ITEP-TH-45/97 R Hidden quantum -matrix in discrete time classical Heisenberg magnet A.Zabrodin∗ October 1997 8 9 9 1 Abstract n We construct local M-operators for an integrable discrete time version of the classical Heisenberg a magnet by convolution of the twisted quantum trigonometric 4×4 R-matrix with certain vectors in its J ”quantum” space. Components of the vectors are identified with τ-functions of the model. Hirota’s 2 bilinear formalism is extensively used. The construction generalizes the known representation of M- 2 operators in continuous timemodels in terms of Lax operators and classical r-matrix. 2 v 5 1 Introduction 1 0 0 The unified treatment [1] of non-linear soliton equations as hamiltonian systems having enough number of 1 conservedquantities in involutionis based onthe (classical)r-matrix. Its roleis to provide a universalform 7 of Poisson brackets for elements of Lax operators. An alternative though less popular point of view on the 9 r-matrix (which we are going to follow) comes from the Zakharov-Shabat approach [1],[2] which consists / t n in representing soliton equations as 2D zero curvature conditions for a pair of matrix functions (called L i and M operators) depending on a spectral parameter. The r-matrix works there as a machine producing - v M-operators from L-operators. l o Inthe paper[3],forthe simplestexampleofthe lattice sine-Gordon(SG)model,we havefoundasimilar s machine in the completely discrete set-up, i.e., in lattice integrable models with discrete time. Remarkably : v enough, it appears to be a quantum R-matrix with the ”quantum” parameter q related to the time lattice i X spacing. The formula for localM-operatorsin the discrete case has as simple structure as in the continuous one (see(1.5)below)withthe R-matrixinplaceofthe r-matrix. This comes asthe resultofacomputation. r a At the moment we do not try to ”explain” why a typical quantum R-matrix takes part in a purely classical problem. In this paper we show that the construction still works for a much more general model – partially anisotropic lattice Heisenberg magnet (HM) in discrete time. (This model contains discrete versions of SG, KdV and other equations as special cases.) Although the final result looks like a straightforward generalizationoftheoneforthelatticeSGmodel,thederivationgetsmuchmoreinvolvedandsomeimportant modifications are necessary. Letus recallthe r-matrix constructionofM-operatorsfor continuousflows. LetL (z)be a classical2×2 l L-operatoron1DlatticewiththeperiodicboundaryconditionL (z)=L (z);z isthespectralparameter. l+N l For the moment we assume the ultralocality – Poisson brackets between elements of the L (z) at different l ∗JointInstitute ofChemical Physics,Kosyginastr. 4,117334, Moscow, RussiaandITEP,117259, Moscow,Russia;e-mail: [email protected] 1 sites are zero. The monodromy matrix T (z) is ordered product of L-operators along the lattice from the l site l to l+N −1: T (z)=L (z)... L (z)L (z). (1.1) l l+N−1 l+1 l Hamiltonians of commuting flows are obtained by expanding logT(z) in z, where T(z) = trT (z) does not l depend on l due to the periodic boundary condition. All these flows admit a zero curvature representation with the generating function of M-operators given by [4],[1] M (z;w)=T−1(w)tr r z/w (T (w)⊗I) . (1.2) l 1 l Here r(z) is a (classical) 4×4 r-matrix, acting in the te(cid:2)ns(cid:0)or pr(cid:1)oduct of two(cid:3) 2-dimensional spaces, tr means 1 trace in the first space, I is the unity matrix. Expanding the r.h.s. of eq.(1.2) in w, one gets M-operators depending on the spectral parameter z. From the hamiltonian point of view, the zero curvature condition follows from Poisson brackets for elements of the L-operator. A similar r-matrix formula exists for M- operators in non-ultralocal models, though, in this case the r-matrix is not skew-symmetric. In general, M (z;w) is non-local. l It is well known [5],[1] how to construct local M-operators from the generating function. Suppose there exists z such that detL (z )=0 for any l. This means that L (z ) is a 1-dimensional projector: 0 l 0 l 0 α(l) β(l) L (z )= , (1.3) l 0 P(l) (cid:12) (cid:11)(cid:10) (cid:12) (cid:12) (cid:12) where α α = 1 , β = β , β (1.4) α 1 2 2 (cid:18) (cid:19) (cid:12) (cid:11) (cid:10) (cid:12) (cid:0) (cid:1) are two-component vectors and P(l)(cid:12)is a scalar normalizati(cid:12)on factor. Components of the vectors as well as the P(l) depend on dynamical variables. It is easy to see that M (z;z ) is a local quantity: l 0 β(l) r(z/z ) α(l−1) 0 M (z)≡M (z;z )= (1.5) l l 0 β(l) α(l−1) (cid:10) (cid:12) (cid:12) (cid:11) (cid:12) (cid:12) (notethatthe normalizationfactorcancels). Thescalarp(cid:10)rodu(cid:12)ctistake(cid:11)ninthe firstspaceonly,sotheresult (cid:12) is a 2×2 matrix with the spectral parameter z. It obeys the zero curvature condition ∂ L (z)=M (z)L (z)−L (z)M (z). (1.6) t l l+1 l l l Let us outline the results of this work. We consider the completely discretized partially anisotropic (XXZ) HM and represent M and L operators M (z), L(z) by formulas of the type (1.5): l l β(l) R(z/z ;q,ξ) βˇ(l−1) 0 M (z)= , (1.7) l β(l) α(l−1) (cid:10) (cid:12) (cid:12) (cid:11) (cid:12) (cid:12) β(l)(cid:10)R(z/(cid:12)(cid:12)z0;q,ξ′)(cid:11)α(l) L(z)= . (1.8) l β(l) α(l−1) (cid:10) (cid:12) (cid:12) (cid:11) (cid:12) (cid:12) In the r.h.s., R(z;q,ξ), R(z;q,ξ′) are quantum 4×(cid:10)4 R-m(cid:12)atrices ((cid:11)to be specified below) with the ”quantum” (cid:12) parameter q and the Drinfeld twist parameters ξ,ξ′ related to the space-time lattice. The vectors α(l) , β(l) are the same as in eq.(1.5), (cid:12) (cid:11) (cid:12) (cid:12)(cid:12) (cid:11) βˇ(l) ≡ 0 (−ξξ′)21 β(l) . (−ξξ′)−21 0 (cid:18) (cid:19) (cid:12) (cid:11) (cid:12) (cid:11) It is worth noting here that the c(cid:12)ontinuous and discrete time mode(cid:12)ls have a common L-operator; formula (1.8) gives its R-matrix representation. To visualize these formulas, it is convenient to use the language of the algebraic Bethe ansatz [6],[7]. The scalar product is taken in the ”quantum” (vertical) space, so one gets a 2×2 matrix in the ”auxiliary” (horizontal) space: 2 β (cid:10) (cid:12) (cid:12) β R(z) α = (cid:10) (cid:12) (cid:12) (cid:11) (cid:12) (cid:12) α (cid:12) (cid:11) (cid:12) TheM-operator(1.7)generatesshiftsofadiscretetimevariablem. Thediscretezerocurvaturecondition M (z)L (z)=L (z)M (z) (1.9) l+1,m l,m l,m+1 l,m gives rise to the HM equations of motion in discrete space-time. Thekeyingredientoftheconstructionistopassfromtheinitialdynamicalvariablestothepairofvectors α(l) , β(l) . In the papers [5] on exact lattice regularization of integrable models components of these vectorswereexpresedintermsofcanonicalvariablesofthe model. Thoseformulaslookedquitecomplicated (cid:12) (cid:11) (cid:12) (cid:11) a(cid:12)nd wer(cid:12)e hardly considered as something instructive. Here we shed some light on their meaning. Using equations of motion of the completely discretized model (derived from the discrete zero curvature condition (1.9)), we show that (suitably normalized) components of the vectors α(l) , β(l) are τ-functions. Theτ-functionisoneofthemostfundamentalobjectsofthetheory(cid:12)(see(cid:11)e.g(cid:12). [8](cid:11),[9]). Itisworthstressing (cid:12) (cid:12) that quantumR-matricesin the classicaldiscrete problems canhardly be noticed until one reformulatesthe non-linear equations and elements of the L-M pair entirely in terms the τ-functions. That is why we are to make a long excursioninto Hirota’s bilinear formalism[10]-[12] (Sect.2). Our guiding principle is Miwa’s interpretation[13]ofthediscretetimeflows,inwhichdiscretizedintegrableequationsaretreatedasmembers ofthesameinfinitehierarchyastheircontinuouscounterparts. Ageneralmethodtoproducediscretesoliton equationswasdevelopedin[14],where,inparticular,thediscreteisotropic(XXX)HMmodelwasproposed. Followingthese ideas,we dealwithbilinear formofthe discrete XXZ HM model. The matrix L-M-pairfor the latter is derived in Sect.3 from basic linear problems for a scalar wave function [12],[14],[15]. Like in the SG model, there are two lattice versions of the classical HM: the discrete HM model on a space-timelattice [14]andthe modelona spacelattice with continuoustime introducedin [16]forthe more general XYZ-case. It is interesting to note that both of them were proposed back in 1982 but the ideas underlying one and another seemed to be ”orthogonal”and did not intersect until the very recent time1. A motivation of this work was just an attempt to understand one of the models in terms of the other one. The manner of exposition here is different from the one in [3], where we started with either Faddeev- Volkov or Izergin-Korepin L-operator for the lattice SG model in terms of lattice fields and passed to the τ-functionbymeansofaspecialsubstitution. Inthispaperwestartdirectlyfromtheτ-function(andbilinear equations it obeys) rather than from lattice spin variables. The reason is twofold. First, the equations of motion in terms of lattice spin variables are too complicated and, anyway, are of no practical use for us. Second, in the discrete case there is no canonical way to introduce spin variables whereas Hirota’s bilinear formalism is gauge-invariant and free of artifacts. Having this in mind, let us remark that by Heisenberg modelwemeanherenothingmorethanaconvenientnamefortheproperlyreduced2DTodalatticehierarchy [22]. (Actually we dealwith a model which is slightly more generalthan the XXZ HM itself.) In this sense a name like partially anisotropic chiral field model would be also appropriate. Needless to say that in the discrete set-up the specific features of each of the two models become irrelevant [8]. What is relevant is the type of reduction of the 2D Toda hierarchy that is the same in both cases. Treatedas a hierarchy, the bilinear equationsimply infinitely many discretevariables(commuting flows) from the very beginning. Generally speaking, any two of them could be chosen as space-time coordinates. However, for our purpose we need a distinguished space-time lattice. In Sect.4 we introduce such a lattice withcoordinatesdenotedbyl,mandderivethematrixL-M-pairfortranslationsl →l+1,m→m+1. These 1Onthequantumlevel,theseideaswerepartiallylinkedtogetherin[17]-[20],wherequantumSGmodelindiscretespace-time (a quantization of Hirota’s discrete SG equation) was constructed; a generalization to affine Toda fieldtheories on the lattice wassuggestedin[21]. 3 LandM operatorsarerepresentedintheform(1.8),(1.7)inSect.5. InSect.6wediscussthecontinuoustime limit of the construction and show how eq.(1.5) is reproduced. As it was already mentioned, the r-matrix in that formula turns out to be not necessarily skew-symmetric. This fact is a signal of non-ultralocality of the model in general case. Sect.7 is a conclusion where we point out a few unsolved problems motivated by our results. 2 Hirota’s bilinear formalism In this section we present the results of the papers [13],[14] in the form convenient for our purpose. We illustrate formulas by the graphicalrepresentationof discrete flows suggested in the review [23] to which we refer for more details. 2.1 General form of 3-term bilinear equations The key object of Hirota’s approach is τ-function. The τ-function τ = τ(a,b,c,...) (as a function of the discretevariablesa,b,c,...)obeysanumberofbilinearpartialdifferenceequations. Toeachdiscretevariable a a complex parameter λ ∈C (”Miwa’s variable”) is associated. One may think of the λ as a continuous a a ”number” which marks the discrete flow. For any two discrete flows a,b we put λ ≡λ −λ , λ =−λ . ab a b ab ba Let us recall how to compose bilinear equations. Each triplet {a,b,c} of discrete flows gives rise to a 3-term bilinear equation [12],[13] for the τ: λ τ(a+1,b,c)τ(a,b+1,c+1) bc + λ τ(a,b+1,c)τ(a+1,b,c+1) ca + λ τ(a,b,c+1)τ(a+1,b+1,c)=0. (2.1) ab Allothervariableswhichtheτ maydependonenterthisequationasparameters. Eachquadruplet{a,b,c,d} gives rise to another 3-term bilinear equation: λ λ τ(a+1,b,c,d+1)τ(a,b+1,c+1,d) ad bc + λ λ τ(a,b+1,c,d+1)τ(a+1,b,c+1,d) bd ca + λ λ τ(a,b,c+1,d+1)τ(a+1,b+1,c,d)=0. (2.2) cd ab Remark Links between eqs.(2.1), (2.2) exist in both directions. On the one hand, eq.(2.1) is a particular case of (2.2) when λ → ∞. (According to [13], this limit means that the dependence on d in the τ disap- d pears.) On the other hand, eq.(2.2), though linearly independent of eqs.(2.1), is an algebraic consequense of equations of the type (2.1) written for the triplets {a,b,c}, {a,b,d} and {a,c,d}. In this sense all what we are going to derive in the sequel follows already from eqs.(2.1). 2.2 Basic bilinear equations for the discrete HM model Let us consider five discrete flows and denote the corresponding variables by p, p′, n, u, v. We denote the τ-function by τ(p,p′,n,u,v) ≡ τp,p′(u,v). In the latter notation the u,v are separated from the others n because they will play the role of chiral space-time coordinates. We call them chiral variables. 4 Bilinear equations for the τ-function of the XXZ HM model are obtained from (2.1), (2.2) by imposing the reduction τp+1,p′+1(u,v)=τp,p′(u,v), (2.3) n n+1 so we are left with four independent variables. Let us choose them to be p, n, u, v. Remark The discrete KdV equation is a particular case λ =∞, so the dependence on n disappears. The n discrete SG model corresponds to further specification λp′ =λp. Using the graphical representationintroduced in [23], we can display the above configurationof discrete flows in the figure (the graph of flows): λn r - r λp′ λ v r v 6n >p′ I λ r (cid:27) r - r λ u u ∞ p p The dashed line is drownhere to indicate that the reduction (2.3) is the same as in the case of the 1D Toda chainin discrete time (a detailed discussionofthis point see in [23]). Indeed, the vectorfield ∂ ≡−∂ +∂ p¯ n p defines the flow p¯corresponding to the dashed line; in these terms the reduction acquires the more familiar Toda-like form τp+1,p¯+1 = τp,p¯. This turns into a differential condition as λp → ∞, λp′ → λn. In this case one gets the isotropic (XXX) HM model in discrete time. Remark Looking at the graph makes it easier to deal with different discrete flows when there are many of them. We refer to [23] for the exact meaning of the graphical elements. Here we only note that keeping in mind solutions of the finite-gap type (see e.g. [2],[24]), the reader may think of this figure as drown on a patchof a Riemann surface with localcoordinateλ−1. Miwa’s variablesare coordinatesof punctures on the surface. The lines are then cuts between the punctures, which give rise to discrete commuting flows on the Jacobian via the Abel map2. The condition (2.3) allows one to get rid of p′ if necessary. Let us give a list of bilinear equations which areobtainedinthiswayfromeqs.(2.1),(2.2). Infrontofeachequationweindicatethetripletorquadruplet which it comes from. The simplest equation involves the variables p,n only: {pp′n}: λp′nτnp+−11(u,v)τnp−+11(u,v)−λpnτnp+1(u,v)τnp−1(u,v)=λp′p τnp(u,v) 2. (2.4) It is the discrete time 1D Toda chain equation in bilinear form [12]. In our ca(cid:0)se it pla(cid:1)ys the role of a constraint on the dynamical variables since the space-time coordinates u,v enter as parameters. Equations of the next group contain shifts of p,n,u only: {pp′u}: λp′uτnp+1(u,v)τnp+−11(u+1,v)−λpuτnp+−11(u,v)τnp+1(u+1,v) = λp′pτnp(u+1,v)τnp+1(u,v), (2.5) {pnu}: λ τp+1(u,v)τp (u+1,v)+λ τp (u,v)τp+1(u+1,v) un n n+1 pu n+1 n = λ τp(u+1,v)τp+1(u,v), (2.6) pn n n+1 2Forthegeneraltheoryoffinite-gapsolutionstoHirota’sdifferenceequation see[25]. 5 {p′nu}: λunτnp(u,v)τnp+1(u+1,v)+λp′uτnp+1(u,v)τnp(u+1,v) = λp′nτnp+1(u,v)τnp−+11(u+1,v), (2.7) {pp′nu}: λpnλp′uτnp+1(u,v)τnp−1(u+1,v)−λp′nλpuτnp+−11(u,v)τnp−+11(u+1,v) = λp′pλunτnp(u,v)τnp(u+1,v). (2.8) Similar equations can be written for p,n,v – it is enough to replace u by v everywhere. Remark Combining eqs.(2.5), (2.6), eq.(2.5) eq.(2.6) − , λp′uτnp+−11(u+1,v) λunτnp+1(u+1,v) and plugging (2.7) into the l.h.s., we obtain eq.(2.8). Equations involving both coordinates u,v read: {nuv}: λ τp(u,v+1)τp (u+1,v)−λ τp(u+1,v)τp (u,v+1) un n n+1 vn n n+1 = λ τp(u+1,v+1)τp (u,v), (2.9) uv n n+1 {pp′uv}: λp′uλpvτnp+1(u,v+1)τnp+−11(u+1,v)−λpuλp′vτnp+−11(u,v+1)τnp+1(u+1,v) = λp′pλuvτnp(u+1,v+1)τnp+1(u,v), (2.10) {puv}: λ τp+1(u,v+1)τp(u+1,v)−λ τp+1(u+1,v)τp(u,v+1) pv n n pu n n = λ τp(u+1,v+1)τp+1(u,v), (2.11) uv n n {p′nuv}: −λp′uλvnτnp+1(u,v+1)τnp(u+1,v)+λp′vλunτnp+1(u+1,v)τnp(u,v+1) = λp′nλuvτnp−+11(u+1,v+1)τnp+1(u,v), (2.12) {p′uv}: λp′vτnp+1(u,v+1)τnp+1(u+1,v)−λp′uτnp+1(u+1,v)τnp+1(u,v+1) = λ τp+1(u+1,v+1)τp (u,v). (2.13) uv n n+1 {pnuv}: −λ λ τp (u,v+1)τp+1(u+1,v)+λ λ τp (u+1,v)τp+1(u,v+1) pu vn n+1 n pv un n+1 n = λ λ τp(u+1,v+1)τp+1(u,v), (2.14) pn uv n n+1 The above list of linearly independent 3-term bilinear equations is by no means complete. The full list contains many other equations which either follow from the already written ones or can be derived from ”higher”analoguesofeqs.(2.1),(2.2)formorethan4variables(which,intheirturn,arealgebraiccorollaries of eqs.(2.1)). Some of them are given in the next subsection. 6 2.3 Some useful corollaries of the basic equations Note that the pair of equations (2.11), (2.12) (respectively (2.13), (2.14)) can be considered as a linear systemforτp+1(u,v+1)τp(u+1,v),τp+1(u+1,v)τp(u,v+1)(respectively,forτp (u,v+1)τp+1(u+1,v), n n n n n+1 n τp (u+1,v)τp+1(u,v+1)). Resolving these systems, we get n+1 n λp′vλunτnp(u+1,v+1)τnp+1(u,v)+λpuλp′nτnp−+11(u+1,v+1)τnp+1(u,v) = Λτp+1(u,v+1)τp(u+1,v), (2.15) n n λp′uλvnτnp(u+1,v+1)τnp+1(u,v)+λpvλp′nτnp−+11(u+1,v+1)τnp+1(u,v) = Λτp(u,v+1)τp+1(u+1,v), (2.16) n n λpvλunτnp+1(u+1,v+1)τnp+1(u,v)+λp′uλpnτnp(u+1,v+1)τnp++11(u,v) = Λτp (u,v+1)τp+1(u+1,v), (2.17) n+1 n λpuλvnτnp+1(u+1,v+1)τnp+1(u,v)+λp′vλpnτnp(u+1,v+1)τnp++11(u,v) = Λτp+1(u,v+1)τp (u+1,v), (2.18) n n+1 where Λ≡λpnλp′n−λunλvn. (2.19) These equations are equally useful in what follows. They are even more informative than eqs.(2.11)-(2.14) sinceallowoneto”fuse”thevariables,i.e. toputλ =λ . (In(2.11)-(2.14)thisleadstotheidentity0=0.) u v In this case τ(u+1,v+1) converts into τ(u+2) and we get: λunλp′uτnp+1(u−1,v)τnp(u+1,v)+λpuλp′nτnp+1(u−1,v)τnp−+11(u+1,v) = λp′nλpn−λ2un τnp(u,v)τnp+1(u,v), (2.20) (cid:0) (cid:1) λunλpuτnp+1(u−1,v)τnp+1(u+1,v)+λpnλp′uτnp++11(u−1,v)τnp(u+1,v) = λp′nλpn−λ2un τnp+1(u,v)τnp+1(u,v). (2.21) (cid:0) (cid:1) The same equations can be written down for v in place of u. At last, we present two other useful equations which are obtained from (2.5)-(2.8), (2.15)-(2.18) and (2.20),(2.21)(togetherwiththeir v-counterparts)by aproceduresimilarto the one explainedinthe remark after eq.(2.8): λpnλp′nτnp−1(u+1,v+1)τnp+1(u,v)−λunλvnτnp(u+1,v+1)τnp(u,v) = Λτp(u+1,v)τp(u,v+1), (2.22) n n 7 λp′uλp′vλpnτnp−1(u+1,v+1)τnp+1(u,v)−λpuλpvλp′nτnp−+11(u+1,v+1)τnp+−11(u,v) = λp′pΛτnp(u+1,v)τnp(u,v+1). (2.23) To conclude the section, let us mention that the complete list of linearly independent equations for the τ-function (even restrictedby the 3-termones) is somewhatlonger. Here we have selected those used in the sequel. 3 Linearization of the discrete HM model The bilinear equations from the previous section can be represented as compatibility conditions for an overdeterminedsystemoflinearproblemsfora”wavefunction”Ψ. Thisiswhatwemeanbythelinearization. For us this is a systematic way to find L-M-pairs. 3.1 Scalar linear problems The bilinear equations (2.4)-(2.14) follow from compatibility of a system of linear equations for a ”wave function” Ψ = Ψp,p′(u,v). The prototype of linear equations for Ψ is [12],[14],[15] (see also [23] for a n review): τ(a,b)τ(a+1,b+1) Ψ(a+1,b)=Ψ(a,b+1)−λ Ψ(a,b), (3.1) ab τ(a+1,b)τ(a,b+1) where a,b stand for any two elementary discrete variables. The reduction condition (2.3) for the Ψ-function reads Ψp+1,p′+1(u,v)=z2Ψp,p′(u,v), (3.2) n n where z is a spectral parameter. Hiding the variable p′ with the help of this condition, we have: τp(u)τp (u+1) {nu}: Ψp(u+1)=Ψp (u)−λ n n+1 Ψp(u). (3.3) n n+1 unτp (u)τp(u+1) n n+1 n τp(u)τp+1(u+1) {pu}: Ψp+1(u)=Ψp(u+1)−λ n n Ψp(u), (3.4) n n puτp+1(u)τp(u+1) n n n τp+1(u)τp (u+1) {p′u}: z2Ψpn+1(u)=Ψpn+1(u+1)−λp′uτnp (u)τnp++11(u+1)Ψpn+1(u) (3.5) n+1 n (and similar equations for v in place of u). In (3.3)-(3.5), v is supposed to be the same everywhere and, therefore, skipped. These equations are basic tools for deriving matrix L-M-pairs. 3.2 Vector linear problem Ψp(u) Combining equations (3.3)-(3.5), one can represent translation of the vector n along the u- Ψp+1(u) (cid:18) n (cid:19) direction in the matrix form: Ψpn(u+1) λpuτpτ+np1((uu))ττnpp+(1u(+u)1) 1 Ψpn(u)    n n   = . (3.6)  Ψpn+1(u+1)  z2λpnτnτpn+p(1u(u)τ)nτp++np+11(1u(u)) z2+λp′uττnnpp++11((uu))ττnnpp++11((uu++11))  Ψpn+1(u)       8 Let us change the gauge passing to the wave function Φ (u) Ψp(u) 1 n =D(u) (3.7)     Φ (u) Ψp+1(u) 2 n     where D(u) is the diagonal matrix D(u)= τp (u)τp+1(u) −12 λp14nz21τnp(u) 0 . n+1 n   (cid:0) (cid:1)  0 λ−pn14z−12τnp+1(u)    Then the linear problem (3.6) acquires the form Φ (u+1) Φ (u) 1 1 =L(u)(z) . (3.8)     Φ (u+1) Φ (u) 2 2     The L-operator L(u)(z) can be compactly written in terms of the three fields τp+1(u,v) τp(u,v) τp+1(u,v) ψ0(u,v)= n , ψ−(u,v)= n , ψ+(u,v)= n+1 . (3.9) τp (u,v) τp (u,v) τp+1(u,v) n+1 n+1 n 1 For brevity we also use the notation φ(u,v)≡ ψ0(u,v) 2 . The L-operator reads φ(u+1(cid:2)) (cid:3) 1 ψ−(u+1) λ zλ2 pu pn φ(u) φ(u)φ(u+1) L(u)(z)= . (3.10)  1 φ(u) φ(u+1)   zλp2nφ(u)φ(u+1)ψ+(u) λp′u +z2   φ(u+1) φ(u)      Note that detL(u)(z)=λpuλp′u−λunz2. (3.11) We call the L(u)(z) chiral L-operator because it shifts the chiral variable of the wave function via (3.8). In the next subsection we study the discrete zero curvature condition with matrices of the type (3.10). 3.3 The zero curvature condition for chiral L-operators Let C=(cid:12)(u,v+1) D=(u(cid:12)+1,v+1) (cid:12) (cid:12) (cid:12) (cid:12) A(cid:12)=(u,v) B=(cid:12)(u+1,v) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) be an elementary cell of the u,v-lattice3. In this notation (borrowed from [17]-[19]) the L-operator (3.10) reads: 3Ingeneralthecoordinateaxesarenotorthogonaltoeachother;inparticular,atλu=λv thelatticecollapsestoa1Done. 9 φ(B) 1 ψ−(B) λ zλ2 pu pn φ(A) φ(A)φ(B) L(u) (z)= . (3.12) B←A  1 φ(A) φ(B)   zλp2nφ(A)φ(B)ψ+(A) λp′u +z2   φ(B) φ(A)      (v) Similarly, we introduce another chiral L-operator,L (z), which is given by the same formula with φ(C) C←A in place of φ(B) and λpv, λp′v in place of λpu, λp′u, respectively. The discrete zero curvature condition L(v) (z)L(u) (z)=L(u) (z)L(v) (z) (3.13) D←B B←A D←C C←A is equivalent to the following non-linear equations of motion for the fields ψ0, ψ±: λp′uψ0(C)−λp′vψ0(B) ψ0(A)ψ0(A)−ψ0(B)ψ0(C) = (cid:0)λ ψ0(B)ψ0(C)ψ0(D) ψ(cid:1)+(cid:0)(C)ψ−(C)−ψ+(B)ψ−(B) (cid:1), (3.14) pn (cid:2) (cid:3) λpvψ0(D)ψ0(C)ψ−(B)+λp′uψ0(A)ψ0(C)ψ−(D) = λpuψ0(D)ψ0(B)ψ−(C)+λp′vψ0(A)ψ0(B)ψ−(D), (3.15) λpvψ0(D)ψ0(C)ψ+(C)+λp′uψ0(A)ψ0(C)ψ+(A) = λpuψ0(D)ψ0(B)ψ+(B)+λp′vψ0(A)ψ0(B)ψ+(A). (3.16) Equations ofmotion for the discrete HM model of XXX type givenin [14] are written for a different choice of dynamical variables. It would be useful to establish a direct correspondence between them. Remark ThediscreteKdVequationintheFaddeev-Volkovform[17],[18]isreproducedfrom(3.15)or(3.16) in the limit λ → ∞ when the n-dependence disappears, so that ψ− = ψ+ = 1 and we are left with the n single field ψ0. The Faddeev-Volkov chiral L-operators for the discrete KdV equation are reproduced from 1 (3.12) in the same limit provided the renormalizadspectral parameter ζ =zλ2 is finite. pn 3.4 Antichiral L-operators Here we introduce another type of L-operatorswhich will be called antichiral. Let us put λ = λ in the chiral L-operator (3.12) and, correspondingly, identify u with n. In this way u n we get the operator which generates the translation A=(u,v,n) −→ A↑ =(u,v,n+1) in the 3D lattice spanned by u,v,n: φ(A↑) 1 ψ−(A↑) λ zλ2 pn φ(A) pnφ(A)φ(A↑) (n)   L (z)= A↑←A  1 φ(A) φ(A↑)   zλp2nφ(A)φ(A↑)ψ+(A) λp′nφ(A↑) +z2 φ(A)      10