Partition function for several Ising model interface structures

TL;DR

This paper analytically calculates the excess free energies of finite Ising cylinders with domain walls, providing insights into finite-size effects and boundary conditions through integral analysis and steepest descent methods.

Contribution

It introduces a procedure to compute free energies for various domain wall configurations in Ising models, including asymptotic analysis and complex plane path determination.

Findings

Exact integral expressions for free energies derived

Asymptotic analysis reveals finite-size effects

Steepest descent paths identified for complex integrals

Abstract

We employ a procedure that enables us to calculate the excess free energies for a finite Ising cylinder with domain walls analytically. This procedure transparently covers all possible configurations of the domain walls under given boundary conditions and allows for a physical interpretation in terms of coarse-grained quantities such as surface and point tensions. The resulting integrals contain all the information about finite-size effects; we extract them by careful asymptotic analysis using the steepest descent method. To this end, we exactly determine the steepest descent path and analyse its features. For the general class of integrals, which are usually found in the study of systems with inclined domain walls, knowledge of the steepest descent path is necessary to detect possible intersections with poles of the integrand in the complex plane.