On the Free Energy of Solvable lattice Models

We conjecture the inversion relations for thermalized solvable interaction round the face (IRF) two dimensional lattice models. We base ourselves on an ansatz for the Baxterization described by the author in the 90's. We solve these inversion relations in the four main regimes of the models, to give the free energy of the models, in these regimes. We use the method of Baxter in the calculation of the free energy of the hard hexagon model. We believe these results to be quite general, shared by most of the known IRF models. Our results apply equally well to solvable vertex models. Using the expression for the free energy we calculate the critical exponent $\alpha$, and from it the dimension of the perturbing (thermal) operator in the fixed point conformal field theory (CFT). We show that it matches either the coset ${\cal O}/{\cal G}$ or ${\cal G}/{\cal O}$, where $\cal O$ is the original CFT used to define the model and $\cal G$ is some unknown CFT, depending on the regime. This agrees with known examples of such models by Huse and Jimbo et al.


Introduction.
Two dimensional solvable lattice models offer a rich ground to study such phenomena as phase transitions, universality and mathematical applications in knot theory. For reviews see [1,2]. These models also enjoy a strong connection with two dimensional conformal quantum filed theory (CFT). See, e.g., the reviews [3,4].
Some time ago the author introduced a method to construct solvable interaction round the face (IRF) from the data of an arbitrary CFT [5]. We call such models IRF (O, h, v) where O is the defining CFT and h and v are two primary fields in the theory.
A long standing question is what is the fixed point CFT of the so defined models and how it is related to the original CFT O. We solve this problem here by calculating the free energy of the thermalized models. From this we deduce the critical exponent α and the dimension of the perturbing field in the fixed point CFT.
To compute the free energy we first need to thermalize the trigonometric ansatz of [5]. This we do by calculating the two inversion relations for the general IRF models. Then we thermalize the models by replacing the sin(u) function in the inversion relations with the function θ 1 (u, q), where θ 1 is the standard elliptic function. This agrees with all the models where the off-critical Boltzmann weights are known and we conjecture that it is true in general. Thus we are in a position to solve exactly models for which the Boltzmann weights are not explicitly known.
We find that in the four main regimes of the IRF model the fixed point CFT is given by a coset of the original theory. Namely, in regimes III and IV the fixed point CFT is consistent with the coset model G/O where G is some unknown CFT. In regimes I and II, the fixed point CFT is O/G. This fixed point RCFT is known exactly, in some cases. For example, in the Andrews-Baxter-Forrester model [6], which is IRF(SU (2) k , [1], [1]), the fixed point field theory was determined to be the unitary minimal models, which are the coset SU (2) k−1 × SU (2) 1 /SU (2) k , in regimes III and IV [7]. In regimes I and II the critical CFT was identified as the parafermionic field theory SU (2) k /U (1), which are the Fateev-Fateev model [8], by Jimbo et al. [9]. Indeed, this agrees with our general result. For the case of O = SU (N ) k , and h = v = fundamental, the fixed point field theory in regime III was shown to be SU (N ) k−1 × SU (N ) 1 /SU (N ) k by Jimbo et al. [10]. Again agreeing with our result for the fixed point CFT.
To compute the free energy, in all the four regimes, we follow the method used by Baxter [1] in the hard hexagon model. Our results for the free energies agree with the hard hexagon case for O = SU (2) 3 , in the four regimes.

The inversion relations.
We wish to study IRF lattice models based on the braiding matrix of conformal field theory (CFT). We fix a conformal field theory O and fixed primary fields in this theory h and v. The IRF model is denoted as IRF(O, h, v) following ref. [5], defined on a square lattice. We assume that the boundary conditions are periodic. Let B i be the braiding matrix in the RCFT which exchanges the field h with the field v [11]. We define the operator < a 1 , a 2 , . . . , a n |B i |a ′ 1 , a ′ 2 , . . . , a ′ n >= B where the matrix B is the braiding matrix and it obeys the braiding relations, The variables on the lattice a m and a ′ m are some primary fields in the RCFT O.
From the braiding matrix one can define the projectors, where n is the number of eigenvalues of B i (called the number of blocks) and λ a are the eigenvalues. The projection operators obey the relations, where ǫ a = ±1 according to whether the product is symmetric or anti-symmetric.
We define the fusion products of the field h as, and whereh is the complex conjugate field of h, n is the number of blocks and the order of the fields is set in a certain way, which allows for the Yang-Baxter equation of the model. The order of the fields appears to be that ψ a+1 is contained in the fusion product of ψ a with the adjoint representation, and similarly forψ a .
(The fact that the number of blocks is the same in both eqs. (the adjoint representation, assuming some quantum group structure). We denote the dimension of ψ a as ∆ a and similarly forψ a the dimension is∆ a . We define the crossing parameters as, where a = 0, 1, . . . , n − 2. We note that ζ a ,ζ a < π/2, which will be important later.
In ref. [5] an ansatz for the trigonometric solution of the Yang Baxter equation (YBE) was given. It is where a = 0, 1, . . . , n−1. Our ansatz is that R h,h i solves the Yang Baxter equation, The two YBE equations (2.12, 2.13) imply that the transfer matrices for R h,h i (u) commutes with each other for different spectral parameters u and the same for The R matrices obey the first inversion relation which follows from eqs. (2.10, and where we changed by an irrelevant factor the normalization of R h,h i (u).
The second inversion relation is crossing. We shall denote again R i (u) by its matrix form. Then the crossing relation is, (as part of our conjectured ansatz), where λ is the crossing parameter, where we used eq. (2.8). The crossing multipliers are where S is the modular matrix [4].
We wish to thermalize now the IRF model. We do not know how to thermalize the Boltzmann weights. So instead we will thermalize the inversion relations. We define the theta function, (This definition differs from the standard one by a factor of 2q 1/4 , which is irrelevant since we will only encounter ratios of theta functions.) Now we conjecture that the thermalization of the first inversion relation, eq.
(2.14), is given by replacing the sin by the theta function θ 1 . We denote θ 1 (u, q 2 ) by θ 1 (u). Then, the thermalization of the R matrix is Finally, the crossing relation eq. (2.18) remains the same for general q except for the crossing multiplier, eq. (2.20), whose explicit expression we will not need here. Note that for q = 0 (the critical limit), θ 1 (u) = sin u so we get the same inversion relations as before.
These conjectures can be indeed verified for many models for which we know the explicit Boltzmann weights, e.g., [1,2].
Next, we wish to define the free energy of the model. It is given by where N is the number of lattice sites and Z is the partition function calculated with R h,h . The free energy is given as usual by, where k B is Boltzmann constant and T is the temperature. Now, since the transfer matrices commute for different spectral parameters u, the inversion relations translate to equations of κ(u) (fixing some q), and In deriving the last equation, we used the fact that the crossing multipliers cancel when calculating the partition function.
Actually, the inversion relations, eqs. (2.28, 2.27), remain the same under We also find it convenient to change the second inversion relation by substi- The second inversion relation then becomes,

Regimes III.
Our aim now is to solve the inversion relations eqs. (2.28, 2.30) and to calculate the free energy. We assume first that the model is in regime III. This is defined by where d = min i ζ i ,ζ i , and q 2 = exp(−ǫ). It is convenient to use the modular transformation of the theta function. This is defined by [1] We find it convenient to redefine, κ(u) = e 2δu/ǫ e 2(n−1)u 2 /ǫ κ(u), Then, the first inversion relation, eq. (2.28, 2.23), becomes, where where w = exp(−4πu/ǫ), (3.9) and we denoted for brevity f (w) for f (w,q). The second inversion relation, eq.
(2.30), becomes,κ We wish to solve now for the free energy, using the two inversion relations. For this purpose we assume that log(k(w)) is analytic in the annulus containing the point w = 1 and the point w = x. We analytically continue k(u) to −λ < u < 0 and to 0 < u < λ. This assumption is justified by considering explicit models, e.g., the hard hexagon model [1].
So we expand, where the summation is convergent in the annulus containing 1 and x . In the neighborhood of 1, eqs. (3.7, 3.8), gives, where m > 0, where Taking the logarithms of eqs. (3.13, 3.7) and equating coefficients, we find, for m ≥ 0. The solution of these two equations is, for m = 0. This completes the calculation of the free energy in regime III. Indeed converges in an annulus containing the points 1 and x, as we assumed. Note that |ζ r | ≤ λ always, for all r, which is needed to show convergence, which we checked in many models, but we do not have a general proof for this fact.
We wish to calculate now the critical exponent α. This is defined as the singularity of the free energy, where T c is the critical temperature. We have thatq = exp(−4π 2 /ǫ). Since c m is divided by 1 − x 2m this means that it becomes a theta function at the modulus x 2 , when summed back. Since λ = π∆ 0 /2, eq. (2.9), we may write this modulus as Under a modular transformation, x 2 becomes, Since q 2 ∝ |T − T c | this implies that and so, The dimension of the perturbing operator at the critical conformal field theory is given by [3], Now, since we assume thatψ 1 is the adjoint operator, in quantum group models, as can be see by considering various models, the dimension of the perturbing field is seen to be, where∆ 0 = ∆ adjoint . ⋆ Such a field appears in the coset theory, where O is the original CFT used to define the model and G is some CFT model, where we take the currents in G and the adjoint representation in O. Thus, we conjecture that the fixed point of Regime III is given by the coset C. In many cases where the fixed point theory was calculated explicitly, this was indeed shown to be the case. For example, the RCFT O of the ABF model [6] is SU (2) k WZW model. The fixed point in Regime III is the k + 1 minimal model [7], defined by the coset

Regime II.
Let us consider regime II. This is defined by The first inversion relation remains the same, eqs. (2.28, 2.23). The second inversion relation becomes, where the theta function is invariant under the shift by π. We define, Then the second inversion relation, eq. (4.2), becomes, where whereq, x, z r andz r are given by eqs. (3.12, 3.14, 3.10, 3.11). We assume now that log[w µκ (w)] (4.7) is analytic in an annulus a < |w| < b containing the points w = 1 and w = w 0 . This is an analytic continuation of w to w < 1 and w > w 0 even though the regime is defined for a subset of these values.
The free energy log[w µκ (w)] is given by the series where a m is some function. Thus it is given by a theta function with the modulus under a modular transformation, as in eq. (3.3), we find where q 2 is proportional to |T − T c |. Thus we find for the exponent α, For a quantum group model∆ 0 is the dimension of the adjoint representation, conjecturally. Thus the dimension of the perturbing thermal operator is An operator with such a dimension appears in the Coset model, where O is the RCFT used to define the model and G is some CFT model.
Thus we conjecture that C is the fixed point theory in regime II. The perturbing operator is the adjoint representation in O and the unit in G. For example, for the ABF model in regime II, where O is SU (2) k WZW model, the fixed point theory is the parafermionic theory [9], Let us get now to regimes IV and I. Regime IV is defined by where d = min i ζ i ,ζ i . We use the inverse modulus [1], where, as before, We define where the crossing parameters were defined in eqs. (2.8, 2.9).
In regime IV we write the two inversion relations as 2πζ r + 2ζ r (π − ζ r ). (5.10) then, the first inversion relation (for regimes I and IV) becomes, The second inversion relation, for regime IV, becomes, where we define the crossing parameter, We denoted for brevity f (w,q) as f (w).
As before, we assume that log(w µκ (w)) is analytic in an annulus a < |w| < b containing w = 1 and w = w 0 . Thus we expand and we find (in regimes I and IV) Similarly we expand for m = 0, and Thus, as before, eq. (3.27), we find where for m = 0 and We note that indeed this series converges in the annulus containing w = 1 and Let us compute now the critical exponent α. The series for c m , eq. (5.23), is an expression for a theta function with the modulus, (we ignore the overall sign). The expression forκ includes factors of θ 4 (u, −q 2 ).
The function θ 4 is defined by and it satisfies the conjugate modulus relation Denoting p = −q 2 , we find that the inverse modulus transformation gives the modulus, This is the same dimension as in Regime III and we conclude that it is the other side of the same fixed point. The RCFT in Regime IV is thus conjectured as the coset model where O is the original CFT used to define the model and G is some unknown CFT.
Let us turn now to regime I. It is defined by 1) The first inversion relation is the same as in regime IV, eq. (5.7). For the second inversion relation we take as in regime II, We define as in regime IV, q,q, x,z r and w, eqs. (5.4-5.6). We further definẽ κ(u) = e 2(n−1)u 2 /ǫ e δu/[ǫ(2λ−π)] κ(u), Then the second inversion relation, eq. (6.2), then becomes where and f (w,q) was defined in eq. (5.3). Now, we need to expand as before and we find, where d 0 is given by eq. (5.16). Also, for m = 0. So, as before, where and as before, for m = 0. The series, eq. (6.10), indeed converges in the annulus containing w = 1 and w = w 0 . Now, we wish to compute the exponent α. The expression for c m is a theta function with the modulus which can be written as where we used the definition of x and λ, eqs. (5.5, 2.9). Thus, the modulus of the theta function has the modulus (q 2 ) 2(1−∆ adjoint ) . (6.15) Now, since the expression for the theta function contains θ 4 , at positive modulus, the same discussion as in section (5) shows that 2 − α = 1 1 − ∆ adjoint , (6.16) and the dimension of the perturbing field is This is the same dimension as in regime II and we conclude that it is the other side of the same phase transition with the critical theory given as in regime II by the coset where O is the original theory used to define the model and G is some unknown CFT.
This concludes the expression for the free energy in all the four regimes.
We can check our results for the hard hexagon model which is IRF(SU (2) 3 , [1]).

Discussion.
Some other two dimensional solvable lattice models are the vertex models.
These are defined by some CFT O and some representation in them h. The Boltzmann weights are elements of End(V × V ) where V are the weights of the representation h. As was discussed in refs. [12,13], the Baxterization of the models is exactly the same formula as the IRF models, eq. (2.10, 2.11). For the case of SU (2) this was described in ref. [14]. Thus, our calculation of the free energy holds equally well for these models, with the parameters ζ i andζ i given by eqs. (2.8, 2.9). We assume here periodic boundary conditions. The free energy then obeys the same inversion relations, eqs. (2.28, 2.30). Thus also the solution for the free energy is the same as described in this paper.
Another interesting point is that in the theories C define integrable models when perturbed by the thermal operator. This may be interesting for the building of new integrable models of massive quantum field theories.