We discuss interfaces in spin glasses. We present new theoretical results and a numerical method to characterize overlap interfaces and the stability of the spin-glass phase in extended disordered systems. We use this definition to characterize the low temperature phase of hierarchical spin-glass models. We use the Replica Symmetry Breaking theory to evaluate the cost for an overlap interface, which in these models is particularly simple. A comparison of our results from numerical simulations with the theoretical predictions shows good agreement.
We study in d=3 dimensions the short range Ising spin glass with Jij=+/-1 couplings at T=0. We show that the overlap distribution is non-trivial in the limit of large system size.
We investigate the distributions of the link overlap, P(Q), in 3-dimensional Ising spin glasses. We use clustering methodology to identify a set of pairs of states from different Gibbs states, and calculate its contribution to P(Q). We find that the distribution over this set does not become trivial as the system size increases.
We compute numerically small window overlaps in the three dimensional Edwards Anderson spin glass. We show that they behave in the way implied by the Replica Symmetry Breaking Ansatz, that they do not qualitatively differ from the full volume overlap and do not tend to a trivial function when increasing the lattice volume. On the contrary we show they are affected by small finite volume effects, and are interesting tools for the study of the features of the spin glass phase.
We use a sample-dependent analysis, based on medians and quantiles, to analyze the behavior of the overlap probability distribution of the Sherrington-Kirkpatrick and 3D Edwards-Anderson models of Ising spin glasses. We find that this approach is an effective tool to distinguish between RSB-like and droplet-like behavior of the spin-glass phase. Our results are in agreement with a RSB-like behavior for the 3D Edwards-Anderson model.
We introduce the concept of Random Multi-Overlap Structures in diluted spin glasses, following the ideas of Aizenman, Sims and Starr for non-diluted models. As a result, we prove the generalized bound and variational principle for the free energy per site, in analogy with the results of Aizenman, Sims and Starr for non-diluted models. We also prove some stability properties of the optimal RMOSt, analogous to the one found by Guerra for non-diluted models.
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 present a complete characterization of the fluctuations and correlations of the squared overlap in the Edwards-Anderson Spin-Glass model in zero field. The analysis reveals that the energy-energy correlations (and thus the specific heat) have a different critical behavior than the fluctuations of the link overlap in spite of the fact that the average energy and average link overlap have the same critical properties. More precisely the link-overlap fluctuations are larger than the specific heat according to a computation at first order in the $6-ε$ expansion. An unexpected outcome is that th…
The lower-critical dimension for the existence of the Ising spin-glass phase is calculated, numerically exactly, as $d_L = 2.520$ for a family of hierarchical lattices, from an essentially exact (correlation coefficent $R^2 = 0.999999$) near-linear fit to 23 different diminishing fractional dimensions. To obtain this result, the phase transition temperature between the disordered and spin-glass phases, the corresponding critical exponent $y_T$, and the runaway exponent $y_R$ of the spin-glass phase are calculated for consecutive hierarchical lattices as dimension is lowered.
Excitations of three-dimensional spin glasses are computed numerically. We find that one can flip a finite fraction of an LxLxL lattice with an O(1) energy cost, confirming the mean field picture of a non-trivial spin overlap distribution P(q). These low energy excitations are not domain-wall-like, rather they are topologically non-trivial and they reach out to the boundaries of the lattice. Their surface to volume ratios decrease as L increases and may asymptotically go to zero. If so, link and window overlaps between the ground state and these excited states become ``trivial''.
Unbiased samples of ground states were generated for the short-range Ising spin glass with Jij=+/-1, in three dimensions. Clustering the ground states revealed their hierarchical structure, which is explained by correlated spin domains, serving as cores for macroscopic zero energy "excitations".
We study the link overlap between two replicas of an Ising spin glass in three dimensions using the Migdal-Kadanoff approximation and scaling arguments based on the droplet picture. For moderate system sizes, the distribution of the link overlap shows the asymmetric shape and large sample-to-sample variations found in Monte Carlo simulations and usually attributed to replica symmetry breaking. However, the scaling of the width of the distribution, and the link overlap in the presence of a weak coupling between the two replicas are in agreement with the droplet picture. We also discuss why it i…
Nearest-neighbor-interaction Ising spin glasses are studied on three different hierarchical lattices, all of them belonging to the Wheatstone-Bridge family. It is shown that the spin-glass lower critical dimension in these lattices should be greater than 2.32. Finite-temperature spin-glass phases are found for a lattice of fractal dimension $D \approx 3.58$ (whose unit cell is obtained from a simple construction of a part of the cubic lattice), as well as for a lattice of fractal dimension close to five.
For the Edwards-Anderson Ising spin-glass model in three and four dimensions (3d and 4d) we have performed high statistics Monte Carlo calculations of those free-energy barriers $F^q_B$ which are visible in the probability density $P_J(q)$ of the Parisi overlap parameter $q$. The calculations rely on the recently introduced multi-overlap algorithm. In both dimensions, within the limits of lattice sizes investigated, these barriers are found to be non-self-averaging and the same is true for the autocorrelation times of our algorithm. Further, we present evidence that barriers hidden in $q$ domi…
We compute and analyze couples of ground states of 3D spin glass systems with the same quenched noise but periodic and anti-periodic boundary conditions for different lattice sizes. We discuss the possible different behaviors of the system, we analyze the average link overlap, the probability distribution of window overlaps (among ground states computed with different boundary conditions) and the spatial overlap and link overlap correlation functions. We establish that the picture based on Replica Symmetry Breaking correctly describes the behavior of 3D Spin Glasses.
We present a combination of heuristic and rigorous arguments indicating that both the pure state structure and the overlap structure of realistic spin glasses should be relatively simple: in a large finite volume with coupling-independent boundary conditions, such as periodic, at most a pair of flip-related (or the appropriate number of symmetry-related in the non-Ising case) states appear, and the Parisi overlap distribution correspondingly exhibits at most a pair of delta-functions at plus/minus the self-overlap. This rules out the nonstandard SK picture introduced by us earlier, and when co…
We investigate the thermodynamic phase transition taking place in the Blume-Capel model in presence of quenched disorder in three dimensions (3D). In particular, performing Exchange Montecarlo simulations, we study the behavior of the order parameters accross the first order phase transition and its related coexistence region. This transition is an Inverse Freezing.
The overlap, or similarity, between liquid configurations is at the core of the mean-field description of the glass transition, and remains a useful concept when studying three-dimensional glass-forming liquids. In liquids, however, the overlap involves a tolerance, typically of a fraction $a/σ$ of the inter-particle distance, associated with how precisely similar two configurations must be for belonging to the same physically relevant "state". Here, we systematically investigate the dependence of the overlap fluctuations and of the resulting phase diagram when the tolerance is varied over a l…
We present results of recent high-statistics Monte Carlo simulations of the Edwards-Anderson Ising spin-glass model in three and four dimensions. The study is based on a non-Boltzmann sampling technique, the multi-overlap algorithm which is specifically tailored for sampling rare-event states. We thus concentrate on those properties which are difficult to obtain with standard canonical Boltzmann sampling such as the free-energy barriers F^q_B in the probability density P_J(q) of the Parisi overlap parameter q and the behaviour of the tails of the disorder averaged density P(q) = [P_J(q)]_av.
Recently, it has been conjectured that the statistics of extremes is of relevance for a large class of correlated system. For certain probability densities this predicts the characteristic large $x$ fall-off behavior $f(x)\sim\exp (-a e^x)$, $a>0$. Using a multicanonical Monte Carlo technique, we have calculated the Parisi overlap distribution $P(q)$ for the three-dimensional Edward-Anderson Ising spin glass at and below the critical temperature, even where $P(q)$ is exponentially small. We find that a probability distribution related to extreme order statistics gives an excellent description …
We introduce a diagrammatic formulation for a cavity field expansion around the critical temperature. This approach allows us to obtain a theory for the overlap's fluctuations and, in particular, the linear part of the Ghirlanda-Guerra relationships (GG) (often called Aizenman-Contucci polynomials (AC)) in a very simple way. We show moreover how these constraints are "superimposed" by the symmetry of the model with respect to the restriction required by thermodynamic stability. Within this framework it is possible to expand the free energy in terms of these irreducible overlaps fluctuations an…
We study the sample-to-sample fluctuations of the overlap probability densities from large-scale equilibrium simulations of the three-dimensional Edwards-Anderson spin glass below the critical temperature. Ultrametricity, Stochastic Stability and Overlap Equivalence impose constraints on the moments of the overlap probability densities that can be tested against numerical data. We found small deviations from the Ghirlanda-Guerra predictions, which get smaller as system size increases. We also focus on the shape of the overlap distribution, comparing the numerical data to a mean-field-like pred…
Temperature chaos is an extreme sensitivity of the equilibrium state to a change of temperature. It arises in several disordered systems that are described by the so called scaling theory of spin glasses, while it seems to be absent in mean field models. We consider a model spin glass on a tree and show that although it has mean field behavior with replica symmetry breaking, it manifestly has ``strong'' temperature chaos. We also show why chaos appears only very slowly with system size.
We report large-scale simulations of the three-dimensional Edwards-Anderson Ising spin glass system using the recently introduced multi-overlap Monte Carlo algorithm. In this approach the temperature is fixed and two replica are coupled through a weight factor such that a broad distribution of the Parisi overlap parameter $q$ is achieved. Canonical expectation values for the entire $q$-range (multi-overlap) follow by reweighting. We present an analysis of the performance of the algorithm and in particular discuss results on spin glass free-energy barriers which are hard to obtain with conventi…
In this paper we obtain two results for the Sherrington-Kirkpatrick (SK) model, and we show that they both emerge from a single approach. First, we prove that the average of the overlap takes positive values when it is non zero. More specificly, the average of the overlap, which is naively expected to take values in the whole interval $[-1,+1]$, becomes positive if we ``first'' apply an external field, so to destroy the gauge invariance of the model, and ``then'' remove it in the thermodynamic limit. This phenomenon emerges at the critical point. This first result is weaker that the one obtain…
Cavity point-to-set correlations are real-space tools to detect the roughening of the free-energy landscape that accompanies the dynamical slowdown of glass-forming liquids. Measuring these correlations in model glass formers remains, however, a major computational challenge. Here, we develop a general parallel-tempering method that provides orders-of-magnitude improvement for sampling and equilibrating configurations within cavities. We apply this improved scheme to the canonical Kob-Andersen binary Lennard-Jones model for temperatures down to the mode-coupling theory crossover. Most signific…
The spin-glass $q$-state Potts model on $d$-dimensional diamond hierarchical lattices is investigated by an exact real space renormalization group scheme. Above a critical dimension $d_l(q)$ for $q > 2$, the coupling constants probability distribution flows to a low-temperature \emph{strange attractor} or to the high-temperature paramagnetic fixed point, according to the temperature is below or above the critical temperature $T_c(q,d)$. The strange attractor was investigated considering four initial different distributions for $q = 3$ and $d = 5$ presenting strong robustness in shape and tempe…
The overlap length of a three-dimensional Ising spin glass on a cubic lattice with Gaussian interactions has been estimated numerically by transfer matrix methods and within a Migdal-Kadanoff renormalization group scheme. We find that the overlap length is large, explaining why it has been difficult to observe spin glass chaos in numerical simulations and experiment.
We analyse numerically thermal fluctuations of the static overlap between equilibrium configurations in a glass-forming liquid approaching the glass transition. We find that the emergence of slow dynamics near the onset temperature correlates with the development of non-Gaussian probability distributions of overlap fluctuations, measured using both annealed and quenched definitions. Below a critical temperature, a thermodynamic field conjugate to the overlap induces a first-order phase transition, whose existence we numerically demonstrate in the annealed case. These results establish that the…
We present numerical simulations of the 4D Edwards Anderson Ising spin glass with binary couplings. Our results, in the midst of strong finite size effects, suggest the existence of a spin glass phase transition. We present a preliminar determination of critical exponents. We discuss spin glass susceptibilities, cumulants of the overlap and energy overlap probability distributions, finite size effects, and the behavior of the disorder dependent and of the integrated probability distribution.
We argue that the lower bound to the barrier energy to flip an up/down spin domain embedded in a down/up spin environment for Ising spin glass is independent of the size of the system. The argument shows the existence of at least one dynamical way through which it is possible to bypass local maxima in the phase space. For an arbitrary case where one flips any cluster of spin of size $l$, we have numerically calculated a lower bound to the exponent $ψ$ characterizing the barrier one has to overcome. In this case $ψ$ corresponding to the lower bound calculated on hierarchical lattice comes out t…
We comment on recent numerical experiments by G.Hed and E.Domany [cond-mat/0608535v2] on the quenched equilibrium state of the Edwards-Anderson spin glass model. The rigorous proof of overlap identities related to replica equivalence shows that the observed violations of those identities on finite size systems must vanish in the thermodynamic limit. See also the successive version cond-mat/0608535v4
We propose and numerically implement a local probe of the static self-induced heterogeneity characterizing glass-forming liquids. The method relies on the equilibrium statistics of the overlap between pairs of configurations measured in mesoscopic cavities with unconstrained boundaries. By systematically changing the location of the probed cavity, we directly detect spatial variations of the overlap fluctuations. We provide a detailed analysis of the statistics of a local estimate of the configurational entropy and we infer an estimate of the surface tension between amorphous states, ingredien…
The locations of multicritical points on many hierarchical lattices are numerically investigated by the renormalization group analysis. The results are compared with an analytical conjecture derived by using the duality, the gauge symmetry and the replica method. We find that the conjecture does not give the exact answer but leads to locations slightly away from the numerically reliable data. We propose an improved conjecture to give more precise predictions of the multicritical points than the conventional one. This improvement is inspired by a new point of view coming from renormalization gr…
We investigate the question of whether chaotic size dependence occurs on hierarichical lattices and demonstrate that it is not present in these systems. Our results show that the metastate for spin glasses on hierarchical lattices is simple.
We show that the dynamical behavior of the 3D Ising spin glass with Gaussian couplings is not compatible with a droplet dynamics. We show that this is implied from the data of reference cond-mat/9904143, that, when analyzed in an accurate way, give multiple evidences of this fact. Our study is based on the analysis of the overlap-overlap correlation function, at different values of the separation r and of the time t.
We discuss the issue of temperature chaos in the Sherrington--Kirkpatrick spin glass mean field model. We numerically compute probability distributions of the overlap among (equilibrium) configurations at two different values of the temperature, both in the spin glass phase. The situation on our medium size systems is clearly non-chaotic, but a weak form of chaos could be emerging on very large lattices.
In the paper [Angelini M C, Parisi G, and Ricci-Tersenghi F, Ensemble renormalization group for disordered systems, Phys. Rev. B 87 134201 (2013)] we introduced a real-space renormalization group called Ensemble Renormalization Group (ERG) and we applied it to the Edwards-Anderson model, obtaining estimates for the critical exponents in good agreement with those from Monte Carlo simulations. Recently the paper [Castellana M, Real-space renormalization-group methods for hierarchical spin glasses, J. Phys. A: Math. Theor. 52 445002 (2019)] re-examined the ERG method from a different perspective,…
The three-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied via large-scale Monte Carlo simulations at low temperatures, deep within the spin-glass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low…
A nearest-neighbor-interaction Ising spin glass, in the presence of an external magnetic field, is studied on different hierarchical lattices that approach the cubic lattice. The magnetic field is considered as uniform, or random (following either a bimodal or a Gaussian probability distribution). In all cases, a spin-glass attractor is found, in the plane magnetic field versus temperature, associated with a low-temperature phase. The physical consequences of this attractor are discussed, in view of the present scenario of the spin-glass problem.
In the simple [hyper]cubic five dimension near neighbor interaction Ising ferromagnet, extensive simulation measurements are made of the link overlap and the spin overlap distributions. These "two replica" measurements are standard in the Spin Glass context but are not usually recorded in ferromagnet simulations. The moments and moment ratios of these distributions (the variance, the kurtosis and the skewness) show clear critical behaviors at the known ordering temperature of the ferromagnet. Analogous overlap effects can be expected quite generally in Ising ferromagnets in any dimension. The …
We study the Blume-Emery-Griffiths spin glass model in presence of an attractive coupling between real replicas, and evaluate the effective potential as a function of the density overlap. We find that there is a region, above the first order transition of the model, where metastable states with a large density overlap exist. The line where these metastable states appear should correspond to a purely dynamical transition, with a breaking of ergodicity. Differently from what happens in p-spin glasses, in this model the dynamical transition would not be the precursor of a 1-step RSB transition, b…
We investigate numerically the time dependence of "window" overlaps in a three-dimensional Ising spin glass below its transition temperature after a rapid quench. Using an efficient GPU implementation, we are able to study large systems up to lateral length $L=128$ and up to long times of $t=10^8$ sweeps. We find that the data scales according to the ratio of the window size $W$ to the non-equilibrium coherence length $ξ(t)$. We also show a substantial change in behavior if the system is run for long enough that it globally equilibrates, i.e. $ξ(t) \approx L/2$, where $L$ is the lattice size. …
Using computer simulations of an atomistic glass-forming liquid, we investigate the fluctuations of the overlap between a fluid configuration and a quenched reference system. We find that large fluctuations of the overlap develop as temperature decreases, consistent with the existence of the random critical point that is predicted by effective field theories. We discuss the scaling of fluctuations near the presumed critical point, comparing the observed behaviour with that of the random-field Ising model. We argue that this critical point directly reveals the existence of an interfacial tensio…
The mean field spin glass model is analyzed by a combination of mathematically rigororous methods and a powerful Ansatz. The method exploited is general, and can be applied to others disordered mean field models such as, e.g., neural networks. It is well known that the probability measure of overlaps among replicas carries the whole physical content of these models. A functional order parameter of Parisi type is introduced by rigorous methods, according to previous works by F. Guerra. By the Ansatz that the functional order parameter is the correct order parameter of the model, we explicitly f…
We consider thermodynamic and dynamic phase transitions in plaquette spin models of glasses. The thermodynamic transitions involve coupled (annealed) replicas of the model. We map these coupled-replica systems to a single replica in a magnetic field, which allows us to analyse the resulting phase transitions in detail. For the triangular plaquette model (TPM), we find for the coupled-replica system a phase transition between high- and low-overlap phases, occuring at a coupling eps*(T), which vanishes in the low-temperature limit. Using computational path sampling techniques, we show that a sin…
We characterize numerically the properties of the phase transition of the three dimensional Ising spin glass with Gaussian couplings and of the low temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three dimensional spin glasses.
In these notes the main theoretical concepts and techniques in the field of mean-field spin-glasses are reviewed in a compact and pedagogical way, for the benefit of the graduate and undergraduate student. One particular spin-glass model is analyzed (the p-spin spherical model) by using three different approaches. Thermodynamics, covering pure states, overlaps, overlap distribution, replica symmetry breaking, and the static transition. Dynamics, covering the generating functional method, generalized Langevin equation, equations for the correlation and the response, the Mode Coupling approximat…
A reduction procedure to obtain ground states of spin glasses on sparse graphs is developed and tested on the hierarchical lattice associated with the Migdal-Kadanoff approximation for low-dimensional lattices. While more generally applicable, these rules here lead to a complete reduction of the lattice. The stiffness exponent governing the scaling of the defect energy $ΔE$ with system size $L$, $σ(ΔE)\sim L^y$, is obtained as $y_3=0.25546(3)$ by reducing the equivalent of lattices up to $L=2^{100}$ in $d=3$, and as $y_4=0.76382(4)$ for up to $L=2^{35}$ in $d=4$. The reduction rules allow the …
The dynamic behaviour of glassy materials displays strong nonequilibrium effects, such as ageing in simple protocols, memory, rejuvenation and Kovacs effects in more elaborated experiments. We show that this phenomenology may be easily understood in the context of the nonequilibrium critical dynamics of non-disordered systems, the main ingredient being the existence of an infinite equilibrium correlation length. As an example, we analytically investigate the behaviour of the 2D XY model submitted to temperature protocols similar to experiments. This shows that typical glassy effects may be obt…
We discuss a phase transition in spin glass models which have been rarely considered in the past, namely the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e. at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that even in a paramagnetic phase the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importan…
We perform large scale simulations of the frustrated Ising lattice gas, a three-dimensional lattice model of a structural glass, using the parallel tempering technique. We evaluate the spin and density overlap distributions, and the corresponding non-linear susceptibilities, as a function of the chemical potential. We then evaluate the relaxation functions of the spin and density self-overlap, and study the behavior of the relaxation times. The results suggest that the spin variables undergo a transition very similar to the one of the Ising spin glass, while the density variables do not show a…
We report the value of the dynamical critical exponent z for the six dimensional Ising spin glass, measured in three different ways: from the behavior of the energy and the susceptibility with the Monte Carlo time and by studying the overlap-overlap correlation function as a function of the space and time. All three results are in a very good agreement with the Mean Field prediction z=4. Finally we have studied numerically the remanent magnetization in 6 and 8 dimensions and we have compared it with the behavior observed in the SK model, that we have computed analytically.
We study analytically M-spin-flip stable states in disordered short-ranged Ising models (spin glasses and ferromagnets) in all dimensions and for all M. Our approach is primarily dynamical and is based on the convergence of a zero-temperature dynamical process with flips of lattice animals up to size M and starting from a deep quench, to a metastable limit. The results (rigorous and nonrigorous, in infinite and finite volumes) concern many aspects of metastable states: their numbers, basins of attraction, energy densities, overlaps, remanent magnetizations and relations to thermodynamic states…
We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: A sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method [M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi. Ensemble renormalization group for disordered systems. $\textit{Phys. Rev. B}$, 87(13):134201, 2013] in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical expo…
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.
In recent times, the theoretical study of the three-dimensional Edwards-Anderson model has produced several rigorous results on the nature of the spin-glass phase. In particular, it has been shown that, as soon as the overlap distribution is non-trivial, ultrametricity holds. However, these theorems are valid only in the thermodynamical limit and are therefore of uncertain applicability for (perennially off-equilibrium) experimental spin glasses. In addition, their basic assumption of non-triviality is still hotly debated. This paper intends to show that the predictions stemming from ultrametr…
