Phase Transition in a Model with Non-Compact Symmetry on Bethe Lattice and the Replica Limit

TL;DR

This paper analyzes a nonlinear vector model with non-compact symmetry on a Bethe lattice, revealing a phase transition and connecting it to supersymmetric models relevant for Anderson localization, highlighting their effectiveness in this context.

Contribution

It demonstrates that non-compact replica models accurately capture Anderson transition features when the replica limit is properly taken.

Findings

Model exhibits a phase transition from ordered to disordered states.

Proper replica limit reduces the model to a supersymmetric form.

Non-compact models correctly describe Anderson transition features.

Abstract

We solve $O(n,1)$ nonlinear vector model on Bethe lattice and show that it exhibits a transition from ordered to disordered state for $0 \leq n < 1$. If the replica limit $n\to 0$ is taken carefully, the model is shown to reduce to the corresponding supersymmetric model. The latter was introduced by Zirnbauer as a toy model for the Anderson localization transition. We argue thus that the non-compact replica models describe correctly the Anderson transition features. This should be contrasted to their failure in the case of the level correlation problem.