- Email: [email protected]
New results for exactly solvable critical RSOS models and vertex models in two dimensions
Physica A 194 (1993) North-Holland
397-405
New results for exactly solvable critical RSOS models and vertex models in two dimensions* Andreas
Kliimpera3’
and Paul A. Pearceb’2
“Institut fiir Theoretische Physik, Universitiit zu Kdn, Ziilpicher Str. 77, W-5ooO K6ln 41, Germany bMathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia
A method for analytically calculating critical properties of two-dimensional exactly solvable models is presented. It is based on the calculation of finite-size corrections of the critical system and on predictions by conformal field theory. It is applied to the restricted solid-onsolid models found by Andrews, Baxter and Forrester. The analysis of the eigenvalues of the finite-size transfer matrices is performed using special functional equations of inversion identity type. The results are given in terms of Rogers dilogarithms. An outlook is given on related problems, e.g. the application to vertex models and the calculation of free energy and correlation lengths of integrable quantum chains at finite temperature.
The critical exponents of statistical mechanical models are related by scaling hypotheses to scaling dimensions x of algebraically decaying two-point functions at criticality,
It is generally assumed that the critical field theory is conformally invariant. In two dimensions this imposes strong constraints on the allowed values of the scaling dimensions x [1,2]. Unitary models with central charge c < 1 belong to a discrete series with exponents given by the Kac tables. For c 2 1 conformal field theory alone does not impose any such restriction on the scaling dimensions x. In this case classifications are only possible provided other symmetries exist, e.g. supersymmetry. In any case, for critical lattice models the underlying field theory can be identified directly by using general scaling predictions for finite-size corrections
* Work supported by the Australian Koln-Aachen-Julich. I Email: [email protected]. ’ Email: [email protected].
Research
037%4371/93/$06.00
Science
@ 1993 - Elsevier
Council
Publishers
and the Sonderforschungsbereich
B.V. All rights
reserved
341,
398
A. Kliimper,
to eigenvalues
of quantum
the thermodynamics. with periodic like E,=
E
chains
The energy
boundary
Ne,, -
P.A. Pearce
conditions
c+
i RSOS and vertex models in 20
or from the low-temperature levels of a quantum are expected
E,
/’
behaviour
ux +
of
on a finite chain
to scale with the system
- =$ E,,
model
size N
I’
where E,, is the ground state energy, E, are the energies of low-lying excited states and u is the sound velocity of the elementary energy-momentum excitations. As a result the central charge c and the scaling dimensions x can be obtained from the amplitudes of the l/N correction terms. These relations have been obtained in ref. [3] and from the logarithmic map of the infinite plane to the semi-infinite torus [4]. There are many applications of (2) where in general the finite-size data have been obtained numerically or in the case of integrable models analytically. Instead of considering a quantum model on a finite geometry at zero temperature one may as well study the free energy density of a quantum system on an infinite chain at low but finite temperature. The prediction of conformal field theory is [5] f(~)
=f(O) - g
T’+
c(T*).
This is a far less popular method for calculating the central charge as a numerical approach to the low-temperature free energy is much more involved than the calculation of a simple ground state energy. However, for integrable models the thermodynamic Bethe ansatz [6,7] provides the means for achieving this aim analytically. Unfortunately, the more interesting quantities x are related to the (exponential) correlation time correlators at finite temperature, 1
lengths
5, of the corresponding
equal
T
;=2=yxy for which even the standard thermodynamic Bethe ansatz does not provide the computational framework. See, however, recent progress in ref. [8] using ideas of ref. [9]. The standard analytic approach to calculating finite-size corrections for exactly solvable models is based on the Bethe ansatz [lO,ll]. It was successfully applied in cases where the ground state distribution of Bethe ansatz numbers did not consist of complex strings, e.g. the six-vertex and the Hubbard model.
A. Kliimper,
P. A. Pearce
I RSOS and vertex models in 20
399
The calculations are quite lengthy and the thorough justification of the method is involved [12]. It is known to fail for instance in the case of higher spin-S XXZ chains. This limitation was the motivation for the search of an alternative method. Another motivation being simplicity, as elegant methods exist for calculating partition functions of lattice models by utilizing special functional equations [13,14]. Generalizations of the inversion identity [15,16] are known for determining the relevant part of the spectrum of the transfer matrix in the thermodynamic limit [17-211 from which also correlation lengths can be obtained. At the heart of this method are functional equations of the type T(U) T(u + A) = 4(u) Z + correction ,
(5)
for the transfer matrix T(U) of the model satisfying the Yang-Baxter equation, u being the spectral variable and A the crossing parameter. Inversion relations like (5) are satisfied for many models. The importance of (5) is that the product of any eigenvalue function at two different arguments yields a known function $J plus correction terms which can be neglected in the thermodynamic limit. It is straightforward to solve (5) for the largest eigenvalue by taking the logarithm, assuming suitable analyticity and using Fourier transforms. The next-largest eigenvalues can be determined by taking into account their zeros which play the role of rapidities. The application of (5) to finite systems usually fails since the correction term is no longer negligible and unfortunately not known in contrast to the first term on the right-hand side. It was pointed out in ref. [16] that the correction term possesses non-trivial analytic properties. Yet it is difficult to utilize these properties. An exceptional case is the Ising model where the correction term is known and the analysis is straightforward. The simplest non-trivial situation arises for the hard square and hard hexagon models for which the correction term is a matrix of the same type as the transfer matrix T, however at the argument u - 2h. This situation was studied in refs. [22,23] where the initial ideas for calculating finite-size corrections were formulated. In this contribution we want to present a procedure and results in a more general framework [24]. We study the RSOS hierarchy of Andrews, Baxter and Forrester [25]. On each site of a square lattice a height variable is placed which may take the values 1,2, . . . , r - 1, with integer r 2 4. For nearest neighbours we have the constraint a - b = ?l where a and b are adjacent spins. At criticality they are Temperley-Lieb interaction models related to the Dynkin diagram of the classical Lie algebra A. The weights of the allowed faces are given by
400
A. Kliimper,
P. A. Pearce
I RSOS and vertex models in 20
(6)
h(u)
where u is the spectral h(u) are defined by
parameter.
The crossing
parameter
A and the function
(7) For these
models
the functional
equation
(5) can be written
T(u) T(u + A) = f(u - A) f(u + A) [I + t’(u)]
in the form
,
(8)
with f(u) = h(u)N. The correction term t’(u) is a matrix whose eigenvalues are not easily computable in general (an exception being the Ising case Y = 4). However, it can be shown that it is the first member of a set of Y- 3 matrices t”(u) which satisfy the closed set of functional equations .t”(u) tY(u + A) = [I + tY-‘(u where
+ A)][1 + t’+‘(U)]
t” = t’-’ = 0. For the derivation
,
(9)
of (8) and (9) the fusion
procedure
can
be used. This yields a hierarchy of matrices T’(U) which are the transfer matrices of models with Boltzmann weights consisting of a column of 4 elementary faces. From the fusion equations [26,27] new functional equations of inversion identity type can be derived [24],
(8) and (9) follow from this after noting
tq(u)
=
TY+‘(W4-‘(~+ A) f(u - 4 f(u + 4
T(u) = T’(u)
and introducing
t”(u)
by
(11)
The scope of the fusion procedure is actually much wider than presented here. It provides the means of constructing new models with different critical properties starting with an elementary model satisfying the Yang-Baxter equation, in our case the ABF model. Of physical interest are p X p face
A. Kliimper, P. A. Pearce I RSOS and vertex models in 20
401
weights which consist of a square array of elementary faces. The above reasoning can be applied directly to this more general setting for which details and results can be found in ref. [24]. Here we present the analysis of (8) and (9) only in the critical regime III/IV of the model. The general idea is to rewrite the functional equations for the eigenvalue functions T(u) and tq(u) in a way which is more suitable for analytic manipulations. Taking logarithms and introducing Fourier transforms we can solve (8) and (9) for the eigenvalues T(u) and t’(u) in terms of functions 1 + t”(u). Using the definitions aY(x):=tY
(’++-A1-q )
W(x) = 1 + CP(x) )
)
we obtain the non-linear
(12)
integral equations
lnuq=ln~q+~*lnY1q-‘+k*lnMq+‘+Dq,
(13)
where the kernel k is given by k(x):=
’
(14)
21r cash(x) ’
and f* g denotes the convolution
(f*g>(d:=
f
Ax-Y)
of functions f and g,
dY)dY.
(15)
-a
The functions e’(x) are given by
q#l, q = 1 ,
(16)
and Dq are constants which are zero for the largest eigenvalue, in which case the integration paths in (13) are straight lines along the real axis. For next-leading eigenvalues the integration paths surround complex zeros of the functions aq(x), aq(x), and the constants 0’ are non-zero. Solving (8) one obtains two contributions to T(u). The bulk behaviour arises from the product of the functions f, the finite-size corrections from 1 + t’(u). Introducing b(x) := *rini_(; x + ; A) ,
(17)
402
A. Kliimper, P. A. Pearce
I RSOS and vertex models in 2 D
we find lnb=k*ln!‘I’+C,
(18)
where C is a constant which is zero for the largest eigenvalue. The analytic solution of (13) seems hopeless as it is a non-linear integral equation. It simplifies somewhat in the large N limit by using the scaling properties of the functions try(x). As the function c’(x) interpolates between 0 at x = 0 and 1 for x-+ ~0 the transition being at x 2: In N, the scaling form we use is a”(x) : = lil& aq(x + In N) ,
A’(X) := 1 -I-a”(x) ,
in the scaling limit and takes the form lnaq=lneq+k*lnAq~‘+k*lnAq”.
(21)
From now on only the largest eigenvalue is considered, i.e. Dy = 0, C= 0. From (18) we see that the leading finite-size corrections are indeed of order 1 IN, 3
In b(x) = 5
r
cash(x) 1 e-’ In A’(y)
dy + P (h)
.
(22)
-1
The amplitude of the correction term is given by an integral of the function In A’. Amazingly just this integral can be computed without solving (21) explicitly! To see this consider the expression r
In u’)’ In A4 - In a4 (In Aq)‘] dx ,
(23)
where the functions In uq can be expressed through (21) leading to
4
I
--z
eeX In A’(X) dx ,
(24)
A. Kliimper,
P. A. Pearce
I RSOS and vertex models in 2 D
403
and all contributions of the kernel k cancel due to symmetry. In fact, the integral appearing in (22) is just (24) and is identical to (23), which can be after changing the variable of integration uq(=)
q=l
1 (ln(l aY(-z)
+ a”) aq
-
$$)
da’ .
The explicit solution of the non-linear integral equation (21) is fortunately needed! Inserting into (22) we obtain cash(x) r-3 In 6(x) = 7 c ]L+(aY(m)) q=l
(25)
not
L+(aY(-91 7
where we have introduced the dilogarithmic function L+(a) which is related to the standard Rogers dilogarithm L(a),
L+(a)=;j(ln(lx+x) - +&-L(f--).
(27)
0
Eq. (26) can be regarded as the final result as the 1 lN corrections are given by definite integrals whose terminals can be obtained explicitly from the functional equations (9) which asymptotically turn into algebraic equations. Furthermore, the evaluation of the required combination of dilogarithms is achieved by using the identity
yielding eventually In 6(x) = &
(I-
A)
After taking the Hamiltonian central charge as c=I----..-
6 ?(T - 1) .
cash(x) .
(29)
limit and comparing with (2) one identifies the
(30)
Therefore the critical properties of the ABF models in regime III/IV are described by the minimal unitary series of conformal field theory. The abovementioned fusion models correspond to other discrete series [24] notably the
404
A. Kliimper,
P.A. Pearce
I RSOS and vertex models in 20
superconformal series for the 2 X 2 fusion. In all these cases a similar analysis to that presented above is applicable and also works for the next-leading eigenvalues. The main modification to this end is a deformation of the integration paths leading to expressions involving certain analytic continuations of the dilogarithms. Using identities generalizing (28) the conformal weights of higher Kac tables are found. In regime I/II the models show different critical behaviour which is described by 2, parafermion theories independent of the fusion level. These methods can in fact also be applied in general to the critical A-D-E models [28]. We next comment on some recent developments relating the presented method to the calculation of thermodynamics for integrable quantum chains taking the above models as example. Following ideas of ref. [9] one can show that the free energy of the quantum chains at finite temperature is given by the largest eigenvalue of the transfer matrix of the related classical RSOS model with row inhomogeneity [8]. This holds in the limit of infinite Trotter number [29]. As a result the thermodynamics are precisely given by eqs. (13) and (18), however with different definitions for the functions c(x) [8]. After some elementary transformations these non-linear integral equations are shown to be identical to the standard equations of the thermodynamic Bethe ansatz [27]. Furthermore, the generalization to next-largest eigenvalues of the quantum transfer matrix is straightforward and a computational framework for correlation lengths at finite temperatures can be set up. This provides a substantial extension of the traditional approach. Thus results for (4) become accessible from lattice calculations. We hope that these remarks show the strong connection of thermodynamics of quantum chains, classical models with inhomogeneity, inversion identities and fusion. We conclude the paper with some speculations on the applicability of inversion identities. First, the presented method should work for other models, e.g. the eight-vertex model. In this case the inversion identity hierarchy will be infinite in general. However, this should not be a serious problem. For the related quantum XYZ chain the thermodynamic equations for the free energy are known since twenty years [7]. In the light of the previous paragraph it should be clear that the eight-vertex model can be treated in a similar way to the ABF models. In any case, one can avoid the treatment of an infinite hierarchy of matrices, which is desirable for instance for numerical purposes [8,30]. It can be achieved by starting from the Bethe ansatz equations. An analysis similar to the above one yields a set of non-linear integral equations for finitely many functions [31,32]. The advantage of this procedure in comparison to refs. [lO,ll] is the exactness of the equations for any finite-system size. All the models mentioned so far are related to SU(2), which is reflected by
A. Kliimper,
the structure fundamental one needs fundamental such a case fact, this is generalized
P. A. Pearce
I RSOS and vertex models in 20
405
of (5). It is related to the tensor product decomposition of two representations of SU(2) which produces a singlet. For SU(N) a generalization. One possibility is a product of N identical representations in order to find a singlet in the decomposition. In a product of N factors on the left-hand side of (5) is expected. In known, e.g., for the models of ref. [33]. Another possibility is a inversion identity hierarchy [34].
References [l] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575; in: Vertex Operators in Mathematics and Physics, J. Lepowsky, S. Mandelstam and I.M. Singer, eds. (Springer, Berlin, 1984). [3] H.W.J. BiBte, J.L. Cardy and M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742. [4] J.L. Cardy, J. Phys. A 17 (1984) L385. [5] I. Affleck, Phys. Rev. Lett. 56 (1986) 746. [6] C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115. [7] M. Takahashi, Prog. Theor. Phys. 46 (1971) 401. [8] A. Kliimper, Ann. Phys. (Leipzig) 1 (1992) 540. 191 T. Koma, Prog. Theor. Phys. 78 (1987) 1213; 81 (1989) 783. [lo] H.J. de Vega and F. Woynarovich, Nucl. Phys. B 251 (1985) 439. [ll] F. Woynarovich and H.-P. Eckle, J. Phys. A 20 (1987) L97. [12] M. Karowski, Nucl. Phys. B 300 [FS22] (1988) 473. [13] Yu.G. Stroganov, Phys. Lett. A 74 (1979) 116. [14] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). (151 N.Yu. Reshetikhin, Lett. Math. Phys. 7 (1983) 205. (161 P.A. Pearce, Phys. Rev. Lett. 58 (1987) 1502. [17] A. Khimper and J. Zittartz, Z. Phys. B 71 (1988) 495. [18] A. Kliimper and J. Zittartz, Z. Phys. B 75 (1989) 371. [19] A. Kliimper, A. Schadschneider and J. Zittartz, Z. Phys. B 76 (1989) 247. [20] A. Khimper, Europhys. Lett. 9 (1989) 815. [21] M.G. Tetel’man, Sov. Phys. JETP 55 (1982) 306. [22] P.A. Pearce and A. Kliimper, Phys. Rev. Lett. 66 (1991) 974. [23] A. Kliimper and P.A. Pearce, J. Stat. Phys. 64 (1991) 13. [24] A. Khimper and P.A. Pearce, Physica A 183 (1992) 304. [25] G.E. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35 (1984) 193. [26] P.P. Kulish, N.Yu. Reshetikhin and E.K. Sklyanin, Lett. Math. Phys. 5 (1981) 393. [27] V.V. Bazhanov and N.Yu. Reshetikhin, Int. J. Mod. Phys. A 4 (1989) 115. [28] P.A. Pearce, Int. J. Mod. Phys. A 7, Suppl. 1B (1992) 791. [29] M. Suzuki and M. Inoue, Prog. Theor. Phys. 78 (1987) 787. [30] A. Kliimper, in preparation. [31] A. Khimper and M.T. Batchelor, J. Phys. A 23 (1990) L189. [32] A. Kliimper, M.T. Batchelor and P.A. Pearce, J. Phys. A 24 (1991) 3111. [33] B. Sutherland, J. Math. Phys. 11 (1970) 3183. [34] A. Kuniba, Thermodynamics of the y(Xl”) Bethe ansatz system with q a root of unity, ANU preprint (1991).
397-405
New results for exactly solvable critical RSOS models and vertex models in two dimensions* Andreas
Kliimpera3’
and Paul A. Pearceb’2
“Institut fiir Theoretische Physik, Universitiit zu Kdn, Ziilpicher Str. 77, W-5ooO K6ln 41, Germany bMathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia
A method for analytically calculating critical properties of two-dimensional exactly solvable models is presented. It is based on the calculation of finite-size corrections of the critical system and on predictions by conformal field theory. It is applied to the restricted solid-onsolid models found by Andrews, Baxter and Forrester. The analysis of the eigenvalues of the finite-size transfer matrices is performed using special functional equations of inversion identity type. The results are given in terms of Rogers dilogarithms. An outlook is given on related problems, e.g. the application to vertex models and the calculation of free energy and correlation lengths of integrable quantum chains at finite temperature.
The critical exponents of statistical mechanical models are related by scaling hypotheses to scaling dimensions x of algebraically decaying two-point functions at criticality,
It is generally assumed that the critical field theory is conformally invariant. In two dimensions this imposes strong constraints on the allowed values of the scaling dimensions x [1,2]. Unitary models with central charge c < 1 belong to a discrete series with exponents given by the Kac tables. For c 2 1 conformal field theory alone does not impose any such restriction on the scaling dimensions x. In this case classifications are only possible provided other symmetries exist, e.g. supersymmetry. In any case, for critical lattice models the underlying field theory can be identified directly by using general scaling predictions for finite-size corrections
* Work supported by the Australian Koln-Aachen-Julich. I Email: [email protected]. ’ Email: [email protected].
Research
037%4371/93/$06.00
Science
@ 1993 - Elsevier
Council
Publishers
and the Sonderforschungsbereich
B.V. All rights
reserved
341,
398
A. Kliimper,
to eigenvalues
of quantum
the thermodynamics. with periodic like E,=
E
chains
The energy
boundary
Ne,, -
P.A. Pearce
conditions
c+
i RSOS and vertex models in 20
or from the low-temperature levels of a quantum are expected
E,
/’
behaviour
ux +
of
on a finite chain
to scale with the system
- =$ E,,
model
size N
I’
where E,, is the ground state energy, E, are the energies of low-lying excited states and u is the sound velocity of the elementary energy-momentum excitations. As a result the central charge c and the scaling dimensions x can be obtained from the amplitudes of the l/N correction terms. These relations have been obtained in ref. [3] and from the logarithmic map of the infinite plane to the semi-infinite torus [4]. There are many applications of (2) where in general the finite-size data have been obtained numerically or in the case of integrable models analytically. Instead of considering a quantum model on a finite geometry at zero temperature one may as well study the free energy density of a quantum system on an infinite chain at low but finite temperature. The prediction of conformal field theory is [5] f(~)
=f(O) - g
T’+
c(T*).
This is a far less popular method for calculating the central charge as a numerical approach to the low-temperature free energy is much more involved than the calculation of a simple ground state energy. However, for integrable models the thermodynamic Bethe ansatz [6,7] provides the means for achieving this aim analytically. Unfortunately, the more interesting quantities x are related to the (exponential) correlation time correlators at finite temperature, 1
lengths
5, of the corresponding
equal
T
;=2=yxy for which even the standard thermodynamic Bethe ansatz does not provide the computational framework. See, however, recent progress in ref. [8] using ideas of ref. [9]. The standard analytic approach to calculating finite-size corrections for exactly solvable models is based on the Bethe ansatz [lO,ll]. It was successfully applied in cases where the ground state distribution of Bethe ansatz numbers did not consist of complex strings, e.g. the six-vertex and the Hubbard model.
A. Kliimper,
P. A. Pearce
I RSOS and vertex models in 20
399
The calculations are quite lengthy and the thorough justification of the method is involved [12]. It is known to fail for instance in the case of higher spin-S XXZ chains. This limitation was the motivation for the search of an alternative method. Another motivation being simplicity, as elegant methods exist for calculating partition functions of lattice models by utilizing special functional equations [13,14]. Generalizations of the inversion identity [15,16] are known for determining the relevant part of the spectrum of the transfer matrix in the thermodynamic limit [17-211 from which also correlation lengths can be obtained. At the heart of this method are functional equations of the type T(U) T(u + A) = 4(u) Z + correction ,
(5)
for the transfer matrix T(U) of the model satisfying the Yang-Baxter equation, u being the spectral variable and A the crossing parameter. Inversion relations like (5) are satisfied for many models. The importance of (5) is that the product of any eigenvalue function at two different arguments yields a known function $J plus correction terms which can be neglected in the thermodynamic limit. It is straightforward to solve (5) for the largest eigenvalue by taking the logarithm, assuming suitable analyticity and using Fourier transforms. The next-largest eigenvalues can be determined by taking into account their zeros which play the role of rapidities. The application of (5) to finite systems usually fails since the correction term is no longer negligible and unfortunately not known in contrast to the first term on the right-hand side. It was pointed out in ref. [16] that the correction term possesses non-trivial analytic properties. Yet it is difficult to utilize these properties. An exceptional case is the Ising model where the correction term is known and the analysis is straightforward. The simplest non-trivial situation arises for the hard square and hard hexagon models for which the correction term is a matrix of the same type as the transfer matrix T, however at the argument u - 2h. This situation was studied in refs. [22,23] where the initial ideas for calculating finite-size corrections were formulated. In this contribution we want to present a procedure and results in a more general framework [24]. We study the RSOS hierarchy of Andrews, Baxter and Forrester [25]. On each site of a square lattice a height variable is placed which may take the values 1,2, . . . , r - 1, with integer r 2 4. For nearest neighbours we have the constraint a - b = ?l where a and b are adjacent spins. At criticality they are Temperley-Lieb interaction models related to the Dynkin diagram of the classical Lie algebra A. The weights of the allowed faces are given by
400
A. Kliimper,
P. A. Pearce
I RSOS and vertex models in 20
(6)
h(u)
where u is the spectral h(u) are defined by
parameter.
The crossing
parameter
A and the function
(7) For these
models
the functional
equation
(5) can be written
T(u) T(u + A) = f(u - A) f(u + A) [I + t’(u)]
in the form
,
(8)
with f(u) = h(u)N. The correction term t’(u) is a matrix whose eigenvalues are not easily computable in general (an exception being the Ising case Y = 4). However, it can be shown that it is the first member of a set of Y- 3 matrices t”(u) which satisfy the closed set of functional equations .t”(u) tY(u + A) = [I + tY-‘(u where
+ A)][1 + t’+‘(U)]
t” = t’-’ = 0. For the derivation
,
(9)
of (8) and (9) the fusion
procedure
can
be used. This yields a hierarchy of matrices T’(U) which are the transfer matrices of models with Boltzmann weights consisting of a column of 4 elementary faces. From the fusion equations [26,27] new functional equations of inversion identity type can be derived [24],
(8) and (9) follow from this after noting
tq(u)
=
TY+‘(W4-‘(~+ A) f(u - 4 f(u + 4
T(u) = T’(u)
and introducing
t”(u)
by
(11)
The scope of the fusion procedure is actually much wider than presented here. It provides the means of constructing new models with different critical properties starting with an elementary model satisfying the Yang-Baxter equation, in our case the ABF model. Of physical interest are p X p face
A. Kliimper, P. A. Pearce I RSOS and vertex models in 20
401
weights which consist of a square array of elementary faces. The above reasoning can be applied directly to this more general setting for which details and results can be found in ref. [24]. Here we present the analysis of (8) and (9) only in the critical regime III/IV of the model. The general idea is to rewrite the functional equations for the eigenvalue functions T(u) and tq(u) in a way which is more suitable for analytic manipulations. Taking logarithms and introducing Fourier transforms we can solve (8) and (9) for the eigenvalues T(u) and t’(u) in terms of functions 1 + t”(u). Using the definitions aY(x):=tY
(’++-A1-q )
W(x) = 1 + CP(x) )
)
we obtain the non-linear
(12)
integral equations
lnuq=ln~q+~*lnY1q-‘+k*lnMq+‘+Dq,
(13)
where the kernel k is given by k(x):=
’
(14)
21r cash(x) ’
and f* g denotes the convolution
(f*g>(d:=
f
Ax-Y)
of functions f and g,
dY)dY.
(15)
-a
The functions e’(x) are given by
q#l, q = 1 ,
(16)
and Dq are constants which are zero for the largest eigenvalue, in which case the integration paths in (13) are straight lines along the real axis. For next-leading eigenvalues the integration paths surround complex zeros of the functions aq(x), aq(x), and the constants 0’ are non-zero. Solving (8) one obtains two contributions to T(u). The bulk behaviour arises from the product of the functions f, the finite-size corrections from 1 + t’(u). Introducing b(x) := *rini_(; x + ; A) ,
(17)
402
A. Kliimper, P. A. Pearce
I RSOS and vertex models in 2 D
we find lnb=k*ln!‘I’+C,
(18)
where C is a constant which is zero for the largest eigenvalue. The analytic solution of (13) seems hopeless as it is a non-linear integral equation. It simplifies somewhat in the large N limit by using the scaling properties of the functions try(x). As the function c’(x) interpolates between 0 at x = 0 and 1 for x-+ ~0 the transition being at x 2: In N, the scaling form we use is a”(x) : = lil& aq(x + In N) ,
A’(X) := 1 -I-a”(x) ,
in the scaling limit and takes the form lnaq=lneq+k*lnAq~‘+k*lnAq”.
(21)
From now on only the largest eigenvalue is considered, i.e. Dy = 0, C= 0. From (18) we see that the leading finite-size corrections are indeed of order 1 IN, 3
In b(x) = 5
r
cash(x) 1 e-’ In A’(y)
dy + P (h)
.
(22)
-1
The amplitude of the correction term is given by an integral of the function In A’. Amazingly just this integral can be computed without solving (21) explicitly! To see this consider the expression r
In u’)’ In A4 - In a4 (In Aq)‘] dx ,
(23)
where the functions In uq can be expressed through (21) leading to
4
I
--z
eeX In A’(X) dx ,
(24)
A. Kliimper,
P. A. Pearce
I RSOS and vertex models in 2 D
403
and all contributions of the kernel k cancel due to symmetry. In fact, the integral appearing in (22) is just (24) and is identical to (23), which can be after changing the variable of integration uq(=)
q=l
1 (ln(l aY(-z)
+ a”) aq
-
$$)
da’ .
The explicit solution of the non-linear integral equation (21) is fortunately needed! Inserting into (22) we obtain cash(x) r-3 In 6(x) = 7 c ]L+(aY(m)) q=l
(25)
not
L+(aY(-91 7
where we have introduced the dilogarithmic function L+(a) which is related to the standard Rogers dilogarithm L(a),
L+(a)=;j(ln(lx+x) - +&-L(f--).
(27)
0
Eq. (26) can be regarded as the final result as the 1 lN corrections are given by definite integrals whose terminals can be obtained explicitly from the functional equations (9) which asymptotically turn into algebraic equations. Furthermore, the evaluation of the required combination of dilogarithms is achieved by using the identity
yielding eventually In 6(x) = &
(I-
A)
After taking the Hamiltonian central charge as c=I----..-
6 ?(T - 1) .
cash(x) .
(29)
limit and comparing with (2) one identifies the
(30)
Therefore the critical properties of the ABF models in regime III/IV are described by the minimal unitary series of conformal field theory. The abovementioned fusion models correspond to other discrete series [24] notably the
404
A. Kliimper,
P.A. Pearce
I RSOS and vertex models in 20
superconformal series for the 2 X 2 fusion. In all these cases a similar analysis to that presented above is applicable and also works for the next-leading eigenvalues. The main modification to this end is a deformation of the integration paths leading to expressions involving certain analytic continuations of the dilogarithms. Using identities generalizing (28) the conformal weights of higher Kac tables are found. In regime I/II the models show different critical behaviour which is described by 2, parafermion theories independent of the fusion level. These methods can in fact also be applied in general to the critical A-D-E models [28]. We next comment on some recent developments relating the presented method to the calculation of thermodynamics for integrable quantum chains taking the above models as example. Following ideas of ref. [9] one can show that the free energy of the quantum chains at finite temperature is given by the largest eigenvalue of the transfer matrix of the related classical RSOS model with row inhomogeneity [8]. This holds in the limit of infinite Trotter number [29]. As a result the thermodynamics are precisely given by eqs. (13) and (18), however with different definitions for the functions c(x) [8]. After some elementary transformations these non-linear integral equations are shown to be identical to the standard equations of the thermodynamic Bethe ansatz [27]. Furthermore, the generalization to next-largest eigenvalues of the quantum transfer matrix is straightforward and a computational framework for correlation lengths at finite temperatures can be set up. This provides a substantial extension of the traditional approach. Thus results for (4) become accessible from lattice calculations. We hope that these remarks show the strong connection of thermodynamics of quantum chains, classical models with inhomogeneity, inversion identities and fusion. We conclude the paper with some speculations on the applicability of inversion identities. First, the presented method should work for other models, e.g. the eight-vertex model. In this case the inversion identity hierarchy will be infinite in general. However, this should not be a serious problem. For the related quantum XYZ chain the thermodynamic equations for the free energy are known since twenty years [7]. In the light of the previous paragraph it should be clear that the eight-vertex model can be treated in a similar way to the ABF models. In any case, one can avoid the treatment of an infinite hierarchy of matrices, which is desirable for instance for numerical purposes [8,30]. It can be achieved by starting from the Bethe ansatz equations. An analysis similar to the above one yields a set of non-linear integral equations for finitely many functions [31,32]. The advantage of this procedure in comparison to refs. [lO,ll] is the exactness of the equations for any finite-system size. All the models mentioned so far are related to SU(2), which is reflected by
A. Kliimper,
the structure fundamental one needs fundamental such a case fact, this is generalized
P. A. Pearce
I RSOS and vertex models in 20
405
of (5). It is related to the tensor product decomposition of two representations of SU(2) which produces a singlet. For SU(N) a generalization. One possibility is a product of N identical representations in order to find a singlet in the decomposition. In a product of N factors on the left-hand side of (5) is expected. In known, e.g., for the models of ref. [33]. Another possibility is a inversion identity hierarchy [34].
References [l] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575; in: Vertex Operators in Mathematics and Physics, J. Lepowsky, S. Mandelstam and I.M. Singer, eds. (Springer, Berlin, 1984). [3] H.W.J. BiBte, J.L. Cardy and M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742. [4] J.L. Cardy, J. Phys. A 17 (1984) L385. [5] I. Affleck, Phys. Rev. Lett. 56 (1986) 746. [6] C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115. [7] M. Takahashi, Prog. Theor. Phys. 46 (1971) 401. [8] A. Kliimper, Ann. Phys. (Leipzig) 1 (1992) 540. 191 T. Koma, Prog. Theor. Phys. 78 (1987) 1213; 81 (1989) 783. [lo] H.J. de Vega and F. Woynarovich, Nucl. Phys. B 251 (1985) 439. [ll] F. Woynarovich and H.-P. Eckle, J. Phys. A 20 (1987) L97. [12] M. Karowski, Nucl. Phys. B 300 [FS22] (1988) 473. [13] Yu.G. Stroganov, Phys. Lett. A 74 (1979) 116. [14] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). (151 N.Yu. Reshetikhin, Lett. Math. Phys. 7 (1983) 205. (161 P.A. Pearce, Phys. Rev. Lett. 58 (1987) 1502. [17] A. Khimper and J. Zittartz, Z. Phys. B 71 (1988) 495. [18] A. Kliimper and J. Zittartz, Z. Phys. B 75 (1989) 371. [19] A. Kliimper, A. Schadschneider and J. Zittartz, Z. Phys. B 76 (1989) 247. [20] A. Khimper, Europhys. Lett. 9 (1989) 815. [21] M.G. Tetel’man, Sov. Phys. JETP 55 (1982) 306. [22] P.A. Pearce and A. Kliimper, Phys. Rev. Lett. 66 (1991) 974. [23] A. Kliimper and P.A. Pearce, J. Stat. Phys. 64 (1991) 13. [24] A. Khimper and P.A. Pearce, Physica A 183 (1992) 304. [25] G.E. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35 (1984) 193. [26] P.P. Kulish, N.Yu. Reshetikhin and E.K. Sklyanin, Lett. Math. Phys. 5 (1981) 393. [27] V.V. Bazhanov and N.Yu. Reshetikhin, Int. J. Mod. Phys. A 4 (1989) 115. [28] P.A. Pearce, Int. J. Mod. Phys. A 7, Suppl. 1B (1992) 791. [29] M. Suzuki and M. Inoue, Prog. Theor. Phys. 78 (1987) 787. [30] A. Kliimper, in preparation. [31] A. Khimper and M.T. Batchelor, J. Phys. A 23 (1990) L189. [32] A. Kliimper, M.T. Batchelor and P.A. Pearce, J. Phys. A 24 (1991) 3111. [33] B. Sutherland, J. Math. Phys. 11 (1970) 3183. [34] A. Kuniba, Thermodynamics of the y(Xl”) Bethe ansatz system with q a root of unity, ANU preprint (1991).