There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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 that 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 forever locally remember 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 reveal dynamically stable ordered phases, and phase transitions among them, that are invisible to equilibrium statistical mechanics and...

Many-Body Localization and Thermalization in Quantum Statistical Mechanics

Annual review of condensed matter physics

Journal of Statistical Mechanics: theory and experiment

Many-body localization (MBL), the disorder-induced localization of interacting particles, signals a breakdown of conventional thermodynamics because MBL systems do not thermalize and show nonergodic time evolution. We experimentally observed this nonergodic evolution for interacting fermions in a one-dimensional quasirandom optical lattice and identified the MBL transition through the relaxation dynamics of an initially prepared charge density wave. For sufficiently weak disorder, the time evolution appears ergodic and thermalizing, erasing all initial ordering, whereas above a critical disorder strength, a substantial portion of the initial ordering persists. The critical disorder value shows a distinctive dependence on the interaction strength, which is in agreement with numerical simulations. Our experiment paves the way to further detailed studies of MBL, such as in noncorrelated disorder or higher dimensions.

https://science.sciencemag.org/content/sci/349/6250/842.full.pdf

Observation of many-body localization of interacting fermions in a quasirandom optical lattice

We study the infinite temperature dynamics of a prototypical one-dimensional system expected to exhibit many-body localization. Using numerically exact methods, we establish the dynamical phase diagram of this system based on the statistics of its eigenvalues and its dynamical behavior. We show that the nonergodic phase is reentrant as a function of the interaction strength, illustrating that localization can be reinforced by sufficiently strong interactions even at infinite temperature. Surprisingly, within the accessible time range, the ergodic phase shows subdiffusive behavior, suggesting that the diffusion coefficient vanishes throughout much of the phase diagram in the thermodynamic limit. Our findings strongly suggest that Wigner--Dyson statistics of eigenvalue spacings may appear in a class of ergodic but subdiffusive systems.

Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice.

Recent studies point towards nontriviality of the ergodic phase in systems exhibiting many-body localization (MBL), which shows subexponential relaxation of local observables, subdiffusive transport and sublinear spreading of the entanglement entropy Here we review the dynamical properties of this phase and the available numerically exact and approximate methods for its study We discuss in which sense this phase could be considered ergodic and present possible phenomenological explanations of its dynamical properties We close by analyzing to which extent the proposed explanations were verified by numerical studies and present the open questions in this field

https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.201600350

The ergodic side of the many‐body localization transition

It is commonly believed that quantum isolated systems satisfying the eigenstate thermalization hypothesis (ETH) are diffusive. We show that this assumption is too restrictive since there are systems that are asymptotically in a thermal state yet exhibit anomalous, subdiffusive thermalization. We show that such systems satisfy a modified version of the ETH ansatz and derive a general connection between the scaling of the variance of the off-diagonal matrix elements of local operators, written in the eigenbasis of the Hamiltonian, and the dynamical exponent. We find that for subdiffusively thermalizing systems the variance scales more slowly with system size than expected for diffusive systems. We corroborate our findings by numerically studying the distribution of the coefficients of the eigenfunctions and the off-diagonal matrix elements of local operators of the random field Heisenberg chain, which has anomalous transport in its thermal phase. Surprisingly, this system also has non-Gaussian distributions of the eigenfunctions, thus, directly violating Berry's conjecture.

Anomalous Thermalization in Ergodic Systems.

How fast does quantum information spread in isolated quantum systems? The authors address this question using out-of-time order correlation functions and find that information spreads behind a power law front in disordered spin chains, with a characteristic exponent, which is related to the exponent of the power law growth of the entanglement entropy. The authors also observe that disorder inhibits information spreading and leads to unsaturated Lieb-Robinson bounds.

Information propagation in isolated quantum systems

Following the field theoretic approach of Basko et al. [Ann. Phys. 321, 1126 (2006)], we study in detail the real-time dynamics of a system expected to exhibit many-body localization. In particular, for time scales inaccessible to exact methods, we demonstrate that within the second-Born approximation the temporal decay of the density-density correlation function is consistent with a finite value for $t\ensuremath{\rightarrow}\ensuremath{\infty}$, as expected in a nonergodic state. This behavior persists over a wide range of disorder and interaction strengths and suggests that the nonergodic phase is surprisingly robust compared to the prediction of Basko et al. We argue that this discrepancy may be resolved by the presence of a nonergodic, but metallic, phase.

Yevgeny Bar Lev

Papers

Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice.

The ergodic side of the many‐body localization transition

Anomalous Thermalization in Ergodic Systems.

Information propagation in isolated quantum systems

Dynamics of many-body localization