We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv:1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m } ∪ {2 m + 1, 2 m + 2, …, 3 m }, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.

中文翻译：

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自由费米子链中两个不相交间隔的缠结熵

我们计算了一个自由费米子链的不连续子系统的纠缠熵。起始点是Jin和Korepin建议的公式，arXiv：1104.1004，用于降低晶格位置P = {1,2，…，m} {2 m + 1，2 m + 2，…，3 m}，适用于此模型。作为该系统渐近分析的第一步，我们考虑将其简化为仅由一个位点分隔的两个不相交的间隔，并严格计算这两个块与链的其余部分之间的互信息。为了计算熵，我们需要使用Riemann-Hilbert方法研究带有Fisher-Hartwig符号的Toeplitz逆矩阵的渐近行为。