Nature abhors a vacuum: A simple rigorous example of thermalization in an isolated macroscopic quantum system

Abstract

We show, without relying on any unproven assumptions, that a low-density free fermion chain exhibits thermalization in the following (restricted) sense. We choose the initial state as a pure state drawn randomly from the Hilbert space in which all particles are in half of the chain. This represents a nonequilibrium state such that the half chain containing all particles is in equilibrium at infinite temperature, and the other half chain is a vacuum. We let the system evolve according to the unitary time evolution determined by the Hamiltonian and, at a sufficiently large typical time, measure the particle number in an arbitrary macroscopic region in the chain. In this setup, it is proved that the measured number is close to the equilibrium value with probability very close to one. Our result establishes the presence of thermalization in a concrete model in a mathematically rigorous manner. The most important theoretical ingredient for the proof of thermalization is the demonstration that a nonequilibrium initial state generated as above typically has a sufficiently large effective dimension. Here, we first give general proof of thermalization based on two assumptions, namely, the absence of degeneracy in energy eigenvalues and a property about the particle distribution in energy eigenstates. We then justify these assumptions in a concrete free-fermion model, where the absence of degeneracy is established by using number-theoretic results. This means that our general result also applies to any lattice gas models in which the above two assumptions are justified. To confirm the potential wide applicability of our theory, we discuss some other models for which the essential assumption about the particle distribution is easily verified, and some non-random initial states whose effective dimensions are sufficiently large.

33 pages, 0 figures. In version 2, Appendix D on finite temperature extension has been added, and upper bounds for N in Theorems 3.1 and 3.2 have been removed. Some important references have been added. There is a 27-minute video that explains the main results of the present work: https://youtu.be/eUIX8WdJftc

References in corpus (10)

• Probing many-body dynamics on a 51-atom quantum simulator
• Many body localization and thermalization in quantum statistical mechanics
• Experimental Observation of a Generalized Gibbs Ensemble
• Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics
• Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem
• Many-body localisation implies that eigenvectors are matrix-product states
• Generalization of von Neumann's Approach to Thermalization
• Thermalization without eigenstate thermalization hypothesis after a quantum quench
• Boltzmann entropy of a freely expanding quantum ideal gas
• Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

Cited by in corpus (8)

• Open-system eigenstate thermalization in a noninteracting integrable model
• Macroscopic Irreversibility in Quantum Systems: Free Expansion in a Fermion Chain
• Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation
• Thermalization in a simple spin-chain model
• Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles
• Absence of thermalization after a local quench and strong violation of the eigenstate thermalization hypothesis
• Basis dependence of eigenstate thermalization
• Rationally independent free fermions with local hopping