Fractal dimension of interfaces in Edwards-Anderson spin glasses for up to six space dimensions

TL;DR

This study investigates the fractal dimension of interfaces in Edwards-Anderson spin glasses across up to six dimensions, revealing a transition at six dimensions where replica symmetry breaking occurs.

Contribution

It provides the first comprehensive analysis of interface fractal dimensions in high-dimensional spin glasses using multiple algorithms, highlighting a critical dimension for replica symmetry breaking.

Findings

Fractal dimension is less than space dimension in ≤5D.

Fractal dimension approaches space dimension at 6D.

Replica symmetry breaking occurs above 6D.

Abstract

The fractal dimension of domain walls produced by changing the boundary conditions from periodic to anti-periodic in one spatial direction is studied using both the strong-disorder renormalization group and the greedy algorithm for the Edwards-Anderson Ising spin-glass model for up to six space dimensions. We find that for five or less space dimensions, the fractal dimension is less than the space dimension. This means that interfaces are not space filling, thus implying replica symmetry breaking is absent in space dimensions fewer than six. However, the fractal dimension approaches the space dimension in six dimensions, indicating that replica symmetry breaking occurs above six dimensions. In two space dimensions, the strong-disorder renormalization group results for the fractal dimension are in good agreement with essentially exact numerical results, but the small difference is significant. We discuss the origin of this close agreement. For the greedy algorithm there is analytical expectation that the fractal dimension is equal to the space dimension in six dimensions and our numerical results are consistent with this expectation.