Entanglement in the open XX chain: Rényi oscillations, hard-edge crossover, and symmetry resolution

Abstract

We derive closed-form asymptotic formulas for the Rényi 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 Szegő 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 $s^{\pm 1/\alpha }$ 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 $-\frac{1}{2}\log \log \ell$ for symmetry-resolved entropies, together with the known halving of the Gaussian width relative to the periodic chain.

26 pages, 18 figures