Erratum: Numerical test of the replica-symmetric Hamiltonian for correlations of the critical state of spin glasses in a field [Phys. Rev. E 105, 054106 (2022)].

This corrects the article DOI: 10.1103/PhysRevE.105.054106.

Numerical test of the replica-symmetric Hamiltonian for correlations of the critical state of spin glasses in a field.

A growing body of evidence indicates that the sluggish low-temperature dynamics of glass formers (e.g., supercooled liquids, colloids, or spin glasses) is due to a growing correlation length. Which is the effective field theory that describes these correlations? The natural field theory was drastically simplified by Bray and Roberts in 1980. More than 40 years later, we confirm the tenets of Bray and Roberts's theory by studying the Ising spin glass in an externally applied magnetic field, both in four spatial dimensions (data obtained from the Janus collaboration) and on the Bethe lattice.

Scaling Law Describes the Spin-Glass Response in Theory, Experiments, and Simulations.

The correlation length ξ, a key quantity in glassy dynamics, can now be precisely measured for spin glasses both in experiments and in simulations. However, known analysis methods lead to discrepancies either for large external fields or close to the glass temperature. We solve this problem by introducing a scaling law that takes into account both the magnetic field and the time-dependent spin-glass correlation length. The scaling law is successfully tested against experimental measurements in a CuMn single crystal and against large-scale simulations on the Janus II dedicated computer.

Aging Rate of Spin Glasses from Simulations Matches Experiments.

Experiments on spin glasses can now make precise measurements of the exponent z(T) governing the growth of glassy domains, while our computational capabilities allow us to make quantitative predictions for experimental scales. However, experimental and numerical values for z(T) have differed. We use new simulations on the Janus II computer to resolve this discrepancy, finding a time-dependent z(T,t_{w}), which leads to the experimental value through mild extrapolations. Furthermore, theoretical insight is gained by studying a crossover between the T=T_{c} and T=0 fixed points.

Improved belief propagation algorithm finds many Bethe states in the random-field Ising model on random graphs.

We first present an empirical study of the Belief Propagation (BP) algorithm, when run on the random field Ising model defined on random regular graphs in the zero temperature limit. We introduce the notion of extremal solutions for the BP equations, and we use them to fix a fraction of spins in their ground state configuration. At the phase transition point the fraction of unconstrained spins percolates and their number diverges with the system size. This in turn makes the associated optimization problem highly non trivial in the critical region. Using the bounds on the BP messages provided by the extremal solutions we design a new and very easy to implement BP scheme which is able to output a large number of stable fixed points. On one hand this new algorithm is able to provide the minimum energy configuration with high probability in a competitive time. On the other hand we found that the number of fixed points of the BP algorithm grows with the system size in the critical region. This unexpected feature poses new relevant questions about the physics of this class of models.

Numerical Construction of the Aizenman-Wehr Metastate.

Chaotic size dependence makes it extremely difficult to take the thermodynamic limit in disordered systems. Instead, the metastate, which is a distribution over thermodynamic states, might have a smooth limit. So far, studies of the metastate have been mostly mathematical. We present a numerical construction of the metastate for the d=3 Ising spin glass. We work in equilibrium, below the critical temperature. Leveraging recent rigorous results, our numerical analysis gives evidence for a dispersed metastate, supported on many thermodynamic states.

Matching Microscopic and Macroscopic Responses in Glasses.

We first reproduce on the Janus and Janus II computers a milestone experiment that measures the spin-glass coherence length through the lowering of free-energy barriers induced by the Zeeman effect. Secondly, we determine the scaling behavior that allows a quantitative analysis of a new experiment reported in the companion Letter [S. Guchhait and R. Orbach, Phys. Rev. Lett. 118, 157203 (2017)].PRLTAO0031-900710.1103/PhysRevLett.118.157203 The value of the coherence length estimated through the analysis of microscopic correlation functions turns out to be quantitatively consistent with its measurement through macroscopic response functions. Further, nonlinear susceptibilities, recently measured in glass-forming liquids, scale as powers of the same microscopic length.

Searching for feasible stationary states in reaction networks by solving a Boolean constraint satisfaction problem.

We analyze the solutions, on single network instances, of a recently introduced class of constraint-satisfaction problems (CSPs), describing feasible steady states of chemical reaction networks. First, we show that the CSPs generalize the scheme known as network expansion, which is recovered in a specific limit. Next, a full statistical mechanics characterization (including the phase diagram and a discussion of the physical origin of the phase transitions) for network expansion is obtained. Finally, we provide a message-passing algorithm to solve the original CSPs in the most general form.

Finite-size corrections to disordered Ising models on random regular graphs.

We derive the analytical expression for the first finite-size correction to the average free energy of disordered Ising models on random regular graphs. The formula can be physically interpreted as a weighted sum over all non-self-intersecting loops in the graph, the weight being the free-energy shift due to the addition of the loop to an infinite tree.

Dynamical transition in the D=3 Edwards-Anderson spin glass in an external magnetic field.

We study the off-equilibrium dynamics of the three-dimensional Ising spin glass in the presence of an external magnetic field. We have performed simulations both at fixed temperature and with an annealing protocol. Thanks to the Janus special-purpose computer, based on field-programmable gate array (FPGAs), we have been able to reach times equivalent to 0.01 s in experiments. We have studied the system relaxation both for high and for low temperatures, clearly identifying a dynamical transition point. This dynamical temperature is strictly positive and depends on the external applied magnetic field. We discuss different possibilities for the underlying physics, which include a thermodynamical spin-glass transition, a mode-coupling crossover, or an interpretation reminiscent of the random first-order picture of structural glasses.

Critical slowing down exponents of mode coupling theory.

A method is provided to compute the exponent parameter λ yielding the dynamic exponents of critical slowing down in mode coupling theory. It is independent from the dynamic approach and based on the formulation of an effective static field theory. Expressions of λ in terms of third order coefficients of the action expansion or, equivalently, in terms of six point cumulants are provided. Applications are reported to a number of mean-field models: with hard and soft variables and both fully connected and dilute interactions. Comparisons with existing results for the Potts glass model, the random orthogonal model, hard and soft-spin Sherrington-Kirkpatrick, and p-spin models are presented.

Field theory of fluctuations in glasses.

We develop a field-theoretical description of dynamical heterogeneities and fluctuations in supercooled liquids close to the (avoided) MCT singularity. Using quasi-equilibrium arguments, we eliminate time from the description and we completely characterize fluctuations in the beta regime. We identify different sources of fluctuations and show that the most relevant ones are associated to variations of "self-induced disorder" in the initial condition of the dynamics. It follows that heterogeneites can be described through a cubic field theory with an effective random field term. The phenomenon of perturbative dimensional reduction ensues, well known in random field problems, which implies an upper critical dimension of the theory equal to 8. We apply our theory to finite size scaling for mean-field systems and we test its prediction against numerical simulations.

Ising spin-glass transition in a magnetic field outside the limit of validity of mean-field theory.

The spin-glass transition in a magnetic field is studied both in and out of the limit of validity of mean-field theory on a diluted one dimensional chain of Ising spins where exchange bonds occur with a probability decaying as the inverse power of the distance. Varying the power in this long-range model corresponds, in a one-to-one relationship, to changing the dimension in spin-glass short-range models. Evidence for a spin-glass transition in a magnetic field is found also for systems whose equivalent dimension is below the upper critical dimension in a zero magnetic field.

Dilute one-dimensional spin glasses with power law decaying interactions.

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 changing 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 find both static and dynamic indications in favor of the so-called replica symmetry breaking theory.

Multivesicular release at developing Schaffer collateral-CA1 synapses: an analytic approach to describe experimental data.

We developed and analytically solved a simple and general stochastic model to distinguish the univesicular from the multivesicular mode of glutamate release. The model solution gives analytical mathematical expressions for average values of quantities that can be measured experimentally. Comparison of these quantities with the experimental measures allows one to discriminate the release mode and to determine the most probable values of model parameters. The model has been validated at glutamatergic CA3-CA1 synapses in the hippocampus from newborn (P1-P5 old) rats. Our results strongly support a multivesicular type of release process requiring a variable pool of immediately releasable vesicles. Moreover, computing quantities that are functions of the model parameters, the mean amplitude of the synaptic response to the release of a single vesicle (q) was estimated to be 5-10 pA, in very good agreement with experimental findings. In addition a multivesicular type of release was supported by the following experimental evidences: 1) a high variability of the amplitude of successes, with a coefficient of variation ranging from 0.12 to 0.73; 2) an average potency ratio a2/a1 between the second and first response to a pair of stimuli >1; and 3) changes in the potency of the synaptic response to the first stimulus when the release probability was modified by increasing or decreasing the extracellular calcium concentration. Our results indicate that at Schaffer collateral-CA1 synapses of the neonatal rat hippocampus a single action potential may induce the release of more than one vesicle from the same release site.

Microscopic description of aging dynamics: fluctuation-dissipation relations, effective temperature, and heterogeneities.

We consider the dynamics of a diluted mean-field spin glass model in the aging regime. The model presents a particularly rich heterogeneous behavior. In order to catch this behavior, we perform a spin-by-spin analysis for a given disorder realization. We confirm the connection between statics and dynamics at the level of single degrees of freedom. Moreover, working with single-site quantities, we can introduce a new response-vs-correlation plot, which clearly shows how heterogeneous degrees of freedom undergo coherent structural rearrangements. We discuss the general scenario which emerges from our work and (possibly) applies to more realistic glassy models. Interestingly enough, some features of this scenario can be understood recurring to thermometric considerations.

Measuring the fluctuation-dissipation ratio in glassy systems with no perturbing field.

A method is presented for measuring the integrated linear response in Ising spin system without applying any perturbing field. Large-scale simulations are performed in order to show how the method works. Very precise measurements of the fluctuation-dissipation ratio are presented for three different Ising models: the two-dimensional ferromagnetic model, the mean-field diluted three-spin model, and the three-dimensional Edwards-Anderson model.

Hiding solutions in random satisfiability problems: a statistical mechanics approach.

A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of statistical mechanics results, we propose random generators of hard and satisfiable instances for the 3-satisfiability problem. The design of the hardest problem instances is based on the existence of a first order ferromagnetic phase transition and the glassy nature of excited states. The analytical predictions are corroborated by numerical results obtained from complete as well as stochastic local algorithms.

Learning to coordinate in a complex and nonstationary world.

We study analytically and by computer simulations a complex system of adaptive agents with finite memory. Borrowing the framework of the minority game and using the replica formalism we show the existence of an equilibrium phase transition as a function of the ratio between the memory lambda and the learning rates Gamma of the agents. We show that, starting from a random configuration, a dynamic phase transition also exists, which prevents agents from reaching optimal coordination. Furthermore, in a nonstationary environment, we show by numerical simulations that the phase transition becomes discontinuous.