Domain walls and Schramm-Loewner evolution in the random-field Ising model

TL;DR

This study investigates domain walls in the two-dimensional random-field Ising model, providing evidence that they are conformally invariant and follow Schramm-Loewner evolution with parameter 6 near the percolation transition.

Contribution

It demonstrates that ground state domain walls in the 2D random-field Ising model exhibit conformal invariance and SLE$_{}$ behavior, linking disordered systems to conformal field theory.

Findings

Domain walls are conformally invariant near the percolation transition.

Ground state domain walls satisfy the domain Markov property.

Evidence supports SLE$_{}$ with =6 for these domain walls.

Abstract

The concept of Schramm-Loewner evolution provides a unified description of domain boundaries of many lattice spin systems in two dimensions, possibly even including systems with quenched disorder. Here, we study domain walls in the random-field Ising model. Although, in two dimensions, this system does not show an ordering transition to a ferromagnetic state, in the presence of a uniform external field spin domains percolate beyond a critical field strength. Using exact ground state calculations for very large systems, we examine ground state domain walls near this percolation transition finding strong evidence that they are conformally invariant and satisfy the domain Markov property, implying compatibility with Schramm-Loewner evolution (SLE$_{\kappa}$) with parameter $\kappa = 6$. These results might pave the way for new field-theoretic treatments of systems with quenched disorder.