Classification of Exactly Solvable Two-Component Models Institute 0

508 Progress of Theoretical Physics, Vol. 68, No.2, August 1982

Classification of Exactly Solvable Two-Component Models Kiyoshi

SOGO,

Mamoru UCHINAMI, Yasuhiro and Miki W ADATI

AKUTSU

Institute 0/ Physics, College 0/ General Education University 0/ Tokyo, Komaba, Tokyo 153 (Received April 6, 1982) In the case of two-component models, all possible solutions for factorization equations of S-matrix are obtained. These solutions give a classification of solvable vertex models. It is found that Yang-Baxter relation is more general than the factorization equation. The six vertex model by Yang, Yang and Sutherland is derived not from the factorization equation but from Yang-Baxter relation. Thus it is concluded that Yang-Baxter relation lies at the basis of all the known solvable two component models. Furthermore, spin Hamiltonians corresponding to our vertex models are presented in explicit forms.

§ 1.

Introduction

Recently there has been renewed interest in the theory of exactly solvable lattice models. It has been found that the theory of commuting transfer matrices is closely connected to a seemingly unrelated subject, that is, the theory of the factorized 5 -matrix. I) Zamolodchikov 2 ) found that Baxter's3) eightvertex model is equivalent to his "Zdactorized 5 -matrix" and that the factorization equations for 5 -matrices of Z4 model are essentially the same as the Yang-Baxter relation 3),4) which is the commutability condition of the eightvertex transfer matrix. Motivated by this discovery, many factorized 5matrices and their equivalent exactly solvable lattice models were discovered 5 ) by solving the factorization equations with different kinds of symmetry. Furthermore, as Baxter pointed out,6) the exactly solvable spin system can be obtained from the transfer matrix. Therefore we now have a powerful method to construct exactly solvable lattice and spin models starting with the factorization equations. The purpose and the outline of this paper are the following. We shall settle the problem: "How many classes of exactly solvable models are there in the two-component models?". By "two-component model" we mean a system whose fundamental constituents take one of two states; particle and antiparticle in field theory, spin up and down in spin system, arrow up and down or right and left in lattice model. Zamolodchikov's Zdactorized 5 -matrix which has additional T- and P-invariance is a special example. In the present paper we do not require these invariances and formulate the factorization equations with the general eight 5 -matrix elements. The formulation of the problem is given

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in § 2_ In § 3, the factorization equations are solved. The solutions are classified into three cases, each of which corresponds to eight-vertex (8V), six-vertex (6V) and seven-vertex (7V) lattice model. Cases 8V, 6V and 7V have three, two and three subcases, respectively. Therefore we find eight solvable lattice models, which we denote 8V(I), 8V(II), 8V(III), 6V(I), 6V(II), 7V(I), 7V(II) and 7V(III). The corresponding spin Hamiltonians are also given. In § 4, the exceptional case of six-vertex model is discussed and the Yang-Baxter relation is established for this case. It is found that for the commuting transfer matrices the Yang-Baxter relation is more fundamental than the factorization equation. The last section is devoted to summary. The details of the calculations are given in the Appendices. § 2.

Yang-Baxter relation for two-component model

For a two-component model which is the generalization of Baxter's (symmetric) eight-vertex model,3) we discuss the structure of the commuting transfer matrices, i.e., the structure of the factorized scattering matrices. We call the two-component model the generalized eight-vertex model. The generalized eight-vertex model is defined by eight non-zero elements corresponding to the vertex configurations shown in Fig. 1, all of which are independent from each other. We introduce the local transition matrix Ln and express it in the form: L 11.~1 (,1) (2·1)

0

Ln(,1)=

(

L22.l1 (,1) where the matrix element L ra.r'a'(,1) is the Boltzmann weight of the vertex

++++ (2)

(3)

(4)

++++ (5)

(8)

Fig.1. Eight nonzero elements; (1) Lll.ll=Sl. (2) L22.22=S2. (3) L12.12=S,. (4) L21,21 =5,. (5) L12,21=Sr,(6) L21.12=5r. (7) L l l •22 = Sa. (8) L22.11=Sa.

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K. Sogo, M. Uchinami, Y. Akutsu and M. Wadati

configuration shown in Fig. 1. We have taken the explicit dependence of Ln upon a spectral parameter A in (2·1). Using the Pauli matrices an j and a j (j 4 =1,2,3,4; an =a 4=I), we can rewrite the local transition matrix Ln(A) in the form: 3 L (A) = l ((Lll,11- L 12 ,12)an +(L ll ,ll + L12,12)an 4 n 2 (L 21 ,12 + L22,ll )an 1 + i (L21,12 - L 22 ,ll )an 2 (L12,21 + L ll ,22)an 1- i(L 12,21- Lll,22)an 2) (L 21,21 - L 22 ,22 )an 3+ (L 21 ,21 + L 22 ,22 )an 4 '

(2·2)

or in the form of direct product:

(2·3) where the nonzero Wij(A) are given by W44= (L ll ,11 + L22,22+ L12,12+ L 21 ,21)/ 4, W33=(Lll,ll + L22,22- L 12 ,12- L 21 ,21)/ 4, W34 = (Lll,ll - L 22,22+ L12,12- L21,21)/ 4,

Wll = (L 12,21 + L21,12+ Lll,22+ L 22,11)/ 4, W22= (L12,21 + L21,12- L ll ,22- L 22 ,ll)/ 4, W12= i( - L12,21 + L 21 ,12+ Lll,22- L22,ll)/ 4, W21 = i (L 12,21 - L 21,12 + L 11,22 - L22,ll )/4 .

(2·4)

Using the form (2·2) of the local transition matrix Ln(A) the transfer matrix T(A) is given by

(2·5) where tr means the trace of 2 x 2 matrix. The transfer matrices T(A) and T(fl.) commute if there exists a non-singular 4 x 4 matrix R(A, fl.) such that (2·6) Condition (2·6) for commuting transfer matrices is called the Yang-Baxter relation. 3 ),4) When the matrix R is chosen to have the same form as Ln, i.e.,

_(Rll'll~A' fl.) R(A, fl.)-

0

R22,ll (A, fl.)

0 RI2,dA, fl.) R21,dA, fl.) 0

0 RI2,21(A, fl.) R21,21 (A, fl.) 0

Ru,"~A,

P)

o R 22,22(A, fl.)

'

(2·7)

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we have 28 equations from (2·6) with (2·2) and (2·7). According to Zamolodchikov Z ) and Takhtadyan and Faddeev,7) we identify Land R as (2·8) where y---> y'.

S::: means the

two-body scattering matrix for the process; a ---> a' and For convenience, we use the following notations:

ZI-SS 12 r,

SU=Sa, Sif=Sr,

SM=St,

S 2ZI-S1a,

(2·9)

SU=Sz.

In terms of the rapidity variables such that

;\ = 81 - 83 =- 813,

fJ. = 81-

8z =- 81Z, ;\ -

fJ. = 8z -

83 =- 823 ,

(2·10)

the 28 equations are given by

SrSrSr= SrSrSr, SaSrSa = SaSrSa ,

(2·11 )

SaSaSr=SaSaSr,

SrSaSa=SrSaSa, (2·12a, b, c) (2·13a)

SISaSr+SaStSt=SaS1Sl+SzSrSa,

(2·13b)

S2SzSa+SaSrSl=SrSaSz+StStSa,

(2·13c)

S2SaSr+SaStSt=SaS2Sz+S1SrSa,

(2·13d)

SrSaSI + StStSa = SaSrSz+ SISISa ,

(2·13a')

S2SrSa+SaS1Sl=SaStSt+S1SaSr,

(2·13b')

SrSaS2+ StStSa = SaSrSI + S2SzSa ,

(2·13c')

SISrSa+SaS2Sz=SaStSt+S2SaSr,

(2·13d')

SISaSt+SaStSr=StSaSl+SrStSa,

(2·14a)

SzSaSt+SaStSr=StSaS2+SrStSa,

(2·14b)

StSaSI +SrStSa=SaStSr+S1SaSt,

(2·14a')

StSaS2+ SrStSa = SaStSr+ S2SaSt ,

(2·14b')

SIStSr+SaSaSt=SrSrSt+StS1Sr,

(2·15a)

SzStSr+SaSaSt=SrSrSt+StSzSr,

(2·15b)

StSaSa+SrStSI = StSrSr+ SrS1St ,

(2·15c)

512

K. Sogo, M. Uchinami, Y Akutsu and M. Wadati StSaSa+SrStS2=StSrSr+SrS2St,

(2·15d)

StSlSr+SrSrSt= SIStSr+SaSaSt,

(2·15a')

StS2Sr+ SrSrSt = S2StSr+ SaSaSt ,

(2·15b')

SrSlSt + StSrSr= SrStSl+ StSaSa ,

(2·15c')

SrS2St+StSrSr= SrStS2+ StSaSa ,

(2·15d')

SISrSI +SaS2Sa=StSrSt+SrSlSr,

(2·16a)

S2SrS2+SaSlSa=StSrSt+SrS2Sr,

(2·16b)

SrSlSr+ StSrSt = SISrSI + SaS2Sa ,

(2·16a')

SrS2Sr+ StSrSt = S2SrS2+SaSlSa ,

(2·16b')

where the arguments of S's are implied to be 812, 813 and 823 in its order. These equations characterize the generalized eight-vertex model with the property of commuting transfer matrices. Equations (2·ll)~(2·16) are identical to the 5matrix factorization equations derived on the basis of Zamolodchikov's symbolic algebra. 1),2) In fact, the following symbolic algebra: Al (81)A 1(82) = 51 (B12)A 1(82)A 1(81)+ SaC 812 )A2( 82)A 2( (1), A2(B1 )A2( (2) = S2( (12)A 2( 82 )A2( (1)+ SaC 812 )A 1 (82)A 1(81), Al (81 )A2( (2) = St(B12 )A 2( (2)A 1(81)+ 5 r(B12)Al (82 )A2( (1), A 2( 81 )Al (82) = St( 812 )Al (82 )A 2( (1) + Sr( 812 )A2( ( 2)A 1(81)

(2·17)

yield the factorization equations (2·ll) ~ (2 ·16) under the assumption of the associativity; (Ai(81)A j (82»Ak(8 3)=Ai(81)(Ai82)Ak(83». The unitarity conditions for the 5 -matrix are SI(8)SI( - 8)+ Sa(8)Sa( - 8)= S2(8)S2( - 8)+ Sa(8)Sa( - 8)= 1, SI(8)Sa( - 8)+Sa(8)S2( - 8)= S2(8)Sa( - 8)+ Sa(8)SI( - 8)=0, Sr(8)Sr( -8)+ St(8)St( -8)= Sr(8)Sr( - 8)+St(8)St( - 8)=1, Sr(8)St( - 8)+St(8)Sr( - 8)= Sr(8)St( -8)+St(8)SA -8)=0. (2·18)

Equations (2·18) are the consistency conditions for the algebra (2·17). In the next section the factorization equations (2·ll) ~ (2 ·16) will be solved systematically. Each set of the solutions corresponds to the solvable vertex model. The spin Hamiltonian related to the model can be constructed through Baxter's formula :6).7)

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513

H = d log T( 8)/ d818~O = T-1l~;I}(0)·(d7{