# Comparison of Gabay-Toulouse and de Almeida-Thouless instabilities for the spin glass XY model in a field on sparse random graphs

## TL;DR

This paper analytically compares the de Almeida-Thouless and Gabay-Toulouse phase transition lines in XY spin glasses on sparse random graphs, highlighting their distinct nature and dependence on field distribution.

## Contribution

It provides the first analytical computation of both critical lines for XY spin glasses on random regular graphs, exploring their differences and crossover behavior.

## Key findings

- Analytical expressions for critical lines on sparse graphs.
- Distinct nature of de Almeida-Thouless and Gabay-Toulouse transitions.
- Crossover behavior depending on field distribution.

## Abstract

Vector spin glasses are known to show two different kinds of phase transitions in presence of an external field: the so-called de Almeida-Thouless and Gabay-Toulouse lines. While the former has been studied to some extent on several topologies (fully connected, random graphs, finite-dimensional lattices, chains with long-range interactions), the latter has been studied only in fully connected models, which however are known to show some unphysical behaviors (e.g. the divergence of these critical lines in the zero-temperature limit). Here we compute analytically both these critical lines for XY spin glasses on random regular graphs. We discuss the different nature of these phase transitions and the dependence of the critical behavior on the field distribution. We also study the crossover between the two different critical behaviors, by suitably tuning the field distribution.