S. SERGEEV
Received 16 February 2006; Accepted 9 May 2006
A lattice model of interactingq-oscillators, proposed by V. Bazhanov and S. Sergeev in 2005 is the quantum-mechanical integrable model in 2 + 1 dimensional space-time. Its layer-to-layer transfer matrix is a polynomial of two spectral parameters, it may be re- garded in terms of quantum groups both as a sum of sl(N) transfer matrices of a chain of lengthMand as a sum of sl(M) transfer matrices of a chain of lengthNfor reducible representations. The aim of this paper is to derive the Bethe ansatz equations for theq- oscillator model entirely in the framework of 2 + 1 integrability providing the evident rank-size duality.
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1. Introduction
Theq-oscillator lattice model was formulated recently in [5,15]. It describes a system of interactingq-oscillators situated in the vertices of two-dimensional lattice, and therefore it is the quantum-mechanical system in 2 + 1 dimensional space-time in the same way, as a chain of interacting particles (or spins) is regarded as a model in 1 + 1 dimensional space-time. Formulation of the q-oscillator model provides a definition of a layer-to- layer transfer matrix as a polynomial of two spectral parameters. This transfer matrix may be interpreted in terms of quantum inverse scattering method and quantum groups, so that both sizes of the two-dimensional lattice may be interpreted as either a length of an effective chain or as symmetry group’s rank. This was called in [5] the “rank-size”
duality. The appearance of a complete set of fundamental transfer matrices forᐁq(sl) series is a signal that the layer-to-layer transfer matrix ofq-oscillator model is closely related to Bethe ansatz in the form of generalized Baxter’s “T-Q” equations. The subject of this paper is the derivation of such equations in the framework of 2 + 1 dimensional integrability.
Below in this introduction we formulate the answer, that is, we give an explicit form of “T-Q” equations in terms of a given layer-to-layer transfer matrix. To do this, we need
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 92064, Pages1–31
DOI10.1155/IJMMS/2006/92064
to repeat the structure of the layer-to-layer transfer matrix forq-oscillator model in more details.
Theq-oscillator model describes a system of interactingq-oscillatorsᏴv,
Ᏼv: xvyv=1−q2+2hv, yvxv=1−qhv, xvqhv=qhv+1xv, yvqhv=qhv−1yv, (1.1) situated in the vertices v of two-dimensional square lattice of sizesN×M. Index v stands for a coordinate of a vertex. Oscillators from different vertices commute (what is called
“locality”), the whole algebra of observables is thusᏴ⊗NM, and the vertex index v corre- sponds to the number of component of the tensor power. In this paper we imply mostly the Fock space representationᏲofq-oscillators.
The two-dimensional lattice may be identified with a layer (or a section) of three- dimensional cubic lattice, further we call it either the layer or the auxiliary lattice.
Auxiliary matricesLα,β[Ᏼv], acting inC2⊗C2⊗Ᏺv, were introduced in [5]. The layer transfer matrix T(u,v) may be constructed as a trace of 2d-ordered product of auxiliary matricesL[Ᏼv]. The transfer matrix is a polynomial of two spectral parameters,
T(u,v)= N n=0
M m=0
unvmtn,m, (1.2)
its coefficients tn,m∈Ᏼ⊗NM form a complete commutative set. MatricesL[Ᏼv] depend on some extraC-valued free parameters, for their generic values the model is inhomoge- neous. The layer transfer matrix (1.2) may be identically rewritten in two ways,
T(u,v)≡N
n=0
unTω(slnN)(v)≡M
m=0
vmTω(slmM)(u), (1.3) where
Tω(slnN)(v)= M m=0
vmtn,m (1.4)
is the 2dtransfer matrix forᐁq(slN) chain of the lengthM, corresponding to the fun- damental representationπωn in the auxiliary space (hereωnstand for the fundamental weights ofAN−1,πω0andπωNare two scalar representations,T0(slN)andTN(slN)may be writ- ten explicitly). The same layer transfer matrix T(u,v) may be rewritten as the sum of ᐁq(slM) transfer matrices
Tω(slmM)(u)=N
n=0
untn,m (1.5)
for the lengthNchain (the last part of (1.3)).
The result of this paper is the derivation of the dual Bethe ansatz equations for the q-oscillator model. They may be formulated as follows. Let normalized transfer matrices
be
τ(slmM)(u)=Tω(slmM)−(−q)mu, τ(slnN)(v)=Tω(slnn)−(−q)nv, (1.6) and letC-numerical parameters ofq-oscillator lattice be inhomogeneous enough.
Then “T-Q” equation for slMis M m=0
(−v)mτ(slmM)(u)Qq2mu=0. (1.7) The statement is that if tn,m take their eigenvalues, then there exist M special values v1,...,vM of v, such that corresponding Q1(u),...,QM(u) in (1.7) are polynomials.
(Parametervin the “u-shift” equation (1.7) is irrelevant since a rescalingQ(u)→uνQ(u) changes it.) Degrees of the polynomials are uniquely defined by certain occupation num- bers of oscillators.
In its turn, equivalent “T-Q” equation for slNis N
n=0
(−u)nτ(slnN)(v)Qq2nv=0. (1.8) If tn,mtake their eigenvalues, then there existN special valuesu1,...,uNof u, such that correspondingQ1(v),...,QN(v) in (1.8) are polynomials. All the other forms of nested Bethe ansatz equations follow from (1.7) or (1.8).
PolynomialsQ(u) andQ(v) may be denoted in the quantum-mechanical way as “wave functions” of statesQ|and|Q:
Q(u)=
Q|u, Q(v)=
v|Q, (1.9)
where|uandv|serve the simple Weyl algebraᐃ: uv=q2vu, u|u = |uu, v|u =q2uv,
v|u=uq2v, v|v=vv|. (1.10) Let
J(u, v)=N
n=0
M m=0
(−q)−nm(−u)n(−v)mtn,m. (1.11) Then (1.7) and (1.8) are correspondingly
Q|J(u, v)|u = v|J(u, v)|Q =0. (1.12) The formulation of theq-oscillator model and definition of T(u,v) are locally 3din- variant, the quantum group interpretation (1.3) is the secondary one. In this paper we will derive (1.11) without any quantum group technique.
To explain our method, we need to comment a little on the classical limit.
In the classical limitq→1, the localq-oscillator generators become the classical dy- namical variables, theq-oscillator model becomes a model of classical mechanics, quan- tum evolution operators become Baecklund transformations for the dynamical variables.
In particular,T(u,v) may be understood as a partition function of a completely inho- mogeneous free fermion six-vertex model on the square lattice (but it should not be re- garded as a model of statistical mechanics). In its turn,J(u,v) becomes a free fermion determinant (the sign (−)nm+n+mcounts the number of fermionic loops). There exists a well-known formula in the theory of two-dimensional free fermion models, relatingT andJ:
T(u,v)=1 2
J(−u,v) +J(u,−v) +J(−u,−v)−J(u,v). (1.13)
In the classical limit, equationJ(u,v)=0 defines the spectral curve. Dynamical variables may be expressed in terms ofθ-functions on the Jacobian of the spectral curve. The se- quence of Baecklund transformations, which is the “discrete time” in the classical model, is a sequence of linear shifts of a point on the Jacobian. The classical model was formu- lated and solved by Korepanov [8].
Classical integrability is based on an auxiliary linear problem. EquationJ(u,v)=0 is the condition of the existence of a solution of the linear problem. Our point is that in quantumq=1 case, the linear problem is still the basic concept of the solvability. Quan- tum J(u, v) is a well-defined determinant of an operator-valued matrix, and J(u, v)|Ψ = 0 is again the condition of the existence of a solution of a quantum linear problem. The polynomial structure of, for example,v|Ψfollows from a more detailed consideration of the quantum linear problem in a special basis of diagonal “quantum Baker-Akhiezer function” (related to a quantum separation of variables).
The structure of the paper is the following. In the first section we recall briefly some basic notions of the classical model [8]: the linear problem, spectral curve, and details of the combinatorial representation of the spectral curve. In the second section we repeat the definition of the quantum model and its integrability [5,15]. In particular, our definition of the spectral parameters differs from that of [5]. Quantum linear problem, derivation of (1.11), and properties of various forms of (1.12) are given in the third section. The fourth section includes an example.
2. The Korepanov model
We start with a short review of the integrable model of classical mechanics in discrete 2 + 1 dimensional space-time [8]. The main purpose of this section is to recall the relation between Korepanov’s linear problem, spectral determinant, and partition function for free fermion model. Another aim is to fix several useful definition and notations.
2.1. Linear problem. Consider a two-dimensional lattice formed by the intersection of straight lines enumerated by the Greek letters. Let the vertices of the lattice are enumer- ated in some way.
ψβ ψβ¼
ψα
ψα¼
Av
ψ¼α=avψα+bvψβ,
ψ¼β=cvψα+dvψβ.
Figure 2.1. Vertex v is formed by intersection ofα- andβ-lines of auxiliary lattice. Vertex linear prob- lem is the pair of relations binding four edge variables.
Consider a particular vertex with a number v formed by the intersection of linesα andβ, as it is shown inFigure 2.1. It was mentioned in the introduction that an auxiliary lattice is a section of three-dimensional lattice, the vertices on the auxiliary lattice are equivalent to the edges of the three-dimensional one. InFigure 2.1, the dashed lines are the lines of auxiliary lattice, while the solid line sprout from the vertex v is the edge of the three-dimensional lattice.
Let four freeC-valued variables Av=
av,bv,cv,dv
(2.1)
be associated with vertex v. In addition, letC-valued variablesψα andψβ be associated with the ingoing edges, and letC-valued variablesψαandψβbe associated with the outgo- ing edges, as it is shown inFigure 2.1(a certain orientation of auxiliary lines is implied).
The local linear problem is a pair of linear relations binding the edge variables. Its stan- dard form, the right-hand side ofFigure 2.1in matrix notations, is
ψα ψβ
=XAv
ψα
ψβ
, whereXAv
def
= av bv
cv dv
. (2.2)
2.2. Korepanov’s equation. Equations of motion in an integrable model arise as an as- sociativity condition of its linear problem. To derive their local form, consider a vertex of three-dimensional lattice (not necessarily the cubic one), and sections of it by two auxiliary planes, as it is shown inFigure 2.2. From three-dimensional point of view, the auxiliary linear variables belong to the faces of 3dlattice, while the dynamical variablesAv
belong to the edges of the 3dlattice. Therefore,Avare distinguished fromAv, but linear variables on outer edgesψα,...,ψγ, in both top and bottom auxiliary planes, are identi- fied. Consider the bottom plane first. The linear problem rule (2.2) may be applied three times for excluding internal edges; as a result one obtains an expression of the “primed”
linear variables in terms of “unprimed”:
⎛
⎜⎝ ψα ψβ ψγ
⎞
⎟⎠=Xα,β A1
·Xα,γ A2
·Xβ,γ A3
·
⎛
⎜⎝ ψα
ψβ ψγ
⎞
⎟⎠, (2.3)
A¼1
A¼2
A¼3
ψγ
ψβ
A3
ψα
A2ψγ¼
A1
ψ¼β ψα¼
A¼1
A¼2
A¼3
ψγ
ψβ
A3
ψα
A2
ψγ¼
A1
ψβ¼ ψα¼
Figure 2.2. Left- and right-hand sides of Korepanov’s equation.
where (cf. the ordering ofα,β,γin column vectors)
Xα,β
A1
=
⎛
⎜⎝
a1 b1 0 c1 d1 0
0 0 1
⎞
⎟⎠, Xα,γ
A2
=
⎛
⎜⎝
a2 0 b2
0 1 0
c2 0 d2
⎞
⎟⎠, Xβ,γ
A3
=
⎛
⎜⎝
1 0 0
0 a3 b3
0 c3 d3
⎞
⎟⎠.
(2.4) The top plane ofFigure 2.2may be considered in the same way,
⎛
⎜⎝ ψα ψβ ψγ
⎞
⎟⎠=Xβ,γ
A3
·Xα,γ
A2
·Xα,β
A1
·
⎛
⎜⎝ ψα ψβ
ψγ
⎞
⎟⎠, (2.5)
where the matricesX#,#are given by (2.4) withAv=(av,bv,cv,dv).
The associativity condition of linear problems (2.3) and (2.5) is the Korepanov equa- tion
Xα,β
A1
·Xα,γ
A2
·Xβ,γ
A3
=Xβ,γ
A3
·Xα,γ
A2
·Xα,β
A1
, (2.6)
relating the set of 12 variablesAvwith the set of 12 variablesAv, v=1, 2, 3. Equation (2.6) describes a single 3dvertex. Equations of motion for three-dimensional integrable model are the collection of (2.6) for all vertices of the 3dlattice.
The Korepanov equation needs a very important comment. MatricesX#,#[Av] by defi- nition (2.4) act in the direct sum of one-dimensional vector spaces labelled by the indices α,β,γ, and so forth. The matrixXα,β[A1] in the block (α,β) coincides withX[A1] (2.2), and in the block (γ,...) it is the unity matrix. In what follows, such “direct sum” imbed- ding of 2×2 matricesXinto higher-dimensional unity matrices will always be implied.
β1
β2
β3
α3 α2 α1
Figure 2.3. A fragment of the auxiliary square lattice.
2.3. Linear problem with periodical boundary conditions. Korepanov’s solution of the equations of motion is based on the solution of the linear problem for the whole auxiliary lattice. Consider the square lattice with the sizesN×M. Let the lines of the lattice be enumerated by
αn, βm, n=1, 2,...,N,m=1, 2,...,M. (2.7) A fragment of the auxiliary lattice is shown inFigure 2.3. Notations for vertex and auxil- iary variables for (n,m)th vertex of the plane are shown inFigure 2.4. The local auxiliary linear problem (2.2) for (n,m)th vertex takes the form
⎛
⎝ψα(mn−1)
ψβ(nm−1)
⎞
⎠=XAn,m
·
⎛
⎝ψα(mn)
ψβ(nm)
⎞
⎠, n=1,...N,m=1,...,M. (2.8)
The linearity of the whole set of (2.8) with respect toψ’s makes it possible to define the quasi-periodical boundary conditions for them:
ψα(mn+M)=uψα(mn), ψβ(nm+N)=vψβ(nm), (2.9) whereuandvareC-valued spectral parameters.
Linear equations (2.8) may be iterated for the whole lattice as follows. Let
ψα(m)=
⎛
⎜⎜
⎜⎜
⎜⎝ ψα(m1)
ψα(m)2
... ψα(m)N
⎞
⎟⎟
⎟⎟
⎟⎠, ψβ(n)=
⎛
⎜⎜
⎜⎜
⎜⎜
⎝ ψβ(n1) ψβ(n2) ... ψβ(nM)
⎞
⎟⎟
⎟⎟
⎟⎟
⎠
. (2.10)
Then the repeated use of (2.8) gives
⎛
⎝ψα(0)
ψβ(0)
⎞
⎠=Xα,β
⎛
⎝ψα(M)
ψβ(N)
⎞
⎠, (2.11)
ψβ(n)m ψ(nβm1)
ψα(m)n
ψα(mn 1)
An,m
Figure 2.4. Notations for (n,m)th vertex of the auxiliary lattice.
where, in terms of matrix imbedding discussed right after (2.6), the (N+M)×(N+M) monodromy matrixXα,βmay be written as
Xα,β=
n
m
Xαn,βm
An,m
, (2.12)
where
n
fndef= f1f2···fN−1fN,
m
fmdef= f1f2···fM−1fM. (2.13)
The boundary conditions (2.9) giveψα(M)=uψα(0),ψβ(N)=vψ(0)β , so that (2.11) becomes
1−Xα,β· u 0 0 v
⎛
⎝ψα(0)
ψβ(0)
⎞
⎠=0. (2.14)
The whole linear problem has a solution if and only if J(u,v)def=det 1−Xα,β· u 0
0 v
(2.15) is zero. EquationJ(u,v)=0 defines the spectral curve for the model, equations of mo- tion (2.6) for the whole three-dimensional lattice have an exact solution in terms ofθ- functions on the Jacobian of the spectral curve [8].
2.4. Free fermion model. The determinant (2.15) has the very well-known combinato- rial representation. Usual way to derive it is to define the determinant in terms of the Grassmanian integration and then to turn from normal symbols to matrix elements.
Let in this subsectionψandψbe the Grassmanian variables with the integration rules dψ=
dψ=0 andψdψ=
ψdψ=1. Then the determinant (2.15) may be written as J(u,v)=
eᏭ[ψ,ψ]ᏰψᏰψ, (2.16)
where the “action” is Ꮽ=
N,M n,m=1
ψ(αmn−1),ψ(nβm−1)·XAn,m
·
⎛
⎝ψα(mn)
ψβ(nm)
⎞
⎠+ψα(mn)ψ(αmn)+ψ(n)βmψ(n)βm
, (2.17) and the measure is
ᏰψᏰψ=
N,M n,m=1
dψ(αmn)d ψα(mn)dψ(n)βmdψβ(n)m. (2.18) Spectral parameters appear in (2.16) via
ψ(0)αn =uψ(αMn), ψ(0)βm=vψ(βNm). (2.19) In terms of the Grassmanian variables, the exponent of a quadratic form is a normal symbol of some operatorL,
exp
ψα,ψβ·XAv
· ψα ψβ
def=
ψα,ψβLα,β
Avψα,ψβ. (2.20)
The fermionic coherent states are defined by
|ψ = |0+|1ψ, ψ| = 0|+ψ1|, (2.21) and the extra summands in (2.17) correspond to the unity operators
1=
|ψeψψdψ dψψ|. (2.22) It is important to note that the indices of the operatorLα,β[Av] (2.20) label copies of two- dimensional vector spaces (2.21)C2x|0+y|1. Thus,Lα,β acts in the tensor product of two-dimensional vector spaces, whileXα,β acts in the tensor sum of one-dimensional vector spaces. In the basis of the fermionic states
nα,nβ
=
|0, 0,|1, 0,|0, 1,|1, 1
, (2.23)
operatorLα,β(2.20) is 4×4 matrix
Lα,β
Av
=
⎛
⎜⎜
⎜⎝
1 0 0 0
0 av bv 0 0 cv dv 0
0 0 0 zv
⎞
⎟⎟
⎟⎠, wherezvdef
=bvcv−avdv. (2.24)
Besides, the Korepanov equation (2.6) is the equality of the exponents of the normal symbol form of the local Yang-Baxter equation
Lα,β
A1
Lα,γ
A2
Lβ,γ
A3
=Lβ,γ
A3
Lα,γ
A2
Lα,β
A3
, (2.25)
since
ψ|Lα,βLα,γLβ,γ|ψ =expψ·Xα,βXα,γXβ,γ·ψ (2.26) and so forth. Turn now to the expression of the determinant (2.15) in terms of operators L. Let 2N+M×2N+M matrixLα,βbe the ordered product of localL’s:
Lα,β=
n
m
Lαn,βm
An,m
. (2.27)
This is related to the monodromy matrix (2.12) by means of (cf. (2.20)) exp
ψα,ψβ·Xα,β· ψα
ψβ
=
ψα,ψβLα,βψα,ψβ
. (2.28)
Define now the boundary matrices forLα,β, D(u)def= 1 0
0 u
, Dα(u)=
n
Dαn(u), Dβ(v)=
m
Dβm(v), (2.29) and let
T(u,v)=Trace
α,β
Dα(u)Dβ(v)Lα,β
. (2.30)
By the construction, T(u,v) is the partition function for a free-fermion lattice model with the inhomogeneous Boltzmann weights—matrix elements ofLαn,βm[An,m]—andu, v-boundary conditions. It is the polynomial ofuandv:
T(u,v)=N
n=0
M m=0
unvmtn,m. (2.31)
Sometimes a pure combinatorial representation of the partition function is very use- ful. Any monomial inT(u,v) (2.30) corresponds to a non-self-intersecting path on the toroidal lattice. A path may go through a vertex in one of five different ways as it is shown inFigure 2.5(or do not go through at all). A factor fvis associated with each variant, these factors are the matrix elements ofLα,β[Av] (2.24). A monomial inT(u,v), corresponding to pathC, is
tC=
along pathC
fv. (2.32)
Any non-self-intersecting path on the toroidal lattice has a homotopy class
w(C)=nA+mB, (2.33)
β
α 1 av cv
zv dv bv
Figure 2.5. Six variants of bypassing the vertex. Vertex factors fvare matrix elements ofL. Note that in the variantzvthe path is not self-intersecting.
whereAis the toroidal cycle along theα-lines, andBis the toroidal cycle along theβ-lines ofFigure 2.3. Then the elementtn,mof (2.31) is
tn,m=
C:w(C)=nA+mB
tC. (2.34)
The determinantJ(u,v) (2.15) is related totn,mvia J(u,v)=N
n=0
M m=0
(−)nm+n+munvmtn,m, (2.35) where the sign (−)nm+n+m counts the number of fermionic loops on the toroidal square lattice. The determinantJmay be expressed in terms ofTand vice versa:
J(u,v)=1 2
T(−u,−v) +T(−u,v) +T(u,−v)−T(u,v), T(u,v)=1
2
J(−u,−v) +J(−u,v) +J(u,−v)−J(u,v).
(2.36)
The last equality is very well known in the two-dimensional free-fermion model as the formula relating the lattice partition function and fermionic determinant.
Now we may finish the collection of notions and definitions of the Korepanov model of classical integrable dynamics on three-dimensional lattice and proceed to the description of their quantum analogues.
3. Quantum model
In the previous section, we did not pay any attention to the structure of vertex variables Av(2.1), they were defined simply as the list of elements ofX(2.2). The key point for the quantization of the model is that it is possible do define a local Poisson structure onAv
[5] such that the transformation
A1⊗A2⊗A3−→A1⊗A2⊗A3, (3.1)
defined by the Korepanov equation (2.6), is a symplectic map. Symplectic structure ad- mits an immediate quantization. We will skip here all the details and proceed directly to the ansatz for quantizedAv,X[Av] andL[Av]. The aim of this section is just to give precise definition of T(u,v) (1.2).
3.1. The quantum Korepanov and tetrahedron equations. The localq-oscillator algebra Ᏼis defined by (1.1). The Fock spaceᏲrepresentation forq-oscillators corresponds to
Spechv
=0, 1, 2,... ∀v. (3.2)
Quantized dynamical variables (2.1) are theq-oscillator generators and a pair ofC-valued parameters,Ꮽv∼(Ᏼv;λv,μv):
Ꮽv=
av=λvqhv,bv=yv,cv= −q−1λvμvxv,dv=μvqhv. (3.3) QuantizedX(2.2) andL(2.24) are given by
XᏭv
= λvqhv yv
−q−1λvμvxv μvqhv
, (3.4)
Lα,β
Ꮽv
=
⎛
⎜⎜
⎜⎜
⎝
1 0 0 0
0 λvqhv yv 0
0 −q−1λvμvxv μvqhv 0
0 0 0 −q−1λvμv
⎞
⎟⎟
⎟⎟
⎠. (3.5)
One may verify directly that the quantum Korepanov equation Xα,β
Ꮽ1
Xα,γ
Ꮽ2
Xβ,γ
Ꮽ3
R=RXβ,γ
Ꮽ3
Xα,γ
Ꮽ2
Xα,β
Ꮽ1
(3.6)
is equivalent to the auxiliary tetrahedron equation (the quantum local Yang-Baxter equa- tion)
Lα,β
Ꮽ1
Lα,γ
Ꮽ2
Lβ,γ
Ꮽ3
R=RLβ,γ
Ꮽ3
Lα,γ
Ꮽ2
Lα,β
Ꮽ1
. (3.7)
Here the intertwining operator R=R123 acts in the tensor product of representation spaces Ᏺ1⊗Ᏺ2⊗Ᏺ3 of three q-oscillatorsᏴ1,2,3. Parametersλv,μv of Ꮽv are the pa- rameters of R. Both (3.6) and (3.7) are equivalent to the following set of six equations:
Rqh2x1=λ2
λ3
qh3x1− q
λ1μ3qh1x2y3
R, Rx2=
x1x3+ q2
λ1μ3qh1+h3x2
R, Rqh2x3=μ2
μ1
qh1x3− q
λ1μ3qh3y1x2
R, Ry2=
y1y3+λ1μ3qh1+h3y2
R,
Rqh1+h2=qh1+h2R, Rqh2+h3=qh2+h3R.
(3.8)
Classical equations (2.6) and (2.25) follow from (3.6) and (3.7) in theq→1 limit of the well-defined automorphismᏭv=RᏭvR−1. For irreducible representations ofᏴv, R is
defined uniquely, its matrix elements for the Fock space representation are given in [5].
Remarkably,Figure 2.2may be used for the graphical representation of both classical and quantum equations, in the quantum case the solid 3dcross inFigure 2.2stands for R.
Integrable model of quantum mechanics may be formulated purely in terms of ma- tricesL(3.5). The intertwiner R is related to evolution operators, it is another subject and we will not consider it here. Below we recall the definition of integrable model of quantum mechanics from [5,15].
3.2. Transfer matrix T. For the square latticeN×Mof the previous section, define the
“monodromy” of quantizedL’s literally by (2.27) Lα,β=
n
m
Lαn,βm
Ꮽn,m
, (3.9)
and its trace (cf. (2.30))
T(u,v)=Trace
α,β
Dα(u)Dβ(v)Lα,β
, (3.10)
where boundary matricesDare defined by (2.29). To distinguish the classical and quan- tum cases, we use the boldface letters for the quantized T and its decomposition (2.31):
T(u,v)=N
n=0
M m=0
unvmtn,m. (3.11)
Since the arguments ofL’s are localq-oscillator generators, T(u,v)∈Ᏼ⊗NM is by defi- nition a layer-to-layer transfer matrix, its graphical representation is again the classical Figure 2.3. The model is integrable since
T(u,v)T(u,v)=T(u,v)T(u,v), (3.12) that is, the coefficients tn,min (3.11) form the set of the integrals of motion.
Now we give the technical proof of the commutativity (3.12). The commutativity of layer-to-layer transfer matrices follows from a proper tetrahedron equation [6]. In addi- tion toLα,β[Ꮽv] (3.5), define
Lα,β
Ꮽ0
=
⎛
⎜⎜
⎜⎜
⎝
1 0 0 0
0 λ0(−q)h0 y0 0 0 q−1λ0μ0x0 μ0(−q)h0 0
0 0 0 q−1λ0μ0
⎞
⎟⎟
⎟⎟
⎠, (3.13)
whereᏭ0∼(Ᏼ0;λ0,μ0). The constant tetrahedron equation forLandL, Lα,α
Ꮽ0Lβ,β
Ꮽ0
Lα,β[Ꮽ]Lα,β[Ꮽ]=Lα,β[Ꮽ]Lα,β[Ꮽ]Lβ,β
Ꮽ0Lα,α
Ꮽ0
, (3.14)
