Entanglement entropy of two disjoint intervals separated by one spin in an XX quantum spin chain

We calculate the entanglement entropy of a non-contiguous subsystem of an XX spin chain with zero magnetic field. Following the success of the Fisher-Hartwig formula for describing the entropy of a single block subsystem in an XX spin chain, Jin and Korepin recently suggested a model to represent the entropy of two disjoint intervals with lattice site index $\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$. As a first step in the asymptotic analysis of this model we consider its simplification to two disjoint intervals separated just by one spin, and calculate the mutual information between these two blocks. Our main tool is to examine the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol. To do this, we shall utilise the Riemann--Hilbert method.