# The Fractal Dimension of Interfaces in Edwards-Anderson and Long-range Ising Spin Glasses: Determining the Applicability of Different Theoretical Descriptions

## TL;DR

This study investigates the fractal dimension of interfaces in spin glasses across different dimensions using renormalization group methods, revealing the transition point where replica symmetry breaks and space-filling excitations emerge.

## Contribution

It applies the strong-disorder renormalization group method to analyze the fractal dimension of domain walls in high-dimensional and long-range spin glasses, clarifying the applicability of theoretical models.

## Key findings

- Replica symmetry breaks in high enough dimensions.
- Space-filling excitations occur for dimensions 6 and above.
- Results inform the appropriate theoretical description of spin glasses.

## Abstract

The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.