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

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$\ddot{o}$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 $\Delta \ne 0$ 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 $\Delta \to 0$ limit and recovers the result for $\Delta =0$.

41 pages, 4 figures. Version 2 corrects a typo in notation for equation (98)