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

We consider the Rényi 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 Painlevé VI differential equation we obtain the asymptotic behaviour, accurate to order ${\cal O}(\ell^{-3})$, of the Rényi 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 $n\to\infty$.