On the stability of many-body localization in d>1d>1

Recent work by De Roeck et al. [Phys. Rev. B 95, 155129 (2017)] has argued that many-body localization (MBL) is unstable in two and higher dimensions due to a thermalization avalanche triggered by rare regions of weak disorder. To examine these arguments, we construct several models of a finite ergodic bubble coupled to an Anderson insulator of non-interacting fermions. We first describe the ergodic region using a GOE random matrix and perform an exact diagonalization study of small systems. The results are in excellent agreement with a refined theory of the thermalization avalanche that includes transient finite-size effects, lending strong support to the avalanche scenario. We then explore the limit of large system sizes by modeling the ergodic region via a Hubbard model with all-to-all random hopping: the combined system, consisting of the bubble and the insulator, can be reduced to an effective Anderson impurity problem. We find that the spectral function of a local operator in the ergodic region changes dramatically when coupling to a large number of localized fermionic states---this occurs even when the localized sites are weakly coupled to the bubble. In principle, for a given size of the ergodic region, this may arrest the avalanche. However, this back-action effect is suppressed and the avalanche can be recovered if the ergodic bubble is large enough. Thus, the main effect of the back-action is to renormalize the critical bubble size.Comment: v3: Published version. Expanded the discussion in Section IV to include a new calculation and figure (Fig. 7

Many-Body Quantum Dynamics and Non-Equilibrium Phases of Matter

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

Classical-Quantum Mixing in the Random 2-Satisfiability Problem

Classical satisfiability (SAT) and quantum satisfiability (QSAT) are complete problems for the complexity classes NP and QMA which are believed to be intractable for classical and quantum computers, respectively. Statistical ensembles of instances of these problems have been studied previously in an attempt to elucidate their typical, as opposed to worst case, behavior. In this paper we introduce a new statistical ensemble that interpolates between classical and quantum. For the simplest 2-SAT/2-QSAT ensemble we find the exact boundary that separates SAT and UNSAT instances. We do so by establishing coincident lower and upper bounds, in the limit of large instances, on the extent of the UNSAT and SAT regions, respectively.Comment: Updated reference

Phase Transitions And Classical-Quantum Mixing in the Satisfiability Problem

During the last few decades, intrigued by the resemblance between some hard combinatorial optimization problems and the physics of glassy systems, physicists have tried to understand why and how these sort of problems (known as NP-complete in the computer science jargon) are hard. To this end, they have employed a plethora of tools from classical statistical mechanics and, among other things, have proven that the random k-satis ability has a phase transition between a "solvable" and "unsolvable" phase which is intrinsically related to an easy-hard-easy transition: instances are easy to decide well within the satis able or unsatisfiable regimes, but are exponentially hard to determine near the critical point. In this thesis, we focus both on classical and quantum complexity theory. After presenting some of the most important results from these fields such as the existence of "hard" problems for classical and quantum computers, we embark on the study of random ensembles of the satis ability problem. We show how these are related to the study of graph theoretical questions and why there exists a phase transition for the random k-SAT and k-QSAT. Motivated by the fact that the "UNSAT-ness" of the latter problem is a purely graph geometric property, we study the classical-quantum mixing in random satis- ability in order to see how this peculiarity emerges. Specifically, we define a new problem which we call (N;M;K)-k-SAT and for k = 2: we implement an exact algorithm proposed by Bravyi and ll in some minor gaps in its construction; we obtain a complete numerical phase diagram; conjecture an analytic expression for the phase boundary in the thermodynamic limit; show that the SAT-UNSAT transition corresponds to an easy-hard-easy transition in the resolution time pattern; and outline a finite size scaling analysis. Finally, to tackle (N;M;K)-k-SAT for k > 2, we present a formalism based on degenerate perturbation theory and report some hints of clustering

Integrable and Chaotic Dynamics of Spins Coupled to an Optical Cavity

We show that a class of random all-to-all spin models, realizable in systems of atoms coupled to an optical cavity, gives rise to a rich dynamical phase diagram due to the pairwise separable nature of the couplings. By controlling the experimental parameters, one can tune between integrable and chaotic dynamics on the one hand and between classical and quantum regimes on the other hand. For two special values of a spin-anisotropy parameter, the model exhibits rational Gaudin-type integrability, and it is characterized by an extensive set of spin-bilinear integrals of motion, independent of the spin size. More generically, we find a novel integrable structure with conserved charges that are not purely bilinear. Instead, they develop “dressing tails” of higher-body terms, reminiscent of the dressed local integrals of motion found in many-body localized phases. Surprisingly, this new type of integrable dynamics found in finite-size spin-1/2 systems disappears in the large-S limit, giving way to classical chaos. We identify parameter regimes for characterizing these different dynamical behaviors in realistic experiments, in view of the limitations set by cavity dissipation