Nature of spin glass order in physical dimensions

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 $L>L^{\ast }$, where $L^{\ast }$ 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 $L^{\ast }$. 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.

10+3 pages, 3 + 3 figures, 2 tables