Modular Hamiltonian in the semi infinite line, Part II: dimensional reduction of Dirac fermions in spherically symmetric regions

In this article, we extend our study on a new class of modular Hamiltonians on an interval attached to the origin on the semi-infinite line, introduced in a recent work dedicated to scalar fields. Here, we shift our attention to fermions and similarly to the scalar case, we investigate the modular Hamiltonians of theories which are obtained through dimensional reduction, this time, of a free massless Dirac field in $d$ dimensions. By following the same methodology, we perform dimensional reduction on both the physical and modular Hamiltonians. This process enables us to establish a correspondence: we identify the modular Hamiltonian in an interval connected to the origin to the one obtained from the reduction of the modular Hamiltonian pertaining to the conformal parent theory on a sphere. Intriguingly, although the resulting one-dimensional theories lack conformal symmetry due to the presence of a term proportional to $1/r$, the corresponding modular Hamiltonians are local functions in the energy density. This phenomenon mirrors the well known behaviour observed in the conformal case and indicates the existence of a residual symmetry, characterized by a subset of the original conformal group. Furthermore, through an analysis of the spectrum of the modular Hamiltonians, we derive an analytic expression for the associated entanglement entropy. Our findings also enable us to successfully recover the conformal anomaly coefficient from the universal piece of the entropy in even dimensions, as well as the universal constant $F$ term in $d=3$, by extending the radial regularization scheme originally introduced by Srednicki to perform the sum over angular modes.