Extensive entropy increase in isolated quantum systems: beyond the eigenstate thermalization hypothesis and beyond instantaneous quenches

A theorem of T. Mori (J. Phys. A 49, 444003 (2016)) established that in any nontrivial instantaneous quantum quench of an isolated many-body system the thermodynamic entropy -- and, when both pre- and post-quench Hamiltonians satisfy the eigenstate thermalization hypothesis (ETH), also the diagonal entropy -- must increase extensively, i.e., proportionally to the system size N. Mori explicitly left open two linked problems: (1) whether the extensive increase of the diagonal entropy is a general feature even for systems violating ETH, such as integrable systems, where extensive increases are observed numerically and confirmed in exact Bethe-ansatz treatments of specific quenches, but no general proof or criterion exists; and (2) whether analogous strict irreversibility, an $O(N)$ entropy increase, holds for arbitrary finite-time (non-quasi-static) adiabatic processes, i.e., isolated systems driven by time-dependent Hamiltonians, for which the quench methods do not extend. The post-quench diagonal entropy of integrable systems is proportional to the Yang-Yang entropy of the steady-state generalized Gibbs ensemble (with a non-universal prefactor), and a weak second law for finite-time macroscopic operations is known (non-decrease of entropy, with no extensive work extraction); however, both open problems -- a general ETH-free criterion for extensive diagonal-entropy increase, and a strict extensive increase under finite-time driving -- remain unresolved.

Background

An isolated quantum many-body system of $N$ constituents evolves unitarily, so the von Neumann entropy $S(\rho)=-\mathrm{Tr}\,\rho\ln\rho$ is exactly conserved; expressing the second law of thermodynamics for such systems therefore requires other entropy notions. Two are standard here. The thermodynamic entropy $S_{\mathrm{TD}}(\rho)$ is the von Neumann entropy of the Gibbs state $e^{-\beta H}/\mathrm{Tr}\,e^{-\beta H}$ whose mean energy matches that of $\rho$. The diagonal entropy $S_{\mathrm{D}}(\rho)$, introduced by Polkovnikov, is the von Neumann entropy of the diagonal ensemble $\rho_{\mathrm{D}}$, the state obtained from $\rho$ by deleting all off-diagonal matrix elements in the eigenbasis of $H$. For any $\rho$ one has $S(\rho)\le S_{\mathrm{D}}(\rho)\le S_{\mathrm{TD}}(\rho)$, an inequality due to Tasaki. A quantum quench is the protocol in which a system prepared in (or near) equilibrium with a Hamiltonian $H_i$ is subjected at time $t=0$ to a new Hamiltonian $H_f$ and subsequently evolves under $H_f$ in isolation.

Mori (2016) proved that in any nontrivial quench -- one whose initial and final equilibrium states are macroscopically distinguishable -- the thermodynamic entropy increases extensively, $S_{\mathrm{TD}}^{(f)}-S_{\mathrm{TD}}^{(i)}=\mathcal{O}(N)$, for any initial state (even a pure state) that represents thermal equilibrium in the sense of being locally indistinguishable from the Gibbs state of $H_i$. For stationary initial states, the same extensive increase was proven for the diagonal entropy, $S_{\mathrm{D}}^{(f)}-S_{\mathrm{D}}^{(i)}=\mathcal{O}(N)$, assuming that both $H_i$ and $H_f$ satisfy the eigenstate thermalization hypothesis (ETH) -- the statement, believed to hold in generic nonintegrable systems, that every individual energy eigenstate already exhibits thermal values of macroscopic observables. Throughout, $\mathcal{O}(N)$ is used as in the source: the increase is at least of order $N$, i.e., bounded below by $cN$ for some $c>0$ that is independent of $N$; as in Mori's proof -- where the large-deviation value $I_f(x_i^{*})$ and the ETH constant $\gamma''$ depend on the quench and on the chosen tolerance but not on $N$ -- the constant $c$ may depend on the specific quench or protocol, on the model, and on the temperature, but must not scale with $N$ (if $c$ were allowed to depend on $N$, any strictly positive increase would satisfy the bound and the statement would be vacuous). The proof rests on the identity $S_{\mathrm{D}}^{(f)}-S_{\mathrm{D}}^{(i)}=D(\rho_{\mathrm{D}}^{(i)}\Vert\rho_{\mathrm{D}}^{(f)})$, where $D(\sigma\Vert\rho)$ is the quantum relative entropy, combined with a large-deviation lower bound $D(\sigma\Vert\rho)\ge N\,I(x_i^{*})+o(N)$ for states with sharply distributed macrovariables; ETH enters only to guarantee that the post-quench diagonal ensemble has a unique typical macrovalue, so that the rate function $I$ at the initial typical value $x_i^{*}$ is strictly positive.

ETH fails, however, in integrable systems, which possess an extensive set of conserved quantities and relax after a quench not to a thermal state but to a generalized Gibbs ensemble (GGE) -- the maximum-entropy state constrained by all local and quasi-local conservation laws. In Bethe-ansatz-solvable models the GGE carries the Yang-Yang entropy, the thermodynamic (large-deviation) entropy of the Bethe quasiparticle rapidity distribution, so integrable steady states do possess a well-defined entropy; the difficulty is that Mori's ETH-based proof of the extensive diagonal-entropy increase does not apply to them. Numerical studies of several integrable models showed extensive increases, and exact treatments of specific integrable quenches confirm them, but no general proof or criterion covering ETH-violating systems is known. A second limitation of Mori's theorem is that it concerns instantaneous quenches and does not cover protocols in which the Hamiltonian $H(t)$ changes continuously over a finite duration. In this literature 'adiabatic' means thermally isolated (no heat exchange with an environment), covering arbitrarily fast driving -- not the quantum-adiabatic (slow-only) sense -- and 'non-quasi-static' means driving outside the quasi-static regime, in which the Hamiltonian changes rapidly enough that extensive irreversibility is expected; neither the source nor the later literature provides a quantitative definition of this class, and supplying one is part of question (2) below. In the quasi-static limit the entropy increase is only $o(N)$, so strict irreversibility is a statement about genuinely non-quasi-static processes; Mori noted that extending the quench result to arbitrary finite-time adiabatic processes would require new ideas. The two questions below ask for exactly these extensions, which together constitute a microscopic foundation of strict irreversibility -- an extensive, ETH-free lower bound on entropy production -- for isolated quantum many-body systems.

Problem Statement

The problem is two linked questions sharing one objective -- a microscopic foundation of strict irreversibility (an extensive lower bound on entropy production) for isolated quantum systems:

(1) ETH-free extensive increase of the diagonal entropy in quantum quenches. Determine whether the extensive increase of the diagonal entropy is a general feature even for systems without the ETH -- integrable systems being the canonical case, and many-body localized systems another ETH-violating class: specifically, whether in any nontrivial quench (macroscopically distinguishable initial and final equilibrium states) acting on a stationary initial state, as in Mori's theorem, the diagonal-entropy increase $S_{\mathrm{D}}(\rho^{(f)})-S_{\mathrm{D}}(\rho^{(i)})$ is bounded below by $cN$ for some constant $c>0$ independent of $N$ (allowed to depend on the quench and on the model). Deliver a general theorem establishing this lower bound, or a counterexample: an ETH-violating system and a nontrivial quench whose diagonal-entropy increase is $o(N)$. For ETH-violating classes in which a Gibbs-type equilibrium setup is not available (e.g., many-body localized or scarred systems), the meaning of the equilibrium initial state and of the nontriviality of the quench must be specified as part of the answer, rather than inherited silently from the integrable setting.

(2) Strict irreversibility for finite-time adiabatic processes. Prove (or disprove) that for an arbitrary non-quasi-static adiabatic (isolated, no heat exchange) process taking a finite time -- unitary evolution $U_\tau$ under a time-dependent Hamiltonian $H(t)$, $0\le t\le\tau$, with $H(0)=H_i$ and $H(\tau)=H_f$ -- the entropy increases extensively, i.e., $S^{(f)}-S^{(i)}$ is bounded below by $cN$ for some constant $c>0$ independent of $N$ (allowed to depend on the protocol). Following the two setups of Mori's quench theorem, the targets are the thermodynamic entropy for an initial state representing thermal equilibrium (setup (I)) and the diagonal entropy for a stationary initial state (setup (II)); the final diagonal ensemble is defined in the eigenbasis of $H(\tau)=H_f$, and any further dephasing or equilibration convention must be stated explicitly. Because quasi-static driving yields only an $o(N)$ increase, protocols that approach the quasi-static limit in the thermodynamic limit (for instance with $\tau$ growing with $N$) lie outside the intended class; the source gives no quantitative definition of 'non-quasi-static', and providing a quantitative characterization of the admissible protocol class (relating $\tau$ and the driving rate to $N$ and to relaxation times) is itself part of the problem -- any proposed theorem or counterexample must specify the class of protocols it covers; that characterization must include every finite-time protocol whose driving rate does not vanish with $N$, in particular $H(t)$ varying on an $N$-independent timescale $\tau=\mathcal{O}(1)$, and a definition that excludes such protocols does not resolve the question. That characterization must also be nondegenerate: the process must be nontrivial, in that the driving changes the state, or its equilibrium description, macroscopically, so that protocols whose entropy production vanishes identically (for instance, because the driving leaves the initial state invariant) lie outside the class; this nontriviality concerns the state produced by the driving rather than the distinguishability of the endpoints, so the cyclic case $H(\tau)=H(0)$ is not excluded.

Both parts are to be answered at the same level of generality as Mori's quench theorem, without assuming the ETH or instantaneous driving: an affirmative answer to question (1) must be a general ETH-free theorem, and an affirmative answer to question (2) must cover the full class of protocols characterized in the answer. The counterexample route of question (1) is a deliberate exception: a single ETH-violating system together with a nontrivial quench exhibiting an $o(N)$ diagonal-entropy increase fully resolves that question. Model-specific positive results, and numerical evidence alone, do not close either question.

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