Download Integrable Models in Classical and Quantum Field Theory

Proceedings of the International Congress of Mathematicians
August 16-24, 1983, Warszawa
L. A. TAKHTAJAN
Integrable Models in Classical
and Quantum Field Theory
Eecent papers of Faddeev, Sklyanin and the author ([25], [27], [28], [33])
contain a quantum version of the inverse scattering method (see also the
reviews and lectures [7]-[9], [15], [23]). This is a new method of exact
solution of the models in 1 + 1 dimensional quantum field theory and in
classical statistical mechanics on a two-dimensional lattice. The profound
papers of Baxter [l]-[3] have also played an important role in the formation of the method. In the present talk I will try to explain the underlying
ideas and basic constructions of this new domain of modern mathematical
physics and also to point out its connections with other parts of mathematics and theoretical physics. Our exposition (as it can be seen from the
references) will be mostly based on the results obtained in the Leningrad
Branch of the Steklov Mathematical Institute.
I. Classical theory
1. Formulation of the inverse scattering method. This method applies
to equations which can be represented as the zero curvature condition
[Po> Pi] = 0 for the connection Vß = djdxß— TJ^Çx, A), p = 0 , 1 ; x0 = t,
xx = x defined in a trivial fibre bundle with the base JB2 (the space-time)
and the fibre Cm (the auxiliary space).^Matrix elements of the matrices
TJ^x, t, A) depend on the classical fields y?(x, t), a = 1, . . . , M, involved
in the equation considered, and on the variable A e C1 called the spectral
parameter. The zero curvature condition must hold for all A.
U331]
1332
Section 13: L. A. Takhtajan
A fundamental role in the method is played by the equation of parallel
translation along the #-axis
dF
dx = TJ(x9 X)F.
(1)
Here we put TJ = Ux and reduce the dependence on t. In the case of periodic
initial data ipa(x+2L) = ipa(x) there naturally emerge the transition
matrix T(x9 y9 A) = exp j U(x'9 X)dx* which is a solution of equation (1)
y
with the initial condition T(x9 y9 A) \Xesy = I and the monodromy matrix
TL(X) for the interval (~L9L)9 TL(K) = T(L9 -L9X). Here I denotes
the unit matrix in C"1 and the integral is understood as a multiplicative
one. The time dependence of these objects is determined by the equation
exp J Updibp = I
(2)
Y
which holds for any closed contour y cz R2, due to the zero curvature
condition. In particular, the functionals trî^UA),"k= 1 , . . . , m— 1, where
tr stands for the trace in Cm9 do not depend on t and play the role of generating functions for the integrals of motion.
In the rapidly decreasing case, where lim yf*(x) = ^ ± , there appear
œ-j-dboo
the reduced monodromy matrix T(X), relating the left and the right Jost
solutions, and the characteristics of the discrete spectrum. In these terms
the dynamics becomes completely transparent and the matrix TJ(x, A)
is uniquely determined by them. This procedure is based on the formalism
of the Eiemann problem, i.e., the problem of analytic factorization of
matrix-valued functions. Here we have a connection with the theory of
functions.
These are the basic/items of the classical inverse scattering method
(see [9], [12], [38]; our exposition follows [9], [12]). It is applied to such
well-known equations as the Korteweg-de Vries equation (KdV), the
nonlinear Schrödinger equation (N3), the Sine-Gordon equation (SG),
the Heisenberg magnet equation (HM), and others.
2. The Hamiltonian approach. The most elegant formulation of the method
is its Hamiltonian formulation originating from paper [37]. Its present
form is based on the concept of a classical r-matrix introduced in [24].
A classical r-matrix is defined as a matrix r(X) in Cm® C™ which enables
Integrarle Models in Classical and Quantum Field Theory
1333
one (if possible) to write down all the Poisson brackets of matrix elements
of U(x9 A) in the following compact form
{U(x,X)®TJ(y,[i)} =[r(A-/i), U(x, A)® L+L® U(x9 (i)lô(x-y).
*
(3)
Here {A®B} denotes the matrix in Cm®Cm of the same structure as
*
A®B with the products of matrix elements replaced by their Poisson
brackets.
As an illustrating example consider the BM model with the equation
of motion
ds
dœ*
dt
8(xt ,*)e R3 >
A 8;
<8,8> == 1.
w
We have ([30])
1
#0
=
dx
Oil)
ZiA.
1
1
8,
2U*
2'IA~
Vi
=
*
•8,
~2Î
2A
8 ••= <s, *>,
(5)
where —- oa, a = 1, 2, 3, are the generators of the fundamental repre2%
sentation of the Lie algebra su(2): aa are called the Pauli matrices. The
Poisson brackets and the Hamiltçnian of the model are
{BM, Sb(y)} = eahc8c(y)à(x-y),
_
1 r / dB
dB\2 J \ dx ' dx j
'
(6)
v
'
—&
where sab0 are the structure constants of su(2). The r-matrix is
r(À) =P/2A,
(8)
2
2
where P is the permutation operator in C ®C .
The r-matrix formalism is based on the following result, though simple,
yet important.
THEOEEM 1. Suppose that relation (3) holds. Then for the transition
matrix of equation (1) we have
. {T(x, y, A) ®T(x9 y, p)} = [rtf-/*), X{a9 y9 A) ®T(x9 y9 /*)],
>
where
—L^y*^x<L.
(9)
1334
Section 13: L. A. Takhtajan
As a corollary we obtain that {tr2^( A), t r l 7 ^ ^ ) } = 0 ; ft, I =1, ...,m —1,
which is the involution property of the families of integrals of motion
including the Hamiltonian of the model. Using Theorem 1 one can establish
complete integrability in the rapidly decreasing case by constructing
a canonical transformation to variables of action-angle type ([9], [12]).
One of the characteristic features of the Hamiltonian formulation
is t h a t the r-matrix replaces the zero curvature condition ([25], [9], [12],
[35]).
THEOREM 2. Suppose that for the matrix U(x, A) in equation (1) the
condition (3) holds. Then for the generic Hamiltonian equation
^ ^ L
= { t r ^ M , W"(x)},
a=l9...,M,
(10)
where \i is a parameter, the zero curvature condition holds with TJx(x, A)
= U(x, A) and
U0(x,X,p)
=t?x[(TL(L,x,iJL)®I)r(p-X)(TL(x^
~L9i*)®I)).
(11)
Here trx stands for the trace in the first factor in C™® Gm.
3 . Symplectic structure associated with an ^-matrix. Formula (3) can also
be interpreted as the way of defining the symplectic structure on the phase
space parametrized by the functions y>a(x), a = 1 , . . . , M. The skewsymmetry of the Poisson brackets is provided by the condition r(X)
= —Pr(—X)P and the relation
{rX2(X-p),
r13(A)] + [r 1 2 (A-^), r 2 3 ( ^ ] + [r13(A), r23(p)] = 0
(12)
guarantees the Jacobi identity. Here P is the permutation matrix in
Cm®Cm and r12 denotes the matrix in Cm®Cm®Cni which acts trivially
in the third factor and coincides with r in the product of the first two
(analogously for r13 and r 23 ). The relation (12) is called the classical YangBaxter equation (or the classical triangle equation) and is quite popular
nowadays'([5], [22]). Thus, in [5] the solutions of (12) associated with
simple Lie algebras were constructed. As a by-product of the* study of
this equation a new object, the Lie-Hamilton group, has appeared in [6].
4. Geometrical interpretation of the r-matrix Poisson brackets. There
exists a very elegant interpretation of the r-matrix in the language of
the representation theory ([9], [11]): the Poisson brackets defined by an
Integrable Models in Classical and Quantum Field Theory
1335
r-matrix of the form (8) are just the Lie-Poisson brackets for an infinitedimensional Lie algebra. More precisely, let g be the finite-dimensional
semisimple Lie algebra. The Lie-Poisson bracket on the phase space g* is
{/,<?}(£) = y°a>c^-§f„
dL ôh
ài
fefif,
(13)
where öabß are the structure constants of g. The brackets (13) can be naturally lifted to define the Poisson brackets for the functionals on </* —the
dual space of the current algebra g. The latter is just the Lie algebra of
the Laurent series in the variable A with coefficients in g. These brackets
are also defined on the dual space of the subalgebra gf+ consisting of the
Laurent series in the negative powers of A; the same holds for the complementary subalgebra gL. Moreover, let Kab be the matrix of the Killing
form in the basis Xa in g, let Kab be its inverse and
n = ^KabXa®Xb.
(14)
a,b
Then we have the following theorem ([9], [11]).
THEOREM
3. The Poisson brächet for the generic element TJ(l) eg\ is
{U(A)®U(V)}
where r(X)
= [r(A—i»)f 17(A)®I + I® U(t*)]9
(15)
=nß.
Introducing the ^-dependence in g (roughly speaking, considering
g = f] ®g)> we obtain from (15) the relation (3).
iceR1
This approach leads to various integrable models if one considers
suitable orbits in g\ (or in f 1). Thus the HMmodel corresponds to g = su(2)
and the simplest orbit consisting of the points U(x9 A) = 8(x)ß9 82(x)
= const I. Here we have a relationship with the representation theory and
the method of orbits.
If g has nontrivial automorphism <r, then the phase space can be reduced
by considering quasiperiodic elements
00
l7(a>,A)= JT AnU{x,K + nco)A-n,
tt=*-oo
32 — Proceedings..., t. II
\
(16)
1336
Section 13: L. A. Takhtajan
where A is the representation of a. If the matrix II commutes with A ® A9
then for V(x9 A) the relation (3) holds, with the r-matrix
ZJ
v
A + wco
'
Thus, in particular, the r-matrix of the SG model is obtained from the
r-matrix of the HM model. For the Lie algebras of An type a second
averaging is possible. Thus one can obtain r-matrices expressed in terms
of elliptic functions.
5. The lattice case. Classical models on a lattice play an intermediate role
in quantizing classical continuous models ("quantization of the auxiliary
space"). The matrix U(x9 A) is replaced by the transition matrix Ln(X)
from the wth lattice site to the (w+l)th. I n the continuous limit Ln(X)
= I+AU(x9 A) + 0(A2)9 where A is the lattice spacing. The monodromy
matrix is given by the ordered product
N
TNW = [JLn(X) = LN(A)...
LX(X),
(18)
where JV is the number of sites of the lattice. Since Ln(X) is the transition
matrix for one lattice site, formula (3) is uniquely transferred to the lattice
case as follows:
{Ln(X)®Lm(p)}
= [ r t f - j i ) , Ln(X)®Lm(p)1ònm
(19)
and in the continuous limit (19) goes back into (3).
All continuous models have their lattice variants with the same rmatrices. The averaging procedure also works for the lattice case. I n
contrast with the continuous case, in the lattice case the matrix Ln(X)
is represented as an ordered product ([11]) and not as in (16). Here we have
a connection with the theory of analytic matrix-valued functions.
As a result of the study of equation (19), in [26] there were introduced
quadratic Poisson brackets algebras. These brackets are nontrivial deformations of the Lie-algebraic Poisson brackets. Here we have an interesting
example of a deformation of algebraic structures.
Integrarle Models in Classical and Quantum Field Theory
1337
II. Quantum theory
We begin with lattice theories, which necessarily a.rise in quantizing
compact models, i.e. models with Poisson brackets algebras associated
with compact Lie algebras (e.g. the HM model). Moreover, introduction
of the lattice plays the role of ultraviolet regularization of continuous
models.
1. The fundamental relation with the quantum JS-matrix. Instead of the
classical fields yf(x) we consider the field operators W^, which act irreducibly on the Hilbert space l)n, the space of quantum states at the nth
lattice site. The complete Hilbert space of the model on a lattice with
N
M sites is $N = f] ®1)n. In quantization it is natural to replace Ln(X)
by the matrix-operator Ln(X) which is a matrix in Cm with matrix elements
belonging to the ring generated by ¥£ and depending on the spectral
parameter A.
The problem of the right generalization of our main relation (19) to
the quantum case is far from being trivial. The study of concrete models
([25], [28], [33]) suggests the following generalization of (19), explicitly
introduced in [28], [33]:
B(A-t*)(Ln(?i) ®Ln(fi)) = (LJp) ®Ln(A))B(A-p).
(20)
Here the tensor product refers only to the auxiliary space Cm and B(A)
is a matrix in Cm®Cm called the quantum B-matrix. If B(A) =P(I —
— ihr (A)) + 0 (%2) as Ti->0, where % is the Planck constant, then, in the quasiclassical limit relation, (20) goes back into (19). The uniqueness of such
a deformation of the Poisson brackets as that given by (19) is an open
question.
As in the classical case, relation (20) leads to a generalization of standard
mathematical objects. In [26] it was used to introduce some nontrivial
deformations of the universal enveloping algebras of Lie algebras, the
so-called SMyanin quadratic algebras.
The quantum monodromy matrix TN(À) is introduced by formula (18),
where matrices Ln(X) are replaced by LU(A). A remarkable property of
the quantum U-matrix is that it gives a compact form of all commutation
relations of the matrix elements of TN(A). The following simple result
holds.
1338
Section 13: L. A. Takhtajan
THEOREM 4 (a
quantum version of Theorem 1). It follcnosfrom the relatim
.(20) that
B(X-p)(TN{X)®TNM)
= (TN(V)®TN(A))B(A-V).
(21)
In particular, one has
ltTTN(A),tvTN(t*)]=0,
(22)
where tr denotes the trace in the auxiliary space Cm.
Thus the operators trT^A) form a commutative family, which is the
family of quantum^ integrals of motion.
From the associative property of the tensor product one obtains
a sufficient condition for admissible JE-matrices
(I®i2(A-^))(i2(A)®I)^
(23)
This relation is called the Yang-Baxter equation (or the triangle equation
and also the factorization equation). It occurs in statistical mechanics [1]
as the commuting condition for transfer-matrices and in the scattering
theory [39] as the factorization condition for ^-matrices (see also [33],
[22]). Lately this equation has become rather popular; its solutions and
methods for constructing them can be found in [4], [21], [22]. This equation
also has connections with algebraic geometry [19].
We shall now consider the case of m = 2 more thoroughly. The simplest
solution of equation (23) has the form
w) ~ i^+aL
(24,
A + rç
and corresponds to the models HM and NS. Here rj is a parameter (the
coupling constant). Other solutions can be obtained by a quantum analogue
of the averaging procedure. Namely, we put 01(A) =B(A)P and ,
00
MW = fi ((r^IWA + ^Ktv»®!),
(25)
n=—oo
i(A)
= Y\ K®I)@{X + nco2)(crïn®I)
(26)
Integrable Models in Classical and Quantum Field Theory
1339
for Im(o)2/co1) > 0 . Then the matrices B(X) ^0t(X)P and B(X) = m(X)P
satisfy the Yang-Baxter equation and correspond to the S G model and
the Xr^-Heisenberg model, respectively (see below).
2. The local vacuum and the algebraic Bethe Ansatz. In addition to the
matrix-operator Ln(X) and the quantum JK-matrix, a local vacuum is
another important object of the quantum inverse scattering method.
This is a vector con e ì)n characterized by the property
a(X) *
coni
0
0(A) t
Ln(X)con
(27)
where a(X) and (5(A) are some functions of A. The reference state QN
N
= fj ® con has an analogous property with respect to the monodromy matrix.
Using the existence of the E-matrix and local vacuum it is possible to
give a general procedure for diagonalizing the operators trTN(X) ([28],
[33]), which is the algebraic background of the method. Namely, the
following statement holds.
THEOREM
5 (the algebraic Bethe Ansatz). Let
TNW =
YAM
BN{X)'
0N(X) BN(X).
(28)
and suppose that there exist a local vacuum and a reference state. Moreover9
suppose that in the basis in C2 ® C29 associated with the basis in C2 where
(27) holds9 the B-matrix has the form
B(X) =
-1 0
0
0
0 6(A) o(A) 0
0 c(X) &(A) 0
.0
0
0
(29)
1
where 6(A)/c(A) is an odd function of A. Then the vectors XFN(XX9..., Xj)
= BN(XX)... BN(Xl)QN (Bethe vectors) are the eigenvectors for the operators
tvTN(X) = AN(X) + JDN(X) with the eigenvalues
+
^ùié^ ^n
1 c{X}-X)
1 1 o(X — L)
(30)
1340
Section 13: L. A. Takhtajau
if the parameters XX9...,
Xx satisfy the system of equations
<5"(A,) - l l
c(A,-A,)'
3-
1
'-'
1
-
(31>
k&
This theorem can also be generalized to the case m > 2.
As we have pointed out before, txTN(X) form a family of commuting
quantum integrals of motion. I t contains also the Hamiltonian of the model.
The simplest expression for the Hamiltonian occurs in the case of fundamental models, where the quantum space is isomorphic to the auxiliary
space. I n this case there exists a point X = X0 for which the operator
trTN(X) is proportional to the cyclic shift operator i n § ^ (the quasimomentum operator). The Hamiltonians with interaction of Jc + 1 nearest neighbors
on the lattice are expressed in terms of the operators
jffÂ==
"^IOgtrTiv(A)lAs=;l«,
*=l,2,--.,tf-l.
I n addition, ô(XQ) = 0, and so the spectrum of Hk is additive. For
nonfundamental models the construction of local Hamiltonians requires
additional tools ([8], [20], [36]).
3 . Characteristic examples. 1. The isotropic HM of spins s, 2s e Z (XXXmodel; see [8], [9], [20], [21], [23], [32], [34]). Here ì)n = C 2 s + 1 and the
matrix Ln(X) looks like
(31)
1
where — 8", a =1,2,3
are generators of an irreducible representation
of sn(2) in \)n and 8% = J8£±*<8£. The ^-matrix has the form (24), where
tj — i. The Bethe vectors are the highest weights [34] with respect to
the action of su(2) in $)N and the system of multiplets associated with
them is complete [16], [32]. In proving this there arise nontrivial combinatorie identities [16]. The Hamiltonian has the form ([20], [21])
Hg**
= s %fs«Sn,
#»+1»,
(32)
Integrable Models in Classical and Quantum Field Theory
1341
where 8N+1 = Sx and f8 is a polynomial of degree 2s, characterized by
2s
the conditions f8(l(l+l)l2-s(s+l))
= £ (Ijh), I = 0 , 1 , . . . , 2s. In the
xx
quasiclassical limit H$ ) goes into the Hamiltonian of t h e lattice HM
model.
2. The lattice ITS model ([13], [15]). Here l)n = ^ ( f i 1 ) and the matrix
Ln(X) is obtained from (31) by a left multiplication by CTQ. The operators
8% are now t h e generators of tho irreducible infinite-dimensional representation of su(2) of spin s = —2\KA, where n is the coupling constant and
A is the lattice spacing. The operators 8% are expressed in a standard way
in terms of t h e usual creation-annihilation operators in J)n. The J2-matrix
has the form (24), where rj = — in and the Hamiltonian is given by (32),
where <$ w , 8n+1} should be replaced by (cr8n, 8n+1}, a being an involution
of su(2). The function fs naturally interpolates the polynomial from (32)
to the case of nonintegral 2s ([36]).
3. The lattice SG model ([14], [15], [28]). Here the field operators
are unitary operators un and vn satisfying Weyl's commutation relation
u v
n n = exP(*V)v7i^»j where y = ß2/8, ß being the coupling constant;
i)n - Se2(R£\2TzZ) if y ^ 2izp\q and \% = C* otherwise. The matrix LJX)
has the form
Ln(X)-
(33)
where
m2A2
/
,
^
mA
\1/2
t
x
9(vi A) = —7- ( 0 0
.
.
— e*v)
and m plays t h e role of mass. The JE-matrix has the form (29), where
....
imiy
sh(A + iy)
7
N
'
shA
Bh(X + iy)
In .contrast to the previous examples, a local vacuum exists only for
the product LnJhl(X)Ln(X) and not for individual matrix Ln(X). The Hamil-
1342
Section 13: L. A. Takhtajan
tonian has a more complicated form than (32), but in the continuous limit
goes into the regularized version of the Hamiltonian of the SG model
^'-
/(T*+T(£)' + £-< I —«h
m
where [cp(x)9 uz(y)] = id(x—y).
4. The anisotropic HM model of spins J (XYZ model; see [2], [3],
[31], [33]). This model is naturally related to the eight-vertex model
of the classical statistical mechanics on the two-dimensional lattice [1],
[33]. Its Hamiltonian is equal to
H(XYZ)
^ ^(JÄ^4+1+J,^^+1+Ja^4+1),
(35)
where the periodic boundary conditions are assumed. The matrices B (X)
and Ln(X) for these models have a more complicated form than (29) and
(31) and are expressed in terms of elliptic functions (see [1], [31], [33]).
The diagonalization procedure for trTN(X) requires more complicated
technical tools, but expressions for the eigenvalues of trTN(X) and the
system of» equations (30) in Theorem 5 retain their algebraic form ([3],
[31], [33]).
4. Thermodynamic limit. Here we shall briefly consider the thermodynamic
limit which is the limit as JV->oo for compact models and as L->oo, A->0
for noncompact ones. The behavior of models in this limit is most interesting from the physical point of view. The main problem here is to define
the ground state — the eigenvector of the Hamiltonian with the minimal
eigenvalue (the lowest energy vector) and to describe the Hilbert space
of states near it — the space of low-lying excitations. There are two possibilities.
1) The ferromagnetic case : the reference state is the ground state (this
occurs for the NS model and the HM model in the case e > 0). Then in
00
the Fock space $F cz g^ adjoining to the vector Q = [J ®coniox XeR1
in a weak sense there exist limits A(X) ='lim a"~N(X)AN(X)9 B(X)
N-+OQ
Integrable Models in Classical and Quantum Field Theory
1343
= lim BN(X). These operators satisfy the commutation relations
N-+OQ
[AW,A((i)l=0,
A
MBM
= „,
[JS(A),5(^)]=0,
\
,-m BMAWi
(36)
c(fi~ A — *0)
A(X)Q = Q.
With the help of these formulae the spectrum and eigenvectors of the
operators A (A) are easily obtained ([25], [27]). The commuting family
log.A (A) has an additive spectrum and contains the limiting Hamiltonian
of the model HF = lim (HN — FQ(N)IN). Here BQ(N) is the ground state
JV-+0O
energy and IN is the unit operator in §>N,
2) The antiferromagnetic case: to the ground state there corresponds
a special distribution of X19 ..., Xx in the Bethe vector (this occurs for the
SG model and the HM model in the case e < 0). As J^-^oo, parameters
Xx,..., Xt become uniformly distributed on the real axis with the density
g (A). This is "the filled Dirac Sea". (This situation occurs in the quantum
field theory and in solid state physios and is called "filling of the vacuum".)
The function Q(X) satisfies a linear integral equation which follows from
the system (30)
oo
27ce(A)+ f 0(i-rìe([i)dii=fw,
(37)
— 00
where
% aX
c( —A)
% aX
a(A)
Excitations above this ground state are characterized by the density
n
1
Q(X, XX, ..., AJ = Q(X) + — • ^?o(X — Xj), where cr(X — [i) is the resolvent
kernel of equation (37), and parameters Xx,..., Xn appear as the holes in
the Dirac Sea.
In these terms the ground state and the excitation creation operator,
respectively, have the form
^rcmna = K m exp J5T f
logBN(X)e(X)dX Q
(38)
1344
Section 13: L. A. Takhtajan
and
B(X) = lim exp
J'logBN(p)a(X-p)äft
(39)
N-+CO
The operator A(X) is defined as in the previous case and A(X) and B(X)
satisfy the commutation relations of the type (36) with c(X) replaced by
c(X) = exp
J logc([j)cs(X—ii)d[i
(40)
As before, the commuting family logA(A) has an additive spectrum
and contains the renormalized Hamiltonian of the model.
Using this method, it is also possible to describe the scattering of
excitations and to calculate the ^-matrices. The in- and out-states are
constructed with the help of operators Z(X) = B(X)A~1(X) (Z(X)
~B(X)Ä~l(X)) satisfying simple commutation relations which follows from
(36) (see [34]).
I n papers [15], [17], [25], [28], [32]-[34] this general scheme was
applied to the detailed investigation of concrete models. I n particular,
in [28] it was used to obtain an exact non-pertubative solution of the SG
model. With proper modifications this approach can be generalized to
the case of models with the auxiliary space Cm, m>2.
Finally, let us mention the latest achievements of the quantum inverse
scattering method: (a) The exact calculation of norms of Bethe vectors
[18], which gives us a hope to obtain explicit expressions for Green's
functions; (b) The quantum variant of the equations of inverse problem
[29], which provides an expression of Heisenberg field-operators W* in
terms of the operators A (A) and -B(A); (c) A new approach towards the
integrability of the quantum 0 (3) nonlinear cr-model [10].
References
[1] Baxter E. J., Partition function of the eight-vertex lattice model, Ann. Phys.
70, No. 1 (1972), pp. 193-228.
[2] Baxter E. J., One-dimensional anisotropic Heisenherg chain, Ann. Phys. 70,
Ko. 2 (1972), pp. 323-337.
[3] Baxter E. J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain, I-III, Ann. Phys. 76, No. 1 (1973), pp. 1-71.
[4] Belavin A. A., Discrete groups and integrability of quantum systems, Funk.
anal, i pril. 14, No. 4 (1980), pp. 18-26.
[5] Belavin A. A. and Drinfeld V. G-., On the solutions of classical Yang-Baxter
equation for simple Lie algebras, Funk. anal, i pril. 16, No. 3 (1982), pp. 1-29.
Integrable Models in Classical and Quantum Field Theory
1345
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LENINGRAD BRANCH OF THE V. A. STEKLOV
MATHEMATICAL INSTITUTE OF THE ACADEMY OF SOIENOES
USSR