Finite size behaviors of critical Ising model on a rectangle with free boundaries

TL;DR

This study uses high-precision numerical methods to analyze the critical behaviors of the 2D Ising model on rectangles with free boundaries, confirming theoretical predictions and revealing geometry-independent corner contributions.

Contribution

The paper provides exact expansions of critical free energy, internal energy, and specific heat, confirming conformal field theory predictions and identifying geometry-independent corner effects.

Findings

Confirmed the conformal field theory prediction of corner free energy with high precision.

Proved corner free energy, internal energy, and specific heat are geometry independent.

Derived high-order corrections to the free energy and other thermodynamic quantities.

Abstract

Using the bond-propagation algorithm, we study the Ising model on a rectangle of size $M \times N$ with free boundaries. For five aspect ratios $\rho=M/N=1,2,4,8,16$, the critical free energy, internal energy and specific heat are calculated. The largest size reached is $M \times N=64\times 10^6$. The accuracy of the free energy reaches $10^{-26}$. Basing on these accurate data, we determine exact expansions of the critical free energy, internal energy and specific heat. With these expansions, we extract the bulk, surface and corner parts of free energy, internal energy and specific heat. The fitted bulk free energy density is given by $f_{\infty}=0.92969539834161021499(1)$, comparing with Onsager's exact result $f_{\infty}=0.92969539834161021506...$. We prove the conformal field theory(CFT) prediction of the corner free energy, in which the central charge of the Ising model is found to be $c=0.5\pm 1\times 10^{-10}$ comparing with the CFT result $c=0.5$. We find that not only the corner free energy but also the corner internal energy and specific heat are geometry independent, i.e., independent of aspect ratio. The implication of this finding on the finite scaling is discussed. In the second order correction of the free energy, we prove the geometry dependence predicted by CFT and find out a geometry independent constant beyond CFT. High order corrections are also obtained.