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Journal of Physics A Mathematical and General

Finite-size scaling amplitudes in a random tiling model

Wenfei Li, Hyunggyu Park, Michael Widom

Published Jun 7, 1990• 16 citations
Publisher

Abstract

The authors study finite-size corrections to the free energy in a two-dimensional random tiling model. This model is equivalent to an asymmetric six-vertex model and exhibits commensurate-incommensurate phase transitions of the Pokrovsky-Talapov type. They calculate finite-size scaling amplitudes of the free energy inside the incommensurate phase, analytically and numerically, by employing a transfer matrix method. Both periodic and free boundary conditions are considered. These amplitudes are consistent with predictions of the theory of conformal invariance with the conformal charge c=1.

DOI
10.1088/0305-4470/23/11/011

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The authors study finite-size corrections to the free energy in a two-dimensional random tiling model. This model is equivalent to an asymmetric six-vertex model and exhibits commensurate-incommensurate phase transitions of the Pokrovsky-Talapov type. They calculate finite-size scaling amplitudes of the free energy inside the incommensurate phase, analytically and numerically, by employing a transfer matrix method. Both periodic and free boundary conditions are considered. These amplitudes are consistent with predictions of the theory of conformal invariance with the conformal charge c=1.
The authors study finite-size-scaling behaviour of a two-dimensional random tiling model. This model exhibits commensurate-incommensurate phase transitions of the Pokrovsky-Talapov type where anisotropic scaling appears. A simple scaling argument near this transition suggests an integral surface critical exponent which, in turn, implies a logarithmic singularity with a universal coefficient in the surface free energy. They confirm the existence of this singularity numerically employing the transfer matrix method in the case of free boundary conditions.
We examine the finite-size-scaling amplitudes of the free energy in incommensurate phases on a torus with periodic and twisted boundary conditions. We show that these amplitudes are equivalent to those of the six-vertex model with electric and magnetic defect lines. The twist angle generates magnetic defect lines, while the electric defect lines are generated by competition between the domain-wall separation and the finite system size. We calculate the amplitudes exactly for the free-fermion model and the spin-(1/2 XXZ chain, and conjectire the form of these amplitudes for a more general model. Numerical calculations employing the Bethe ansatz confirm our conjecture.
Finite-size corrections to the energy levels of the asymmetric six-vertex model transfer matrix are considered using the Bethe ansatz solution for the critical region. The nonuniversal complex anisotropy factor is related to the bulk susceptibilities. The universal Gaussian coupling constant g is also related to the bulk susceptibilities as g=2${\mathit{H}}^{\mathrm{\ensuremath{-}}1/2}$/\ensuremath{\pi}, H being the Hessian of the bulk free energy surface viewed as a function of the two fields. The modular covariant toroidal partition function is derived in the form of the modified Coulombic partition function which embodies the effect of incommensurability through two mismatch parameters. The effect of twisted boundary conditions is also considered. \textcopyright{} 1996 The American Physical Society.
We show that for conformally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.
We investigate the thermodynamic and critical properties of an interacting-domain-wall model which is derived from the triangular-lattice antiferromagnetic Ising model with the anisotropic nearest- and next-nearest-neighbor interactions. The model is equivalent to the general five-vertex model. Diagonalizing the transfer matrix exactly by the Bethe Ansatz method, we obtain the phase diagram displaying the commensurate and incommensurate (IC) phases separated by the Pokrovksy-Talapov transitions. The phase diagram exhibits commensurate phases where the domain-wall density q is locked at the values of 0, 1/2, and 1. The IC phase is a critical state described by the Gaussian fixed point. The effective Gaussian coupling constant is obtained analytically and numerically for the IC phase using the finite-size-scaling predictions of the conformal-field theory. It takes the value 1/2 in the noninteracting limit and also at the boundaries of q=0 or 1 phase and the value 2 at the boundary of q=1/2 phase, while it varies smoothly throughout the IC region.
We derive the exact values of the universal amplitudes associated with the finite-size-scaling behavior of the free energy and interfacial free energies of the Potts model on a torus, by means of the extended-scaling method. These amplitudes vary continuously with the value of the aspect ratio of the torus. Our results make it possible to apply the type of finite-size-scaling techniques that are standard in the context of transfer-matrix calculations to Monte Carlo simulations, and thus enhance the effectiveness of these simulations in establishing the universality class of a phase transition.