On Small Perturbations of a Spin Glass System

TL;DR

This paper demonstrates that small, undetectable perturbations to a spin glass Hamiltonian can significantly change the system's overlap structure, leading to ultrametricity emergence at the thermodynamic limit.

Contribution

It reveals that perturbations of order O(log N), previously thought negligible, can induce ultrametricity in spin glasses, showing a new sensitivity in the system's behavior.

Findings

Perturbations of order O(log N) can alter the overlap structure.

Undetectable perturbations at the free energy level can have large effects.

Ultrametricity emerges due to small Hamiltonian perturbations.

Abstract

We show through a simple example that perturbations of the Hamiltonian of a spin glass which cannot be detected at the level of the free energy can completely alter the behavior of the overlap. In particular, perturbations of order O(log N), with N the size of the system, suffice to have ultrametricity emerge in the thermodynamical limit.