Methods for understanding classical disordered spin systems with interactions conforming to some idealized graphical structure are well developed. The equilibrium properties of the Sherrington-Kirkpatrick model, which has a densely connected structure, have become well understood. Many features generalize to sparse Erdos-Renyi graph structures above the percolation threshold, and to Bethe lattices when appropriate boundary conditions apply. In this paper we consider spin states subject to a combination of sparse strong interactions with weak dense interactions, which we term a composite model. The equilibrium properties are examined through the replica method, with exact analysis of the high temperature paramagnetic, spin glass and ferromagnetic phases by perturbative schemes. We present results of a replica symmetric variational approximations where perturbative approaches fail at lower temperature. Results demonstrate novel reentrant behaviors from spin glass to ferromagnetic phases as temperature is lowered, including transitions from replica symmetry broken to replica symmetric phases. The nature of high temperature transitions is found to be sensitive to the connectivity profile in the sparse sub-graph, with regular connectivity a discontinuous transition from the paramagnetic to ferromagnetic phases is apparent.
We study the interplay between hysteresis and equilibrium behavior in capillary condensation of fluids in mesoporous disordered materials via a mean-field density functional theory of a disordered lattice-gas model. The approach reproduces all major features observed experimentally. We show that the simple van der Waals picture of metastability fails due to the appearance of a complex free-energy landscape with a large number of metastable states. In particular, hysteresis can occur both with and without an underlying equilibrium transition, thermodynamic consistency is not satisfied along the…
The equilibrium and non--equilibrium disorder induced phase transitions are compared in the random-field Ising model (RFIM). We identify in the demagnetized state (DS) the correct non-equilibrium hysteretic counterpart of the T=0 ground state (GS), and present evidence of universality. Numerical simulations in d=3 indicate that exponents and scaling functions coincide, while the location of the critical point differs, as corroborated by exact results for the Bethe lattice. These results are of relevance for optimization, and for the generic question of universality in the presence of disorder.
It is still under debate whether the classical Einstein relation in disordered organic semiconductors is valid. We investigated Einstein relation in disordered organic semiconductors theoretically. The results show that, the classic Einstein relation deviate dramatically with disorder and electric field, even in the case of thermal equilibrium.
We discuss some aspects of the non-equilibrium relaxational dynamics which occur after a quench at a disordered critical point. In particular, we focus on the violation of the fluctuation dissipation theorem for local as well as non-local observables and on persistence properties.
The hopping transport of charge carries in 3-dimensional disordered semiconductors is analyzed in this work. It is assumed that the density of localized states is a decreasing function of energy and contribution of thermoactivated jumps into the transport is negligible. The possibility of equilibrium energetic distribution for this case is shown. This distribution is characterized by field--dependent effective temperature.
Rare regions, i.e., rare large spatial disorder fluctuations, can dramatically change the properties of a phase transition in a quenched disordered system. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. Stronger effects can be observed at zero-temperature quantum phase transitions, at nonequilibrium phase transitions, and in systems with correlated disorder. In some cases, rare regions can actually completely destroy the sharp phase transition by smearing. This top…
Mean-field models of glasses that present a random first order transition exhibit highly non-trivial fluctuations. Building on previous studies that focused on the critical scaling regime, we here obtain a fully quantitative framework for all equilibrium conditions. By means of the replica method we evaluate Gaussian fluctuations of the overlaps around the thermodynamic limit, decomposing them in thermal fluctuations inside each state and heterogeneous fluctuations between different states. We first test and compare our analytical results with numerical simulation results for the p-spin spheri…
By connecting realistic spin glass models at low temperature to the highly disordered model at zero temperature, we argue that ordinary Edwards-Anderson spin glasses below eight dimensions have at most a single pair of physically relevant pure states at nonzero low temperature. Less likely scenarios that evade this conclusion are also discussed.
We present a theoretical study of capillary condensation of fluids adsorbed in mesoporous disordered media. Combining mean-field density functional theory with a coarse-grained description in terms of a lattice-gas model allows us to investigate both the out-of-equilibrium (hysteresis) and the equilibrium behavior. We show that the main features of capillary condensation in disordered solids result from the appearance of a complex free-energy landscape with a large number of metastable states. We detail the numerical procedures for finding these states, and the presence or absence of transitio…
We consider a chain of one-dimensional dipole moments connected to two thermal baths with different temperatures. The system is in nonequilibrium steady state and heat flows through it. Assuming that fluctuation of the dipole moment is a small parameter, we develop an analytically solvable model for the problem. The effect of disorder is introduced by randomizing the positions of the dipole moments. We show that the disorder leads to Anderson-like transition from conducting to a thermal insulating state of the chain. It is shown that considered chain supports both ballistic and diffusive heat …
We study the dynamic and metastable properties of the fully connected Ising $p$-spin model with finite number of variables. We define trapping energies, trapping times and self correlation functions and we analyse their statistical properties in comparison to the predictions of trap models.
In this article we discuss several aspects of the stochastic dynamics of spin models. The paper has two independent parts. Firstly, we explore a few properties of the multi-point correlations and responses of generic systems evolving in equilibrium with a thermal bath. We propose a fluctuation principle that allows us to derive fluctuation-dissipation relations for many-time correlations and linear responses. We also speculate on how these features will be modified in systems evolving slowly out of equilibrium, as finite-dimensional or dilute spin-glasses. Secondly, we present a formalism that…
We study the energy absorption in real time of a disordered quantum spin chain subjected to coherent monochromatic periodic driving. We determine characteristic fingerprints of the well-known ergodic (Floquet-ETH for slow driving/weak disorder) and many-body localized (Floquet-MBL for fast driving/strong disorder) phases. In addition, we identify an intermediate regime, where the energy density of the system -- unlike the entanglement entropy a local and bounded observable -- grows logarithmically slowly over a very large time window.
We study the effects of the coupling to an Ohmic quantum reservoir on the static and dynamical properties of a family of disordered SU(2) spin models in a transverse magnetic field using a method of direct spin summation. The tendency to form a glassy phase increases with the strength of the coupling of the system to the environment. We study the influence of the environment on the features of the phase diagram of the various models as well as the stability of the possible phases.
We consider the dynamics of the disordered, one-dimensional, symmetric zero range process in which a particle from an occupied site $k$ hops to its nearest neighbour with a quenched rate $w(k)$. These rates are chosen randomly from the probability distribution $f(w) \sim (w-c)^{n}$, where $c$ is the lower cutoff. For $n > 0$, this model is known to exhibit a phase transition in the steady state from a low density phase with a finite number of particles at each site to a high density aggregate phase in which the site with the lowest hopping rate supports an infinite number of particles. In the …
We study the low temperature out of equilibrium Monte Carlo dynamics of the disordered Ising $p$-spin Model with $p=3$ and a small number of spin variables. We focus on sequences of configurations that are stable against single spin flips obtained by instantaneous gradient descent from persistent ones. We analyze the statistics of energy gaps, energy barriers and trapping times on sub-sequences such that the overlap between consecutive configurations does not overcome a threshold. We compare our results to the predictions of various trap models finding the best agreement with the step model wh…
We study the unitary relaxation dynamics of disordered spin chains following a sudden quench of the Hamiltonian. We give analytical arguments, corroborated by specific numerical examples, to show that the existence of a stationary state depends crucially on the spectral and localization properties of the final Hamiltonian, and not on the initial state. We test these ideas on integrable one-dimensional models of the Ising or XY class, but argue more generally on their validity for more complex (nonintegrable) models.
We show that an appropriate description of the non-equilibrium dynamics of disordered systems is obtained through a strong disorder renormalization procedure in {\it configuration space}, that we define for any master equation with transitions rates $W ({\cal C} \to {\cal C}')$ between configurations. The idea is to eliminate iteratively the configuration with the highest exit rate $W_{out} ({\cal C})= \sum_{{\cal C}'} W ({\cal C} \to {\cal C}')$ to obtain renormalized transition rates between the remaining configurations. The multiplicative structure of the new generated transition rates sugg…
We study a model in which particles interact with short-ranged attractive and long-ranged repulsive interactions, in an attempt to model the equilibrium cluster phase recently discovered in sterically stabilized colloidal systems in the presence of depletion interactions. At low packing fraction particles form stable equilibrium clusters which act as building blocks of a cluster fluid. We study the possibility that cluster fluids generate a low-density disordered arrested phase, a gel, via a glass transition driven by the repulsive interaction. In this model the gel formation is formally descr…
We study the dynamics of macroscopic observables such as the magnetization and the energy per degree of freedom in Ising spin models on random graphs of finite connectivity, with random bonds and/or heterogeneous degree distributions. To do so we generalize existing implementations of dynamical replica theory and cavity field techniques to systems with strongly disordered and locally tree-like interactions. We illustrate our results via application to the dynamics of e.g. $\pm J$ spin-glasses on random graphs and of the overlap in finite connectivity Sourlas codes. All results are tested again…
We study the interplay of disorder with pumping and decay in coupled qubit-cavity arrays, the Jaynes-Cummings-Hubbard model. We find that relatively weak disorder can wash out the bistability present in the clean pumped system, and that moreover, the combination of disorder in on-site energies and decay can lead to effective phase disorder. To explore these questions, we present a non- equilibrium generalisation of Stochastic-Mean-Field theory, providing a simple tool to address such questions. This technique is developed for rather general forms of light-matter coupling, driving, dissipation,…
We study spectral properties and the dynamics after a quench of one-dimensional spinless fermions with short-range interactions and long-range random hopping. We show that a sufficiently fast decay of the hopping term promotes localization effects at finite temperature, which prevents thermalization even if the classical motion is chaotic. For slower decays, we find that thermalization does occur. However, within this model, the latter regime falls in an unexpected universality class, namely, observables exhibit a power-law (as opposed to an exponential) approach to their thermal expectation v…
To describe the equilibrium properties of disordered systems and the possible emergence of various 'phases' at low temperature, we adopt here the 'broken ergodicity' point of view advocated in particular by Palmer [Adv. Phys. 31, 669 (1982)] : the aim is then to construct the valleys of configurations that become separated by diverging barriers and to study their relative weights, as well as their internal properties. To characterize the slow non-equilibrium dynamics of disordered systems, we have recently introduced in [C. Monthus and T. Garel, J. Phys. A 41, 255002 (2008) and arxiv:0804.1847…
We explore the quench dynamics of a two-dimensional, weakly interacting disordered Bose gas for various relative strengths of interactions and disorder. This allows us to identify two well distinct out-of-equilibrium regimes. When interactions are smaller than the disorder, the gas experiences multiple scattering and exhibits a short-range spatial coherence. At short time this coherence is only smoothly affected by interactions, via a diffusion process of the particles' energies. When interactions are larger than the disorder, scattering ceases and the gas behaves more and more like a fluid, u…
The theory of large deviations is already the natural language for the statistical physics of equilibrium and non-equilibrium. In the field of disordered systems, the analysis via large deviations is even more useful to describe within a unified perspective the typical events and the rare events that occur on various scales. In the present pedagogical introduction, we revisit various emblematic classical and quantum disordered systems in order to highlight the common underlying mechanisms from the point of view of large deviations.
In equilibrium, disorder conspires with topological defects to redefine the ordered states of matter in systems as diverse as crystals, superconductors and liquid crystals. Far from equilibrium, however, the consequences of quenched disorder on active condensed matter remain virtually uncharted. Here, we reveal a state of strongly disordered active matter with no counterparts in equilibrium: a dynamical vortex glass. Combining high-content microfluidic experiments and theory, we show how colloidal flocks collectively cruise through disorder without relaxing the topological singularities of the…
We analyze the statistics of daily price change of stock market in the framework of a statistical physics model for the collective fluctuation of stock portfolio. In this model the time series of price changes are coded into the sequences of up and down spins, and the Hamiltonian of the system is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. Through the analysis of Dow-Jones industrial portfolio consisting of 30 stock issues by this model, we find a non-equilibrium fluctuation mode on the point slightly below the boundary between ordered and disord…
We consider an XXZ spin-1/2 chain in the presence of several types of disorder that do not break the XY symmetry of the system. We calculate the complete asymptotic form of the spin-correlation functions at zero temperature at the transition between liquid and disordered phase that occurs for a special value of anisotropy in the limit of small disorder. Apart from a universal power law decay of correlations, we find additional logarithmic corrections due to marginally irrelevant operator of disorder.
The so-called 'Level 2.5' general result for the large deviations of the joint probability of the density and of the currents for Markov Jump processes is applied to the case of $N$ independent particles on a ring with random transition rates. We first focus on the Directed Trap model, where the contractions needed to obtain the large deviations properties of the density alone and of the current alone can be explicitly written in each disordered sample, and where the deformed Markov operator needed to evaluate the generating function of the current can be also explicitly analyzed via its highe…
We briefly review different approaches used recently to describe collective effects in the strong pinning model of disordered charge and spin density waves, in connection with the CRTBT very low temperature heat relaxation experiments.
We present a solution for the nonequilibrium dynamics of an interacting disordered system. The approach adapts the combination of the equilibrium dynamical mean field theory (DMFT) and the equilibrium coherent potential approximation (CPA) methods to the nonequilibrium many-body formalism, using the Kadanoff-Baym-Keldysh complex time contour, for the dynamics of interacting disordered systems away from equilibrium. We use our time domain solution to obtain the equilibrium density of states of the disordered interacting system described by the Anderson-Hubbard model, bypassing the necessity for…
Dephasing of spins is a major roadblock to scaling up the size of quantum computing systems. We explore the possibility of utilizing highly disordered environments which are in the Many-Body Localized phase to arrest this dephasing. We embedded 2 `special' spins in such a highly disordered environment of Heisenberg spins to act as the target qubits and use the long-time value of the spin-spin correlator $\langle \vecσ_i \cdot \vecσ_j\rangle$ as an order parameter to quantify the transition between the thermal and MBL phases of this system. It is seen that the dephasing between spins, as encode…
Out of equilibrium phases of matter exhibiting order in individual eigenstates, such as many-body localised spin glasses and discrete time crystals, can be characterised by inherently dynamical quantities such as spatiotemporal correlation functions. In this work, we introduce dynamical potentials which act as generating functions for such correlations and capture eigenstate phases and order. These potentials show formal similarities to their equilibrium counterparts, namely thermodynamic potentials. We provide three representative examples: a disordered, many-body localised XXZ chain showing …
We study the kinetic roughening of a driven domain wall between spin-up and spin-down domains for a model with non-conserved order parameter and quenched disorder. To understand the scaling behavior of this interface we construct an equation of motion and study it theoretically.
The influence of Rashba spin-orbit interaction on the spin dynamics of a topologically disordered hopping system is studied in this paper. This is a significant generalization of a previous investigation, where an ordered (polaronic) hopping system has been considered instead. It is found, that in the limit, where the Rashba length is large compared to the typical hopping length, the spin dynamics of a disordered system can still be described by the expressions derived for an ordered system, under the provision that one takes into account the frequency dependence of the diffusion constant and …
This paper describes the observation of non-equilibrium field effects at room temperature in four disordered insulating systems: granular Al, discontinuous Au, amorphous NbSi and amorphous InOx thin films. The use of wide enough gate voltage ranges and a cautious analysis of the data allow us to uncover memory dips, the advocated hallmark of the electron glass, in the four systems. These memory dips are found to relax slowly over days of measurements under gate voltage changes, reflecting the impossibility for the systems to reach an equilibrium state within experimentally accessible times. Ou…
We present a new field theoretic approach for finite dimensional site disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. We analyze this field theory using the variational replica formalism. For a variety of interactions we find a spin glass phase (with continuous replica symmetry breaking) and explicitly discuss a three dimensional system where this occurs. This approximation also suggests that any ferromagnetic transition occur within the spin glass phase.
A long-standing problem of the low-energy dynamics of a disordered XY spin chain is re-examined. The case of a rigid chain is studied where the quantum effects can be treated quasiclassically. It is shown that as the frequency decreases, the relevant excitations change from localized spin waves to two-level systems to soliton-antisoliton pairs. The linear-response correlation functions are calculated. The results apply to other periodic glassy systems such as pinned density waves, planar vortex lattices, stripes, and disordered Luttinger liquids.
We examine the validity of the recently proposed semi-Poisson level spacing distribution function P(S), which characterizes `critical quantum chaos', in 2D disordered systems with spin-orbit coupling. At the Anderson transition we show that the semi-Poisson P(S) can describe closely the critical distribution obtained with averaged boundary conditions, over Dirichlet in one direction with periodic in the other and Dirichlet in both directions. We also obtain a sub-Poisson linear number variance $Σ_{2}(E)\approx χ_{0}+ χE$, with asymptotic value $χ\approx0.07$. The obtained critical statistics, …
A model for magnetoresistance in positionally disordered organic materials is presented and solved using percolation theory. The model describes the effects of spin flips on hopping transport by considering the effect of spin dynamics on an effective density of hopping sites. Faster spin-flip transitions open up `spin-blocked' pathways to become viable conduction channels and hence produces magnetoresistance. The magnetoresistance can be found analytically in several regimes, including when the spin-flip time is slower than the hopping time. The ratio of hopping time to the hyperfine precessio…
We present a numerical study of the diffusion of energy at high temperature in strongly disordered chains of interacting classical spins evolving deterministically. We find that quenched randomness strongly suppresses transport, with the diffusion constant becoming reduced by several orders of magnitude upon the introduction of moderate disorder. We have also looked for but not found signs of a classical many-body localization transition at any nonzero strength of the spin-spin interactions.
We study the spin-wave dispersion of localized spins in a disordered double-exchange model using the perturbation theory with respect to the strength of the disorder potential. We calculate the dispersion upto the next-leading order, and extensively examine the case of one-dimension. We show that in that case, disorder yields anomalous gapped-like behavior at the Fermi wavenumber of the conduction electrons.
Here is the first part of the summary of my work on random Ising model using real-space renormalization group (RSRG), also known as a Migdal-Kadanoff one. This approximate renormalization scheme was applied to the analysis thermodynamic properties of the model, and of probabilistic properties of a pair correlator, which is a fluctuating object in disordered systems. PACS numbers: 02.50.-r, 05.20.-y, 05.70.Fh, 64.60.ae, 75.10.-b, 75.10.Hk, 87.10.+e Keywords: statistical physics, magnetism, spin glass, neural networks
Site dilution of spin-gapped antiferromagnets leads to localized free moments, which can order antiferromagnetically in two and higher dimensions. Here we show how a weak magnetic field drives this order-by-disorder state into a novel disordered-free-moment phase, characterized by the formation of local singlets between neighboring moments and by localized moments aligned antiparallel to the field. This disordered phase is characterized by the absence of a gap, as it is the case in a Bose glass. The associated field-driven quantum phase transition is consistent with the universality of a super…
We study short-range quantum spin systems with Gaussian disorder. We obtain quantum mechanical extensions of the Ghirlanda-Guerra identities. We discuss properties of overlap spin operators with these identities.
Low temperature properties of antiferromagnetic two-leg spin-1/2 ladders with bond randomness and site dilution (or doping with nonmagnetic impurities) are studied using the real-space renormalization-group technique. We find that for non zero doping concentrations the systems are driven into a phase dominated by large effective spins, i.e. the Large Spin phase. The susceptibility follows a universal Curie-like 1/T behavior at low temperature,regardless of the doping concentrations (as long as it is nonzero) and the strength of bond randomness. Very similar behavior has been found in ladders t…
A recent theory of magnetoresistance in positionally disordered organic semiconductors is extended to include exchange and dipolar couplings between polarons. Analytic results are discovered when the hyperfine, exchange, and dipolar interactions have little time to operate between hopping events. We find an angle-of-field dependence of the magnetoresistance that agrees with previous experiments and numerical simulations. In addition we report new magnetoresistive behavior that critically depends upon the amount of anisotropy in the dipolar interaction.
We define a replica field theory describing finite dimensional site disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. A general analysis of this field theory is made using the Gaussian variational or Hartree Fock method, and illustrated with several specific examples. Irrespective of the form of interaction between the spins this approximation predicts a spin glass phase. We discuss the replica symmetric phase at length, explicitly identifying the correlator that diverges at the spin glass transition. …
Real-Time Out-of-Equilibrium Quantum Dynamics in Disordered Materials
Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST（加泰罗尼亚纳米科学和技术研究所（ICN2，CSIC和BIST））；ICREA, Institució Catalana de Recerca i Estudis Avançats（加泰罗尼亚研究与高级研究学院（ICREA））
We discuss ergodicity breaking in frustrated disordered systems with no apparent broken symmetry of the Hamiltonian and present a way how to amend it in the low-temperature phase. We demonstrate this phenomenon on mean-field models of spin glasses. We use replicas of the spin variables to test thermodynamic homogeneity of ergodic equilibrium systems. We show that replica-symmetry breaking reflects ergodicity breaking and is used to restore an ergodic state. We then present explicit asymptotic solutions for the Ising, Potts and $p$-spin glasses. Each of the models shows a different low-temperat…
Aging phenomena have been studied in very different materials like polymers, supercooled liquids or disordered orientational crystals. We recall here the main features of aging in spin glasses, and use this example of magnetic systems as a guideline for the description of the others. A particular attention is put on the sensitivity of aging to the cooling rate and to temperature variations. This allows us to point out differences between temperature specific processes, yielding ``rejuvenation and memory effects'' as known in spin glasses, and domain growth processes, giving cumulative contribu…
We study the equilibrium and non-equilibrium properties of strongly interacting bosons on a lattice in presence of a random bounded disorder potential. Using a Gutzwiller projected variational technique, we study the equilibrium phase diagram of the disordered Bose Hubbard model and obtain the Mott insulator, Bose glass and superfluid phases. We also study the non equilibrium response of the system under a periodic temporal drive where, starting from the superfluid phase, the hopping parameter is ramped down linearly in time, and back to its initial value. We study the density of excitations c…
In disordered systems, the hopping conductivity regime is usually realized at low temperatures where spin-related phenomena differ strongly from the case of delocalized carriers. We develop the unified microscopic theory of current induced spin orientation, spin-galvanic and spin-Hall effects for the two-dimensional hopping regime. We show that the corresponding susceptibilities are proportional to each other and determined by the interplay between the drift and the diffusion spin currents. Estimations are made for realistic semiconductor heterostructures using the percolation theory. We show …
We study antiferromagnetic two-leg spin-1/2 ladders with strong bond randomness, using the real space renormalization group method. We find the low-temperature spin susceptibility of the system follows non-universal power laws, and the ground state spin-spin correlation is short-ranged. Our results suggest that there is no phase transition when the bond randomness increases from zero; for strong enough randomness the system is in a Griffith region with divergent spin susceptibility and short-range spin-spin correlation.
