Second law with strict irreversibility: extensive entropy increase for non-ETH quantum quenches and for finite-time adiabatic processes

Mori (J. Phys. A 49, 444003 (2016)) proved that in any nontrivial instantaneous quantum quench of an isolated many-body quantum system the thermodynamic entropy increases extensively, $S^{(f)}-S^{(i)}=O(N)$, for any initial state locally indistinguishable from thermal equilibrium (even a pure state), and that the diagonal entropy increases extensively for any stationary initial state when both the pre-quench and post-quench Hamiltonians satisfy the eigenstate thermalization hypothesis (ETH). The paper explicitly poses the two open problems that define this record: (A) whether the extensive increase of the diagonal entropy after a quench is a general feature even for systems without the ETH, motivated by numerical observations of extensive increase in several integrable models (other non-ETH classes, such as many-body localized systems, are natural test cases studied subsequently but are not named explicitly in the paper); and (B) whether the strict-irreversibility statement $S^{(f)}-S^{(i)}=O(N)$ can be proved or disproved for an arbitrary adiabatic process of finite duration (an isolated system driven by a time-dependent Hamiltonian $H(t)$). The subsequent literature (2016-2026) contains numerical confirmations and model-specific structural results in both directions, but no general proof, sharp criterion, or counterexample for either question has been found; both remain open.

Background

The subject is the microscopic, quantitative content of the second law of thermodynamics for isolated quantum many-body systems. A quantum quench is the sudden change of a Hamiltonian $H_i\to H_f$ at time $t=0$, after which the system evolves unitarily; more generally, an isolated system may be driven by a time-dependent Hamiltonian $H(t)$ over a finite duration $\tau$ with no heat exchange with an environment (called an adiabatic process in Mori's terminology, which includes arbitrarily fast driving and is not restricted to slow changes). Because the von Neumann entropy $-\mathrm{Tr}\,\rho\ln\rho$ is invariant under any unitary evolution, irreversibility must be quantified by other entropies. Two are central here. The thermodynamic entropy $S_{TD}(\rho)$ is the von Neumann entropy of the Gibbs state $\rho_G \propto e^{-\beta H}$ whose mean energy equals that of $\rho$. The diagonal entropy $S_D(\rho)$, introduced by Polkovnikov (Ann. Phys. 326, 486 (2011)), is the von Neumann entropy of the diagonal ensemble $\rho_D$, obtained by deleting the off-diagonal matrix elements of $\rho$ in the basis of the relevant Hamiltonian's energy eigenstates; it is the Shannon entropy of the energy distribution and changes under driving. The eigenstate thermalization hypothesis (ETH) is the statement that every energy eigenstate of a nonintegrable Hamiltonian looks like a thermal state at the matching energy density; it is the standard mechanism of thermalization of isolated nonintegrable systems. Mori (J. Phys. A 49, 444003 (2016)) proved the 'second law with strict irreversibility' for instantaneous quenches: in any nontrivial quench (one in which the equilibrium macrostate of the initial state under $H_f$ differs from its initial macrostate), $S_{TD}$ increases by $O(N)$, and, if both $H_i$ and $H_f$ satisfy ETH, $S_D$ also increases by $O(N)$ for any stationary initial state (one commuting with $H_i$, hence already diagonal in the initial energy eigenbasis), i.e. $S_D(\rho^{(f)})-S_D(\rho^{(i)}) \ge N\gamma''+o(N)$ for a model-dependent $\gamma''>0$. The proof uses ensemble equivalence (measure equivalence implies macrostate equivalence; Mori, Phys. Rev. E 94, 020101 (2016)): ETH guarantees that the typical macrostate of the final diagonal ensemble is unique, so the initial macrostate sits at a point where the final large-deviation rate function (the exponential decay rate, with system size, of the probability of atypical macrostates) is positive, giving an extensive penalty. Both pillars of the proof fail outside its regime. First, integrable systems possess an extensive number of local and quasilocal conserved charges, do not satisfy ETH, and after a quench relax instead to a generalized Gibbs ensemble (GGE), the maximum-entropy state subject to all those charges; in Bethe-ansatz-solvable chains (the Bethe ansatz being the exact solution method for one-dimensional integrable models) the thermodynamic entropy of the GGE macrostate is the Yang-Yang entropy, the entropy of the thermodynamic distribution of occupied rapidities. Many-body localized (MBL) systems, which remain ergodicity-broken even at high energy due to disorder, form a second class without ETH. In both classes the typical macrostate of the diagonal ensemble need not be unique, so the rate-function mechanism collapses. Second, for a finite-duration drive $H(t)$ there is no single energy eigenbasis and no instantaneous-switch ensemble comparison at all; Mori's analysis has no direct extension. The magnitude question is nevertheless physically sharp: thermodynamics alone gives only $S^{(f)} \ge S^{(i)}$, and the extensive character of the increase is the precise microscopic signature of irreversibility, as Mori emphasizes; he called the finite-time extension particularly important for understanding the origin of irreversibility.

Problem Statement

An isolated quantum system of $N$ degrees of freedom with a local Hamiltonian undergoes a thermodynamically irreversible process: either an instantaneous quench $H_i \to H_f$ at $t=0$, or a drive by a time-dependent Hamiltonian $H(t)$, $0 \le t \le \tau$, of finite duration $\tau$, with no heat exchange with an environment. Irreversibility is quantified by the thermodynamic entropy $S_{TD}(\rho)$, the von Neumann entropy of the Gibbs state matching the mean energy of $\rho$, and by the diagonal entropy $S_D(\rho)$, the von Neumann entropy of the diagonal ensemble $\rho_D$ (the state dephased in the relevant energy eigenbasis). A process is nontrivial when the equilibrium (typical) macrostate of the initial state with respect to the final Hamiltonian differs from its initial macrostate. The second law with strict irreversibility is the statement that $S^{(f)}-S^{(i)} \ge cN$ for some $c>0$ independent of $N$ in the thermodynamic limit (as for Mori's constant $\gamma''$, $c$ may depend on the Hamiltonian and on the details of the process, but not on $N$); it is currently proven only for instantaneous quenches, with the eigenstate thermalization hypothesis (ETH) the essential hypothesis for the diagonal-entropy part. Settle the following two regimes, in each case by a rigorous proof, an explicit counterexample, or a sharp criterion delimiting when extensive increase holds.

(A) Systems without ETH. Prove, refute, or characterize: after a nontrivial quench from a stationary initial state (one commuting with $H_i$), the diagonal entropy increases extensively in generic systems without the ETH, for instance interacting integrable models relaxing to a generalized Gibbs ensemble, noninteracting integrable systems, quantum-chaotic quadratic Hamiltonians (free-fermion models whose single-particle dynamics is chaotic), and many-body localized systems. The target is the general question for systems without ETH; the listed subclasses are illustrative test cases rather than individually required deliverables. The relations between the diagonal entropy and the Yang-Yang (conserved-mode) entropy of the post-quench steady state are known numerically to be model- and initial-state-dependent (the diagonal entropy equals one half of the complete-GGE Yang-Yang entropy for pair-structured XXZ quenches, and the ratio to the stationary thermodynamic entropy lies generically between 1/2 and 1 for free-fermion quenches without pair structure).

(B) Finite-time processes. Prove or disprove: for an arbitrary adiabatic process taking a finite time that is not quasi-static (in Mori's sense, an isolated system driven by a time-dependent Hamiltonian $H(t)$ with no heat exchange with an environment, where $H(t)$ is a generic drive and not one fine-tuned to track the instantaneous eigenstates), the strict-irreversibility statement holds, i.e. $S^{(f)}-S^{(i)} \ge cN$ for some $c>0$ independent of $N$ (possibly depending on the process). Following the source, the question asks to extend both proven quench theorems, so the statement covers both the thermodynamic entropy $S_{TD}$ and the diagonal entropy $S_D$ (the latter defined with respect to the energy eigenbasis of the final Hamiltonian $H(\tau)$). Engineered protocols such as counterdiabatic driving (auxiliary controls added to cancel diabatic transitions, known as shortcuts to adiabaticity) suppress entropy production at finite time by tuning $H(t)$ to track the instantaneous eigenstates; being fine-tuned in this way, they fall outside the generic class of drives quantified over here.

The common objective of (A) and (B) is to establish the quantitative, extensive content of the second law for isolated many-body quantum dynamics outside the ETH instantaneous-quench regime in which it is proven; following the source, the two regimes are separate open problems.

References

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