient for the onset of synchronization. In particular, we have seen that for ${\epsilon}/\sigma=8$ we have $\bar{q} = 1.66(6)$. We now show numerical evidence that in the large coupling limit, ${\epsilon}/\sigma \to \infty$, we have $\bar{q} = 1^+$, that is the onset of synchronization do coincide with the emergence of giant connected components in the graph topology. ![Stationary values $\Delta$ of the system as function of $q$ for $T=20\pi/\sigma$. Top: Behavior of $\Delta$ as a function of $q$ for two different sizes ($N_1=10^3$, empty circles, and $N_2=2\times 10^3$, full squares) and different values of ${\epsilon}/\sigma$. From the left to the right: $\sigma^2=1/8$ (black), $\sigma^2=1/4$ (red), $\sigma^2=1/2$ (green), $\sigma^2=1$ (blue) and $\sigma^2=2$ (brown). Bottom: Ratios $\Delta(N_1)/\Delta(N_2)$ as a function of $q$. The two horizontal dotted lines mark the ratios $\sqrt{2}$ (disordered phase) and 1 (synchronized phase). The colours coding for the variance is the same as in the top panel. The vertical dashed line $q=1$ marks the emergence of giant connected components. Data has been averaged over $\Omega=10$ different realizations.[]{data-label="FigSpread"}](Spread.eps){width="0.95\linewidth"} Next we show that the phase transition thresholds depend on the distribution of frequencies $\omega$ when the coupling $\varepsilon$ is fixed, which demonstrates that the critical point is not dependent on the topology (and the percolation threshold for the giant connected component), but only on the dynamics. In the following, we analyze numerically the synchronization transition at in the region $q \approx 1$ through the finite size analysis of the averaged parameter $\Delta$ at large rewiring times $T$ as the ratio ${\epsilon}/\sigma$ is progressively increased. Fixing ${\epsilon}$, we increase $\sigma$ between $1/\sqrt{8}$ and $\sqrt{2}$. In order to evaluate $\Delta$ at large enough rewiring times, in agreement with the scaling relation (\[scaling\]) we fix $T$ such that the dimensionless parameter $T \sigma = 20 \pi$ [@NOTE2], and compare the order parameter at two different system sizes $N_1 < N_2$. As already remarked in Sec. \[scales\], we can distinguish the synchronized from the disordered phase by the ratio $\Delta(N_1)/\Delta(N_2)$. This is, for instance, how in Sec. \[sims\] we have estimated $\bar{q}=1.66(6)$ for ${\epsilon}/\sigma=8$ from the data of Fig. \[FigSpread\] (blue symbols). More in general, numerical simulations, reported in Fig. \[FigSpread\], clearly indicates that, as $\sigma$ is lowered and the strong coupling regime is approached, the synchronization threshold approaches the onset of giant connected components, i.e. $\bar{q} \to 1$. These results indicate that for finite couplings, in the regime $1<q<\bar{q}$, giant connected components may be unable to synchronize when large enough rewiring times $T$ are considered. This effect is indeed due to the interaction between the giant component topology and the quenched disorder. For $q \gtrapprox 1$ the giant component should be characterized by a large number of bridges (i.e. links whose deletion would split the giant component in two disconnected parts). When these bridges insist on nodes characterized by extreme natural frequencies (i.e. lying in the tail of the distribution $P(\omega)$) which do escape partial synchronization, they act as effective obstacles to information spreading, splitting the topologically connected giant component into different synchronized subcomponents which are, however, not mutually synchronized. This mechanism, which clearly prevent macroscopic synchronization to emerge in the slow switching regime, is however going to become less and less important as the connectivity $q$ is increased and the number of bridges in the giant connected component is reduced, allowing for a more efficient information flow, eventually leading to global synchronization as $q > \bar{q}$. Fast switching, on the other hand, allows information to travel through the network by rearranging the giant cluster quickly enough, preventing instantaneous bridges from acting as effective roadblocks. Therefore, for $1<q<\bar{q}$, a transition to macroscopic synchronization is eventually observed as the rewiring time is decreased. A precise analytical estimate of $T_c(q)$ in this regime, however, is beyond the scope of this work. Critical behavior ----------------- We have finally verified that, as expected, the phase transition to synchronization belongs to the usual Kuramoto model class. In Fig. \[critp\] we report numerical simulations for both (relatively) slow and fast rewiring times $T$, showing that the average order parameter follows the usual Kuramoto model scaling, $\Delta \sim \sqrt{q -q_c(T)}$ for $q > q_c$ [@Strogatz2000], with $q_c(T)$ being the ($T$ dependent) critical connectivity parameter. Numerical results suggest this to be true for any finite $q_c$, as expected given that synchronization seems to be essentially guided by the properties of the globally connected time-averaged connectivity matrix. ![Log-log plot for the critical behavior of $\Delta\sim (q-q_c)^\beta$ in a network with $N=10^4$, ${\epsilon}=8$ and $\sigma=1$ for faster ($T=0.31$, black circles) and slower ($T=9.42$, red squares) switching times. The blue dashed curve marks the Kuramoto exponent $\beta=1/2$. Data has been averaged over $\Omega=20$ independent realization of the natural frequencies.[]{data-label="critp"}](crit_p_updated.eps){width="0.95\linewidth"} Conclusions {#sec5} =========== We discussed a time-varying network of heterogeneous Kuramoto phase oscillators characterized by links being randomly switched on and off with a Poissonian probability distribution. The network dynamics exhibits three well defined time scales associated respectively to local synchronization, local desynchronization and effective rewiring, whose separation is controlled by model parameters. Numerical simulations and analytical arguments show that this system is able to achieve statistically stable macroscopic synchronization even for arbitrarily small net connectivity (i.e., for a sparse and infrequent coupling among the oscillators), provided sufficiently fast switching and strong couplings are considered. In the formal fast switching limit, $T \to 0$, we have argued that the synchronization dynamics is fully captured by the time-averaged connectivity matrix, suggesting that results from the averaging theorem can be applied to our Kuramoto setup. At finite $T$, on the other hand, our analytical arguments, based on the comparison between the different timescales at play, are indeed able to predict with a satisfactory precision the synchronization transition line in the small connectivity regime. This switching-induced synchronization maintains the same qualitative characteristics of its static counterpart, such as the Kuramoto order parameter scaling $\Delta \sim \sqrt{q-q_c}$ at $q \gtrapprox q_c$ [@acebron]. For larger connectivity values, beyond the onset of giant connected components ($q>1$), we have finally shown that the interaction between instantaneous connectivity topology and quenched disorder may prevent the onset of synchronization for sufficiently slow rewiring times and large but finite couplings. Our findings are primarily intended as a theoretical contribution to the field of synchronization in time-varying complex networks, and in particular non-equilibrium synchronization models with alternative mechanisms giving rise to synchronization. However, we can still envisage several lines of potential applications for our results. For instance, one can think of engineered systems of heterogeneous (and possibly mobile) units [@mobileoscillators] – a simple paradigma for the “Internet of Things” [@iot] – where maintaining constant connectivity could be costly, yet the system is still required to exhibit synchronization or other collective properties. Our model could help develop alternatives to constant interactions, able to generate the same collective dynamics albeit a sparse and seldom connectivity. Also, as mentioned earlier, this setup could be seen as a crude model for social interactions. As such, our model could be used to qualitatively model the emergence of consensus in a community where different individuals are only interacting with a few of their contacts at any time. Finally, these results open several avenues of future work. Rather than rewiring links at random, one can consider a network where links are rewired preferentially to nodes with a similar instantaneous dynamical state, thus favoring interactions with “like-minded” individuals. This set-up could be used, for instance, to investigate qualitatively the “echo-chambers” phenomenon in social media, which as been recently suggested to be a possible source of an increased polarization in political opinions. To this regard, one can also wish consider different distribution of quenched frequencies, such as uniformly distributed ones, which do not favour middle natural frequencies (i.e. opinions) as the Gaussian one. Acknowledgments {#acknowledgments .unnumbered} =============== We wish to thank D. Fanelli and M. Lucas for fruitful discussions. This work has been supported by H2020-MSCAITN-2015 Project COSMOS No. 642563. ZL also acknowledges support from “Slovenian research agency” via P1-0383 and J5-8236". FR acknowledges support from H2020 MSCA grant agreement No. 702981. [99]{} A. Barrat, M. Barthélemy, A. Vespignani, [*Dynamical Processes on Complex Networks*]{} (Cambridge University Press, Cambridge, 2008). Newman M [*Networks: an Introductio*]{} (Oxford University Press, Oxford 2010). L. da Fontoura Costa [*et al.*]{}, Adv. in Phys. [**60**]{}, 329 (2011). A. L. Barabasi, [*Network Science*]{} (Cambridge University Press, 2016). M. Porter and J. Gleeson, [*Dynamical Systems on Networks*]{} (Springer Verlag, Berlin 2016). A. Pikovsky, M. Rosenblum, and J. Kurths. [*Synchronization: A universal concept in nonlinear sciences*]{} (Cambridge University Press, Cambridge, 2001). A. Arenas, A. Díaz-Guilera, J.Kurths, Y. Moreno, C. Zhou, Phys. Rep. [**469**]{} 93 (2008). Z. Levnajić, A. Pikovsky, Physical Review E [**82**]{}, 056202 (2010). A. Pikovsky, M. Rosenblum, Chaos [**25**]{}, 097616 (2015). F. A. Rodrigues, T. K. DM. Peron, P. Ji, J. Kurths, Phys Rep [**610**]{} 1 (2016). J. A. Acebrón, Rev. Mod. Phys. [**77**]{}, 137 (2005). B. Ermentrout, J. Math. Biol. [**29**]{} 571 (1991). M. Ballerini, et al., Proc. Natl. Acad. Sci. USA [**105**]{}, 1232 (2008). H. Markram, J. Lübke, M. Frotscher, B. Sakmann, Science [**275**]{} 213 (1997). Y. L. Maistrenko [*et al.*]{}, Phys. Rev. E [**75**]{} 066207 (2007). G. Pini, A. Brutschy, M.Frison, A.Roli, M. Dorigo, M.Birattari, Swarm Intelligence [**5**]{} 283 (2011). V. Sekara, A. Stopczynski, and S. Lehmann, Proc. Natl. Acad. Sci USA [**113**]{} 9977 (2016). H. Hua, S. Myers, V. Colizza, A. Vespignani, Proc. Natl. Acad. Sci USA [**106**]{} 1318 (2009). I.V. Belykh, V.N. Belykh, M. Hasler, Phys D [**195**]{} 188 (2004). D. J. Stilwell, E. M. Bollt, and D. G. Roberson SIAM J. Appl. Dyn. Syst., [**5**]{} 140 (2006). R. E. Amritkar, and Chin-Kun Hu, Chaos [**16**]{}, 015117 (2006). Z. Li, L. Jiao, J. Lee, Physica A [**387**]{} 1369 (2008). M. Lucas, D. Fanelli, T. Carletti, and J. Petit, Europhys. Lett. [**121**]{} 50008 (2018). S.H. Strogatz, Phys D. [**143**]{} 1 (2000). F. Verhulst, [*Nonlinear Differential Equations and Dynamical Systems*]{} (Springer Science & Business Media, 1990). P. So, B. C. Cotton, and E. Barreto, Chaos [**18**]{} 037114 (2008). Kuramoto, Y., 1975, [*International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics No. 30*]{} Springer, New York, p. 420. (2005) Kuramoto, Y., [*Chemical Oscillations, Waves and Turbulence*]{}, Springer, New York (1984) Note that the instantaneous dynamics itself is not fully symmetric, as two nodes $i$ and $j$ sharing the same link can still have different instantaneous degrees. S.N. Dorogovtsev and A.V. Goltsev, Review of Modern Physics, Vol.[**80**]{} 1275 (2008). H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud, Phys Rev E, [**77**]{}, 046113 (2008). F. Ginelli, Eur. Phys. J. Spec. Top. [**225**]{} 2099 (2016) M Puzzo, A. De Virgiliis, TS Grigera, arXiv:1810.02141 (2018). E. Ott, and T. Antonsen, Chaos [**18**]{}, 037113 (2008). J. Petit, B. Lauwens, D. Fanelli and T. Carletti, Phys. Rev. Lett. [**119**]{}, 148301 (2017) We have verified that larger choices for $T \sigma$ do not change qualitatively the overall picture. N. Fujiwara, J. Kurths, A. Diaz-Guilera, Phys. Rev. E [**83**]{}, 025101(R) (2011); N. Fujiwara, J. Kurths, A. Diaz-Guilera, Chaos [**26**]{}, 094824 (2016). J. Gubbi, R. Buyya, S. Marusic and M. Palaniswami, Future Gener. Comput. Syst. [**29**]{}, 1645–1660 (2013)

--- abstract: 'The early strength evolution of self-consolidating concrete (SCC) is studied by a set of non-standard mechanical tests for compressive, tensile, shear and bending failure. The results are applicable in an industrial environment for process control, e.g. of slip casting with adaptive molds in robotic fabrication. A procedure for collapsing data to a master evolution curve is presented that allows to distinguish two regimes in the evolution. In the first, the material is capable of undergoing large localized plastic deformation, as expected from thixotropic yield stress fluids. This is followed by a transition to cohesive frictional material behavior dominated by crack growth. The typical differences in tensile and compressive strength of hardened concrete are observed to originate at the transition. Finally, the evolution of a limit surface in principal stress space is constructed and discussed.' address: 'Institute for Building Materials, ETH Zurich, Stefano-Franscini-Platz 3, CH-8093 Zurich, Switzerland.' author: - 'Linus K. Mettler' - 'Falk K. Wittel' - 'Robert J. Flatt' - 'Hans J. Herrmann' title: Evolution of Strength and Failure of SCC during Early Hydration --- , Mechanical Properties (C), Hydration (A), Rheology (A), Strength (C), Concrete (E) Introduction ============ Novel robotic fabrication techniques for non-standard concrete structures rely on the interplay of control systems and the evolution of material performance during setting and early hydration. One example is an innovative process for slip-forming of concrete pillars with variable cross-section using a flexible actuated mold \[1\]. The mold is significantly smaller than the structures produced and is continuously raised or “slipped” by a robot. The morphological design space is significantly extended with respect to traditional slip casting by actively deforming the hydrating concrete as it leaves the mold. However, the trade-off between deformability and strength restricts the time window for shaping to only a few minutes. The process is constrained by two types of failure: (1) local liquefaction due to the thixotropic nature of the material, leading to collapse of the structure and (2) crack formation due to the rapid rise of the plastic yield stress with respect to the current tensile strength. The technique has so far mostly employed a polymer fiber reinforced self-consoli-dating concrete (SCC), but has also successively been tested with steel fibers and standard steel reinforcement. The process involves first the preparation of one large batch of heavily retarded SCC which is consecutively accelerated in smaller portions prior to casting \[2\]. For a review on retarders and superplasticizers as well as their impact on hydration, readers may refer to \[3-5\]. As the evolution of mechanical performance is affected by various factors (e.g. details of the mixing protocol, variations in raw materials, temperature, humidity, etc.), continuous monitoring by means of simple on-line rheological measurements is necessary at all times during the process. The interpretation of such measurements requires knowledge of the evolution of the failure envelope, i.e. the failure criterion for general multi-axial stress states. Both the fluid and solid properties of concrete have been studied extensively in the context of conventional casting techniques \[6\]. On one hand studies on the setting and strength development in fresh concrete typically report first measurements at an age of several hours \[7\]. On the other hand, studies on the workability of concrete focus on the thixotropy of fresh cement pastes typically for time spans up to one hour \[8,9\]. In this time scale, yield strength is determined first by colloidal interactions and later on by the strength of the first soft, then increasingly rigid percolating network with CSH bonds. Explanatory models based on bonded particles suggest a linear stress-time relation after mixing up to 25 minutes after mixing \[9\]. However, in studies over longer time periods an exponential relation was observed \[10,11\]. For the problem described above, it is crucial to describe the rheological and mechanical properties of the material during the early phase of its transition from a non-Newtonian yield stress fluid to a cohesive frictional material. To assess this early setting regime by mechanical testing for different stress states, “non-standard” mechanical tests were devised covering a wide range of hydration states within the described bounds of liquefaction and fracture. They capture the strength evolution of the initially ductile material to a point where failure is brittle and starts to become dependent on hydrostatic stress. A set of 6 different test setups is described, the respective results are discussed – for simplicity without fiber reinforcement – and condensed into failure envelopes for plane stress evolving with the advancing state of hydration. Material and methods ==================== The objective of this study is to determine the evolution of strength for arbitrary stress states in fresh SCC, starting even before the onset of setting. Material handling and test setups are designed to minimize unintentional thixotropic structural breakdown and to withstand gravity early on. All experiments were carried out in a single campaign. This work is limited to a single concrete mixture, designed to meet the requirements for the process of adaptive slip-forming. Material and specimen preparation --------------------------------- The SCC comprises (in grams per liter of the final mixture) CEM I 52.5 R Portland cement (981.73 g/l), sand aggregates of up to 4 mm in diameter (740.34 g/l), the mineral admixtures fly ash (164.67 g/l) and silica fume (92.89 g/l), as well as a superplasticizer (4 g/l) as a water reducing admixture and water (371 g/l). The SCC is heavily retarded by means of a sucrose solution (30% D(+)sucrose (99.7%) and 70% water, 2.7 g/l) and subsequently accelerated (60 g/l) when taken from the retarded batch, to obtain a constant workability and strength evolution over the entire timespan of the slip-forming process. The slump diameter is typically 20 to 22 cm and the final density of the mixture amounts to 2400 kg/m$^3$. Special attention was given to maintaining a consistent mixing procedure for all experiments using a rotating pan mixer at predefined mixing speeds (40 or 80 rpm) and intervals (in min) interrupted by cleaning of edges and blades (step denoted cl). After homogenizing the dry material at 80 rpm for 5 min (denoted by 80/5) all fluid parts except the accelerator were added and mixed 40/5 and 80/1. The sequence cl-40/1-cl-80/3 followed, before accelerator was added and finally mixed with 80/5. Between 5 to 10 min after mixing the samples were cast into the respective molds and put to rest. Because of the thixotropic structural buildup at rest, it is crucial to cast all samples within the shortest time span possible (cf. \[8-9,12\]). Test setup and procedure ------------------------ All samples were stored and tested under controlled climatic conditions (50% relative humidity at 20$^\circ$C) for a period of up to 12 hours after mixing, with experiments starting as early as possible with each test method. The penetration test (Fig.1a) employed a force gauge, recording forces up to 1000 N with 0.2 N accuracy. For all other setups (Fig.1b-f) a universal testing machine, equipped with a 10 kN load cell with 0.1 N accuracy was used. Note that due to the fresh state of the material, an accurate determination of the degree of hydration through calorimetry and other means proved unfeasible. Rate effects, such as thixotropy or viscoelasticity, are not addressed here. Instead, a constant displacement rate $v$ of 1 mm/s was employed, which lies in a regime where the dependence on the test rate was observed to vanish. In addition, the chosen rate allowed for swift execution of each measurement and thus depicted a snapshot in the material evolution where ongoing hydration and flocculation during testing were negligible. A total of 6 different mechanical test setups were employed, named (a)-(f) and shown in Fig.1a-f with dimensions being summarized in Tab.1. Specimen size effects, e.g. arising from the presence of aggregates, were precluded by initial tests with varying dimensions. Each test series of (b)-(f) was accompanied by penetration tests (a) on the same material. The specimen geometries and loading conditions are described in the following. #### Test (a) - Penetration: {#test-a---penetration .unnumbered} A rigid cylinder of diameter $d$ and height $b$ was driven at constant velocity $v$ into a basin of concrete of depth $h$. The resistance experienced by the indenter was recorded as a function of the penetration depth. The total displacement imposed was of the order of $d$ which is sufficient to reach a steady state flow for fresh concrete. The test was fully automatized using a tri-axial robot and 3 concrete basins, each accommodating 17 separate penetration measurement points. The penetration test forms the basis of the subsequent analysis. It is performed on all material samples in parallel to the other tests (b)-(f). The test is based on ASTM C403/C403M-08 \[13\] and its use for this application has already been reported in \[14\]. #### Test (b) - Punch-through: {#test-b---punch-through .unnumbered} A cylindrical indenter with diameter $d$ cuts a hole into a flat cylindrical concrete sample at constant $v$. The associated reaction force was recorded versus depth. The sample of diameter $D$ and height $h$ was vertically and radially supported by a mold comprising a hole of diameter slightly larger than $d$. The sample is thin in the sense that $h\ll d\pi$, yet thick enough to contain a sufficient amount of aggregates to assume a homogeneous material. This setup is inspired by the standard shear punch test, e.g. ASTM D732-10 \[15\]. #### Test (c) - Shear: {#test-c---shear .unnumbered} The setup for measuring early shear strength is similar to the Jenike shear cell (ASTM D6128-14 \[16\]). The upper compartment (gray hatched area in Fig.1c) was horizontally displaced at constant rate $v$, measuring its resistance. A steel cable translated the vertical movement of the testing machine via a pulley to the sample. Contrary to a classical shear cell, no normal force was applied and two horizontal plates, attached to the lower and upper compartments respectively, produced a notch for cleaner shear conditions at the fracture plane. An array of pins near the lower and upper surfaces of the sample (not shown in Fig.1c) ensured a rigid body motion of the two concrete blocks even when the sample was fresh. Note that rollers at the top suppressed the separation of the compartments. #### Test (d) - Compression: {#test-d---compression .unnumbered} Cylindrical concrete samples were vertically compressed by a plate moving at rate $v$. The sample diameter $d$ was chosen large enough to avoid size effects due to inhomogeneous particle distribution at walls, and $h/d$ was big enough to allow a slip layer to form diagonally to the loading direction. Samples were cast on a base plate using pipe segments lined with a thin sheet of Teflon treated with a mold-releasing agent. Just before testing the sample was demolded and the Teflon sheet carefully detached to preserve the thixotropic structural buildup in the sample at early stages of hydration. #### Test (e) - Tension: {#test-e---tension .unnumbered} Tensile testing was implemented using a mold of two separable parts. Unlike conventional tensile specimens, fresh concrete cannot be clamped. In order to enhance shear force transmission from the mold to the sample, several rows of pins were attached laterally to the box walls, decreasing in size towards the center of the specimen. A round notch of radius $R$ reduced the effective cross section of the sample to $A_{eff}=dh$ with width $d$ and height $h$. Note that the notch segments were free to move after casting and therefore did not exert any significant loads on the material during testing. At time of testing the right half (gray in Fig.1e) was horizontally separated from its fixed counterpart (white) at constant velocity $v$ using a similar cable-and-pulley configuration as in test (c). #### Test (f) - Bending: {#test-f---bending .unnumbered} To obtain the flexural strength, four-point bending tests on prismatic samples of size $L\cdot b\cdot h$ similar to ASTM C78/C78M \[17\] were performed as soon as the material was able to withstand the gravitational load at the supports (distance $L_1$). The loading frame (distance $L_2$) moved at v while the reaction force of the sample was recorded. Tests are valid only if a tensile crack formed at the lower side of the sample within $L_2$, where the bending moment is maximal and homogeneous. Note that contrary to tests (a)-(e) four-point bending was only applicable to specimens from a relatively advanced state of hydration on. \[tab1\] ------ ------------ ------------ ------------ ------------ -------------------- ----------------------- -------- test $b$ \[mm\] $d$ \[mm\] $h$ \[mm\] $D$ \[mm\] $L/L_{1,2}$ \[mm\] $A_{eff}$ \[mm$^2$\] $N$ (a) 4 18.8 40 - - $(d/2)^2\cdot \pi$ $>$500 (b) - 60 20 120 - $dh \pi$ 50 (c) 60$^*$ 60 50 150 - $bd$ 41 (d) - 63 100 63 - $(d/2)^2\cdot \pi$ 93 (e) - 60 (R=31) 60$^*$ 80 300 $dh$ 43 (f) 60$^*$ - 60 - 225 / 180 / 80 $2/3(bh^2)/(L_1-L_2)$ 16 ------ ------------ ------------ ------------ ------------ -------------------- ----------------------- -------- Experimental Observations ========================= For each sample the resistance force was recorded as a function of displacement. Additionally, the deformation patterns were observed visually in order to distinguish between “plastic” and “brittle” failure, when possible. Force-Displacement Curves ------------------------- Force-displacement curves measured at different age since mixing, like the examples shown in Fig.2 for each test at different hydration states, are the basis for obtaining the strength evolution in time. As typical for such tests, the increasing slope in the first millimeters of each test is due to the displacement until initial contact with the specimen is established (tests (a),(b),(d)) or until horizontal load cables are pre-tensioned (tests (c),(e)). Large plastic deformations at the contact zones of fresh samples in bending test (f) lead to an overestimation of the deflection. The resulting compliance of the system superimposes the elastic regime of the material, thus impeding stiffness measurements, but has no effect on the magnitude of the maximum force. Failure Patterns ---------------- Qualitative observations of the failure evolution and emerging patterns give an important insight into the materials behavior for the different setups and times after mixing. The penetration test (a) exhibits an increasing difficulty to reach steady state flow (i.e. constant resistance force at constant velocity) as hydration progresses (see Fig.2a). Instead, the force tends to increase with penetration depth as expected for solid-like behavior to a point where cracks suddenly propagate radially from the edge of the indentation through the basin. At this point, the test method ceases to provide data independent from the geometry of the boundary of the basin. Cross-sections of penetration samples made of colored concrete layers, see Fig.3, illustrate the deformation pattern of fresh concrete reaching steady state. As the indenter penetrates, an undisturbed concrete “plug” of conical shape outlined by a thin region of high shear forms in front of the disk, increasing the effective surface of the indenter. A steady shear flow is induced around the plug, provided that the material is fresh enough to flow. The punch and shear tests (b), (c) induce a homogeneous shear-dominated stress state along an interface inside the sample, inducing local shear failure. However, at a distance from this interface the shear stress decreases significantly. Small cracks are confined to the proximity of the interface even if they wanted to extend further into the bulk of the material. It is therefore possible that an alleged shear crack in fact consists of a connection of many small diagonal tensile cracks. Furthermore, the presence of aggregates at the sheared interface prevents the formation of a clean crack, thus contributing to the dissipation of energy. At early times, cylinder compression tests (d) exhibit a shear band formation in a plane of maximum shear, i.e. at 45$^\circ$ to the loading direction (Fig.4a), dividing the sample into two relatively undeformed parts. The measured force reaches a maximum as the band is fully formed and remains nearly constant as the upper part of the specimen slides along the diagonal shear plane. Hence very young samples exhibit steady state flow at a localized shear band similar to the penetration test. As the material becomes increasingly solid, the specimen accommodate the deformation more globally. A multitude of vertical cracks propagate through the sample as a result of circumferential tensile stresses arising from the increase in cross section (see Fig.4b). The peak force roughly coincides with the formation of such cracks, which dramatically lower the remaining strength of the sample thereafter. A plug of conical shape is found in front of the indenter. This failure pattern is qualitatively very similar to fully hydrated concrete samples under uniaxial compression \[18\]. Similar to compression, the tensile test (e) of relatively fresh samples shows a tendency to form a slip layer at 45$^\circ$C to the loading direction (see Fig.5a). This is consistent with the pronounced softening branch in Fig.2e even without fiber reinforcement. However, the mechanism of failure evolves quickly towards a single crack orthogonal to the loading direction, separating the sample (Fig.5b). Accordingly, the decline in measured force is abrupt (Fig.2e). The behavior of the bending test (f) is similar (Fig.2f), although this test is applicable only at later times due to high localized stresses at the contact zones. Data Processing =============== A comprehensive constitutive model ranging from the early state of the yield fluid to the solidified state of the frictional cohesive material is missing up to date. Without a priori knowledge of a suitable constitutive model, the definition of material parameters such as yield stress or strength is ambiguous \[19\]. The choice of experimentally identified parameters originates from the demand for finding a predictive model for the collapse of structures fabricated by adaptive slip-forming. In this section the derivation of strength evolution based on the measured data is described. The procedure includes a practicable definition of strength, a correction for gravity effects and a correction for inconsistencies in the evolution of different batches of material. Strength -------- The key property is the strength for multi-axial stress states and its evolution. Strength is defined for each test setup, at a certain time with respect to the end of the mixing procedure, as the maximum measured resistance force normalized by the initial reference area of the sample. For practical purposes the rheological behavior is neglected by fixing the displacement rate at a value that minimizes rate dependencies. Note that heat release due to hydration plays a minor role, since the time frame of hydration in this study is within the “period of slow reaction” \[20\] and hence is not accessible by calorimetry. The reference area corresponding to each test is listed in Tab.1. It intuitively represents the cross-section of the sample suffering the highest homogeneous stress. However, the choice is not always clear, as in case of the penetrometer test (a) \[21\]. For this test, penetration forces are normalized by the bearing area of the disk as suggested in \[15\], despite its tendency to form a cone-shaped plug. The bending test (f) does not exhibit a homogeneous stress state either. Instead, the flexural strength is defined as the maximum tensile stress along the beam cross-section. It is expected to relate qualitatively to the tensile strength of the material, being slightly larger than the tensile strength. Corrections for Gravity ----------------------- Testing starts as soon as possible without the samples collapsing prematurely. The gravitational loads are thus of the same order as the early strength and should be accounted for in the interpretation of results, in particular for the compression and bending tests (d), (f). Failing to do so would imply a substantial underestimation of early strength. In detail, compressive strength calculations include an additional gravitational load of $\rho gh$. The flexural strength or “modulus of rupture” is calculated as $MoR=1.5F_{max}\cdot (L_1-L_2)/(bh^2)+0.75\rho g L_1^2/h$, where the second term incorporates the contribution of gravitational load to the bending moment. Reference Time Correction Method -------------------------------- When summarizing all strength measurement points versus time since mixing in one plot, it becomes evident that despite of extraordinary diligence, significant deviations occur across experiments carried out with different batches of material in the same campaign. Fig.6a illustrates the extent of scatter in penetration and compression test data. These inconsistencies are thought to be mainly due to variations in the raw materials as well as environmental changes over the course of the test campaign, e.g. temperature and humidity during mixing. For a meaningful interpretation these effects need to be corrected for, while preserving the validity of the results. The correction procedure is based on the observation that despite the overall fluctuation, penetration data (test (a)) from within one mixture (i.e. measured at different locations of the same basin of concrete) follows approximately an exponential evolution law between 10 kPa and 1 MPa. Consequently, each penetration series/batch $n$ can be fitted with an exponential curve (e.g. dashed line in Fig.6a). Equivalently an exponential “reference curve” can be constructed from an average of all penetration test series (solid line). Each measurement point $i$ of series $n$ is compared to the reference curve and shifted in time as illustrated in Fig.6a (inset): - Consider the measurement $(t_i^n,S_i^n)$. According to the exponential fit to series $n$ (dashed line), the estimated penetration strength at time $t_i^n$ is $S_{p,i}^n$. - The reference penetration curve reaches the same strength $S_{p,i}^n$ at a different time $t_i^{ref}$. - Hence the data point is shifted by $\Delta t_i$ to the new location $(t_i^{ref}, S_i^n)$ in order to compensate for the difference in evolution of set $n$. - The routine is not only applicable to the correction of penetration data, but is used for all test setups (a)-(f) by considering any measurement $(t_i^n, S_i^n)$ in connection with the corresponding penetration strength $S_{p,i}^n$ of the same series $n$. A mixture does not always evolve like the reference, but its current state can be determined from the reference data simply by a penetration measurement that reveals the current state $t^{ref}$. The corrected strength data are shown in Fig.6b with respect to the reference time $t^{ref}$. Note that while deviations across different series of experiments are greatly reduced, the method preserves scatter inherent to the respective test setup as well as systematic deviations from the exponential fit. Discussion of merged results ============================ Finally, results from the different setups need to be combined into one coherent framework for strength evolution of the material. Its sensitivity to the environment necessitates a large set of experiments along with the described correction method. Only after obtaining data collapse, the strength evolution with the transition from the initially ductile failure at a shear band to the prevalence of cracks becomes observable. Evolution of Strength --------------------- The compression (test (d)) and penetration (test (a)) strength data in Fig.6b initially grow exponentially in good approximation, expressed by the relation $S=a\exp(bt)$, with different prefactors $a$ but similar exponent $b$ (as indicated by parallel lines). However, with progressing hydration the compressive strength departs from the exponential path to a significantly slower growth rate. This apparent change of regime coincides with the transition of the samples’ failure pattern from a localized shear zone to distributed cracks (as depicted in Fig.4). The corrected data of all tests is summarized in Fig.7. Distinct scaling regimes are particularly pronounced when comparing compressive with tensile tests (e), (f) (Fig.7a). The initially fresh samples exhibit largely the same strength in both uniaxial compression and tension, i.e. their strength appears to be independent of the hydrostatic component of stress. Thereafter the growth in tensile strength slows down dramatically as cracks appear. The change in failure mode of the compression test (d) occurs 60 minutes later. The estimated transition times for all tests are listed in Tab.2. In the following the two regimes are discussed separately. \[tab1\] ------------------------ -------------- ---------------- time \[min\] stress \[kPa\] (test (b)) Punch 250.2 81.6 (test (c)) Shear 219.4 44.1 (test (d)) Compression 227.4 137.5 (test (c)) Tension 167.5 13.1 ------------------------ -------------- ---------------- \[tab1\] ------------------------ ------------------------- ----------------------- ------- ----- $a$ \[kPa\] $b$ \[min$^{-1}$\] $R^2$ $N$ (test (a)) Penetration 0.188 \[-0.011,+0.011\] 0.033 \[$\pm$0.0003\] 0.992 310 (test (b)) Punch 0.035 \[-0.010,+0.015\] 0.031 \[$\pm$0.002\] 0.982 23 (test (c)) Shear 0.098 \[-0.036,+0.056\] 0.028 \[$\pm$0.003\] 0.942 28 (test (d)) Compression 0.070 \[-0.009,+0.010\] 0.033 \[$\pm$0.001\] 0.993 56 (test (e)) Tension 0.161 \[-0.075,+0.138\] 0.026 \[$\pm$0.005\] 0.927 14 ------------------------ ------------------------- ----------------------- ------- ----- #### First “Plastic” Regime: {#first-plastic-regime .unnumbered} The exponential fit parameters of the first regime are summarized in Tab.3. Again, the similarity of exponents $b$ and the disparity in the prefactors $a$ is evident. This conceptually corresponds to a scaling between the curves, where the scaling factor of each test is closely related to the different stress states in the samples and the choice of reference area $A_0$ used for the conversion of force to strength (Tab.1). With the knowledge of the three-dimensional stress state of each test it should thus be possible to deduce the shape of the yield surface from the scaling factors. Moreover, the similarity in exponents $b$ in Tab.3 suggests that the yield surface just expands equally in all directions in the first regime at an exponential rate. As a consequence, the first regime of tests (a)-(e) (Fig.8a) can be collapsed to a single evolution curve of “equivalent uniaxial strength” (Fig.8b), when data of each test is scaled with the following constant conversion factors. The punch/shear strengths are increased by a factor of $\sqrt{3}$ according to the von Mises equivalent stress under pure shear \[22\]. This assumption is supported by the ductile nature of the material in this regime and the equivalence of tensile and compressive strength data. For the penetration test neither the natural choice of $A_0$ nor the stress state are trivial. Instead the penetration strength is scaled to the effective surface area of the plug cone estimated from Fig.3, thus decreased by the factor $A_{cone}/A_0=(\sin \phi)^{-1}$. The compression and tensile strength remain unscaled. Note that pure shear conditions are assumed for the punch test, thereby neglecting the additional compressive stress component on the sheared surface caused by the radial confinement of the sample. Consequently, the “equivalent strength” derived from the punch tests is consistently underestimated (cf. Fig.8b), indicating that a significant confining pressure is prevalent in the test. #### Second “Brittle” Regime: {#second-brittle-regime .unnumbered} The strength gain in the second regime, albeit slower, can again be approximated by an exponential function for about one order of magnitude of strength in the observed time span (Fig.7 and coefficients in Tab.4). The exponents of the evolution law for the punch, compression and tension tests are similar, implying that in this regime the ratio between compressive and tensile strength remains roughly constant (at 5.4 in this case). The shear strength exhibits a significant amount of scatter due to frictional forces in the test setup and thus remains inconclusive. Of course a further slowdown in strength gain is anticipated for all tests until the concrete is fully hydrated and strength converges to a final value. This trend is indicated by the four-point bending test data (see Fig.7a), which is qualitatively related to the tensile strength. It is important to observe that a very large portion of the well-known difference in compressive and tensile strength of hardened concrete emerges already at the early transition from a pressure-insensitive thixotropic yield stress fluid to a brittle cohesive frictional solid. It is not a putative difference in rate (or exponent) of compressive and tensile strength growth that gives rise to the pressure sensitivity of the final material, but rather their different times of transition from the fluid to the solid regime. The difference in “brittle” compression and tension may always be existent, except that in fresh concrete the low plastic yield strength undercuts these failure modes. The solid regime thus emerges only after the plastic strength, growing at a higher rate, surpasses brittle strength – a transition which occurs at different times for compression and tension because of the inherent difference in brittle compressive and tensile strength. \[tab1\] ------------------------ ----------------------------- ---------------------- ------- ----- $a$ \[kPa\] $b$ \[min$^{-1}$\] $R^2$ $N$ (test (b)) Punch 4.0 \[-1.08,+1.48\] 0.012 \[$\pm$0.001\] 0.967 27 (test (c)) Shear 13.9 \[-9.33,+28.29\] 0.005 \[$\pm$0.0044\] 0.381 13 (test (d)) Compression 3.7 \[-0.91,+1.22\] 0.016 \[$\pm$0.001\] 0.967 37 (test (e)) Tension 1.3 \[-0.28,+0.36\] 0.014 \[$\pm$0.001\] 0.963 29 ------------------------ ----------------------------- ---------------------- ------- ----- Strength Envelope ----------------- Finally, the strength envelope, generally represented by a convex surface in 6-dimensional stress space, is constructed. The envelope joins all points of failure in the stress space, thus delimiting the field of stable stress configurations within the envelope from the failure field outside the envelope. Owing to the isotropic behavior of the (unreinforced) concrete and the ensuing rotational symmetries of the elasticity tensor, the strength envelope can be expressed in only 3 dimensions, e.g. in terms of the principal stresses. Each strength measurement represents a point on the strength envelope for the given state of hydration (or reference time since mixing). The above test setups, with the exception of the penetration and punch tests for which the tri-axial stress state is not a priori known, impose plane stress conditions where at least one principal stress component is negligible. Therefore, it is sufficient to examine failure in the $\sigma_1-\sigma_2$-plane, where the strength envelope reduces to a closed convex curve. Such curves can be estimated for a given time after mixing by taking strength values from the exponential fits to tensile, compressive and shear test data as shown in Fig.9 for 100 to 300 min after mixing. Curves are drawn through these points in the $\sigma_1-\sigma_2$-plane by means of a spline interpolation in cylindrical coordinates ${r,\phi}$. The early strength data, up to an order of a few kPa, correspond well with the von Mises criterion. Thereafter strength becomes increasingly pressure sensitive, thus the curves extend far into the pressure zone. Clearly, the accuracy of the resulting envelope could be greatly improved by collecting more experimental data under various bi-axial stress states. Friction Angle -------------- Additionally, it is possible to calculate the cohesion $c$ and internal friction angle $\phi$ based on the compressive and tensile strength evolution e.g. for the Coulomb-Mohr and Drucker-Prager models. The Drucker-Prager yield surface describes an infinite cone in principal stress space, where $c$ is the intercept of the cone with the principal stress axes and $2\phi$ describes the opening angle (aperture) of the cone. The cohesion, a measure of overall strength particularly under tension, increases roughly exponentially in the observed time frame. As expected, $c$ continues to grow at a slower rate after the regime transition. The friction angle, closely related to the difference in compressive and tensile strength, increases rapidly from 0 to 48 degrees at the transition, and is set to grow only slightly thereafter. Summary and Conclusions ======================= The strength evolution during early hydration is of increasing importance with the growing recognition for new technologies in concrete fabrication such as concrete printing or adaptive slip casting. It was shown by evaluating more than 500 individual samples in six different non-standard experimental setups at times of up to 12 hours after mixing, that two principal regimes govern the strength evolution: A first one, where the material is ductile and where shear, compressive and tensile strengths are well-represented by the von Mises criterion. At the transition, typically around 3 hours after mixing for the SCC studied, cracks begin to prevail and failure is increasingly dependent on the hydrostatic component of stress, as seen e.g. by the divergence between compressive and tensile strengths. The second regime is characterized by significantly slower strength increase, but at a similar rate for tension and compression. In both regimes, strength scales exponentially with time over several orders of magnitude. This differs with respect to observations on small time scales accessible to rheometers, where a linear relation is found \[9\]. To make the findings applicable to more general states of stress, failure envelopes in plane stress space at different concrete ages were constructed, which would finally converge to the final strength envelope known for hardened high-strength concrete, reported for biaxial stress states e.g. in \[23-24\]. Despite variations in the strength evolution of different material batches, the current failure envelope is accessible by a simple penetration measurement. There are limitations to this study that go beyond the choice of a single concrete mix. Considerable differences in evolution occurred across samples due to variations in raw materials and temperature. A meaningful interpretation of the collected data necessitated correction procedures, such as the time shift method proposed herein. By shifting solely in time (and not in force), however, it is assumed that the penetration force is a state variable uniquely describing the material. Thus it is ignored for now, that flocculation and hydration can evolve differently – therefore calling for at least two state variables and evolution laws \[25\] – and that the same strength does not necessarily imply an identical state of the material. Instead, samples were casted systematically at the same time after mixing and it was assumed that flocculation and hydration depend equally on temperature (or that the effect of temperature on flocculation is very small compared to hydration). However, the influence of flocculation on the yield stress is limited to time scales that are significantly below the ones addressed in this study \[9\]. The validity of these assumptions was demonstrated only within the parameter space of the experiments that keeps in mind the technological design window of the adaptive slip-casting process. Acknowledgements {#acknowledgements .unnumbered} ================ The support by the ETH Zurich under ETHIIRA grant no. ETH-13 12-1 “Smart Dynamic Casting”, as well as from the European Research Council advanced grant no. 319968-FlowCCS is acknowledged. We thank David Walker, Lukas Bodenmann and Stefan Aebersold, who contributed to this work by preparatory studies and also express our gratitude to Ena Lloret, Nicolas Roussel and Peter Fischer for numerous useful discussions on the topic. References {#references .unnumbered} ========== - E. Lloret, A.R. Shahab, L.K. Mettler, R.J. Flatt, F. Gramazio, M. Kohler, S. Langenberg, Complex concrete structures Merging existing casting techniques with digital fabrication, Comput. Aided Design 60 (2014) 40–49. - A.R. Shahab, E. Lloret, P. Fischer, F. Gramazio, M. Kohler, R.J. Flatt, Smart dynamic casting or how to exploit the liquid to solid transition in cementitious materials, extended abstract, 7th International RILEM Conference on SCC, Paris (2013). - G. Gelardi, S. Mantellato, D. Marchon, M. Palacios, A. Eberhardt, R.J. Flatt, Chapter 9 – Chemistry of chemical admixtures, in Science and Technology of Concrete Admixtures, Ed. P.-C. Aïtcin and R.J. Flatt, Woodhead Publishing (2016), 149-218. - D. Marchon, R.J. Flatt, Chapter 8 – Mechanisms of cement hydration, in Science and Technology of Concrete Admixtures, Ed. P.-C. Aïtcin and R.J. Flatt, Woodhead Publishing (2016), 129-145. - D. Marchon, R.J. Flatt, Chapter 12 – Impact of chemical admixtures on cement hydration, in Science and Technology of Concrete Admixtures, Ed. P.-C. Aïtcin and R.J. Flatt, Woodhead Publishing (2016), 279-304. - K. Kovler, N. Roussel, Properties of fresh and hardened concrete, Cem. Concr. Res. 41 (2011) 775-792. - R.C.A. Pinto, A.K. Schindler: Unified modeling of setting and strength development, Cem. Concr. Res. 40 (2010) 58-65. - N. Roussel, A thixotropy model for fresh fluid concretes: Theory, validation and applications, Cem. Concr. Res. 36 (2006) 1797-1806. - N. Roussel, G. Ovarlez, S. Garrault, C. Brumaud: The origins of thixotropy of fresh cement pastes, Cem. Concr. Res. 42 (2012) 148-157. - A. Perrot, A. Pierre, S. Vitaloni, V. Picandet : Prediction of lateral form pressure exerted by concrete at low casting rates, Mater. Struct. 48 (2015) 2315-2322. - K.V. Subramaniam, X. Wang: An investigation on microstructure evolution in cement paste through setting using ultrasonic and rheological measurements, Cem. Concr. Res. 40 (2010) 33-44. - G. Ovarlez, X. Chateau, Influence of shear stress applied during flow stoppage and rest period on the mechanical properties of thixotropic suspensions, Phys. Rev. E 77 (2008) 061403. - Standard Test Method for Time of Setting of Concrete Mixtures by Penetration Resistance, ASTM C403-/C403M-08 (2008). - E. Lloret, L.K. Mettler, A.R. Shahab, F. Gramazio, M. Kohler, R.J. Flatt, Smart Dynamic Casting: A robotic fabrication system for complex structures, 1st Concrete Innovation Conference, Oslo (2014). - Standard Test Method for Shear Strength of Plastics by Punch Tool, ASTM D732-10 (2010). - Standard Test Method for Shear Testing of Bulk Solids Using the Jenike Shear Cell, ASTM D6128-14 (2014). - Standard Test Method for Flexural Strength of Concrete (using simple beam with third-point loading), ASTM C78-/C78M-10 (2013). - J.G.M. van Mier: Fracture Processes of Concrete, CRC Press, 1997. - P.C.F. Moller, J. Mewis, D. Bonn, Yield stress and thixotropy: on the difficulty of measuring yield stress in practice, Soft Matter 2 (2006) 274-283. - J.W. Bullard, H.M. Jennings, R.A. Livingston, A. Nornat, G.W. Scherer, J.S. Schweitzer, K.L. Scrivener, J.J. Thomas, Mechanisms of cement hydration, Cem. Concr. Res. 41 (2011) 1208-1223. - D. Lootens, P. Jousset, L. Martinie, N. Roussel, R.J. Flatt, Yield stress during setting of cement pastes from penetration tests, Cem. Concr. Res. 39 (2009) 401-408. - J. Lubliner, Plasticity theory (Dover books on engineering), Dover Publications (2008). - H. Kupfer, Das Verhalten des Betons unter zweiachsiger Beanspruchung (in German), report, Lehrstuhl für Massivbau, Technical University Munich, no. 78 (1969) 124 ff. - A. Nimura, Experimental research on failure criteria of ultra-high strength concrete under biaxial stress (in Japanese), Architectural Institute of Japan, Structures II vol. C (1991) 473-474. - P. Coussot, Rheometry of Pastes, suspensions, and Granular Materials: Applications in Industry and Environment, J. Whiley & Sons Inc., Hoboken, New Jersey (2005).