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YANG–BAXTER SYMMETRY IN INTEGRABLE MODELS:

arXiv:hep-th/9303052v1 9 Mar 1993PAR LPTHE 93/07UPRF −93 −370February 1993YANG–BAXTER SYMMETRY IN INTEGRABLE MODELS:NEW LIGHT FROM THE BETHE ANSATZ SOLUTION⋆C. DestriDipartimento di Fisica, Universit`a di Parma,and INFN, Gruppo Collegato di Parma†H.J.

de VegaLaboratoire de Physique Th´eorique et Hautes Energies‡, Paris§ABSTRACT⋆Work supported by the CNR/CNRS exchange program† E–mail: destri@parma.infn.itmail address:Dipartimento di FisicaViale delle Scienze, 43100 Parma, ITALIA‡ Laboratoire Associ´e au CNRS UA 280§ E–mail: devega@lpthe.jussieu.frmail address:L.P.T.H.E., Tour 16, 1er ´etage, Universit´e Paris VI,4, Place Jussieu, 75252, Paris cedex 05, FRANCE

We show how any integrable 2D QFT enjoys the existence of infinitely manynon–abelian conserved charges satisfying a Yang–Baxter symmetry algebra. Thesecharges are generated by quantum monodromy operators and provide a represen-tation of q−deformed affine Lie algebras.

We review and generalize the work of deVega, Eichenherr and Maillet on the bootstrap construction of the quantum mon-odromy operators to the sine–Gordon (or massive Thirring) model, where suchoperators do not possess a classical analogue.Within the light–cone approachto the mT model, we explicitly compute the eigenvalues of the six–vertex alter-nating transfer matrix τ(λ) on a generic physical state, through algebraic Betheansatz. In the thermodynamic limit τ(λ) turns out to be a two–valued periodicfunction.

One determination generates the local abelian charges, including energyand momentum, while the other yields the abelian subalgebra of the (non–local)YB algebra. In particular, the bootstrap results coincide with the ratio betweenthe two determinations of the lattice transfer matrix.1.

IntroductionWhen a physical model is integrable it always possess extra conserved quan-tities not related to manifest symmetries but presumably with hidden dynamicalsymmetries. For 2D lattice models and 2D quantum field theory (QFT), integra-bility is a consequence of the Yang-Baxter equation (YBE).In lattice vertex models, using the R−matrix elements to define the local sta-tistical weights, the monodromy matrix Tab(λ) for a lattice line obeys the YBalgebra:R(λ −µ) [T(λ) ⊗T(µ)] = [T(µ) ⊗T(λ)] R(λ −µ)(1.1)where λ stands for the spectral parameter.

Its trace, t(λ) ≡Pa Taa(λ), providesa commuting family of operators,[ t(λ), t(µ) ] = 0for any lattice size. In general the transfer matrix t(λ) is conserved, since, among all2

commuting charges, it generates also the Hamiltonian. It is now clear from eq.

(1.1),that the Tab(λ) do not commute with t(λ) and are generally not conserved. Thatis, for finite lattice size we have only an infinite abelian symmetry generated byt(λ) (or by its series expansion coefficients).In ref.

[1], a bootstrap construction of monodromy matrices Tab(u) was proposedin a class of integrable QFT. These Tab(u) are conserved and obey a YB algebraanalogous to eq.(1.1).

Hence, this class of QFT enjoy an infinite YB non–abeliansymmetry generated by the Tab(u). (This construction is valid in the infinite space).A classically conserved limit of Tab(u) exists in the class of models considered inref.

[1] where the R−matrix is a rational function of u.In the present paper we first show that the bootstrap construction of conservedTab(u) generalizes to integrable models with trigonometric R−matrices such as thesine-Gordon or massive Thirring model. In such cases the classical limit is abelian,as shown explicitly in sec.3.The main aim of this work is then to investigate and clarify, from a microscopicpoint of view, the problem of unveiling the existence of the infinite YB symmetry ofthe sG–mT model.

In other words, since lattice models provide regularized versionof QFT, we seek an explicit connection between the lattice and the bootstrap YBalgebras. For this purpose we adopt the so–called light-cone approach, which isa general method to precisely derive QFT’s as scaling limits of integrable latticemodels [2,3].

One starts from a diagonal-to-diagonal lattice with lines at angles2Θ. The light-cone evolution operators UR and UL are introduced (eq.

(5.2)) whichdefine the lattice hamiltonian and momentum (eq.(5.3)). They can be expressedin terms of the values at λ = ±Θ of the alternating transfer matrix t(λ, Θ) (eqs.(5.5)–(5.6)).

Through algebraic Bethe Ansatz the ground state and excited statesare constructed in the thermodynamic limit. When the ground state is antifer-romagnetic, it corresponds to a Dirac sea of interacting pseudoparticles.

Excitedstates around it describe particle-like physical excitations. In this way, the sG–mTmodel is obtained from the six-vertex model (with anisotropy γ)[2].

QFT like mul-3

ticomponent Thirring models, sigma models and others follow from various vertexmodels [3].In order to investigate the operators present in such QFT, it is important tolearn how the monodromy operators Tab(λ, Θ) act on physical states. In the presentpaper, we explicitly compute the eigenvalues of the alternating six–vertex transfermatrix t(λ, Θ) ≡Pa Taa(λ, Θ), on a generic n−particle state, in the thermody-namic limit.

The explicit formulae are given by eqs. (6.32),(6.37) and (6.38).

Theeigenvalues of t(λ, Θ) turn out to be iπ−periodic and multi–valued functions of λ,each determination of t(λ, Θ) being a meromorphic function of λ. We call tII(λ, Θ)and tI(λ, Θ) the determinations associated with the periodicity strips closer to thereal axis (see fig.

4). The ground–state contribution exp[−iG(λ)V ] is exponentialon the lattice size, as expected, whereas the excited states contributions are finiteand express always in terms of hyperbolic functions [see sec.

6].We then compare these Bethe Ansatz eigenvalues with the eigenvalues of thebootstrap transfer matrix τ(u) ≡Pa Taa(u). Remarkably enough, we find thefollowing simple relation between the two results, for 0 < γ < π/2 (repulsiveregime),τ(u) = tIIγπu −iγ2, ΘtIγπu −iγ2, Θ−1(1.2)where tII(λ, Θ) and tI(λ, Θ) have been normalized to one on the ground state.

Thus, we succeed in connecting the bootstrap transfer matrix τ(u) of the sG-mT model with the alternating transfer matrix t(λ, Θ) of the six vertex model.In the thermodynamic limit τ(u) coincide with the jump between the two maindeterminations of t(λ, Θ) . Notice the renormalization of the rapidity by γ/π andthe precise overall shift by iγ/2 in the argument in order the equality to hold.We find in addition that t(λ, Θ), for 0 < Im λ < γ/2, generates the hamiltonianand momentum togheter with an infinite number of higher–dimension and higher–spin conserved abelian charges, through expansion in powers of e±πλ/γ.

We seetherefore that the same bare operator generates two kinds of conserved quantities.4

Energy and momentum as well the higher–spin abelian charges are local in thebasic fields which interpolate physical particles, whereas the infinite set of chargesobtained from the jump from tII(λ, Θ) to tI(λ, Θ) are nonlocal in the same fields.The fact that local and nonlocal charges come from different sides of a naturalboundary, clearly shows that they carry independent information. That is, onecannot produce the nonlocal charges from the sole knowledge of the local charges.We also recall that the monodromy matrix T(λ, Θ) can be written in terms of thelattice fermi fields of the mT model [2], so that local and non local charges doadmit explicit expressions in terms of local field operators.As a byproduct of this analysis, we find an explicit relation for the light-conelattice hamiltonian and momentum P± in terms of the continuum hamiltonian andmomentum p± plus an infinite series of higher conserved continuum charges I±j ,playing the rˆole of irrelevant operators,P± = (P±)V + p± + m4∞Xj=1ma42jI±j(1.3)where (P±)V stands for the ground state contribution.We expect eqs.

(1.2) - (1.3), and the discussion in-between, to be valid formany other integrable models provided the appropiate rapidity renormalizationand imaginary shift are introduced.The next natural step after finding the connection (1.2) between transfer ma-trices would be to relate the bare and renormalized monodromies Tab(λ, Θ) andTab(u) . This is necessarily more involved.

For γ ̸= 0 they obey the same YB alge-bra but with different anisotropy parameters γ and ˆγ ≡γ1−γ/π, respectively. Therational case (γ = 0) is evidently simpler and it is the only case where classicallyconserved monodromies are present.The quantum monodromy operators Tab(u) generate a Fock representation ofthe q−deformed affine Lie algebra Uq( ˆG) corresponding to the given R−matrix.More precisely, by expanding Tab(u) in powers of z = eu around z = 0 and z = ∞,5

one obtains non–abelian non-local conserved charges representing the algebra Uq( ˆG)on the Fock space of in– and out–particles. This connects our approach based onthe YB symmetry, to the q−deformed algebraic approach of ref.[4].

Uq( ˆG) is a Hopfalgebra endowed with an universal R−matrix, which reduces to the R−explicitlyentering the YB algebra, upon projection to the finite–dimensional vector spacespanned by the indexes of Tab(u) [5]. In particular, the two expansions around z = 0and z = ∞generate the two Borel subalgebras of Uq( ˆG).

A single monodromymatrix T (u) is sufficent for this purpose, since this field–theoretic representationhas level zero [6]. This fact receives a new explanation in the light–cone approach,since Uq( ˆG) emerges as true symmetry only in the infinite–volume limit above theantiferromagnetic ground state (with no need to take the continuum limit), butits action is uniquely defined already on finite lattices, and all finite–dimensionalrepresentations have level zero.Besides the conserved operators Tab(u), Zamolodchikov-Faddeev non-conservedoperators Zα(θ) act by creating particles on physical states.

Their algebra withthe Tab(u) is determined by the two body S-matrix:Tab(u)Zβ(θ) =XcαZα(θ)Tac(u)Scαbβ(u + θ)(1.4)The ZF operators provide a representation of the dynamical symmetry of q−deformedvertex operators in the sense of [7].6

2. Bootstrap construction of quantum monodromy operators.We briefly review in this section the work of refs.

[1] where the exact (renormal-ized) matrix elements of a quantum monodromy matrix Tab(u) (u is the generallycomplex spectral parameter) were derived using a bootstrap–like approach for aclass of integrable local QFT’s. In such theories there is no particle productionand the S−matrix factorizes.

The two–body S−matrix then satisfies the Yang–Baxter (YB) equations. Moreover, in the models considered in refs.

[1] (the O(N)nonlinear sigma model, the SU(N) Thirring model and the 0(2N) Gross–Neveumodel), thanks to scale invariance there exist classically conserved monodromymatrices. In general, the quantum Tab(u) can be constructed by fixing its actionon the Fock space of physical in and out many–particle states.

The starting pointare the following three general principles:a) Tab(u), a, b = 1, 2, . .

. , n, exist as quantum operators and are conserved.b) Tab(u) fulfil a quantum factorization principle.c) Tab(u) is invariant under P, T and the internal symmetries of the theory.The quantum factorization principle referred above under b) is nowadays calledthe ”coproduct rule”.

This means that there exists the following relation betweenthe action of Tab(u) on k−particles states and its action on one–particle statesTab(u) |θ1α1, θ2α2, . .

. , θkαk⟩in =Xa1a2...ak−1Taa1(u) |θ1α1⟩Ta1a2(u) |θ2α2⟩.

. .

Tak−1b(u) |θkαk⟩(2.1)Tab(u) |θ1α1, θ2α2, . .

. , θkαk⟩out =Xa1a2...ak−1Ta1b(u) |θ1α1⟩Ta2a1(u) |θ2α2⟩.

. .

Taak−1(u) |θkαk⟩(2.2)where θj and αj (1 ≤j ≤k) label the rapidities and the internal quantum numbersof the particles, respectively, in the asymptotic in and out states.Hence it isunderstood that θi > θj for i > j.Although Tab(u) acts differently on in and out states, the assumption of con-servation is nonetheless consistent. All the eigenvalues of a maximal commuting7

subset of {Tab(u), a, b = 1, 2, . .

. , n, u ∈|C} are identical for in and out stateswith given rapidities.

Indeed the two in and out forms of the action on the inter-nal quantum numbers are related by the unitary permutation |α1, α2, . .

. , αk⟩→|αk, αk−1, .

. .

, α1⟩.Furthermore, principles (a) and (c) imply that Tab(u) acts in a trivial way onthe physical vacuum state |0⟩:Tab(u) |0⟩= δab |0⟩(2.3)This also fixes the normalization of Tab(u) in agreement with the classical limit [1].An immediate consequence of point (b) is that when Tab(u) is expanded inpowers of the spectral parameter u, it generates an infinite set of noncommutingand nonlocal conserved charges. This is the clue to the matching of the quantummonodromy matrix with its classical counterpart which is written nonlocally interms of the local fields.The main result in refs.

[1] was to derive from (a), (b) and (c) the explicitmatrix elements of Tab(u) on one–particle states. This result can be written as⟨θα| Tab(u)θ′β= δ(θ −θ′)Saαbβ (κ(u) + θ)(2.4)where Saαbβ (θ −θ′) stands for the S−matrix of two–body scatteringθb, θ′βin =Xaαθa, θ′αout Saαbβ (θ −θ′)(2.5)and κ(u) is an odd function of u.

Notice that this requires the presence in the modelof particles with indices a, b, ... as internal state labels. In the simplest situationthese new labels coincide with those of the original particles.The appearanceof a nontrivial “renormalization” u →κ(u) is to be expected when there exist adefinition of the spectral parameter outside the bootstrap itself.

This is the case8

of the models of refs. [1], which posses Lax pairs and auxiliary problems which fixthe definition of u.

Here we shall adopt the purely bootstrap viewpoint and fix thedefinition of u so that κ(u) = u. In principle, an extra u−and θ−dependent phasefactor may appear in the r.h.s.

of eq.(2.4). However, no phase showed up in thespecific models of refs.

[1], when nonperturbative checks were performed using theoperator product expansion. Eq.

(2.4) can be written in a more suggestive way asTab(u) |θβ⟩=Xα|θα⟩Saαbβ (u + θ)(2.6)This equation, when combined with eqs. (2.1) and (2.2), completely defines thequantum monodromy operators in the Fock space.

From the YB equations satisfiedby the S−matrix it then follows that Tab(u) fulfils the YB algebraˆR(u −v) [T (u) ⊗T (v)] = [T (u) ⊗T (v)] ˆR(u −v)(2.7)where ˆRaαbβ (u) = Sαabβ (u). It should be stressed that the conservation of Tab(u)implies that this YB algebra is a true non–abelian infinite symmetry algebra of therelativistic local QFT.

On the contrary the rˆole of the YB algebra in integrablevertex and face models on finite lattices or in nonrelativistic quantum models isthat of a dynamical symmetry underlying the Quantum Inverse Scattering Method.In these latter cases, only the transfer matrix, namelyτ(u) =XaTaa(u)(2.8)is conserved. Since [τ(u), τ(v)] = 0, the transfer matrix just generates an abeliansymmetry.The dynamical symmetry underlying the integrable QFT includes in additionnon–conserved operators Zα(θ) which create the particle eigenstates out of the9

vacuum. In the bootstrap framework they can be introduced `a la Zamolodchikov–Faddeev, by setting|θ1α1, θ2α2, .

. .

, θkαk⟩in = Zαk(θk)Zαk−1(θk−1) . .

. Zα1(θ1) |0⟩|θ1α1, θ2α2, .

. .

, θkαk⟩out = Zα1(θ1)Zα2(θ2) . .

. Zαk(θk) |0⟩(2.9)with the fundamental commutation rulesZα2(θ2)Zα1(θ1) =Xβ1β2Sβ1β2α1α2(θ1 −θ2)Zβ1(θ1)Zβ2(θ2)(2.10)Combining now eqs.

(2.1), (2.2), (2.6) and (2.10), we obtain the algebraic relationbetween monodromy and Zamolodchikov–Faddeev operators:Tab(u)Zβ(θ) =XcαZα(θ)Tac(u)Scαbβ(u + θ)(2.11)Together with eqs. (2.7) and (2.10), these relations close the complete dynami-cal algebra of an integrable QFT.

For the XXZ spin chain in the regime |q| < 1,the ZF operators have been identified in ref. [7] with special vertex operators(or representation intertwiners of the relevant q−deformed affine Lie algebra).They are uniquely characterized by being solutions of the q−deformed Knizhnik–Zamolodchikov equation and by their normalization [5].3.

Yang-Baxter symmetry in the sine-Gordon modelIn refs. [1] the infinite YB symmetry was explicitely considered and exhibitedfor classically scale–invariant models like the O(N) nonlinear sigma, the SU(N)Thirring and the 0(2N) Gross–Neveu models.

Indeed, it is this scale–invariancewhich guarantees the conservation also of the nondiagonal elements of the classicalmonodromy matrix.An important example of integrable QFT which does notbelong to this category is the sine–Gordon (sG) model. The presence of mass at10

the classical level implies that only its transfer matrix is conserved.Then theclassical symmetry algebra is commuting (as Poisson brakets) and admits a basisof local conserved charges. It is well known that these charges survive quantization,leading to factorization of the scattering with corresponding YB equations satisfiedby the two–body S−matrix [8].It appears therefore natural to apply the general bootstrap methods of theprevious section also to the sG model.

The quantum Tab(u) (with a, b = ±) de-fined in this way is assumed to be conserved from the outset, and does not reduceto its classical counterpart in the classical limit. Let us consider first soliton andantisoliton states.They have mass M and a charge α = +1 (−1) for solitons(antisolitons).

Let us recall that the solitons (antisolitons) are the fermions (an-tifermions) of the massive Thirring model, which is equivalent to the sG model asQFT in 2D Minkowski space.Eq. (2.6) in the one-particle soliton/antisoliton sector takes now the formT±±(u) |θ±⟩= S(u + θ) |θ±⟩T±±(u) |θ∓⟩= ST (u + θ) |θ∓⟩T±∓(u) |θ±⟩= SR(u + θ) |θ∓⟩T±∓(u) |θ∓⟩= 0(3.1)where S(θ) is the soliton/soliton scattering amplitude, while ST (θ) and SR(θ) are,respectively the transmission and reflection amplitudes of the soliton/antisolitonscattering.

Explicitly they read [ 8 ]:S(θ) = exp i∞Z0dkksinh(π/ˆγ −1)k/2sinh(πk/2ˆγ)sin kθ/πcosh k/2ST(θ) = S(iπ −θ)SR(θ) = S(θ)sin ˆγsin [ˆγ(1 −θ/iπ)](3.2)11

where ˆγ is related to the usual sG coupling constant β byˆγπ = 8πβ2 −1(3.3)The action of Tab(u) on multiparticle soliton/antisoliton states is obtained by sim-ply inserting eqs. (3.1) into eqs.

(2.1) and (2.2). Notice that S(θ), ST(θ) and SR(θ)posses essential singularities at β2 = 0.

That is,S(θ)β→0= exp −i16πβ2∞Z0dkk2 tanh(k/2)sin kθ/π ≡Sc(θ)(3.4)Hence the quantum monodromy matrix is singular in the free boson limit β = 0. Ofcourse it is regular, although trivial (Tab(u) = δab), in the free fermion limit ˆγ = π.In the classical limit instead, we must replace the scattering amplitudes with theanalogous quantities computed for soliton field configurations in the classical sGmodel, namelyS(θ) = ST (iπ −θ) = Sc(θ) ,SR(θ) = 0(there is no soliton reflection at the classical level).

Hence Tab(u) becomes diagonaland the YB algebra becomes abelian. This is consistent with the fact that theintegrals of motion of the classical sG equation are all in involution.Besides solitons and antisolitons states, the sG-MTM model possess breathersstates for ˆγ > π.

These particles are labeled by and index n running from 1 to[ˆγ/π] −1 (where [..] stands for integer part) and have massesmn = 2M sin(nπ22ˆγ )(3.5)[n = 1 corresponds to the fundamental particle associated to the sG-field ]. Ap-plying the general formula (2.4) to these particle states yields⟨θn| Tab(u)θ′m= δ(θ −θ′)δnmδabSn(u −θ)(3.6)where |θn⟩stands for a nth.

breather state with rapidity θ and Sn(θ −θ′) is the12

soliton- breather S-matrix. That is [ 8 ] ,Sn(θ) =sinh θ + i cos nπ22ˆγsinh θ −i cos nπ22ˆγn−1Yl=1sin2(n2 −l π22ˆγ −π4 + iθ2 )sin2(n2 −l π22ˆγ −π4 −iθ2 )(3.7)In particular,S1(θ) =sinh θ + i cos π22ˆγsinh θ −i cos π22ˆγ(3.8)We conclude that Tab(u) has a rather trivial action on breather statesTab(u) = δabSB(u)(3.9)where SB(u) is a diagonal operator with eigenvalues Sn(u−θ) on the nth breatherstate.

In conclusion, we have uncovered the infinite YB symmetry of the sG-mTmodel providing the explicit form of its conserved operators on all the asymptoticstates.It is instructive to study the u →∞limit of the YB operator Tab(u). We findfrom eqs.

(3.1)-(3.2) for u →∞,Tab(u) = exp(ia4π2β2 σ3) δab + 2ie4i π2β2 exp(−8πuβ2 )Qb(1 −δab) + O(e−16πuβ2 )(3.10)Here β is the usual sine-Gordon coupling constant and Qa acts on one-particlesoliton/antisoliton states asQa = e−γθπ σa(3.11)This is a SU(2)q generator for the spin 1/2 representation. (For spin 1/2, SU(2)and SU(2)q generators coincide).

Using eq. (3.10) and the coproduct relations (2.1)and (2.2) , we find that eq.

(3.10) holds as it stands on two (or more) particle states13

but now withσ3 = σ(1)3+ σ(2)3,Qa = e−ia 4π2β2 σ(1)3 Q(2)a+ Q(1)a eia 4π2β2 σ(2)3(3.12)Analogous relations hold for multiparticle states. This tells us that Qa and σ3 arerelated to SU(2)q generators withq = e8iπ2β2(3.13)asJ+ = Q+,J−= Q†−,Jz = σ3(3.14)Alternatively, we can make the identification q = e−8iπ2β2 with:J+ = Q†+,J−= Q−,Jz = σ3(3.15)A nonlocal charge equivalent to Qa studied in ref.[9].

The fact that YB generatorsfor u = ∞yield SU(2)q generators in this way is typical of periodic boundary con-ditions [10]. For fixed boundary conditions (that is scattering of particles betweentwo walls) the connection is much cleaner [11].4.

Bethe Ansatz at the bootstrap levelThe maximal abelian subalgebra of the YB algebra (2.7) is generated by thetransfer matrix τ(u) (eq.(2.8)). With respect to this subalgebra, the remaining ele-ments of Tab(u) act as generalized raising and lowering operators.

This observationprovides the basis for the so–called Algebraic Bethe Ansatz, which is a purely al-gebraic method to construct the eigenvectors and the eigenvalues of τ(u) [12]. Thecrucial starting point is the identification of the highest weight states annihilatedby the raising operators.

Since particles are conserved in an integrable QFT model,one can restrict the problem to states with a fixed number, say k, of particles.14

In the case of the sG model the highest weight states are the ferromagneticstates containing only solitons, that is the states |θ1+, θ2+, . .

. , θk+⟩with the high-est possible value Jz = k/2 of the z−projection of the SU(2)q spin in the sectorwith k particles.On such states the monodromy matrix Tab(u) is indeed up-per triangular (compare eqs.

(2.1) and (2.2) with eqs. (3.1)).

The rapidities θnof the solitons are arbitrary and act as fixed parameters in the problem, sincethey are left unchanged by the action of Tab(u). Then the BA in–eigenstates ofτ(u) = T++(u) + T−−(u) with k −m solitons and m antisolitons can be writtenB(u1)B(u2) .

. .

B(um) |θ1+, θ2+, . .

. , θk+⟩inwhere B(u) ≡T+−(u + iπ/2) act as lowering operators of Jz and the distinctnumbers u1, u2, .

. .

, um must satisfy the BA equationskYn=1sinh ˆγ[i/2 + (uj + θn)/π]sinh ˆγ[i/2 −(uj + θn)/π] = −mYr=1sinh ˆγ[+i + (uj −ur)/π]sinh ˆγ[−i + (uj −ur)/π](4.1)The eigenvalues ξ(u) of τ(u) on the BA states readξ(u) =( kYn=1S(u + θn))[ ξ+(u) + ξ−(u)]ξ+(u) =mYj=1sinh ˆγ[i/2 + (u −uj)/π]sinh ˆγ[i/2 −(u −uj)/π]ξ−(u) =" kYn=1sinh ˆγ(u + θn)/πsinh ˆγ[i −(u + θn)/π]# mYj=1sinh ˆγ[3i/2 + (u −uj)/π]sinh ˆγ[−i/2 + (u −uj)/π](4.2)Up to the common factor Qkn=1 S(u + θn), ξ±(u) is just the contribution comingfrom T±±(u). It is clear, moreover, that the presence of breathers introduce nofurther complications, due to the diagonal action (3.9) of the monodromy matrixon breather states.15

Eqs. (4.1) and (4.2) follow directly from the YB algebra (2.7) satisfied byconstruction by the bootstrap monodromy matrix.

This algebraic Bethe Ansatzcan be generalized to a whole class of integrable field theories where the bootstrapconstruction of sec. 2 applies.

Furthermore, let us observe that the diagonaliza-tion of the bootstrap transfer matrix represents the basic step in the so–calledThermodynamic BA, which is a way to obtain off–shell exact results on the inte-grable relativistic QFT at hand. In fact, the transfer matrix τ(u), as trace of themonodromy matrix (eq.

(2.8)), is directly related to the multiscattering amplitudessuffered by each particle in a system of k particles confined on a ring, namelyτ(−θj) = Sjk . .

. Sj j+1Sj j−1 .

. .

Sj1(4.3)where j = 1, 2, . .

. , k and the two–body matrices Sij are defined bySij |θ1α1, .

. .

, θkαk⟩=Xβiβj Yn̸=i,jδβnαnSαjαkβiβj (θi −θj) |θ1β1, . .

. , θkβk⟩By periodicity, eqs.

(4.3) and (4.2) determine the quantization of the momentumof each particle in the standard wayξ(−θj) exp (i mjL sinh θj) = 1(4.4)where L is the length of the ring. Together with the BA equations (4.1) for the rootsu1, u2, .

. .

, um (the so–called magnon parameters), these new equations provide thebasis for the TBA [15].16

5. Light-cone lattice regularization.In order to obtain a first–principles, microscopic understanding of the boot-strap picture presented above, we now consider the integrability–preserving latticeregularization of an integrable relativistic QFT defined by the so–called light-coneapproach [2,3] to vertex models.In this approach one starts from the discretized Minkowski 2D space–timeformed by a regular diagonal lattice of right–oriented and left–oriented straightlines (see fig.1).

These represent true world–lines of “bare” objects (pseudo–particles) which are thus naturally divided in left– and right–movers. The right–movers have all the same positive rapidity Θ, while the left–movers have rapidity−Θ.

One can regard Θ as a cut–offrapidity, which will be appropriately takento infinity in the continuum limit. Furthermore, we shall denote by V the Hilbertspace of states of a pseudo–particle (we restrict here to the case in which V is thesame for both left– and right–movers and has finite dimension n, although moregeneral situations can be considered).The dynamics of the model is fixed by the microscopic transition amplitudesattached to each intersection of a left– and a right–mover, that is to each vertexof the lattice.

This amplitudes can be collected into linear operators Rij, the localR−matrices, acting non–trivially only on the space Vi ⊗Vj of ith and jth pseudo–particles. Rij thus represent the relativistic scatterings of left–movers on right–movers and depend on the rapidity difference Θ −(−Θ) = 2Θ, which is constantthroughout the lattice.

Moreover, by space–time translation invariance any otherparametric dependence of Rij must be the same for all vertices. We see thereforethat attached to each vertex there is a matrix R(2Θ)cdab, where a, b, c, d are labelsfor the states of the pseudo–particles on the four links stemming out of the vertex,and take therefore n distinct values (see fig.

1). This is the general frameworkof a vertex model.

The difference with the standard statistical interpretation isthat the Boltzmann weights are in general complex, since we should require theunitarity of the matrix R. In any case, the integrability of the model is guaranteed17

whenever R(λ)cdab satisfy the Yang–Baxter equationsRij(λ)Rjk(λ + µ)Rij(µ) = Rjk(µ)Rij(λ + µ)Rjk(λ)(5.1)For periodic boundary conditions, the one–step light–cone evolution operatorsUL(Θ) and UR(Θ), which act on the ”bare” space of states HN = (⊗V)2N , (N isthe number of sites on a row of the lattice, that is the number of diagonal lines),are built from the local R−matrices Rij as [2]UR(Θ) = U(Θ)V ,UL(Θ) = U(Θ)V −1U(Θ) = R12R34 . .

. R2N−1 2N(5.2)where V is the one-step space translation to the right.

UR ( UL ) evolves states byone step in right (left) light–cone direction. UR and UL commute and their productU = URUL is the unit time evolution operator.

The graphical representation of Uis given by the section of the diagonal lattice with fat lines in fig. 1.

If a stands forthe lattice spacing, the lattice hamiltonian H and total momentum P are naturallydefined throughU = e−iaH ,URU−1L= eiaP(5.3)The action of other fundamental operators is naturally defined on the same Hilbertspace HN. These are the n2 Yang-Baxter operators for 2N sites, which are con-ventionally grouped into the n × n monodromy matrix T(λ) = {Tab(λ), a, b =1, .

. .

, n}. One usually regards the indices a, b of Tab as horizontal indices fixingthe out– and in–states of a reference pseudo–particle.

Then T(λ) is defined ashorizontal coproduct of order 2N of the local vertex operators Lj(λ) = R0j(λ)P0j,where 0 label the reference space and Pij is the transposition in Vi⊗Vj . ExplicitlyT(λ) = L1(λ)L2(λ) .

. .

L2N(λ)The inhomogenuous generalization T(λ, ⃗ω ) then readsT(λ, ⃗ω ) = L1(λ + ω1)L2(λ + ω2) . .

. L2N(λ + ω2N)and has the graphical representation of fig.

2. The formal structure of this ex-18

pression is identical to that of eq.(2.1). In fact Lj(λ + ωj) can be regarded as thescattering of the jth pseudo–particle carrying formal rapidity ωj with the referencepseudo–particle carrying formal rapidity −λ.

In the same way, thanks to eq. (2.6),the single particle terms in eq.

(2.1) represent the scattering of the correspondingparticle on a reference particle carrying physical rapidity −u. In the case of ourdiagonal lattice of right– and left–moving pseudoparticles, there exists a specific,physically relevant choice of the inhomogeneities, namelyωk = (−1)kΘ ,k = 1, 2, .

. .

2N(5.4)leading to the definition of the alternating monodromy matrixT(λ, Θ) ≡T(λ, {ωk = (−1)kΘ})(5.5)In fact, the evolution operators UL(Θ) and UR(Θ) can be expressed in terms of thealternating transfer matrix t(λ, Θ) = tr0T(λ, Θ) as [3]UR(Θ) = t(Θ, Θ) ,UL(Θ) = t(−Θ, Θ)−1(5.6)At any rate, no matter how the ωk are chosen, the monodromy matrix T(λ, ⃗ω )fulfils the YB algebraR(λ −µ) [T(λ, ⃗ω ) ⊗T(µ, ⃗ω )] = [T(µ, ⃗ω ) ⊗T(λ, ⃗ω )] R(λ −µ)(5.7)just as the quantum T (u) satisfies the YB algebra (2.7). We see that the “bare” YBalgebra involves the finite–dimensional operators Tab(λ, ⃗ω ) and, correspondingly,the “bare” R−matrix R(λ) defines it.Notice that T(λ, Θ) fails to be conserved on the lattice only because of bound-ary effects.

Indeed from fig. 3, which graphically represents the insertion of T(λ, Θ)in the lattice time evolution, one readily sees that U and T(λ, Θ) fail to commute19

only because of the free ends of the horizontal line. For all vertices in the bulk,the graphical interpretation of the YB equations (5.1), namely that lines can befreely pulled through vertices, allows to move T(λ, Θ) up or down, that is to freelycommute it with the time evolution.

The problem lays at the boundary: if periodicboundary conditios are assumed, then the free horizontal ends of T(λ, Θ) cannotbe dragged along with the bulk, unless they are tied up, to form the transfer matrixt(λ, Θ). After all, for p.b.c., the boundary is actually equivalent to any point ofthe bulk and thus t(λ, Θ) commutes with U, as obvious also from eqs.

(5.6) andthe general fact that [t(λ, Θ), t(µ, Θ)] = 0. One might think that the thermody-namic limit N →∞, by removing infinitely far away the troublesome free endsof T(λ, Θ), will allow for its conservation and thus for the existence of an exactYB symmetry with bare R−matrix.

The situation however is not so simple: firstof all one must fix the Fock sector of the N →∞non–separable Hilbert space inwhich to take the thermodynamic limit. Different choices leads to different phaseswith dramatically different dynamics.Then the non–local structure of T(λ, Θ)must be taken into account.

It is evident, for instance, that in the spin–wave Focksector above ferromagnetic reference states T(λ, Θ) can never be conserved. In-deed, the working itself of the Quantum Inverse Scattering Method, where energyeigenstates are built applyind non–diagonal elements of T(λ, Θ) on a specific fer-romagnetic reference state, of course depends on T(λ, Θ) not commuting with thehamiltonian!From the field–theoretic point of view, the most interesting phase is the antifer-romagnetic one, in which the ground state plays the rˆole of densily filled interactingDirac sea (this holds for all known integrable lattice vertex models [2,3,12]).

Thecorresponding Fock sector is formed by particle–like excitations which become rela-tivistic massive particles within the scaling limit proper of the light–cone approach[3]. This consists in letting a →0 and Θ →∞in such a way that the physicalmass scaleµ = a−1e−κΘ(5.8)20

stays fixed. Here κ is a model–dependent parameter which for the so–called rationalclass of integrable model (to this class belong the models considered in ref.

[1])takes the general form [13]κ = 2π th s(5.9)where h is the dual Coxeter number of the underlying Lie algebra, s equals 1, 2 or 3for simply, doubly and triply laced algebras, respectively, and t = 1 (t = 2) for non–twisted (twisted) algebras. For the class of model characterized by a trigonometricR−matrix (with anisotropy parameter γ) the expression (5.9) for κ is to be dividedby γ [13].The ground state or (physical vacuum) and the particle–like excitations ofthis antiferromagnetic phase are extremely more complicated than those of theferromagnetic phase.

It is therefore very hard to control, in the limit N →∞,the action of the alternating monodromy matrix T(λ, Θ) on the particle–like BAeigenstates of the alternating transfer matrix t(λ, Θ). On the other hand, sincethese particles enjoy a factorized scattering, one can proceed according to thegeneral tenets of the bootstrap approach described in sec.2.

In this way oneconstructs the bootstrap monodromy matrix T (u) and it is natural to search foran explicit connection between T (u) and T(λ, Θ). It is a connection like this thatwould provide the microscopic interpretation of the bootstrap results.In order to study the infinite volume limit of T(λ, Θ) on the physical Fock space(that is, finite energy excitations around the antiferromagnetic vacuum), one needsto compute scalar products of Bethe Ansatz states to derive relations like (2.4) or(2.6) with T(λ, Θ) instead of T (λ, Θ) in the l.h.s.

Since this kind of calculationsare indeed possible but rather involved, we start by computing the eigenvaluesof t(λ, Θ) on a generic state of the physical Fock space. Then, we shall comparethese eigenvalues with those of τ(u).

This will tell us whether the bare and therenormalized YB algebras have a common abelian subalgebra. Notice that thisfact alone would provide a microscopic basis for the TBA, which originally reliessolely on the bootstrap.21

We shall consider once more the sG model as example, although the same resultwould apply to any integrable QFT admitting a light–cone lattice regularization.This class of models contains also the O(N) nonlinear sigma model and the SU(N)Thirring model considered from the bootstrap viewpoint in refs. [1].The integrable light–cone lattice regularization of the sG–mT model is pro-vided the six-vertex model [2].

Therefore, the space V is two–dimensional and theunitarized local R−matrices can be writtenRjk(λ) =1 + c2+ 1 −c2σzj σzk + b2(σxj σxk + σyj σyk)b =sinh λsinh(iγ −λ) ,c =sinh iγsinh(iγ −λ)(5.10)where γ is commonly known as anisotropy parameter.The standard Algebrized BA can be applied to the diagonalization of the al-ternating transfer matrix t(λ, Θ) with the following results [2,3,10]. The BA statesare writtenΨ(⃗λ ) = B(λ1)....B(λM)Ω(5.11)where ⃗λ ≡(λ1, λ2, .

. .

, λM), B(λi) = T+−(λi +iγ/2, Θ) and Ωis the ferromagneticground-state (all spins up). They are eigenvectors of t(λ, Θ)t(λ, Θ)Ψ(⃗λ ) = Λ(λ;⃗λ )Ψ(⃗λ )(5.12)provided the λi are all distinct roots of the “bare” BA equationssinh[iγ/2 + λj −Θ]sinh[iγ/2 −λj + Θ]sinh[iγ/2 + λj + Θ]sinh[iγ/2 −λj −Θ]N= −MYk=1sinh[+iγ + λj −λk]sinh[−iγ + λj −λk](5.13)The eigenvalues Λ(λ;⃗λ ) are the sum of a contribution coming from A(λ) = T++(λ, Θ)22

and one coming from D(λ) = T−−(λ, Θ),Λ(λ;⃗λ ) = ΛA(λ;⃗λ ) + ΛD(λ;⃗λ )(5.14)HereΛA(λ;⃗λ ) = exph−iG(λ,⃗λ )iΛD(λ;⃗λ ) = e−iN[φ(λ−iγ/2−Θ,γ/2)+φ(λ−iγ/2+Θ,γ/2)] exp [iG(λ −iγ,⃗λ )](5.15)andG(λ,⃗λ ) ≡MXj=1φ(λ −λj, γ/2) ,φ(λ, γ) ≡i log sinh(iγ + λ)sinh(iγ −λ)(5.16)G(λ,⃗λ ) is manifestly a periodic function of λ with period iπ. Notice also thatΛD(±Θ,⃗λ) = 0.

That is, only ΛA(±Θ,⃗λ) contributes to the energy and momentumeigenvalues:E(Θ) = a−1MXj=1[φ(Θ + λj, γ/2) + φ(Θ −λj, γ/2) −2π]P(Θ) = a−1MXj=1[φ(Θ + λj, γ/2) −φ(Θ −λj, γ/2)](5.17)The ground state and the particle–like excitations of the light–cone six–vertexmodel are well known [2,12]: the ground state corresponds to the unique solutionof the BAE with N/2 consecutive real roots (notice that the energy in eq. (5.17)is negative definite, so that the ground state is obtained by filling the interactingDirac sea).

In the limit N →∞this yields the antiferromagnetic vacuum. Holesin the sea appear as physical particles.

A hole located at ϕ carries energy and23

momentum, relative to the vacuum,e(ϕ) = 2a−1 arctancosh πϕ/γsinh πΘ/γ,p(ϕ) = −2a−1 arctan sinh πϕ/γcosh πΘ/γ(5.18)In the scaling limit a →0, Θ →∞with e(0) held fixed, we then obtain (e, p) =m(cosh θ, sinh θ) withm ≡4a−1 exp(−πΘ/γ) ,θ ≡−πϕ/γ(5.19)identified, respectively, as physical mass and physical rapidity of a sG soliton (mTfermion) or antisoliton (antifermion). Complex roots of the BAE are also possible.They correspond to magnons, that is to different polarization states of several sGsolitons (mT fermions) , or to breather states (in the attractive regime γ > π/2).

Inthe rest of this paper, we shall restrict our attention to the repulsive case γ < π/2,where the complex roots corresponding to the breathers are absent.6. Thermodynamic limit of the transfer matrixWe proceed now to evaluate the function G(λ,⃗λ ) in the infinite volume limit(N →∞at fixed lattice spacing) for the antiferroelectric ground state (the physicalvacuum) and for the excited states, in the repulsive regime γ < π/2.For the vacuum, the density of roots ⃗λV results to be [10]ρ(λ)V = N+∞Z−∞dk2πeikλcos kΘcosh kγ/2(6.1)Using the integral representationφ(λ, γ/2) = P+∞Z−∞dkik eikλsinh k [π/2 −γ]sinh kπ/2(6.2)24

which is valid for |Im λ| < γ2, and eq. (6.1), we obtain for G(λ)V ≡G(λ,⃗λV )G(λ)V = −iN P+∞Z−∞dkk eikλ cos kΘ sinh k(π −γ)/2cosh kγ/2 sinh kπ/2,|Im λ| < γ/2(6.3)When π −γ/2 > |Im λ| > γ2 we need another integral representation for φ(λ, γ/2),φ(λ, γ/2) = −P+∞Z−∞dkik eikλ+(πk/2)sign(Im λ) sinh kγ/2sinh kπ/2(6.4)We then find using eqs.

(5.16),(6.1) and (6.4),G(λ)V = iN P+∞Z−∞dkk eikλ+(πk/2)sign(Im λ)cos kΘ sinh kγ/2cosh kγ/2 sinh kπ/2(6.5)when π −γ/2 > |Im λ| > γ/2. That is, the function G(λ)V is discontinuous onthe lines Im λ = ±γ/2.

As we shall see this fact holds true also for all excitedstates. On the other hand G(λ,⃗λ ) is periodic with period iπ, so that there existtwo main analytic determinations of its infinite volume limit G(λ), that we shallcall henceforth GI(λ) and GII(λ).G(λ) = GI(λ) , for the strip I :−γ/2 < Im λ < γ/2G(λ) = GII(λ) , for the strip II :−π + γ/2 < Im λ < −γ/2(6.6)GI(λ)V and GII(λ)V have, respectively, the integral representations (6.3) and (6.5).The functions GI(λ)V and GII(λ)V analytically continued in λ are meromorphicfunctions.

Of course, they do not coincide with G(λ) except for the strips indicatedin eq. (6.6) .

For Im λ outside these two strips, G(λ) can be expressed in terms of25

GI(λ)V or GII(λ)V using the iπ periodicity as follows:G(λ) = GI(λ −inπ)fornπ −γ/2 < Im λ < nπ + γ/2G(λ) = GII(λ −inπ)for(n −1)π + γ/2 < Im λ < nπ −γ/2(6.7)where n ∈ZZ. The reflection principle also holds here:G(λ) = ¯G(¯λ)We find from eqs.

(6.3) and (6.5) the following expression for the difference betweenthe meromorphic functions GI(λ)V and GII(λ)V :GII(λ)V −GI(λ)V = −2iNArg tanhcosh (πΘ/γ)cosh (πλ/γ)(6.8)The discontinuities of G(λ) through the other cuts follow by iπ periodicity and thereflection principle.In addition, when λ and λ −iγ lay both in strip II (which is indeed possiblefor γ < π/2), we can relate the functions G(λ)V and G(λ −iγ)V as follows :GII(λ)V + GII(λ −iγ)V = −iN P+∞Z−∞dkk eikλ−πk/2(1 + ekγ) cos kΘ sinh kγ/2cosh kγ/2 sinh kπ/2= N [φ(λ −iγ/2 −Θ, γ/2) + φ(λ −iγ/2 + Θ, γ/2)](6.9)We find an analogous relation when λ lays in strip I and λ −iγ in strip IIGI(λ)V + GII(λ −iγ)V = N [φ(λ −iγ/2 −Θ, γ/2) + φ(λ −iγ/2 + Θ, γ/2)]+ iN logcosh πΘγ −i sinh πλγcosh πΘγ + i sinh πλγ(6.10)Let us now consider excited states. We start with a two hole state (the number of26

holes is always even when N is even). The density of roots is then [10]ρ(λ) = ρ(λ)V + ρ(λ −ϕ1)h + ρ(λ −ϕ2)h −δ(λ −ϕ1) −δ(λ −ϕ2)(6.11)where ϕ1 and ϕ2 are the hole positions andρ(λ)h =+∞Z−∞dk2πeikλ sinh[k2(π −2γ)]sinh kπ2 + sinh[k2(π −2γ)](6.12)The function G(λ) takes then the formG(λ) = G(λ)V + G(λ −ϕ1)h + G(λ −ϕ2)h(6.13)We find from eqs.

(5.16), (6.2), (6.11), and (6.12)GI(λ)h = i P+∞Z−∞dkkeikλ2 cosh kγ2= −2 arctantanh πλ2γGII(λ)h = −i P+∞Z−∞dk2keikλ+k(π/2)sign(Im λ) sinh kγ2cosh kγ2 sinh[k2(π −γ)]= −2 arctantanh πλ2γ+ i log S(πλ/γ −sign(Im λ)iπ/2)(6.14)where S(θ) is recognized as the soliton–soliton scattering amplitude (3.2) upon theidentificationˆγ ≡γ1 −γπ(6.15)The function S(θ) enjoys the crossing propertyS(iπ −θ) = ˆb(θ) S(θ)(6.16)whereˆb(θ) =sinh( ˆγθπ )sinh ˆγπ(iπ −θ)(6.17)Notice that ˆb(θ) ˆb(iπ −θ) = 1 . We see from eq.

(6.14) that G(λ)h has cuts on thelines Im λ = ±γ/2, with discontinuityi log S(πγ λ ∓iπ2).27

Next consider states containing complex roots. There are four kinds of complexroots [14] associated to excited states close to the N →∞antiferromagneticvacuum, in the regime γ < π/2:a) Close roots with |Im λ| < γ .

They appear as quartets : λ = (σ ± iη, σ ±i[γ −η]), where 0 < η < γ , or as two strings: λ = σ ± iγ/2.b) Wide roots with |Im λ| > γ. They appear in pairs λ = σ±iη, γ/2 < η < π/2,or as self–conjugate single roots with |Im λ| = π/2.The presence of such complex roots produces a backflow in the real rootsdensity.

For a close pair we have [11]ρη(λ)c = −12π[p(λ −σ −iη) + p(λ −σ + iη)] ,η < γ < π/2(6.18)while for a wide pair [11]ρη(λ)w = −12πddλ[φγ(λ −σ, η −γ) −φγ(λ −σ, η)] ,η > γ < π/2(6.19)whereφγ(λ, η) ≡φλ1 −γ/π,η1 −γ/π(6.20)A self–conjugate root at σ + iπ/2 gives insteadρ(λ)sc = 12ρπ/2(λ)w(6.21)Let us denote by Gη(λ)c and Gη(λ)w the contribution of a closed pair and of awide pair to the function G(λ), respectively. For self–conjugate roots one simplyhas G(λ)sc = 12Gπ/2(λ)w. We find from eqs.

(5.16) and (6.18) :Gη(λ)c =+∞Z−∞dµ φ(λ−µ, γ/2)ρη(µ)c+φ(λ−σ−iη, γ/2)+φ(λ−σ+iη, γ/2) (6.22)28

Then the integral representations (6.2) for φ(λ, γ/2) and the density (6.18) yieldGIη(λ)c = 2 arctantanh π2γ[λ −σ −iη]+ 2 arctantanh π2γ[λ −σ + iη](6.23)It is easy to see that the total contribution for a quartet vanishes when |Im λ| < γ/2and that the two–string contributions equal ±π in this region:GIη(λ)c + GIγ−η(λ)c = 0 mod 2π,GIγ/2(λ)c = i log(−1) ,|Im λ| < γ/2(6.24)Hence, quartets and two–strings do not contribute to the energy and momentum.Let us now consider the more interesting strips of type II. There, using theintegral representation (6.4) for φ(λ, γ/2), we obtainGIIη (λ)c = 2 arctantanh π2γ [λ −σ −iη]+ 2 arctantanh π2γ[λ −σ + iη]−i log Sπγ [λ −σ + iη] −iπ2−i log Sπγ [λ −σ −iη] −iπ2(6.25)Then, the total contribution for quartets and two strings results inGIIη (λ)c + GIIγ−η(λ)c = i logˆbπγ [λ −σ + iη] −iπ2ˆbπγ [λ −σ −iη] −iπ2GIIγ/2(λ)c = i log−ˆbπγ [λ −σ](6.26)where we used eq.

(6.16).Let us finally consider the wide pairs. Their contribution is given byGη(λ)w =+∞Z−∞dµ φ(λ−µ, γ/2)ρη(µ)w+φ(λ−σ−iη, γ/2)+φ(λ−σ+iη, γ/2) (6.27)Use of eqs.

(6.2) , (6.4) and (6.19) now yieldsGIη(λ)w = 0mod 2πGIIη (λ)w = φγ(λ −σ −iγ/2, γ −η) + φγ(λ −σ −iγ/2, η)(6.28)We see from eq. (6.28) that wide pairs do not contribute to the energy and momen-tum.29

We are now in position to analyze the excitation spectrum of the infinite–volume transfer matrix t(λ, Θ). Let us begin with λ lying in strip I.

From eqs. (5.15) and (6.10), we find for the vacuumΛA(λ)V = exph−iGI(λ)ViΛD(λ)V = exph−iGI(λ)Vi cosh πΘ/γ + i sinh πλ/γcosh πΘ/γ −i sinh πλ/γN(6.29)The extra factor in ΛD(λ) tends to zero (infinity) for Im λ positive (negative) whenN →∞.

Since this vacuum contribution is present in any physical particle–likeexcitation, we see that the ΛD(λ) will always behave like ΛD(λ)V . Let us recall thatΛD(±Θ) = 0 for any finite N, giving no contribution to energy and momentum.Hence, choosing 0 < Im λ < γ/2, we are guaranteed that the two limits λ →±Θand N →∞commute.

The reduced strip 0 < Im λ < γ/2 is therefore the mostnatural one to define the renormalized type I transfer matrixtI(λ) = limN→∞t(λ, Θ) expiGI(λ)V(−)Jz−N/2(6.30)where Jz = N/2 −M is to be identified with the soliton (or fermion) charge ofthe continuum sG–mT model.The last sign factor in eq. (6.30) corresponds tosquare–root branch choice suitable to obtain the relationtI(±Θ) = exp{−ia[P± −(P±)V ]}(6.31)where P± ≡(H ± P)/2 (see eqs.

(5.3), (5.6), (5.17)) and (P±)V stands for thevacuum contribution. Notice that the Θ−dependence of tI(λ) has been completelycanceled out, since it is present only in the vacuum contribution.

In fact, fromeqs. (6.13), (6.14), (6.24) and (6.28), we read the eigenvalue ΛI(λ) of tI(λ) on ageneric particle state:ΛI(λ) = exp"−2ikXn=1arctaneπλ/γ+θn#=kYn=1cothπλ2γ + θn2 + iπ4(6.32)where θn ≡−πϕn/γ are the physical particle rapidities.

Suppose now we expand30

log ΛI(λ) in powers of z = e−π|λ|/γ around λ = ±∞,±i log ΛI(λ) =∞Xj=0z2j+1 (−1)jj + 1/2kXn=1e±(2j+1)θn(6.33)One has to regard the coefficents of the expansion parameter z as the eigenvaluesof the conserved abelian charges generated by the transfer matrix. The additivityof the eigenvalues implies the locality of the charges.

In terms of operators we canwrite, around λ = ±∞,±i log tI(λ) =∞Xj=04zm2j+1I±j(6.34)where I±0 = p± is the continuum light–cone energy–momentum and the I±j , j ≥1,are local conserved charges with dimension 2j + 1 and Lorentz spin ±(2j + 1).Their eigenvalues(−1)jj + 1/2kXn=1hm4 e±θni2j+1coincide with the values on multisoliton solutions of the higher integrals of motionof the sG equation [16]. It is remarkable that these eigenvalues are free of quantumcorrections although the corresponding operators in terms of local fields certainlyneed renormalization.

Let us stress that explicit expressions for these conservedcharges can be obtained by writing the local R−matrices in terms of fermi opera-tors, as in ref.[2]. Notice also that, combining eqs.

(6.31) with (6.34), and recallingthe scaling law (5.19), we can writeP± −(P±)V = p± + m4∞Xj=1ma42jI±j(6.35)That is, the light-cone lattice hamiltonian and momentum can be expressed in aprecise way as the continuum hamiltonian and momentum plus an infinite seriesof continuum higher conserved charges, playing the rˆole of irrelevant operators.31

We now come back to the problem of comparing the light–cone results withthe bootstrap predictions. As we have just seen, there is no chance to match thebootstrap predictions for λ in strip I, since ΛA(λ) and ΛD(λ) cannot be renor-malized by a common factor (see eq.

(6.29)).Indeed, the structure of the sumΛA(λ) + ΛD(λ), that is of the eigenvalue Λ(λ) of t(λ, Θ), will never match that ofthe eigenvalue ξ(u) of the bootstrap transfer matrix τ(u) (eq.(4.2)). The situationis more favourable when both λ and λ −iγ lay in strip II.

In this case eq. (6.9)applies and we find thatΛA(λ)V = ΛD(λ)V = exph−iGII(λ)Vi(6.36)In order to consider all other excited states, it is important to recall that in theinfinite volume limit the complex roots and the holes are coupled by equationswith the BA structure [14].

These “higher–level” BAE follow from the originalBAE, eq. (5.13), by summing up the Dirac sea of real roots in much the same wayas we have done here for the function G(λ).

The result can be cast in the mostsymmetrical form by parametrizing the complex roots as follows [14]:a) σ = γπu, for two–stringsb) σ + iη = γπ(u + iπ/2) and σ −iη = γπ(¯u −iπ/2), for quartets and wide pairs.c) σ + iπ/2 = γπ(u + iπ/2) for self–cojugate roots.Then the equations satisfied by the new complex root parameters {uj, j =1, 2, . .

. , m} exactly coincide with the bootstrap BAE (5.13), upon the naturalidentification of −πϕn/γ with the physical rapidity θn of the nth hole (or particle)where 1 ≤n ≤k .

By construction, the number m of higher–level roots is equalto the number of two–strings and self–conjugated roots plus twice the number ofquartets and wide pairs. Notice that a self–conjugate root in the bare BAE is alsoself–conjugate in the higher–level BAE.Then, combining eqs.

(6.26), (6.28), (6.36) and using the new u−parametrizationfor the complex roots, we obtain the general form of the A and D contributions to32

the eigenvalue of t(λ, Θ) on the N →∞limit of the BA states for λ in strip II:ΛA(λ) = −e−iGII(λ)V( kYn=1S(xn) coth xn2) mYj=1sinh ˆγ[i/2 + (πγ (λ + iγ/2) −uj)/π]sinh ˆγ[i/2 −(πγ (λ + iγ/2) + uj)/π](6.37)andΛD(λ) = −e−iGII(λ)V( kYn=1S(xn)ˆb(xn) coth xn2) mYj=1sinh ˆγ[3i/2 + (πγ (λ + iγ/2) −uj)/π]sinh ˆγ[−i/2 + (πγ (λ + iγ/2) −uj)/π](6.38)where for definiteness we chose the strip II , −π + γ/2 < Im λ < −γ/2 and setxn = πγ (λ + iγ/2) + θn. These last two expressions can be connected with that forthe eigenvalues of the bootstrap transfer matrix τ(u), eq.

(4.2), provided we identifyu with πγ (λ + iγ/2). We find indeed from eqs.

(4.2), (5.14), (6.37), (6.38):Λ(λ) = −e−iGII(λ)V ξ(u)kYn=1cothu + θn2(6.39)with λ in strip II . In analogy with eq.

(6.30), we now define the type II renormalizedtransfer matrixtII(λ) = limN→∞t(λ, Θ) expiGII(λ)V(−)Jz−N/2(6.40)Then, taking into account eq. (6.32), eq.

(6.39) can be rewrittenξ(u) = ΛII γπu −iγ2ΛI γπu −iγ2(6.41)Notice that the dependence on the cutoffrapidity Θ has completely disappearedfrom the r.h.s.of eq. (6.41).This holds true both for the explicit dependencein the vacuum function G(λ)V and for the implicit dependence through the bareBAE, which are now replaced by the Θ−independent higher–level ones.

In other33

words, the eigenvalues of the bootstrap transfer matrix can be recovered from thelight–cone regularization already on the infinite diagonal lattice, with no need totake the continuum limit. This should cause no surprise, since after all a factor-ized scattering can be defined also on the infinite lattice, with physical rapiditiesreplaced by lattice rapidities (see eq.(5.18)).

The bootstrap construction of thequantum monodromy operators Tab(u) then proceeds just like on the continnum.In this case, some q0−deformation of the two dimensional Lorentz algebra shouldact as a symmetry on the physical states. This q0 becomes unit when Θ →∞.7.

Final remarksIn the previous section we have established the precise relation (6.41) betweenthe BA eigenvalues of the bootstrap and microscopic lattice transfer matrices sG–mT–6V model, when |Im u| < π/2 and γ < π/2. With the implicit understandingthat the thermodynamic limit N →∞is taken in the ground state representation,such a relation extends to the operators themselves:τ(u) = tIIγπu −iγ2tIγπu −iγ2−1(7.1)where τ(u) is the bootstrap operator (2.8) .

The relation (7.1) between τ(u) andt(λ, Θ) is remarkably simple, specially taking into account the long chain of stepsinvolved in their totally independent constructions. For t(λ, Θ) we have:1.

Defined the light-cone lattice with the alternating parameter Θ.2. Found the antiferroelectric ground state.3.

Considered general finite energy excitations around it.4. Let the volume N become infinity.In the other hand, τ(u) follows solely from the bootstrap principles (a)–(c) ofsec.

2.34

Notice that the bootstrap construction by itself does not provide any rela-tionship between Tab(u) and the local fundamental fields entering the lagrangianwhich supposedly corresponds to the given factorized scattering model. On theother hand Tab(λ, Θ) can be explicitly written in terms of the bare fermi field ofthe mT–model [2], so that eq.

(7.1) represents a relevant piece of information forthe search of such a relationship.It is clear, however, that a direct extensionof eq. (7.1) to the full monodromy matrix would not work: indeed, suppose thatoperators ˜Tab(u) are consistently defined by the relation˜Tab(u) = T IIabγπu −iγ2, ΘtIγπu −iγ2−1(7.2)then certainly the trace Pa ˜Taa(u) coincides with τ(u), due to eq.

(7.1), but ˜Tab(u)cannot be identified with Tab(u) because it still satisfies a bare YB algebra, withanisotropy γ rather than ˆγ. [In the YB algebra (2.7) the R-matrix elements, asgiven by eq.

(3.2), depend on ˆγ ]. It is presumable therefore that eq.

(7.2) doesnot provide a consistent renormalization for the complete monodromy matrix. Itshould be noted, in this respect, that all the models considered in refs.

[1], where theexistence of a classsical analogue of Tab(u) allows to relate it to local curvature–freedivergenceless non–abelian currents, correspond to rational forms of the R−matrix.But then one would find no finite renormalization like γ →ˆγ when taking theN →∞limit in the light–cone lattice regularization of these models. Since bothbare and bootstrap R−matrices are rational and depend non–trivially only onthe spectral parameter, it is always possible to rescale the latter so that bareand boootstrap YB algebras coincide.

In other words, in these rational models,there exist a thermodynamic limit in which the microscopically defined latticemonodromy matrix is conserved. Notice that this lattice monodromy matrix canbe written in term of lattice non–abelian currents [3] in a way which representsan integrable regularization of the classical monodromy matrix.The picture istherefore fully consistent for the rational models.Evidently, the situation appears to be more subtle in a trigonometric integrablemodel like the sG–mT–6V model considered here in detail.

At the microscopic35

level the model enjoys a dynamical YB symmetry characterized by the anisotropyγ, which underlies the BA solution based on the ferromagnetic reference state Ω.At the “renormalized” level, when the reference state is the physical antiferromag-netic ground state of the infinite lattice (and still in the presence of the UV cutoffprovided by the lattice spacing), the model aquires a true YB symmetry charac-terized by the anisotropy ˆγ. Eq.

(7.1) shows that the Cartan subalgebras of thesetwo YB algebras are essentially identical, strongly supporting both the bootstrapand the light–cone lattice constructions. It would be very interesting to relate thecomplete monodromy matrices, that is to find general YB–algebraic arguments toprovide a microscopic interpretation for the bootstrap monodromy.

The recentwork reported in refs. [7], which relies on the q−deformed affine algebra approachto the YB symmetry, seems very promising in this respect, although it is restrictedto the regime |q| < 1 (while |q| = 1 in the sG–mT–6V model).REFERENCES1.

H.J. de Vega, H. Eichenherr and J.M.

Maillet, Nucl. Phys.

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C. Destri and H.J. de Vega, Nucl.

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C. Destri and H.J. de Vega, J. Phys.

A22 (1989) 1329.4. D Bernard and A LeClair, Comm.

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B. Frenkel and N. Yu. Reshetikhin, Comm.

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B. Davies, O. Foda, M. Jimbo and A. Nakayashiki,Comm. Math.

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Phys. 21 (1974) 1046.FIGURE CAPTIONSFig.1.

Light–cone lattice representing a discretized portion of Minkowski space–time. An R−matrix of probability amplitudes is attached to each vertex.The bold lines correspond to the action, at a given time, of the one–stepevolution operator U.Fig.2.

Graphical representation of the inhomogeneous monodromy matrix.Theangles between the horizontal and the vertical lines are site–dependent in anarbitrary way.Fig.3. Insertion of the alternating monodromy matrix in the light–cone lattice.Fig.4.

The two main determinations, GI(λ) and GII(λ) are defined by G(λ) withλ in strips I and II, respectively.37

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