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

2022/02/15 by Saenz, Axel, Tracy, Craig A., Widom, Harold · 1 citation
#60K35 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2202.07695

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 Δ, also known as the XXZ spin-1/2 chain, on the one-dimensional lattice ℤ with domain wall initial conditions. We use the Bethe Ansatz to solve the Schrödinger 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 Δ=0 case, the distribution F₂ arises. In the Δ≠ 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 Δ→ 0 limit and recovers the result for Δ= 0.

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