The Sherrington-Kirkpatrick model with random $\pm1$ couplings is programmed on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type graph. The performance of the optimizer compares and correlates to simulated annealing. When considering the effect of the static noise, which degrades the performance of the annealer, one can estimate an improvement on the comparative scaling of the two methods in favor of the D-Wave machine. The optimal choice of parameters of the embedding on the Chimera graph is shown to be associated to the emergence of the spin-glass critical temperature of the embedded problem.
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 analyze the properties of the energy landscape of {\it finite-size} fully connected $p$-spin-like models. In the thermodynamic limit the high temperature phase is described by the schematic Mode Coupling Theory of super-cooled liquids. In this limit the barriers between different basins are infinite below the critical dynamical temperature the ergodicity is broken on in infinite times. We show that {\it finite-size} fully connected $p$-spin-like models, where activated processes are possible, do exhibit properties typical of real super-cooled liquid when both are near the critical glass tra…
Using the results of large scale numerical simulations we study the probability distribution of the pseudo critical temperature for the three-dimensional Edwards-Anderson Ising spin glass and for the fully connected Sherrington-Kirkpatrick model. We find that the behavior of our data is nicely described by straightforward finite-size scaling relations.
We study the 3D Edwards-Anderson spin glasses, by analyzing spin-spin correlation functions in thermalized spin configurations at low T on large lattices. We consider individual disorder samples and analyze connected clusters of very correlated sites: we analyze how the volume and the surface of these clusters increases with the lattice size. We qualify the important excitations of the system by checking how large they are, and we define a correlation length by measuring their gyration radius. We find that the clusters have a very dense interface, compatible with being space filling.
We consider the problem of temperature chaos in mean-field spin-glass models defined on random lattices with finite connectivity. By means of an expansion in the order parameter we show that these models display a much stronger chaos effect than the fully connected Sherrington-Kirkpatrick model with the exception of the Bethe lattice with bimodal distribution of the couplings.
We study numerically the local low-energy excitations in the 3-d Edwards-Anderson model for spin glasses. Given the ground state, we determine the lowest-lying connected cluster of flipped spins with a fixed volume containing one given spin. These excitations are not compact, having a fractal dimension close to two, suggesting an analogy with lattice animals. Also, their energy does not grow with their size; the associated exponent is slightly negative whereas the one for compact clusters is positive. These findings call for a modification of the basic hypotheses underlying the droplet model.
In this paper we calculate the mean number of metastable states for spin glasses on so called random thin graphs with couplings taken from a symmetric binary distribution $\pm J$. Thin graphs are graphs where the local connectivity of each site is fixed to some value $c$. As in totally connected mean field models we find that the number of metastable states increases exponentially with the system size. Furthermore we find that the average number of metastable states decreases as $c$ in agreement with previous studies showing that finite connectivity corrections of order $1/c$ increase the numb…
We introduce a set of theoretical ideas that form the basis for an analytical framework capable of describing nonequilibrium dynamics in glassy systems. We test the resulting scenario by comparing its predictions with numerical simulations of short-range spin glasses. Local fluctuations and responses are shown to be connected by a generalized local out-of-equilibrium fluctuation-dissipation relation. Scaling relationships are uncovered for the slow evolution of heterogeneities at all time scales.
We present results from Monte Carlo simulations to test for ultrametricity and clustering properties in spin-glass models. By using a one-dimensional Ising spin glass with random power-law interactions where the universality class of the model can be tuned by changing the power-law exponent, we find signatures of ultrametric behavior both in the mean-field and non-mean-field universality classes for large linear system sizes. Furthermore, we confirm the existence of nontrivial connected components in phase space via a clustering analysis of configurations.
The stability to dislocations of the elastic phase, or ``Bragg glass'', of a randomly pinned elastic medium in two dimensions is studied using the minimum-cost-flow algorithm for a disordered fully-packed loop model. The elastic phase is found to be unstable to dislocations due to the quenched disorder. The energetics of dislocations are discussed within the framework of renormalization group predictions as well as in terms of a domain wall picture.
The properties of discrete two-dimensional spin glasses depend strongly on the way the zero-temperature limit is taken. We discuss this phenomenon in the context of the Migdal-Kadanoff renormalization group. We see, in particular, how these properties are connected with the presence of a cascade of fixed points in the renormalization group flow. Of particular interest are two unstable fixed points that correspond to two different spin-glass phases at zero temperature. We discuss how these phenomena are related with the presence of entropy fluctuations and temperature chaos, and universality in…
We investigate the low temperature phase of three-dimensional Edwards-Anderson model with Bernoulli random couplings. We show that at a fixed value $Q$ of the overlap the model fulfills the clustering property: the connected correlation functions between two local overlaps decay as a power whose exponent is independent of $Q$ for all $0\le |Q| < q_{EA}$. Our findings are in agreement with the RSB theory and show that the overlap is a good order parameter.
We apply for the first time a new one-loop topological expansion around the Bethe solution to the spin-glass model with field in the high connectivity limit, following the methodological scheme proposed in a recent work. The results are completely equivalent to the well known ones, found by standard field theoretical expansion around the fully connected model (Bray and Roberts 1980, and following works). However this method has the advantage that the starting point is the original Hamiltonian of the model, with no need to define an associated field theory, nor to know the initial values of the…
At the mean-field level, on fully connected lattices, several disordered spin models have been shown to belong to the universality class of "structural glasses", with a "random first-order transition" (RFOT) characterized by a discontinuous jump of the order parameter and no latent heat. However, their behavior in finite dimensions is often drastically different, displaying either no glassiness at all or a conventional spin-glass transition. We clarify the physical reasons for this phenomenon and stress the unusual fragility of the RFOT to short-range fluctuations, associated e.g. with the mer…
These lecture notes focus on the mean field theory of spin glasses, with particular emphasis on the presence of a very large number of metastable states in these systems. This phenomenon, and some of its physical consequences, will be discussed in details for fully-connected models and for models defined on random lattices. This will be done using the replica and cavity methods. These notes have been prepared for a course of the PhD program in Statistical Mechanics at SISSA, Trieste and at the University of Rome "Sapienza". Part of the material is reprinted from other lecture notes, and when t…
While in the fully-connected limit the solution of the spin-glass model is known, with the existence of a complex transition on a critical line in the temperature-external field phase diagram, in finite dimensions we don't know if a transition is present or not and, if present, if its nature is the same as the one in infinite dimensions. This work contains a review of the real-space Renormalization Group methods that have been applied to Spin Glasses, highlighting both their point of strength and weakness in characterizing the finite dimensional behaviour of the model.
The distribution of solutions of the Thouless-Anderson-Palmer equation is studied by extensive numerical experiments for fully connected 3-body interaction Ising spin glass models in a level of annealed calculation. A recent study predicted that when the equilibrium state of the system is characterized by one-step replica symmetry breaking, the distribution is described by a Becchi-Rouet-Stora-Tyutin (BRST) supersymmetric solution in the relatively low free energy region, whereas the BRST supersymmetry is broken for higher values of free energy (Crisanti et al., Phys. Rev. B 71 (2005) 094202).…
We prove that the Aizenman-Contucci relations, well known for fully connected spin glasses, hold in diluted spin glasses as well. We also prove more general constraints in the same spirit for multi-overlaps, systematically confirming and expanding previous results. The strategy we employ makes no use of self-averaging, and allows us to generate hierarchically all such relations within the framework of Random Multi-Overlap Structures. The basic idea is to study, for these structures, the consequences of the closely related concepts of stochastic stability, quasi-stationarity under random shifts…
We review the Random First Order Transition Theory of the glass transition, emphasizing the experimental tests of the theory. Many distinct phenomena are quantitatively predicted or explained by the theory, both above and below the glass transition temperature $T_g$. These include: the viscosity catastrophe and heat capacity jump at $T_g$, and their connection; the non-exponentiality of relaxations and their correlation with the fragility; dynamic heterogeneity in supercooled liquids owing to the mosaic structure; deviations from the Vogel-Fulcher law, connected with strings or fractral cooper…
We have investigated the interrelation between the spin glasses and the structural glasses. Spin glasses in this case are random magnets without reflection symmetry (e.g. $p$ - spin interaction spin glasses and Potts glasses) which contain quenched disorder, whereas the structural glasses are here exemplified by the homopolymeric globule, which can be viewed as a liquid of connected molecules on nano scales. It is argued that the homopolymeric globule problem can be mapped onto a disorder field theoretical model whose effective Hamiltonian resembles the corresponding one for the spin glass mod…
Spin-glass and chiral-glass orderings in three-dimensional Heisenberg spin glasses are studied both by equilibrium and off-equilibrium Monte Carlo simulations. Fully isotropic model is found to exhibit a finite-temperature chiral-glass transition without the conventional spin-glass order. Although chirality is an Ising-like quantity from symmetry, universality class of the chiral-glass transition appears to be different from that of the standard Ising spin glass. In the off-equilibrium simulation, while the spin autocorrelation exhibits only an interrupted aging, the chirality autocorrelation …
We introduce a diluted version of the one dimensional spin-glass model with interactions decaying in probability as an inverse power of the distance. In this model varying the power corresponds to change the dimension in short-range models. The spin-glass phase is studied in and out of the range of validity of the mean-field approximation in order to discriminate between different theories. Since each variable interacts only with a finite number of others the cost for simulating the model is drastically reduced with respect to the fully connected version and larger sizes can be studied. We fin…
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.…
A huge number of independent true ground-state configurations is calculated for two-, three- and four-dimensional +- J spin-glass models. Using the genetic cluster-exact approximation method, system sizes up to N=20^2,8^3,6^4 spins are treated. A ``ballistic-search'' algorithm is applied which allows even for large system sizes to identify clusters of ground states which are connected by chains of zero-energy flips of spins. The number of clusters n_C diverges with N going to infinity. For all dimensions considered here, an exponential increase of n_C appears to be more likely than a growth wi…
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…
We investigate the performance of the recently proposed stationary Fokker-Planck sampling method considering a combinatorial optimization problem from statistical physics. The algorithmic procedure relies upon the numerical solution of a linear second order differential equation that depends on a diffusion-like parameter D. We apply it to the problem of finding ground states of 2d Ising spin glasses for the +-J-Model. We consider square lattices with side length up to L=24 with two different types of boundary conditions and compare the results to those obtained by exact methods. A particular v…
We study the energy minima of the fully-connected $m$-components vector spin glass model at zero temperature in an external magnetic field for $m\ge 3$. The model has a zero temperature transition from a paramagnetic phase at high field to a spin glass phase at low field. We study the eigenvalues and eigenvectors of the Hessian in the minima of the Hamiltonian. The spectrum is gapless both in the paramagnetic and in the spin glass phase, with a pseudo-gap behaving as $λ^{m-1}$ in the paramagnetic phase and as $\sqrtλ$ in the spin glass phase. Despite the long-range nature of the model, the eig…
We study analytically and numerically the statics and the off-equilibrium dynamics of spin models over finitely connected random graphs. We identify a threshold value for the connectivity beyond which the loop structure of the graph becomes thermodynamically relevant. Glauber dynamics simulations show that this loop structure is responsible for the onset of dynamical features of a local character (dynamical heterogeneities and spontaneous time scale separation), consistently with previous (experimental and numerical) studies of glasses and spin glasses in their approach to the low temperature …
We study topology of frustration in $d$-dimensional Ising spin glasses with $d\ge 2$ with nearest-neighbor interactions. We prove the following: For any given spin configuration, the domain walls on the unfrustration network are all transverse to a frustrated loop on the unfrustration network, where a domain wall is defined to be a connected element of the collection of all the $(d-1)$-cells which are dual to the bonds having an unfavorable energy, and the unfrustration network is the collection of all the unfrustrated plaquettes. These domain walls are topologically nontrivial because they ar…
To identify emerging microscopic structures in low temperature spin glasses, we study self-sustained clusters (SSC) in spin models defined on sparse random graphs. A message-passing algorithm is developed to determine the probability of individual spins to belong to SSC. Results for specific instances, which compare the predicted SSC associations with the dynamical properties of spins obtained from numerical simulations, show that SSC association identifies individual slow-evolving spins. This insight gives rise to a powerful approach for predicting individual spin dynamics from a single snaps…
The spectral properties of disordered fully-connected graphs with a special type of the node-node interactions are investigated. The approximate analytical expression for the ensemble-averaged spectral density for the Hamiltonian defined on the fully-connected graph is derived and analysed both for the electronic and vibrational problems which can be related to the contact process and to the problem of stochastic diffusion, respectively. It is demonstrated how to evaluate the extreme eigenvalues and use them for finding the lower bound estimates of the critical parameter for the contact proces…
By quenched-randomly mixing local units of different spatial dimensionalities, we have studied Ising spin-glass systems on hierarchical lattices continuously in dimensionalities 1 =< d =< 3. The global phase diagram in temperature, antiferromagnetic bond concentration, and spatial dimensionality is calculated. We find that, as dimension is lowered, the spin-glass phase disappears to zero temperature at the lower-critical dimension d_c=2.431. Our system being a physically realizable system, this sets an upper limit to the lower-critical dimension in general for the Ising spin-glass phase. As di…
Aging Phase Diagram and Exact Asymptotic Energies of Mixed Spherical Spin Glasses
Universität zu Köln（科隆大学）；Università di Salerno（萨莱诺大学）；CNR-Nanotec, Rome Unit（意大利国家研究委员会纳米技术研究所罗马分部）；Sapienza Università di Roma（罗马第一大学）；INFN, Sezione di Roma（意大利国家核物理研究所罗马分部）
We investigate partition-function zeros of the many-body interacting spherical spin glass, the so-called $p$-spin spherical model, with respect to the complex temperature in the thermodynamic limit. We use the replica method and extend the procedure of the replica symmetry breaking ansatz to be applicable in the complex-parameter case. We derive the phase diagrams in the complex-temperature plane and calculate the density of zeros in each phase. Near the imaginary axis away from the origin, there is a replica symmetric phase having a large density. On the other hand, we observe no density in t…
Vector spin glasses are known to show two different kinds of phase transitions in presence of an external field: the so-called de Almeida-Thouless and Gabay-Toulouse lines. While the former has been studied to some extent on several topologies (fully connected, random graphs, finite-dimensional lattices, chains with long-range interactions), the latter has been studied only in fully connected models, which however are known to show some unphysical behaviors (e.g. the divergence of these critical lines in the zero-temperature limit). Here we compute analytically both these critical lines for XY…
We introduce a finite dimensional anharmonic soft spin glass in a field and show how it allows the construction a field theory at zero temperature and the corresponding loop expansion. The mean field level of the model coincides with a recently introduced fully connected model, the KHGPS model, and it has a spin glass transition in a field at zero temperature driven by the appearance of pseudogapped non-linear excitations. We analyze the zero temperature limit of the theory and the behavior of the bare masses and couplings on approaching the mean field zero temperature critical point. Focusing…
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…
Extremal Optimization (EO), a new local search heuristic, is used to approximate ground states of the mean-field spin glass model introduced by Sherrington and Kirkpatrick. The implementation extends the applicability of EO to systems with highly connected variables. Approximate ground states of sufficient accuracy and with statistical significance are obtained for systems with more than N=1000 variables using $\pm J$ bonds. The data reproduces the well-known Parisi solution for the average ground state energy of the model to about 0.01%, providing a high degree of confidence in the heuristic.…
We consider the statistical properties over disordered samples of the overlap distribution $P_{\cal J}(q)$ which plays the role of an order parameter in spin-glasses. We show that near zero temperature (i) the {\it typical} overlap distribution is exponentially small in the central region of $-1<q<1$: $ P^{typ}(q) = e^{\bar{\ln P_{\cal J}(q)}} \sim e^{- βN^θ ϕ(q)} $, where $θ$ is the droplet exponent defined here with respect to the total number $N$ of spins (in order to consider also fully connected models where the notion of length does not exist); (ii) the rescaled variable $v = - (\ln P_{\…
By neutron scattering, we have studied the spin correlations and spin fluctuations in frustrated Laves hydrides, where magnetic disorder sets in the topologically frustrated Mn lattice. Below the transition towards short range magnetic order, static spin clusters coexist with fluctuating and alsmost uncorrelated spins. The magnetic response shows a complexe lineshape, connected with the presence of the magnetic inhomogeneities. Its analysis shows the existence of two different processes, relaxation and local excitations, for the spin fluctuations below the transition. The paramagnetic fluctuat…
The thermodynamic and retrieval properties of fully connected Blume-Emery-Griffiths networks, storing ternary patterns, are studied using replica mean-field theory. Capacity-temperature phase diagrams are derived for several values of the pattern activity. It is found that the retrieval phase is the largest in comparison with other three-state neuron models. Furthermore, the meaning and stability of the so-called quadrupolar phase is discussed as a function of both the temperature and the pattern activity. Where appropriate, the results are compared with the diluted version of the model.
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 use of parameters measuring order-parameter fluctuations (OPF) has been encouraged by the recent results reported in \cite{RS} which show that two of these parameters, $G$ and $G_c$, take universal values in the $\lim_{T\to 0}$. In this paper we present a detailed study of parameters measuring OPF for two mean-field models with and without time-reversal symmetry which exhibit different patterns of replica symmetry breaking below the transition: the Sherrington-Kirkpatrick model with and without a field and the Ising p-spin glass (p=3). We give numerical results and analyze the consequences…
We analyse biased ensembles of trajectories for the random-field Ising model on a fully-connected lattice, which is described exactly by mean-field theory. By coupling the activity of the system to a dynamical biasing field, we find a range of dynamical phase transitions, including spontaneous symmetry breaking into ordered states. For weak bias, the phase behaviour is controlled by extrema of the free energy, which may be local minima or saddle points. For large bias, the system tends to states of extremal activity, which may differ strongly from free energy minima. We discuss connections of …
We discuss generation of series expansions for Ising spin-glasses with a symmetric $\pm J$ (i.e. bimodal) distribution on d-dimensional hypercubic lattices using linked-cluster methods. Simplifications for the bimodal distribution allow us to go to higher order than for a general distribution. We discuss two types of problem, one classical and one quantum. The classical problem is that of the Ising spin glass in a longitudinal magnetic field, $h$, for which we obtain high temperature series expansions in variables $\tanh(J/T)$ and $\tanh(h/T)$. The quantum problem is a $T=0$ study of the Ising…
The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied at zero temperature for arbitrary using a probabilistic approach. A recursive scheme is found determining the complete time evolution of the order parameters, taking into account all feedback correlations. It is based upon the evolution of the distribution of the local field, the structure of which is determined in detail. As an illustrative example, explicit analytic formula are given for the first few time steps of the dynamics. Furthermore, equilibrium fixed-point equations are derived and com…
We study order-parameter fluctuations (OPF) in disordered systems by considering the behavior of some recently introduced paramaters $G,G_c$ which have proven very useful to locate phase transitions. We prove that both parameters G (for disconnected overlap disorder averages) and $G_c$ (for connected disorder averages) take the respective universal values 1/3 and 13/31 in the $T\to 0$ limit for any {\em finite} volume provided the ground state is {\em unique} and there is no gap in the ground state local-field distributions, conditions which are met in generic spin-glass models with continuous…
We have studied numerically the states reached in a quench from various temperatures in the one-dimensional fully-connected Kotliar, Anderson and Stein Ising spin glass model. This is a model where there are long-range interactions between the spins which falls off a\ s a power $σ$ of their separation. We have made a detailed study in particular of the energies of the states reached in a quench from infinite temperature and their overlaps, including the spin glass susceptibility. In the regime where $σ\le 1/2$, where th\ e model is similar to the Sherrington-Kirkpatrick model, we find that the…
Using a probabilistic approach we study the parallel dynamics of fully connected Q-Ising neural networks for arbitrary Q. A Lyapunov function is shown to exist at zero temperature. A recursive scheme is set up to determine the time evolution of the order parameters through the evolution of the distribution of the local field. As an illustrative example, an explicit analysis is carried out for the first three time steps. For the case of the Q=3 model these theoretical results are compared with extensive numerical simulations. Finally, equilibrium fixed-point equations are derived and compared w…
In this paper we investigate the superspin glass behavior of a concentrated assembly of interacting maghemite nanoparticles and compare it to that of canonical atomic spin glass systems. ac versus temperature and frequency measurements show evidence of a superspin glass transition taking place at low temperature. In order to fully characterize the superspin glass phase, the aging behavior of both the thermo-remanent magnetization (TRM) and ac susceptibility has been investigated. It is shown that the scaling laws obeyed by superspin glasses and atomic spin glasses are essentially the same, aft…
In these notes we review first in some detail the concept of random overlap structure (ROSt) applied to fully connected and diluted spin glasses. We then sketch how to write down the general term of the expansion of the energy part from the Boltzmann ROSt (for the Sherrington-Kirkpatrick model) and the corresponding term from the RaMOSt, which is the diluted extension suitable for the Viana-Bray model. From the ROSt energy term, a set of polynomial identities (often known as Aizenman-Contucci or AC relations) is shown to hold rigorously at every order because of a recursive structure of these …
The quantum excitations in glasses have long presented a set of puzzles for condensed matter physicists. A common view is that they are largely disordered analogs of elementary excitations in crystals, supplemented by two level systems which are chemically local entities coming from disorder. A radical revision of this picture argues that the excitations in low temperature glasses are deeply connected to the energy landscape of the glass when it vitrifies: the excitations are not low excited states built on a single ground state but locally defined resonances, high in the energy spectrum of a …
