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arXiv:cond-mat/9511081 · Journal of Physics A Mathematical and General

Surface free energies, interfacial tensions and correlation lengths of the ABF models

David O’Brien, Paul A. Pearce

Published Apr 7, 1997• 22 citations
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Abstract

The surface free energies, interfacial tensions and correlation lengths of the Andrews - Baxter - Forrester models in regimes III and IV are calculated with fixed boundary conditions. The interfacial tensions are calculated between arbitrary phases and are shown to be additive. The associated critical exponents are given by with in regime III and with in regime IV. Our results are obtained using general commuting transfer matrix and inversion relation methods that may be applied to other solvable lattice models.

DOI
10.1088/0305-4470/30/7/017

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The surface free energies, interfacial tensions and correlation lengths of the Andrews - Baxter - Forrester models in regimes III and IV are calculated with fixed boundary conditions. The interfacial tensions are calculated between arbitrary phases and are shown to be additive. The associated critical exponents are given by with in regime III and with in regime IV. Our results are obtained using general commuting transfer matrix and inversion relation methods that may be applied to other solvable lattice models.
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We compute by numerical transfer-matrix methods the surface free energy \ensuremath{\tau}(T), the surface stiffness coefficient \ensuremath{\kappa}(T), and the step free energy s(T) for Ising ferromagnets with (\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}L) square-lattice and (\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}M) cubic-lattice geometries, into which an interface is introduced by imposing antiperiodic or plus/minus boundary conditions in one transverse direction. These quantities occur in expansions of the angle-dependent surface tension for either rough or smooth interfaces. The finite-size scaling behavior of the interfacial correlation length provides the means of investigating \ensuremath{\tau}(T), \ensuremath{\kappa}(T), and s(T). The resulting transfer-matrix estimates are fully consistent with previous series and Monte Carlo studies, although current computational technology does not permit transfer-matrix studies of sufficiently large systems to show quantitative improvement over the previous estimates.
Estimates for the critical interfacial tension amplitudes of the three-state Potts model in two dimensions via direct Monte Carlo sampling are presented for the first time. Both the square and triangular lattices are analyzed with use of finite-size scaling considerations. Also presented for the first time are estimates for the correlation-length amplitudes as derived from high-temperature series. These numerical results allow investigation of the ratio U=\ensuremath{\tau}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/kB{k}_{B}kB​T as T\ensuremath{\rightarrow}Tc{T}_{c}Tc​. The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.
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 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 α, 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 O/G or G/O, where O is the original CFT used to define the model and G is some unknown CFT, depending on the regime. This agrees with known examples of such models by Huse and Jimbo et al.
The authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next-nearest neighbour (diagonal) interactions on the special surface corresponding to regimes III and IV of the generalised hard hexagon model. The interfacial tension, correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row-to-row and corner transfer matrices. The associated critical exponents are found to be mu =v= 5 / 4 , beta = 3 / 32 in regime III and mu '=v'= 5 / 2 , beta '= 1 / 4 in regime IV. In particular, their results confirm the recent proposal by Huse that regime III is the first-order coexistence surface (separating the disordered fluid phase from the square ordered solid phases) and that the regime III/regime IV boundary is a line of tricritical points.