The infinite XXZ quantum spin chain revisited: Structure of low lying spectral bands and gaps

TL;DR

This paper analyzes the spectral structure of the infinite XXZ quantum spin chain, revealing detailed band and gap structures at low energy, especially under strong anisotropy, and provides a comprehensive mathematical introduction to the model.

Contribution

It offers a detailed spectral analysis of the infinite XXZ chain, particularly characterizing low-energy bands and gaps in the Ising phase for strong anisotropy, with a self-contained presentation.

Findings

Droplet bands are separated from higher spectral bands at strong anisotropy.

The spectrum's band and gap structure is explicitly determined at low energy.

The presentation serves as an accessible introduction to the mathematical theory of XXZ chains.

Abstract

We study the structure of the spectrum of the infinite XXZ quantum spin chain, an anisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves the number of down spins (or particle number), allowing to represent it as a direct sum of N-particle interacting discrete Schr\"odinger-type operators restricted to the fermionic subspace. In the Ising phase of the model we use this representation to give a detailed determination of the band and gap structure of the spectrum at low energy. In particular, we show that at sufficiently strong anisotropy the so-called droplet bands are separated from higher spectral bands uniformly in the particle number. Our presentation of all necessary background is self-contained and can serve as an introduction to the mathematical theory of the Heisenberg and XXZ quantum spin chains.