Massively parallel implementation and approaches to simulate quantum dynamics using Krylov subspace techniques

Brenes M., Varma V., We have developed an application and implemented parallel algorithms in order to provide a computational framework suitable for massively parallel supercomputers to study the unitary dynamics of quantum systems. We use renowned parallel libraries such as PETSc/SLEPc combined with high-performance computing approaches in order to overcome the large memory requirements to be able to study systems whose Hilbert space dimension comprises over 9 billion independent quantum states. Moreover, we provide descriptions of the parallel approach used for the three most important stages of the simulation: handling the Hilbert subspace basis, constructing a matrix representation for a generic Hamiltonian operator and the time evolution of the system by means of the Krylov subspace methods. We employ our setup to study the evolution of quasidisordered and clean many-body systems, focussing on the return probability and related dynamical exponents: the large system sizes accessible provide novel insights into their thermalization properties. Program summary: Program Title: DSQMKryST Program Files doi: http://dx.doi.org/10.17632/f6vty3wkwj.1 Licensing provisions: BSD 3-clause Programming language: C++ Supplementary material: https://github.com/mbrenesn/DSQMKryST External routines/libraries: PETSc (https://www.mcs.anl.gov/petsc/), SLEPc (http://slepc.upv.es), Boost C++ (http://www.boost.org) Nature of problem: Unitary dynamics of quantum mechanical many-body systems Solution method: Krylov subspace techniques (Arnoldi procedure) with a massively parallel, distributed memory approach

Can we study the many-body localisation transition?

Panda R.K., We present a detailed analysis of the length- and timescales needed to approach the critical region of MBL from the delocalised phase, studying both eigenstates and the time evolution of an initial state. For the eigenstates we show that in the delocalised region there is a single length, which is a function of disorder strength, controlling the finite-size flow. Small systems look localised, and only for larger systems do resonances develop which restore ergodicity in the form of the eigenstate thermalisation hypothesis. For the transport properties, we study the time necessary to transport a single spin across a domain wall, showing how this grows quickly with increasing disorder, and compare it with the Heisenberg time. For a sufficiently large system the Heisenberg time is always larger than the transport time, but for a smaller system this is not necessarily the case. We conclude that the properties of the MBL transition cannot be explored using the system sizes or times available to current numerical and experimental studies.

Soap films with gravity and almost-minimal surfaces

Maggi F., Stuvard S., Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.

Impact of jamming criticality on low-temperature anomalies in structural glasses

Franz S., Maimbourg T., Parisi G., We present a mechanism for the anomalous behavior of the specific heat in low-temperature amorphous solids. The analytic solution of a mean-field model belonging to the same universality class as high-dimensional glasses, the spherical perceptron, suggests that there exists a cross-over temperature above which the specific heat scales linearly with temperature, while below it, a cubic scaling is displayed. This relies on two crucial features of the phase diagram: (i) the marginal stability of the free-energy landscape, which induces a gapless phase responsible for the emergence of a power-law scaling; and (ii) the vicinity of the classical jamming critical point, as the cross-over temperature gets lowered when approaching it. This scenario arises from a direct study of the thermodynamics of the system in the quantum regime, where we show that, contrary to crystals, the Debye approximation does not hold.

Energy transport in a disordered spin chain with broken U(1) symmetry: Diffusion, subdiffusion, and many-body localization

Schulz M., Taylor S.R., Hooley C.A., We explore the physics of the disordered XYZ spin chain using two complementary numerical techniques: exact diagonalization (ED) on chains of up to 17 spins, and time-evolving block decimation (TEBD) on chains of up to 400 spins. Our principal findings are as follows. First, we verify that the clean XYZ spin chain shows ballistic energy transport for all parameter values that we investigated. Second, for weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Third, for disorder strengths above this critical value, energy transport becomes increasingly subdiffusive. Fourth, the many-body localization transition moves to significantly higher disorder strengths as the XY anisotropy is increased. We discuss these results, and their relation to our current physical picture of subdiffusion in the approach to many-body localization.

Return probability for the Anderson model on the random regular graph

Bera S., De Tomasi G., Khaymovich I.M., We study the return probability for the Anderson model on the random regular graph and give evidence of the existence of two distinct phases: a fully ergodic and nonergodic one. In the ergodic phase, the return probability decays polynomially with time with oscillations, being the attribute of the Wigner-Dyson-like behavior, while in the nonergodic phase the decay follows a stretched exponential decay. We give a phenomenological interpretation of the stretched exponential decay in terms of a classical random walker. Furthermore, comparing typical and mean values of the return probability, we show how to differentiate an ergodic phase from a nonergodic one. We benchmark this method first in two random matrix models, the power-law random banded matrices, and the Rosenzweig-Porter matrices, which host both phases. Second, we apply this method to the Anderson model on the random regular graph to give further evidence of the existence of the two phases.

Statistics of orthogonality catastrophe events in localised disordered lattices

Cosco F., Borrelli M., Laine E.M., Pascazio S., We address the phenomenon of statistical orthogonality catastrophe in insulating disordered systems. In more detail, we analyse the response of a system of non-interacting fermions to a local perturbation induced by an impurity. By inspecting the overlap between the pre- and post-quench many-body ground states we fully characterise the emergent statistics of orthogonality events as a function of both the impurity position and the coupling strength. We consider two well-known one-dimensional models, namely the Anderson and Aubry-André insulators, highlighting the arising differences. Particularly, in the Aubry-André model the highly correlated nature of the quasi-periodic potential produces unexpected features in how the orthogonality catastrophe occurs. We provide a quantitative explanation of such features via a simple, effective model. We further discuss the incommensurate ratio approximation and suggest a viable experimental verification in terms of charge transfer statistics and interferometric experiments using quantum probes.

Many-Body Localization Dynamics from Gauge Invariance

Brenes M., Dalmonte M., Heyl M., We show how lattice gauge theories can display many-body localization dynamics in the absence of disorder. Our starting point is the observation that, for some generic translationally invariant states, the Gauss law effectively induces a dynamics which can be described as a disorder average over gauge superselection sectors. We carry out extensive exact simulations on the real-time dynamics of a lattice Schwinger model, describing the coupling between U(1) gauge fields and staggered fermions. Our results show how memory effects and slow, double-logarithmic entanglement growth are present in a broad regime of parameters - in particular, for sufficiently large interactions. These findings are immediately relevant to cold atoms and trapped ion experiments realizing dynamical gauge fields and suggest a new and universal link between confinement and entanglement dynamics in the many-body localized phase of lattice models.

Ergodic and localized regions in quantum spin glasses on the Bethe lattice

Mossi G., By considering the quantum dynamics of a transverse-field Ising spin glass on the Bethe lattice, we find the existence of a many-body localized (MBL) region at small transverse field and low temperature. The region is located within the thermodynamic spin glass phase. Accordingly, we conjecture that quantum dynamics inside the glassy region is split into a small MBL region and a large delocalized (but not necessarily ergodic) region. This has implications for the analysis of the performance of quantum adiabatic algorithms. This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

Entanglement critical length at the many-body localization transition

Pietracaprina F., Parisi G., Mariano A., Pascazio S., We study the details of the distribution of the entanglement spectrum (eigenvalues of the reduced density matrix) of a disordered spin chain exhibiting a many-body localization (MBL) transition. In the thermalizing region we identify the evolution under increasing system size of the eigenvalue distribution function, whose thermodynamic limit is close to (but possibly different from) the Marchenko-Pastur distribution. From the analysis we extract a correlation length Ls(h) determining the minimum system size to enter the asymptotic region. We find that Ls(h) diverges at the MBL transition. We discuss the nature of the subleading corrections to the entanglement spectrum distribution and to the entanglement entropy.

Local integrals of motion in many-body localized systems

Imbrie J., Ros V., We review the current (as of Fall 2016) status of the studies on the emergent integrability in many-body localized models. We start by explaining how the phenomenology of fully many-body localized systems can be recovered if one assumes the existence of a complete set of (quasi)local operators which commute with the Hamiltonian (local integrals of motions, or LIOMs). We describe the evolution of this idea from the initial conjecture, to the perturbative constructions, to the mathematical proof given for a disordered spin chain. We discuss the proposed numerical algorithms for the construction of LIOMs and the status of the debate on the existence and nature of such operators in systems with a many-body mobility edge, and in dimensions larger than one. (Figure presented.).

Energy diffusion in the ergodic phase of a many body localizable spin chain

Varma V.K., Lerose A., Pietracaprina F., Goold J., The phenomenon of many-body localization in disordered quantum many-body systems occurs when all transport is suppressed despite the fact that the excitations of the system interact. In this work we report on the numerical simulation of autonomous quantum dynamics for disordered Heisenberg chains when the system is prepared with an initial inhomogeneity in the energy density profile. Using exact diagonalisation and a dynamical code based on Krylov subspaces we are able to simulate dynamics for up to L = 26 spins. We find, surprisingly, the breakdown of energy diffusion even before the many-body localization transition whilst the system is still in the ergodic phase. Moreover, in the ergodic phase we also find a large region in parameter space where the energy dynamics remains diffusive but where spin transport has been previously evidenced to occur only subdiffusively: this is found to be true for initial states composed of infinitely many hydrodynamic modes (square-wave energy profile) or just the single longest mode (sinusoidal profile). This suggestive finding points towards a peculiar ergodic phase where particles are transported slower than energy, reminiscent of the situation in amorphous solids and of the gapped phase of the anisotropic Heisenberg model.

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses

Baldwin C.L., Laumann C.R., Pal A., The two primary categories for eigenstate phases of matter at a finite temperature are many-body localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that, in the paradigmatic quantum p-spin models of the spin-glass theory, eigenstates violate the ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the nonergodic phase at a small transverse field from an ergodic phase at a large transverse field. The nonergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in the spin-glass theory. We show that the nonergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.

On the quantum spin glass transition on the Bethe lattice

Mossi G., Parolini T., Pilati S., We investigate the ground-state properties of a disorderd Ising model with uniform transverse field on the Bethe lattice, focusing on the quantum phase transition from a paramagnetic to a glassy phase that is induced by reducing the intensity of the transverse field. We use a combination of quantum Monte Carlo algorithms and exact diagonalization to compute Rényi entropies, quantum Fisher information, correlation functions and order parameter. We locate the transition by means of the peak of the Rényi entropy and we find agreement with the transition point estimated from the emergence of finite values of the Edwards-Anderson order parameter and from the peak of the correlation length. We interpret the results by means of a mean-field theory in which quantum fluctuations are treated as massive particles hopping on the interaction graph. We see that the particles are delocalized at the transition, a fact that points towards the existence of possibly another transition deep in the glassy phase where these particles localize, therefore leading to a many-body localized phase.

Signatures of many-body localization in the dynamics of two-site entanglement

Iemini F., Russomanno A., Rossini D., We are able to detect clear signatures of dephasing - a distinct trait of many-body localization (MBL) - via the dynamics of two-site entanglement, quantified through the concurrence. Using the protocol implemented by M. Schreiber et al. [Science 349, 842 (2015)SCIEAS0036-807510.1126/science.aaa7432], we show that in the MBL phase the average two-site entanglement decays in time as a power law, while in the Anderson localized phase it tends to a plateau. The power-law exponent is not universal and displays a clear dependence on the interaction strength. This behavior is also qualitatively different from the ergodic phase, where the two-site entanglement decays exponentially. All the results are obtained by means of time-dependent density matrix renormalization-group simulations and further corroborated by analytical calculations on an effective model. Two-site entanglement has been measured in cold atoms: our analysis paves the way for the first direct experimental test of many-body dephasing in the MBL phase.

Extreme lattices: Symmetries and decorrelation

Andreanov A., We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings in d-dimensional Euclidean space . Specifically, we ascertain statistics of the densities and kissing numbers as well as the numbers of distinct symmetries of the packings for dimensions 8 through 13 using the stochastic Voronoi algorithm. The extreme lattices in a fixed dimension of space d (d≥8) are dominated by typical lattices that have similar packing properties, such as packing densities and kissing numbers, while the best and the worst packers are in the long tails of the distribution of the extreme lattices. We also study the validity of the recently proposed decorrelation principle, which has important implications for sphere packings in general. The degree to which extreme-lattice packings decorrelate as well as how decorrelation is related to the packing density and symmetry of the lattices as the space dimension increases is also investigated. We find that the extreme lattices decorrelate with increasing dimension, while the least symmetric lattices decorrelate faster.

Holographic disorder driven superconductor-metal transition

Areán D., Pando Zayas L., Landea I., We implement the effects of disorder on a holographic superconductor by introducing a random chemical potential on the boundary. We demonstrate explicitly that increasing disorder leads to the formation of islands where the superconducting order is enhanced and subsequently to the transition to a metal. We study the behavior of the superfluid density and of the conductivity as a function of the strength of disorder. We find explanations for various marked features in the conductivities in terms of hydrodynamic quasinormal modes of the holographic superconductors. These identifications plus a particular disorder-dependent spectral weight shift in the conductivity point to a signature of the Higgs mode in the context of disordered holographic superconductors. We observe that the behavior of the order parameter close to the transition is not mean-field type as in the clean case; rather we find robust agreement with exp(-A|T-Tc|-ν), with ν=1.03±0.02 for this disorder-driven smeared transition.

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.