Nature of spin glass order in physical dimensions

TL;DR

This study investigates the nature of spin glass order in a diluted Heisenberg model with long-range interactions, revealing finite size effects and features consistent with TNT and replica symmetry breaking theories.

Contribution

It provides new insights into spin glass behavior in low dimensions, especially regarding finite size effects and the emergence of TNT features in long-range models.

Findings

Correlation length varies with random field as expected from Imry-Ma and droplet theories.

Finite size effects produce TNT features in spin glasses.

Large system sizes diminish TNT features for certain parameters.

Abstract

We have studied the diluted Heisenberg spin glass model in a 3-component random field for the commonly-used one-dimensional long-range model where the probability that two spins separated by a distance $r$ interact with one another falls as $1/r^{2 \sigma}$, for two values of $\sigma$, $0.75$ and $0.85$. No de Almeida-Thouless line is expected at these $\sigma$ values. The spin glass correlation length $\xi_{\text{SG}}$ varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when $\xi_{\text{SG}}$ becomes comparable to the system size $L$, there are departures which we attribute to the features deriving from the TNT picture of spin glasses. For the case $\sigma =0.85$ these features go away for system sizes with $L >L^*$, where $L^*$ is large ($\approx 4000-8000$ lattice spacings). In the case of $\sigma = 0.75$ we have been unable to study large enough systems to determine its value of $L^*$. We sketch a renormalization group scenario to explain how these features could arise. On this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce TNT features and some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.