Quantum thermalization and average entropy of a subsystem

Abstract

Page's seminal result on the average von Neumann (VN) entropy does not immediately apply to realistic many-body systems which are restricted to physically relevant smaller subspaces. We investigate here the VN entropy averaged over the pure states in the subspace ${\mathcal{H}}_E$ corresponding to a narrow energy shell centered at energy $E$. We find that the average entropy is ${\overline{S}}_1\simeq \ln d_1$, where $d_1$ represents first subsystem's effective number of states relevant to the energy scale $E$. If $d_E=\dim ({\mathcal{H}}_E)$ and $D$ ($D_1$) is the Hilbert space dimension of the full system (first subsystem), we estimate that $d_1\simeq D_1^{\gamma }$, where $\gamma =\ln (d_E)/\ln (D)$ for nonintegrable (chaotic) systems and $\gamma <\ln (d_E)/\ln (D)$ for integrable systems. This result can be reinterpreted as a volume-law of entropy, where the volume-law coefficient depends on the density-of-states for nonintegrable systems, and remains below the maximal possible value for integrable systems. We numerically analyze a spin model to substantiate our main results.

9 pages, 3 figures; final version

References in corpus (25)

• Quantum entanglement
• Chaos and Quantum Thermalization
• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Average Entropy of a Subsystem
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
• Canonical Typicality
• Foundation of Statistical Mechanics under experimentally realistic conditions
• Thermalization and prethermalization in isolated quantum systems: a theoretical overview
• Generalized Thermalization in an Integrable Lattice System
• Entanglement of Exact Excited States of AKLT Models: Exact Results, Many-Body Scars and the Violation of Strong ETH
• Does a single eigenstate encode the full Hamiltonian?
• Average Entropy of a Subsystem
• Eigenstate thermalization within isolated spin-chain systems
• Entanglement entropy of excited states
• The microscopic origin of thermodynamic entropy in isolated systems
• Entanglement and matrix elements of observables in interacting integrable systems
• Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians
• Eigenstate Entanglement Entropy in Random Quadratic Hamiltonians
• The Page Curve for Fermionic Gaussian States
• Typical entanglement entropy in the presence of a center: Page curve and its variance
• Bound states and entanglement in the excited states of quantum spin chains
• Eigenstate Gibbs Ensemble in Integrable Quantum Systems
• Exact Entanglement Studies of Strongly Correlated Systems: Role of Long-Range Interactions and Symmetries of the System
• Typical entanglement for Gaussian states