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Anisotropic vertex models
Volume 92A, number 2
PHYSICS LETTERS
25 October 1982
ANISOTROPIC VERTEX MODELS R.Z. BARlEY and Yu.V. KOZHINOV The Kazan Physico-Technical Institute of the Academy of Sciencesof the USSR, Kazan, 420029, USSR Received 14 July 1982
The exact solution is obtained for eigenvalues and eigenvectors of the transfer matrix of the two-dimentional vertex model with different numbers of states on the horizontal and vertical edges.
At present a more general exactly solved model of statistical mechanics is the eight-vertex Baxter model [1], including as special cases practically all models solved before (see ref. [21).l’his model is defined on a square lattice with periodic boundary conditions. There is a “spin” variable taking q discrete values on each edge. Each site of the lattice has a Boltzmann weight, depending on the values of the four spin variables arranged on the edges, surrounding the given site L,1~,where i, j (a, j3) are values of variables on the vertical (horizontal) edges. Calculation of the free energy of the system is reduced to diagonalization of the transfer matrix
::~
T~?~ = VNaNI1N_laN_1 {‘} {a} 1N01 iN_laN
...
(1)
.
l~a 2
The condition of commutation of two transfer matrix T and T’ with different sets of Boltzmann weights occupies the central place in the Baxter solution,
E
R7172L~1P141~2 = ala2 1171 172
~
(2)
L~171L~172R~1~2. 1a2 7172
k1,71,’y2 11a1
Relations (2) are called the Yang—Baxter equations. They play a key role in the theory ofthe factorized S-matrix [3,4] and in the quantum method of the inverse problem [5]. The solution of eqs. (2) where q = 2 was obtained by Baxter [1] (see also ref. [6]). Generalisation of the case q > 2 were considered in refs. [7—11].It is supposed that spin variables on vertical and horizontal edges take the same number of values (q,, = In this paper we shall examine the generalisation of the Baxter model for the case q,~,* q~,concretely q~ 4, c’i~ = 3. We will consider that there are two spin variables left and right on the vertical edge and suppose, that number of left (right) spins directed up by transition from one row to another is conserved. The latter restriction is the necessary condition of application of Bethe ansatz. So each site, besides the basic state ~4, has three excited ones, ~t, t tt. If in a given row the numbers of these excitations are n1, n2, n3, then under transition from one line to another n1 + n~ and ~ 2 ÷n are conserved. It is noted that in models considered before with three types of excitations [7] it was supposed that all three numbers were conserved separately. More general matrices, satisfying the formulated requirements above have the following non-zero elements: ~,
/a
32
21
33
43
11
21
23
43
12
32
L,~,, L21 L32 L11 L21 L33 L43 L12 L32 L23 L43 3 Following the quantum method of the inverse problem [5] we will examine L as a 3 X 3 matrix, whose elements are spin operators of n lattice sites. Then eqs. (2) are rewritten in a compact form, ,
,
,
,
,
,
,
,
,
.
R(X,X’)L~(X)®L~(X’)=L~(X’)®L~(A)R,
(4)
o 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
79
Volume 92A, number 2
PHYSICS LETTERS
25 October 1982
where X is a spectral parameter. An analogous relation holds for the monodromy matrix 9(X)=LN(X)...Ll(X)
(5)
through which the transfer matrix (1) is expressed, T(X)=Tr [9(A)]
(6)
.
Supposing that all elements (3) differ from zero, we obtained the following general solution of eqs. (2): 13’Lii =czKx_1L~1=aKx—’L~I=B(f1D)’Li~=x~L~=aL~=a13D(xB)’L~=A(13B)~(x+Gi)’Li~ =
[B(x +G2)}—’L~=aA[B(1 +K_1G3x)]_1L~=a~3[B(l +K_1G4x)]_1L~
=
cry2A 1D[y1C1B(x
+ E1)]_1L~= -y1A2D[72C2B(x + E2)]_1L~ =
L~I,
A1(fryi)_1L~ yj’Lj~=ClB[a1372D]_1L~] =y~~L~ =L~, y1A2B—1L~j=j3’y1B1L~)j=72C2D_1L~=a1372B~L~ =L~,
(7)
where 2/CB2,
A=A
K=:AD
1A2,
C=C1C2,
E1=A1D/C1B,
E2A2D/C2B,
1 (A
1). C1 =DB— 1/C1 +A2/C2—A), G2—D/BC, G3=AD/B, G4=DB~(A1/C1+A2/C2— C— So 22 Boltzman weights (3) were expressed through 5 arbitrary functions of the spectral parameter A(x, a, 13, 71’ 72) and 6 constants (A 1 A2, C1, C2, B, D). The functions ‘~iand 72 do not appear in the expression of the free energy and may be chosen arbitrarily. Without violating the generality we may suppose 7i = 72 = 1, 2~’<= —(A 172 a, j3 = const., x =‘~jisin(X+ K)/sin(X ,~), e 1C2/A2C1) Then for the R-matrix elements we have the following solution: ,
—
Rll=R~=1, RII
R~=sin(X’—A+2K)Isin(A—X’+2K), R~iR~R~,
R}~-R~J= e2i(?~’)R~= ei(?~’)sin 2K/sin(A
R~ =
=
(l/c43)(B/D)R~ = (1/a2K)R~=
—
A’
+
2i~),
sin(A A’) sin(X—X +2k) _
,
R~ = j3~R~= aj3(D/B)R~ (8) .
Solution (8) under a = j3 = BID = A/c = 1 is graduated analogous of R-matrix of ref. [12]. This solution itself may serve for the construction of a commutating family of transfer matrices in graduated space. Under a = AD/BC = 1 for the foundation of eigenvectors and eigenvalues of the transfer matrix we may use the quantum method of the inverse problem [5,7]. As a result for the eigenvalues we have the following solution: ‘~ sin(X—A+2,~) A(X 12]~‘ sin (A 1, X,~A) = [a(C/A)’ 1. A) [~i(x)1Nj~(Ai, A,~A) + (—1~’[L~(X)]N}, —
~
...
,
—
where
~
-..,
2 sin(Ak A) sin(uj jj sin(X 2~) IcW sin(A—vj) ~ k=1SmO~k_A+21~)f~~1 sin(v ‘~
—
—
IA\m/2
—
—
A + 2K)
+
/=1
1-—A)
Two sets of values ~A1}and {v1} are defined from the conditions 1A(A = (~_1)n_ 1 A,1 Aj), 80
(9)
Volume 92A, number 2
n/2 n sin& - Uj) =(_l)rn-1 l-I A k=l sin& -ui +2~)
c
0
PHYSICS LETTERS
fi I=1 1+j
Sin(fJj - Ul t Sin(IJl -
25 October 1982
2K)
Uj t 2K)’
(10)
Eqs. (9) and (10) completely determine the thermodynamic properties of the system. Analogously, we may consider vertex models having 4 and 24-l states accordingly on the horizontal and vertical edges. Commutating transfer matrices contain as the first integrals the hamiltonians of quantum n sublattice systems. Consideration of this problem will be the subject of a forthcoming paper. References [l] R.J. Baxter, Ann. Phys. 70 (1972) 193. [2] [3] [4] [5] [6] [7] [8] [9]
R.J. Baxter, Philos. Tran. Roy. Sot. (London) A289 (1978) 315. C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. A.B. Zamolodchikov and A.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. Z.A. Takhtajan and L.D. Faddeev, Usp. Mat. Nauk 34 (1979) 13. A.B. Zamolodchikov, Commun. Math. Phys. 69 (1979) 165. P.P. Kulish and N.Yu. Reshetikhin, Zh. Eksp. Teor. Fiz. 80 (1981) 214. A.A. Belavin, Func. Anal. Appl. 14 (1980) 18. P.P. Kulish and E.K. Sklyanin, Zap. Nauch. Semin. LOMI 95 (1980) 125. [lo] C.L. Schultz, Phys. Rev. Lett. 46 (1981) 629. [ll] J.H.H. Perk and C.L. Schultz, Phys. Lett. 84A (1981) 407. [12] P.P. Kulish, Zap. Nauch. Semin. LOMI 109 (1981) 83.
81
PHYSICS LETTERS
25 October 1982
ANISOTROPIC VERTEX MODELS R.Z. BARlEY and Yu.V. KOZHINOV The Kazan Physico-Technical Institute of the Academy of Sciencesof the USSR, Kazan, 420029, USSR Received 14 July 1982
The exact solution is obtained for eigenvalues and eigenvectors of the transfer matrix of the two-dimentional vertex model with different numbers of states on the horizontal and vertical edges.
At present a more general exactly solved model of statistical mechanics is the eight-vertex Baxter model [1], including as special cases practically all models solved before (see ref. [21).l’his model is defined on a square lattice with periodic boundary conditions. There is a “spin” variable taking q discrete values on each edge. Each site of the lattice has a Boltzmann weight, depending on the values of the four spin variables arranged on the edges, surrounding the given site L,1~,where i, j (a, j3) are values of variables on the vertical (horizontal) edges. Calculation of the free energy of the system is reduced to diagonalization of the transfer matrix
::~
T~?~ = VNaNI1N_laN_1 {‘} {a} 1N01 iN_laN
...
(1)
.
l~a 2
The condition of commutation of two transfer matrix T and T’ with different sets of Boltzmann weights occupies the central place in the Baxter solution,
E
R7172L~1P141~2 = ala2 1171 172
~
(2)
L~171L~172R~1~2. 1a2 7172
k1,71,’y2 11a1
Relations (2) are called the Yang—Baxter equations. They play a key role in the theory ofthe factorized S-matrix [3,4] and in the quantum method of the inverse problem [5]. The solution of eqs. (2) where q = 2 was obtained by Baxter [1] (see also ref. [6]). Generalisation of the case q > 2 were considered in refs. [7—11].It is supposed that spin variables on vertical and horizontal edges take the same number of values (q,, = In this paper we shall examine the generalisation of the Baxter model for the case q,~,* q~,concretely q~ 4, c’i~ = 3. We will consider that there are two spin variables left and right on the vertical edge and suppose, that number of left (right) spins directed up by transition from one row to another is conserved. The latter restriction is the necessary condition of application of Bethe ansatz. So each site, besides the basic state ~4, has three excited ones, ~t, t tt. If in a given row the numbers of these excitations are n1, n2, n3, then under transition from one line to another n1 + n~ and ~ 2 ÷n are conserved. It is noted that in models considered before with three types of excitations [7] it was supposed that all three numbers were conserved separately. More general matrices, satisfying the formulated requirements above have the following non-zero elements: ~,
/a
32
21
33
43
11
21
23
43
12
32
L,~,, L21 L32 L11 L21 L33 L43 L12 L32 L23 L43 3 Following the quantum method of the inverse problem [5] we will examine L as a 3 X 3 matrix, whose elements are spin operators of n lattice sites. Then eqs. (2) are rewritten in a compact form, ,
,
,
,
,
,
,
,
,
.
R(X,X’)L~(X)®L~(X’)=L~(X’)®L~(A)R,
(4)
o 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
79
Volume 92A, number 2
PHYSICS LETTERS
25 October 1982
where X is a spectral parameter. An analogous relation holds for the monodromy matrix 9(X)=LN(X)...Ll(X)
(5)
through which the transfer matrix (1) is expressed, T(X)=Tr [9(A)]
(6)
.
Supposing that all elements (3) differ from zero, we obtained the following general solution of eqs. (2): 13’Lii =czKx_1L~1=aKx—’L~I=B(f1D)’Li~=x~L~=aL~=a13D(xB)’L~=A(13B)~(x+Gi)’Li~ =
[B(x +G2)}—’L~=aA[B(1 +K_1G3x)]_1L~=a~3[B(l +K_1G4x)]_1L~
=
cry2A 1D[y1C1B(x
+ E1)]_1L~= -y1A2D[72C2B(x + E2)]_1L~ =
L~I,
A1(fryi)_1L~ yj’Lj~=ClB[a1372D]_1L~] =y~~L~ =L~, y1A2B—1L~j=j3’y1B1L~)j=72C2D_1L~=a1372B~L~ =L~,
(7)
where 2/CB2,
A=A
K=:AD
1A2,
C=C1C2,
E1=A1D/C1B,
E2A2D/C2B,
1 (A
1). C1 =DB— 1/C1 +A2/C2—A), G2—D/BC, G3=AD/B, G4=DB~(A1/C1+A2/C2— C— So 22 Boltzman weights (3) were expressed through 5 arbitrary functions of the spectral parameter A(x, a, 13, 71’ 72) and 6 constants (A 1 A2, C1, C2, B, D). The functions ‘~iand 72 do not appear in the expression of the free energy and may be chosen arbitrarily. Without violating the generality we may suppose 7i = 72 = 1, 2~’<= —(A 172 a, j3 = const., x =‘~jisin(X+ K)/sin(X ,~), e 1C2/A2C1) Then for the R-matrix elements we have the following solution: ,
—
Rll=R~=1, RII
R~=sin(X’—A+2K)Isin(A—X’+2K), R~iR~R~,
R}~-R~J= e2i(?~’)R~= ei(?~’)sin 2K/sin(A
R~ =
=
(l/c43)(B/D)R~ = (1/a2K)R~=
—
A’
+
2i~),
sin(A A’) sin(X—X +2k) _
,
R~ = j3~R~= aj3(D/B)R~ (8) .
Solution (8) under a = j3 = BID = A/c = 1 is graduated analogous of R-matrix of ref. [12]. This solution itself may serve for the construction of a commutating family of transfer matrices in graduated space. Under a = AD/BC = 1 for the foundation of eigenvectors and eigenvalues of the transfer matrix we may use the quantum method of the inverse problem [5,7]. As a result for the eigenvalues we have the following solution: ‘~ sin(X—A+2,~) A(X 12]~‘ sin (A 1, X,~A) = [a(C/A)’ 1. A) [~i(x)1Nj~(Ai, A,~A) + (—1~’[L~(X)]N}, —
~
...
,
—
where
~
-..,
2 sin(Ak A) sin(uj jj sin(X 2~) IcW sin(A—vj) ~ k=1SmO~k_A+21~)f~~1 sin(v ‘~
—
—
IA\m/2
—
—
A + 2K)
+
/=1
1-—A)
Two sets of values ~A1}and {v1} are defined from the conditions 1A(A = (~_1)n_ 1 A,1 Aj), 80
(9)
Volume 92A, number 2
n/2 n sin& - Uj) =(_l)rn-1 l-I A k=l sin& -ui +2~)
c
0
PHYSICS LETTERS
fi I=1 1+j
Sin(fJj - Ul t Sin(IJl -
25 October 1982
2K)
Uj t 2K)’
(10)
Eqs. (9) and (10) completely determine the thermodynamic properties of the system. Analogously, we may consider vertex models having 4 and 24-l states accordingly on the horizontal and vertical edges. Commutating transfer matrices contain as the first integrals the hamiltonians of quantum n sublattice systems. Consideration of this problem will be the subject of a forthcoming paper. References [l] R.J. Baxter, Ann. Phys. 70 (1972) 193. [2] [3] [4] [5] [6] [7] [8] [9]
R.J. Baxter, Philos. Tran. Roy. Sot. (London) A289 (1978) 315. C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. A.B. Zamolodchikov and A.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. Z.A. Takhtajan and L.D. Faddeev, Usp. Mat. Nauk 34 (1979) 13. A.B. Zamolodchikov, Commun. Math. Phys. 69 (1979) 165. P.P. Kulish and N.Yu. Reshetikhin, Zh. Eksp. Teor. Fiz. 80 (1981) 214. A.A. Belavin, Func. Anal. Appl. 14 (1980) 18. P.P. Kulish and E.K. Sklyanin, Zap. Nauch. Semin. LOMI 95 (1980) 125. [lo] C.L. Schultz, Phys. Rev. Lett. 46 (1981) 629. [ll] J.H.H. Perk and C.L. Schultz, Phys. Lett. 84A (1981) 407. [12] P.P. Kulish, Zap. Nauch. Semin. LOMI 109 (1981) 83.
81