Disconnectivity graphs are used to visualize the minima and the lowest energy barriers between the minima of complex systems. They give an easy and intuitive understanding of the underlying energy landscape and, as such, are excellent tools for understanding the complexity involved in finding low-lying or global minima of such systems. We have developed a classification scheme that categorizes highly-degenerate minima of spin glasses based on similarity and accessibility of the individual states. This classification allows us to condense the information pertained in different dales of the energy landscape to a single representation using color to distinguish its type and a bar chart to indicate the average size of the dales at their respective energy levels. We use this classification to visualize disconnectivity graphs of small representations of different tile-planted models of spin glasses. An analysis of the results shows that different models have distinctly different features in the total number of minima, the distribution of the minima with respect to the ground state, the barrier height and in the occurrence of the different types of minimum energy dales.
We propose damage spreading (DS) as a tool to investigate the topological features related to the ruggedness of the free energy landscape. We argue that DS measures the positiveness of the largest Lyapunov exponent associated to the basins of attraction visited by the system during its dynamical evolution. We discuss recent results obtained in the framework of mode-coupling theory and comment how to extend them to the study of realistic glasses. Preliminary results are presented for purely repulsive soft-sphere glasses.
We analyse the relationship between dynamics and configuration space structure of Ising spin glass systems. The exact knowledge of the structure of the low--energy landscape is used to study the relaxation of the system by random walk in the configuration space. The influence of the size of the valleys, clusters and energy barriers and the connectivity between them on the spin correlation function is shown.
Large numbers of ground states of two-dimensional Ising spin glasses with periodic boundary conditions in both directions are calculated for sizes up to 40^2. A combination of a genetic algorithm and Cluster-Exact Approximation is used. For each quenched realization of the bonds up to 40 independent ground states are obtained. For the infinite system a ground-state energy of e=-1.4015(3) is extrapolated. The ground-state landscape is investigated using a finite-size scaling analysis of the distribution of overlaps. The mean-field picture assuming a complex landscape describes the situation bet…
We use confocal microscopy to directly visualize the dynamics of aging colloidal glasses. We prepare a colloidal suspension at high density, a simple model system which shares many properties with other glasses, and initiate experiments by stirring the sample. We follow the motion of several thousand colloidal particles after the stirring and observe that their motion significantly slows as the sample ages. The aging is both spatially and temporally heterogeneous. Furthermore, while the characteristic relaxation time scale grows with the age of the sample, nontrivial particle motions continue …
A major challenge in the modeling of ionically conducting glasses is to understand how the large variety of possible chemical compositions and specific structural properties influence ionic transport quantities. Here we revisit and extend a theoretical approach for alkali borophosphate glasses, where changes of conductivity activation energies with the borate to phosphate mixing ratio are related to modifications of the ionic site energy landscape. The landscape modifications are caused by varying amounts of different units forming the glassy network, which lead to spatial redistributions of t…
connected spin-glass models with a discontinuous transition. In the thermodynamic limit the equilibrium properties in the high temperature phase are described by the schematic Mode Coupling Theory of super-cooled liquids. We show that {\it finite-size} fully connected spin-glass models do exhibit properties typical of Lennard-Jones systems when both are near the critical glass transition, where thermodynamics is ruled by energy minima distribution. Our study opens the way to consider activated processes in real glasses through finite-size corrections (i.e. calculations beyond the saddle point …
We numerically analyse the landscape governing the evolution of the vibrational dynamics of hard disk glasses as the density increases towards jamming. We find that the dynamics becomes slow, spatially correlated, and starts to display aging dynamics across an avoided Gardner transition, with a phenomenology that resembles three dimensional observations. We carefully analyse the behaviour of single glass samples, and find that the emergence of aging dynamics is controlled by the apparition of a complex organisation of the landscape that splits into a remarkable hierarchy of minima as jamming i…
The equation of state for a liquid in equilibrium, written in the potential energy landscape formalism, is generalized to describe out-of-equilibrium conditions. The hypothesis that during aging the system explores basins associated to equilibrium configurations is the key ingredient in the derivation. Theoretical predictions are successfully compared with data from molecular dynamics simulations of different aging processes, such as temperature and pressure jumps.
Recent work suggests that Heisenberg spin glasses may belong to the same universality class than structural glasses. Indeed, finding a lattice equivalent for supercooled liquids would probably allow easier numerical and analytical studies, that may help to answer long-standing questions on the glass transition. Supercooled liquids have many peculiar behaviors that should be found in the paramagnetic phase of Heisenberg spin glasses if the analogy between the two systems holds. It is with this motivation that we undertake a study of the paramagnetic phase of Heisenberg spin glasses. We shall em…
An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field. We calculated the distribution of the average value of a Gaussian random field in a finite domain. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of …
An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field, generated by randomly located and oriented dipoles and quadrupoles. Correlation properties of energetic disorder directly dictates the mobility dependence on the applied electric field. Electrostatic disorder is significantly modified in the vicinity of the electrode that affects injection properties. Correlated Gaussian field forms clusters. We suggest a simple method to estimate an asymptotics of the cluster distribution on size for deep clusters where a value of the field on eac…
We present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. Using parallel tempering (replica exchange) Monte Carlo we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. The correlation between static and dynamic observables for 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature is examined. Our results show that autocorrelation times are directl…
For a model glass-former we demonstrate via computer simulations how macroscopic dynamic quantities can be inferred from a PEL analysis. The essential step is to consider whole superstructures of many PEL minima, called metabasins, rather than single minima. We show that two types of metabasins exist: some allowing for quasi-free motion on the PEL (liquid-like), the others acting as traps (solid-like). The activated, multi-step escapes from the latter metabasins are found to dictate the slowing down of dynamics upon cooling over a much broader temperature range than is currently assumed.
Fossil amber offers the unique opportunity of investigating an amorphous material which has been exploring its energy landscape for more than 110 Myears of natural aging. By applying different x-ray scattering methods to amber before and after annealing the sample to erase its thermal history, we identify a link between the potential energy landscape and the structural and vibrational properties of glasses. We find that hyperaging induces a depletion of the vibrational density of states in the THz region, also ruling the sound dispersion and attenuation properties of the corresponding acoustic…
Spin glasses are notoriously difficult to study both analytically and numerically due to the presence of frustration and metastability. Their highly non-convex landscapes require collective updates to explore efficiently. Currently, most state-of-the-art algorithms rely on stochastic spin clusters to perform non-local updates, but such "cluster algorithms" lack general efficiency. Here, we introduce a non-equilibrium approach for simulating spin glasses based on classical dynamics with memory. By simulating various classes of 3d spin glasses (Edwards-Anderson, partially-frustrated, and fully-f…
Despite the extreme simplicity in their definition, spin glasses disclose a wide variety of non-trivial behaviors that are not yet fully understood. In this thesis we try to shed light on some of them, focusing on one hand on the search of phase transitions under perturbations of the Hamiltonian, and on the other hand on the zero-temperature properties and responses to external stimuli. After introducing spin glasses through a historiographical review, and useful concepts on phase transitions and numerics, the results of two massive Monte Carlo campaigns on three-dimensional systems are shown.…
Glassy behavior is one of the main open problems in condensed matter physics. In this thesis, we approach the problem by studying spin-glasses and colloids, using several complementary strategies. From the point of view of model building, we propose a new model, the hypercube model, which is mean-field but defined in a metric space. We show that, also in Mean Field, the spin-glass non-equilibrium dynamics is a domain-growth process very similar to its three-dimensional counterpart. We also consider spin-glasses in three spatial dimensions, focusing on temperature chaos. Thanks to the low-tempe…
Glasses are amorphous solids whose constituent particles are caged by their neighbors and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a "roughness transition" to fractal basins. This brings about isostaticity at jamming and marginality of glassy states near jamming. Critical exponents for the basin width, the weak force distribution, and the spat…
There is a growing belief that the mode coupling theory is the proper microscopic theory for the dynamics of the undercooled liquid above a critical temperature T_c. In addition, there is some evidence that the system leaves the saddlepoints of the energy landscape to settle in the valleys at this critical temperature. Finally, there is a microscopic theory for the entropy at the calorimetric glass transition T_g by Mezard and Parisi, which allows to calculate the Kauzmann temperature from the atomic pair potentials. The dynamics of the frozen glass phase is at present limited to phenomenologi…
We study the problem of glassy relaxations in the presence of an external field in the highly controlled context of a spin-glass simulation. We consider a small spin glass in three dimensions (specifically, a lattice of size L=8, small enough to be equilibrated through a Parallel Tempering simulations at low temperatures, deep in the spin glass phase). After equilibrating the sample, an external field is switched on, and the subsequent dynamics is studied. The field turns out to reduce the relaxation time, but huge statistical fluctuations are found when different samples are compared. After t…
We study the distribution of overlaps of glassy minima, taking proper care of residual symmetries of the system. Ensembles of locally stable, low lying glassy states are efficiently generated by rapid cooling from the liquid phase which has been equilibrated at a temperature $T_{run}$. Varying $T_{run}$, we observe a transition from a regime where a broad range of states are sampled to a regime where the system is almost always trapped in a metastable glassy state. We do not observe any structure in the distribution of overlaps of glassy minima, but find only very weak correlations, comparable…
We have studied the topology of the energy landscape of a spin-glass model and the effect of frustration on it by looking at the connectivity and disconnectivity graphs of the inherent structure. The connectivity network shows the adjacency of energy minima whereas the disconnectivity network tells us about the heights of the energy barriers. Both graphs are constructed by the exact enumeration of a two-dimensional square lattice of a frustrated spin glass with nearest-neighbor interactions up to the size of 27 spins. The enumeration of the energy-landscape minima as well as the analytical mea…
We consider damage spreading transitions in the framework of mode-coupling theory. This theory describes relaxation processes in glasses in the mean-field approximation which are known to be characterized by the presence of an exponentially large number of meta-stable states. For systems evolving under identical but arbitrarily correlated noises we demonstrate that there exists a critical temperature $T_0$ which separates two different dynamical regimes depending on whether damage spreads or not in the asymptotic long-time limit. This transition exists for generic noise correlations such that …
Calorimetric experiments on network glasses provide information on the ergodicity (landscape) temperature of supercooled liquids and can be compared with a recent theory developed by Hall and Wolynes [PRL90, 085505 (2003)]
The population annealing algorithm introduced by Hukushima and Iba is described. Population annealing combines simulated annealing and Boltzmann weighted differential reproduction within a population of replicas to sample equilibrium states. Population annealing gives direct access to the free energy. It is shown that unbiased measurements of observables can be obtained by weighted averages over many runs with weight factors related to the free energy estimate from the run. Population annealing is well suited to parallelization and may be a useful alternative to parallel tempering for systems …
The boson peak (BP) is a universal feature in the Raman and inelastic scattering spectra of both disordered and crystalline materials. The current paradigm presents the boson peak as the result of a Ioffe-Regel crossover between ballistic (phonon) and diffusive-type excitations, where the loss of coherence of phonons is described as a purely harmonic process due to structural disorder. This "harmonic disorder" paradigm for the BP has never been challenged or tested at the atomistic level. Here, through a set of atomistically-resolved characterizations of amorphous metallic alloys, we uncover a…
A ``ballistic-search'' algorithm is presented which allows the identification of clusters (or funnels) of ground states in Ising spin glasses even for moderate system sizes. The clusters are defined to be sets of states, which are connected in state-space by chains of zero-energy flips of spins. The technique can also be used to estimate the sizes of such clusters. The performance of the method is tested with respect to different system sizes and choices of parameters. As an application the ground-state funnel structure of two-dimensional +or- J spin glasses of systems up to size L=20 is analy…
The correlations of the free-energy landscape of mean-field spin glasses at different temperatures are investigated, concentrating on models with a first order freezing transition. Using a ``potential function'' we follow the metastable states of the model in temperature, and discuss the possibility of level crossing (which we do not find) and multifurcation (which we find). The dynamics at a given temperature starting from an equilibrium configuration at a different temperature is also discussed. In presence of multifurcation, we find that the equilibrium is never achieved, leading to aging b…
We use record dynamics (RD), a coarse-grained description of the ubiquitous relaxation phenomenology known as "aging", as a diagnostic tool to find universal features that distinguish between the energy landscapes of Ising spin models and the ferromagnet. According to RD, a non-equilibrium system after a quench relies on fluctuations that randomly generate a sequence of irreversible record-sized events (quakes or avalanches) that allow the system to escape ever-higher barriers of meta-stable states within a complex, hierarchical energy landscape. Once these record events allow the system to ov…
A method is presented, which allows to sample directly low-temperature configurations of glassy systems, like spin glasses. The basic idea is to generate ground states and low lying excited configurations using a heuristic algorithm. Then, with the help of microcanonical Monte Carlo simulations, more configurations are found, clusters of configurations are determined and entropies evaluated. Finally equilibrium configuration are randomly sampled with proper Gibbs-Boltzmann weights. The method is applied to three-dimensional Ising spin glasses with +- J interactions and temperatures T<=0.5. The…
We study mean field systems whose free energy landscape is dominated by marginally stable states. We review and develop various techniques to describe such states, elucidating their physical meaning and the interrelation between them. In particular, we give a physical interpretation of the two-group replica symmetry breaking scheme and confirm it by establishing the relation to the cavity method and to the counting of solutions of the Thouless-Anderson-Palmer equations. We show how these methods all incorporate the presence of a soft mode in the free energy landscape and interpret the occurrin…
In this paper we expose the results of our recent work on the dynamical TAP approach to mean field glassy systems. Our aim is to clarify the connection between free energy landscape and out of equilibrium dynamics in solvable models. Frequently qualitative explanations of glassy behaviour are based on the ``Free Energy Landscape Paradigm''. If its relevance for equilibrium properties is clear, the relationship between free energy landscape and out of equilibrium dynamics is not well understood yet. In this paper we clarify this relationship for the class of spin glass models which reproduce ph…
Spin glasses have competing interactions that lead to a rough energy landscape which is highly susceptible to small perturbations. These chaotic effects strongly affect numerical simulations and, as such, gaining a deeper understanding of chaos in spin glasses is of much importance. The use of thermal boundary conditions is an effective approach to study chaotic phenomena. Here, we generalize population annealing Monte Carlo, combined with thermal boundary conditions, to study bond chaos due to small perturbations in the spin-spin couplings of the three-dimensional Edwards-Anderson Ising spin …
Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity Ns of the inherent structures generically has a lognormal distribution. In addition, the large volume limit of ln<Ns>/<lnNs> differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for…
Rare Trajectories in a Prototypical Mean-field Disordered Model: Insights into Landscape and Instantons
Duke University（杜克大学）；Institute for Cross-disciplinary Physics and Complex Systems IFISC (CSIC-UIB)（交叉物理与复杂系统研究所（CSIC-UIB））；Bocconi University（博科尼大学）；Institute for Data Science and Analytics, Bocconi University（博科尼大学数据科学与分析研究所）；Institute of Complex Systems (ISC) - CNR, Rome unit（意大利国家研究委员会复杂系统研究所罗马分部）；Sapienza Università di Roma（罗马大学）
We show that a system with competing interactions on different length scales, as relevant for the formation of stripes in doped Mott insulators, undergoes a self-generated glass transition which is caused by the frustrated nature of the interactions and not related to the presence of quenched disorder. An exponentially large number of metastable configurations is found, leading to a slow, landscape dominated long time relaxation and a break up of the system into a disordered inhomogeneous state.
Metastable states in Ising spin-glass models are studied by finding iterative solutions of mean-field equations for the local magnetizations. Two different equations are studied: the TAP equations which are exact for the SK model, and the simpler `naive-mean-field' (NMF) equations. The free-energy landscapes that emerge are very different. For the TAP equations, the numerical studies confirm the analytical results of Aspelmeier et al., which predict that TAP states consist of close pairs of minima and index-one (one unstable direction) saddle points, while for the NMF equations saddle points w…
We study the linear excitations around typical energy minima of a mean-field disordered model with continuous degrees of freedom undergoing a Random First Order Transition (RFOT). Contrary to naive expectations, the spectra of linear excitations are ungapped and we find the presence of a pseudogap corresponding to localized excitations with arbitrary low excitation energy. Moving to deeper minima in the landscape, the excitations appear increasingly localized while their abundance decreases. Beside typical minima, there also exist rare ultra-stable minima, with an energy gap and no localised e…
We carefully investigate the two fundamental assumptions in the Stillinger-Weber analysis of the inherent structures (IS's) in the energy landscape and come to conclude that they cannot be validated. This explains some of the conflicting results between their conclusions and some recent rigorous and exact results. Our analysis shows that basin free energies, and not IS's, are useful for understanding glasses.
Experimental evidence for the Gardner transition, theoretically predicted to arise deep in the glassy state of matter, is scarce. At this transition, the energy landscape sensed by the particles forming the glass is expected to become more complex. In the present work, we report the dielectric response of two typical glass formers with well-pronounced Johari-Goldstein beta relaxation following this response down to unprecedented low temperatures, far below the glass transition. As the Johari-Goldstein process is believed to arise from the local structure of the energy landscape, its investigat…
Population annealing is an efficient sequential Monte Carlo algorithm for simulating equilibrium states of systems with rough free energy landscapes. The theory of population annealing is presented, and systematic and statistical errors are discussed. The behavior of the algorithm is studied in the context of large-scale simulations of the three-dimensional Ising spin glass and the performance of the algorithm is compared to parallel tempering. It is found that the two algorithms are similar in efficiency though with different strengths and weaknesses.
In this paper, we review several important features of the out-of-equilibrium dynamics of spin glasses. Starting with the simplest experiments, we discuss the scaling laws used to describe the isothermal aging observed in spin glasses after a quench down to the low temperature phase. We report in particular new results on the sub-aging behaviour of spin glasses. We then discuss the rejuvenation and memory effects observed when a spin glass is submitted to temperature variations during aging, from the point of view of both energy landscape pictures and of real space pictures. We highlight the f…
We show that the competition between interactions on different length scales, as relevant for the formation of stripes in doped Mott insulators, can cause a glass transition in a system with no explicitly quenched disorder. We analytically determine a universal criterion for the emergence of an exponentially large number of metastable configurations that leads to a finite configurational entropy and a landscape dominated viscous flow. We demonstrate that glassines is unambiguously tied to a new length scale which characterizes the typical length over which defects and imperfections in the stri…
The free energy landscape of mean field marginal glasses is ultrametric. We demonstrate that this feature remains in finite three dimensional systems by finding sets of minima which are nearby in configuration space. By calculating the distance between these nearby minima, we produce a small region of the distance metric. This metric exhibits a clear hierarchical structure and shows the signature of an ultrametric space. That such a hierarchy exists for the jamming energy landscape provides direct evidence for the existence of a marginal phase along the zero temperature jamming line.
The low-temperature quasi-universal behavior of amorphous solids has been attributed to the existence of spatially-localized tunneling defects found in the low-energy regions of the potential energy landscape. Computational models of glasses can be studied to elucidate the microscopic nature of these defects. Recent simulation work has demonstrated the means of generating stable glassy configurations for models that mimic metallic glasses using the swap Monte Carlo algorithm. Building on these studies, we present an extensive exploration of the glassy metabasins of the potential energy landsca…
The multidimensional topography of the collective potential energy function of a so-called strong glass former (silica) is analyzed by means of classical molecular dynamics calculations. Features qualitatively similar to those of fragile glasses are recovered at high temperatures : in particular an intrinsic characteristic temperature $T_c\simeq 3500$K is evidenced above which the system starts to investigate non-harmonic potential energy basins. It is shown that the anharmonicities are essentially characterized by a roughness appearing in the potential energy valleys explored by the system fo…
It is a persistent problem in condensed matter physics that glasses exhibit vibrational and thermal properties that are markedly different from those of crystals. While recent works have advanced our understanding of vibrational excitations in glasses at the harmonic approximation limit, efforts in understanding finite-temperature anharmonic processes have been limited. It is well known that phonons in crystals couple through phonon-phonon interactions, an extremely efficient mechanism for anharmonic decay that is also important in glasses. Here, however, we show that an additional anharmonic …
Cooperative events requiring anomalously large fluctuations are a defining characteristic for the onset of glassy relaxation across many materials. The importance of such intermittent events has been noted in systems as diverse as superconductors, metallic glasses, gels, colloids, and granular piles. Here, we show that prohibiting the attainment of new record-high energy fluctuations -- by explicitly imposing a ``lid'' on the fluctuation spectrum -- impedes further relaxation in the glassy phase. This lid allows us to directly measure the impact of record events on the evolving system in exten…
Population annealing is a Monte Carlo algorithm that marries features from simulated annealing and parallel tempering Monte Carlo. As such, it is ideal to overcome large energy barriers in the free-energy landscape while minimizing a Hamiltonian. Thus, population annealing Monte Carlo can be used as a heuristic to solve combinatorial optimization problems. We illustrate the capabilities of population annealing Monte Carlo by computing ground states of the three-dimensional Ising spin glass with Gaussian disorder, whilst comparing to simulated annealing and parallel tempering Monte Carlo. Our r…
Dynamics near the surface of glasses is generally much faster than in the bulk. Neglecting static perturbations of structure at the surface, we use random first order transition theory to show the free energy barrier for activated motion near a free surface should be half that of the bulk at the same temperature. The increased mobility allows the surface layers to descend much further on the energy landscape than the bulk ordinarily does. The simplified RFOT calculation however predicts a limiting value for the configurational entropy a vapor deposited glass may reach as a function of depositi…
As shown by early studies on mean-field models of the glass transition, the geometrical features of the energy landscape provide fundamental information on the dynamical transition at the Mode-Coupling temperature $T_d$. We show that active particles can serve as a useful tool for gaining insight into the topological crossover in model glass-formers. In such systems the landmark of the minima-to-saddle transition in the potential energy landscape, taking place in the proximity of $T_d$, is the critical slowing down of dynamics. Nevertheless, the critical slowing down is a bottleneck for numeri…
In this paper, we review the general features of the out-of-equilibrium dynamics of spin glasses. We use this example as a guideline for a brief description of glassy dynamics in other disordered systems like structural and polymer glasses, colloids, gels etc. Starting with the simplest experiments, we discuss the scaling laws used to describe the isothermal aging observed in spin glasses after a quench down to the low temperature phase (these scaling laws are the same as established for polymer glasses). We then discuss the rejuvenation and memory effects observed when a spin glass is submitt…
We numerically investigate the aging dynamics of a monatomic Lennard-Jones glass, focusing on the topology of the potential energy landscape which, to this aim, has been partitioned in basins of attraction of stationary points (saddles and minima). The analysis of the stationary points visited during the aging dynamics shows the existence of two distinct regimes: i) at short times, t<t_c, the system visits basins of saddles whose energies and orders decrease with t; ii) at long times, t>t_c, the system mainly lies in basins pertaining to minima of slowly decreasing energy. The dynamics for t>t…
