The critical 2-dimensional Ising model with fixed boundaries

TL;DR

This paper investigates the critical 2D Ising model with various boundary conditions, using numerical methods to analyze finite size effects and conjecture exact formulas for boundary-related terms, confirming conformal field theory predictions.

Contribution

It introduces a numerical approach with Bond Propagation algorithms to accurately analyze boundary effects in the 2D Ising model and proposes conjectures for exact boundary-related terms.

Findings

Exact conjectures for corner logarithmic terms in free energy.

Numerical verification of boundary effects with high precision.

Agreement of corner free energy terms with conformal field theory.

Abstract

The critical 2-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, fixed double antiferromagnetic. Using Bond Propagation algorithms with surface fields, we obtained the free energy, internal energy and specific heat numerically on square lattices with square shape and various combinations of the four types boundary conditions. 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 and corner logarithmic terms in the internal energy and specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.