Droplet States in the XXZ Heisenberg Chain

TL;DR

This paper analyzes the ground states of the ferromagnetic XXZ Heisenberg chain with fixed down spins, identifying droplet states, their energies, and spectral gaps, revealing an isolated band structure for large droplets.

Contribution

It provides exact solutions for droplet states in the XXZ chain and demonstrates the existence of spectral gaps and isolated bands for large droplet sizes.

Findings

Exact energy and states for droplet configurations

Existence of a spectral gap above droplet states

Droplet bands become isolated as droplet size increases

Abstract

We consider the ground states of the ferromagnetic XXZ chain with spin up boundary conditions in sectors with a fixed number of down spins. This forces the existence of a droplet of down spins in the system. We find the exact energy and the states that describe these droplets in the limit of an infinite number of down spins. We prove that there is a gap in the spectrum above the droplet states. As the XXZ Hamiltonian has a gap above the fully magnetized ground states as well, this means that the droplet states (for sufficiently large droplets) form an isolated band. The width of this band tends to zero in the limit of infinitely large droplets. We also prove the analogous results for finite chains with periodic boundary conditions and for the infinite chain.