Domain Walls in the Heisenberg-Ising Spin-1/2 Chain

TL;DR

This paper derives formulas for the distribution of the left-most up-spin in the Heisenberg-Ising spin-1/2 chain with domain wall initial conditions, using Bethe Ansatz and antisymmetrization techniques, proposing a conjectural series expansion for general anisotropy.

Contribution

It introduces a new conjectural series expansion formula for the spin distribution in the XXZ chain with domain wall initial conditions, extending known results at zero anisotropy.

Findings

Derived formulas for the distribution of the left-most up-spin.

Proposed a conjectural series expansion for non-zero anisotropy.

Reproduces known results in the zero anisotropy limit.

Abstract

In this paper we obtain formulas for the distribution of the left-most up-spin in the Heisenberg-Ising spin-1/2 chain with anisotropy parameter $\Delta$, also known as the XXZ spin-1/2 chain, on the one-dimensional lattice $\mathbb{Z}$ with domain wall initial conditions. We use the Bethe Ansatz to solve the Schr$\"o$dinger equation and a recent antisymmetrization identity of Cantini, Colomo, and Pronko (arXiv:1906.07636) to simplify the marginal distribution of the left-most up-spin. In the $\Delta=0$ case, the distribution $F_2$ arises. In the $\Delta \neq 0$ case, we propose a conjectural series expansion type formula based on a saddle point analysis. The conjectural formula turns out to be a Fredholm series expansion in the $\Delta \rightarrow 0$ limit and recovers the result for $\Delta = 0$.