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We have studied numerically the appearance of multiscaling behavior in the three-dimensional ferromagnetic Ising site diluted model, in the form of a multifractal distribution of the decay exponents for the spatial correlation functions at the critical temperature. We have computed the exponents of the long-distance decay of higher moments of the correlation function, up to the 10th power, by studying three different quantities: global susceptibilities, local susceptibilities and correlation functions. We have found very clear evidences for multiscaling behavior., Comment: 19 pages and 5 figures. Final version of the paper

Rejuvenation and memory, long considered the distinguishing features of spin glasses, have recently been proven to result from the growth of multiple length scales. This insight, enabled by simulations on the Janus~II supercomputer, has opened the door to a quantitative analysis. We combine numerical simulations with comparable experiments to introduce two coefficients that quantify memory. A third coefficient has been recently presented by Freedberg et al. We show that these coefficients are physically equivalent by studying their temperature and waiting-time dependence., Comment: 10 pages, 8 figures

We unveil the multifractal behavior of Ising spin glasses in their low-temperature phase. Using the Janus II custom-built supercomputer, the spin-glass correlation function is studied locally. Dramatic fluctuations are found when pairs of sites at the same distance are compared. The scaling of these fluctuations, as the spin-glass coherence length grows with time, is characterized through the computation of the singularity spectrum and its corresponding Legendre transform. A comparatively small number of site pairs controls the average correlation that governs the response to a magnetic field. We explain how this scenario of dramatic fluctuations (at length scales smaller than the coherence length) can be reconciled with the smooth, self-averaging behavior that has long been considered to describe spin-glass dynamics., Comment: 15 pages, 15 figures

The extended principle of superposition has been a touchstone of spin glass dynamics for almost thirty years. The Uppsala group has demonstrated its validity for the metallic spin glass, CuMn, for magnetic fields $H$ up to 10 Oe at the reduced temperature $T_\mathrm{r}=T/T_\mathrm{g} = 0.95$, where $T_\mathrm{g}$ is the spin glass condensation temperature. For $H > 10$ Oe, they observe a departure from linear response which they ascribe to the development of non-linear dynamics. The thrust of this paper is to develop a microscopic origin for this behavior by focusing on the time development of the spin glass correlation length, $\xi(t,t_\mathrm{w};H)$. Here, $t$ is the time after $H$ changes, and $t_\mathrm{w}$ is the time from the quench for $T>T_\mathrm{g}$ to the working temperature $T$ until $H$ changes. We connect the growth of $\xi(t,t_\mathrm{w};H)$ to the barrier heights $\Delta(t_\mathrm{w})$ that set the dynamics. The effect of $H$ on the magnitude of $\Delta(t_\mathrm{w})$ is responsible for affecting differently the two dynamical protocols associated with turning $H$ off (TRM, or thermoremanent magnetization) or on (ZFC, or zero field-cooled magnetization). In this paper, we display the difference between the zero-field cooled $\xi_{\text {ZFC}}(t,t_\mathrm{w};H)$ and the thermoremanent magnetization $\xi_{\text {TRM}}(t,t_\mathrm{w};H)$ correlation lengths as $H$ increases, both experimentally and through numerical simulations, corresponding to the violation of the extended principle of superposition in line with the finding of the Uppsala Group., Comment: 23 pages and 18 figures

Memory and rejuvenation effects in the magnetic response of off-equilibrium spin glasses have been widely regarded as the doorway into the experimental exploration of ultrametricity and temperature chaos (maybe the most exotic features in glassy free-energy landscapes). Unfortunately, despite more than twenty years of theoretical efforts following the experimental discovery of memory and rejuvenation, these effects have thus far been impossible to simulate reliably. Yet, three recent developments convinced us to accept this challenge: first, the custom-built Janus II supercomputer makes it possible to carry out "numerical experiments" in which the very same quantities that can be measured in single crystals of CuMn are computed from the simulation, allowing for parallel analysis of the simulation/experiment data. Second, Janus II simulations have taught us how numerical and experimental length scales should be compared. Third, we have recently understood how temperature chaos materializes in aging dynamics. All three aspects have proved crucial for reliably reproducing rejuvenation and memory effects on the computer. Our analysis shows that (at least) three different length scales play a key role in aging dynamics, while essentially all theoretical analyses of the aging dynamics emphasize the presence and the crucial role of a single glassy correlation length., Comment: Main text:13 pages, 4 figures SM: 7 pages, 8 figures

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 forty years later, we confirm the tenets of Bray and Roberts 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., Comment: 8 pages, 4 figures, accepted in PRE

A recent 3-XORSAT challenge required to minimize a very complex and rough energy function, typical of glassy models with a random first order transition and a golf course like energy landscape. We present the ideas beyond the quasi-greedy algorithm and its very efficient implementation on GPUs that are allowing us to rank first in such a competition. We suggest a better protocol to compare algorithmic performances and we also provide analytical predictions about the exponential growth of the times to find the solution in terms of free-energy barriers., Comment: 7 pages, 7 figure, EPL format + SM (2 pages)

The synergy between experiment, theory, and simulations enables a microscopic analysis of spin-glass dynamics in a magnetic field in the vicinity of and below the spin-glass transition temperature $T_\mathrm{g}$. The spin-glass correlation length, $\xi(t,t_\mathrm{w};T)$, is analysed both in experiments and in simulations in terms of the waiting time $t_\mathrm{w}$ after the spin glass has been cooled down to a stabilised measuring temperature $T

We find a dynamic effect in the non-equilibrium dynamics of a spin glass that closely parallels equilibrium temperature chaos. This effect, that we name dynamic temperature chaos, is spatially heterogeneous to a large degree. The key controlling quantity is the time-growing spin-glass coherence length. Our detailed characterization of dynamic temperature chaos paves the way for the analysis of recent and forthcoming experiments. This work has been made possible thanks to the most massive simulation to date of non-equilibrium dynamics, carried out on the Janus~II custom-built supercomputer., Comment: Version accepted for publication in Communication Physics 10 pages, 9 figures

The correlation length $\xi$, 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., Comment: Revised version, including supplemental material

Spin Glasses (SG) are paradigmatic models for physical, computer science, biological and social systems. The problem of studying the dynamics for SG models is NP hard, i.e., no algorithm solves it in polynomial time. Here we implement the optical simulation of a SG, exploiting the N segments of a wavefront shaping device to play the role of the spin variables, combining the interference at downstream of a scattering material to implement the random couplings between the spins (the J ij matrix) and measuring the light intensity on a number P of targets to retrieve the energy of the system. By implementing a plain Metropolis algorithm, we are able to simulate the spin model dynamics, while the degree of complexity of the potential energy landscape and the region of phase diagram explored is user-defined acting on the ratio the P/N = \alpha. We study experimentally, numerically and analytically this peculiar system displaying a paramagnetic, a ferromagnetic and a SG phase, and we demonstrate that the transition temperature T g to the glassy phase from the paramagnetic phase grows with \alpha. With respect to standard in silico approach, in the optical SG interaction terms are realized simultaneously when the independent light rays interferes at the target screen, enabling inherently parallel measurements of the energy, rather than computations scaling with N as in purely in silico simulations.

For a long time, the predictive limits of perturbative quantum field theory have been limited by our inability to carry out loop calculations to arbitrarily high order, which become increasingly complex as the order of perturbation theory is increased. This problem is exacerbated by the fact that perturbation series derived from loop diagram (Feynman diagram) calculations represent asymptotic (divergent) series which limits the predictive power of perturbative quantum field theory. Here, we discuss an ansatz which could overcome these limits, based on the observations that (i) for many phenomenologically relevant field theories, one can derive dispersion relations which relate the large-order growth (the asymptotic limit of "infinite loop order") with the imaginary part of arbitrary correlation functions, for negative coupling ("unstable vacuum"), and (ii) one can analyze the imaginary part for negative coupling in terms of classical field configurations (instantons). Unfortunately, the perturbation theory around instantons, which could lead to much more accurate predictions for the large-order behavior of Feynman diagrams, poses a number of technical as well as computational difficulties. Here, we study, to further the above mentioned ansatz, correlation functions in a one-dimensional (1D) field theory with a quartic self-interaction and an O(N) internal symmetry group, otherwise known as the 1D N-vector model. Our focus is on corrections to the large-order growth of perturbative coefficients, i.e., the limit of a large number of loops in the Feynman diagram expansion. We evaluate, in momentum space, the two-loop corrections for the two-point correlation function, and its derivative with respect to the momentum, as well as the two-point correlation function with a wigglet insertion. Also, we study the four-point function., Comment: 27 pages; RevTeX

We study the problem of glassy relaxations in the presence of an external field in the highly controlled context of a spin-glass simulation. We consider a small spin glass in three dimensions (specifically, a lattice of size L=8, small enough to be equilibrated through a Parallel Tempering simulations at low temperatures, deep in the spin glass phase). After equilibrating the sample, an external field is switched on, and the subsequent dynamics is studied. The field turns out to reduce the relaxation time, but huge statistical fluctuations are found when different samples are compared. After taking care of these fluctuations we find that the expected linear regime is very narrow. Nevertheless, when regarded as a purely numerical method, we find that the external field is extremely effective in reducing the relaxation times., Comment: 22 pages, 10 figures; Published version

We consider the spatial correlation function of the two-dimensional Ising spin glass under out-equilibrium conditions. We pay special attention to the scaling limit reached upon approaching zero temperature. The field-theory of a non-interacting field makes a surprisingly good job at describing the spatial shape of the correlation function of the out-equilibrium Edwards-Anderson Ising model in two dimensions., Comment: 20 pages + 5 Figures

Recent high precision experimental results on spin-glass films ask for a detailed understanding of the domain-growth dynamics of two-dimensional spin glasses. To achieve this goal, we numerically simulate the out-equilibrium dynamics of the Ising spin glass for a time that spans close to twelve orders of magnitude (from picoseconds to order of a second), in systems large enough to avoid finite-size effects. We find that the time-growth of the size of the glassy domains is excellently described by a single scaling function. A single time-scale $\tau(T)$ controls the dynamics. $\tau(T)$ diverges upon approaching the $T=0$ critical point. The divergence of $\tau(T\to 0)$ is Arrhenius-like, with a barrier height that depends very mildly on temperature. The growth of this barrier-height is best described by critical dynamics. As a side product we obtain an impressive confirmation of universality of the equilibrium behavior of two-dimensional spin-glasses., Comment: 21 pages, 9 figures. Updated references. Added DOI and Journal ref

The Mpemba effect occurs when a hot system cools faster than an initially colder one, when both are refrigerated in the same thermal reservoir. Using the custom built supercomputer Janus II, we study the Mpemba effect in spin glasses and show that it is a non-equilibrium process, governed by the coherence length \xi of the system. The effect occurs when the bath temperature lies in the glassy phase, but it is not necessary for the thermal protocol to cross the critical temperature. In fact, the Mpemba effect follows from a strong relationship between the internal energy and \xi that turns out to be a sure-tell sign of being in the glassy phase. Thus, the Mpemba effect presents itself as an intriguing new avenue for the experimental study of the coherence length in supercooled liquids and other glass formers., Comment: Version accepted for publication in PNAS. 6 pages, 7 figures

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., Comment: Version accepted for publication in PRL. 12 pages, 9 figures

We study numerically various properties of the free energy barriers in the Edwards-Anderson model of spin glasses in the low-temperature region both in three and four spatial dimensions. In particular, we investigated the dependence of height of free energy barriers on system size and on the distance between the initial and final states (i.e. the overlap distance). A related quantity is the distribution of large local fluctuations of the overlap in large three-dimensional samples at equilibrium. Our results for both quantities (barriers and large deviations) are in agreement with the prediction obtained in the framework of mean field theory. In addition, our result supports $D_{lc}=2.5$ as the lower critical dimension for the model., Comment: 11 pages, 4 figures. To appear in PNAS

We have introduced a variational method to improve the computation of integrated correlation times in the Parallel Tempering Dynamics, obtaining a better estimate (a lower bound, at least) of the exponential correlation time. Using this determination of the correlation times, we revisited the problem of the characterization of the chaos in temperature in finite dimensional spin glasses by means of the study of correlations between different chaos indicators computed in the static and the correlation times of the Parallel Tempering dynamics. The sample-distribution of the characteristic time for the Parallel Tempering dynamics turns out to be fat-tailed and it obeys finite-size scaling., Comment: New appendix added. Version to appear in JSTAT

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)]. 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, non-linear susceptibilities, recently measured in glass-forming liquids, scale as powers of the same microscopic length., Comment: 6 pages, 4 figures

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., Comment: 5 pages, 5 figures

The unifying feature of glass formers (such as polymers, supercooled liquids, colloids, granulars, spin glasses, superconductors, ...) is a sluggish dynamics at low temperatures. Indeed, their dynamics is so slow that thermal equilibrium is never reached in macroscopic samples: in analogy with living beings, glasses are said to age. Here, we show how to relate experimentally relevant quantities with the experimentally unreachable low-temperature equilibrium phase. We have performed a very accurate computation of the non-equilibrium fluctuation-dissipation ratio for the three-dimensional Edwards-Anderson Ising spin glass, by means of large-scale simulations on the special-purpose computers Janus and Janus II. This ratio (computed for finite times on very large, effectively infinite, systems) is compared with the equilibrium probability distribution of the spin overlap for finite sizes. The resulting quantitative statics-dynamics dictionary, based on observables that can be measured with current experimental methods, could allow the experimental exploration of important features of the spin-glass phase without uncontrollable extrapolations to infinite times or system sizes., Comment: Version accepted for publication in PNAS. Reuploaded to fix a typo in the list of authors

Temperature chaos plays a role in important effects, like for example memory and rejuvenation, in spin glasses, colloids, polymers. We numerically investigate temperature chaos in spin glasses, exploiting its recent characterization as a rare-event driven phenomenon. The peculiarities of the transformation from periodic to anti-periodic boundary conditions in spin glasses allow us to conclude that temperature chaos is non-local: no bounded region of the system causes it. We precise the statistical relationship between temperature chaos and the free-energy changes upon varying boundary conditions., Comment: 15 pages, 8 figures. Version accepted for publication in JSTAT

We use finite size scaling to study Ising spin glasses in two spatial dimensions. The issue of universality is addressed by comparing discrete and continuous probability distributions for the quenched random couplings. The sophisticated temperature dependency of the scaling fields is identified as the major obstacle that has impeded a complete analysis. Once temperature is relinquished in favor of the correlation length as the basic variable, we obtain a reliable estimation of the anomalous dimension and of the thermal critical exponent. Universality among binary and Gaussian couplings is confirmed to a high numerical accuracy., Comment: 11 pages,7 figures

The propagation and divergence properties of beams carrying orbital angular momentum (OAM) play a crucial role in many applications. Here we present a general study on the divergence of optical beams with OAM. We show that the mean absolute value of the OAM imposes a lower bound on the value of the beam divergence. We discuss our results for two different definitions of the divergence, the so called rms or encircled-energy. The bound on the rms divergence can be expressed as a generalized uncertainty principle, with applications in long-range communication, microscopy and 2D quantum systems., Comment: RevTex, published version

We study the characterization of multipartite entanglement for the random states of an $n$-qbit system. Unable to solve the problem exactly we generalize it, changing complex numbers into real vectors with $N_c$ components (the original problem is recovered for $N_c=2$). Studying the leading diagrams in the large-$N_c$ approximation, we unearth the presence of a phase transition and, in an explicit example, show that the so-called entanglement frustration disappears in the large-$N_c$ limit., Comment: 12 pages, 15 figures

We revisited, by means of numerical simulations, the one dimensional bond diluted Levy Ising spin glasses outside the limit of validity of mean field theories. In these models the probability that two spins at distance $r$ interact (via a disordered interactions, $J_{ij}=\pm 1$) decays as $r^{-\rho}$. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for $\rho=5/3$ and $\rho=9/5$. We have obtained strong evidence for divergences of the previous observables at a non zero critical temperature. We discuss the behavior of the critical exponents, especially when approaching the value $\rho=2$, corresponding to a critical threshold beyond which the model has no phase transition. Finally, we numerically study the model right at the threshold value $\rho=2$., Comment: 9 pages and 16 figures

Recent theoretical advances offer an exact, first-principle theory of jamming criticality in infinite dimension as well as universal scaling relations between critical exponents in all dimensions. For packings of frictionless spheres near the jamming transition, these advances predict that nontrivial power-law exponents characterize the critical distribution of (i) small inter-particle gaps and (ii) weak contact forces, both of which are crucial for mechanical stability. The scaling of the inter-particle gaps is known to be constant in all spatial dimensions $d$ -- including the physically relevant $d=2$ and 3, but the value of the weak force exponent remains the object of debate and confusion. Here, we resolve this ambiguity by numerical simulations. We construct isostatic jammed packings with extremely high accuracy, and introduce a simple criterion to separate the contribution of particles that give rise to localized buckling excitations, i.e., bucklers, from the others. This analysis reveals the remarkable dimensional robustness of mean-field marginality and its associated criticality., Comment: 12 pages, 4 figures

We observe numerically the properties of the infinite-temperature inherent structures of m-component vector spin glasses in three dimensions. An increase of m implies a decrease of the amount of minima of the free energy, down to the trivial presence of a unique minimum. For little m correlations are small and the dynamics are quickly arrested, while for larger m low-temperature correlations crop up and the convergence is slower, to a limit that appears to be related with the system size., Comment: Version accepted in Phys. Rev. B, 10 pages, 11 figures

We present a numerical method to generate explicit realizations of the tree of states in mean-field spin glasses. The resulting study illuminates the physical meaning of the full replica symmetry breaking solution and provides detailed information on the structure of the spin-glass phase. A cavity approach ensures that the method is self-consistent and permits the evaluation of sophisticated observables, such as correlation functions. We include an example application to the study of finite-size effects in single-sample overlap probability distributions, a topic that has attracted considerable interest recently., Comment: Version accepted for publication in JSTAT

We perform equilibrium parallel-tempering simulations of the 3D Ising Edwards-Anderson spin glass in a field. A traditional analysis shows no signs of a phase transition. Yet, we encounter dramatic fluctuations in the behaviour of the model: Averages over all the data only describe the behaviour of a small fraction of it. Therefore we develop a new approach to study the equilibrium behaviour of the system, by classifying the measurements as a function of a conditioning variate. We propose a finite-size scaling analysis based on the probability distribution function of the conditioning variate, which may accelerate the convergence to the thermodynamic limit. In this way, we find a non-trivial spectrum of behaviours, where a part of the measurements behaves as the average, while the majority of them shows signs of scale invariance. As a result, we can estimate the temperature interval where the phase transition in a field ought to lie, if it exists. Although this would-be critical regime is unreachable with present resources, the numerical challenge is finally well posed., Comment: 42 pages, 19 figures. Minor changes and added figure (results unchanged)

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 ultrametricity are already well satisfied for the lattice sizes where numerical simulations are possible (i.e., up to $V = 32^3$ spins) and are, therefore, relevant at experimental scales. To this end we introduce a three-replica correlation function, which evinces the ultrametric properties of the system and is shown to scale in the same way as the overlap correlation function., Comment: 5 pages, 4 figures

We show in numerical simulations that a system of two coupled replicas of a binary mixture of hard spheres undergoes a phase transition in equilibrium at a density slightly smaller than the glass transition density for an unreplicated system. This result is in agreement with the theories that predict that such a transition is a precursor of the standard ideal glass transition. The critical properties are compatible with those of an Ising system. The relations of this approach to the conventional approach based on configurational entropy are briefly discussed., Comment: 5 pages, 3 figures, version accepted for publication in the Physical Review E

We study the flow of the renormalized model parameters obtained from a sequence of simple transformations of the 1D Anderson model with long-range hierarchical hopping. Combining numerical results with a perturbative approach for the flow equations, we identify three qualitatively different regimes at weak disorder. For a sufficiently fast decay of the hopping energy, the Cauchy distribution is the only stable fixed-point of the flow equations, whereas for sufficiently slowly decaying hopping energy the renormalized parameters flow to a delta peak fixed-point distribution. In an intermediate range of the hopping decay, both fixed-point distributions are stable and the stationary solution is determined by the initial configuration of the random parameters. We present results for the critical decay of the hopping energy separating the different regimes., Comment: 8 pages, 6 figures

We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to L=40 using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain Tc = 1.1019(29) for the critical temperature, \nu = 2.562(42) for the thermal exponent, \eta = -0.3900(36) for the anomalous dimension and \omega = 1.12(10) for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield \alpha = -5.69(13), \beta = 0.782(10) and \gamma = 6.13(11). We also compute several universal quantities at Tc., Comment: 9 pages, 5 figures

This paper describes the architecture, the development and the implementation of Janus II, a new generation application-driven number cruncher optimized for Monte Carlo simulations of spin systems (mainly spin glasses). This domain of computational physics is a recognized grand challenge of high-performance computing: the resources necessary to study in detail theoretical models that can make contact with experimental data are by far beyond those available using commodity computer systems. On the other hand, several specific features of the associated algorithms suggest that unconventional computer architectures, which can be implemented with available electronics technologies, may lead to order of magnitude increases in performance, reducing to acceptable values on human scales the time needed to carry out simulation campaigns that would take centuries on commercially available machines. Janus II is one such machine, recently developed and commissioned, that builds upon and improves on the successful JANUS machine, which has been used for physics since 2008 and is still in operation today. This paper describes in detail the motivations behind the project, the computational requirements, the architecture and the implementation of this new machine and compares its expected performances with those of currently available commercial systems., Comment: 28 pages, 6 figures

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 FPGAs, we have been able to reach times equivalent to 0.01 seconds 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., Comment: Version accepted in PRE. 23 pages, 23 figures

Temperature chaos has often been reported in literature as a rare-event driven phenomenon. However, this fact has always been ignored in the data analysis, thus erasing the signal of the chaotic behavior (still rare in the sizes achieved) and leading to an overall picture of a weak and gradual phenomenon. On the contrary, our analysis relies on a large-deviations functional that allows to discuss the size dependencies. In addition, we had at our disposal unprecedentedly large configurations equilibrated at low temperatures, thanks to the Janus computer. According to our results, when temperature chaos occurs its effects are strong and can be felt even at short distances., Comment: 5 pages, 5 figures

We present strong numerical evidence for the existence of a localization-delocalization transition in the eigenstates of the 1-D Anderson model with long-range hierarchical hopping. Hierarchical models are important because of the well-known mapping between their phases and those of models with short range hopping in higher dimensions, and also because the renormalization group can be applied exactly without the approximations that generally are required in other models. In the hierarchical Anderson model we find a finite critical disorder strength Wc where the average inverse participation ratio goes to zero; at small disorder W < Wc the model lies in a delocalized phase. This result is based on numerical calculation of the inverse participation ratio in the infinite volume limit using an exact renormalization group approach facilitated by the model's hierarchical structure. Our results are consistent with the presence of an Anderson transition in short-range models with D > 2 dimensions, which was predicted using renormalization group arguments. Our finding should stimulate interest in the hierarchical Anderson model as a simplified and tractable model of the Anderson localization transition which occurs in finite-dimensional systems with short-range hopping., Comment: 7 pages, 7 figures

A recent interesting paper [Yucesoy et al. Phys. Rev. Lett. 109, 177204 (2012), arXiv:1206:0783] compares the low-temperature phase of the 3D Edwards-Anderson (EA) model to its mean-field counterpart, the Sherrington-Kirkpatrick (SK) model. The authors study the overlap distributions P_J(q) and conclude that the two models behave differently. Here we notice that a similar analysis using state-of-the-art, larger data sets for the EA model (generated with the Janus computer) leads to a very clear interpretation of the results of Yucesoy et al., showing that the EA model behaves as predicted by the replica symmetry breaking (RSB) theory., Comment: Version accepted for publication in PRL. 1 page, 1 figure

We describe Janus, a massively parallel FPGA-based computer optimized for the simulation of spin glasses, theoretical models for the behavior of glassy materials. FPGAs (as compared to GPUs or many-core processors) provide a complementary approach to massively parallel computing. In particular, our model problem is formulated in terms of binary variables, and floating-point operations can be (almost) completely avoided. The FPGA architecture allows us to run many independent threads with almost no latencies in memory access, thus updating up to 1024 spins per cycle. We describe Janus in detail and we summarize the physics results obtained in four years of operation of this machine; we discuss two types of physics applications: long simulations on very large systems (which try to mimic and provide understanding about the experimental non-equilibrium dynamics), and low-temperature equilibrium simulations using an artificial parallel tempering dynamics. The time scale of our non-equilibrium simulations spans eleven orders of magnitude (from picoseconds to a tenth of a second). On the other hand, our equilibrium simulations are unprecedented both because of the low temperatures reached and for the large systems that we have brought to equilibrium. A finite-time scaling ansatz emerges from the detailed comparison of the two sets of simulations. Janus has made it possible to perform spin-glass simulations that would take several decades on more conventional architectures. The paper ends with an assessment of the potential of possible future versions of the Janus architecture, based on state-of-the-art technology., Comment: 19 pages, 3 figures

Spin glasses are a longstanding model for the sluggish dynamics that appears at the glass transition. However, spin glasses differ from structural glasses for a crucial feature: they enjoy a time reversal symmetry. This symmetry can be broken by applying an external magnetic field, but embarrassingly little is known about the critical behaviour of a spin glass in a field. In this context, the space dimension is crucial. Simulations are easier to interpret in a large number of dimensions, but one must work below the upper critical dimension (i.e., in d<6) in order for results to have relevance for experiments. Here we show conclusive evidence for the presence of a phase transition in a four-dimensional spin glass in a field. Two ingredients were crucial for this achievement: massive numerical simulations were carried out on the Janus special-purpose computer, and a new and powerful finite-size scaling method., Comment: 10 pages, 6 figures

A method is provided to compute the parameter exponent $\lambda$ 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 $\lambda$ in terms of third order coefficients of the action expansion or, equivalently, in term 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 Potts glass model, ROM, hard and soft-spin Sherrington-Kirkpatrick and p-spin models are presented., Comment: 4 pages, 1 figure