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arXiv:2401.15263 · Physical review. E

Macroscopic irreversibility in quantum systems: Free expansion in a fermionic chain

Hal Tasaki

Published May 30, 2025• 5 citations
arXivHTMLPDF

Abstract

We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).

DOI
10.1103/8cdf-sth8

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We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).
For a class of translation-invariant free-fermion systems including those with uniform nearest-neighbor hopping) on a d-dimensional L×⋯×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.
We study macroscopic thermalization in the ferromagnetic Ising model with a weak generic (highly nonlocal and many-body) random quantum perturbation. We prove that a single typical perturbation makes every initial state ∣Φ(0)⟩|Φ(0)\rangle∣Φ(0)⟩ in a specified energy shell thermalize: a measurement of a large class of macroscopic observables in the time-evolved state ∣Φ(t)⟩=e−iH~Lt∣Φ(0)⟩|Φ(t)\rangle=e^{-i\widetilde H_Lt}|Φ(0)\rangle∣Φ(t)⟩=e−iHL​t∣Φ(0)⟩ yields their thermal equilibrium values, within a prescribed precision and with overwhelming probability, at sufficiently large, typical times ttt. The result covers the full finite-temperature range, including the ordered phase, where we use plus boundary conditions. With periodic boundary conditions in the ordered phase, every energy eigenstate in the shell is shown to be an almost balanced macroscopic Schrödinger-cat state. Moreover, every pure initial state becomes such a cat at sufficiently large, typical times, with each branch in the corresponding macroscopic thermal equilibrium. The proof combines the theory of Roos, Sugimoto, Teufel, Tumulka, and Vogel with rigorous Ising large-deviation estimates.
In the present note, we discuss a simple example of a macroscopic quantum many-body system in which the approach to thermal equilibrium from an arbitrary initial state in the microcanonical energy shell is proved without relying on any unproven assumptions. The model, which is equivalent to a free fermion chain, is designed to be a toy model for a weakly heat-conducting one-dimensional solid. We take a phenomenological point of view and perceive that the system is in thermal equilibrium when the measured coarse-grained energy distribution is uniform. The result on thermalization reported here is a variation (and an improvement) of our previous result on the irreversible expansion in a free fermion chain. As far as we know, this is the first concrete and rigorous realization of the philosophy on the foundation of equilibrium statistical mechanics proposed by von Neumann in 1929, and further developed recently by Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zangh\`ıand the present author, namely, to characterize thermal equilibrium from a macroscopic viewpoint and to make use of the strong ETH to control the long-time dynamics. This note will be the most technical part of my longer article on thermalization, "What is thermal equilibrium and how do we get there?". I am making this document public at this stage since I have already announced (and will announce) the results at some of my talks.
By applying the theory recently developed by Roos, Sugimoto, Teufel, Tumulka, and Vogel [1], we construct a rigorous example of an isolated macroscopic quantum system in which every initial state chosen from a microcanonical energy shell exhibits thermalization solely by the unitary time evolution determined by the Hamiltonian. We consider the Hamiltonian H^L\hat{H}_LH^L​ of the standard two-dimensional ferromagnetic Ising model with the plus boundary conditions and perturb it with a small self-adjoint operator λLV^Lλ_L\hat{V}_LλL​V^L​ drawn randomly from the space of self-adjoint operators on the whole Hilbert space. The Roos-Sugimoto-Teufel-Tumulka-Vogel theory then guarantees that, for all but an exponentially small fraction of the random perturbations, the model exhibits strong ETH (energy eigenstate thermalization hypothesis), i.e., every energy eigenstate in a microcanonical energy shell is in thermal equilibrium in the sense of large deviation, as precisely stated in the main text. It is then standard that the strong ETH implies the following result for thermalization. Suppose that the initial state Ψ(0)⟩Ψ(0)\rangleΨ(0)⟩ is chosen from the microcanonical energy shell and may be very far from thermal equilibrium. Then the time-evolved state ∣Ψ(t)⟩=e−i(H^L+λLV^L)t∣Ψ(0)⟩|Ψ(t)\rangle=e^{-i(\hat{H}_L+λ_L\hat{V}_L)t}|Ψ(0)\rangle∣Ψ(t)⟩=e−i(H^L​+λL​V^L​)t∣Ψ(0)⟩ is in thermal equilibrium at sufficiently long and typical times ttt, in the sense that the measurement result of the magnetization density almost certainly coincides with the corresponding spontaneous magnetization. When we state ``typical'' and ``almost certainly'' above, we mean that the corresponding exceptional fractions are exponentially small in the system size LLL.
Having analytical instances of the eigenstate thermalization hypothesis (ETH) is of obvious interest, both for fundamental and applied reasons. This is generally a hard task, due to the belief that nonlinear interactions are basic ingredients of the thermalization mechanism. In this article we prove that random Gaussian-free fermions satisfy ETH in the multiparticle sector, by analytically computing the correlations and entanglement entropies of the theory. With the explicit construction at hand, we finally comment on the differences between fully random Hamiltonians and random Gaussian systems, providing a physically motivated notion of randomness of the microscopic quantum state.
The Eigenstate Thermalization Hypothesis (ETH) has played a major role in understanding thermodynamic phenomena in closed quantum systems. However, its connection to the timescale of thermalization for open system dynamics has remained elusive. This paper establishes a rigorous link between ETH and fast thermalization to the global Gibbs state. Specifically, we demonstrate fast thermalization for a system coupled weakly to a bath of quasi-free Fermions that we refresh periodically. To describe the joint evolution, we derive a finite-time version of Davies' generator with explicit error bounds and resource estimates. Our approach exploits a critical feature of ETH: operators in the energy basis can be modeled by independent random matrices in a near-diagonal band. This gives quantum expanders at nearby eigenstates of the Hamiltonian and reduces the problem to a one-dimensional classical random walk on the energy eigenstates. Our results explain finite-time thermalization in chaotic open quantum systems.