Boundary emptiness formation probabilities in the six-vertex model at $\Delta = -\frac12$

TL;DR

This paper introduces a new family of overlaps for the XXZ Hamiltonian and derives explicit formulas for boundary emptiness formation probabilities at the special point .5, connecting quantum spin chains and six-vertex model correlations.

Contribution

It defines boundary emptiness formation probabilities as ratios of overlaps and provides explicit closed-form formulas at .5, a significant step in understanding these correlations.

Findings

Explicit formulas for overlaps at .5

Closed-form expressions for boundary emptiness probabilities

Connection between XXZ model overlaps and six-vertex model correlations

Abstract

We define a new family of overlaps $C_{N,m}$ for the XXZ Hamiltonian on a periodic chain of length $N$. These are equal to the linear sums of the groundstate components, in the canonical basis, wherein $m$ consecutive spins are fixed to the state ${\uparrow}$. We define the boundary emptiness formation probabilities as the ratios $C_{N,m}/C_{N,0}$ of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter $N$. At the combinatorial point $\Delta = -\frac12$, we obtain closed-form expressions in terms of simple products of ratios of integers.