Ising surface defects can get dirty

TL;DR

This paper studies the effects of surface disorder in 3D Ising systems near criticality, revealing stable boundary conditions and a novel 'dirty' boundary state through theoretical analysis.

Contribution

It introduces a non-trivial 'dirty' boundary condition in disordered critical systems and analyzes its stability and spectral properties using the replica method.

Findings

The ordinary boundary condition remains stable under surface disorder.

A new 'dirty' boundary condition can be achieved by tuning disorder or temperature.

Logarithmic structures in the defect spectrum are identified and analyzed.

Abstract

Real critical systems, such as uniaxial ferromagnets in the 3d Ising universality class, are constrained by boundaries and subject to random couplings. We consider the Wilson-Fisher fixed point in $4-\epsilon$ dimensions subject to a random magnetic field localized on a two-dimensional surface, which becomes co-dimension 1 in the physical $\epsilon\to1$ limit. Using the replica method for the disordered field, we find that the ordinary boundary condition is stable under disorder but also discover a non-trivial ``dirty" boundary condition which can be reached by tuning the disorder strength or the local temperature. We also investigate the logarithmic structure of the defect spectrum and how it emerges via the replica formalism.