Entanglement Entropy of Fermi Liquids via Multi-dimensional Bosonization

TL;DR

This paper demonstrates that the logarithmic violation of the area law in entanglement entropy originates from the Fermi surface discontinuity and persists in Fermi liquids with interactions, using multi-dimensional bosonization and perturbative analysis.

Contribution

It extends the understanding of entanglement entropy in Fermi liquids by showing the persistence of logarithmic violations with interactions using bosonization and perturbation theory.

Findings

Logarithmic violations originate from Fermi surface discontinuity.

Interactions do not alter the leading entanglement entropy scaling.

The approach relates violations in 1D and high-dimensional free fermions.

Abstract

The logarithmic violations of the area law, i.e. an "area law" with logarithmic correction of the form $S \sim L^{d-1} \log L$, for entanglement entropy are found in both 1D gapless system and for high dimensional free fermions. The purpose of this work is to show that both violations are of the same origin, and in the presence of Fermi liquid interactions such behavior persists for 2D fermion systems. In this paper we first consider the entanglement entropy of a toy model, namely a set of decoupled 1D chains of free spinless fermions, to relate both violations in an intuitive way. We then use multi-dimensional bosonization to re-derive the formula by Gioev and Klich [Phys. Rev. Lett. 96, 100503 (2006)] for free fermions through a low-energy effective Hamiltonian, and explicitly show the logarithmic corrections to the area law in both cases share the same origin: the discontinuity at the Fermi surface (points). In the presence of Fermi liquid (forward scattering) interactions, the bosonized theory remains quadratic in terms of the original local degrees of freedom, and after regularizing the theory with a mass term we are able to calculate the entanglement entropy perturbatively up to second order in powers of the coupling parameter for a special geometry via the replica trick. We show that these interactions do not change the leading scaling behavior for the entanglement entropy of a Fermi liquid. At higher orders, we argue that this should remain true through a scaling analysis.