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Exact solution of the two-sublattice vertex model
Volume 138, number 4,5
PHYSICS LETTERS A
26 June 1989
EXACT S O L U T I O N OF T H E TWO-SUBLATTICE VERTEX M O D E L R.Z. BARIEV Kazan Physico-Technical Institute of the Academy of Sciences of the USSR, Kazan 420029, USSR Received 16 November 1988; revised manuscript received 20 April 1989; accepted for publication 24 April 1989 Communicated by A.A. Maradudin
An exact solution is obtained for the eigenvalues and eigenvectors of the diagonal-to-diagonal transfer matrix of the two-dimensional vertex model with two sublattices. On the basis of this solution the exact expression for the free energy of the model is derived. This model is the classical analogue of the one-dimensional Hubbard model.
For two decades the one-dimensional Hubbard model has attracted the attention of specialists working in both solid state physics and mathematical physics. The exact solution of this model was obtained by Lieb and Wu in 1968 [ 1 ]. However, the algebraic structure of the model was not well understood for a long time [2,3]. Recently the exact integrability of the one-dimensional Hubbard model was established by Shastry [ 4 ] (see also refs. [ 5,6 ] ). In ref. [4] Shastry proposed the new two-dimensional vertex model and showed that a one-parameter family of the row-to-row transfer matrices of this model commutes with the Hamiltonian of the Hubbard model. Thus the algebraic structure of the model was understood. However, the problem of diagonalizing the transfer matrix is quite nontrivial from either the coordinate-space Bethe ansatz point of view or the quantum inverse scattering method [2,7]. Therefore Shastry [ 8 ] has not succeeded in an explicit diagonalization, but presented a conjecture for the general eigenvalue from which the results of Lieb and Wu [ 1 ] follow. In this paper we consider the two-dimensional vertex model proposed by Shastry using the method of the diagonal-to-diagonal transfer matrix [ 9 ]. We obtained an exact solution for eigenvalues and eigenvectors of this transfer matrix with the help of the generalized coordinate-space Bethe ansatz. Thus we diagonalized the Shastry model. Following Shastry we shall consider a system of two zero-field six-vertex models which are presented
by the solid and dashed lines on fig. 1. We shall place an arrow on each edge of the lattices that points toward one of the lattice points. At each of the vertices allow for six standard configurations obeying the ice rule with Boltzmann weights a, b and c (see fig. 2). The solid circles are interaction vertices, coupling the two models by giving a weight e h (e -h) to parallel (antiparallel) arrows. The diagonal-to-diagonal transfer matrix T d is de-
Fig. 1.
Z//Y a
// 8
c
Fig. 2.
173
Volume 138, number 4,5
PHYSICS LETTERS A
scribed by its matrix elements that are the products of the Boltzmann weights o f the diagonal row of vertices, N
T~({a,s)l{a',s'}) = I]
~'~*
S'~kS~*
26 June 1989
~e[ (x,. r, ) ..... (xo, r . ) ] =
E A~,":S °~ P
X H
W[(vQ.rQj)I(~Q .....
Q,+,-=Q,,
~,+,)](kp,,kp,+, ) ~u,~, (kj,) (xQ,),
p,+,=p,,
(6)
k=l
where the summation is carried out over all permutations [p~ ..... p,] of the numbers 1, 2, ..., n.
Xexp[ ½h(a2kS2k + 0"~kS~k) ] , a2U+l=al,
S2N+I=Sl,
(1)
where non-zero elements of the R-matrix are
RT_{S_]=RI{=a,
R-l-',=R_'t_l=b,
R - I _ I - - R - 1 - 1 , =C,
(2)
a, s = + 1 ( - 1 ) for arrows directed up (down). Let us assume that ~'/[ ( X l , r l ) . . . . . ( X m , T m ) l ( x , , + ~ , r m + , ) , . . . ,
(X,, r,,)]
(3) is the amplitude o f the row state when the arrows directed downward are located at the edges (x~, z~ ) ..... (xm, r~) of the first sublattice and the edges (xm+ ~, Zm+l), ..., (X,, Z,) of the second sublattice, r~= 1, 2 respectively, depending on whether the ith edge is tilted or vertical. The situation o f the downward directed arrows on each sublattice is defined by (1, 1 ) ~ ( x l , z l ) < ( x z , r:)<...<(Xm, r , , ) ~ ( N , 2) ,
( 1, 1 ) ~ (x,.+ ~, rm+ ~) < (x.,+2, z,.+2) < . . . < (x~, %) ~< (N, 2) .
(4)
The inequality (x~, r,) < (xj, rj) designates that x~
W[(x.~)j(,,~,)](k,, k2) = W(kl, k2),
forx=x', r=r'=2
= 1,
otherwise,
t//Ik) (X) = ( - 1 (~ __b-e-Zh)eikx ,
~k~ (x) = e~' . 2(k) = e x p ( - h - i p ) { r c o s ( p + i h ) + [ 1 + 6 2 cosZ(p+ih) J11/2tJ
6=b/a,
e=c/a,
p=k/2.
(8)
The eigenvalue appropriate to (6) is
A=a2Uexp(Nh) f i 2(kj) .
(9)
/=1
The original m o m e n t of the Bethe ansatz usage is the introduction o f the factors (7). It allows us to consider amplitude (6) in the case when two coordinates coincide. Consider the eigenstate equations on the boundary of the subregions (5). We shall require that the amplitude (6) remains an eigenfunction of T d with the eigenvalue (9) in the case when two downward directed arrows coexist on the same site o f the lattice. This leads to the conditions on the coefficients AJQ'"a°° and allows us to determine the factors W ( k , pl...pn k2), 2
,4 'a'a ...... ----.P,Pi+ 1... =
(5)
where [Q~ .... , Q , ] is a permutation of the numbers 1, 2 ..... n. In each of the subregions (5) the amplitude (3) may be chosen as a generalized coordinatespace Bethe ansatz,
~ c~.13= 1
~ SA,d,+, ( k p,,kp,+,
.... '~ t..... A...t,i+,
(10)
W(kl, k:) = (2t22 e2a+e -2h) (2122 + 1 ) -~ , ~, =,t ( k;), where the nonzero elements of the S-matrix are as follows, 1I 22 Sll(kl, k2) =$22(k~, k2) = - I ,
174
(7)
~ k ) (X) is the amplitude of the one-particle state,
(1, I)<~(XQ,,rQi)<~(XQ2, ZQ=) <~...~< (XQ., rQ.) ~< (N, 2 ) ,
,
Volume 138, number 4,5
k2)--s~l (k,,
Sl~(k,,
PHYSICS LETTERS A
k2)
= ( 2 2 - 2 ~ ) ( 2 2 - 2 ~ +&2) -1 , S1921(kl, k2) = S 2 1 1 ( k l , k 2 ) = - z ~ 1 2 ( 2 2 - 2 1 JvAl2) -1 ,
U=2c2sh(2h)/ab.
A~2 = - - U[ 1 + ( 2 , 2 2 ) - ' ] ,
To prove the conjecture (6) we must test also the eigenstate equations when the number o f arrows directed downward coexistent on the same site is equal to 3 and 4. We have carried out this procedure. The last problem we have to attend to is the fulfillment o f the periodic boundary conditions. As a result, we have A,J,...,~. A.J2...a..~, p I...pn _ -- "~ p2...pnp 1 exp(ik.~ N)
( 11 )
Thus, we have obtained expressions for the eigenvalues of the transfer matrix in terms of the set of distinct numbers k~. To find the k~ we have the system of equations (10), ( 11 ). A necessary and sufficient condition of compatibility o f (10) is the satisfaction o f the Baxter-Yang relations [ 7,9-11 ]. In our case the S-matrix satisfies these relations and we may use the q u a n t u m method o f the inverse problem [2,7]. As a result we have
26 June 1989
a standard way go to a system of integral equations which can be solved in the thermodynamic limit by Fourier transform. Details of these calculations will be published elsewhere. As a result we get the following expression for the free energy of the considered model, oo
t
Jo(x)R(x) dx
- flf=ln a + ½[hl + 2 3 x [ l ~ l
U/)]'
0
if
6 < e -2'h' ,
(13)
where
R(x) =
b-2k-~ c h [ 2 h ( 2 k - 1) ] J2k-~(X), k=l
(14)
J~(x) are the Bessel functions of the first kind. The expression for the case 6 > e -21hl is obtained from ( 13 ) by interchange of [7 and e - 21hl. The linear term of the expansion in powers of 6in eqs. (13) and (14) corresponds to the energy o f the ground state of the one-dimensional Hubbard model.
References
- A t - U/2 ?=~~ - A t + U/2'
exp(2ipjN) = ( - 1 )n-m+l f i 2 J - - 2 7 1
(_ l )Ofl
-2z' -
j = l ~ j - - 2 j -1 - - A l +
f i Ar - A t - U r=l Ar-At+ U"
t-u/2
UI2 (12)
r#l
Eqs. (12) completely describe the eigenstates of the model under consideration. This completes the diagonalization o f the transfer matrix. Thus, we have obtained the eigenvalue spectrum and eigenvectors o f the diagonal-to-diagonal transfer matrix of the model through the solution o f the system o f equations (12). F r o m eqs. (12) one may in
[ 1] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 ( 1968) 1445. [2] P.P. Kulish and E.K. Sklyanin, in: Lecture notes in physics, Vol. 151 (Springer, Berlin, 1982) p. 61. [3] R.Z. Bariev and Yu.V. Kozhinov, Physica A 127 (1984) 316. [4] B.S. Shastry, Phys. Rev. Lett. 56 (1986) 1529, 2453. [ 5 ] M. Wadati, E. Olmedilla and Y. Acutsu, J. Phys. Soc. Japan 56 (1987) 1340, 2298. [6] E. Olmedilla and M. Wadati, Phys. Rev. Lett. 60 (1988) 1595. [7] L.D. Faddeev and L.A. Takhtajan, Usp. Mat. Nauk 34 (1979) 13. [8] B.S. Shastry, J. Stat. Phys. 50 (1988) 57. [9] R.Z. Bariev, Teor. Mat. Fiz. 49 (1981) 261 [Theor. Math. Phys. 49 (1982) 1021]. [ 10 ] A.A. Belavin, Phys. Lett. B 87 (1979) 117. [ 11 ] R.J. Baxter, Stud. Appl. Math. 50 ( 1971 ) 51.
175
PHYSICS LETTERS A
26 June 1989
EXACT S O L U T I O N OF T H E TWO-SUBLATTICE VERTEX M O D E L R.Z. BARIEV Kazan Physico-Technical Institute of the Academy of Sciences of the USSR, Kazan 420029, USSR Received 16 November 1988; revised manuscript received 20 April 1989; accepted for publication 24 April 1989 Communicated by A.A. Maradudin
An exact solution is obtained for the eigenvalues and eigenvectors of the diagonal-to-diagonal transfer matrix of the two-dimensional vertex model with two sublattices. On the basis of this solution the exact expression for the free energy of the model is derived. This model is the classical analogue of the one-dimensional Hubbard model.
For two decades the one-dimensional Hubbard model has attracted the attention of specialists working in both solid state physics and mathematical physics. The exact solution of this model was obtained by Lieb and Wu in 1968 [ 1 ]. However, the algebraic structure of the model was not well understood for a long time [2,3]. Recently the exact integrability of the one-dimensional Hubbard model was established by Shastry [ 4 ] (see also refs. [ 5,6 ] ). In ref. [4] Shastry proposed the new two-dimensional vertex model and showed that a one-parameter family of the row-to-row transfer matrices of this model commutes with the Hamiltonian of the Hubbard model. Thus the algebraic structure of the model was understood. However, the problem of diagonalizing the transfer matrix is quite nontrivial from either the coordinate-space Bethe ansatz point of view or the quantum inverse scattering method [2,7]. Therefore Shastry [ 8 ] has not succeeded in an explicit diagonalization, but presented a conjecture for the general eigenvalue from which the results of Lieb and Wu [ 1 ] follow. In this paper we consider the two-dimensional vertex model proposed by Shastry using the method of the diagonal-to-diagonal transfer matrix [ 9 ]. We obtained an exact solution for eigenvalues and eigenvectors of this transfer matrix with the help of the generalized coordinate-space Bethe ansatz. Thus we diagonalized the Shastry model. Following Shastry we shall consider a system of two zero-field six-vertex models which are presented
by the solid and dashed lines on fig. 1. We shall place an arrow on each edge of the lattices that points toward one of the lattice points. At each of the vertices allow for six standard configurations obeying the ice rule with Boltzmann weights a, b and c (see fig. 2). The solid circles are interaction vertices, coupling the two models by giving a weight e h (e -h) to parallel (antiparallel) arrows. The diagonal-to-diagonal transfer matrix T d is de-
Fig. 1.
Z//Y a
// 8
c
Fig. 2.
173
Volume 138, number 4,5
PHYSICS LETTERS A
scribed by its matrix elements that are the products of the Boltzmann weights o f the diagonal row of vertices, N
T~({a,s)l{a',s'}) = I]
~'~*
S'~kS~*
26 June 1989
~e[ (x,. r, ) ..... (xo, r . ) ] =
E A~,":S °~ P
X H
W[(vQ.rQj)I(~Q .....
Q,+,-=Q,,
~,+,)](kp,,kp,+, ) ~u,~, (kj,) (xQ,),
p,+,=p,,
(6)
k=l
where the summation is carried out over all permutations [p~ ..... p,] of the numbers 1, 2, ..., n.
Xexp[ ½h(a2kS2k + 0"~kS~k) ] , a2U+l=al,
S2N+I=Sl,
(1)
where non-zero elements of the R-matrix are
RT_{S_]=RI{=a,
R-l-',=R_'t_l=b,
R - I _ I - - R - 1 - 1 , =C,
(2)
a, s = + 1 ( - 1 ) for arrows directed up (down). Let us assume that ~'/[ ( X l , r l ) . . . . . ( X m , T m ) l ( x , , + ~ , r m + , ) , . . . ,
(X,, r,,)]
(3) is the amplitude o f the row state when the arrows directed downward are located at the edges (x~, z~ ) ..... (xm, r~) of the first sublattice and the edges (xm+ ~, Zm+l), ..., (X,, Z,) of the second sublattice, r~= 1, 2 respectively, depending on whether the ith edge is tilted or vertical. The situation o f the downward directed arrows on each sublattice is defined by (1, 1 ) ~ ( x l , z l ) < ( x z , r:)<...<(Xm, r , , ) ~ ( N , 2) ,
( 1, 1 ) ~ (x,.+ ~, rm+ ~) < (x.,+2, z,.+2) < . . . < (x~, %) ~< (N, 2) .
(4)
The inequality (x~, r,) < (xj, rj) designates that x~
W[(x.~)j(,,~,)](k,, k2) = W(kl, k2),
forx=x', r=r'=2
= 1,
otherwise,
t//Ik) (X) = ( - 1 (~ __b-e-Zh)eikx ,
~k~ (x) = e~' . 2(k) = e x p ( - h - i p ) { r c o s ( p + i h ) + [ 1 + 6 2 cosZ(p+ih) J11/2tJ
6=b/a,
e=c/a,
p=k/2.
(8)
The eigenvalue appropriate to (6) is
A=a2Uexp(Nh) f i 2(kj) .
(9)
/=1
The original m o m e n t of the Bethe ansatz usage is the introduction o f the factors (7). It allows us to consider amplitude (6) in the case when two coordinates coincide. Consider the eigenstate equations on the boundary of the subregions (5). We shall require that the amplitude (6) remains an eigenfunction of T d with the eigenvalue (9) in the case when two downward directed arrows coexist on the same site o f the lattice. This leads to the conditions on the coefficients AJQ'"a°° and allows us to determine the factors W ( k , pl...pn k2), 2
,4 'a'a ...... ----.P,Pi+ 1... =
(5)
where [Q~ .... , Q , ] is a permutation of the numbers 1, 2 ..... n. In each of the subregions (5) the amplitude (3) may be chosen as a generalized coordinatespace Bethe ansatz,
~ c~.13= 1
~ SA,d,+, ( k p,,kp,+,
.... '~ t..... A...t,i+,
(10)
W(kl, k:) = (2t22 e2a+e -2h) (2122 + 1 ) -~ , ~, =,t ( k;), where the nonzero elements of the S-matrix are as follows, 1I 22 Sll(kl, k2) =$22(k~, k2) = - I ,
174
(7)
~ k ) (X) is the amplitude of the one-particle state,
(1, I)<~(XQ,,rQi)<~(XQ2, ZQ=) <~...~< (XQ., rQ.) ~< (N, 2 ) ,
,
Volume 138, number 4,5
k2)--s~l (k,,
Sl~(k,,
PHYSICS LETTERS A
k2)
= ( 2 2 - 2 ~ ) ( 2 2 - 2 ~ +&2) -1 , S1921(kl, k2) = S 2 1 1 ( k l , k 2 ) = - z ~ 1 2 ( 2 2 - 2 1 JvAl2) -1 ,
U=2c2sh(2h)/ab.
A~2 = - - U[ 1 + ( 2 , 2 2 ) - ' ] ,
To prove the conjecture (6) we must test also the eigenstate equations when the number o f arrows directed downward coexistent on the same site is equal to 3 and 4. We have carried out this procedure. The last problem we have to attend to is the fulfillment o f the periodic boundary conditions. As a result, we have A,J,...,~. A.J2...a..~, p I...pn _ -- "~ p2...pnp 1 exp(ik.~ N)
( 11 )
Thus, we have obtained expressions for the eigenvalues of the transfer matrix in terms of the set of distinct numbers k~. To find the k~ we have the system of equations (10), ( 11 ). A necessary and sufficient condition of compatibility o f (10) is the satisfaction o f the Baxter-Yang relations [ 7,9-11 ]. In our case the S-matrix satisfies these relations and we may use the q u a n t u m method o f the inverse problem [2,7]. As a result we have
26 June 1989
a standard way go to a system of integral equations which can be solved in the thermodynamic limit by Fourier transform. Details of these calculations will be published elsewhere. As a result we get the following expression for the free energy of the considered model, oo
t
Jo(x)R(x) dx
- flf=ln a + ½[hl + 2 3 x [ l ~ l
U/)]'
0
if
6 < e -2'h' ,
(13)
where
R(x) =
b-2k-~ c h [ 2 h ( 2 k - 1) ] J2k-~(X), k=l
(14)
J~(x) are the Bessel functions of the first kind. The expression for the case 6 > e -21hl is obtained from ( 13 ) by interchange of [7 and e - 21hl. The linear term of the expansion in powers of 6in eqs. (13) and (14) corresponds to the energy o f the ground state of the one-dimensional Hubbard model.
References
- A t - U/2 ?=~~ - A t + U/2'
exp(2ipjN) = ( - 1 )n-m+l f i 2 J - - 2 7 1
(_ l )Ofl
-2z' -
j = l ~ j - - 2 j -1 - - A l +
f i Ar - A t - U r=l Ar-At+ U"
t-u/2
UI2 (12)
r#l
Eqs. (12) completely describe the eigenstates of the model under consideration. This completes the diagonalization o f the transfer matrix. Thus, we have obtained the eigenvalue spectrum and eigenvectors o f the diagonal-to-diagonal transfer matrix of the model through the solution o f the system o f equations (12). F r o m eqs. (12) one may in
[ 1] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 ( 1968) 1445. [2] P.P. Kulish and E.K. Sklyanin, in: Lecture notes in physics, Vol. 151 (Springer, Berlin, 1982) p. 61. [3] R.Z. Bariev and Yu.V. Kozhinov, Physica A 127 (1984) 316. [4] B.S. Shastry, Phys. Rev. Lett. 56 (1986) 1529, 2453. [ 5 ] M. Wadati, E. Olmedilla and Y. Acutsu, J. Phys. Soc. Japan 56 (1987) 1340, 2298. [6] E. Olmedilla and M. Wadati, Phys. Rev. Lett. 60 (1988) 1595. [7] L.D. Faddeev and L.A. Takhtajan, Usp. Mat. Nauk 34 (1979) 13. [8] B.S. Shastry, J. Stat. Phys. 50 (1988) 57. [9] R.Z. Bariev, Teor. Mat. Fiz. 49 (1981) 261 [Theor. Math. Phys. 49 (1982) 1021]. [ 10 ] A.A. Belavin, Phys. Lett. B 87 (1979) 117. [ 11 ] R.J. Baxter, Stud. Appl. Math. 50 ( 1971 ) 51.
175