Many-body localization beyond eigenstates in all dimensions

Chandran A., Pal A., Laumann C.R., Isolated quantum systems with quenched randomness exhibit many-body localization (MBL), wherein they do not reach local thermal equilibrium even when highly excited above their ground states. It is widely believed that individual eigenstates capture this breakdown of thermalization at finite size. We show that this belief is false in general and that a MBL system can exhibit the eigenstate properties of a thermalizing system. We propose that localized approximately conserved operators (l∗-bits) underlie localization in such systems. In dimensions d>1, we further argue that the existing MBL phenomenology is unstable to boundary effects and gives way to l∗-bits. Physical consequences of l∗-bits include the possibility of an eigenstate phase transition within the MBL phase unrelated to the dynamical transition in d=1 and thermal eigenstates at all parameters in d>1. Near-term experiments in ultracold atomic systems and numerics can probe the dynamics generated by boundary layers and emergence of l∗-bits.

Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System

Žnidarič M., We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. By employing a density matrix renormalization group technique for the study of the stationary states of the boundary-driven Lindblad equation we are able to study extremely large systems (400 spins). We find both a diffusive and a subdiffusive phase depending on the strength of the disorder and on the anisotropy parameter of the Heisenberg chain. Studying finite-size effects, we show numerically and theoretically that a very large crossover length exists that controls the passage of a clean-system dominated dynamics to one observed in the thermodynamic limit. Such a large length scale, being larger than the sizes studied before, explains previous conflicting results. We also predict spatial profiles of magnetization in steady states of generic nondiffusive systems.

Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions

Pietracaprina F., Ros V., In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-body) or many-body localized phase. This approximation, which consists of summing over the amplitudes of only the shortest paths in the locator expansion, is known to overestimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean-field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is 0.9% off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder hc=4.0±0.3 is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent ν=1. In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean-field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium.

The many-body localized phase of the quantum random energy model

Baldwin C.L., Laumann C.R., Pal A., The random energy model (REM) provides a solvable mean-field description of the equilibrium spin-glass transition. Its quantum sibling (the QREM), obtained by adding a transverse field to the REM, has similar properties and shows a spin-glass phase for sufficiently small transverse field and temperature. In a recent work, some of us have shown that the QREM further exhibits a many-body localization-delocalization (MBLD) transition when viewed as a closed quantum system, evolving according to the quantum dynamics. This phase encloses the familiar equilibrium spin-glass phase. In this paper, we study in detail the MBLD transition within the forward-scattering approximation and replica techniques. The predictions for the transition line are in good agreement with the exact diagonalization numerics. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.

Large-N-approximated field theory for multipartite entanglement

Facchi P., Florio G., Parisi G., Pascazio S., We try to characterize the statistics of multipartite entanglement of the random states of an n-qubit system. Unable to solve the problem exactly we generalize it, replacing complex numbers with real vectors with Nc components (the original problem is recovered for Nc=2). Studying the leading diagrams in the large-Nc approximation, we unearth the presence of a phase transition and, in an explicit example, show that the so-called entanglement frustration disappears in the large-Nc limit.

Total correlations of the diagonal ensemble herald the many-body localization transition

Goold J., Gogolin C., Clark S.R., Eisert J., The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work we introduce the total correlations, a concept from quantum information theory capturing multipartite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular, we show that the total correlations has sublinear dependence on the system size in delocalized, ergodic phases, whereas we find that it scales extensively in the localized phase developing a pronounced peak at the transition. We exemplify the power of our approach by means of an exact diagonalization study of a Heisenberg spin chain in a disordered field. By a finite size scaling analysis of the peak position and crossover point from log to linear scaling we collect evidence that ergodicity is broken before the MBL transition in this model.

Corrigendum to Integrals of motion in the many-body localized phase [Nucl. Phys. B 891, (2015), 420-465] doi:10.1016/j.nuclphysb.2014.12.014

Ros V., Müller M., We correct a small error in our article Integrals of motion in the many body localized phase[1]. The cor-rection does not alter the main result regarding the convergence of the perturbative expansion for integrals of motion in forward approximation, but reduces the estimate of the radius of convergence by a numerical factor of roughly ≃1.79.

Random Coulomb antiferromagnets: From diluted spin liquids to Euclidean random matrices

Rehn J., Sen A., Andreanov A., Damle K., Moessner R., We study a disordered classical Heisenberg magnet with uniformly antiferromagnetic interactions which are frustrated on account of their long-range Coulomb form, i.e., J(r)∼-Alnr in d=2 and J(r)∼A/r in d=3. This arises naturally as the T→0 limit of the emergent interactions between vacancy-induced degrees of freedom in a class of diluted Coulomb spin liquids (including the classical Heisenberg antiferromagnets in checkerboard, SCGO, and pyrochlore lattices) and presents a novel variant of a disordered long-range spin Hamiltonian. Using detailed analytical and numerical studies we establish that this model exhibits a very broad paramagnetic regime that extends to very large values of A in both d=2 and d=3. In d=2, using the lattice-Green-function-based finite-size regularization of the Coulomb potential (which corresponds naturally to the underlying low-temperature limit of the emergent interactions between orphans), we find evidence that freezing into a glassy state occurs only in the limit of strong coupling, A=∞, while no such transition seems to exist in d=3. We also demonstrate the presence and importance of screening for such a magnet. We analyze the spectrum of the Euclidean random matrices describing a Gaussian version of this problem and identify a corresponding quantum mechanical scattering problem.

Holographic p-wave superconductor with disorder

Areán D., Farahi A., Pando Zayas L.A., Salazar Landea I., Abstract: We implement the effects of disorder on a holographic p-wave superconductor by introducing a random chemical potential which defines the local energy of the charge carriers. Since there are various possibilities for the orientation of the vector order parameter, we explore the behavior of the condensate in the parallel and perpendicular directions to the introduced disorder. We clarify the nature of various branches representing competing solutions and construct the disordered phase diagram. We find that moderate disorder enhances superconductivity as determined by the value of the condensate. Though we mostly focus on uncorrelated noise, we also consider a disorder characterized by its spectral properties and study in detail its influence on the spectral properties of the condensate and charge density. We find fairly universal responses of the resulting power spectra characterized by linear functions of the disorder power spectrum.

Localized systems coupled to small baths: From Anderson to Zeno

Huse D.A., Nandkishore R., Pietracaprina F., Ros V., We investigate what happens if an Anderson localized system is coupled to a small bath, with a discrete spectrum, when the coupling between system and bath is specially chosen so as to never localize the bath. We find that the effect of the bath on localization in the system is a nonmonotonic function of the coupling between system and bath. At weak couplings, the bath facilitates transport by allowing the system to "borrow" energy from the bath. But, above a certain coupling the bath produces localization because of an orthogonality catastrophe, whereby the bath "dresses" the system and hence suppresses the hopping matrix element. We call this last regime the regime of "Zeno localization" since the physics of this regime is akin to the quantum Zeno effect, where frequent measurements of the position of a particle impede its motion. We confirm our results by numerical exact diagonalization.

Quantum annealing: The fastest route to quantum computation?

Laumann C.R., Moessner R., In this review we consider the performance of the quantum adiabatic algorithm for the solution of decision problems. We divide the possible failure mechanisms into two sets: small gaps due to quantum phase transitions and small gaps due to avoided crossings inside a phase. We argue that the thermodynamic order of the phase transitions is not predictive of the scaling of the gap with the system size. On the contrary, we also argue that, if the phase surrounding the problem Hamiltonian is a Many-Body Localized (MBL) phase, the gaps are going to be typically exponentially small and that this follows naturally from the existence of local integrals of motion in the MBL phase.

Integrals of motion in the many-body localized phase

Ros V., Müller M., We construct a complete set of quasi-local integrals of motion for the many-body localized phase of interacting fermions in a disordered potential. The integrals of motion can be chosen to have binary spectrum {0, 1}, thus constituting exact quasiparticle occupation number operators for the Fermi insulator. We map the problem onto a non-Hermitian hopping problem on a lattice in operator space. We show how the integrals of motion can be built, under certain approximations, as a convergent series in the interaction strength. An estimate of its radius of convergence is given, which also provides an estimate for the many-body localization-delocalization transition. Finally, we discuss how the properties of the operator expansion for the integrals of motion imply the presence or absence of a finite temperature transition.

Many-body mobility edge in a mean-field quantum spin glass

Laumann C.R., Pal A., The quantum random energy model provides a mean-field description of the equilibrium spin glass transition. We show that it further exhibits a many-body localization-delocalization (MBLD) transition when viewed as a closed quantum system. The mean-field structure of the model allows an analytically tractable description of the MBLD transition using the forward-scattering approximation and replica techniques. The predictions are in good agreement with the numerics. The MBLD transition lies at energy density significantly above the equilibrium spin glass transition, indicating that the closed system dynamics freezes well outside of the traditional glass phase. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.

Anderson localization on the Bethe lattice: Nonergodicity of extended states

De Luca A., Altshuler B.L., Kravtsov V.E., Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f(α) defined in Eq. (3) remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization; i.e., the ergodicity can be reached only in the absence of disorder. The one-particle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated. © 2014 American Physical Society.

Holographic superconductor with disorder

Areán D., Farahi A., Pando Zayas L., Landea I., We study the effects of disorder on a holographic superconductor by introducing a random chemical potential on the boundary. We consider various realizations of disorder and find that the critical temperature for superconductivity is enhanced. We also present evidence for a precise form of renormalization in this system. Namely, when the random chemical potential is characterized by a Fourier spectrum of the form k-2α we find that the spectra of the condensate and the charge density are again power laws, whose exponents are accurately and universally governed by linear functions of α. © 2014 American Physical Society.

Satisfiability-unsatisfiability transition in the adversarial satisfiability problem

Bardoscia M., Nagaj D., Adversarial satisfiability (AdSAT) is a generalization of the satisfiability (SAT) problem in which two players try to make a Boolean formula true (resp. false) by controlling their respective sets of variables. AdSAT belongs to a higher complexity class in the polynomial hierarchy than SAT, and therefore the nature of the critical region and the transition are not easily parallel to those of SAT and worthy of independent study. AdSAT also provides an upper bound for the transition threshold of the quantum satisfiability problem (QSAT). We present a complete algorithm for AdSAT, show that 2-AdSAT is in P, and then study two stochastic algorithms (simulated annealing and its improved variant) and compare their performances in detail for 3-AdSAT. Varying the density of clauses α we claim that there is a sharp SAT-UNSAT transition at a critical value whose upper bound is αc 1.5, suggesting a much stricter upper bound for the QSAT transition than those previously found. © 2014 American Physical Society.

Ergodicity breaking in a model showing many-body localization

De Luca A., We study the breaking of ergodicity measured in terms of return probability in the evolution of a quantum state of a spin chain. In the non-ergodic phase a quantum state evolves in a much smaller fraction of the Hilbert space than would be allowed by the conservation of extensive observables. By the anomalous scaling of the participation ratios with system size we are led to consider the distribution of the wave function coefficients, a standard observable in modern studies of Anderson localization. We finally present a criterion for the identification of the ergodicity-breaking (many-body localization) transition based on these distributions which is quite robust and well suited for numerical investigations of a broad class of problems. © Copyright EPLA, 2013.

Random perfect lattices and the sphere packing problem

Andreanov A., Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d=19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily. Their number, however, grows superexponentially with the dimension, so to get an idea of their properties we propose to study a randomized version of the generating algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Monte Carlo simulation. We therefore study the distribution of packing fractions and kissing numbers of these ensembles and show how as the temperature is decreased the best known packers are easily recovered. We find that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A d and D d), and we propose two hypotheses between which we cannot distinguish in this paper: one in which they improve the Minkowsky bound φ∼2 -(0.84 ±0.06 )d, and a competitor in which their packing fraction decreases superexponentially, namely, φ∼d -ad but with a very small coefficient a=0.06±0.04. We also find properties of the random walk which are suggestive of a glassy system already for moderately small dimensions. We also analyze local structure of network of perfect lattices conjecturing that this is a scale-free network in all dimensions with constant scaling exponent 2.6±0.1. © 2012 American Physical Society.

Quantum adiabatic algorithm and scaling of gaps at first-order quantum phase transitions

Laumann C.R., Moessner R., Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling of the Hamiltonian gap at quantum first-order transitions, generally expected to be exponentially small in the size of the system. However, we show that a quantum antiferromagnetic Ising chain in a staggered field can exhibit a first-order transition with only an algebraically small gap. In addition, we construct a simple classical translationally invariant one-dimensional Hamiltonian containing nearest-neighbor interactions only, which exhibits an exponential gap at a thermodynamic quantum first-order transition of essentially topological origin. This establishes that (i) the QAA can be successful even across first-order transitions but also that (ii)it can fail on exceedingly simple problems readily solved by inspection, or by classical annealing. © 2012 American Physical Society.

Statistical mechanics of classical and quantum computational complexity

Laumann C., Moessner R., The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers ("NP-complete") and those which are likely hard for quantum computers ("QMA-complete") by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances-typically those that involve optimization-by studying random ensembles of them. These pose unusual and interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In these lecture notes we present an introduction to this set of ideas with older work on classical satisfiability and recent work on quantum satisfiability as primary examples. We also touch on the connection of computational hardness with the physical notion of glassiness. © 2012 Springer-Verlag Berlin Heidelberg.