Stability of energy landscape for Ising models

TL;DR

This paper investigates the stability of the energy landscape of Ising models under perturbations to couplings, fields, and graph structure, providing conditions for ground state stability and probabilistic bounds on energy gaps.

Contribution

It introduces sufficient conditions for stability of ground states under perturbations and estimates the probability of energy gap bounds in Gaussian random Ising models.

Findings

Ground states are stable under certain perturbations with preserved energy order.

Probabilistic bounds on energy gaps between original and perturbed Hamiltonians.

Concrete example demonstrating stability under graph perturbations.

Abstract

In this paper, we explore the stability of the energy landscape of an Ising Hamiltonian when subjected to two kinds of perturbations: a perturbation on the coupling coefficients and external fields, and a perturbation on the underlying graph structure. We give sufficient conditions so that the ground states of a given Hamiltonian are stable under perturbations of the first kind in terms of order preservation. Here by order preservation we mean that the ordering of energy corresponding to two spin configurations in a perturbed Hamiltonian will be preserved in the original Hamiltonian up to a given error margin. We also estimate the probability that the energy gap between ground states for the original Hamiltonian and the perturbed Hamiltonian is bounded by a given error margin when the coupling coefficients and local external magnetic fields of the original Hamiltonian are i.i.d. Gaussian random variables. In the end we show a concrete example of a system which is stable under perturbations of the second kind.