Title: Mutual information of two intervals in quantum XX spin chain - a Riemann-Hilbert approach
Abstract: In this talk we consider the quantum XX spin chain in its ground state and in the thermodynamic limit. In 2007, A.R. Its, B.-Q. Jin and V.E. Korepin calculated the asymptotic behaviour of the entanglement entropy of an interval of length $n$ (i.e. a block of $n$ consecutive particles) as $n\to\infty$. It is a very natural question what happens if we consider a more complicated subsystem of particles, for instance, a union of two intervals?
In my talk I will present our most recent result on the case when the subsystem is such a union, where the first interval has length $m$, the second has length $n$, and the two intervals are separated by a gap of fixed length 1. Namely, we calculate the mutual information between the two intervals as $m,n\to\infty$, and hence compute the limiting entropy of the mentioned subsystem. We will see that this problem leads to a rather complicated mathematical problem, namely, to the estimation of a certain inner product involving a Toeplitz matrix whose symbol possesses Fisher-Hartwig singularities. Using techniques from the theory of integrable operators we connect this problem first to the famous Fokas-Its-Kitaev Riemann-Hilbert problem, and then to the $R$-Riemann-Hilbert problem appearing in the celebrated 2011 paper of P. Deift, A.R. Its and I. Krasovsky, in which they solved the Fisher-Hartwig conjecture.
A joint work with A.R. Its, V.E. Korepin, F. Mezzadri, J. Virtanen.
Alexi Morin-Duchesne
