Entanglement Entropy of Free Fermions with a Random Matrix as a One-Body Hamiltonian

Abstract

We consider a quantum system of large size $N$ and its subsystem of size $L$ assuming that $N$ is much larger than $L$, which can also be sufficiently large, i.e., $1\ll L\lesssim N$. A widely accepted mathematical version of this heuristic inequality is the asymptotic regime of successive limits: first the macroscopic limit $N\to \infty$, then an asymptotic analysis of the entanglement entropy as $L\to \infty$. In this paper, we consider another version of the above heuristic inequality: the regime of asymptotically proportional $L$ and $N$, i.e., the simultaneous limits $L\to \infty ,N\to \infty ,L/N\to \lambda >0$. Specifically, we consider the system of free fermions which is in its ground state and such that its one-body Hamiltonian is a large random matrix, that is often used to model the long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-ranged hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give a streamlined proof of Page's formula for the entanglement entropy of the black hole radiation for a wide class of typical ground states, thereby proving the universality of the formula.

References in corpus (31)

• Quantum entanglement
• Area laws for the entanglement entropy - a review
• Many-body localization, thermalization, and entanglement
• Average Entropy of a Subsystem
• Information in Black Hole Radiation
• Calculation of reduced density matrices from correlation functions
• Quantum entanglement in condensed matter systems
• Reduced density matrices and entanglement entropy in free lattice models
• The entropy of Hawking radiation
• Notes on Some Entanglement Properties of Quantum Field Theory
• Eigenstate Thermalization Hypothesis
• Violation of the entropic area law for Fermions
• Hawking Radiation and Black Hole Thermodynamics
• Open-System Dynamics of Entanglement
• Entanglement in indistinguishable particle systems
• Universality in volume law entanglement of pure quantum states
• Universality of Wigner random matrices: a Survey of Recent Results
• Criticality and entanglement in random quantum systems
• Eigenstate Entanglement Entropy in Random Quadratic Hamiltonians
• The Page Curve for Fermionic Gaussian States
• Random pure states: quantifying bipartite entanglement beyond the linear statistics
• On the Area Law for Disordered Free Fermions
• A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy
• Entanglement Typicality
• Large Block Properties of the Entanglement Entropy of Disordered Fermions
• A uniform area law for the entanglement of eigenstates in the disordered XY chain
• How much delocalisation is needed for an enhanced area law of the entanglement entropy?
• The absence of the selfaveraging property of the entanglement entropy of disordered free fermions in one dimension
• Entropy fluctuation formulas of fermionic Gaussian states
• On the Mott formula for the a.c. conductivity and binary correlators in the strong localization regime of disordered systems
• Entanglement entropy and particle number cumulants of disordered fermions