Entanglement entropies of inhomogeneous Luttinger liquids

TL;DR

This paper develops a framework for calculating entanglement entropy scaling in inhomogeneous Luttinger liquids, addressing the challenges posed by position-dependent parameters and UV cutoff contributions, validated through numerical checks.

Contribution

It introduces a method to compute entanglement entropy in inhomogeneous Luttinger liquids with position-dependent parameters, overcoming the limitations of conformal invariance.

Findings

The framework accurately predicts entanglement entropy in inhomogeneous systems.

Numerical validation shows excellent agreement with the spin-1/2 XXZ chain results.

Proper handling of UV cutoff contributions is crucial for correct entropy calculations.

Abstract

We develop a general framework to compute the scaling of entanglement entropy in inhomogeneous one-dimensional quantum systems belonging to the Luttinger liquid universality class. While much insight has been gained in homogeneous systems by making use of conformal field theory techniques, our focus is on systems for which the Luttinger parameter $K$ depends on position, and conformal invariance is broken. An important point of our analysis is that contributions stemming from the UV cutoff have to be treated very carefully, since they now depend on position. We show that such terms can be removed either by considering regularized entropies specifically designed to do so, or by tabulating numerically the cutoff, and reconstructing its contribution to the entropy through the local density approximation. We check our method numerically in the spin-1/2 XXZ spin chain in a spatially varying magnetic field, and find excellent agreement.