Entanglement entropies of an interval in the free Schr\"odinger field theory at finite density

TL;DR

This paper analyzes the entanglement entropy of an interval in a non-relativistic free fermionic Schrödinger field at finite density, deriving analytic expressions and revealing non-monotonic behavior of the entropic C function.

Contribution

It provides the first analytic expressions for entanglement entropy in a Lifshitz non-relativistic model and explores how the dynamical exponent affects entanglement properties.

Findings

Entanglement entropy is a finite, monotonically increasing function of a phase space parameter.

Analytic expansions of entanglement entropy are derived for small and large phase space regions.

The relativistic entropic C function is shown to be non-monotonous in this non-relativistic setting.

Abstract

We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schr\"odinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent $z=2$. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic $C$ function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent $z$, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.