Droplet states in quantum XXZ spin systems on general graphs

TL;DR

This paper investigates droplet states in quantum XXZ spin systems on various graphs, analyzing their spectral properties, formation, and stability, especially on strips and Euclidean lattices, using mathematical tools like adjacency matrices and Laplacians.

Contribution

It provides a detailed description of droplet states in XXZ models on general graphs and establishes spectral gap existence and exponential decay properties of eigenstates.

Findings

Droplet states form near the spectrum's bottom in the Ising phase.

Spectral gaps exist above the droplet regime on certain graphs.

Eigenstates are exponentially small perturbations of classical droplets.

Abstract

We study XXZ spin systems on general graphs. In particular, we describe the formation of droplet states near the bottom of the spectrum in the Ising phase of the model, where the Z-term dominates the XX-term. As key tools we use particle number conservation of XXZ systems and symmetric products of graphs with their associated adjacency matrices and Laplacians. Of particular interest to us are strips and multi-dimensional Euclidean lattices, for which we discuss the existence of spectral gaps above the droplet regime. We also prove a Combes-Thomas bound which shows that the eigenstates in the droplet regime are exponentially small perturbations of strict (classical) droplets.