The droplet-scaling versus replica symmetry breaking debate in spin glasses revisited

TL;DR

This paper revisits the debate between droplet-scaling and replica symmetry breaking in spin glasses, proposing that the replicon exponent's behavior depends on system size and crossover length, clarifying previous conflicting results.

Contribution

It introduces a size-dependent perspective on the replicon exponent, linking it to the droplet-scaling picture and providing a formula that aligns with observed data.

Findings

The replicon exponent $eta$ is zero only for systems larger than a crossover length $L^*$.

For smaller systems, $eta$ approximates $2 heta$, consistent with reported values.

The crossover length $L^*$ can be very large, affecting interpretations of spin glass phases.

Abstract

Simulational studies of spin glasses in the last decade have focussed on the so-called replicon exponent $\alpha$ as a means of determining whether the low-temperature phase of spin glasses is described by the replica symmetry breaking picture of Parisi or by the droplet-scaling picture. On the latter picture, it should be zero, but we shall argue that it will only be zero for systems of linear dimension $L > L^*$. The crossover length $L^*$ may be of the order of hundreds of lattice spacings in three dimensions and approach infinity in 6 dimensions. We use the droplet-scaling picture to show that the apparent non-zero value of $\alpha$ when $L < L^*$ should be $2 \theta$, where $\theta$ is the domain wall energy scaling exponent, This formula is in reasonable agreement with the reported values of $\alpha$.