Entanglement in the open XX chain: R\'enyi oscillations, hard-edge crossover, and symmetry resolution

TL;DR

This paper derives asymptotic formulas for Re9nyi entanglement entropies in the open XX chain, revealing oscillation behaviors and crossover phenomena near the band edge using advanced mathematical techniques.

Contribution

It introduces a novel analytical approach to evaluate entanglement entropies in open XX chains, connecting Toeplitz-plus-Hankel determinants to Riemann--Hilbert results for asymptotic analysis.

Findings

Derived explicit formulas for oscillatory amplitudes and phases.

Identified power-law and logarithmic behaviors in the crossover regime.

Confirmed symmetry-resolved entropy offset and Gaussian width halving.

Abstract

We derive closed-form asymptotic formulas for the R\'enyi entanglement entropies of the open XX spin-$1/2$ chain by mapping the underlying determinant of the boundary correlation matrix (which has Toeplitz-plus-Hankel structure) to a Hankel determinant with a positive weight whose large-size asymptotics follow from known Riemann--Hilbert results. An explicit evaluation of the Szeg\H{o} function yields the leading $2k_F$ oscillatory amplitude and phase. A single variable $s = 2\ell \sin(k_F/2)$ organizes the hard-edge crossover as the Fermi momentum approaches the band edge: the oscillation envelope obeys $s^{\pm1/\alpha}$ power laws and $\ln s$ is the natural leading logarithm for a clean data collapse. For detached blocks the oscillatory amplitude is numerically consistent with a factorization through the conformal cross-ratio. The same framework recovers the open-boundary-condition (OBC) equipartition offset $-\tfrac{1}{2}\log\log\ell$ for symmetry-resolved entropies, together with the known halving of the Gaussian width relative to the periodic chain.