We review recent developments in the study of out-of-equilibrium topological states of matter in isolated systems. The phenomenon of many-body localization, exhibited by some isolated systems usually in the presence of quenched disorder, prevents systems from equilibrating to a thermal state where the delicate quantum correlations necessary for topological order are often washed out. Instead, many-body localized systems can exhibit a type of eigenstate phase structure wherein their entire many-body spectrum is characterized by various types of quantum order, usually restricted to quantum ground states. After introducing many-body localization and explaining how it can protect quantum order, we then explore how the interplay of symmetry and dimensionality with many-body localization constrains its role in stabilizing topological phases out of equilibrium.
We review recent developments in the study of out-of-equilibrium topological states of matter in isolated systems. The phenomenon of many-body localization, exhibited by some isolated systems usually in the presence of quenched disorder, prevents systems from equilibrating to a thermal state where the delicate quantum correlations necessary for topological order are often washed out. Instead, many-body localized systems can exhibit a type of eigenstate phase structure wherein their entire many-body spectrum is characterized by various types of quantum order, usually restricted to quantum ground states. After introducing many-body localization and explaining how it can protect quantum order, we then explore how the interplay of symmetry and dimensionality with many-body localization constrains its role in stabilizing topological phases out of equilibrium.
Closed quantum systems with quenched randomness exhibit many-body localized regimes wherein they do not equilibrate, even though prepared with macroscopic amounts of energy above their ground states. We show that such localized systems can order, in that individual many-body eigenstates can break symmetries or display topological order in the infinite-volume limit. Indeed, isolated localized quantum systems can order even at energy densities where the corresponding thermally equilibrated system is disordered, i.e., localization protects order. In addition, localized systems can move between ordered and disordered localized phases via nonthermodynamic transitions in the properties of the many-body eigenstates. We give evidence that such transitions may proceed via localized critical points. We note that localization provides protection against decoherence that may allow experimental manipulation of macroscopic quantum states. We also identify a ``spectral transition'' involving a sharp change in the spectral statistics of the many-body Hamiltonian.
Topological ordered states are exotic quantum states of matter that defy the usual description in terms of symmetry breaking and local order parameters. The type or order they feature is of non-local, topological nature, and it allows such systems to present unusual and interesting behaviours, like robust degeneracy, and anyonic excitations. Here we focus on the behaviour of such systems away from equilibrium. After a sudden change of the Hamiltonian - the so called quantum quench - quantum many body systems show thermalization, unless the quench is integrable. We show how in topologically ordered systems out of equilibrium behaviour do not follow the usual thermalization behaviour.
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.
Department of Physics, Harvard University, Cambridge MA 02138, USAClosed quantum systems with quenched randomness exhibit many-body localized regimes whereinthey do not equilibrate even though prepared with macroscopic amounts of energy above their groundstates. We show that such localized systems can order in that individual many-body eigenstatescan break symmetries or display topological order in the in nite volume limit. Indeed, isolatedlocalized quantum systems can order even at energy densities where the corresponding thermallyequilibrated system is disordered, i.e.: localization protects order. In addition, localized systemscan move between ordered and disordered localized phases via non-thermodynamic transitions inthe properties of the many-body eigenstates. We give evidence that such transitions may proceedvia localized critical points. We note that localization provides protection against decoherence thatmay allow experimental manipulation of macroscopic quantum states. We also identify a ‘spectraltransition’ involving a sharp change in the spectral statistics of the many-body Hamiltonian.
We derive general constraints on the existence of many-body localized (MBL) phases in the presence of global symmetries, and show that MBL is not possible with symmetry groups that protect multiplets (e.g., all non-Abelian symmetry groups). Based on simple representation theoretic considerations, we derive general Mermin-Wagner--type principles governing the possible alternative fates of nonequilibrium dynamics in isolated, strongly disordered quantum systems. Our results rule out the existence of MBL symmetry-protected topological phases with non-Abelian symmetry groups, as well as time-reversal symmetry-protected electronic topological insulators, and in fact all fermion topological insulators and superconductors in the 10-fold way classification. Moreover, extending our arguments to systems with intrinsic topological order, we rule out MBL phases with non-Abelian anyons as well as certain classes of symmetry-enriched topological orders.
Recent work shows that highly excited many-body localized eigenstates can exhibit broken symmetries and topological order, including in dimensions where such order would be forbidden in equilibrium. In this paper we extend this analysis to discrete symmetry-protected order via the explicit examples of the Haldane phase of one-dimensional spin chains and the topological Ising paramagnet in two dimensions. We comment on the challenge of extending these results to cases where the protecting symmetry is continuous.