the quantum method of the inverse problem and the heisenberg xyz model

Baxter made use of the ideas of Kramers— Wannier [13] and, in particular, of Onsager [14] on the transfer matrix, and of Lieb’s solution {15]—[18] of the special case of the eight-vertex

Uspekhi Mat Nauk 34:5 (1979), 13-63

THE QUANTUM METHOD OF THE INVERSE PROBLEM AND THE HEISENBERG XYZ MODEL

L A Takhtadzhan and L D Faddeev

Contents

§1.Classical statistical physics on a two-dimensional lattice and

quantum mechanics on a chain - - - 15

§2.Connection with the inverse problem method - - - 25

§3.The six-vertex model - << nh + 29

§4.Generating vectors and permutation relations - - - - - - 37

§5.The general Bethe Ansatz - 43

(3) o,=1Q 0 @ - OI (=1, 2, 3, 4; n=1, , ),

where the o/ are Pauli operators, which in the orthonormal basis of C?

have the following form:

6 d=e=( 1), -omate(? I), nota (5 2),

and o* = Jis the identity operator in C? In (3) it is assumed that the n-th

factor in the tensor product is o/ and the remaining factors are identity

operators, so that of, acts non-trivially only in 6, We also assume that

(6) ove1=0} g=1, 2, 3),

that is, periodic boundary conditions hold The energy operator thus

introduced is a matrix of order 2% x 2”

The quantum-mechanical model of magnetism thus described was proposed

by Werner Heisenberg in 1928 [1] and was actively investigated by many

scholars beginning with Hans Bethe In the present-day literature on mathe-

matical physics it is known-by the jargon name “the XYZ model”, which is

used in the situation of general position J, #J, #J, The particular cases

J, =J, #J, and J, =J, =J, =J are called the XXZ and the XXX models,

respectively

Heisenberg’s model turned out to be very fruitful in the theory of magnetism

and there is an extensive physical literature devoted to it In recent years it has

attracted the attention of mathematical physicists, since it turned out that the

problem of finding the eigenvalues and eigenvectors for H can be solved

exactly, in a certain sense, and requires beautiful mathematical constructions

The first step on the way to a solution of this problem was taken by Bethe in

1931 [2], when he examined the completely isotropic case of the XXX model

and found the eigenvalues and eigenvectors of its Hamiltonian Bethe’s exact

solution counts as one of the fundamental results in the theory of spin models,

and the method proposed by him, the famous “‘Ansatz’’ (substitution) of ˆ

Bethe, has been applied successfully to other many-particle models in one-

dimensional mathematical physics (the term is due to Lieb and Mattis, see [3])

There is physical interest in the study of asymptotic properties of the model

as N > 9, and particularly, of the ground state of excitation — the state with

the least energy and the states with almost the least energy The case J > 0 in

the XXX model corresponds to ferromagnetism In this case the ground state is

very simply constructed and the excitations can be found with the help of

Bethe’s ansatz When J < 0, we are concerned with antiferromagnetism The

ground state for this case was constructed in 1938 by Hulthén [4], who started

from Bethe’s formulae and calculated the asymptotic expression for the energy

of the ground state as N > ee, Twenty-four years later des Cloiseaux and Pearson in [5] constructed excitations upon the antiferromagnetic ground state and found the asymptotic expression for their energy The generalization of Bethe’s method to the XXZ model presents no difficulties of principle Yang

and Yang ({6] —[8] ) showed in 1968 that Bethe’s Ansatz in its classical form is

applicable to this model too

-The papers [6] —[8] contain a detailed investigation of the problems arising

in the justification of the limit passages as N-> © in the Bethe—Hulthén

method, and raised this group of problems to a higher mathematical level At the same time it became clear that the solution of the completely anisotropic XYZ model requires to all appearances new technical ideas, and the question

of its solubility remained open right up to 1972

In 1972 Rodney Baxter in his remarkable papers [9] —[10] (the results were announced by him in 1971 in [11]—[12]) gave a solution for the XYZ model

He discovered a link between the quantum X YZ model and a problem of two-

dimensional classical physics, the so-called eight-vertex model (its exact

definition will be given in the main text), — in fact, it had been his principal aim in his paper to investigate this Baxter made use of the ideas of Kramers—

Wannier [13] and, in particular, of Onsager [14] on the transfer matrix, and of

Lieb’s solution {15]—[18] of the special case of the eight-vertex model, the so-called six-vertex model connected with the quantum XXZ model In [9] — [10] he obtained a system of transcendental equations generalising the system derived from Bethe’s method, and with its help he calculated the energy of the

ground state of the X YZ model In the subsequent series of papers [19] —

[21] Baxter, by means of a very complicated and non-trivial generalization of

Bethe’s Ansatz, was able to construct the eigenvectors and to find the eigen-

values of the transfer matrix, and so to solve completely Heisenberg’s XYZ |

model In 1973 Johnson, Krinsky, and McCoy [22], using Baxter’s results,

calculated the energy of the excitations of the XYZ model

Although Baxter’s work is rightly considered one of the most important

achievements in statistical physics since the time of the famous paper of

Onsager [14], only a few specialists understand his method Many have made use of his results only, whereas the method itself remained totally without attention This can be explained partly by the unusual difficulty of his work, partly by the mass of cleverly-constructed devices based on deep technical intuition

We became acquainted with Baxter’s work in the following way In reading the paper of Luther [23], in which Baxter’s results and [22] are applied to the

investigation of the spectrum of the Sine—Gordon quantum model we noticed that a number of Baxter’s formulae resembled the formulae, already familiar to

us, from the inverse problem method This and other considerations promoted

us, together with E K Sklyanin, to create a quantum version of the inverse problem method (see the survey [24]), which we applied to a complete

solution of the quantum-field theoretical Sine—Gordon model [25] As will

become clear later, one of Baxter’s formulae played an important part in this,

and so it was only natural to turn to the XYZ model and to investigate it with

the help of the inverse problem method The results thus achieved form the

content of this survey It seems to us that the inverse problem method allows a

simplification and algebraization of Baxter’s formulae and to give a complete

proof of them

We came to the conclusion that the inverse problem method is a natural

unification of the basic achievements of one-dimensional classical physics, the

ideas of Kramers—Wannier, Onsager, and Baxter, with Bethe’s Ansatz and leads

to a natural understanding of completely-integrable quantum systems

The inverse problem method for the solution of classical non-linear

equations, which began in 1967 with the pioneering work of Kruskal and

others [26], was further developed in [27]—[{37] and now attracts the

attention of a large number of experts in mathematical physics (see the sur-

veys [38]—[41] and [42], with the detailed bibliography there) Put very

briefly, its basic propositions are as follows

Related to a non-linear evolution equation there is an auxiliary spectral pro-

blem and the Cauchy data occur in it as coefficients The spectral

characteristics of this problem (the elements of the mondromy matrix) are com-

plex functions of the Cauchy data However, and herein lies the success of the

method, the Hamiltonian of the system and the Poisson brackets can be

expressed explicitly in terms of the spectral characteristics The equations of

motion in terms of the spectral characteristics become linear Thus, the

transition from the Cauchy data to new variables — the spectral characteristics —

is a transformation to action — angle variables [29] In the quantum version of

the inverse problem method ([24] —[25], [43] —[44]) the coefficients of the

associated spectral problem are constructed by the correspondence principle and

can be expressed in terms of the Schrodinger canonical operators The elements

of the monodromy matrix are now operators in the Hilbert state space of the

quantum system, and the quantum Hamiltonian can be expressed in terms of

them In this state space there is a particular vector such that all eigenvectors

of the Hamiltonian are generated from it by means of the elements of the

monodromy matrix The existence of the generating vector and the simple

commutativity relations for the elements of the monodromy matrix show that

the latter are quantum analogues of variables of action-angle type These general

propositions will be illustrated in detail in the text by the examples of the XXZ

and the general XYZ model

Now a few words about the contents of the paper In the first section we give

the classical results on the transfer matrix of two-dimensional lattice models,

leading up to Kramers—Wannier—Onsager Here also we obtain Baxter’s para-

meterization for Boltzmann weights and we comment on the formulae that arise

there In §2 we show how the constructions of the preceding section fit

naturally into the quantum method of the inverse problem, and we deduce a

formula connecting the Hamiltonian of the XYZ model with the transfer

matrix of the eight-vertex model The third section is devoted to the solution

of the special case of the six-vertex model and the XYZ quantum model con- nected with it On this comparatively simple model we explain the appearance

of the generating vector and the algebrization of Bethe’s Ansatz In §4 we construct a family of states generalizing the generating vector in §3 to the case

of the eight-vertex model Here too we obtain a series of permutation relations

for the operator elements of the monodromy matrix In §5, using this family

of generating vectors and the permutation relations, we construct an algebraic

generalization of Bethe’s Ansatz for finding the eigenvalues and eigenvectors

of the transfer matrix for the eight-vertex model In §6 we calculate the energy

of the ground state of the XYZ model This section may also serve as an intro- duction to the method of integral equations in the theory of spin systems, which was first proposed by Hulthen in [4] In the conclusion we sum up briefly and formulate some interesting mathematical problems arising in con- nection with our account In Appendix 1 we give a summary of definitions and formulae from the theory of Jacobi elliptic functions and theta-functions Appendix 2 is devoted to a geometrical interpretation of the Baxter—Yang relations from §1, which was recently given by Cheredrick [45]

In our exposition we follow the tradition of contemporary mathematical

physics and draw attention mainly to the algebraic aspects of the problem at hand; therefore, our arguments often take on a formal character, especially in questions of convergence of series, of asymptotic estimates, and the like, A thorough airing of these questions would take us too far from the basic problem — to convey to a reader who is a mathematician the beautiful structures that arise in theoretical physics

We express our thanks to P P Kulish and E K Sklyanin for useful comments and remarks and to A G Reiman for a discussion of the results of

[45] We should also like to express our respect and our gratitude to Rodney J Baxter, whose papers we have read with so much pleasure

The authors dedicate this paper to Academician N N Bogolyubov on the

occasion of his seventieth birthday

§1 Classical statistical physics on a two-dimensional lattice and quantum

mechanics on a chain

We take a square lattice of order M X N on a two-dimensional torus, that is,

a plane lattice with M + 1 rows and N + 1 columns in which the extreme rows

and columns are identified We give a definite direction, an arrow, to each edge

of the lattice, that is, to each vertical or horizontal section joining adjacent

nodes

Four edges come together at each node of the lattice, and so there are 16

distinct types of combinations of arrows at a node We ascribe to each possible

combination a positive number e, (j = 1, ., 16) that does not depend on

the number of the node (of the energy of the combination) With each dis-

position of arrows on the lattice-configuration we associate a total energy,

which is defined as the sum of the energies of the nodes,

16 j=

where Ni is the number of nodes with a combination of arrows of type j in the

given configuration As a result we obtain a model of interacting arrows

situated along the edges of the lattice

As an alternative definition, instead of arrows one may use a spin variable a

with the two values + 1, where + 1 corresponds to the arrows going to the right

or upwards, and — 1 to the arrows going left or downwards In this way there

correspond four quantities a, a’, y, and +’ taking the values + 1 to each com-

bination of arrows at a node (Fig.1) The variables

Fig 1

aand a’ correspond to the vertical, y and ¥’ to the horizontal edges

- The model described is called the general 16-vertex model and plays an

important part in classical statistical physics Interesting combinatorial

problems are also connected with it (see [46])

The statistical sum is defined as the quantity

(4.2) Z= > exp {—fZ},

where the summation is over all configurations of arrows on the lattice, and E

is the total energy of a configuration defined by (1.1) The statistical sum, as a

function of the parameters B, &, ., 8 determines all the thermodynamic

properties of the model The parameter ổ is inversely proportional to the -

temperature, and the vy = exp{—fe;} (j = 1, ., 16) are called the

Boltzmann weights

The problem of finding an exact expression f for the statistical sum of a model

on a finite lattice is very complicated Fortunately, of greater physical interest

is the simpler problem of finding an asymptotic value of Z for large M and N,

more accurately, of calculating the Sorcalied thermodynamic limit

» N00

The quantity f is called the specific free energy

For a further investigation of the statistical sum Z it is useful to rewrite (1.2)

in a more appropriate form We denote by R& (+7, 7’) the Boltzmann weight for

the combination of arrows in Fig 1, and by {a7} ={a,, ., @y}an arbitrary configuration of vertical arrows between the neighbouring rows of the lattice numbered j andj + 1 We note that {a™+!} = {a} by virtue of the periodicity condition

We now transform (1.2) with the outer summation over the vertical, and the inner over the horizontal configurations As a result we get the expression

(1.4) Z= > > LỆ (a) eee TM), ta Sp T*,

tay aM;

where T is an operator in the space Gy = C?Ÿ, In the orthonormal basis

(1.5) ftay = Ca, @ «++ Dla ys

its matrix elements are numbered by multi-indices {a}, {a’} and have the form (1.6) Tai, {a} = ¬"» Uh Ran (Yn› Yn+d)›

2 1 VN 8=

where it is also assumed that yy, = ¥1; Sp in (1.4) denotes the trace of Gy The operator T is called the transfer matrix of our model It is an operator

on 9: x: the state space of the quantum system of N spins In the expression

Re (y, 7’) the indices a, a’ and +, 7’ play different parts The first are naturally

ố

Fig 2

“quantum” indices, and the second “auxiliary” Regarding the latter as the

matrix indices of the 2 X 2 matrix R# , which number its rows and columns,

we can rewrite (1.6) in the form

(4.7) Pray, (a'y = tt (Rab « Hay) =tr(Rếi Tay),

where tr signifies the trace in the auxiliary space C?

The relations (1.4)—(1.7) also establish a link between problems of quantum mechanics on one-dimensional chains and problems of classical statistical physics on two-dimensional lattices For with their help we have reduced the problem of calculating the statistical sum (1.2) to a problem in quantum mechanics, the calculation of the eigenvalues of the transfer matrix T, an operator in 9 For the simpler problem of calculating the free energy it is sufficient to know the asymptotic value as N > © of the eigenvalue A,,,,(),

of greatest modulus, of 7 It is obvious that if the limit (1.3) exists and does

not depend on the order of the limit operations over M and N, then

(1.8) —Bf = Jim Flog] Amax ()

Baxter’s main result [9], which we reproduce in §6, consists in the calculation

of f for the important special case of the system in question, the so-called eight-

vertex model, which we shall now describe

We consider only those configurations of arrows for which the number of

arrows at each node of the lattice is even The eight admissible combinations

are drawn in Fig.2 The model of interacting arrows that arises is also called the

eight-vertex model In the expression (1.2) for the statistical sum the

summation is now only over such configurations of arrows

In other words, for the eight-vertex model the Boltzmann weights u,,

7=9, , 16 are zero, that is, ej = 00 (j = 9, ., 16) (we assume that

B>0)

We assume also that the interaction is invariant under simultaneous inversion

of the directions of all the arrows on the lattice This means that

&; = Đạy &y = &4, &s = Đạy Bị — 6

Vy = Ug, Vy = Ug, Us = Ug, Vz = Ve

For just such a model Baxter calculated the free energy and constructed a

generalization of Bethe’s ansatz for finding the eigenvalues and eigenvectors of

its transfer matrix In the present survey we consider only this model

We introduce coefficients

P 9) Uy = (Đ; +-ty), w= (D;— ty),

0y -} (Đ—Ạ), - =-(0 +),

in terms of which the Boltzmann weights Re (y, 7’) corresponding to the com-

bination in Fig 1 can be written in the following form:

4 (1.10) Re (yy y= > U/0yy'Oaœ'‹

Here we use the Pauli operators o/ G=1, 2, 3, 4), which were defined in the

oom where o/ ,, and of, denote their matrix elements in the basis

)

To write out (1.7) in invariant form it is convenient to introduce operator

matrices £, of order 2 X 2, defined by

é

(1.11 ) #,= Mu gi = (+308 ~=—-w 0}, — iw, 03

P 2 fo" @ Om W407, + iw,o3 w0h—w,o8 )

The matrices £, are of order 2 X 2 on the ring 8, of operators in Gy The

product of the £, and the trace tr are understood as operations in the algebra

of matrices over Sy

Comparing (1.7), (1.10), and (1.11) we see that the transfer matrix T can

be written in the form

(4.12) Ty=tr (Ly eee ⁄.)

Side-by-side with transfer matrix Ty it is natural to look at the monodromy

three-dimensional projective space over the field of complex numbers

The experience of exactly-solvable two-dimensional lattice models, such as

Ising’s model solved by Onsager, and the six-vertex model solved by Lieb sug-

gests that it is very fruitful to find conditions under which the transfer matrices with different values of the coefficients w; commute We investigate under what conditions on the w, this is possible

Let us look at two sets of coefficients w; and w; ,and let Z and 7’ be their

monodromy matrices The corresponding transfer matrices T and T’ commute

if there is a non-singular 4 X 4 numerical matrix # such that

The Kronecker product in (1.15) is to be understood as a tensor product in

the matrix algebra over 8 For two 2 X 2 matrices A and # overan arbitrary ring, their tensor product 4 © # is the 4 X 4 matrix with the blocks

20 L A Takhtadzhan and L D Faddeev

Now (1.15) is true if the analogous relation is satisfied for the matrices £3

It holds true also for matrices £, and La, since it follows from (1.11) that the

matrix elements of £, commute with each other for distinct values of the

index n

We now find conditions under which the local formula (1.17) holds good

Without loss of generality we look for a numerical 4 X 4 matrix # of the form

where &j,; is a completely antisymmetric tensor and e,, = 1 We find that

for (1.17) to be true it is necessary and sufficient that for all permutations

¢, k, 1, n) of (1, 2, 3, 4) the following relation holds:

In (1.21) there are six independent equations If we regard them as a system of

linear homogeneous equations for the four unknowns w{, w3, w3, and w4, we _-

find that the system is soluble if for all admissible values of j, k, ƒ and n we have

Thus, we have shown that (1.17) is valid if the coefficients w; and Uy have

the form (1.23) Now the coefficients of the #-matrix are defined by (1.25),

The quantum method of the inverse problem and the Heisenberg XYZ model

where for fixed u,, U2, U3, Ug, the parameter u" is a function of u and u’ The choice of normalizing factors p, p’, p" is immaterial by virtue of the

homogeneity of the equations (1.21) To determine u” as a function of u and u’

we substitute (1.23) and (1.25) in (1.21) and differentiate with respect u and u' As a result we find that

1 au’ 1 au"

where

4 (1.27) g (u) = const H (u—uy}"1

(see Appendix I) With a suitable choice of the constants in (1.27) we obtain

Ug (Ug —Uy) — Us (Ug— 4) 8? (v, 2)

Ls — Uy — (u.—4) Sn? (v, 1) ,

where sn (uv, /) is the Jacobi elliptic sine function with modulus /,

P= (uy — U4) (Ua— Us)

ca eam ean 80%, 0), đn(0, Y 4, sar b

(1.32) Wy? We + ys y= en (€, Ù “dn (6, by *”*” sn(@, Ð ° The convenience of (1.32) consists in the fact that for two matrices Z, and

£;, constructed from w, and w: with identical values for the parameters [ and land differing values of v and uv there is a matrix & satisfying (1.17) whose

elements depend only on v—v’ And so two transfer matrices with differing

values of the parameter v commute for fixed [ and J

The parametrization (1.32) was found by Baxter in his famous paper [9], from which the arguments above are taken

We draw the reader’s attention to the fact that (1 32), in fact, coincides with the expressions for the moments of a completely asymmetrical gyroscope (see, for example, [47])

22 + A Takhtadzhan and L D Faddeey

We emphasize that the relations (1.32) do not constitute a restriction on the

coefficients w, For given w, the parameters /, Ệ, and v are defined by the

To find the coefficients ø,` we substitute th i

*, in “dx D sing the addition theorems for the Jacobi elliptic functions (see

pendix I), we obtain the following expression for the soluti

(1.21), the coefficients w;’: © Solution of the system

(1.34) www + tú” = en (ø0—ơˆ-Lÿ, Ù) „ dn(ø—ơ'-{, Ð) 4,8n(0—ơ-Lễ, Ì

= sn (A + , ): sn (À — n, &) : sn (2n, È) : & sn(2n, k)sn(A — n, k)sn(À-+n, È)

For the coefficients w;" we obtain, respectively,

(1.87) (wy-+ w5) + (wy —w}) : (wf + wh) : (wf — wi) =

=sn(h—p-+2n, k):sn(4—p, k):sn(2n, A): ksn(2n, k)sn(A—p, k) x

x sn (4—p+2n, k), where p = iv'/(1 +k)

Henceforth, when it is not necessa ; ry to indicate the mod indi i

simply sn (A) instead of sn (A, k) caus ke we waite

Thus, the w, in (1.11) are defined b , t j efined by (1.36) for any value of the common

normalizing factor For fixed k and 7 we denote by £, (A) the corresponding

matrix £, Then (1.17) takes the form

(1.38) 220, b) (Za (À) @ Za (0) =(Ln (p) @ Ln (d)) Kd, p),

where the #-matrix is

The quantum method of the inverse problem and the Heisenberg XYZ

za 0 0 đ

0

d00a and the coefficients a, b, c, and d are given by

4.40 {er p) = sa (h—p+2n), b(A, p) =sn 2m,

Since the relation (1.38) is homogeneous, it holds for any choice of common

normalizing factors in (1.36) and (1.37) It follows from (1.39)—(1.40) that the

#-matrix is singular only for a discrete set of values of X—y, therefore, the

validity of the equation

(4.44) - (TQ), T (wl = 0

for all values of \ and p follows from the principle of analytic continuation

Having at our disposal the formulae (1.1 1), (1.36), (1.39), and (1.40), we

can now Verify (1.28) directly with the aid of the addition theorems for the

Jacobi theta-functions (see Appendix I) It is clear, however, that the task of

finding a suitable parameterization (1.36) for the coefficients w; has the same non-trivial place in the reasoning advanced above Therefore, instead of deriving and proving (1.36)—(1 40) at once we have first reproduced Baxter’s arguments leading to the parameterization (1.36)

Our relation (4.42) (dy p)(Z7 (0) @ Z (w) =(Z (w) @ Z7 (A) R (Ay B) plays an essential part in what follows We mention that Baxter himself in [9] used it only to prove (1.41) There he also gave another proof of this formula, not depending on (1.42) Baxter’s formula (1.42) remained in the shade and

_ did not attract the attention of other researchers

We have put (1.42) at the basis of our discussion It is for us an important

constituent part of the quantum version of the inverse problem method We

made an announcement of this in joint papers with Sklyanin [25], [48] and worked on field-theoretical examples of the non-linear Schrodinger equation [43] and the Sine—Gordon equation [25] Here we show that this method works in the case of the XYZ model

We now look at the degenerate case when the modulus / of the elliptic functions tends to zero In this case the Jacobi elliptic functions become ordinary trigonometric functions (see Appendix U and the parameterization (1.36) has the form

(1.43) (tạ -E tạ) : (0ạ — 03) : (0y † 02) : (Uy — 2) =

= sin (À + 1) : sin (À — T)) : sin 2n : Ô

From (1.9) we find with the parameterization (1.43) that vu; =ug = 0, that is,

the cases 7, and 8, in Fig 2 are forbidden This model, which for obvious reasons

24 L A Takhtadzhan and L D Faddeev

is called the six-vertex lattice model, was solved by ' ; y Lieb in 1967 by means of Lieb i

the classical Ansatz of Bethe In §3 we explai plain the invers i i

the simpler example of this model © problem method in

To simplify the formulae in § §4 and 5 it is helpful to choose a concrete

EON) So the projective parameterization (1.36) Taking

" n) O(A — n) O(A + 7) as the normalizing factor in (1.3 i

formula (see Appendix I) ° in @.90) and using the

where H(z) and @(u) are the Jacobi theta-functions see A i

obtain the following expressions: (See Appendix), we

e(À, ») = 8 (2n) H(A—p) 8 (A—p-+ 2m), đ(À, 4) = H (2n) H(A —p) H(A—p-+ 2m)

Using (1.45) and (1.46) it is easy to verif y (1.38) by means of th iti

theorem for the Jacobi theta-functions ý oo nee men

To conclude this section we point out that (1.38) ha: s already occurred, in i

a somewhat different form, in the literature of one-dimensional mathematical

physics For this way of writing (1.38) we introduce the following notation

ince the spin operators o/, act non-trivially only in the quantum space of spins

at the n-th node, we may, allowing a certain liberty, forget the number # and

identify dị with the Pauli operators o/ Then the matrix

(1.48) Lia, iB (A) = Ui0i;0ag,

where the summation is over the repeated indice s The i

(1.49) S$ (A) = Lia, 39 (bE ne

Then the matrix elements of the 2-matrix take the form

(4.50) Bis, an (h— Bb) = Sik (hp)

The quantum method of the inverse problem and the Heisenberg XYZ model

In the notation (1.48)—( 50) we can write (1.38) in the following form:

(1.51) S15 (A—p) SR (A) SHE (B) = SH (v)SIE A) Som (À —R):

The relation (1.51) was first proposed by T N Yang in 1967 [49] ina dis-

cussion of many-particle factorizing S-matrices, therefore, it is natural to call (1.38) the Baxter—Yang relation Later it was used in the papers by Karovskii

and others [50], A B and Al B Zamolodchikov [51] —[52] in which

conditions were investigated for the factorizibility of S-matrices in various models of the quantum field theory in two-dimensional space-time

Our attention to the cited form of (1.38) was drawn by A B Zamolodchikov after we had met him

He showed in [51] that (1.51) can be interpreted as a condition for the associativity of the algebra with the formal generators A,(A) satisfying the

relations

(4.52) Aa (4) Ai (1) = 53 (AB) An (2) As (A)

The generators A;(A) in [51] played the role of operators generating the ‘in’

and ‘out’ states of a many-particle system

In a recent paper [45] I V Cherednik proposed a realization of relations of

the type (1.52) in terms of fibrations over Abelian varieties over the field of complex numbers In Appendix IT we give an account of Cherednik’s results

for our case, which corresponds to Abelian varieties of dimension 1, that is, elliptic

curves

We turn now to the second section, in which, we hope, the apparent artificiality of our constructions will become more comprehensible, or at least

more conventional

§2, Connection with the inverse problem method

The attentive reader will probably have guessed that the term “monodromy matrix” for 7(A) was not chosen accidentally

The equations (1.11)—(1.13) show that the linear problem

is connected with our model in a natural way Here £,(A) is given by the

expression (1.11) and (1.36), and W,, isa two-dimensional column with

We consider the matrix solution ¥,, (A) of (2.1) with the boundary condition

Iy 0

(2.2) W0) =(q" ry)›

where Iy is the identity operator in @y As is known, the monodromy matrix

TJ x(a) of the problem (2.1) on a chain of length N is defined as the value of the solution ¥,,(A) when n = N + 1, that is,

L A Takhtadzhan and L D Faddeev

Systems of type (2.1) constitute the basis of the inverse problem method for

the solution of non-linear classical equations on a chain The coefficients of the

local transition matrix ¥,(A) in this case are, generally speaking, complex-

valued functions of the characteristic canonical variables and the spectral para-

meter A The system (2.1) is called the auxiliary spectral problem As an

example we look at a finite periodic Toda chain [53] whose equations of

motion have the form

(2.4) zy, e%H-en_exm—srt (n=, ., WV; Iyer=2))

The local transition matrix £, (}) in the case of a Toda chain has the follow-

ing form

(2.5) L, (0) = (72 Tạ) (a=1, - , M),

where the p„ are impulses, canonically conjugate in the coordinates X„

The monodromy matrix Z x„(À)is of order 2 X 2 and its elements are

functions on the phase space of of our system The Poisson brackets on of

have the form

It turns out that the Poisson brackets for the elements of the monodromy

matrix can be calculated explicitly In particular,

(2.7) {tr 7 „ À), tr.7„(p)}=0

Thus, with the help of £,(4)we have constructed on of a family of commuting

flows The remarkable fact is that among them there is also the flow generated

Y the system (2.4) For the Hamiltonian ở of the Toda chain then has the

orm

(2.8) h= 3 (+ Pat exn—an)

nei

and can be expressed via the trace of the monodromy matrix by the method

characteristic of the inverse problem:

(2.9) h= > (Wop par Tvl) -

LG nàng

—N—3T yc tr wy (Mla=o-

The trace of the monodromy matrix for a Toda chain of length N asa

The quantum method of the inverse problem and the Heisenberg XYZ model 27 function of A is a polynomial of degree N with the highest coefficient 1 The

relation (2.7) shows that the coefficients of the polynomial tr 7 y(A) are

involution integrals of the motion The trace of the monodromy matrix is a generating function for commuting integrals of motion of the finite periodic Toda chain One can prove that these coefficients are functionally independent

as functions on the phase space o# Their number is equal to half the

dimension of e# Hence, by Liouville’s standard theorem (see, for example,

[54]), there follows that the Hamiltonian system of the Toda chain is com- pletely integrable, just as for any other system in which one of these integrals of motion can be taken as the Hamiltonian Thus, (2.7) indicates that the finite periodic Toda chain is completely integrable

The inverse problem method for a Toda chain was applied by Manakov in

1973 [55] and independently by Flaschka in 1974 [56] A matrix £,(4) of

type (2.5) can be abstracted from [55], in which for the proof of complete integrability a spectrum problem in a somewhat different form is used

In concluding this short digression into the classical method of the inverse problem we mention that variables of action-angle type for a finite periodic Toda chain, whose existence is guaranteed by Liouville’s theorem, can be found

by the methods of algebraic geometry (see the surveys [40] —[41])

Turning to our original theme we see that the study of the eight-vertex model leads quite naturally to the auxiliary spectral problem (2.1) However, in

contrast to the example just studied, this is a quantum problem, since the ele-

ments of £,(4) are linear operators in § y The transfer matrix T(A) has the

property

and is a generating function for a family of commuting operators Each of these can be regarded as a Hamiltonian for a quantum system with many integrals of motion The remarkable thing is that the Hamiltonian for the XYZ model is contained within this family

For, as Baxter has shown in [10],

A connection between Heisenberg’s XYZ and the eight-vertex model was noted

by Sutherland in 1970 [57] He showed that under certain relations between the Boltzmann weights u, and the parameters J, Jy, J, the Hamiltonian of the

XYZ model commutes with the transfer-matrix of the eight-vertex model

We emphasize here that the formulae (2.12) does not constitute a restriction

28 L A Takhtadzhan and L D, Faddeev

on J,, Jy, J,, but give a parameterization of them up to an insignificant

common factor

Let us Prove (2.11)—(2.13) In the process of proof the special role of the

value =H will become clear In (2.1 1)—(2.13) it is naturally assumed that for

the weights w, the parameterization chosen is (1.36) with the common normal-

(2.15) T\œy,a' (n) =snX 2n ômejÖays§ - «+ Ôage¿-

Thus, sn 2nT(n) is an operator of cyclic displacement in §y We now

differentiate (1.7) with respect to A and put A= With due regard for (2.14)

we obtain the following expression:

Pa = + (—U¡— Wa Wat), Pez (Wy + W2+ 0y -}- Uy),

The quantum method of the inverse problem and the Heisenberg XYZ model

we can rewrite (2.17) as follows:

where the dash signifies the derivative with respect to A

The coefficients Pj are easily calculated with the help of the formulae for

differentiation of Jacobi’s elliptic functions (see Appendix I) As a result we get

where 7 is an arbitrary complex number

We end with some preliminary conclusions Following Baxter, for the

operator H — the Hamiltonian of the Heisenberg model — we have found the auxiliary spectral problem (2.1) By its very statement it is a quantum problem

We have introduced the monodromy matrix 7 (A) and by (2.11) have

established a connection between H and the trace of the monodromy matrix,

the transfer-matrix 7 (i) The choice of the parameterization (1.36) for the coefficients w,; allowed us to write down the permutation relations between all the elements of £,() and £,(p) in the compact form (1.38) As a consequence

of the commutativity of the elements of £,(A) and £,,(y) for m #m an analogous relation (1.42) is valid for monodromy matrices

In the following section we shall show that the quantum method for the inverse problem works in the comparatively simple example of the XXZ model

We shall use throughout a significant simplification arising from the fact that

w, =wz, and that the coefficient d in the #-matrix is zero The general case

w, #w, will be examined in §§4—5

§3 The six-vertex model

In the preceding section we have established, among other things, a con- nection between the Hamiltonian of the Heisenberg XXZ model and the

transfer-matrix of the six-vertex lattice model Here we take up the problem of

finding the eigenvalues and eigenvectors of the transfer-matrix of the six-vertex model We solve it with the help of the quantum method of the inverse

problem At the end of the section we apply the results obtained to the

analogous problem for the XXZ model

We recall that for the six-vertex model 0; = vg = 0, that is, w; =w, In

(1.36) this case corresponds to k = 0 Then the common parameterization of

the coefficients w; (1.36) becomes (1.43) We take the specific realization of

(1.43), putting the common normalizing factor in (1.43) equal to 1:

(3-1) A +w,=sin(A+n), w—w3;=sin(A—n), w= w= sin 2n

The local transition matrix £,(A) has the form

_ ( Đạøậ-+Ouạg$ tì (0°—¡03)

and satisfies (1.38), in which the #-matrix is given by the expression

becomes the unit matrix

The monodromy matrix 7 y(A) is defined by (1.13), that is,

In this notation the transfer-matrix Ty (A) is

(3.7) Ly (A) Str F y (A) = Ay (A) + Dy (A)

From this point on we again omit the index N in the relevant objects

The monodromy matrix 7 (A) satisfies (1.42), that is,

(3-8) 2A, u)(Z (Ã) @ Z7 (w)=(Z (w) @ Z (A)) #(A, p)

We have already noted above that this equality contains permutation

relations between all the operator matrix elements A(A), B(A), C(A), and D(A),

of 7 (4).We write down explicitly the ones that are basic

The quantum method of the inverse problem and the Heisenberg XYZ model 31

[A (4), 4(w)]=0, IÐ(4), Ø(w)]=0,

[C (A), C(w)J=0, (D(A), D(u)] =0, B(A)A (pu) =5(A, pw) B(u) A (A) +e (A, w) A (yu) B(A), B(u) D(A) = b(A, pw) B(A) D(w) +e (A, w) D(A) B(u), e(A, p)[C (A), B(u)]= (A, pw) (A (u) D(A)—A (A) D(y)), { ¢(, w)[D(A), A(u)]=5(A, 2) (B(u) C (A) — B (M € (w))

We now proceed to the construction of a special condition, a generating vector in §y We rewrite £,,(A) in the form

_ { #n(À) Br

(3.40) Ln (A) = ( ye Ôn) )-

(Note that from (3.1)—(3.2) it follows that in our case the operators ổ„ and Vn

do not depend on A) The operators a, , 8, , 7, and 6, act non-trivially only in

the local space §,, therefore, without changing the notation we can restrict

them to ð„ Let e) be the vector in bn, (see (4))

(3.13) a(A) = sin(A + y), 6(A) = sin(A — n)

We call e, a local vacuum It annihilates the lower left element of £,(4)and for all X is an eigenvector for its diagonal elements

From the local formulae (3.12) there follow analogous formulae for the elements of 7(A) with respect to Q

We show that the eigenvectors of T(A) can be put in the form (3.16) Dry, eoey An) = H BẠ)9,

I1

where the À; satisfy the system of transcendental equations

n tt

(3.18) A(A; Agy coe, An) ==a% (A) |] ay + 8 (A) U aa:

The reader familiar with problems that can be solved by Bethe’s Ansatz (see

the surveys [3], [46] and the original papers [2], [6] —[8] ) will recognize

here the basic formulae of this method New are the compact formula (3.16)

for the eigenvectors and the simple algebraic derivation of (3.16)—(3.18),

which depends only on the permutation relations (3.9) and on the existence of

a generating vector & It is natural to call the B(A;) generating operators of

elementary excitations of the generating state 2 This algebraic version of

Bethe’s Ansatz was proposed by us together with Sklyanin in [25], where an

exact solution of the Sine—Gordon quantum model was obtained with its help

We mention that in (3.16)—(3.18) the permissible values of the parameter

nare 0,1, ,N, because for 1 > N the expression (3.16) vanishes identically

For by using the commutativity of the operators a,, ổ;, +y,, and 6; for various j

and the fact that B? = 0 we can easily verify that the product (3.16) vanishes

forn >N We also point out that since 7 (A) is periodic in \ with period 27,

in (3.17) it must be understood that | Re A; | <7, that is, the solutions A, are

considered only in the fundamental domain

We come now to the proof of (3.16)—(3.18) We rewrite the necessary

permutation relations in (3.9) in the more convenient form

From the explicit form of the coefficients b and e in (3.4) it follows that

b(u, À) _ (Ay BY) (3.24) e(u, a) (Ay BD

With the help of (3.19—(3.20) we can transform the expression

(3.22) (A (A) + D(A) [[ 80a) 9,

i>{

taking A(A) and D(A) through B(A;) to © and using (3.15) and (3.21) Then

there arise 2” terms, which naturally come together in n + 1 expression of the

where A and A, are numerical coefficients The structure of the operator

factors in (3.23) is obtained if on commutation we take account of only the

first terms on the right-hand sides of (3.19) and (3.20) As a result we conclude immediately that that coefficient A(A; 1, -» An) is given by (3.18) The structure of the operator factors in (3.24) for j = 1 is obtained if upon commutation of A(A) and D(A) with B(A, ) we use the second terms in (3.19) and (3.20) and upon further commutation of the resulting A(X, ) and D(Aj, ) with BQ¿), !> 2, we again take into account only the first terms in (3.19) and (3.20) The coefficient A, (A; A1,.-., A,) then takes the form (using (3.21))

2 (aC) Il away "TỦ a) I a, ta tty) *

For calculating the coefficients A;, 7 = 2, , n), we must combine a larger number of terms However, it is not necessary to make these calculations, since

by virtue of the fact that the B(A;) commute the expression (3.22) is a symmetric function of A,, , A,, consequently, the coefficients A; are

obtained from A, by a suitable permutation of the A,; Thus, the coefficients

Aj have the form

(3.25) Aj (M5 May sees An) =

term vanishes separately, that is,

(3.26) AAs Igy ey An) = 0 0=1, ,n)

Now (3.16)—(3.18) are proved

We note that a direct proof of (3.25) must depend on the addition formula for the coefficients b and c We have already used these addition theorems in

proving (1.38) It should also be mentioned that in the derivation of (3.25) we

have used the condition that all the A, are distinct We shall not consider in detail here the case of repeated A, in (3.16), which requires a special investi-

gation, since it would lead us too far from the basic theme of our survey Comparison of (3.16), the operator representation for the eigenvectors of

T(A), with the coordinate representation in [6] —-[8] is very useful If we write

34 L A Takhtadzhan and L D Faddeev

out (3.16) in coordinate form, that is, in terms of the local operators

Gœ„, Ổ„, +„› and ồ„, then it differs only by a constant factor depending on À¿

from the coordinate representations in [6] —[8] and, in contrast to the latter,

it does not vanish when A, = A; Hence, (3.16) can be used for repeated A) It

should also be pointed out that, as in the case of the classical Bethe Ansatz, not

all solutions A, of the system of transcendental equations (3.17) for

0<n<N correspond to eigenvectors of the form (3.16) with eigenvalues

(3.18), because for certain solutions A, the corresponding eigenvector

®(A,, -, À„) may vanish It can be shown that the eigenvector (3.16) con-

structed from a solution of (3.17) for n = N vanishes for almost all 7

However, in the situation of the general position with respect to the A, a vector

of the form (3.16) does not vanish for m = N, but is proportional to

õ=e@ @s., =(1}›

that is, it is an eigenvector of the operator T(A) and the same eigenvalue as the

generating vector 82 To prove this latter assertion it is sufficient to note that

the vector e, annihilates 8, and is an eigenvector of a, (A) and 6, (A) for all A,

and then to repeat the derivation of (3.15) Thus, the investigation of this

simple example shows that the vanishing of ®(A,, , A,,) for certain solutions

À, of (3.17) is connected with the degeneracy of the spectrum of T(A)

We move now to the investigation of the Heisenberg XXZ model The

Hamiltonian H of this model takes the form

N

(B27) HZ Y Cheha toner + Boron) n=1

where it is assumed that the periodic boundary conditions (6) hold, and is con-

nected with the transfer-matrix T(A) of the six-vertex model by (2.1 1)—(2.13),

that is,

(3.28) H = —sin 2nG InT (A) ant Jy»

A=cos 2n

‘Now (3.28) shows that the eigenvectors D(A, - +» Ay) OF the commuting

family of operators T(A) are eigenvectors also for the energy operator H The

eigenvalues of H can be found, from the eigenvalues (3.18) with the help of

(3.28) We give the corresponding expressions

For consistency of the resulting formulae with the analogous formulae in

[2] and [6] —[8], we introduce, in place of X, a variable k (not to be confused

with the modulus of the elliptic functions in § §1 and 2)

a — Sin (À+ n)

and write

(3.30) W (ky, «oe, A= I B(u) 9

The quantum method of the inverse problem and the Heisenberg

Then, after elementary transformations of (3.17)—(3.18), we obtain the follow- ing results

The solution Y of the eigenvalue problem of the Hamiltonian 1

and the eigenvalues E(k,, , &„) have the form

(3.34) EB (kus vee) Rn) = A+ 2 D (A008 In)

¡=1

The formulae (3.32)—(3.34) were obtained by Yang and Yang in [6] —[8]

with the help of the classical Bethe Ansatz We have shown how the quantum method of the inverse problem leads naturally to the classical results (3.32)— (3.34) for the XXZ model

In the study of the XXZ model it turns out to be helpful to use the variable

X and the system (3.17), as well as the variable k just introduced and the

system (3.32)—(3.33) In (3.17) the function c¢(A, 2) depends only on A~ ứ and, as will be explained in §6, is transformed as N > © into an integral equation with a kernel depending on a variety of arguments This last circum- stance is very convenient On the other hand, in terms of the k, the expression for the energy E(k;, , k„) takes the physically intuitive form (3.34), the sum of the energies of the elementary excitations in the state Q It should also

be noted that, in contrast to the system of equations (3.17), which in the natural range of the parameter n, namely, | Re n | <7, makes sense for n #0,

+ n/2, + m, the system (3.33) has a meaning for all values of A In particular, putting A = 1 in (3.27) and (3.32)—(3.34) we obtain the classical Bethe

formulae for the XXX model [2]

So far we have been looking at the case of arbitrary complex values of the parameter A Only real A are of physical interest, because only for such A is

H a Hermitian operator Now (3.28) shows that it is natural to consider the

following ranges of the real parameter A:L-1<A<1, 0 421,

Wl A<-1 In], the parameter 7 takes the real values 0 < <‡7; in Il, n=in',O<n’ <~; in Ul, n= zj2+in,0<'<®

For 7 in I we put

(3.35) £ (=i, (iM),

for n in I

36 L.A Takhtadzhan and L, D Faddeev

(3.36) £,, (0) = £n (A)

and for 7 in III

Then for the operator matrix elements of the local transition matrices £,,(A)

thus defined we obtain from (3.1)

(3.38) Zt ar(h)= Zn, 22), £2,124) = + £n, 01 (A),

where the minus sign corresponds to the values of 7 in I and II, the plus sign in

Ill, and the tilde and * denote complex and Hermitian conjugation,

respectively For a matrix % on the ring By we denote by # the matrix with

the elements %), = £}, Then (3.38) can be rewritten in the form

(3.39) £, (kA) = L(A) 0?

for n in I and II and _

for 7 in II

Let 7(A) be the monodromy matrix constructed from the local transition

matrices ¥,(A) (3.35)—(3.37) Since the elements of £,(4) commute for

Thus, in the case of real A the monodromy matrix satisfies the supple-

mentary relations (3.42)—(3.43), that is,

where the minus sign corresponds to values of n in I and II, and the plus sign to

lil

Since H in this case is Hermitian, the eigenvectors of the form (3.16) for

distinct eigenvalues (3.18), are orthogonal However, these states are non-

normalized In the case of real A, for the calculation of scalar products of

vectors of the form (3.16) it is convenient to use (3.44) and the permutation -

relation for B(A) and C(p) in (3.9),

~

The quantum method of the inverse problem and the Heisenberg XYZ

Setting out from (3.45), we can construct from the generating operator B (A) for non-normalized elementary excitations on the state 2 a generating operator

of normalized eigenstates for the Hamiltonian H We do not go into further details here

In conclusion we summarize the contents of this section Using (3.8) and the existence and properties (3.14)—(3.15) of the generating vector §2, we have obtained an algebrization of Bethe’s Ansatz to find the eigenvalues and eigen- yectors of the transfer matrix T(A) for the six-vertex model, the trace of the monodromy matrix J (A) The eigenvectors of T(A) are obtained by applying

to 2 the generating operators B(A,), the matrix elements of 7 (A) for values of

À satisfying the system of transcendental equations (3.17) The eigenvalues

then are given by (3.18) We have shown how the results obtained for the six-

yertex model lead to the classical results (3.32)—(3.34) for the Heisenberg XXZ model We have looked at the physically interesting case of real A In this case

TJ (a)allows the involution (3.42)—(3.43), and B(A) turns out to be conjugate to

C(A), the annihilation operator for elementary excitations on 2 We have pro- posed in this case a scheme for constructing normalized eigenvectors of H, using (3.44) and (3.45)

§4 Generating vectors and permutation relations

We devote this section to the analysis of the general case, the eight-vertex lattice model The local transition matrix £,(A) corresponding to it has the

(44) — #«()={ werkt wah wih w, 0} + iw.0}, to, OỆ — t0a0ï, — trek) _ (On 0) Ba) ) yn (A) Sn (A) where the w, are defined by (1.45):

1ø -Ƒ ipạ = @ (2) 8 (A—) H (A+),

w,— 1ø = Ð (2n) H (A— 1) 8 (A+ y),

w, + we =H (2) 0 (A—4) 8 (A+),

w, — w, = H (2n) H (A—n) H(A +9)

The matrices £,(A) satisfy the Baxter—Yang relation in which the #-matrix has

the form (1.39), (1.46), namely

d

0

0 7

a

a(A, p) = 9 (2n) 0 (A—p) H(A—p +t 2m),

A b(A, p) =H (2n) 0 (A—p) 8 (A—p+ 2m),

38 L A Takhtadzhan and L, D Faddeev

The monodromy matrix 7 (4) is defined by (1.3), and as a2 X 2 matrix

over „has the form

A(a) BỘ,

It satisfies (1.42)

(4.6) #(đ, u) (7 (4) @ Z (w)=(Z (w) @ Z (A)2(À mì,

which contains all permutation relations between the operators A(A), B(A),

C(A), and D(A)

(4.1)—(4.2) show that, in contrast to the case of the six-vertex model, the

operator +, (A) is non-degenerate for almost all \ Hence, we cannot obtain a

local vacuum, either for Z,(A) or for a finite product of such matrices More-

over, in this case the coefficients d(A, #) is non-zero, so that the permutation

relations for the A(A), B(A), C(A), and D(A) no longer have the simple form

(3.9) Consequently, the method developed in §3 is not directly applicable to

the case of the eight-vertex model However, we can generalize the

considerations in §3 appropriately, and this leads to a solution of the eight-

vertex model Instead of the generating vector (1.14), we shall use a family of

generating vectors and instead of the relations (3.9) a series of permutation

relations for various linear combinations of A(A), B(A), C(A), and D(A)

Let us proceed to the construction of the family of generating vectors We

use the following argument: the monodromy matrix 7 (A) can be calculated if

we replace the local transition matrices £,(4) by matrices £,(A) gauge-

equivalent to £,(A):

(4.7) Li, (d) = Mads (A) Zn (A) Mạ (À),

where the M,,(A) are arbitrary non-singular 2 X 2 numerical matrices Indeed,

the new monodromy matrix

‘N

differs from 7 (A) only by the simple linear transformation

(4.9) FT! (M=My (A) T (A) M, (A)

It turns out that the gauge transformations M, (A) can be chosen in such a

way that each matrix £;,(A) has a local vacuum independent of d that is

annihilated for all \ by its lower left element This condition defines the first

column of M,,(A) The second column is defined, if necessary, so that the dia-

gonal matrix elements of £;,(A) act on the local vacuum in the simplest

possible way The corresponding formulae may be extracted from Baxter

[19]—[20] It becomes clear that there is a whole family of gauge trans-

formations MẶOs, t) depending on an integer / and arbitrary complex

parameters s and t We write

(4.10) Mnsi-1 (458, )= Ma ( 3, 0), m=(5 Wh

where

nh

(4.14) =e H+ 2th), k= 1 z‡ = 9(s + 2kn— À), AC) @ (¢-++ 2kyn +A) 1

g(u) = H(u) @(u), and 7, =" 5 ' + 2kn —K Here K is a half-period of @(u) (see Appendix I)

The corresponding local transition matrices £;,(4) are denoted by

In terms of 74(4) we write the monodromy matrix 7 '(A)

= Mri (A; 8, 1) Ln (0) Mays (Ai 5; 2={

(4.13) 7w (A)= Mi (ÂN Z (4) Mi),

with the matrix elements A‘, (A), BY (A), Cf-(A), and Djy (A)

A local vacuum w/, € 5, with

(4.14) vi (A) of =0,

exists and is given by the formula (4.15) œ@t =H(s+2(®+))n—n) # + 9(s +2 (ø +0) n—n) #n-

In contrast to the case of the six-vertex model, ws, is not an eigenvector of

of (A) and jf (A) However, its transformation under these operators is simple:

al (A) @ =h (A+ y) ol 4,

68 {ssc of =k of

where (4.17) b (u) =9 (0) g (u) =9 (0) H (u) 9 (u)

(4.11)—(4.12) and (4.14)—(4.17) are easily checked on the basis of the

addition theorems for the Jacobi theta-functions

From the local formulae (4.12)—(4.17) we find that the vectors

(4.18) Qy= 0! @ @ oh

satisfy the relations