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Timescale for macroscopic equilibration in isolated quantum systems: a rigorous derivation for free fermions

Takashi Hara, Tatsuhiko Koike

TL;DR

The paper establishes a rigorous $O(L)$ timescale for macroscopic equilibration in isolated quantum systems of translation-invariant free fermions on a $d$-dimensional lattice, by analyzing a coarse-grained density observable and partitioning the Hilbert space into equilibrium and nonequilibrium subspaces. The authors reduce the many-body problem to a single-particle framework, derive a key time-averaged bound on the squared density deviation, and then translate this bound into a bound on the nonequilibrium time fraction that vanishes in the thermodynamic limit. The core mechanism relies on a Fourier-transform-inspired decomposition of time correlations, a reduction to energy-difference dynamics, and careful estimates of densities of states via $|\Omega_{\boldsymbol{m}}(E)|$ and related quantities, yielding an explicit operator norm bound that scales as $\delta_{1/2}(\tau,L)$ with $\tau$ of order $L$. This result provides the first rigorous macroscopic equilibration timescale for a realistic short-range quantum system and clarifies the role of macroscopic observables and energy-level structure in dynamical equilibration. The findings illuminate the interplay between system size, observation scale, and dynamical relaxation in isolated quantum matter, and pave the way for extending such results to broader classes of many-body systems. $O(L)$ is thus identified as the natural equilibration timescale in these freely evolving quantum lattices.

Abstract

For a class of translation-invariant free-fermion systems (including those with uniform nearest neighbor hopping) on a $d$-dimensional $L \times \cdots \times L$ hypercubic lattice, we prove that, starting from an arbitrary pure initial state, the system equilibrates with respect to the coarse-grained density within a timescale of order $L$. This scaling is optimal, since there exist initial states whose equilibration requires time of order $L$. Our result establishes $O(L)$ as the equilibration timescale, as is expected in normal macroscopic systems with a conserved quantity, such as total number of particles.

Timescale for macroscopic equilibration in isolated quantum systems: a rigorous derivation for free fermions

TL;DR

The paper establishes a rigorous timescale for macroscopic equilibration in isolated quantum systems of translation-invariant free fermions on a -dimensional lattice, by analyzing a coarse-grained density observable and partitioning the Hilbert space into equilibrium and nonequilibrium subspaces. The authors reduce the many-body problem to a single-particle framework, derive a key time-averaged bound on the squared density deviation, and then translate this bound into a bound on the nonequilibrium time fraction that vanishes in the thermodynamic limit. The core mechanism relies on a Fourier-transform-inspired decomposition of time correlations, a reduction to energy-difference dynamics, and careful estimates of densities of states via and related quantities, yielding an explicit operator norm bound that scales as with of order . This result provides the first rigorous macroscopic equilibration timescale for a realistic short-range quantum system and clarifies the role of macroscopic observables and energy-level structure in dynamical equilibration. The findings illuminate the interplay between system size, observation scale, and dynamical relaxation in isolated quantum matter, and pave the way for extending such results to broader classes of many-body systems. is thus identified as the natural equilibration timescale in these freely evolving quantum lattices.

Abstract

For a class of translation-invariant free-fermion systems (including those with uniform nearest neighbor hopping) on a -dimensional hypercubic lattice, we prove that, starting from an arbitrary pure initial state, the system equilibrates with respect to the coarse-grained density within a timescale of order . This scaling is optimal, since there exist initial states whose equilibration requires time of order . Our result establishes as the equilibration timescale, as is expected in normal macroscopic systems with a conserved quantity, such as total number of particles.
Paper Structure (24 sections, 54 equations)

This paper contains 24 sections, 54 equations.

Table of Contents

  1. 1. Introduction.---
  2. 2. The setup and the main result.---
  3. 2.1. Free fermions on a lattice.---
  4. 2.2. Nonequilibrium subspace.---
  5. 2.3. Main Result.---
  6. Remarks.---
  7. 3. Our key estimate for time average.---
  8. Proof of Theorem 1$'$, based on Proposition 2.---
  9. 4. Proof of Proposition 2.---
  10. 4.1. Reduction to a single particle problem.---
  11. Remarks.---
  12. 4.2. Key formula for $\bigl[{ (\widehat{\varDelta\rho}_H(t))^2 }\bigr]_\tau$.---
  13. Remark.---
  14. 4.3. Our key estimate.---
  15. 4.4. Estimating ${\left\vert \Omega_{\boldsymbol{m}}(E) \right\vert}$ for $d=1$.---
  16. ...and 9 more sections