Universal corrections to scaling for block entanglement in spin-1/2 XX chains

TL;DR

This paper derives precise asymptotic formulas for the Rénnyi entropies in the spin-1/2 XX chain, revealing detailed corrections and oscillatory behaviors for large block sizes using advanced mathematical methods.

Contribution

It provides the first detailed asymptotic analysis of Rénnyi entropies in the XX chain, including corrections up to order rac{}{}(\u00f6) and oscillatory terms, using Fisher-Hartwig and Painleve9 techniques.

Findings

Asymptotic formulas for Re9nyi entropies with order rac{}{}() accuracy

Explicit leading terms for n=1,2,3,10

Rich oscillatory structure in correction terms

Abstract

We consider the R\'enyi entropies $S_n(\ell)$ in the one dimensional spin-1/2 Heisenberg XX chain in a magnetic field. The case n=1 corresponds to the von Neumann ``entanglement'' entropy. Using a combination of methods based on the generalized Fisher-Hartwig conjecture and a recurrence relation connected to the Painlev\'e VI differential equation we obtain the asymptotic behaviour, accurate to order ${\cal O}(\ell^{-3})$, of the R\'enyi entropies $S_n(\ell)$ for large block lengths $\ell$. For n=1,2,3,10 this constitutes the 3,6,10,48 leading terms respectively. The o(1) contributions are found to exhibit a rich structure of oscillatory behaviour, which we analyze in some detail both for finite $n$ and in the limit $n\to\infty$.