Maximally Random Discrete-Spin Systems with Symmetric and Asymmetric Interactions and Maximally Degenerate Ordering

TL;DR

This paper investigates discrete-spin systems with maximally random interactions, revealing that in dimensions greater than one, they tend to a maximally degenerate ordered state at all finite temperatures, with a detailed analysis of entropy and disordering behavior.

Contribution

It introduces a renormalization-group analysis of maximally random discrete-spin systems with symmetric and asymmetric interactions, highlighting their universal degenerate ordering in higher dimensions.

Findings

Systems renormalize to a maximally degenerate ordered state in d>1.

Entropy shows a maximum at the short-range disordering temperature.

Disordered at all temperatures in one-dimensional systems.

Abstract

Discrete-spin systems with maximally random nearest-neighbor interactions that can be symmetric or asymmetric, ferromagnetic or antiferromagnetic, including off-diagonal disorder, are studied, for the number of states $q=3,4$ in $d$ dimensions. We use renormalization-group theory that is exact for hierarchical lattices and approximate (Migdal-Kadanoff) for hypercubic lattices. For all d>1 and all non-infinite temperatures, the system eventually renormalizes to a random single state, thus signaling qxq degenerate ordering. Note that this is the maximally degenerate ordering. For high-temperature initial conditions, the system crosses over to this highly degenerate ordering only after spending many renormalization-group iterations near the disordered (infinite-temperature) fixed point. Thus, a temperature range of short-range disorder in the presence of long-range order is identified, as previously seen in underfrustrated Ising spin-glass systems. The entropy is calculated for all temperatures, behaves similarly for ferromagnetic and antiferromagnetic interactions, and shows a derivative maximum at the short-range disordering temperature. With a sharp immediate contrast of infinitesimally higher dimension 1+\epsilon, the system is as expected disordered at all temperatures for d=1.