Critical two-dimensional Ising model with free, fixed ferromagnetic, fixed antiferromagnetic, and double antiferromagnetic boundaries

Abstract

Original citation:Wu, Xintian and Izmailyan, N. (2015) Publisher:American Physical Society Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. The critical two-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size N × N and 30 < N < 1000. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well. CURVE is the Institutional Repository for Coventry University

The bulk, surface and corner free energies of the anisotropic triangular Ising model

“…We obtain the critical behaviour of the bulk, surface and the two corner free energies, and find logarithmic singularities corresponding to the exponents α being 2, 1, 0, 0, respectively, in agreement with the square lattice results. The results for the corner free energies agree with the predictions of conformal invariance [18, eqn 4.4] and later numerical results [20,21].…”

Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

“…Since the fan network is the plane rectangular lattice with three free boundary conditions and one with Dirichlet-Neumann boundary conditions, we have four corners for the fan network and one can expect the corner free energy f corner contribution in the exact asymptotic expansions of the free energy Equation (43). However, it is easy to see from Equation (43) that, in the exact asymptotic expansions of the free energy for the spanning tree on the finite fan network, the corner free energy f corner is equal to zero.…”

“…In that case, the cobweb becomes an infinitely long cylinder with circumference N and the fan network becomes an infinitely long strip of width N and with free boundary condition on both sides of the strip. The asymptotic expansion of the free energy for the cobweb and fan networks can be obtained from Equations (34) and (43), respectively. Using the facts that…”

“…Again, using Equations (54) and (56), the asymptotic expansion of the free energy for the fan network on the infinitely long strip with width N and with free boundary condition on both sides of the strip can be obtained from Equation (43)…”

“…Using Equations (55), (56), (59) and (60), one can obtain the asymptotic behavior of θ 2 (τ ) and η(τ ) as τ → 0 (or N → ∞). Then, from Equations (34) and (43), one can obtain the asymptotic expansion of the free energy for the cobweb and fan networks on the infinitely long strip in the following form…”