Boundary Rényi oscillations in the open XX chain

Abstract

For the ground state of the semi-infinite XX chain with Fermi momentum $k_F$, we derive the extracted $2k_F$ contribution to the Rényi entropy of its first $\ell$ sites at fixed $\alpha >1$. Its phase $\sin [(2\ell +1)k_F]$ matches the Friedel oscillation at the entangling bond and becomes the even--odd alternation at half filling. An exact Hankel determinant reduction and Riemann--Hilbert analysis give every term ${\mathcal{L}}^{-(2q-1)/\alpha }$ for positive integers $q$ with $2q-1<\alpha$, where $\mathcal{L}=4\ell \sin k_F$, with an $O({\mathcal{L}}^{-1})$ error within this extracted component for $k_F$ away from $0$ and $\pi$. At integer $\alpha =p\ge 2$, exact factorization gives a finite sum over all roots with error $O({\mathcal{L}}^{-1-1/p})$. We also determine the $2k_F$ terms of $\log Z_{\alpha }(\varphi )$, where $Z_{\alpha }(\varphi )=\mathrm{Tr}({\rho }_A^{\alpha }{e}^{i\varphi Q_A})$, ${\rho }_A$ is the block's reduced density matrix, and $Q_A$ is its fermion number. At fixed nonzero flux $|\varphi |<\pi$ with $2q-1\mp \varphi /\pi \ne \alpha$, the two exponential terms have different decay powers. Exact factorization sharpens the integer result. These pointwise estimates do not justify projection to fixed charge. The uncharged sequence and the two leading charged terms agree with earlier generalized Fisher--Hartwig predictions. Numerical calculations at $p=4$ test the predicted subleading coefficients without an adjustable normalization.

arXiv commentv2: 28 pages, 3 figures. Substantially revised