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Quantum Entanglement of Fermionic Local Operators

Masahiro Nozaki, Tokiro Numasawa, Shunji Matsuura

TL;DR

This work analyzes how local operator excitations modify entanglement in a four-dimensional free massless fermion, using the replica trick to compute excess Rényi entropies ΔS_A^{(n)} for a half-space subsystem. By deriving the replica-space fermion propagators and performing analytic continuation, it shows that ΔS_A^{(n)} vanishes before the light-cone crosses the entangling surface (t<l) and saturates to finite, spin-dependent values afterward, due to the directional propagation of left- and right-moving quasi-particles. The results are corroborated by explicit reduced density matrices for several local operators (ψ, ψ†ψ, and their combinations), highlighting spin-direction effects via the operator-specific matrix elements of γ^tγ^1. A quasi-particle interpretation, including a spin-dependent exotic anti-commutation structure, accounts for the late-time entanglement and connects the replica results to a physically intuitive picture of entangled fermionic excitations across the entangling surface.

Abstract

In this paper we study the time evolution of (Renyi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Renyi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.

Quantum Entanglement of Fermionic Local Operators

TL;DR

This work analyzes how local operator excitations modify entanglement in a four-dimensional free massless fermion, using the replica trick to compute excess Rényi entropies ΔS_A^{(n)} for a half-space subsystem. By deriving the replica-space fermion propagators and performing analytic continuation, it shows that ΔS_A^{(n)} vanishes before the light-cone crosses the entangling surface (t<l) and saturates to finite, spin-dependent values afterward, due to the directional propagation of left- and right-moving quasi-particles. The results are corroborated by explicit reduced density matrices for several local operators (ψ, ψ†ψ, and their combinations), highlighting spin-direction effects via the operator-specific matrix elements of γ^tγ^1. A quasi-particle interpretation, including a spin-dependent exotic anti-commutation structure, accounts for the late-time entanglement and connects the replica results to a physically intuitive picture of entangled fermionic excitations across the entangling surface.

Abstract

In this paper we study the time evolution of (Renyi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Renyi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.

Paper Structure

This paper contains 27 sections, 109 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Replica Trick
  4. Locally Excited States
  5. Replica method
  6. Propagator
  7. Computation of Proapgator
  8. Analytic Continuation to Real Time
  9. Dominant Propagators
  10. Propagators in the Cartesian Coordinate
  11. $\Delta S^{(n)}_A$ for Various Local Operators
  12. $\Delta S^{(n)}_A$ for $\psi'_a$
  13. The Excess of R$\acute{e}$nyi Entanglement Entropies
  14. Reduced Density Matrix
  15. $\Delta S^{(n)}_A$ for $\bar{\psi}'\psi'$
  16. ...and 12 more sections

Figures (7)

  • Figure 1: The location of an local operator and the subsystem $A$ in the Minkowski spacetime.
  • Figure 2: $n$-sheeted manifold with operator insertion
  • Figure 3: The location of operators in Euclidean space.
  • Figure 4: The schematic description of dominant propagators. $S_{ab}(r, r_2, \theta-\theta_2)$ and $S_{ab}(r_2, r, \theta_2-\theta)$ correspond to the right arrow and the left arrow respectively.
  • Figure 5: The schematic description of $A_1$ and $A_2$.
  • ...and 2 more figures