Casimir Terms and Shape Instabilities for Two-Dimensional Critical Systems

TL;DR

This paper uses conformal field theory to analyze the universal free energy contributions and shape instabilities of two-dimensional critical systems with specific geometries, revealing new Casimir-induced boundary instabilities.

Contribution

It introduces a novel analysis of shape instabilities and Casimir effects in 2D critical systems with complex geometries using conformal field theory.

Findings

Identifies a Casimir instability toward sharp corners on the boundary.

Shows the rectangle is unstable against small curvature at fixed area and edge length.

Derives consequences for transfer matrix and corner transfer matrix elements.

Abstract

We calculate the universal part of the free energy of certain finite two- dimensional regions at criticality by use of conformal field theory. Two geometries are considered: a section of a circle ("pie slice") of angle \phi and a helical staircase of finite angular (and radial) extent. We derive some consequences for certain matrix elements of the transfer matrix and corner transfer matrix. We examine the total free energy, including non- universal edge free energy terms, in both cases. A new, general, Casimir instability toward sharp corners on the boundary is found; other new instability behavior is investigated. We show that at constant area and edge length, the rectangle is unstable against small curvature.