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

Naoto Shiraishi, Hal Tasaki

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.

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.
Paper Structure (21 sections, 9 theorems, 123 equations)

This paper contains 21 sections, 9 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Setting and main assumptions
  4. Initinal state and its effective dimension
  5. Time evolution and thermalization
  6. Nature of the final state
  7. Free fermion on the chain
  8. Energy eigenstates and eigenvalues
  9. Justification of Assumption \ref{['a:nondeg']}
  10. Justification of Assumption \ref{['a:P2N']}
  11. Discussion
  12. Models with degenerate energy eigenvalues
  13. Models with degenerate energy eigenvalues
  14. Models satisfying Assumption \ref{['a:P2N']}
  15. Interacting fermions on double-lattice
  16. ...and 6 more sections

Key Result

Theorem 1.1

When $N$ (or $L$) is sufficiently large and $\rho=N/L\le1/5$, the following is true with probability larger than $1-e^{-(\rho/3)N}$ (where the probability is that for the choice of $|\Phi(0)\rangle$). There exists a sufficiently long time $T>0$ and a subset (a collection of intervals) $G\subset[0,T] with probability larger than $1-e^{-(\rho/4)N}$ (where the probability is that for quantum measurem

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1: Nondegeneracy of $E_{\boldsymbol{k}}$ for most $\theta$
  • Theorem 3.2: Nondegeneracy of $E_{\boldsymbol{k}}$ for small $|\theta|\ne0$
  • Lemma 3.3
  • Lemma 3.4
  • Lemma B.1
  • Theorem C.1