Abstract Equilibrium statistical mechanics rests on the assumption of chaotic dynamics of a system modulo the conservation laws of local observables: extremization of entropy immediately gives Gibbs’ ensemble (GE) for energy conserving systems and a generalized version of it (GGE) when the number of local conserved quantities is more than one. Through the last decade, statistical mechanics has been extended to describe the late-time behaviour of periodically driven (Floquet) quantum matter starting from a generic state. The structure built on the fundamental assumptions of ergodicity and identification of the relevant conservation laws in this inherently non-equilibrium setting. More recently, it has been shown that the statistical mechanics of Floquet systems has a much richer structure due to the existence of emergent conservation laws: these are approximate but stable conservation laws arising due to the drive, and are not present in the undriven system. Extensive numerical and analytical results support perpetual stability of these emergent (though approximate) conservation laws, probably even in the thermodynamic limit. This banks on the recent finding of a sharp threshold for Floquet thermalization in clean, interacting non-integrable Floquet systems. This indicates to the possibility of stable Floquet phases of matter in disorder-free systems. This review intends to give a self-contained theoretical overview of these developments for a broad physics audience. We conclude by briefly surveying the current experimental scenario.
Abstract Equilibrium statistical mechanics rests on the assumption of chaotic dynamics of a system modulo the conservation laws of local observables: extremization of entropy immediately gives Gibbs’ ensemble (GE) for energy conserving systems and a generalized version of it (GGE) when the number of local conserved quantities is more than one. Through the last decade, statistical mechanics has been extended to describe the late-time behaviour of periodically driven (Floquet) quantum matter starting from a generic state. The structure built on the fundamental assumptions of ergodicity and identification of the relevant conservation laws in this inherently non-equilibrium setting. More recently, it has been shown that the statistical mechanics of Floquet systems has a much richer structure due to the existence of emergent conservation laws: these are approximate but stable conservation laws arising due to the drive, and are not present in the undriven system. Extensive numerical and analytical results support perpetual stability of these emergent (though approximate) conservation laws, probably even in the thermodynamic limit. This banks on the recent finding of a sharp threshold for Floquet thermalization in clean, interacting non-integrable Floquet systems. This indicates to the possibility of stable Floquet phases of matter in disorder-free systems. This review intends to give a self-contained theoretical overview of these developments for a broad physics audience. We conclude by briefly surveying the current experimental scenario.
The ergodicity postulate, a foundational pillar of Gibbsian statistical mechanics predicts that a periodically driven (Floquet) system in the absence of any conservation law heats to a featureless `infinite temperature' state. Here, we find--for a clean and interacting generic spin chain subject to a {\it strong} driving field--that this can be prevented by the emergence of {\it approximate but stable} conservation-laws not present in the undriven system. We identify their origin: they do not necessarily owe their stability to familiar protections by symmetry, topology, disorder, or even high energy costs. We show numerically, {\it in the thermodynamic limit,} that when required by these emergent conservation-laws, the entanglement-entropy density of an infinite subsystem remains zero over our entire simulation time of several decades in natural units. We further provide a recipe for designing such conservation laws with high accuracy. Finally, we present an ensemble description, which we call the strongly driven ensemble incorporating these constraints. This provides a way to control many-body chaos through stable Floquet-engineering. Strong signatures of these conservation-laws should be experimentally accessible since they manifest in all length and time scales. Variants of the spin model we have used, have already been realized using Rydberg-dressed atoms.
A periodically driven quantum many-body system is believed to approach a locally infinite-temperature-like state as dictated by the ergodicity hypothesis, when all the couplings/parameters are of the same order. When the drive is stronger than a threshold value, a large class of systems is shown to break this mandate, exhibiting Dynamical Freezing (DF): ergodicity is broken due to the emergence of local operators with permanent memory of their initial values (PRX 11, 021008, 2021). However, an exception to DF has recently been noted in a particular setting (PRX 16, 011041, 2026). Here we identify the condition for ergodicity-breaking in a system under strong drive, and show how to stabilize DF by exact, local engineering. We further show that the Floquet Hamiltonian Heff=(i/T)ln[U(T)] is non-local when ordered by spin-flips, in a non-DF system, while it turns quasi-local under our simple, local DF engineering that keeps the Hamiltonian H(t) still non-integrable at each t. This perplexing non-universality is absent in equilibrium, where non-integrability is believed to imply ergodicity and thermalization universally. Our work paves a way to engineering novel ergodicity-broken interacting Floquet phases of matter in various quantum platforms, and also endows DF-assisted quantum error mitigation and state stabilization techniques (see Nature 653, 1027 (2026)) with unprecedented versatility and flexibility.
Isolated, many-body quantum systems, evolving under their intrinsic dynamics, exhibit a multitude of exotic phenomena and raise foundational questions about statistical mechanics. A flurry of theoretical work has been devoted to understanding how these systems reach thermal equilibrium in the absence of coupling to an external bath and, when thermalization does not occur, investigating the emergent non-equilibrium phases of matter. With the advent of synthetic quantum systems, such as ultra-cold atoms in optical lattices or trapped ions, these questions are no longer academic and can be directly studied in the laboratory. This dissertation explores the non-equilibrium phenomena that stem from the interplay between interactions, disorder, symmetry, topology, and external driving. First, we study how strong disorder, leading to many-body localization, can arrest the heating of a Floquet system and stabilize symmetry-protected topological order that does not have a static analogue. We analyze its dynamical and entanglement properties, highlight its duality to a discrete time crystal, and propose an experimental implementation in a cold-atom setting.Quenched disorder and the many-body localized state are crucial ingredients in protecting macroscopic quantum coherence. We explore the stability of many-body localization in two and higher dimensions and analyze its robustness to rare regions of weak disorder.We then study a second example of non-thermal behavior, namely integrability. We show that a class of random spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram, which includes regions of integrability, classical chaos, and of a novel integrable structure whose conservation laws are reminiscent of the integrals of motion found in a many-body localized phase.The third group of disordered, non-ergodic systems we consider, spin glasses, have fascinating connections to complexity theory and the hardness of constraint satisfaction. We define a statistical ensemble that interpolates between the classical and quantum limits of such a problem and show that there exists a sharp boundary separating satisfiable and unsatisfiable phases.
Clean, interacting, periodically driven systems are believed to thermalize to featureless, infinite-temperature states that preclude the formation of stable ordered states. Here, we show stable quantum order, distinct from prethermal order, and its phase transitions in the ground state of the average-energy operator. Decomposing the time evolution into geometric and average-energy dynamics, we apply Geometric Floquet theory to identify the Floquet ground state. We derive a general variational principle to compute the Floquet ground state in large interacting periodically driven systems, and use it to develop a matrix product state algorithm. In driven nonintegrable chains of more than 100 interacting spins, our numerical simulations exhibit a gapped, area-law entangled Floquet ground state in extended parameter regimes; the ground-state manifold can support both dressed equilibrium order (e.g., Floquet antiferromagnets) and inherently nonequilibrium order (e.g., ground-state time crystals). This quantum order is protected by a finite average-energy gap, which closes at phase transition points upon changing model parameters. We decompose Floquet phases of matter into their equilibrium and nonequilibrium elements: whereas the Floquet ground state at any fixed time is a representative of an equilibrium phase of matter, the nonequilibrium content is entirely produced by the drive holonomy and is geometric. Finally, we identify quasilocal counterdiabatic driving as a mechanism for the stability against heating of gapped Floquet ground states in nonintegrable quantum systems, and argue why Floquet ground states evade the Eigenstate Thermalization Hypothesis. Our work enables the study of interacting nonequilibrium matter deep into the many-body regime, and lays the foundations for the engineering of strongly correlated ordered quantum states.
Statistical mechanics characterizes systems in or near equilibrium using in terms of a handful of "state" variables, e.g. temperature, rather than infinitely many degrees of freedom. Statistical physics describes the expansion of the early universe, aspects of black holes, and most fruitfully, phases of matter and their properties. Quantum considerations have improved this understanding over time and revealed new phenomena, especially in complicated "strongly correlated" systems. Topological phases of matter, e.g., are of both fundamental and practical interest: these phases cannot be distinguished locally, unlike ice and water, which also allows them to store and process quantum information in a "fault-tolerant" manner, recently proposed for application to quantum computation. However, above zero temperature, thermal effects can overwrite this information.Recent experiments on isolated systems have raised fundamental questions and revealed new routes to quantum computing. We now know that entanglement, generated dynamically as a quantum state evolves, "hides" local information about the past, producing familiar equilibrium states, described by a temperature. However, many systems do not thermalize: strong disorder can lead to MBL, which supports numerous phenomena forbidden in equilibrium and can protect quantum information at infinite temperature. In particular, both MBL and thermal systems are robust phases of matter, with a novel, athermal phase transition between them. This thesis begins with an overview of MBL and thermalization, followed by an overview of exactly soluble quantum systems. We then turn to an important result in the field by this author: we introduce the first nontrivial example of an integrable Floquet model and comment on its solution and salient features. We then discuss how integrable models can provide insight into quantum thermalization, e.g. in terms of entanglement growth and demonstrating that conserved charges diffuse. We then investigate thermalization away from the integrable limit, also known as "quantum chaos." We review the standard techniques in this field and, briefly, several important results, before reproducing work by this author establishing definitively the long-conjectured result that the onset of thermalization in the presence of a conserved charge is governed by diffusion of said charge. We then investigate the interplay of conventional and topological order with nonequilibrium phase structure, with applications to quantum computation in mind. We review localization-protected quantum order in several models. We then investigate two models with non-Abelian symmetry, and show that MBL in such models can only realize if the symmetry breaks spontaneously to an Abelian subgroup. Finally, we conclude by examining open quantum systems, where we find several counterintuitive results that show that baths can, in some cases, enhance localization in certain systems, which may have use in realizing quantum computation.
Predicting the stationary behavior of observables in isolated many-body quantum systems is a central challenge in quantum statistical mechanics. While one can often use the Gibbs ensemble, which is simple to compute, there are many scenarios where this is not possible and one must instead use another ensemble, such as the diagonal, microcanonical or generalized Gibbs ensembles. However, these all require detailed information about the energy or other conserved quantities to be constructed. Here we propose a general and computationally easy approach to determine the stationary probability distribution of observables with few outcomes. Interpreting coarse measurements at equilibrium as noisy communication channels, we provide general analytical arguments in favor of the applicability of a maximum entropy principle for this class of observables. We show that the resulting theory accurately predicts stationary probability distributions without detailed microscopic information like the energy eigenstates. Extensive numerical experiments on 7 non-weakly interacting spin-1/2 Hamiltonians demonstrate the broad applicability and robustness of this framework in both quantum integrable and chaotic models.