Entanglement-Ergodic Quantum Systems Equilibrate Exponentially Well

Abstract

One of the outstanding problems in non-equilibrium physics is to precisely understand when and how physically relevant observables in many-body systems equilibrate under unitary time evolution. General equilibration results show that equilibration is generic provided that the initial state has overlap with sufficiently many energy levels. But results not referring to typicality which show that natural initial states actually fulfill this condition are lacking. In this work, we present stringent results for equilibration for systems in which Rényi entanglement entropies in energy eigenstates with finite energy density are extensive for at least some, not necessarily connected, sub-system. Our results reverse the logic of common arguments, in that we derive equilibration from a weak condition akin to the eigenstate thermalization hypothesis, which is usually attributed to thermalization in systems that are assumed to equilibrate in the first place. We put the findings into the context of studies of many-body localization and many-body scars.Over recent years the study of the relaxation to equilibrium of complex many-body systems has attracted great attention. This interest can be motivated from at least two points of view. From a foundational viewpoint, it is desirable to understand how statistical equilibrium ensembles emerge within the framework of unitary quantum mechanics -without introducing any external probability measures. It is then necessary to first explain how systems undergoing unitary evolution attain equilibrium at all. A key ingredient to explain this behavior has been found to be the dynamical build-up of entanglement from low-entangled initial states and results showing equilibration under quite general conditions have been derived [1][2][3][4][5][6][7][8][9][10][11][12]. The increase of entanglement over time is a generic feature of complex quantum systems and leads to an increase of the entropy of subsystems over time reminiscent to the second law of thermodynamics.From a more concrete perspective, the recent interest in the study of non-equilibrium dynamics is motivated by the fact that such dynamics can now be realized in well-controlled experiments, for example in ion traps or optical lattices [13][14][15][16][17][18]. Moreover, the discovery of many-body localized systems [19], which equilibrate [20] but fail to thermalize [18], shows that there remains much to be understood about the equilibration behavior of complex quantum systems. Despite the great progress in understanding the equilibration behavior of manybody systems, rigorous results showing that systems with natural initial states equilibrate to high precision based on concrete physical properties have been lacking.In this article we aim to fill this gap, by taking a new perspective to the problem. To do this, we carefully reconsider the entanglement content of energy eigenstates in complex, interacting many-body systems and devise a working-definition of "entanglement-ergodic" systems whose energy eigenstates at finite energy densi...

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