Exact ground states of one-dimensional long-range random-field Ising magnets

TL;DR

This paper computes exact ground states of a one-dimensional long-range random-field Ising model using graph algorithms, analyzing phase transitions and critical behavior for various interaction ranges.

Contribution

It provides the first exact ground state calculations for this model at large system sizes across different interaction parameters.

Findings

Identified critical random-field strengths for phase transitions.

Determined critical exponents consistent with theoretical predictions.

Found evidence for a non-zero critical field at  interaction decay.

Abstract

We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p \sim r^{-1-\sigma}$, where $r$ is the distance between these spins and $\sigma$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to $L = 2^{19}$ via graph theoretical algorithms for four different values of $\sigma \in \{0.25,0.4,0.5,1.0\}$ while varying the strength $h$ of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values $h_c(\sigma)$ as well as the corresponding critical exponents are obtained. The results agree well with theory and interestingly we find for $\sigma = 1/2$ the data is compatible with a critical random-field strength $h_c > 0$.