# Open XXZ chain and boundary modes at zero temperature

## TL;DR

This paper analyzes the open XXZ spin chain at zero temperature, revealing boundary-induced degeneracies, localized boundary excitations, and their effects on local magnetization and spin autocorrelations, with exact solutions for even and odd chain lengths.

## Contribution

It provides an exact Bethe ansatz analysis of boundary modes and their impact on ground state degeneracy, local magnetization, and autocorrelation functions in the open XXZ chain.

## Key findings

- Ground state is doubly degenerate for even length chains with equal boundary fields.
- Boundary roots localize excitations at the edges, affecting local magnetization.
- Explicit expression for boundary autocorrelation function at large times.

## Abstract

We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.