Replica Symmetry Breaking Phase Research Articles

About the de Almeida–Thouless line in neural networks

In this work we present a rigorous and straightforward method to detect the onset of the instability of replica-symmetric theories in information processing systems, which does not require a full replica analysis as in the method originally proposed by de Almeida and Thouless for spin glasses. The method is based on an expansion of the free-energy obtained within one-step of replica symmetry breaking (RSB) around the RS value. As such, it requires solely continuity and differentiability of the free-energy and it is robust to be applied broadly to systems with quenched disorder. We apply the method to the Hopfield model and to neural networks with multi-node Hebbian interactions, as case studies. In the appendices we test the method on the Sherrington–Kirkpatrick and the Ising P-spin models, recovering the AT lines known in the literature for these models, as a special limit, which corresponds to assuming that the transition from the RS to the RSB phase can be obtained by varying continuously the order parameters. Our method provides a generalization of the AT approach, which does not rely on this limit and can be applied to systems with discontinuous phase transitions, as we show explicitly for the spherical P-spin model, recovering the known RS instability line.

Transient replica symmetry breaking in Brillouin random fiber lasers

Replica symmetry breaking (RSB), as a featured phase transition between paramagnetic and spin glass state in magnetic systems, has been predicted and validated among random laser-based complex systems, which involves numerous random modes interplayed via gain competition and exhibits disorder-induced frustration for glass behavior. However, the dynamics of RSB phase transition involving micro-state evolution of a photonic complex system have never been well investigated. Here, we report experimental evidence of transient RSB in a Brillouin random fiber laser (BRFL)-based photonic system through high-resolution unveiling of random laser mode landscape based on heterodyne technique. Thanks to the prolonged lifetime of activated random modes in BRFLs, an elaborated mapping of time-dependent statistics of the Parisi overlap parameter in both time and frequency domains was timely resolved, attributing to a compelling analogy between the transient RSB dynamics and the random mode evolution. These findings highlight that BRFL-based systems with the flexible harness of a customized photonic complex platform allow a superb opportunity for time-resolved transient RSB observation, opening new avenues in exploring fundamentals and application of complex systems and nonlinear phenomena.

Intensity pseudo-localized phase in the glassy random laser

Evidence of an emergent pseudo-localized phase characterizing the low-temperature replica symmetry breaking phase of the complex disordered models for glassy light is provided in the mode-locked random laser model. A pseudo-localized phase corresponds to a state in which the intensity of light modes is neither equipartited among all modes nor strictly condensed on few of them. Such a hybrid phase, recently characterized as a finite size effect in other models, such as the discrete non-linear Schrödinger equation, in the low temperature phase of the glassy random laser appears to be robust in the limit of large size.

Belief propagation on the random k-SAT model

Corroborating a prediction from statistical physics, we prove that the belief propagation message passing algorithm approximates the partition function of the random k-SAT model well for all clause/variable densities and all inverse temperatures for which a modest absence of long-range correlations condition is satisfied. This condition is known as “replica symmetry” in physics language. From this result we deduce that a replica symmetry breaking phase transition occurs in the random k-SAT model at low temperature for clause/variable densities below but close to the satisfiability threshold.

The Ising Antiferromagnet and Max Cut on Random Regular Graphs

The Ising antiferromagnet is an important statistical physics model with close connections to the {\sc Max Cut} problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the {\sc Max Cut} of random regular graphs predicted by Zdeborova and Boettcher [Journal of Statistical Mechanics 2010]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound.

Replica symmetry breaking in coherent and incoherent random lasing modes.

We investigate intensity fluctuations of a weakly scattering optofluidic random laser having coherent and incoherent emission fractions. The coherent part comprises random spikes, whereas the incoherent part forms a broad pedestal in the emission spectra. Evaluating the fractional ratio of the coherent and incoherent parts of the emission, a replica symmetry breaking phase transition is observed independently in both coherent and incoherent parts of the intensity. Also, the incoherent component has higher non-zero correlation values compared to those of the coherent part, implying a larger contribution to mode coupling. Moreover, survival function analysis reveals a significant contribution of the incoherent part on determining the decay profile of lasing intensity.

Evaluation of Pearson correlation coefficient and Parisi parameter of replica symmetry breaking in a hybrid electronically addressable random fiber laser.

The hybrid electronically addressable random (HEAR) laser is a novel type of random fiber laser that presents the remarkable property of selection of the fiber section with lasing emission. Here we present a joint analysis of the correlations between intensity fluctuations at distinct wavelengths and replica symmetry breaking (RSB) behavior of the HEAR laser. We introduce a modified Pearson coefficient that simultaneously comprises both the Parisi overlap parameter and standard Pearson correlation coefficient. Our results highlight the contrast between the correlations and presence or not of RSB phenomenon in the spontaneous emission behavior well below threshold, replica-symmetric ASE regime slightly below threshold, and RSB phase with random lasing emission above threshold. In particular, in the latter we find that the onset of RSB behavior is accompanied by a stochastic dynamics of the lasing modes, leading to competition for gain intertwined with correlation and anti-correlation between modes in this complex photonic phase.

Dynamical mean-field theory and aging dynamics

Dynamical mean-field theory (DMFT) replaces the many-body dynamical problem with one for a single degree of freedom in a thermal bath whose features are determined self-consistently. By focusing on models with soft disordered p-spin interactions, we show how to incorporate the mean-field theory of aging within DMFT. We study cases with only one slow time-scale, corresponding statically to the one-step replica symmetry breaking phase, and cases with an infinite number of slow time-scales, corresponding statically to the full replica symmetry breaking (FRSB) phase. For the former, we show that the effective temperature of the slow degrees of freedom is fixed by requiring critical dynamical behavior on short time-scales, i.e. marginality. For the latter, we find that aging on an infinite number of slow time-scales is governed by a stochastic equation where the clock for dynamical evolution is fixed by the change of the effective temperature, hence obtaining a dynamical derivation of the stochastic equation at the basis of the FRSB phase. Our results extend the realm of the mean-field theory of aging to all situations where DMFT holds.

Jamming and replica symmetry breaking of weakly disordered crystals

We discuss the physics of crystals with small polydispersity near the jamming transition point. For this purpose, we introduce an effective single-particle model taking into account the nearest neighbor structure of crystals. The model can be solved analytically by using the replica method in the limit of large dimensions. In the absence of polydispersity, the replica symmetric solution is stable until the jamming transition point, which leads to the standard scaling of perfect crystals. On the contrary, for finite polydispersity, the model undergoes the full replica symmetry breaking (RSB) transition before the jamming transition point. In the RSB phase, the model exhibits the same scaling as amorphous solids near the jamming transition point. These results are fully consistent with the recent numerical simulations of crystals with polydispersity. The simplicity of the model also allows us to derive the scaling behavior of the vibrational density of states that can be tested in future experiments and numerical simulations.

Random laser spectroscopy and replica symmetry breaking phase transitions in a solvent-rich polymer thin film waveguide

We present the statistical analysis of random lasing intensity fluctuations in a 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminos-styryl)-4H-pyran (DCM) doped polyvinyl alcohol (PVA) thin film waveguide subjected to a constant heat treatment. The microscopic changes occurring in the density of the polymer thin film during the various stages of solvent evaporation are probed using the changes in the statistics of random laser (RL) emission intensities. In the solvent rich wet film, stronger RL emission intensity fluctuations are observed compared to the dried film, leading to a relatively slower decay of the survival function distribution of the emission intensities for a mode at the gain maxima of the averaged spectrum. The mode interactions in the wet and dried films, studied using covariance between lasing mode intensities, are found to be different. Further, replica symmetry breaking studies indicate the changes in the mode interactions with the microscopic modifications in the system during solvent evaporation at a constant pump energy above the lasing threshold. The statistical analyses of the RL emission intensity fluctuations are proposed as a spectroscopic tool to probe material properties.

One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses.

The de Almeida-Thouless (AT) line is the phase boundary in the temperature-magnetic field plane of an Ising spin glass at which a continuous (i.e., second-order) transition from a paramagnet to a replica-symmetry-breaking (RSB) phase occurs, according to mean-field theory. Here, using field-theoretic perturbative renormalization group methods on the Bray-Roberts reduced Landau-Ginzburg-type theory for a short-range Ising spin glass in space of dimension d, we show that at nonzero magnetic field the nature of the corresponding transition is modified as follows: (a) For d-6 small and positive, with increasing field on the AT line, first, the ordered phase just below the transition becomes the so-called one-step RSB, instead of the full RSB that occurs in mean-field theory; the transition on the AT line remains continuous with a diverging correlation length. Then at a higher field, a tricritical point separates the latter transition from a quasi-first-order one, that is one at which the correlation length does not diverge, and there is a jump in part of the order parameter, but no latent heat. The location of the tricritical point tends to zero as d→6^{+}. (b) For d≤6, we argue that the quasi-first-order transition could persist down to arbitrarily small nonzero fields, with a transition to full RSB still expected at lower temperature. Whenever the quasi-first-order transition occurs, it is at a higher temperature than the AT transition would be for the same field, preempting it as the temperature is lowered. These results may explain the reported absence of a diverging correlation length in the presence of a magnetic field in low-dimensional spin glasses in some simulations and in high-temperature series expansions. We also draw attention to the similarity of the "dynamically frozen" state, which occurs at temperatures just above the quasi-first-order transition, and the "metastate-average state" of the one-step RSB phase, and discuss the issue of the number of pure states in either.

The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

The ferromagnetic XY model on sparse random graphs in a randomly oriented field is analyzed via the belief propagation algorithm. At variance with the fully connected case and with the random field Ising model on the same topology, we find strong evidence of a tiny region with replica symmetry breaking (RSB) in the limit of very low temperatures. This RSB phase is robust against different choices of the external field direction, while it rapidly vanishes when increasing the graph mean degree, the temperature or the directional bias in the external field. The crucial ingredients to have such a RSB phase seem to be the continuous nature of vector spins, mostly preserved by the -invariant random field, and the strong spatial heterogeneity, due to graph sparsity. We also uncover that the ferromagnetic phase can be marginally stable despite the presence of the random field. Finally, we study the proper correlation functions approaching the critical points to identify the ones that become more critical.

Replica Symmetry Breaking in FRET‐Assisted Random Laser Based on Electrospun Polymer Fiber

AbstractSpin‐glass theory has been widely introduced to describe the statistical behaviors in complex physical systems. By analogy between disorder photonics and other complex systems, the glassy behavior, especially the replica symmetry breaking (RSB) phenomenon, has been observed in random lasers. However, previous studies only analyzed the statistical properties of the random laser systems with single gain material. Here, the first experimental evidence of the glassy behavior in a random laser with complex energy level structure is reported. This novel random laser is demonstrated based on the electrospun polymer fibers with the assistance of Förster resonance energy transfer (FRET). The electrospinning technology employed in the experiment herein promises high‐volume production of random laser devices with multiple energy levels, enabling the comprehensive investigation of lasing properties in multi‐energy level random laser system. Clear paramagnetic phase and spin‐glass phase are observed in the FRET‐assisted random laser under different pump energies. The RSB phase transition is verified to occur at the laser threshold, which is robust among the random lasers with different donor–acceptor ratio. The finding of RSB in FRET‐assisted random laser provides a new statistical analysis method toward the laser system with complex energy level, for example, quantum cascade laser.

Charting the Replica Symmetric Phase

Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous `cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al.: PNAS 2007]. In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the $k$-XORSAT model and the diluted $k$-spin model for even $k$. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention [Decelle et al.: Phys. Rev. E 2011].

Replica symmetry breaking in trajectory space for the trap model

We study the localization in the one-dimensional trap model in terms of statistical mechanics of trajectories. By numerically investigating overlap between trajectories of two particles on a common disordered potential, we find that there is a phase transition in the path ensemble. We characterize the low temperature phase as a replica symmetry breaking phase in trajectory space.

Parisi Formula, Disorder Chaos and Fluctuation for the Ground State Energy in the Spherical Mixed p-Spin Models

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

Some properties of the phase diagram for mixed p-spin glasses

In this paper we study the Parisi variational problem for mixed $p$-spin glasses with Ising spins. Our starting point is a characterization of Parisi measures whose origin lies in the first order optimality conditions for the Parisi functional, which is known to be strictly convex. Using this characterization, we study the phase diagram in the temperature-external field plane. We begin by deriving self-consistency conditions for Parisi measures that generalize those of de Almeida and Thouless to all levels of Replica Symmetry Breaking (RSB) and all models. As a consequence, we conjecture that for all models the Replica Symmetric (RS) phase is the region determined by the natural analogue of the de Almeida-Thouless condition. We show that for all models, the complement of this region is in the RSB phase. Furthermore, we show that the conjectured phase boundary is exactly the phase boundary in the plane less a bounded set. In the case of the Sherrington-Kirkpatrick model, we extend this last result to show that this bounded set does not contain the critical point at zero external field.

Complex spherical 2 + 4 spin glass: A model for nonlinear optics in random media

A disordered mean field model for multimode laser in open and irregular cavities is proposed and discussed within the replica analysis. The model includes the dynamics of the mode intensity and accounts also for the possible presence of a linear coupling between the modes, due, e.g., to the leakages from an open cavity. The complete phase diagram, in terms of disorder strength, source pumping and non-linearity, consists of four different optical regimes: incoherent fluorescence, standard mode locking, random lasing and the novel spontaneous phase locking. A replica symmetry breaking phase transition is predicted at the random lasing threshold. For a high enough strength of non-linearity, a whole region with nonvanishing complexity anticipates the transition, and the light modes in the disordered medium display typical discontinuous glassy behavior, i.e., the photonic glass has a multitude of metastable states that corresponds to different mode-locking processes in random lasers. The lasing regime is still present for very open cavities, though the transition becomes continuous at the lasing threshold.

Replica Plefka expansion of Ising systems

The replica extension theory has been developed for the purpose of investigating spin glasssystems in the replica symmetry breaking phase. The replica extension theory is a replicatheory for a specific sample and has the scheme of the one-step replica symmetry breakingansatz. In this paper, using the Plefka expansion, we propose the perturbative expansion ofGibbs free energies of Ising systems with the replica extension theory, and we derive theexplicit form of the second- and third-order approximations. Our Plefka expansionsystematically provides perturbative approximations taking the one-step replica symmetrybreaking into account. In some numerical experiments, we show that our perturbativeapproximations are quantitatively superior to conventional perturbative approximationsthat do not take the one-step replica symmetry breaking into account in the spin glassphase.

Statistical and algebraic analysis of a family of random Boolean equations

The statistical and algebraic properties of a family of random Boolean equations arestudied in this paper. Based on studying the mechanism of influence propagation fromsome variable fixed to 1, the giant strongly connected component and magnetization ofsolutions are investigated, which exhibit the splitting phenomenon of solution space and aferromagnetic transition. Furthermore, by analyzing the semi-group property of thesolution space and the influence of propagation from some variable fixed to 0, the scaleof generating elements is calculated, which undergoes linear, polynomial andexponential phases. Compared with the analysis by statistical mechanics, it is suggestedthat these different phases of algebraic complexity correspond to the structuralcomplexity of the solution space as replica symmetry, one-step replica symmetrybreaking and further-step replica symmetry breaking phases. It is supposed that thestructural complexity of solution space can be interpreted from the viewpoint ofalgebra.