Modular Hamiltonians for the massless Dirac field in the presence of a defect

TL;DR

This paper derives the modular Hamiltonians for a massless Dirac field with a point defect, revealing phase-dependent bi-local terms and showing entanglement entropy independence from scattering details.

Contribution

It provides explicit modular Hamiltonians for the Dirac field with a defect, including bi-local terms and analysis of modular flows and entanglement entropy.

Findings

Modular Hamiltonians include local and scattering-dependent bi-local terms.

Two phases preserve either vector or axial symmetry.

Entanglement entropy is independent of the scattering matrix.

Abstract

We study the massless Dirac field on the line in the presence of a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. Considering this system in its ground state, we derive the modular Hamiltonians of the subregion given by the union of two disjoint equal intervals at the same distance from the defect. The absence of energy dissipation at the defect implies the existence of two phases, where either the vector or the axial symmetry is preserved. Besides a local term, the densities of the modular Hamiltonians contain also a sum of scattering dependent bi-local terms, which involve two conjugate points generated by the reflection and the transmission. The modular flows of each component of the Dirac field mix the trajectory passing through a given initial point with the ones passing through its reflected and transmitted conjugate points. We derive the two-point correlation functions along the modular flows in both phases and show that they satisfy the Kubo-Martin-Schwinger condition. The entanglement entropies are also computed, finding that they do not depend on the scattering matrix.