Entanglement entropies of an interval in the free Schr\"odinger field theory on the half line

TL;DR

This paper analyzes the entanglement entropies of an interval near a boundary in a free Schrödinger field theory, revealing oscillatory behavior and providing analytic expressions for different regimes, with extensions to Lifshitz models.

Contribution

It provides the first detailed analytic and numerical study of boundary entanglement entropies in free Schrödinger fields, including Lifshitz generalizations and related charge cumulants.

Findings

Entanglement entropy exhibits oscillations due to Friedel oscillations.

Analytic formulas match numerical results in small and large regimes.

Parity of Lifshitz exponent influences entanglement properties.

Abstract

We study the entanglement entropies of an interval adjacent to the boundary of the half line for the free fermionic spinless Schr\"odinger field theory at finite density and zero temperature, with either Neumann or Dirichlet boundary conditions. They are finite functions of the dimensionless parameter given by the product of the Fermi momentum and the length of the interval. The entanglement entropy displays an oscillatory behaviour, differently from the case of the interval on the whole line. This behaviour is related to the Friedel oscillations of the mean particle density on the half line at the entangling point. We find analytic expressions for the expansions of the entanglement entropies in the regimes of small and large values of the dimensionless parameter. They display a remarkable agreement with the curves obtained numerically. The analysis is extended to a family of free fermionic Lifshitz models labelled by their integer Lifshitz exponent, whose parity determines the properties of the entanglement entropies. The cumulants of the local charge operator and the Schatten norms of the underlying kernels are also explored.