1001.0035.txt

arXiv:1001.0035v2  [hep-th]  25 Mar 2010Reconstruction of Baxter Q-operator fromSklyanin SOV
for cyclic representations ofintegrable quantum models
G. Niccoli
DESY,Notkestr. 85, 22603 Hamburg, GermanyDESY 09-227
Abstract
In [1], the spectrum (eigenvaluesand eigenstates) of a latt ice regularizationsof the Sine-
Gordon model has been completely characterized in terms of p olynomial solutions with
certain propertiesof the Baxter equation. This characteri zation for cyclic representations
hasbeenderivedbytheuseofthe SeparationofVariables(SO V)methodofSklyaninand
bythedirectconstructionoftheBaxter Q-operatorfamily. Here,wereconstructtheBaxter
Q-operatorandthesamecharacterizationofthespectrumbyo nlyusingtheSOVmethod.
This analysis allows us to deduce the main features required for the extension to cyclic
representationsofotherintegrablequantummodelsofthis kindofspectrumcharacteriza-
tion.
Keywords: Integrable Quantum Systems;Separation of Variables;Baxt erQ-operator; PACScode 02.30.IK2
1. Introduction
Theintegrabilityofa quantummodelisbydefinitionrelated to theexistenceofamutuallycommu-
tativefamily Qofself-adjointoperators Tsuchthat
(A) [T,T′] = 0,
(B) [T,U] = 0,
(C) if [ T,O] = 0,∀T,T′∈ Q,
∀T∈ Q,
∀T∈ Q,thenO=O(Q),(1.1)
whereUis the unitary operatordefining the time-evolutionin the mo del; note that the property(C)
stays for the completeness of the family Q. In the framework of the quantum inverse scattering
method [2, 3, 4] the Lax operator L(λ)is the mathematical tool which allows to define the transfer
matrix:
T(λ) = trC2M(λ),M(λ)≡/parenleftbiggA(λ)B(λ)
C(λ)D(λ)/parenrightbigg
≡LN(λ)...L1(λ),(1.2)
aoneparameterfamilyofmutualcommutativeself-adjointo perators. Theintegrabilityofthemodel
follows from T(λ)if the properties (B) and (C) of definition (1.1) can be proven for it. In some
quantummodeltheintegrabilityisderivedbyprovingtheex istenceofafurtherone-parameterfamily
ofself-adjointoperatorsthe Q-operatorwhichbydefinitionsatisfiesthefollowingproper ties:
[Q(λ),Q(µ)] = 0,[T(λ),Q(µ)] = 0,∀λ,µ∈C, (1.3)
plusthe Baxterequationwith thetransfermatrix:
T(λ)Q(λ) =a(λ)Q(q−1λ)+d(λ)Q(qλ). (1.4)
This is in particular the case for those models (like Sine-Go rdon [1]) for which the time-evolution
operatorUis expressedin terms of Q. A naturalquestionarises: Is the integrablestructure of t hese
quantummodelscompletelycharacterizedbythetransferma trixT(λ)?
Note that a standard procedure1to prove the existence of Q(λ)is by a direct construction of an
operatorsolutionoftheBaxterequation(1.4). Moreover,t hecoefficients a(λ)andd(λ)aswellasthe
analytic and asymptotics properties of Q(λ)are some model dependent features which are derived
bythe construction. Let usrecall thatthe generalstrategy [11, 12,13,14, 15] ofthisconstructionis
to findagaugetransformation2such that the action of each gaugetransformedLax matrixon Q(λ)
becomesupper-triangular. Thenthe Q-operatorassumesa factorized localformandthe problemof
its existence in such a form is reduced to the problem of the ex istence of some model dependent
specialfunction3.
1Itis worth recalling that there are also others constructio ns of theQ-operator. An interesting example is presented in the
series of works [5,6,7] by V.V. Bazhanov, S.L.Lukyanov and A .B.Zamolodchikov on the integrable structure of conformal
field theories. In [6,7] the Q-operator is obtained asatransfer-matrix byatrace proced ure ofafundamental L-operator with
q-oscillator representation for the auxiliary space (see al so [8, 9]). This construction can be extended to massive inte grable
quantum field theories as itwas argued by thesame authors in [ 10].
2Itleaves unchanged thetransfer matrix while modifies the mo nodromy matrix M(λ)defined in (1.2) .
3Thequantumdilogarithm functions [16,17,18,19,20,21,22,23,24,25]forexample a ppear intheSinh-Gordon model
[26],in their non-compact form,and in the Sine-Gordon model[1],in their cyclicform.3
It is worth pointing out that on the one hand the construction of these special functionsfor general
modelscanrepresenta concretetechnicalproblem4andthat ontheotherhandtheexistenceofsuch
functionsis onlya sufficientcriterionforthe existence of Q(λ). It is then a relevantquestionif it is
possibletobypassthiskindofconstructionprovidingadif ferentproofofthe existenceof Q(λ).
Given an integrable quantum model the first fundamental task to solve is the exact solution of its
spectral problem , i.e. the determination of the eigenvalues and the simultan eous eigenstates of the
operator family Q, defined in (1.1). There are several methods to analyze this s pectral problem as
thecoordinate Bethe ansatz [27, 28, 29], the TQmethod [28], the algebraic Bethe ansatz (ABA)
[2, 3, 4], the analyticBethe ansatz [30] and the separation of variables (SOV) meth od of Sklyanin
[31, 32, 33]; this last one seems to be more promising. Indeed , on the one hand it resolves the
problems related to the reduced applicability of other meth ods (like ABA) and on the other hand
it directly implies the completeness of the characterizati on of the spectrum which instead for other
methodshastobeproven.
For cyclic representations [34] of integrable quantum mode ls the SOV method should lead to the
characterizationoftheeigenvaluesandthesimultaneouse igenstatesofthetransfermatrix T(λ)bya
finite5systemofBaxter-likeequations. However,itisworthpoint ingoutthatsuchacharacterization
of the spectrum is not the most efficient; this is in particula r true in view of the analysis of the
continuum limit. Here the main question reads: Is it possibl e to define a set of conditions under
whichtheSOVcharacterizationofthespectrumcanbereform ulatedintermsofafunctionalBaxter
equation? In fact, this is equivalent to ask if we can reconst ruct theQ-operator from the finite
system of Baxter-like equations. In this case the solution o f the spectral problem is reduced to the
classification of the solutions of the Baxter equation which satisfy some analytic and asymptotic
propertiesfixedbythe operators TandQ.
The lattice Sine-Gordonmodelis used asa concreteexamplew herethese questionsaboutquantum
integrability find a complete and affirmative answer. Indeed , in section 3, we show that the SOV
characterization of the transfer matrix spectrum is exactl y equivalent to a functional equation of
the form detD(Λ) = 0, whereD(λ)(see (3.21)) is a one-parameter family of quasi-tridiagonal
matrices. In section 4, we show that this functional equatio n is indeed equivalent to the Baxter
functional equation and, in section 5, we use these results t o reconstruct the Baxter Q-operator
with the same level of accuracy obtained by the direct constr uction presented in [1]. It is worth
pointingoutthat these resultsallowusto provethat thetra nsfermatrix T(λ)(plustheΘ-chargefor
even chain) describes the family Qof complete commuting self-adjoint charges which implies t he
quantum integrabilityof the model accordingto definition ( 1.1). So that in the Sine-Gordonmodel
theBaxter Q-operatorplaysonlytheroleofa usefulauxiliaryobject.
Let us point out that one of the main advantages of the spectru m characterization derived for the
Sine-Gordonmodelisthe possibilityto proveanexactrefor mulationin termsof non-linearintegral
4TheSine-Gordon model at irrational values of the coupling β2is asimple case where this kind of problem emerges.
5Thenumber of equations in thesystem is finite and related to t hedimension of thecyclic representation.4
equations6(NLIE).Thiswill bethe subjectof a futurepublicationwher ethe NLIEcharacterization
will lead us by the implementation of the continuum limit to t he description of the Sine-Gordon
spectrum in all the interesting regimes. These results will be shown to be consistent with those
obtained previously in the literature7[37, 38, 39, 40, 41, 42] (see [43, 44] for reviews). Note that
the methodbasedon thereformulationofthe spectralproble min termsofNLIEhasbeenalso used
recently [49] to derive the Sinh-Gordonspectrum in finite vo lume and to characterize the spectrum
in theinfraredandultravioletlimits.
The analysis of the Sine-Gordon model allows us to infer the m ain features required to extend this
kindofspectrumcharacterizationtocyclicrepresentatio nsofotherintegrablequantummodels. This
is particularly relevant for those models for which a direct construction of the Baxter Q-operator
encounterstechnicaldifficulties.
Acknowledgments. I would like to thank J. Teschner for stimulating discussion s and suggestions on a prelimi-
nary versionof this workand J.-M.Maillet for the interests hown.
I gratefullyacknowledge support from the ECbythe Marie Cur ie Excellence GrantMEXT-CT-2006-042695.
2. The Sine-Gordon model
Weusethissectiontorecallthemainresultsderivedin[1]o nthedescriptionintermsofSOVofthe
lattice Sine-Gordonmodel. This will be used as the starting point to introducea characterizationof
the spectrumof the transfermatrix T(λ)which will lead to the constructionof the Q-operatorfrom
SOV.
2.1 Definitions
Thelattice Sine-Gordonmodelcanbecharacterizedbythefo llowingLaxmatrix8:
LSG
n(λ) =κn
i/parenleftigg
iun(q−1
2κnvn+q+1
2κ−1
nv−1
n)λnvn−λ−1
nv−1
n
λnv−1
n−λ−1
nvniu−1
n(q+1
2κ−1
nvn+q−1
2κnv−1
n)/parenrightigg
,(2.5)
whereλn≡λ/ξnfor anyn∈ {1,...,N}withξnandκnparameters of the model. For any n∈
{1,...,N}thecoupleofoperators( un,vn)defineaWeyl algebra Wn:
unvm=qδnmvmun,whereq=e−πiβ2. (2.6)
We will restrictourattentiontothecase inwhich qisarootofunity,
β2=p′
p, p,p′∈Z>0, (2.7)
6Thistype of equations werebefore introduced in adifferent framework in [35,36]
7See [45, 46] for a related model analyzed in the framework of A BA and [47, 48] for the corresponding finite volume
continuum limit.
8Thelattice regularization of the Sine-Gordon model that we consider here goes back to [4,50] and is related to formula-
tions which have morerecently been studied in [51,52,53].5
withp≡2l+ 1odd andp′even so that qp= 1. In this case each Weyl algebra Wnadmits a
finite-dimensional representation of dimension p. In fact, we can represent the operators un,vnon
thespace ofcomplex-valuedfunctions ψ:SN
p→Cas
un·ψ(z1,...,z N) =unznψ(z1,...,z n,...,z N),
vn·ψ(z1,...,z N) =vnψ(z1,...,q−1zn,...,z N).(2.8)
whereSp={q2n;n= 0,...,2l}is a subset of the unit circle; note that Sp={qn;n= 0,...,2l}
sinceq2l+2=q.
Themonodromymatrix M(λ)definedin(1.2)intermsoftheLax-matrix(2.5)satisfiesthe quadratic
relations:
R(λ/µ)(M(λ)⊗1)(1⊗M(µ)) = (1⊗M(µ))(M(λ)⊗1)R(λ/µ), (2.9)
wheretheauxiliary R-matrixisgivenby
R(λ) =
qλ−q−1λ−1
λ−λ−1q−q−1
q−q−1λ−λ−1
qλ−q−1λ−1
. (2.10)
The elements of M(λ)generate a representation RNof the so-called Yang-Baxter algebra char-
acterized by the 4Nparametersκ= (κ1,...,κ N),ξ= (ξ1,...,ξ N),u= (u1,...,u N)and
v= (v1,...,v N); in the present paper we will restrict to the case un= 1,vn= 1,n= 1,...,N.
The commutation relations (2.9) are at the basis of the proof of the mutual commutativity of the
T-operators.
Inthecase ofa latticewith Nevenquantumsites, we havealso tointroducetheoperator:
Θ =N/productdisplay
n=1v(−1)1+n
n, (2.11)
whichplaystheroleofa gradingoperator inthe Yang-Baxteralgebra:
Proposition 6 of [1] Θcommuteswiththetransfermatrixandsatisfiesthefollowin gcommutation
relationswith theentriesofthemonodromymatrix:
ΘC(λ) =qC(λ)Θ,[A(λ),Θ] = 0, (2.12)
B(λ)Θ =qΘB(λ),[D(λ),Θ] = 0. (2.13)
Moreover,the Θ-chargeallowstoexpressthe asymptoticsofthetransferma trixas:
lim
logλ→∓∞λ±NT(λ) =/parenleftiggN/productdisplay
a=1κaξ±1
a
i/parenrightigg
/parenleftbig
Θ+Θ−1/parenrightbig
. (2.14)6
Let us denotewith ΣTthe spectrum(the set of the eigenvaluefunctions t(λ)) of the transfer matrix
T(λ). By the definitions(1.2) and (2.5), then ΣTis contained9inC[λ2,λ−2](N+eN−1)/2, where we
haveusedthenotatione N= 0forNoddand1forNeven.
Notethat inthecase of Neven,the Θ-chargenaturallyinducesthegrading ΣT=/uniontextl
k=0Σk
T,where:
Σk
T≡/braceleftigg
t(λ)∈ΣT: lim
logλ→∓∞λ±Nt(λ) =/parenleftiggN/productdisplay
a=1κaξ±1
a
i/parenrightigg
(qk+q−k)/bracerightigg
.(2.15)
This simply follows by the asymptotics of T(λ)and by its commutativity with Θ. In particular,
anyt(λ)∈Σk
Tis aT-eigenvalue corresponding to simultaneous eigenstates of T(λ)andΘwith
Θ-eigenvalues q±k.
2.2 CyclicSOVrepresentations
TheseparationofvariablesmethodofSklyaninisbasedonth eobservationthatthespectralproblem
forT(λ)simplifies considerablyif one worksin an auxiliaryreprese ntationwherethe commutative
familyofoperators B(λ)isdiagonal.
InthecaseoftheSine-Gordonmodelthevectorspace10CpNunderlyingtheSOVrepresentationcan
beidentifiedwiththespaceoffunctions Ψ(η)definedforηtakenfromthediscreteset
BN≡/braceleftbig
(qk1ζ1,...,qkNζN); (k1,...,k N)∈ZN
p/bracerightbig
, (2.16)
onthesefunctions B(λ)actsasa multiplicationoperator,
BN(λ)Ψ(η) =ηeN
Nbη(λ)Ψ(η), b η(λ)≡N/productdisplay
n=1κn
i[N]/productdisplay
a=1(λ/ηa−ηa/λ) ; (2.17)
where[N]≡N−eNandη1,...,η[N]are the zerosof bη(λ). In the case of even Nit turns out that
we needa supplementaryvariable ηNinordertobeable toparameterizethe spectrumof B(λ).
In[1]wehaveproventhatforgeneralvaluesoftheparameter sκandξoftheoriginalrepresentation
it is possible to construct these SOV representationsand mo reoverwe have defined the map which
fixestheSOVparameter ηintermsoftheparameters κandξ.
In these SOV representations the spectral problem for T(λ)is reduced to the following discrete
system ofBaxter-likeequationsin thewave-function Ψt(η) =/a\}bracketle{tη|t/a\}bracketri}htofaT-eigenstate |t/a\}bracketri}ht:
t(ηr)Ψ(η) =a(ηr)T−
rΨ(η)+d(ηr)T+
rΨ(η)∀r∈ {1,...,[N]},(2.18)
9Herewith C[x,x−1]Mwearedenoting the linear space ofthe Laurentpolynomials o f degreeMin thevariable x∈C.
10It is always possible to provide the structure of Hilbert spa ce to this finite-dimensional linear space. In particular, t he
scalar product in the SOVspace is naturally introduced by th e requirement that the transfer matrix is self-adjoint in th e SOV
representation. Appendix B addresses this issue.7
whereT±
raretheoperatorsdefinedby
T±
rΨ(η1,...,η N) = Ψ(η1,...,q±1ηr,...,η N),
whilethe coefficients a(λ)andd(λ)are definedby:
a(λ) =N/productdisplay
n=1κn
iλn(1−iq−1/2λnκn)(1−iq−1/2λn
κn),d(λ) =qNa(−λq).(2.19)
Inthecase of Nevenwe haveto addto thesystem(2.18) thefollowingequatio ninthevariable ηN:
T+
NΨ±k(η) =q±kΨ±k(η), (2.20)
fort(λ)∈Σk
Twithk∈ {0,...,l}.NotethatthecyclicityoftheseSOVrepresentationsisexpr essed
bytheidentificationof (T±
j)pwith theidentityforany j∈ {1,...,N}.
3. SOV characterization of T-eigenvalues
Let usintroducetheoneparameterfamily D(λ)ofp×pmatrix:
D(λ)≡
t(λ)−d(λ) 0 ··· 0 −a(λ)
−a(qλ)t(qλ)−d(qλ) 0 ··· 0
0......
... ···...
... ···...
...... 0
0... 0−a(q2l−1λ)t(q2l−1λ)−d(q2l−1λ)
−d(q2lλ) 0 ... 0−a(q2lλ)t(q2lλ)
(3.21)
wherefornow t(λ)isjust anevenLaurentpolynomialofdegree N+eN−1inλ.
Lemma 1. Thedeterminant detpDisanevenLaurentpolynomialofmaximaldegree N+eN−1in
Λ≡λp.
Proof.Let us start observingthat D(λq)is obtainedby D(λ)exchangingthe first and p-th column
andafterthefirst and p-throw,so that
det
pD(λq) = det
pD(λ)∀λ∈C, (3.22)
whichimpliesthat detpDisfunctionof Λ. Let usdevelopthedeterminant:
det
pD(Λ) =p/productdisplay
h=1a(λqh)+p/productdisplay
h=1a(−λqh)−qNa(λ)a(−λ) det
2l−1D(1,2l+1),(1,2l+1)(λ)
−qNa(λq)a(−λq) det
2l−1D(1,2),(1,2)(λ)+t(λ)det
2lD1,1(λ), (3.23)8
whereD(h,k),(h,k)(λ)denotes the (2l−1)×(2l−1)sub-matrix of D(λ)obtained removing the
rowsandcolumns handkwhileDh,k(λ)denotesthe 2l×2lsub-matrixof D(λ)obtainedremoving
therowhandcolumn k. Theinteresttowardthisdecompositionof detpD(Λ)isduetothefact that
the matrices D(1,2),(1,2)(λ),D(1,2l+1),(1,2l+1)(λ)andD1,1(λ)aretridiagonal matrices. Following
thesamereasoningusedinLemma4toprovethat det2lD1,1(λ)isanevenfunctionof λwecanalso
showthatthisistruefor det2l−1D(1,2),(1,2)(λ)anddet2l−1D(1,2l+1),(1,2l+1)(λ). Fromtheparityof
these functionsthe parityof detpD(Λ)followsbyusing(3.23).
Beinga(λ),d(λ)andt(λ)Laurentpolynomialofdegree Ninλ,inthecaseof Neventhestatement
ofthelemmaisalreadyproven;so wehavejust toshowthat:
lim
logΛ→∓∞Λ±Ndet
pD(Λ) = 0 (3.24)
forNoddwhichfollowsobservingthat:
lim
logΛ→∓∞Λ±Ndet
pD(Λ) =i±pNN/productdisplay
n=1κp
nξ±p
ndet
p/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δh,k+1−q(1∓1)N/2δh,k−1/vextenddouble/vextenddouble/vextenddouble.
(3.25)
The interesttowardthe function detpD(Λ)isdueto the fact thatit allowsthefollowingcharacteri-
zationofthe T-spectrum:
Lemma 2. ΣTis the set of all the functions t(λ)∈C[λ2,λ−2](N+eN−1)/2which satisfy the system
of equations:
det
pD(ηp
a) = 0∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN, (3.26)
plusin thecaseof Neven:
lim
logΛ→∓∞Λ±Ndet
pD(Λ) = 0. (3.27)
Proof.The requirement that the system of equations (2.18) admits a non-zerosolution leads to the
equations(3.26),while theequation(3.27) foreven Nsimplyfollowsbyobservingthat:
lim
logΛ→∓∞Λ±Ndet
pD(Λ) = det
p/vextenddouble/vextenddouble/vextenddoubleq(1∓1)N/2δi,j−1+q−(1∓1)N/2δi,j+1−(qk+q−k)δi,j/vextenddouble/vextenddouble/vextenddouble
×(−1)N/productdisplay
n=1/parenleftbig
iκnξ±
n/parenrightbigp= 0. (3.28)
Note that the above characterization of the T-spectrum ΣTrequires as input the knowledge of BN,
i.e. the lattice of zeros of the operator B(λ). It is so interesting to notice that this characterization9
has in fact a reformulation which is independent from the kno wledge of BN. To explain this let us
notethatLemma1allowsto introducethefollowingmap:
Dp,N:t(λ)∈C[λ2,λ−2](N+eN−1)/2→ Dp,N(t(λ))≡det
pD(Λ)∈C[Λ2,Λ−2](N+eN−1)/2.
(3.29)
Intermsofthismapwecanintroduceafurthercharacterizat ionofthespectrumofthetransfermatrix
T(λ).
Theorem 1. The spectrum ΣTof the transfer matrix T(λ)coincides with the kernel NDp,N⊂
C[λ2,λ−2](N+eN−1)/2ofthe map Dp,N.
Proof.The inclusion NDp,N⊂ΣTis trivial thanks to Lemma 2, vice-versa if t(λ)∈ΣTthen
the function detpD(Λ)is zero in N+eNdifferent values of Λ2which thanks to Lemma 1 implies
detpD(Λ)≡0,i.e.ΣT⊂ NDp,N.
That is the set of eigenvalues of the transfer matrix T(λ)is exactly characterized as the subset of
C[λ2,λ−2](N+eN−1)/2whichcontainsallthesolutionsofthefunctionalequation detpD(Λ) = 0. In
thenextsectionwewill showthat thisfunctionalequationi s nothingelse thattheBaxterequation.
Remark 1. Let us note that the same kind of functional equation detD(Λ) = 0 also appears
in [54, 55, 56]. There it recasts, in a compact form, the funct ional relations which result from the
truncatedfusionsoftransfermatrixeigenvalues. Itissor elevanttopointoutthatforthe BBS-model11
in the SOV representation the non-triviality condition of t he solutions of the system of Baxter-like
equations has been shown [60] to be equivalent to the truncat ion identity in the fusion of transfer
matrixeigenvalues.
4. Baxterfunctional equation
The main consequence of the previous analysis is that it natu rally leads to the complete character-
ization of the transfer matrix spectrum in terms of polynomi al solutions of the Baxter functional
equation.
Theorem2. Lett(λ)∈ΣTthent(λ)definesuniquelyuptonormalizationapolynomial Qt(λ)that
satisfiestheBaxterfunctionalequation:
t(λ)Qt(λ) =a(λ)Qt(λq−1)+d(λ)Qt(λq)∀λ∈C. (4.30)
Proof.The fact that given a t(λ)∈C[λ2,λ−2](N+eN−1)/2there exists up to normalizationat most
one polynomial Qt(λ)that satisfies the Baxter functional equationhas been prove nin Lemma 2 of
[1]. So we have to prove only the existence of Qt(λ)∈C[λ]. An interesting point about the proof
givenhereisthatit isa constructiveproof.
11TheBBS-model [12, 57,58,59] has been analyzed in the SOVapp roach in aseries of works [60,61,62].10
Let usnoticethatthe condition t(λ)∈ΣT≡ NDp,Nimpliesthatthe p×pmatrixD(λ)hasrank2l
foranyλ∈C\{0}. Letusdenotewith
Ci,j(λ) = (−1)i+jdet
2lDi,j(λ) (4.31)
the(i,j)cofactorof the matrix D(λ); then the matrix formedout of these cofactorshasrank 1, i.e.
all thevectors:
Vi(λ)≡(Ci,1(λ),Ci,2(λ),...,Ci,2l+1(λ))T∈Cp∀i∈ {1,...,2l+1}(4.32)
areproportional:
Vi(λ)/Ci,1(λ) =Vj(λ)/Cj,1(λ)∀i,j∈ {1,...,2l+1},∀λ∈C. (4.33)
Theproportionality(4.33)oftheeigenvectorsV i(λ)implies:
C2,2(λ)/C2,1(λ) =C1,2(λ)/C1,1(λ) (4.34)
which,byusingtheproperty(A.69),canberewrittenas:
C1,1(λq)/C1,2l+1(λq) =C1,2(λ)/C1,1(λ). (4.35)
Moreover,thefirst elementinthe vectorialcondition D(λ)V1(λ) =0¯reads:
t(λ)C1,1(λ) =a(λ)C1,2l+1(λ)+d(λ)C1,2(λ). (4.36)
Let us note that from the form of a(λ),d(λ)andt(λ)∈ΣTit follows that all the cofactors are
Laurentpolynomialofmaximaldegree122lNinλ:
Ci,j(λ) = Ci,jλ−2lN+ai,j4lN−(ai,j+bi,j)/productdisplay
h=1(λ(i,j)
h−λ). (4.37)
In Lemma 5, we show that the equations (4.35) and (4.36) imply that if C 1,1(λ)has a common
zero with C 1,2(λ)then this is also a zero of C 1,2l+1(λ)and that the same statement holds ex-
changing C 1,2(λ)with C 1,2l+1(λ). So we can denote with C1,1C1,1(λ),C1,2l+1C1,2l+1(λ)and
C1,2C1,2(λ)the polynomials of maximal degree 4lNobtained simplifying the common factors in
C1,1(λ), C1,2l+1(λ)and C1,2(λ). Then,byequation(4.35),theyhavetosatisfythe relation s:
C1,2l+1(λ) =q¯N1,1C1,1(λq−1),C1,2(λ) =q−¯N1,1C1,1(λq)andC1,2l+1=ϕC1,1,(4.38)
whereϕ≡C1,1/C1,2and¯N1,1is the degree of the polynomial C1,1(λ). So that equation (4.36)
assumestheformofa Baxterequationin thepolynomial C1,1(λ):
t(λ)C1,1(λ) = ¯a(λ)C1,1(λq−1)+¯d(λ)C1,1(λq), (4.39)
12Theai,jandbi,jare non-negative integers and λ(i,j)
h/ne}ationslash= 0for anyh∈ {1,...,4lN−(ai,j+bi,j)}.11
with coefficients ¯a(λ)≡q¯N1,1ϕa(λ)and¯d(λ)≡q−¯N1,1ϕ−1d(λ). Note that the consistence of the
aboveequationimpliesthat ϕisap-rootoftheunity. Indeed,denotingwith ¯D(Λ)thematrixdefined
asin(3.21) butwithcoefficients ¯a(λ)and¯d(λ), equation(4.39) implies:
0 = det
p¯D(Λ)≡(ϕp−1)/parenleftiggp/productdisplay
h=1a(λqh)−ϕ−pp/productdisplay
h=1a(−λqh)/parenrightigg
. (4.40)
The expansionfor detp¯D(Λ)in (4.40) is derivedby using the expansion(3.23) for detp¯D(Λ), the
formulae13:
det
2lD1,1(λ) = det
2lD1,1(λ), (4.41)
det
2l−1D(1,2),(1,2)(λ) = det
2l−1D(1,2),(1,2)(λ), (4.42)
det
2l−1D(1,2l+1),(1,2l+1)(λ) = det
2l−1D(1,2l+1),(1,2l+1)(λ), (4.43)
andthecondition t(λ)∈ΣT. Finally,if wedefine:
Qt(λ)≡λaC1,1(λ), (4.44)
whereq−a=q¯N1,1ϕwitha∈ {0,..,2l},we getthestatementofthetheorem.
Remark2. Theprevioustheoremimpliesthatforany t(λ)∈ΣTthepolynomialsolution Qt(λ)of
theBaxterequationcanberelatedtothedeterminantofatri diagonalmatrixoffinitesize p−1. Note
thatthe spectrumoftheSine-Gordonmodelinthecase ofirra tionalcoupling ¯β2shouldbededuced
fromβ2=p′/prational in the limit β2→¯β2. In particular, this implies that underthis limit ( p→
+∞)thedimensionoftherepresentationdivergesaswellasthe sizeofthetridiagonalmatrixwhose
determinant is associated to the solution Qt(λ)of the Baxter equation. It is then relevant to point
out that in the case of the quantum periodic Toda chain the sol utions of the corresponding Baxter
equationareexpressedintermsofdeterminantsofsemi-infi nitetridiagonalmatrices[63,13, 64].
It is worth noticing that the set of polynomials Qt(λ), introducedin the previoustheorem,admitsa
moreprecisecharacterization:
Theorem 3. Lett(λ)∈ΣTthent(λ)defines uniquely up to normalization a polynomial solution
Qt(λ)oftheBaxterfunctionalequation(4.30) ofmaximaldegree 2lN.
Inthecase Nodd,it results:
Qt(0)≡Q0/\e}atio\slash= 0,andlim
λ→∞λ−2lNQt(λ)≡Q2lN/\e}atio\slash= 0. (4.45)
In the case Neven, the condition (4.45) selects t(λ)∈Σ0
Twhile fort(λ)∈Σk
Twithk∈ {1,...,l}
we havethecharacterization Q0=Q2lN= 0and:
lim
λ→0Qt(λq)
Qt(λ)=q±k,lim
λ→∞Qt(λq)
Qt(λ)=q−(N±k). (4.46)
13They follow from the tridiagonality of these matrices and by using Lemma3.12
Proof.Thankstoformula(A.74),thecofactor C 1,1(λ)∈C[λ,λ−1]2lNiseveninλandso it admits
theexpansions:
C1,1(λ) = C1,1λ−2lN+2˜a1,12lN−(˜a1,1+˜b1,1)/productdisplay
i=1(λ(1,1)
i−λ)(λ(1,1)
i+λ).(4.47)
Let us note now that by using the properties(A.69) and (A.74) , the relation (4.34) can be rewritten
as:
C1,1(λq)C1,1(λ) =qNC1,2(λ)C1,2(−λ). (4.48)
Usingthat andthegeneralrepresentation(4.37)forthe cof actor C 1,2(λ), weget:
a1,2= 2˜a1,1≡2a,b1,2= 2˜b1,1≡2b,C2
1,2=C2
1,1q−2(N+b)(4.49)
and:/parenleftig
λ(1,1)
i/parenrightig2
=/parenleftig
λ(1,2)
i/parenrightig2
≡¯λ2
i,/parenleftig
λ(1,2)
i+2lN−(a+b)/parenrightig2
=/parenleftbig¯λi/q/parenrightbig2(4.50)
with¯λi/\e}atio\slash= 0for anyi∈ {1,...,2lN−(a+b)}withaandb∈Z≥0. Note that the equation (4.49)
andthefactthat ϕ≡C1,1/C1,2isap-rootofthe unityimply ϕ=qb+N. Thenwecanwrite:
C1,1(λ) = Cλ−2lN+2a2lN−(a+b)/productdisplay
i=1(¯λi+λ)(¯λi−λ), (4.51)
C1,2(λ) =qaCλ−2lN+2a2lN−(a+b)/productdisplay
i=1(¯λi+λ)((−1)H(x−i)¯λi−λq), (4.52)
whereC≡C1,1andH(n)≡ {0forn <0,1forn≥0}is the Heaviside step function. Here, x
isanon-negativeintegerwhichisfixedtozerothankstoform ula(4.38). Thenthesolution Qt(λ)of
theBaxter equation(4.30) belongsto C[λ]2lNandhastheform:
Qt(λ)≡λa2lN−(a+b)/productdisplay
i=1(¯λi−λ). (4.53)
Let usshow nowthe remainingstatementsof thetheoremconce rningthe asymptoticsof Qt(λ). To
thisaimwe computethe limits:
lim
logλ→∓∞λ±2lNC1,1(λ) = det
2l/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δi,j+1+q(1∓1)N/2δi,j−1−(qk+q−k)δeN,1δi,j/vextenddouble/vextenddouble/vextenddouble
i/ne}ationslash=1,j/ne}ationslash=1
×N/productdisplay
h=1(κhξ±1
h
i)2l= (δeN,1(1+(2l+1)δk,0)−1)N/productdisplay
h=1(κhξ∓1
h
i)2l,(4.54)
whichimply:
a=b= 0, (4.55)13
forNoddandNevenwitht(λ)∈Σ0
T,i.e. thecondition(4.45). Inthe remainingcases, Nevenand
t(λ)/∈Σ0
T,the sameformulaimplies:
a/\e}atio\slash= 0,b/\e}atio\slash= 0, (4.56)
sothatQ0=Q2lN= 0,whiletheasymptoticsbehaviors(4.46)simplyfollowtaki ngtheasymptotics
oftheBaxterequationsatisfied by Qt(λ).
5.Q-operator: Existence andcharacterization
Let us denote with Σtthe eigenspace of the transfer matrix T(λ)corresponding to the eigenvalue
t(λ)∈ΣT,then:
Definition 1. LetQ(λ)betheoperatorfamily definedby:
Q(λ)|t/a\}bracketri}ht ≡Qt(λ)|t/a\}bracketri}ht ∀|t/a\}bracketri}ht ∈Σtand∀t(λ)∈ΣT, (5.57)
withQt(λ)the element of C[λ]2lNcorresponding to t(λ)∈ΣTby the injection defined in the
previoustheorem.
Under the assumptions ξandκreal or imaginarynumbers, which assure the self-adjointne ssof the
transfermatrix T(λ)forλ∈R,thefollowingtheoremholds:
Theorem4. Theoperatorfamily Q(λ)isaBaxter Q-operator:
(A)Q(λ)satisfieswith T(λ)thecommutationrelations:
[Q(λ),T(µ)] = [Q(λ),Q(µ)] = 0∀λ,µ∈C, (5.58)
plusthe Baxterequation:
T(λ)Q(λ) =a(λ)Q(λq−1)+d(λ)Q(λq)∀λ∈C. (5.59)
(B)Q(λ)isa polynomialofdegree 2lNinλ:
Q(λ)≡2lN/summationdisplay
n=0Qnλn,
with coefficients Qnself-adjointoperators.
(C)Inthecase Nodd,the operator Q2lN=idandQ0isaninvertibleoperator.
(D)Inthecase Neven,Q(λ)commuteswiththe Θ-chargeandtheoperator Q2lNistheorthogonal
projectionontothe Θ-eigenspacewith eigenvalue1. Q0hasnon-trivialkernel coincidingwith
theorthogonalcomplementto the Θ-eigenspacewith eigenvalue1.14
Proof.Note that the self-adjointness of the transfer matrix T(λ)implies that Q(λ)is well defined,
indeed its action is defined on a basis. The property (A) is a tr ivial consequence of Definition 1.
Notethat theinjectivityofthemap t(λ)∈ΣT→Qt(λ)∈C[λ]2lNimplies:
(Qt(λ))∗=Qt(λ∗)∀λ∈C (5.60)
being(a(λ))∗=d(λ∗)and(t(λ))∗=t(λ∗). So we get the Hermitian conjugation property
(Q(λ))†=Q(λ∗), i.e. the self-adjointness of the operators Qn. The properties (C) and (D) of
the operators Q0andQ2lNdirectly follow from the asymptotics of the eigenfunction Qt(λ)while
thecommutativityof Q(λ)andΘisa directconsequenceofthecommutativityof T(λ)andΘ.
6. Conclusion
Intheprevioussectionwehaveshownthatbyonlyusingthech aracterizationofthespectrumofthe
transfer matrix obtained by the SOV method we were able to rec onstruct the Q-operator. It is also
interestingto pointoutastheresultsderivedin [1]togeth erwiththoseofthepresentarticleyield:
Theorem5. Thefamily Qwhichcharacterizesthequantumintegrabilityofthelatti ceSine-Gordon
model(see definition(1.1)) isdescribedby thetransfermat rixT(λ)fora chainwith Noddnumber
of siteswhile by T(λ)plustheΘ-chargefora chainwith Nevennumberof sites.
Proof.LetusstartnoticingthatProposition3andTheorem4of[1]a rederivedonlyusingtheSOV
method (i.e. without any assumption about the existence of t heQ-operator). So only using SOV
analysis we have derived that for Nodd the transfer matrix T(λ)has simple spectrum while for
Neven this is true for T(λ)plus theΘ-charge; i.e. they define a complete family of commuting
observables and so satisfy the properties(A) and (C) of the d efinition (1.1). In this article we have
moreover shown that the Q-operator is defined as a function of the transfer matrix whic h implies
the property(B) of (1.1) recalling that in [1] the time-evol utionoperator Uhas been expressed as a
functionofthe Q-operator.
Let us shortly point out the main features required in abstra ct to extend to cyclic representationsof
other integrable quantum models the same kind of spectrum ch aracterization derived here for the
lattice Sine-Gordonmodel.
R1.The model admits an SOV description and the spectrum of the tr ansfer matrix can be charac-
terizedbyasystem ofBaxter-likeequationsin the T-wave-function Ψ(η) =/a\}bracketle{tη|t/a\}bracketri}ht:
t(ηr)Ψ(η) =a(ηr)Ψ(η1,...,q−1ηr,...,η N)+d(ηr)Ψ(η1,...,qη r,...,η N),(6.61)
where(η1,...,ηN)∈BNwithBNtheset ofzerosofthe B-operatorintheSOV representation.
Here,theparameter qisa rootofunitydefinedasin (2.6) and(2.7).15
Note that for cyclic representationsof an integrable quant um model the set BNis a finite subset of
CN. So the coefficients a(ηr)andd(ηr)are specified only in a finite number of points where they
satisfy thefollowingaveragevaluerelations14:
A(ηp
r) =p/productdisplay
k=1a(qkηr),D(ηp
r) =p/productdisplay
k=1d(qkηr). (6.62)
HereA(Λ)andD(Λ)are the average values of the operator entries A(λ)andD(λ)of the mon-
odromy matrix. Let us recall that the operator entries of the monodromymatrix are expected to be
polynomials(orLaurentpolynomials)inthespectralparam eterλsothecorrespondingaverageval-
uesarepolynomials(orLaurentpolynomials)in Λ≡λp. Itisthennaturaltointroducethefunctions
a(λ)andd(λ)aspolynomial(orLaurentpolynomial)solutionsofthefoll owingaveragerelations:
A(Λ)+γB(Λ) =p/productdisplay
k=1a(qkλ),D(Λ)+δB(Λ) =p/productdisplay
k=1d(qkλ), (6.63)
whereB(Λ)istheaveragevalueoftheoperator B(λ)andγandδare constanttobe fixed.
R2.Let usdenotewith Zf(λ)the set ofthezerosofthefunctions f(λ), then:
∃λ0∈Za(λ):λ0/∈ ∪2l−1
h=0Zd(λqh). (6.64)
R3.Theaveragevaluesofthefunctions aanddarenotcoincidinginallthezerosofthe B-operator:
A(ηp
a)/\e}atio\slash=D(ηp
a)∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN. (6.65)
The requirement R1yields the introduction of the p×pmatrixD(λ), defined as in (3.21), by
the functions a(λ)andd(λ)solutions of (6.63). This should allow us to reformulate the spectral
problem for the transfer matrix as the problem to classify al l the solutions t(λ)to the functional
equationdetpD(Λ) = 0ina modeldependentclassoffunctions.
The requirement R2implies that the rank of the matrix D(λ)is almost everywhere 2l. Indeed, the
condition (6.64) implies C 1,p(λ0)/\e}atio\slash= 0, independently from the function t(λ). Being the cofactor
C1,p(λ)acontinuousfunctionofthespectralparametertheabovest atementontherankofthematrix
D(λ)follows. Underthisconditionwecanfollowtheprocedurepr esentedinTheorem2toconstruct
the solutionsof the Baxter equation. Then the self-adjoint nessof the transfer matrix Tallows us to
proceedasinsection5to showthe existenceofthe Q-operatorasa functionof T.
The requirement R3is a sufficient criterion15to show the simplicity of the spectrum of Twhich
should imply that the full integrable structure of the quant um model should be described by the
14Theequations in (6.62) are trivial consequences of the SOVr epresentation and of the cyclicity.
15It is worth noticing that in the case of the Sine-Gordon model the criterion R3does not apply to the representations
withun=vn= 1. Nevertheless, we have shown the simplicity of Tby using some model dependent properties of the
coefficients a(λ)andd(λ), see section 5of [1].16
transfermatrixassoonastheproperty(B)indefinition(1.1 )isshownforthemodelunderconsider-
ation.
Following the schema here presented, in a future publicatio n we will address the analysis of the
spectrumfortheso-called α-sectorsoftheSine-Gordonmodel(see[1]). Theuseofthisapproachi s
in particularrelevantin these sectorsof theSine-Gordonm odelbecausea direct constructionof the
Q-operatorleadstosometechnicaldifficulty.
A. Properties ofthecofactors C i,j(λ)
Let usconsideran M×Mtridiagonal matrix16O:
O≡
z1y10··· 0 0
x1z2y20··· 0
0x2z3y3...
.........
...... 0
0...0xM−2zM−1yM−1
0 0...0xM−1zM
(A.66)
i.e. a matrix with non-zero entries only along the principal diagonal and the next upper and lower
diagonals.
Lemma 3. The determinantof atridiagonalmatrix is invariantundert he transformation ̺αwhich
multiplies for αthe entries above the diagonal and for α−1the entries below the diagonal leaving
theentriesonthediagonalunchanged.
Proof.Letusnotethatthedeterminantofa tridiagonalmatrixadmi tsthefollowingexpansion:
det
MO=z1det
M−1O1,1+x1y1det
M−2O(1,2),(1,2), (A.67)
wherewe haveused thesame notationsintroducedafterformu la(3.23). By usingit, we getthat the
actionof̺αreads:
det
M̺α(O) =z1det
M−1̺α(O)1,1+x1y1det
M−2̺α(O)(1,2),(1,2). (A.68)
Then the statement follows by induction noticing that the tr ansformation ̺αleaves always un-
changedthedeterminantofa 2×2matrix.
16An interesting analysis of the eigenvalue problem for tridi agonal matrices is presented in [65].17
Lemma 4. Thefollowingpropertieshold:
Ch+i,k+i(λ) =Ch,k(λqi)∀i,h,k∈ {1,...,2l+1}, (A.69)
and:
C1,1(λ) =C1,1(−λ)andC2,1(λ) =qNC1,2(−λ). (A.70)
Proof.Note that by the definition (4.31) of the cofactors C i,j(λ)the equations (A.69) are simple
consequencesof qp= 1andareprovenexchangingrowsandcolumnsin thedeterminan ts.
Let us provenow that the cofactor C 1,1(λ) = det 2lD1,1(λ)is an even function of λ. The tridiago-
nalityofthematrix D1,1(λ)allowsusto usethepreviouslemma:
C1,1(λ)≡det
2l/vextenddouble/vextenddoublet(λqh)δh,k−a(λqh)δh,k+1−qNa(−λqh+1)δh,k−1/vextenddouble/vextenddouble
h>1,k>1
= det
2l/vextenddouble/vextenddoublet(λqh)δh,k−qNa(λqh)δh,k+1−a(−λqh+1)δh,k−1/vextenddouble/vextenddouble
h>1,k>1
= det
2l/vextenddouble/vextenddoublet(λqh)δh,k−d(−λqk)δk,h−1−a(−λqk)δk,h+1/vextenddouble/vextenddouble
h>1,k>1
≡det
2l(D1,1(−λ))T=C1,1(−λ). (A.71)
To provenowthesecondrelationin (A.70) weexpandthe cofac tors:
C2,1(λ) =2l+1/productdisplay
h=2a(λqh)+d(λ) det
2l−1D(1,2),(1,2)(λ), (A.72)
C1,2(λ) =2l/productdisplay
h=1d(λqh)+a(λq) det
2l−1D(1,2),(1,2)(λ). (A.73)
Byusingthesamestepsshownin(A.71),thetridiagonalityo fthematrixD (1,2),(1,2)(λ)impliesthat
its determinant is an even function of λfrom which the statement C 2,1(λ) =qNC1,2(−λ)follows
recallingthat d(λ) =qNa(−λq).
Remark 3. Inthisarticlewe needonlytheproperties(A.70);however, it isworthpointingoutthat
theyarespecialcasesofthefollowingpropertiesofthe cof actors:
Ci,j(λ) =qN(i−j)Cj,i(−λ)∀i,j∈ {1,...,2l+1}. (A.74)
Theproofof(A.74)canbedonesimilarlytothatof(A.70) but we omitit forsimplicity.
Let ususe onceagainthenotation Zfforthe set ofthezerosofafunction f(λ), then:
Lemma 5. Theequations(4.35)and(4.36) imply:
ZC1,1∩ZC1,2≡ZC1,1∩ZC1,2l+1. (A.75)18
Proof.The inclusions/parenleftbig
ZC1,1∩ZC1,2/parenrightbig
\Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
ZC1,1∩ZC1,2l+1/parenrightbig
\Zd⊂
ZC1,1∩ZC1,2triviallyfollowbyequation(4.36).
Let us observe now that C 1,2(λq−1)has no common zero with a(λ)and that C 1,2l+1(λq)has no
common zero with d(λ). These statements simply follow from (A.73), (A.69)and(A. 72) when we
recall that a(λ)has no common zero with/producttext2l−1
h=0d(λqh)and thatd(λ)has no common zero with/producttext2l+1
h=2a(λqh). So,if/parenleftbig
ZC1,1∩ZC1,2/parenrightbig
∩Zaisnotemptyand λ0∈/parenleftbig
ZC1,1∩ZC1,2/parenrightbig
∩Za,theequation
(4.35) computed in λ=q−1λ0implies C 1,2l+1(λ0) = 0being C 1,2(λ0q−1)/\e}atio\slash= 0, i.e.λ0∈
ZC1,1∩ZC1,2l+1. Similarly,if/parenleftbig
ZC1,1∩ZC1,2l+1/parenrightbig
∩Zdisnotemptyand λ0∈/parenleftbig
ZC1,1∩ZC1,2l+1/parenrightbig
∩Zd,
the equation (4.35) computed in λ=λ0implies C 1,2(λ0) = 0being C 1,2l+1(λ0q)/\e}atio\slash= 0, i.e.λ0∈
ZC1,1∩ZC1,2. So that (4.35) implies the inclusions/parenleftbig
ZC1,1∩ZC1,2/parenrightbig
∩Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
ZC1,1∩ZC1,2l+1/parenrightbig
∩Zd⊂ZC1,1∩ZC1,2inthiswaycompletingthe proofofthelemma.
B. Scalarproduct inthe SOV space
Here is described as a natural structure of Hilbert space can be provided to the linear space of the
SOV representationbypreservingtheself-adjointnessoft hetransfermatrix.
B.1 CyclicrepresentationsoftheWeylalgebra
Here,we considerthecyclicrepresentationsoftheWeyl alg ebraW(n)
qinthecase:
up
n=vp
n= 1forβ2=p′/pwithp′evenandp= 2l+1odd. (B.76)
At anysitenofthechain,weintroducethe quantumspace Rnwithvn-eigenbasis:
vn|k,n/a\}bracketri}ht=qk|k,n/a\}bracketri}ht ∀|k,n/a\}bracketri}ht ∈Bn={|k,n/a\}bracketri}ht,∀k∈ {−l,...,l}}. (B.77)
Note that the eigenvaluesof vndescribe the unit circle Sp={qk:k∈ {−l,...,l}},indeedql+1=
q−l. OnRnisdefinedap-dimensionalrepresentationoftheWeyl algebrabysetting :
un|k,n/a\}bracketri}ht=|k+1,n/a\}bracketri}ht ∀k∈ {−l,...,l} (B.78)
with thecyclicitycondition:
|k+p,n/a\}bracketri}ht=|k,n/a\}bracketri}ht. (B.79)
B.2 Representationin the SOVbasis
The analysis developed in [1] define recursively the eigenba sis{|¯η1qh1,...,¯ηNqhN/a\}bracketri}ht}of theB-
operator in the original representation, i.e. as linear com binations of the elements of the basis
{|h1,...,hN/a\}bracketri}ht ≡/circlemultiplytextN
n=1|hn,n/a\}bracketri}ht}, where|hn,n/a\}bracketri}htare the elements of the vn-eigenbasis defined in
(B.77). To writethischangeofbasisin amatrixformlet usin troducethe followingnotations:
|yj/a\}bracketri}ht ≡ |¯η1qh1,...,¯ηNqhN/a\}bracketri}htand|xj/a\}bracketri}ht ≡ |h1,...,hN/a\}bracketri}ht (B.80)19
where:
j:=h1+N/summationdisplay
a=2(2l+1)(a−1)(ha−1)∈ {1,...,(2l+1)N}, (B.81)
notethat thisdefinesa oneto onecorrespondencebetween N-tuples(h1,...,hN)∈ {1,...,2l+1}N
and integers j∈ {1,...,(2l+1)N}, which just amountsto chose an orderingin the elementsof th e
two basis. Underthisnotation,wehave:
|yj/a\}bracketri}ht=W|xj/a\}bracketri}ht=(2l+1)N/summationdisplay
i=1Wi,j|xi/a\}bracketri}ht, (B.82)
where we are representing |xj/a\}bracketri}htas the vector |j/a\}bracketri}htin the natural basis in C(2l+1)NandW=||Wi,j||
is a(2l+1)N×(2l+1)Nmatrix. The matrix Wis defined by recursion in terms of the kernel K
constructedinappendixCof[1], letususethenotation:
K({h1,...,hN},k1,{k2,...,kN})≡KN(η|χ2;χ1), (B.83)
whereweareconsideringthecase N−M = 1. Thenthe recursionreads:
W(N)
i,j=2l+1/summationdisplay
k2,...,kN=1K({h1(j),...,hN(j)},h1(i),{k2,...,kN})W(N−1)
¯h(i),a(k2,...,kN),(B.84)
where we have introduced the index (N)and(N−1)in the matrices Wto make clear the step
of the recursion. Here, (h1(j),...,hN(j))is the unique N-tuples corresponding to the integer j∈
{1,...,(2l+ 1)N}andh1(i)is the first entry in the unique N-tuples corresponding to the integer
i∈ {1,...,(2l+1)N}. Moreover,wehavedefined:
¯h(i) := 1+i−h1(i)
2l+1∈ {1,...,(2l+1)(N−1)}anda(k2,...,kN) =k2+N/summationdisplay
a=3(2l+1)(a−2)(ka−1),
(B.85)
Remarks:
a)Underthechangeofbasis {|xj/a\}bracketri}ht} → {|yj/a\}bracketri}ht}thegenericoperatorX transformsforsimilarity:
XSOV≡W−1XW, (B.86)
so fromtheactionofthezerooperators ηaandtheshift operators T±
aontheB-eigenbasis |yj/a\}bracketri}ht:
ηa|yj/a\}bracketri}ht= ¯ηaqha(j)|yj/a\}bracketri}htandT±
a|yj/a\}bracketri}ht=|yj±(2l+1)(a−1)/a\}bracketri}ht (B.87)
we havethat:
(ηa)SOV= ¯ηa||qha(j)δi,j||and/parenleftbig
T±
a/parenrightbig
SOV=||δi,j±(2l+1)(a−1)||. (B.88)20
Fromtheaboveexpressionwe have17:
(ηa)†
SOV= (ηa)∗
SOVand/parenleftbig
T±
a/parenrightbig†
SOV=/parenleftbig
T∓
a/parenrightbig
SOV. (B.89)
b) The known transformation properties of the entries of the monodromy matrix in the original
representationimply:
/parenleftbiggDSOV(λ)CSOV(λ)
BSOV(λ)ASOV(λ)/parenrightbigg
=/parenleftigg
G−1(ASOV(λ∗))†G−G−1(BSOV(λ∗))†G
−G−1(CSOV(λ∗))†G G−1(DSOV(λ∗))†G/parenrightigg
,(B.90)
withGisapositiveself-adjointmatrixdefinedby G:=W†W.
c)Thequantumdeterminantrelationis invariantundersimi laritytransformationsandso we have:
a(λ)d(λq−1) =ASOV(λ)DSOV(λq−1)−BSOV(λ)CSOV(λq−1), (B.91)
Lemma 6. Thebasis {|yj/a\}bracketri}ht}isnotanorthogonalbasisw.r.t. thenaturalscalarproduct on{|xj/a\}bracketri}ht}.
Proof.Note that the condition {|yj/a\}bracketri}ht}is an orthogonal basis is equivalent to the statement Gis
a diagonal matrix (with positive diagonal entries). Let us r ecall that the Hermitian conjugation
propertyofB(λ)togetherwiththeYang-Baxtercommutationrelationsimply :
[B†(λ),B(µ)] = [B(µ),C(λ∗)] =q−q−1
λ∗/µ−µ/λ∗(A(λ∗)D(µ)−A(µ)D(λ∗))/\e}atio\slash= 0 (B.92)
that is the operator B(λ)is not a normal operator. Now let us show that the non-normali tyofB(λ)
impliesthat Gisnotdiagonal. Indeed,wecanwrite:
[B†(λ),B(µ)] =/parenleftbig
W†/parenrightbig−1(BSOV(λ))†GBSOV(µ)W−1−WBSOV(µ)G−1(BSOV(λ))†W†
=W(G−1(BSOV(λ))†GBSOV(µ)−BSOV(µ)G−1(BSOV(λ))†G)W−1.(B.93)
Notenowthatifweassume Gdiagonal,then Gcommutesbothwith BSOV(λ)andwith(BSOV(λ))†,
being all diagonal matrices in the SOV representation, whic h implies the absurd [B†(λ),B(µ)] =
0.
B.3 Scalarproductin theSOVspace
The self-adjointness of the family T(λ)implies that the transfer matrix eigenstates are orthogona l
undertheoriginalscalar product:
δi,j= (|ti/a\}bracketri}ht,|tj/a\}bracketri}ht), (B.94)
we have chosen the orthonormal ones. Note that the above equa tion naturally induces a scalar
productintheSOV representationobtainedunderchangeofb asis:
(|b/a\}bracketri}ht,|a/a\}bracketri}ht)SOV≡(G|b/a\}bracketri}ht,|a/a\}bracketri}ht) (B.95)
17Here, weare using the standard notation for the adjoint X†≡(X∗)t.21
thatisascalarproductforwhichtheadjointofavector |a/a\}bracketri}htisthenaturaladjointtimesthematrix G:
|b/a\}bracketri}ht†SOV≡ /a\}bracketle{tb|Gwith/a\}bracketle{tb|=/parenleftig
(|b/a\}bracketri}ht)t/parenrightig∗
, (B.96)
andsoforthegenericoperator Xwehave:
X†SOV≡G−1X†G. (B.97)
It istrivialtonoticethat:
Lemma 7. The family of operators TSOV(λ)is self-adjoint w.r.t. †SOVand the eigenstates
|tj/a\}bracketri}htSOV≡W−1|tj/a\}bracketri}htare orthonormal w.r.t. the scalar product defined in (B.95). Moreover, it
results:
/parenleftigg
(ASOV(λ∗))†SOV(BSOV(λ∗))†SOV
(CSOV(λ∗))†SOV(DSOV(λ∗))†SOV/parenrightigg
=/parenleftbiggDSOV(λ)−CSOV(λ)
−BSOV(λ)ASOV(λ)/parenrightbigg
.(B.98)
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