Scaling behavior of domain walls at the T=0 ferromagnet to spin-glass transition

TL;DR

This paper investigates the scaling behavior of domain walls in 2D disordered Ising models at zero temperature, identifying critical disorder parameters and analyzing how domain-wall properties scale with system size near the transition.

Contribution

It introduces a novel graph-theoretical approach using minimum-weight perfect-matching algorithms to study domain walls and provides finite-size scaling analysis of critical exponents at the transition.

Findings

Critical disorder parameters for ferromagnet to spin-glass transition identified.

Exponents for domain-wall energy and length remain constant in the spin-glass phase.

Results consistent with conformal field theory and SLE predictions.

Abstract

We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length L, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows to tune the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic groundstate. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the the average domain-wall energy (~L^\theta) and the average domain-wall length (~L^df). Performing a finite-size scaling analysis for systems with a side length up to L=512, we find that both exponents remain constant in the spin-glass phase, i.e. \theta ~- 0.28 and df~1.275. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of Stochastic Loewner evolutions (SLEs) via df-1=3/[4(3+\theta)]. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.