Fermion Parity Resolution of Entanglement

Abstract

Entanglement is analyzed in the Majorana fermion conformal field theory (CFT) in the vacuum, in the fermion state, and in states built from conformal interfaces. In the boundary-state approach, the Hilbert space admits two factorizations for a single interval, producing distinct entanglement spectra determined by spin structures. Although R\'enyi and relative entropies are shown to be insensitive to these structures, symmetry-resolved entanglement naturally reveals their differences. The Majorana fermion's $\mathbb{Z}_2^F$ symmetry, generated by the fermion-parity operator $(-1)^F$, distinguishes bosonic from fermionic sectors, motivating the notion of fermion-parity resolution. While $\mathbb{Z}_2^F$ is naturally a symmetry of the vacuum and fermion reduced density matrices, the Hilbert space factorization is shown to stabilize this symmetry in conformal interface states. When an unpaired Majorana zero mode is present, fermion-parity-resolved entropies display equipartition at all orders in the UV cutoff; in its absence, the breaking of equipartition is quantified by Ramond-sector data. This behavior persists across all states considered. Connections with symmetry-protected topological phases of matter are outlined. All results are compared with twist field computations.