We review some recent developments in the statistical mechanics of isolated quantum systems.We provide a brief introduction to quantum thermalization, paying particular attention to the 'Eigenstate Thermalization Hypothesis' (ETH), and the resulting 'single-eigenstate statistical mechanics'. We then focus on a class of systems which fail to quantum thermalize and whose eigenstates violate the ETH: These are the many-body Anderson localized systems; their long-time properties are not captured by the conventional ensembles of quantum statistical mechanics. These systems can locally remember forever information about their local initial conditions, and are thus of interest for possibilities of storing quantum information. We discuss key features of many-body localization (MBL), and review a phenomenology of the MBL phase. Single-eigenstate statistical mechanics within the MBL phase reveals dynamically-stable ordered phases, and phase transitions among them, that are invisible to equilibrium statistical mechanics and can occur at high energy and low spatial dimensionality where equilibrium ordering is forbidden.

We suggest that if a localized phase at nonzero temperature $T>0$ exists for strongly disordered and weakly interacting electrons, as recently argued, it will also occur when both disorder and interactions are strong and $T$ is very high. We show that in this high-$T$ regime the localization transition may be studied numerically through exact diagonalization of small systems. We obtain spectra for one-dimensional lattice models of interacting spinless fermions in a random potential. As expected, the spectral statistics of finite-size samples cross over from those of orthogonal random matrices in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. However, these data show deviations from simple one-parameter finite-size scaling: the apparent mobility edge ``drifts'' as the system's size is increased. Based on spectral statistics alone, we have thus been unable to make a strong numerical case for the presence of a many-body localized phase at nonzero $T$

We use exact diagonalization to explore the many-body localization transition in a random-field spin-1/2 chain. We examine the correlations within each many-body eigenstate, looking at all high-energy states and thus effectively working at infinite temperature. For weak random field the eigenstates are thermal, as expected in this nonlocalized, "ergodic" phase. For strong random field the eigenstates are localized, with only short-range entanglement. We roughly locate the localization transition and examine some of its finite-size scaling, finding that this quantum phase transition at nonzero temperature might be showing infinite-randomness scaling with a dynamic critical exponent z → ∞.

The e8'ects of thermal fluctuations, quenched disorder, and anisotropy on the phases and phase transitions in type-II superconductors are examined, focusing on linear and nonlinear transport properties. In zero magnetic Geld there are two crossovers upon approaching T" first the "Ginzburg" crossover from mean-field behavior to the universality class of an uncharged superAuid, and then, much closer to T, for strongly type-II systems, a crossover to the universality class of a charged superAuid. The primary focus of this paper is on the behavior in the presence of a penetrating magnetic field. In a clean system the vortex-lattice phase can melt due to thermal fluctuations; we estimate the phase boundary in a variety of regimes. Pinning of vortices due to impurities or other defects destroys the long-range correlations of the vortex lattice, probably replacing it with a new vortex-glass phase that has spin-glasslike ofF-diagonal long-range order and is truly superconducting, in contrast to conventional theories of "Aux creep. " The properties of this vortex-glass phase are examined, as well as the critical behavior near the transition from the vortex-glass to the vortex-Quid phase. The crossover from lattice to vortex-glass behavior for weak pinning is also examined. Linear and nonlinear conductivity measurements and other experiments on the high-T, superconductors Y-Ba-Cu-0 and Bi-Sr-Ca-Cu-0 are discussed in light of the results. The latter is found to exhibit strongly two-dimensional behavior over large portions of its phase diagram.

Condensed matter physicists have long sought a realistic two-dimensional (2D) magnetic system whose ground state is a spin liquid-a zero temperature state in which quantum fluctuations have melted away any form of magnetic order. The nearest-neighbor S = 1 2Heisenberg model on the kagome lattice has seemed an ideal candidate, but in recent years some approximate numerical approaches to it have yielded instead a valence bond crystal. We have used the density matrix renormalization group to perform very accurate simulations on numerous cylinders with circumferences up to 12 lattice spacings, finding instead of the valence bond crystal a singlet-gapped spin liquid with substantially lower energy that appears to have Z2 topological order. Our results, through a combination of very low energy, short correlation lengths and corresponding small finite size effects, a new rigorous energy bound, and consistent behavior on many cylinders, provide strong evidence that the 2D ground state of this model is a gapped spin liquid.

We consider isolated quantum systems with all of their many-body eigenstates localized. We define a sense in which such systems are integrable, and discuss a method for finding their localized conserved quantum numbers ("constants of motion"). These localized operators are interacting pseudospins and are subject to dephasing but not to dissipation, so any quantum states of these pseudospins can in principle be recovered via (spin) echo procedures. We also discuss the spreading of entanglement in many-body localized systems, which is another aspect of the dephasing due to interactions between these localized conserved operators. PACS numbers:Isolated quantum many-body systems with shortrange interactions and static randomness may be in a many-body localized phase where they do not thermally equilibrate under their own dynamics. While this possibility was pointed out long ago by Anderson [1], such localization of highly-excited states in systems with interactions did not receive a lot of attention until after Basko, et al.[2] forcefully brought the subject into focus. Isolated systems in the many-body localized phase have strictly zero thermal conductivity [2], so if some energy is added to the system locally, it only excites localized degrees of freedom and does not diffuse, even when the system's energy density corresponds to a nonzero (even infinite [3]) temperature.We expect that the many-body eigenstates of a system's Hamiltonian in the localized phase are product states of localized degrees of freedom, with some shortrange "area-law" entanglement between the "bare" local degrees of freedom. One goal of this paper is to explore how one can define suitably "dressed" localized pseudospin operators in terms of which the many-body eigenstates within the localized phase are indeed precisely product states with zero entanglement. When the Hamiltonian is then expressed in terms of these dressed localized pseudospins it has exponentially decaying long-range interactions, and it is these long-range interactions that cause decoherence and dephasing of local observables in the many-body insulator. These interactions also cause the spreading of entanglement for a nonentangled initial product state of the bare spins, as has been seen and explored in Refs. [4][5][6][7][8].To be concrete, assume we have a system of N spin-1/2's {σ i } on some lattice (say, in one, two or three dimensions). For an example, see, e.g., Ref. [9]. Our system has a specific random Hamiltonian H that contains only short-range interactions and strong enough static random fields on each spin so that, with probability one in the limit of large N , all 2 N many-body eigenstates of this H are localized. The construction we present below should be readily generalizable to local operators with more than two states. It should also be generalizable to systems where the dominant strong randomness is instead the spin-spin interactions rather than random fields. In those cases, the pseudospins we will construct may instead be localized domain wall operators [...

A fundamental assumption in statistical physics is that generic closed quantum many-body systems thermalize under their own dynamics. Recently, the emergence of many-body localized systems has questioned this concept and challenged our understanding of the connection between statistical physics and quantum mechanics. Here we report on the observation of a many-body localization transition between thermal and localized phases for bosons in a two-dimensional disordered optical lattice. With our single-site-resolved measurements, we track the relaxation dynamics of an initially prepared out-of-equilibrium density pattern and find strong evidence for a diverging length scale when approaching the localization transition. Our experiments represent a demonstration and in-depth characterization of many-body localization in a regime not accessible with state-of-the-art simulations on classical computers.

When a system thermalizes it loses all local memory of its initial conditions. This is a general feature of open systems and is well described by equilibrium statistical mechanics. Even within a closed (or reversible) quantum system, where unitary time evolution retains all information about its initial state, subsystems can still thermalize using the rest of the system as an effective heat bath. Exceptions to quantum thermalization have been predicted and observed, but typically require inherent symmetries [1,2] or noninteracting particles in the presence of static disorder [3][4][5][6]. The prediction of many-body localization (MBL), in which disordered quantum systems can fail to thermalize in spite of strong interactions and high excitation energy, was therefore surprising and has attracted considerable theoretical attention [3, 7-10]. Here we experimentally generate MBL states by applying an Ising Hamiltonian with long-range interactions and programmably random disorder to ten spins initialized far from equilibrium. We observe the essential signatures of MBL: memory retention of the initial state, a Poissonian distribution of energy level spacings, and entanglement growth in the system at long times. Our platform can be scaled to higher numbers of spins, where detailed modeling of MBL becomes impossible due to the complexity of representing such entangled quantum states. Moreover, the high degree of control in our experiment may guide the use of MBL states as potential quantum memories in naturally disordered quantum systems [11]. i (γ = x, y, z) as well as arbitrary spin correlation functions along any direction.J i,j is a tunable, long-range coupling that falls off approximately algebraically as J i,j ∝ J max /|i − j| α [17], where J max is typically 2π(0.5 kHz). Here we tune α arXiv:1508.07026v1 [quant-ph]