Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

TL;DR

This paper extends conformal field theory methods to analyze entanglement and relative entropies in low-lying excited states of inhomogeneous one-dimensional free-fermionic systems, revealing universal scaling behaviors.

Contribution

It generalizes the CFT approach to excited states in inhomogeneous systems and identifies a universal scaling function expressed through a new variable related to the curved metric.

Findings

Universal scaling function matches homogeneous systems when expressed in the new variable.

Exact numerical results in trapped Fermi gases confirm the theoretical predictions.

The new scaling variable depends on subsystem size and inhomogeneity, derived from the curved metric.

Abstract

Conformal field theories in curved backgrounds have been used to describe inhomogeneous one-dimensional systems, such as quantum gases in trapping potentials and non-equilibrium spin chains. This approach provided, in a elegant and simple fashion, non-trivial analytic predictions for quantities, such as the entanglement entropy, that are not accessible through other methods. Here, we generalise this approach to low-lying excited states, focusing on the entanglement and relative entropies in an inhomogeneous free-fermionic system. Our most important finding is that the universal scaling function characterising these entanglement measurements is the same as the one for homogeneous systems, but expressed in terms of a different variable. This new scaling variable is a non-trivial function of the subsystem length and system's inhomogeneity that is easily written in terms of the curved metric. We test our predictions against exact numerical calculations in the free Fermi gas trapped by a harmonic potential, finding perfect agreement.