L. Brightmore | Searcharxiv

arXiv2019

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion

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, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, 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.

math-ph↗

arXiv2013

A matrix model with a singular weight and Painleve' III

We investigate the matrix model with weight $w(x):=\exp(-z^2/2x^2 + t/x - x^2/2)$ and unitary symmetry. and unitary symmetry. In particular we study the double scaling limit as $N \to \infty$ and $(\sqrt{N} t, Nz^2 ) \to (u_1,u_2)$, where $N$ is the matrix dimension and the parameters $(u_1,u_2)$ remain finite. Using the Deift-Zhou steepest descent method we compute the asymptotics of the partition function when $z$ and $t$ are of order $O\bigl(N^{-1/2}\bigr)$. In this regime we discover a phase transition in the $(z,N)$-plane characterised by the Painleve' III equation. This is the first time that Painleve' III appears in studies of double scaling limits in Random Matrix Theory and is associated to the emergence of an essential singularity in the weighting function. The asymptotics of the partition function is expressed in terms of a particular solution of the Painleve' III equation. We derive explicitly the initial conditions in the limit $Nz^2\rightarrow u_2$ of this solution.

math-ph↗