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A new completely integrable bi-plane 2D vertex model
Physics Letters A 174 ( 1993) 407-410 North-Holland
PHYSICS LETTERS A
A new completely integrable bi-plane 2D vertex model A.E. Borovick, S.I. Kulinich, V.Yu. P o p k o v a n d Yu.M. S t r z h e m e c h n y B.I. Verkin Institute for Low Temperature Physics and Engineering, Ukrainian Academy o f Sciences, 47 Lenin Av., Kharkov 310164, Ukraine Received 5 March 1992; revised manuscript received 14 October 1992; accepted for publication 3 February 1993 Communicated by A.P. Fordy
In this paper we proposea new class of completelyintegrablebi-planevertex models. The correspondingYang-Baxterequation has been proved to be valid. The Bethe ansatz has also been found, enabling us to completely investigate the relevant phase diagram.
The construction of a rigorous theory of phase transitions is a theoretical and mathematical problem of fundamental importance. Until recently only systems of no more than two dimensions were permitted a complete investigation [ 1 ]. All these systems can be solved with the use of the transfermatrix technique, which requires the validity of the triangle (Yang-Baxter) equation. When investigating 3D systems one has two principally different ways of generalizing the transfermatrix technique. One is to generalize the triangle equation onto three dimensions which consists in finding 3D systems whose local weight functions are solutions to the tetrahedron equation. The only 3D statistical mechanical model that to date has been solved virtually to the end is the Zamolodchikov model [2 ]. Regretfully, this model is far from reality (the weight function of the elementary cell for certain configurations of sites can be negative). The other way is to look for such 3D systems that are sets of interacting 2D systems each of which represents a completely integrable 2D model. The local weight functions of such a 3D system are again solutions of the triangle equation but on a class of matrices with dimensionality 4 K, where K is the number of interacting planes. While in the former case (of the tetrahedron equation) a large but finite number of equations has to be satisfied, the number of equations to hold in the latter is infinite. However, if the triangle equation turns out to have a block nature, ElsevierScience Publishers B.V.
the number of equations reduces to be finite, rendering the problem completely solvable. Within the latter approach, the consideration of "bi-plane" systems seems to be the most natural first step. Important and interesting results in this direction have been obtained by Shastry. A very constructive idea has been elaborated in ref. [ 3 ], viz. two sixvertex planes (sublattices) were coupled in a specific way to form a completely integrable "bi-plane" system. The Bethe ansatz for the Shastry model was obtained by Bariev [ 4 ]. This paper is devoted to a new completely solvable model of two interacting six-vertex planes. The partition function of a bi-plane model is M
Z=
~
N
I-I I-I e x p [ - ( l / k T ) ( ~ o t +~o2+~t2)] .
all staten i-- I j= 1
Here ~i~ (~02) is the energy of site ij in the first (second) plane in one of the possible site configurations; c~2 is the interaction energy between the nearest sites of different planes. Each "bi-site" (see fig. 1 ) in such a model can be assigned with a Boltzmann weight of the form V alot2;plp~ ~ l ~ ; ~ _-- ( V I I~¢qpl , ~ ( V 2) ~
exp( - ~12/kT)
The subscripts and superscripts {a,-, fl,} define the site configuration in plane i ( i = 1, 2; a ( # ) = + 1 ). V ~ is the Boltzmann weight of the site proper within plane i. In our model each of the two interacting planes is a six-vertex one [ 1 ] with 407
Volume 174, number 5,6
PHYSICS LETTERSA
22 March 1993
us to rewrite the local weight function for the "bi-site" in the invariant form
I
t,().) = e x p ( h a 3 r 3 ) L ~ ( X ) L ~ ( 2 ) e x p ( - h r ~ o ~ ) , where 4 m
L'~(')= ~
waa~(r~)a~(r~)
a=l
is the local weight function of the first (second) plane with the trigonometric parameterization for the sixvertex single-plane model: a = - w 4 + w s = s i n O . + q ) ; b=w4-w3=sin(2-q); c=2wl=2W2=sin2rl. The local operators L~ (° satisfy the Yang-Baxter equations for the single-plane six-vertex model (see, e.g., ref. [5] )
1. ¢
Fig. 1. Bi-sitevertex diagram.
R~(*) ( ; t - a ) [L~(~)(2) ®L~(~) (/~) ] ( V , ) ++++ = ( V ' ) .
= a.,
( .V ' )
.~ = ( V ' ) + = b ,
( V ' ) +7. = ( V ' ) ¥ + = c . For the case under consideration the interaction energy between nearest sites of different planes is
= [L~(~)(lt)®L~(')(2)] R " ( ° ( ) t - l . t )
(3)
(® implies the tensor product of the auxiliary spaces). In eq. (3) 4
R°(')(,~)=eg,~ ) ~ ~ a ( 2 ) a ~ ( t g ) ® a ~ , ( z ~ , ) , ~t2 =hkT(flla'2 -Ctlfl'2) .
(1)
We must emphasize here that the energy ct2 from eq. ( l ) differs from that of Shastry's model where e~z =hkT(fllfl2 +ff~fl'2). Besides, Shastry's model is a bi-plane model for which in both intei'acting planes the "free fermion condition", viz. d - ( a2 + b Z - c 2 ) /2ab=O ,
(2)
must be satisfied (our model does not require this a priori ). Let us assign to the set of indexes {Otl; a2} the quantum space C'w which is a direct product of the two quantum spaces C 2u ( i = l, 2) associated with each of the interacting planes. Operators acting in the space C ¢' are a',~=l®
... ®a'~® n-- I times
z~=l®
... N+n-- 1 times
... ® I , 2N--n times
®a'~® ... ® I N--n times
(a '~ are the Pauli operators). The set of indexes {ill; //2} generates an auxiliary space C2®C 2 where the Pauli operators a~ and r~ act. This space embedding of our model is equivalent to Shastry's [ 3]. It allows 408
0~1
• ,,(,~) = w,,(2 + q ) , is a numerical matrix and P~7) is the permutation operator (associated with the a (first) or r (second) plane) of spaces i and i'. Now the partition function Z of the bi-plane system can be rewritten in the following form for the periodic boundary conditions, Z = S p I-I T , double rows
T=Tr
I-I L, double sites
(Sp and Tr denote the trace in quantum and auxiliary spaces, respectively). Two lemmas are valid: L e m m a I. R"(') ( ; t - U) [f~(') (;~) ® f~(*) (u) l = [E~o(/~)®E~(*)(2)lRO(*)(~-/~) where ~a L~().) = e x p ( h o ' o3 r ~3 ) L .a ( 2 ) ,
L~ T, ( 2 ) - L- - , ( 2T)
exp( - h z o sa , )3 ,
i.e. L,(2) =/:~(2)/S~(2).
,
Volume 174, number 5,6
PHYSICS LETTERS A
Lemma 2.
22 March 1993
Yt (2) = e h sin(;t+r/) , ~l (2) = e - h sin(2-~/) ,
exp(htr~'r~)L°~(2) L,~(/z)
?2(2) = e -h s i n ( 2 + r / ) , =/.~,~(#) L~°(2) exp(hoo~ro~) . Taking the direct product of the Yang-Baxter equations (3) for the local operators L ° and L ~ one has to insert e x p ( h a ] r ] ) and exp(-hT3oo~) in the proper places. Further it is necessary to " m o v e through" (using lemmas 1 and 2) the relevant exponent combinations, making sure that the following equation is valid, R(2, # ) [ L . ( 2 ) ® L . ( / z )
] = [t.(#)®t.(2)
]R(2,/~), (4)
with the numerical 16× 16 matrix R a~
R = P'~.o,P ~.o, e x p ( h tr~r ~ ) ( ~,~t ~'~,tr~ tr~,) X(#~=, ~ , ~ r , r , ) e x p ( - h t 3 o a ~ , ) .
m
sin(p-q) K(p; q ) = s i n ( p - q + 2 ~ / ) " In eqs. (5), 2j,/tj can be found from the Bethe ansatz equations (sin(2¢ sin ( 2 j -+ r/) r/)) ~ exp[2h(N-2n2)]= \sin(/tj-r/)/(sin(#J+F/)~ e x p [ - 2 h ( N - 2 n . )
]= ~l ~
Thus, we have proved the validity of the Yang-Baxter equation and obtained the transfer-matrix eigenvalues for the general case zt # 0. However, a free energy calculation can be performed analytically only for the "free fermion" (eq. ( 2 ) ) case. With zl=0 we can present the nonequilibrium free energy lim [ ( 1 / 2 N ) lnlAI] 2N~oo
in the following form,
kT -'¢' in(COS h ~v~-+ c o s k ) f = - 4--nnt !
\cosh ~j- - c o s
dk
xo
1
+ ~
Gi = y ~ ( 2 ) e x p ( - 2 h n 2 ) k=lI-IX(2k; 2)
2j)) , t~-~-' l x(2t; x(2j; 2t
N
f=-kT
Within the framework of the quantum inverse scattering problem [ 5 ], making use of eq. (4), we obtain an expression for the transfer-matrix eigenvalues A provided the first and second planes support the nt and n2 "excitations", respectively, A = G~ G2. Here
~2(;t) = e h s i n ( ; t - r / ) ,
ln(COsh ~i- +cos k'~ \c-ff~sh~u~- - c o s k] dk
0
+d~l~(3.) exp(2hn2)
l ¢4 11 x(;t; 2~) '
(5a)
k~l
,n I G2=?~'(;t) exp(2hn~) I-I k=~ x(#k; 2) '~ 1 +82~(2) e x p ( - 2 h n ~ ) I-I k=~ X(2;#~) '
+n(I ~i~ I + I ~- I - 2 A ) ] ,
(6)
where ~,= lim (5b)
(nJN), A=ln(b/a),
N~oo
~F = A _ + 2 h ( 1 - 2 ~ )
~2
with ~ With the help of lemmas 1 and 2 it can be shown that our model with free boundaries is gauge equivalent to a pair of six-vertex planes with an essentially nonlocal interaction between the plane boundaries only. Due to the nonlocal nature of these interactions their contribution will be comparable with the energy of the whole system. Therefore such a realization of the starling settings is by far less reasonable in the sense of applications. We are grateful to the referee for noting this property of our model.
Minimizing eq. (6) over ~ and ~2 we obtain a set of transcendent equations which could be investigated to a satisfactory degree of completeness only using a computer. As a result, we can obtain the phase diagram with the following phases (see fig. 2): (a) Disordered phase: each plane is in the paras2 One can see that eq. (6) is invariant with respect to the substitutions: a-~-ct, ~"-'~2; ct--,-or, h-~-h. 409
Volume 174, number 5,6
PHYSICS LETTERS A
12 0.5
22 March 1993
(c) Intermediate phase: one o f the planes is in the ordered state, while the other is in a state which is that o f a paraelectric in an external field. Phase ( a ) is the only one existing in the case o f noninteracting planes. The other two planes a p p e a r due to the interaction and have no analogs in the sixvertex model. We report here only the R-matrix, the Bethe ansatz, a n d the phase diagram. A m o r e detailed study o f the new bi-plane models as well as a generalization to the case o f K interacting planes, on the road to truly 3D models, will be published elsewhere. We are indebted to E.K. Sklyanin, M.A. Strzhemechny, V.M. Tsukernik and A.A. Zvyagin for helpful discussions.
Fig. 2. Schematic phase diagram of the model. electric state (all types o f vertexes are realized in both planes) ~3. ( b ) Two-sublattice ferrielectric phase: although both planes are in the ferroelectric state, the polarization vectors o f different planes are mutually perpendicular. w3 We follow the Baxter terminology [ 1] for the definition of the states.
410
References [ 1] R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press, New York, 1982 ). [2] A.B. Zamolodchikov, Commun. Math. Phys. 79 ( 1981 ) 489; R.J. Baxter and G.R.W. Quispel, J. Stat. Phys. 58 (1990) 411. [3] B.S. Shastry, Phys. Rev. Lett. 56 (1986) 1529, 2453; J. Stat. Phys. 50 (1988) 57. [4] R.Z. Barley, Teor. Mat. Fiz. 82 (1990) 311. [5] L.D. Faddeev, Soy. Sci. Rev. Math. Phys. C I (1981) 107.
PHYSICS LETTERS A
A new completely integrable bi-plane 2D vertex model A.E. Borovick, S.I. Kulinich, V.Yu. P o p k o v a n d Yu.M. S t r z h e m e c h n y B.I. Verkin Institute for Low Temperature Physics and Engineering, Ukrainian Academy o f Sciences, 47 Lenin Av., Kharkov 310164, Ukraine Received 5 March 1992; revised manuscript received 14 October 1992; accepted for publication 3 February 1993 Communicated by A.P. Fordy
In this paper we proposea new class of completelyintegrablebi-planevertex models. The correspondingYang-Baxterequation has been proved to be valid. The Bethe ansatz has also been found, enabling us to completely investigate the relevant phase diagram.
The construction of a rigorous theory of phase transitions is a theoretical and mathematical problem of fundamental importance. Until recently only systems of no more than two dimensions were permitted a complete investigation [ 1 ]. All these systems can be solved with the use of the transfermatrix technique, which requires the validity of the triangle (Yang-Baxter) equation. When investigating 3D systems one has two principally different ways of generalizing the transfermatrix technique. One is to generalize the triangle equation onto three dimensions which consists in finding 3D systems whose local weight functions are solutions to the tetrahedron equation. The only 3D statistical mechanical model that to date has been solved virtually to the end is the Zamolodchikov model [2 ]. Regretfully, this model is far from reality (the weight function of the elementary cell for certain configurations of sites can be negative). The other way is to look for such 3D systems that are sets of interacting 2D systems each of which represents a completely integrable 2D model. The local weight functions of such a 3D system are again solutions of the triangle equation but on a class of matrices with dimensionality 4 K, where K is the number of interacting planes. While in the former case (of the tetrahedron equation) a large but finite number of equations has to be satisfied, the number of equations to hold in the latter is infinite. However, if the triangle equation turns out to have a block nature, ElsevierScience Publishers B.V.
the number of equations reduces to be finite, rendering the problem completely solvable. Within the latter approach, the consideration of "bi-plane" systems seems to be the most natural first step. Important and interesting results in this direction have been obtained by Shastry. A very constructive idea has been elaborated in ref. [ 3 ], viz. two sixvertex planes (sublattices) were coupled in a specific way to form a completely integrable "bi-plane" system. The Bethe ansatz for the Shastry model was obtained by Bariev [ 4 ]. This paper is devoted to a new completely solvable model of two interacting six-vertex planes. The partition function of a bi-plane model is M
Z=
~
N
I-I I-I e x p [ - ( l / k T ) ( ~ o t +~o2+~t2)] .
all staten i-- I j= 1
Here ~i~ (~02) is the energy of site ij in the first (second) plane in one of the possible site configurations; c~2 is the interaction energy between the nearest sites of different planes. Each "bi-site" (see fig. 1 ) in such a model can be assigned with a Boltzmann weight of the form V alot2;plp~ ~ l ~ ; ~ _-- ( V I I~¢qpl , ~ ( V 2) ~
exp( - ~12/kT)
The subscripts and superscripts {a,-, fl,} define the site configuration in plane i ( i = 1, 2; a ( # ) = + 1 ). V ~ is the Boltzmann weight of the site proper within plane i. In our model each of the two interacting planes is a six-vertex one [ 1 ] with 407
Volume 174, number 5,6
PHYSICS LETTERSA
22 March 1993
us to rewrite the local weight function for the "bi-site" in the invariant form
I
t,().) = e x p ( h a 3 r 3 ) L ~ ( X ) L ~ ( 2 ) e x p ( - h r ~ o ~ ) , where 4 m
L'~(')= ~
waa~(r~)a~(r~)
a=l
is the local weight function of the first (second) plane with the trigonometric parameterization for the sixvertex single-plane model: a = - w 4 + w s = s i n O . + q ) ; b=w4-w3=sin(2-q); c=2wl=2W2=sin2rl. The local operators L~ (° satisfy the Yang-Baxter equations for the single-plane six-vertex model (see, e.g., ref. [5] )
1. ¢
Fig. 1. Bi-sitevertex diagram.
R~(*) ( ; t - a ) [L~(~)(2) ®L~(~) (/~) ] ( V , ) ++++ = ( V ' ) .
= a.,
( .V ' )
.~ = ( V ' ) + = b ,
( V ' ) +7. = ( V ' ) ¥ + = c . For the case under consideration the interaction energy between nearest sites of different planes is
= [L~(~)(lt)®L~(')(2)] R " ( ° ( ) t - l . t )
(3)
(® implies the tensor product of the auxiliary spaces). In eq. (3) 4
R°(')(,~)=eg,~ ) ~ ~ a ( 2 ) a ~ ( t g ) ® a ~ , ( z ~ , ) , ~t2 =hkT(flla'2 -Ctlfl'2) .
(1)
We must emphasize here that the energy ct2 from eq. ( l ) differs from that of Shastry's model where e~z =hkT(fllfl2 +ff~fl'2). Besides, Shastry's model is a bi-plane model for which in both intei'acting planes the "free fermion condition", viz. d - ( a2 + b Z - c 2 ) /2ab=O ,
(2)
must be satisfied (our model does not require this a priori ). Let us assign to the set of indexes {Otl; a2} the quantum space C'w which is a direct product of the two quantum spaces C 2u ( i = l, 2) associated with each of the interacting planes. Operators acting in the space C ¢' are a',~=l®
... ®a'~® n-- I times
z~=l®
... N+n-- 1 times
... ® I , 2N--n times
®a'~® ... ® I N--n times
(a '~ are the Pauli operators). The set of indexes {ill; //2} generates an auxiliary space C2®C 2 where the Pauli operators a~ and r~ act. This space embedding of our model is equivalent to Shastry's [ 3]. It allows 408
0~1
• ,,(,~) = w,,(2 + q ) , is a numerical matrix and P~7) is the permutation operator (associated with the a (first) or r (second) plane) of spaces i and i'. Now the partition function Z of the bi-plane system can be rewritten in the following form for the periodic boundary conditions, Z = S p I-I T , double rows
T=Tr
I-I L, double sites
(Sp and Tr denote the trace in quantum and auxiliary spaces, respectively). Two lemmas are valid: L e m m a I. R"(') ( ; t - U) [f~(') (;~) ® f~(*) (u) l = [E~o(/~)®E~(*)(2)lRO(*)(~-/~) where ~a L~().) = e x p ( h o ' o3 r ~3 ) L .a ( 2 ) ,
L~ T, ( 2 ) - L- - , ( 2T)
exp( - h z o sa , )3 ,
i.e. L,(2) =/:~(2)/S~(2).
,
Volume 174, number 5,6
PHYSICS LETTERS A
Lemma 2.
22 March 1993
Yt (2) = e h sin(;t+r/) , ~l (2) = e - h sin(2-~/) ,
exp(htr~'r~)L°~(2) L,~(/z)
?2(2) = e -h s i n ( 2 + r / ) , =/.~,~(#) L~°(2) exp(hoo~ro~) . Taking the direct product of the Yang-Baxter equations (3) for the local operators L ° and L ~ one has to insert e x p ( h a ] r ] ) and exp(-hT3oo~) in the proper places. Further it is necessary to " m o v e through" (using lemmas 1 and 2) the relevant exponent combinations, making sure that the following equation is valid, R(2, # ) [ L . ( 2 ) ® L . ( / z )
] = [t.(#)®t.(2)
]R(2,/~), (4)
with the numerical 16× 16 matrix R a~
R = P'~.o,P ~.o, e x p ( h tr~r ~ ) ( ~,~t ~'~,tr~ tr~,) X(#~=, ~ , ~ r , r , ) e x p ( - h t 3 o a ~ , ) .
m
sin(p-q) K(p; q ) = s i n ( p - q + 2 ~ / ) " In eqs. (5), 2j,/tj can be found from the Bethe ansatz equations (sin(2¢ sin ( 2 j -+ r/) r/)) ~ exp[2h(N-2n2)]= \sin(/tj-r/)/(sin(#J+F/)~ e x p [ - 2 h ( N - 2 n . )
]= ~l ~
Thus, we have proved the validity of the Yang-Baxter equation and obtained the transfer-matrix eigenvalues for the general case zt # 0. However, a free energy calculation can be performed analytically only for the "free fermion" (eq. ( 2 ) ) case. With zl=0 we can present the nonequilibrium free energy lim [ ( 1 / 2 N ) lnlAI] 2N~oo
in the following form,
kT -'¢' in(COS h ~v~-+ c o s k ) f = - 4--nnt !
\cosh ~j- - c o s
dk
xo
1
+ ~
Gi = y ~ ( 2 ) e x p ( - 2 h n 2 ) k=lI-IX(2k; 2)
2j)) , t~-~-' l x(2t; x(2j; 2t
N
f=-kT
Within the framework of the quantum inverse scattering problem [ 5 ], making use of eq. (4), we obtain an expression for the transfer-matrix eigenvalues A provided the first and second planes support the nt and n2 "excitations", respectively, A = G~ G2. Here
~2(;t) = e h s i n ( ; t - r / ) ,
ln(COsh ~i- +cos k'~ \c-ff~sh~u~- - c o s k] dk
0
+d~l~(3.) exp(2hn2)
l ¢4 11 x(;t; 2~) '
(5a)
k~l
,n I G2=?~'(;t) exp(2hn~) I-I k=~ x(#k; 2) '~ 1 +82~(2) e x p ( - 2 h n ~ ) I-I k=~ X(2;#~) '
+n(I ~i~ I + I ~- I - 2 A ) ] ,
(6)
where ~,= lim (5b)
(nJN), A=ln(b/a),
N~oo
~F = A _ + 2 h ( 1 - 2 ~ )
~2
with ~ With the help of lemmas 1 and 2 it can be shown that our model with free boundaries is gauge equivalent to a pair of six-vertex planes with an essentially nonlocal interaction between the plane boundaries only. Due to the nonlocal nature of these interactions their contribution will be comparable with the energy of the whole system. Therefore such a realization of the starling settings is by far less reasonable in the sense of applications. We are grateful to the referee for noting this property of our model.
Minimizing eq. (6) over ~ and ~2 we obtain a set of transcendent equations which could be investigated to a satisfactory degree of completeness only using a computer. As a result, we can obtain the phase diagram with the following phases (see fig. 2): (a) Disordered phase: each plane is in the paras2 One can see that eq. (6) is invariant with respect to the substitutions: a-~-ct, ~"-'~2; ct--,-or, h-~-h. 409
Volume 174, number 5,6
PHYSICS LETTERS A
12 0.5
22 March 1993
(c) Intermediate phase: one o f the planes is in the ordered state, while the other is in a state which is that o f a paraelectric in an external field. Phase ( a ) is the only one existing in the case o f noninteracting planes. The other two planes a p p e a r due to the interaction and have no analogs in the sixvertex model. We report here only the R-matrix, the Bethe ansatz, a n d the phase diagram. A m o r e detailed study o f the new bi-plane models as well as a generalization to the case o f K interacting planes, on the road to truly 3D models, will be published elsewhere. We are indebted to E.K. Sklyanin, M.A. Strzhemechny, V.M. Tsukernik and A.A. Zvyagin for helpful discussions.
Fig. 2. Schematic phase diagram of the model. electric state (all types o f vertexes are realized in both planes) ~3. ( b ) Two-sublattice ferrielectric phase: although both planes are in the ferroelectric state, the polarization vectors o f different planes are mutually perpendicular. w3 We follow the Baxter terminology [ 1] for the definition of the states.
410
References [ 1] R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press, New York, 1982 ). [2] A.B. Zamolodchikov, Commun. Math. Phys. 79 ( 1981 ) 489; R.J. Baxter and G.R.W. Quispel, J. Stat. Phys. 58 (1990) 411. [3] B.S. Shastry, Phys. Rev. Lett. 56 (1986) 1529, 2453; J. Stat. Phys. 50 (1988) 57. [4] R.Z. Barley, Teor. Mat. Fiz. 82 (1990) 311. [5] L.D. Faddeev, Soy. Sci. Rev. Math. Phys. C I (1981) 107.