Fluctuations in the non-linear supersymmetric hyperbolic sigma model with long-range interactions

TL;DR

This paper investigates a class of non-linear supersymmetric hyperbolic sigma models with long-range interactions, demonstrating that the associated random field exhibits arbitrarily small fluctuations at large interaction strengths, indicating symmetry breaking.

Contribution

It proves that the random field in these models has asymptotically small fluctuations for large interactions, a strong form of Lorentz boost symmetry breaking.

Findings

Random field fluctuations become arbitrarily small at large interactions

Symmetry breaking is demonstrated in the models

Results hold uniformly in box size

Abstract

We consider a class of non-linear supersymmetric hyperbolic sigma models with long-range interactions on boxes in $\mathbb{Z}^d$ and on a hierarchical lattice. We prove that the random field associated to a marginal in horospherical coordinates has asymptotically arbitrarily small fluctuations for large enough interactions, uniformly in the size of the boxes. This can be viewed as a strong version of spontaneous breaking of the Lorentz boost symmetry.