Critical Behavior in Rectangles with Mixed Boundaries

TL;DR

This paper analyzes the critical behavior of the 2D Ising model in rectangular domains with mixed boundary conditions, revealing complex density profiles and corner effects through conformal transformations and simulations.

Contribution

It provides exact density profiles for the critical Ising model with mixed boundary conditions in rectangles, introducing a Corner-Operator-Expansion to describe corner effects.

Findings

Density profiles exhibit rich behavior near corners.

Conformal transformations relate known half-plane results to rectangles.

Simulation results agree with analytical predictions without adjustable parameters.

Abstract

Density profiles are investigated arising in a critical Ising model in two dimensions which is confined to a rectangular domain with uniform or mixed boundary conditions and arbitrary aspect ratio. For the cases in which the two vertical sides of the rectangle have up-spin boundary conditions + and the two horizontal sides with either down-spin boundary conditions $-$ or with free-spin boundary conditions $f$, exact results are presented for the density profiles of the energy and the order parameter which display a surprisingly rich behavior. The new results follow by means of conformal transformations from known results in the half plane with $+-+-+$ and $+f+f+$ boundary conditions. The corners with mixed boundary conditions lead to interesting behavior, even in the limit of a half-infinite strip. The behavior near these corners can be described by a ``Corner-Operator-Expansion'', which is discussed in the second part of the paper. The analytic predictions agree very well with simulations, with no adjustable parameters.