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Fermion Parity Resolution of Entanglement

Christian Northe

TL;DR

This work investigates entanglement in the Majorana fermion CFT through the boundary-state formalism, focusing on how spin structures and conformal interfaces influence entanglement spectra. It introduces fermion-parity (Z2^F) symmetry resolution, showing that an unpaired Majorana zero mode enforces complete equipartition between bosonic and fermionic sectors across vacuum, fermion, and conformal-interface states, while absence of the zero mode yields Ramond-sector controlled deviations. Rényi entropies align with twist-field results and can be insensitive to spin structures, whereas symmetry-resolved quantities and charged moments reveal rich parity-structured entanglement and the stabilization of Z2^F by factorization. The findings connect to symmetry-protected topological phases, clarify the role of Majorana zero modes in entanglement, and provide a framework for analyzing interface-induced entanglement signatures in fermionic CFTs and related models.

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 Rényi 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.

Fermion Parity Resolution of Entanglement

TL;DR

This work investigates entanglement in the Majorana fermion CFT through the boundary-state formalism, focusing on how spin structures and conformal interfaces influence entanglement spectra. It introduces fermion-parity (Z2^F) symmetry resolution, showing that an unpaired Majorana zero mode enforces complete equipartition between bosonic and fermionic sectors across vacuum, fermion, and conformal-interface states, while absence of the zero mode yields Ramond-sector controlled deviations. Rényi entropies align with twist-field results and can be insensitive to spin structures, whereas symmetry-resolved quantities and charged moments reveal rich parity-structured entanglement and the stabilization of Z2^F by factorization. The findings connect to symmetry-protected topological phases, clarify the role of Majorana zero modes in entanglement, and provide a framework for analyzing interface-induced entanglement signatures in fermionic CFTs and related models.

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 Rényi and relative entropies are shown to be insensitive to these structures, symmetry-resolved entanglement naturally reveals their differences. The Majorana fermion's symmetry, generated by the fermion-parity operator , distinguishes bosonic from fermionic sectors, motivating the notion of fermion-parity resolution. While 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.

Paper Structure

This paper contains 28 sections, 115 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Primary State Entanglement in the Boundary State Approach
  4. Majorana Fermion BCFT
  5. Subsystem Entropies for Global Primary States
  6. Rényi Entropies for the Global Vacuum State
  7. Rényi Entropies for the Global Fermion State
  8. Relative Entropy
  9. Fermion Parity Resolution of Entanglement
  10. Subsystem Fermion Parity and Symmetry Resolution
  11. Fermion Parity-Resolved Rényi Entropies for Global Primary States
  12. Ramond Boundary States, Charged Moments and Equipartition
  13. Conformal Interfaces in the Majorana Theory
  14. General Majorana Theory Conformal Interfaces
  15. Factorizing Interfaces
  16. ...and 13 more sections

Figures (3)

  • Figure 1: Left: $n=3$ copies of \ref{['cutSphere']} are glued cyclically and mapped onto an annulus by the uniformization map $\xi^{1/n}$. The excized disks are mapped to the inner and outer circle. The boundaries are no longer open and boundary states are imposed on the disks. The interval loci (orange) split the annulus into three equivalent regions. Right: Conformal equivalence with a cylinder slab coordinatized by either \ref{['twCoordn']} or \ref{['wCoordn']}.
  • Figure 2: $n=3$ replica geometry for the RDM \ref{['RDMI']}. Left: on the (uniformized) complex plane. Right: on the strip.
  • Figure 3: A factorizing interface splits the entangling interval into two disconnected segments, each attached to a physical boundary resulting from the interface. The boundaries $\alpha,\beta$ describe entangling edges, as before.