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.
Highlights
In generic spacetime dimensions, is the modular Hamiltonian of half space x > 0 for a Lorentz invariant quantum field theory in its vacuum
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
The first example of non-local modular Hamiltonian has been found by Casini and Huerta [11] for the massless Dirac field in its ground state and on the infinite line, when the subsystem A is the union of disjoint intervals, by employing the lattice results for this operator obtained by Peschel [12]
Summary
We review the basic properties of the massless Dirac field with a point-like defect. The defect, localised in the origin x = 0 of the infinite line R without loss of generality, splits the line in two half-lines (edges). In order to treat these two edges in a symmetric way, we find it convenient to adopt the following coordinates (x, i) : x > 0, i = 1, 2. Where x indicates the distance from the defect and i labels the edges, as shown in figure 1
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