Percolation and Schramm-Loewner evolution in the 2D random-field Ising model

TL;DR

This study investigates the percolation properties and conformal invariance of ground states in the 2D random-field Ising model, finding evidence of SLE$_6$ behavior in certain regimes despite the absence of a zero-field percolation transition.

Contribution

It provides the first evidence that ground-state domain walls in the 2D RFIM exhibit Schramm-Loewner evolution with $=6$, suggesting conformal invariance in disordered systems.

Findings

No evidence of a zero-field percolation transition in ground states.

Ground-state domain walls are consistent with SLE$_6$.

Finite samples show critical percolation-like behavior in a specific parameter regime.

Abstract

The presence of random fields is well known to destroy ferromagnetic order in Ising systems in two dimensions. When the system is placed in a sufficiently strong external field, however, the size of clusters of like spins diverges. There is evidence that this percolation transition is in the universality class of standard site percolation. It has been claimed that, for small disorder, a similar percolation phenomenon also occurs in zero external field. Using exact algorithms, we study ground states of large samples and find little evidence for a transition at zero external field. Nevertheless, for sufficiently small random field strengths, there is an extended region of the phase diagram, where finite samples are indistinguishable from a critical percolating system. In this regime we examine ground-state domain walls, finding strong evidence that they are conformally invariant and satisfy Schramm-Loewner evolution ($SLE_{\kappa}$) with parameter $\kappa = 6$. These results add support to the hope that at least some aspects of systems with quenched disorder might be ultimately studied with the techniques of SLE and conformal field theory.