Shape dependent finite-size effect of critical two-dimensional Ising model on a triangular lattice

TL;DR

This study uses high-precision calculations to analyze how the shape of finite triangular lattices influences the critical properties of the 2D Ising model, revealing shape-dependent logarithmic corrections and non-universal corner effects.

Contribution

It provides detailed, shape-dependent finite-size analysis of the critical 2D Ising model on triangular lattices, confirming some conformal field theory predictions and identifying non-universal corner effects.

Findings

Logarithmic corrections in free energy for certain shapes

Universal corner free energy confirmed for some geometries

Corner internal energy and specific heat are shape-dependent

Abstract

Using the bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangle, rhombus, trapezoid, hexagon and rectangle. The critical free energy, internal energy and specific heat are calculated. The accuracy of the free energy reaches $10^{-26}$. Based on accurate data on several finite systems with linear size up to N=2000, we extract the bulk, surface and corner parts of the free energy, internal energy and specific heat accurately. We confirm the conformal field theory prediction of the corner free energy to be universal and find logarithmic corrections in higher order terms in the critical free energy for the rhombus, trapezoid, and hexagon shaped systems, which are absent for the triangle and rectangle shaped systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the free energy and the specific heat are different. The corner internal energy and corner specific heat for angles $\pi/3$, $\pi/2$ and $2\pi/3$ are obtained, as well as higher order corrections. Comparing with the corner internal energy and corner specific heat previously found on a rectangle of the square lattice (Phys. Rev. E. 86 041149 (2012)), we conclude that the corner internal energy and corner specific heat for the rectangle shape are not universal.