Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems

TL;DR

This paper derives analytical expressions for the entanglement Hamiltonian, spectrum, and contour in inhomogeneous 1D critical systems using conformal field theory, validated by numerical data from the rainbow chain.

Contribution

It provides the first analytical study of entanglement properties in inhomogeneous 1D critical systems with curved spacetime methods, confirmed by numerical results.

Findings

Analytical formulas for entanglement Hamiltonian and spectrum in inhomogeneous systems.

Excellent agreement between theory and numerical data for the rainbow chain.

Introduction of a contour function related to the entanglement Hamiltonian.

Abstract

Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.