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

TL;DR

This paper derives explicit modular Hamiltonians for a massless Dirac fermion on a half-line with boundary conditions that preserve either vector or axial symmetry, analyzing their flows, correlations, and entanglement entropy.

Contribution

It provides the first explicit forms of modular Hamiltonians for the Dirac field with boundary conditions, revealing phase-dependent symmetries and modular flows.

Findings

Explicit modular Hamiltonians for two phases

Modular flows preserve different symmetries

Entanglement entropy computed for both phases

Abstract

We study the modular Hamiltonians of an interval for the massless Dirac fermion on the half-line. The most general boundary conditions ensuring the global energy conservation lead to consider two phases, where either the vector or the axial symmetry is preserved. In these two phases we derive the corresponding modular Hamiltonian in explicit form. Its density involves a bi-local term localised in two points of the interval, one conjugate to the other. The associated modular flows are also established. Depending on the phase, they mix fields with different chirality or charge that follow different modular trajectories. Accordingly, the modular flow preserves either the vector or the axial symmetry. We compute the two-point correlation functions along the modular flow and show that they satisfy the Kubo-Martin-Schwinger condition in both phases. The entanglement entropies are also derived.