Routes to thermalization in isolated non-integrable quantum systems: shrinking eigenstate-observable support versus persistent rare non-thermal eigenstates with weakly biased diagonal weights

After a global quantum quench, an isolated many-body system thermalizes only if its diagonal ensemble reproduces microcanonical averages of local observables. Biroli, Kollath, and Läuchli (Phys. Rev. Lett. 105, 250401 (2010)) proved that the generic vanishing of the width of the eigenstate-expectation distribution (a 'weak' eigenstate thermalization hypothesis) does not imply thermalization, and posed the mechanism-level question of which of two routes is realized in the thermodynamic limit for non-integrable systems: either the support of the distribution of eigenstate expectation values shrinks onto the microcanonical value, so that rare non-thermal eigenstates disappear altogether (the later-termed strong eigenstate thermalization hypothesis), or rare non-thermal eigenstates persist while the diagonal weights become sufficiently unbiased that thermalization still follows. They further asked for the correct quantitative requirement on the weight distribution that guarantees thermalization, and for the finite-size 'thermalization length' and its dependence on the distance from integrability, proposing as diagnostics the tracking of the support of the observable distribution with system size and of the intensive Kullback-Leibler entropy between the diagonal and thermal distributions. The problem is to settle these three linked components: the thermodynamic-limit route determination for generic non-integrable systems, the correct quantitative requirement on the diagonal weights that guarantees thermalization, and the finite-size characterization near integrability. Later literature partially executes the numerical program for specific spin chains (finding support shrink with double-exponentially decaying athermal fraction) and exhibits persistent rare non-thermal eigenstates in special non-integrable models (scars, confinement, models with nonlocal conserved quantities), but a general route determination, a weight criterion, and the systematic finite-size scaling remain open.

Background

A quantum quench is a sudden global change of a Hamiltonian parameter, after which an isolated many-body system evolves unitarily. Thermalization means that the long-time averages of all local few-body observables coincide with Gibbs (equivalently, microcanonical) averages at the same intensive energy and particle density. Because unitary evolution has no attracting fixed point, equilibration is a dephasing phenomenon: assuming no degeneracies, the long-time value of an observable $O$ is governed by the diagonal ensemble, $O_D=\sum_\alpha O_\alpha |c_\alpha|^2$, where $|\alpha\rangle$ are the eigenstates of the post-quench Hamiltonian with energies $E_\alpha$, $O_\alpha=\langle\alpha|O|\alpha\rangle$ are eigenstate expectation values, and $|c_\alpha|^2=|\langle\alpha|\psi_0\rangle|^2$ are the squared overlaps of the initial state with the eigenbasis. Under very general conditions the energy per particle has vanishing fluctuations in the diagonal ensemble, so the weights $|c_\alpha|^2$ concentrate on a narrow energy shell.

The eigenstate thermalization hypothesis (ETH), introduced by Deutsch and Srednicki and verified numerically by Rigol, Dunjko, and Olshanii, states that for generic non-integrable interacting systems $O_\alpha$ equals, to high accuracy, a smooth function of the eigenenergy — the microcanonical average at $E_\alpha$ — for every eigenstate in a narrow energy window. If ETH holds, the diagonal and microcanonical ensembles agree and thermalization follows. The subtlety exposed by Biroli, Kollath, and Läuchli (BKL) is that for a finite system the $O_\alpha$ always fluctuate within an energy shell, so ETH must be formulated as a statement about how the distribution of $O_\alpha$ evolves with system size $L$ in the thermodynamic limit. BKL proved (via a bound relating eigenstate fluctuations to grand-canonical fluctuations) that generically the width of the $O_\alpha$ distribution within a narrow energy window vanishes as $L\to\infty$; this 'weak ETH' only implies that the fraction of non-thermal eigenstates vanishes. It does not imply thermalization, because the support of the distribution — the range of values attained by any eigenstate — may fail to shrink: rare states with atypical $O_\alpha$ exist at an exponential-in-$L$ entropic cost, and if the weights $|c_\alpha|^2$ assign them large probability, the diagonal ensemble retains memory of the initial state. BKL demonstrated this mechanism exactly for an integrable harmonic-oscillator chain, where the large-deviation function of the diagonal ensemble differs from the microcanonical entropy and the diagonal and microcanonical distributions peak at different values, and numerically for the non-integrable one-dimensional Bose-Hubbard model (bosons hopping with amplitude $J$ and on-site repulsion $U$), where at large $U_f/J$ the weights correlate strongly with atypical $O_\alpha$ near band edges.

To discriminate the mechanisms, BKL proposed to measure the Kullback-Leibler (KL) entropy, an information-theoretic dissimilarity between two probability distributions, $S_{KL}(\rho_I|\rho_{II})=\sum_C \rho_I(C)\log[\rho_I(C)/\rho_{II}(C)]$, of the thermal (Gibbs) distribution with respect to the diagonal ensemble in the energy shell, in BKL's convention: 'rather uniform sampling' of the shell by the weights corresponds to a zero intensive $S_{KL}$ (i.e. $S_{KL}/L\to 0$) in the thermodynamic limit. BKL also stressed that thermalization is an emergent property of sufficiently large systems, coining a 'finite-size thermalization length' whose dependence on the distance from integrability (e.g. the value of $U/J$ relative to the integrable limits $U=0$ and $J=0$) was posed as an open problem.

The subsequent literature fixed the terminology: the support-shrinking route is the strong ETH — every eigenstate in the microcanonical shell is thermal for every local few-body observable — while the alternative allows rare non-thermal eigenstates to persist provided the diagonal weights do not bias the shell distribution. The strong-ETH question acquired additional structure through the discovery of special non-integrable systems that violate it: quantum many-body scars (special non-thermal eigenstates embedded in otherwise chaotic spectra of constrained Rydberg-chain models, causing weak ergodicity breaking and persistent revivals), confinement-induced rare states in the Ising chain with a longitudinal field, and non-integrable models with highly nonlocal conserved quantities that thermalize without satisfying ETH. Rigorous results exist for random (typical) translation-invariant Hamiltonians and for typical high-entropy initial states, but not for any fixed generic non-integrable local Hamiltonian. Reviews by D'Alessio, Kafri, Polkovnikov, and Rigol codified this state of affairs.

Problem Statement

The problem is to determine the mechanism by which isolated non-integrable many-body quantum systems thermalize after a quench, at fixed particle density in the thermodynamic limit and for local few-body observables $O$. It has three linked components sharing this single objective. (1) Route determination: establish which of the two routes to thermalization is realized — either (i) the support of the distribution of the eigenstate expectation values $O_\alpha$ within narrow energy windows shrinks onto the microcanonical value as the system size $L\to\infty$, so that rare non-thermal eigenstates disappear altogether (the strong eigenstate thermalization hypothesis), or (ii) rare non-thermal eigenstates persist in the spectrum while the diagonal weights $|c_\alpha|^2$ sample the energy shell sufficiently uniformly that their bias on the microcanonical distribution vanishes in the thermodynamic limit (equivalently, the intensive Kullback-Leibler entropy between the diagonal and thermal distributions tends to zero). This includes delimiting for which classes of non-integrable Hamiltonians each route holds, given the known special non-integrable models hosting persistent rare states (scars, confinement-induced states, models with nonlocal conserved quantities). (2) Weight conditions: formulate the correct — quantitative, and ideally necessary and sufficient — requirement on the distribution of the diagonal-ensemble weights $|c_\alpha|^2$ that guarantees thermalization in the thermodynamic limit, going beyond the heuristic assumption of 'rather uniform sampling'; established facts to be reconciled are that the strong ETH is sufficient, and that it is not necessary in specific constructions. (3) Finite-size thermalization length: characterize the system-size scale $L_{\mathrm{th}}(\delta)$ at which thermal behavior emerges, and its dependence on the distance $\delta$ from the nearest integrable point, including the systematic finite-size scaling of the width and support of the $O_\alpha$ distributions and of the intensive Kullback-Leibler entropy. The scale must be defined operationally in the answer — for instance as the smallest size $L$ such that $\sup_O|O_D-O_{\mathrm{mc}}|\le\varepsilon$ holds for all local few-body observables $O$ and all larger sizes, for a fixed tolerance $\varepsilon$ — with the tolerance, the observable set and norm, and the measure of $\delta$ (for example the dimensionless coupling ratio of the chosen family) stated explicitly and held fixed throughout; any consistent operationalization of this form is acceptable. Component (1) identifies the eigenstate-level mechanism, component (2) the initial-state conditions under which that mechanism yields thermal steady states, and component (3) the finite-size scale on which the mechanism becomes operative; a complete answer must address all three components — including the delimitation, required by component (1), of the classes of non-integrable Hamiltonians for which each route holds — whether by rigorous results or by systematic numerical scaling to the thermodynamic limit. A numerical answer must in addition state the system sizes used, perform an error-controlled extrapolation to the thermodynamic limit of the support of the $O_\alpha$ distribution and of the intensive Kullback-Leibler entropy $S_{KL}/L$, and give an explicit bound on their finite-size drift; every answer must separate the generic class of non-integrable local Hamiltonians from the known special classes (scars, confinement-induced states, nonlocal conserved quantities). The strong-ETH route requires controlling the tails (large-deviation cost) of the $O_\alpha$ distribution rather than only its typical fluctuations; the rare-states route requires simultaneous asymptotic control of the rare-state statistics and of the initial-state overlaps.

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