Energy gap of quantum spin glasses: a projection quantum Monte Carlo study
Università di Camerino（卡梅里诺大学）；INFN, Sezione di Perugia（意大利国家核物理研究所佩鲁贾分部）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；CNR-INO（意大利国家研究委员会光学研究所）；Università di Trento（特伦托大学）
L. Brodoloni, G. E. Astrakharchik, S. Giorgini, S. Pilati
The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap $Δ$ encountered at quantum phase transitions. We investigate the scaling of $Δ$ with system size $N$ for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as $N$ increases. This indicates that the unfavorable super-algebraic scaling of $Δ$, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to $Δ\propto N^{-1/3}$. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.
We develop a parallel rejection algorithm to tackle the problem of low acceptance in Monte Carlo methods, and apply it to the simulation of the hopping conduction in Coulomb glasses using Graphics Processing Units, for which we also parallelize the update of local energies. In two dimensions, our parallel code achieves speedups of up to two orders of magnitude in computing time over an equivalent serial code. We find numerical evidence of a scaling relation for the relaxation of the conductivity at different temperatures.
We start from recently published numerical data by Hatano and Gubernatis cond-mat/0008115 to discuss properties of convergence to equilibrium of optimized Monte Carlo methods (bivariate multi canonical and parallel tempering). We show that these data are not thermalized, and they lead to an erroneous physical picture. We shed some light on why the bivariate multi canonical Monte Carlo method can fail.
We study universality in three-dimensional Ising spin glasses by large-scale Monte Carlo simulations of the Edwards-Anderson Ising spin glass for several choices of bond distributions, with particular emphasis on Gaussian and bimodal interactions. A finite-size scaling analysis suggests that three-dimensional spin glasses obey universality.
The non-equilibrium dynamics of the three-dimensional Edwards-Anderson spin-glass model with different bond distributions is investigated by means of Monte Carlo simulation. A numerical method is used to determine the critical temperature and the scaling exponents of the correlation and the integrated response functions. The results obtained agree with those calculated in equilibrium simulations and suggest that the universality class does not depend on the exact form of the bond distribution.
A new Monte Carlo algorithm for 2-dimensional spin glasses is presented. The use of clusters makes possible global updates and leads to a gain in speed of several orders of magnitude. As an example, we study the 2-dimensional +/-J Edwards-Anderson model. The new algorithm allows us to equilibrate systems of size 100^2 down to temperature T = 0.1. Our main result is that the correlation length diverges as an exponential and not as a power law as T -> Tc = 0.
We report the results of Monte Carlo simulations on several spin glass models at low temperatures. By using the parallel tempering (Exchange Monte Carlo) technique we are able to equilibrate down to low temperatures, for moderate sizes, and hence the data should not be affected by critical fluctuations. Our results for short range models are consistent with a picture proposed earlier that there are large scale excitations which cost only a finite energy in the thermodynamic limit, and these excitations have a surface whose fractal dimension is less than the space dimension. For the infinite ra…
We report a very detailed relation about the study, by a constrained Monte Carlo dynamic, of a 4D EA spin glass model (J=+/- 1). In particular we concentrate our attention on the behaviour of the system under different bounds imposed in order to optimize the dynamic and try to understand the low energy states structure. We find that in the thermodynamic limit this structure is clearly ultrametric as already established for the SK model.
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…
We applied a multicanonical algorithm (entropic sampling) to a two-dimensional and a three-dimensional Lennard-Jones system with quasicrystalline and glassy ground states. Focusing on the ability of the algorithm to locate low lying energy states, we compared the results of the multicanonical simulations with standard Monte Carlo simulated annealing and molecular dynamics methods. We find slight benefits to using entropic sampling in small systems (less than 80 particles), which disappear with larger systems. This is disappointing as the multicanonical methods are designed to surmount energy b…
We present results of Monte Carlo simulations, using parallel tempering, on the three- and four-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings at low temperatures with free boundary conditions. Our results suggest that large-scale low-energy excitations may be space filling. The data implies that the energy of these excitations increases with increasing system size for small systems, but we see evidence in three dimensions, where we have a greater range of sizes, for a crossover to a regime where the energy is independent of system size, in accordance with replica symmet…
Recently a cluster Monte Carlo algorithm has been used very successfully in the two-dimensional Edwards-Anderson (EA) model. We show that this algorithm and a variant thereof can also be used successfully in models with a non-zero spin glass transition temperature. The application of such algorithms to the site-diluted EA model in three dimensions is discussed and the efficiency of the two algorithms is compared among each other and to parallel tempering. Finally, we give evidence for a spin glass transition in the three-dimensional site-diluted EA model with Gaussian couplings at a site occup…
In this paper I report results for simulations of the three-dimensional gauge glass and the four-dimensional XY spin glass using the parallel tempering Monte Carlo method at low temperatures for moderate sizes. The results are qualitatively consistent with earlier work on the three- and four-dimensional Edwards-Anderson Ising spin glass. I find evidence that large-scale excitations may cost only a finite amount of energy in the thermodynamic limit. The surface of these excitations is fractal, but I cannot rule out for the XY spin glass a scenario compatible with replica symmetry breaking where…
The autoregressive neural networks are emerging as a powerful computational tool to solve relevant problems in classical and quantum mechanics. One of their appealing functionalities is that, after they have learned a probability distribution from a dataset, they allow exact and efficient sampling of typical system configurations. Here we employ a neural autoregressive distribution estimator (NADE) to boost Markov chain Monte Carlo (MCMC) simulations of a paradigmatic classical model of spin-glass theory, namely the two-dimensional Edwards-Anderson Hamiltonian. We show that a NADE can be train…
The structure of Ge$_{22}$Ga$_3$Sb$_{10}$S$_{65}$ and Ge$_{15}$Ga$_{10}$Sb$_{10}$S$_{65}$ glasses was investigated by neutron diffraction (ND), X-ray diffraction (XRD), and extended X-ray absorption fine structure (EXAFS) measurements at the Ge, Ga and Sb K-edges. Experimental data sets were fitted simultaneously in the framework of the reverse Monte Carlo (RMC) simulation technique. Short range order parameters were determined from the obtained large-scale configurations. It was found that the coordination numbers of Ge, Sb and S are around the values predicted by the Mott-rule (4, 3 and 2, r…
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…
Recently a local mean field theory for both eqilibrium and transport properties of the Coulomb glass was proposed [A. Amir et al., Phys. Rev. B 77, 165207 (2008); 80, 245214 (2009)]. We compare the predictions of this theory to the results of dynamic Monte Carlo simulations. In a thermal equilibrium state we compare the density of states and the occupation probabilities. We also study the transition rates between different states and find that the mean field rates underestimate a certain class of important transitions. We propose modified rates to be used in the mean field approach which take …
We have studied the Parisi overlap distribution for the three dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures T around 0.7Tc and system sizes upto L=32, we found a P(q) as expected for the full Parisi replica symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behaviour in Monte Carlo simulations is due to the fact that all existing simulations have been done at temperatures too c…
Results of Monte Carlo simulations of XY and Heisenberg spin glass models in three dimensions are presented. A finite size scaling analysis of the correlation length of the spins and chiralities of both models shows that there is a single, finite-temperature transition at which both spins and chiralities order.
We study the four-dimensional Ising spin glass with Gaussian and bond-diluted bimodal distributed interactions via large-scale Monte Carlo simulations and show via an extensive finite-size scaling analysis that four-dimensional Ising spin glasses obey universality.
Local minima also known as inherent structures are expected to play an essential role for the behavior of spin glasses. Here, we propose techniques to efficiently sample these configurations in Monte Carlo simulations. For the Sherrington-Kirkpatrick and the three-dimensional Edwards-Anderson model their spectra are determined and compared to analytical results.
In this work we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions $C_{ij}\equiv< S_i S_j>$. We argue that more than one extensive (i.e., O(N)) eigenvalue in this matrix signals replica symmetry breaking. Monte-Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte-Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consis…
The specific heat capacity $c_v$ of glass formers undergoes a hysteresis when subjected to a cooling-heating cycle, with a larger $c_v$ and a more pronounced hysteresis for fragile glasses than for strong ones. Here, we show that these experimental features, including the unusually large magnitude of $c_v$ of fragile glasses, are well reproduced by kinetic Monte Carlo and equilibrium study of a distinguishable particle lattice model (DPLM) incorporating a two-state picture of particle interactions. The large $c_v$ in fragile glasses is caused by a dramatic transfer of probabilistic weight from…
Fluoride glasses BaMnMF7 (M = Fe, V, assuming isomorphous replacement) have been structurally modelled. The neutron patterns were simulated by the reverse Monte Carlo (RMC) and the Rietveld for disordered materials (RDM) methods. The best fit is obtained from one of the seven tested crystalline models : the BaMnGaF7 structure-type in which the glasses crystallize. However, by RMC applied to enlarged RDM models, the difference between the best and the worst fit is quite small (DRp ~ 0.5%), suggesting that two interference functions are not enough for the characterization of such multicomponent …
A kinetic Monte Carlo (KMC) method is used to study the structural properties and dynamics of a supercooled binary Lennard-Jones liquid around the glass transition temperature. This technique permits us to explore the potential energy surface and barrier distributions without suffering the exponential slowing down at low temperature that affects molecular dynamics simulations. In agreement with previous studies we observe a distinct change in behaviour around $T=0.45$, close to the dynamical transition temperature $T_c$ of mode coupling theory (MCT). Below this temperature the number of differ…
Mean-field models of 2-spin Ising spin glasses with interaction matrices taken from ensembles which are invariant under O(N) transformations are studied. A general study shows that the nature of the spin glass transition can be deduced from the eigenvalue spectrum of the interaction matrix. A simple replica approach is derived to carry out the average over the O(N) disorder. The analytic results are confirmed by extensive Monte Carlo simulations for large system sizes and by exact enumeration for small system sizes.
An one-step replica-symmetry-breaking solution for finite connectivity spin-glass models with K body interaction is constructed at finite temperature using the replica method and thermodynamic constraints. In the absence of external fields, this construction provides a general extension of replica symmetric solution at finite replica number to one-step replica-symmetry-breaking solution. It is found that this result is formally equivalent to that of the one-step replica-symmetry-breaking cavity method. To confirm the validity of the obtained solution, Monte Carlo simulations are performed for …
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 study the effects of small temperature as well as disorder perturbations on the equilibrium state of three-dimensional Ising spin glasses via an alternate scaling ansatz. By using Monte Carlo simulations, we show that temperature and disorder perturbations yield chaotic changes in the equilibrium state and that temperature chaos is considerably harder to observe than disorder chaos.
We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $β=1/T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state and the specific heat. We confirm that $T_c=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $ν$. We find $-1/ν=-0.295(30)$, whi…
We present results of Monte Carlo simulations of the three-dimensional Edwards-Anderson Ising spin glass in the presence of a (random) field. A finite-size scaling analysis of the correlation length shows no indication of a transition, in contrast to the zero-field case. This suggests that there is no Almeida-Thouless line for short-range Ising spin glasses.
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 …
We present a quantum statistical analysis of a microscopic mean-field model of structural glasses at low temperatures. The model can be thought of as arising from a random Born von Karman expansion of the full interaction potential. The problem is reduced to a single-site theory formulated in terms of an imaginary-time path integral using replicas to deal with the disorder. We study the physical properties of the system in thermodynamic equilibrium and develop both perturbative and non-perturbative methods to solve the model. The perturbation theory is formulated as a loop expansion in terms o…
We analyze the steady state regime of systems interpolating between spin glasses and ferromagnets under a tapping dynamics recently introduced by analogy with the dynamics of mechanically perturbed granular media. A crossover from a second order to first order ferromagnetic transition as a function of the spin coupling distribution is found. The flat measure over blocked states introduced by Edwards for granular media is used to explain this scenario. Annealed calculations of the Edwards entropy are shown to qualitatively explain the nature of the phase transitions. A Monte-Carlo construction …
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.
I describe the classic circle-packing problem on a sphere, and the analytic and numerical approaches that have been used to study it. I then present a very simple Markov-chain Monte Carlo algorithm, which succeeds in finding the best solutions known today. The behavior of the algorithm is put into the context of the statistical physics of glasses.
We study the fragility of spin glasses to small temperature perturbations numerically using population annealing Monte Carlo. We apply thermal boundary conditions to a three-dimensional Edwards-Anderson Ising spin glass. In thermal boundary conditions all eight combinations of periodic versus antiperiodic boundary conditions in the three spatial directions are present, each appearing in the ensemble with its respective statistical weight determined by its free energy. We show that temperature chaos is revealed in the statistics of crossings in the free energy for different boundary conditions.…
Using large-scale Monte Carlo simulations that combine parallel tempering with specialized cluster updates, we show that Ising spin glasses with Levy-distributed interactions share the same universality class as Ising spin glasses with Gaussian or bimodal-distributed interactions. Corrections to scaling are large for Levy spin glasses. In order to overcome these and show that the critical exponents agree with the Gaussian case, we perform an extended scaling of the two-point finite size correlation length and the spin glass susceptibility. Furthermore, we compute the critical temperature and c…
We numerically study yielding in two-dimensional glasses which are generated with a very wide range of stabilities by swap Monte-Carlo simulations and then slowly deformed at zero temperature. We provide strong numerical evidence that stable glasses yield via a nonequilibrium discontinuous transition in the thermodynamic limit. A critical point separates this brittle yielding from the ductile one observed in less stable glasses. We find that two-dimensional glasses yield similarly to their three-dimensional counterparts but display larger sample-to-sample disorder-induced fluctuations, stronge…
We present experimental results attempting to fingerprint nonanalyticities in the magnetization curves of spin glasses found by Katzgraber et al. [Phys. Rev. Lett. 89, 257202 (2002)] via zero-temperature Monte Carlo simulations of the Edwards-Anderson Ising spin glass. Our results show that the singularities at zero temperature due to the reversal-field memory effect are washed out by the finite temperatures of the experiments. The data are analyzed via the first order reversal curve (FORC) magnetic fingerprinting method. The experimental results are supported by Monte Carlo simulations of the…
Spin-glass (SG) and chiral-glass (CG) orderings in three dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a low…
Starting from a Cluster Variational Method, and inspired by the correctness of the paramagnetic Ansatz (at high temperatures in general, and at any temperature in the 2D Edwards-Anderson model) we propose a novel message passing algorithm --- the Dual algorithm --- to estimate the marginal probabilities of spin glasses on finite dimensional lattices. We show that in a wide range of temperatures our algorithm compares very well with Monte Carlo simulations, with the Double Loop algorithm and with exact calculation of the ground state of 2D systems with bimodal and Gaussian interactions. Moreove…
Spin systems with long-range interactions are "non-extensive" if the strength of the interactions falls off sufficiently slowly with distance. It has been conjectured for ferromagnets, and more recently for spin glasses, that, everywhere in the non-extensive regime, the free energy is exactly equal to that for the infinite range model in which the characteristic strength of the interaction is independent of distance. In this paper we present the results of Monte Carlo simulations of the one-dimensional long-range spin glasses in the non-extensive regime. Using finite-size scaling, our results …
Spin-glass and chiral-glass orderings in three-dimensional Heisenberg spin glasses are studied with and without randaom magnetic anisotropy by dynamical Monte Carlo simulations. In isotropic case, clear evidence of a finite-temperature chiral-glass transition is presented. While the spin autocorrelation exhibits only an interrupted aging, the chirality autocorrelation persists to exhibit a pronounced aging effect reminisecnt of the one observed in the mean-field model. In anisotropic case, asymptotic mixing of the spin and the chirality is observed in the off-equilibrium dynamics.
We extract a pair correlation function (PCF) from probability distributions of the spin-overlap parameter q. The distributions come from Monte Carlo simulations. A measure, w, of the thermal fluctuations of magnetic patterns follows from the PCFs. We also obtain rms deviations (over different system samples) delta_p away from average probabilities for q. For the linear system-sizes $L$ we have studied, (i) w and delta_p are independent of L in the Edwards-Anderson model but scale as 1/L and sqrt{L}, respectively, in the Sherrington-Kirkpatrik model.
We perform careful numerical simulations of slow Monte-Carlo annealings in the dense 3-body spin glass model and compare with the predictions from different theories: thresholds states, isocomplexity, following state. We conclude that while isocomplexity and following state both provide excellent agreement the numerical data, the influence of threshold states -- that is still the most commonly considered theory -- can be excluded from our data.
The aim of this work is to prove that it is possible to realise an optical system which produces as output a light intensity that can be expressed in the same mathematical form of the spin glass Hamiltonian. The optical system under study is controlled through an adaptive optics device composed by millions of switchable mirrors in an ON and OFF position. These user controlled mirrors are playing the role of the spin states. Furthermore, such optical system can be used to run simulations and, applying the same Metropolis Monte Carlo algorithm used in computer simulations, extract the quantities…
We use a non-equilibrium simulation method to study the spin glass transition in three-dimensional Ising spin glasses. The transition point is repeatedly approached at finite velocity $v$ (temperature change versus time) in Monte Carlo simulations starting at a high temperature. The normally problematic critical slowing-down is not hampering this kind of approach, since the system equilibrates quickly at the initial temperature and the slowing-down is merely reflected in the dynamic scaling of the non-equilibrium order parameter with $v$ and the system size. The equilibrium limit does not have…
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 …
Amorphous solids exhibit quasi-universal low-temperature anomalies whose origin has been ascribed to localized tunneling defects. Using an advanced Monte Carlo procedure, we create {\it in silico} glasses spanning from hyperquenched to ultrastable glasses. Using a multidimensional path-finding protocol, we locate tunneling defects with energy splittings smaller than $k_{B}T_Q$, with $T_Q$ the temperature below which quantum effects are relevant ($T_Q \approx 1 \,$K in most experiments). We find that as the stability of a glass increases, its energy landscape as well as the manner in which it i…
We test for the existence of a spin-glass phase transition, the de Almeida-Thouless line, in an externally-applied (random) magnetic field by performing Monte Carlo simulations on a power-law diluted one-dimensional Ising spin glass for very large system sizes. We find that an Almeida-Thouless line only occurs in the mean field regime, which corresponds, for a short-range spin glass, to dimension d larger than 6.
We develop a systematic expansion method of physical quantities for the SK model and the finite-dimensional $\pm J$ model of spin glasses in non-equilibrium states. The dynamical probability distribution function is derived from the master equation using a high temperature expansion. We calculate the expectation values of physical quantities from the dynamical probability distribution function. The theoretical curves show satisfactory agreement with Monte Carlo simulation results in the appropriate temperature and time regions. A comparison is made with the results of a dynamics theory by Cool…
Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the dro…
We investigate the ferromagnetic-glassy transitions which separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice +-J (Edwards-Anderson) Ising model with bond distribution $P(J) = p δ(J - 1) + (1-p) δ(J + 1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly re…
Using a non-local Monte Carlo algorithm, we study the aging of a fragile glass, being able to follow it up to equilibibrium down to 0.89 Tmc (Tmc is the Mode-Coupling temperature) and up to unprecedentedly large waiting times at lower temperatures. We show that the fluctuation-dissipation ratio is independent of the dynamics chosen and is compatible with a phase transition, and that the scaling behaviour of the aging part of the correlation supports the full-aging scenario.
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.
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,…
