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arXiv:2409.09395

Macroscopic thermalization by unitary time evolution in the weakly perturbed two-dimensional Ising model --- An application of the Roos-Sugimoto-Teufel-Tumulka-Vogel theory

Hal Tasaki

Published Sep 14, 2024• 0 citations
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Abstract

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 $\hat{H}_L$ of the standard two-dimensional ferromagnetic Ising model with the plus boundary conditions and perturb it with a small self-adjoint operator $λ_L\hat{V}_L$ 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 $Ψ(0)\rangle$ is chosen from the microcanonical energy shell and may be very far from thermal equilibrium. Then the time-evolved state $|Ψ(t)\rangle=e^{-i(\hat{H}_L+λ_L\hat{V}_L)t}|Ψ(0)\rangle$ is in thermal equilibrium at sufficiently long and typical times $t$, 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 $L$.

DOI
10.48550/arxiv.2409.09395

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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.
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.
We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.
We consider an isolated macroscopic quantum system. Let H\mathcal{H}H be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between EEE and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy EEE corresponds to a subspace Heq{\mathcal{H}}_{eq}Heq​ of H\mathcal{H}H such that dim Heq/dim H\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}dim Heq​/dim H is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to Heq{\mathcal{H}}_{eq}Heq​. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times ttt. This result is closely related to von Neumann's quantum ergodic theorem of 1929.
Abstract We say of an isolated macroscopic quantum system in a pure state ψ\psi ψ ψ that it is in macroscopic thermal equilibrium (MATE) if ψ\psi ψ ψ lies in or close to a suitable subspace Heq\mathcal {H}_\textrm{eq}Heq​ H eq of Hilbert space. It is known that every initial state ψ0\psi _0ψ0​ ψ 0 will eventually reach and stay there most of the time (“thermalize”) if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation HθfFH_\theta ^\textrm{fF}HθfF​ H θ fF of the Hamiltonian H0fFH_0^\textrm{fF}H0fF​ H 0 fF of N≫1N\gg 1N≫1 N ≫ 1 free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of H0fFH_0^\textrm{fF}H0fF​ H 0 fF . Here, we first point out that also for degenerate Hamiltonians all ψ0\psi _0ψ0​ ψ 0 thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for H0fFH_0^\textrm{fF}H0fF​ H 0 fF . Inspired by the fact that there is one eigenbasis of H0fFH_0^\textrm{fF}H0fF​ H 0 fF for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given H0H_0H0​ H 0 that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of H0H_0H0​ H 0 lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, H=H0+λVH=H_0+\lambda VH=H0​+λV H = H 0 + λ V with λ≪1\lambda \ll 1λ≪1 λ ≪ 1 , for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.
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).