Entanglement Entropy of Fermions from Wigner Functions: Excited States and Open Quantum Systems

TL;DR

This paper introduces a Wigner function-based method to calculate entanglement entropies in fermionic systems, providing exact formulas for open quantum systems and analyzing entanglement scaling in momentum Fock states.

Contribution

The authors develop a novel formalism using Wigner characteristics to compute entanglement entropies analytically for non-interacting fermions, including excited and open systems, bypassing complex manifold calculations.

Findings

Entanglement entropy scales logarithmically or linearly depending on momentum discontinuities.

Classification of Fock states by momentum blocks predicts critical states.

Analysis of entanglement dynamics reveals coherent and incoherent processes.

Abstract

We formulate a new ``Wigner characteristics'' based method to calculate entanglement entropies of subsystems of Fermions using Keldysh field theory. This bypasses the requirements of working with complicated manifolds for calculating R\'{e}nyi entropies for many body systems. We provide an exact analytic formula for R\'{e}nyi and von-Neumann entanglement entropies of non-interacting open quantum systems, which are initialised in arbitrary Fock states. We use this formalism to look at entanglement entropies of momentum Fock states of one-dimensional Fermions. We show that the entanglement entropy of a Fock state can scale either logarithmically or linearly with subsystem size, depending on whether the number of discontinuities in the momentum distribution is smaller or larger than the subsystem size. This classification of states in terms number of blocks of occupied momenta allows us to analytically estimate the number of critical and non-critical Fock states for a particular subsystem size. We also use this formalism to describe entanglement dynamics of an open quantum system starting with a single domain wall at the center of the system. Using entanglement entropy and mutual information, we understand the dynamics in terms of coherent motion of the domain wall wavefronts, creation and annihilation of domain walls and incoherent exchange of particles with the bath.