We investigate the p-state mean-field model of the
Potts glass ($2\le p \le 4$) below the continuous phase transition to a glassy phase. We find that apart from a solution with a first hierarchical level of replica-symmetry breaking (1RSB), locally stable close to the transition point, there is a continuous full replica-symmetry breaking (FRSB) solution. The latter is marginally stable and has a higher free energy than the former. We argue that the true equilibrium is reached only by FRSB, being globally thermodynamically homogeneous, whereas 1RSB is only locally homogeneous.
The low-temperature instability of one-step replica symmetry breaking (1RSB) phase in 3-state Potts spin glass is obtained explicitly. The temperature of the instability is higher than the temperature where the 1RSB entropy becomes negative. The conjecture of the possibility of the low-temperature full RSB is supported.
We perform the replica symmetry breaking (RSB) in the vicinity of the point of instability of the replica symmetric solution in the model of axial quadrupolar glass. It is shown that the solution with the first stage RSB is stable against the second stage RSB. Although there is no reflection symmetry the 1RSB solution bifurcates continuously from the RS one. These facts mean that our model can not be associated with one of two classes usually considered in spin glass theory.
We define modulated replica symmetry breaking (RSB)-schemes which combine tree- and wave-like structures. A modulated scheme and unmodulated RSB are applied at 1-step level to a semiconductor model with antiferromagnetic Korenblit-Shender interaction. By comparison of the free energies we find evidence that a T=0 phase transition in the ferrimagnetic phase leads to a transition between the different RSB-schemes. An embedding factor of Parisi block matrices with sublattice-asymmetrical size is employed as a new variational parameter in the modulated scheme.
We prove the impossibility of recent attempts to decouple the Replica Symmetry Breaking (RSB) picture for finite-dimensional spin glasses from the existence of many thermodynamic (i.e., infinite-volume) pure states while preserving another signature RSB feature --- space filling relative domain walls between different finite-volume states. Thus revisions of the notion of pure states cannot shield the RSB picture from the internal contradictions that rule out its physical correctness in finite dimensions at low temperature in large finite volume.
Bethe Lattice Spin Glasses (BLSG) are models with finite connectivity which undergo a Replica Symmetry Breaking (RSB) phase transition in field, even at zero temperature. The order parameter describing this phase is a function related to the non-trivial organization of the states. We compute numerically the spin glass order parameter at zero temperature in BLSG with coordination number $z=3$ and couplings $J=\pm1$, near the transition. The method is based on a universal formula which relates the order parameter to the joint probability distribution of the energy gaps of the lowest lying states…
We investigate the replica symmetry broken (RSB) phase of spin glass (SG) models in a random field defined on Bethe lattices at zero temperature. From the properties of the RSB solution we deduce a closed equation for the extreme values of the cavity fields. This equation turns out not to depend on the parameters defining the RSB, and it predicts that the spontaneous RSB does not take place homogeneously on the whole system. Indeed, there exist spins having the same effective local field in all local ground states, exactly as in the replica symmetric (RS) phase, while the spontaneous RSB manif…
We investigate the nature of the Bose glass phase of the disordered Bose-Hubbard model in $d>2$ and demonstrate the existence of a glass-like replica symmetry breaking (RSB) order parameter in terms of particle number fluctuations. Starting from a strong-coupling expansion around the atomic limit, we study the instability of the Mott insulator towards the formation of a Bose glass. We add some infinitesimal RSB, following the Parisi hierarchical approach in the most general form, and observe its flow under the momentum-shell renormalization group scheme. We find a new fixed point with one-step…
We study the Blume-Emery-Griffiths spin glass model in presence of an attractive coupling between real replicas, and evaluate the effective potential as a function of the density overlap. We find that there is a region, above the first order transition of the model, where metastable states with a large density overlap exist. The line where these metastable states appear should correspond to a purely dynamical transition, with a breaking of ergodicity. Differently from what happens in p-spin glasses, in this model the dynamical transition would not be the precursor of a 1-step RSB transition, b…
A Potts glass model proposed by Nishimori and Stephen[H. Nishimori and M. J. Stephen, Phys. Rev. B 27, 5644 (1983)] is analyzed by means of the replica mean field theory. This model is a discrete model, has a gauge symmetry, and is called the Potts gauge glass model. By comparing the present results with the results of the conventional Potts glass model, we find the coincidences and differences between the models. We find a coincidence that the property for the Potts glass phase in this model is coincident with that in the conventional model at the mean field level. We find a difference that, …
In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in a strong sense in the mean field limit. We also suggest how the computation could be generalized in an approximate way to finite dimensi…
We study chaotic size dependence of the low temperature correlations in the SK spin glass. We prove that as temperature scales to zero with volume, for any typical coupling realization, the correlations cycle through every spin configuration in every fixed observation window. This cannot happen in short-ranged models as there it would mean that every spin configuration is an infinite-volume ground state. Its occurrence in the SK model means that the commonly used `modified clustering' notion of states sheds little light on the RSB solution of SK, and conversely, the RSB solution sheds little l…
A continuous 3-state Potts model with an analog of spherical constraints is proposed and is shown to have an exact solution in the case of infinite-ranged interactions. "Spherical" 3-state Potts spin glass model is solved using the known properties of a large random matrix. For this model the results are identical to those obtained by the replica approach for replica symmetric solution.
We consider the p-spin spherical spin-glass model in the presence of an external magnetic field as a general example of a mean-field system where a one step replica symmetry breaking (1-RSB) occurs. In this context we compute the complexity of the Thouless-Anderson-Palmer states, performing a quenched computation. We find what is the general connection between this method and the standard static 1-RSB one, formulating a clear mapping between the parameters used in the two different calculations. We also perform a dynamical analysis of the model, by which we confirm the validity of our results.
We introduce a new disordered system, the Super-Potts model, which is a more frustrated version of the Potts glass. Its elementary degrees of freedom are variables that can take M values and are coupled via pair-wise interactions. Its exact solution on a completely connected lattice demonstrates that for large enough M it belongs to the class of mean-field systems solved by a one step replica symmetry breaking Ansatz. Numerical simulations by the parallel tempering technique show that in three dimensions it displays a phenomenological behaviour similar to the one of glass-forming liquids. The …
We show that for a particular choice of the coupling parameters the Ashkin-Teller spin-glass neural network model with the Hebb learning rule and one condensed pattern yields the same thermodynamic properties as the four-state anisotropic Potts-glass neural network model. This equivalence is not seen at the level of the Hamiltonians.
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…
In this paper, we investigate the finite connectivity spin-glass problem. Our work is focused on the expansion around the point of infinite connectivity of the free energy of a spin glass on a graph with Poissonian distributed connectivity: we are interested to study the first-order correction to the infinite connectivity result for large values or the connectivity $z$. The same calculations for one and two replica symmetry breakings were done in previous works; the result for the first-order correction was divergent in the limit of zero temperature and it was suggested that it was an artifact…
We perform numerical simulations, including parallel tempering, on the Potts glass model with binary random quenched couplings using the JANUS application-oriented computer. We find and characterize a glassy transition, estimating the location of the transition and the value of the critical exponents. We show that there is no ferromagnetic transition in a large temperature range around the glassy critical temperature. We also compare our results with those obtained recently on the "random permutation" Potts glass.
We use Monte Carlo simulations to investigate the dynamical properties of the infinite range 10 states Potts glass. By analyzing the spin autocorrelation function for system sizes up to N=2560, we show that strong finite size effects are present around the predicted dynamical transition temperature. The autocorrelation function shows strong self-averaging at high temperatures, whereas close to the dynamical transition they show the lack of self-averaging.
In this short note, we present supporting evidence for the replica symmetric approach to the random bond q-state Potts models. The evidence is statistically strong enough to reject the applicability of the Parisi replica symmetry breaking scheme to this class of models. The test we use is a generalization of one formerly proposed by Dotsenko et al. and consists in measuring scaling laws of disordered-averaged moments of the spin-spin correlation functions. Numerical results, obtained via Monte Carlo simulations for several values of q, are shown to be in fair agreement with the replica symmetr…
Expansion for the free energy functionals of the Potts spin glass models with 3, 4 and 5 states up to the fourth order in $δq_{αβ}$ around the replica symmetric solution (RS) is investigated using a special quadrupole-like representation. The temperature dependence of the 1RSB order parameters is obtained in the vicinity of the point $T=T_c$ where the RS solution becomes unstable. The crossover from continuous to jumpwise behavior with increasing of number of states is derived analytically. The comparison is made of the free energy expansion for the Potts spin glass with that for other models.
We present the results of Monte Carlo simulations of two different 10-state Potts glasses with random nearest neighbor interactions on a simple cubic lattice. In the first model the interactions come from a \pm J distribution and in the second model from a Gaussian one, and in both cases the first two moments of the distribution are chosen to be equal to J_0=-1 and Delta J=1. At low temperatures the spin autocorrelation function for the \pm J model relaxes in several steps whereas the one for the Gaussian model shows only one. In both systems the relaxation time increases like an Arrhenius law…
We report the equivalence of the ``Glassy Potts model'', recently introduced by Marinari et al. and the ``Chiral Potts model'' investigated by Nishimori and Stephen. Both models do not exhibit any spontaneous magnetization at low temperature, differently from the ordinary glass Potts model. The phase transition of the glassy Potts model is easily interpreted as the spin glass transition of the ordinary random Potts model.
We compute the Functional Renormalization Group (FRG) disorder- correlator function R(v) for d-dimensional elastic manifolds pinned by a random potential in the limit of infinite embedding space dimension N. It measures the equilibrium response of the manifold in a quadratic potential well as the center of the well is varied from 0 to v. We find two distinct scaling regimes: (i) a "single shock" regime, v^2 ~ 1/L^d where L^d is the system volume and (ii) a "thermodynamic" regime, v^2 ~ N. In regime (i) all the equivalent replica symmetry breaking (RSB) saddle points within the Gaussian variati…
The one-step replica symmetry breaking (RSB) is used to study a two-sublattice fermionic infinite-range Ising spin glass (SG) model in a transverse field $Γ$. The problem is formulated in a Grassmann path integral formalism within the static approximation. In this model, a parallel magnetic field $H$ breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions $V$ increase the difference between the RSB solutions of each …
A constructive scheme for determining pure states (clusters) at very low temperature in the 3-spins glass model on a random lattice is provided, in full agreement with Parisi's one step replica symmetry breaking (RSB) scheme. Proof is based on the analysis of an exact decimation procedure. When the number c of couplings per spin is smaller than some critical value c_d, all spins are eliminated at the end of decimation (RS phase). In the range c_d<c<c_s, a reduced Hamiltonian is left; each ground state (GS) of the latter is a "seed" from which a cluster of GS of the original Hamiltonian can be …
The spin-glass $q$-state Potts model on $d$-dimensional diamond hierarchical lattices is investigated by an exact real space renormalization group scheme. Above a critical dimension $d_l(q)$ for $q > 2$, the coupling constants probability distribution flows to a low-temperature \emph{strange attractor} or to the high-temperature paramagnetic fixed point, according to the temperature is below or above the critical temperature $T_c(q,d)$. The strange attractor was investigated considering four initial different distributions for $q = 3$ and $d = 5$ presenting strong robustness in shape and tempe…
We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q>=4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between …
The two-dimensional q-state Potts model is subjected to a Z_q symmetric disorder that allows for the existence of a Nishimori line. At q=2, this model coincides with the +/- J random-bond Ising model. For q>2, apart from the usual pure and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q=3, tracing out the phase diagram and…
We obtain phase diagrams of regular and irregular finite connectivity spin-glasses. Contact is firstly established between properties of the phase diagram and the performances of low density parity check codes (LDPC) within the Replica Symmetric (RS) ansatz. We then study the location of the dynamical and critical transition of these systems within the one step Replica Symmetry Breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that, away from the Nishimori line, in the low temperature region, the location of …
It was pointed out by de Arcangelis et al. [Europhys. Lett. 14 (1991), 515] that the correct understanding of the percolation phenomenon of the Fortuin-Kasteleyn cluster in the Edwards-Anderson model is important since a dynamical transition, which is characterized by a parameter called the Hamming distance or damage, and the percolation transition are related to a transition for a signal propagating between spins. We show analytically the percolation thresholds of the Fortuin-Kasteleyn cluster for a Potts gauge glass model, which is an extended model of the Edwards-Anderson model, on random g…
uch experimental and theoretical efforts have been devoted in the past twenty years to search for a genuine thermodynamic reentrant phase transition from a ferromagnetic to either a paramagnetic or spin glass phase in disordered ferromagnets. So far, no real system or theoretical model of a short-range spin glass system has been shown to convincingly display such a reentrant transition. We present here results from Migdal-Kadanoff real-space renormalization-group calculations that provide for the first time strong evidence for ferromagnetic to paramagnetic reentrance in Potts spin glasses on h…
The chiral spin-glass Potts system with q=3 states is studied in d=2 and 3 spatial dimensions by renormalization-group theory and the global phase diagrams are calculated in temperature, chirality concentration p, and chirality-breaking concentration c, with determination of phase chaos and phase-boundary chaos. In d=3, the system has ferromagnetic, left-chiral, right-chiral, chiral spin-glass, and disordered phases. The phase boundaries to the ferromagnetic, left- and right-chiral phases show, differently, an unusual, fibrous patchwork (microreentrances) of all four (ferromagnetic, left-chira…
We use Monte Carlo simulations to study the static and dynamical properties of a Potts glass with infinite range Gaussian distributed exchange interactions for a broad range of temperature and system size up to N=2560 spins. The results are compatible with a critical divergence of the relaxation time tau at the theoretically predicted dynamical transition temperature T_D, tau \propto (T-T_D)^{-Δ} with Delta \approx 2. For finite N a further power law at T=T_D is found, tau(T=T_D) \propto N^{z^\star} with z^\star \approx 1.5 and for T>T_D dynamical finite-size scaling seems to hold. The order p…
Potts glasses are prototype models that have been used to understand the structural glass transition. However, in finite space dimensions a glass transition remains to be detected in the 10-state Potts glass. Using a one-dimensional model with long-range power-law interactions we present evidence that a glass transition below the upper critical dimension can exist for short-range systems at low enough temperatures. Gaining insights into the structural glass transition for short-range systems using spin models is thus potentially possible, yet difficult.
In this paper we study the critical behaviour of the fully-connected p-colours Potts model at the dynamical transition. In the framework of Mode Coupling Theory (MCT), the time autocorrelation function displays a two step relaxation, with two exponents governing the approach to the plateau and the exit from it. Exploiting a relation between statics and equilibrium dynamics which has been recently introduced, we are able to compute the critical slowing down exponents at the dynamical transition with arbitrary precision and for any value of the number of colours p. When available, we compare our…
We introduce a Potts model with quenched, frustrated disorder, that enjoys of a gauge symmetry that forbids spontaneous magnetization, and allows the glassy phase to extend from $T_c$ down to T=0. We study numerical the 4 dimensional model with $q=4$ states. We show the existence of a glassy phase, and we characterize it by studying the probability distributions of an order parameter, the binder cumulant and the divergence of the overlap susceptibility. We show that the dynamical behavior of the system is characterized by aging.
Finite-size effects in the mean-field Ising spin glass and the mean-field three-state Potts glass are investigated by Monte Carlo simulations. In the thermodynamic limit, each model is known to exhibit a continuous phase transition into the ordered state with a full and a one-step replica-symmetry breaking (RSB), respectively. In the Ising case, Binder parameter g calculated for various finite sizes remains positive at any temperature and crosses at the transition point, while in the Potts case g develops a negative dip without showing a crossing in the g>0 region. By contrast, non-self averag…
We study the critical behavior of the short-range p-state Potts spin glass in three and four dimensions using Monte Carlo simulations. In three dimensions, for p = 3, a finite-size scaling analysis of the correlation length shows clear evidence of a transition to a spin-glass phase at T_c = 0.273(5) for a Gaussian distribution of interactions and T_c = 0.377(5) for a bimodal distribution. These results indicate that the lower critical dimension of the 3-state Potts glass is below three. By contrast, the correlation length of the ten-state (p = 10) Potts glass in three dimensions remains small …
A field-theory approach is used to investigate the ''spin-glass effects'' on the critical behaviour of systems with weak temperature-like quenched disorder. The renormalization group (RG) analysis of the effective Hamiltonian of a model with replica symmetry breaking (RSB) potentials of a general type is carried out in the two-loop approximation. The fixed-point (FP) stability, recently found within the one-step RSB RG treatment, is further explored in terms of replicon eigenvalues. We find that the traditional FPs, which are usually considered to describe the disorder-induced universal critic…
We study the random energy model with a hierarchical structure known as the generalized random energy model (GREM). In contrast to the original analysis by the microcanonical ensemble formalism, we investigate the GREM by the canonical ensemble formalism in conjunction with the replica method. In this analysis, spin-glass-order parameters are defined for respective hierarchy level, and all possible patterns of replica symmetry breaking (RSB) are taken into account. As a result, we find that the higher step RSB ansatz is useful for describing spin-glass phases in this system. For investigating …
We studied the phase transition of the $\pm J$ Heisenberg model with and without a random anisotropy on four dimensional lattice $L\times L\times L\times (L+1)$ $(L\leq 9)$. We showed that the Binder parameters $g(L,T)$'s for different sizes do not cross even when the anisotropy is present. On the contrary, when a strong anisotropy exists, $g(L,T)$ exhibits a steep negative dip near the spin-glass phase transition temperature $T_{\rm SG}$ similarly to the $p-$state infinite-range Potts glass model with $p \geq 3$, in which the one-step replica-symmetry-breaking (RSB) occurs. We speculated that…
We present the results of Monte Carlo simulations of two different Potts glass models with short range random interactions. In the first model a \pm J-distribution of the bonds is chosen, in the second model a Gaussian distribution. In both cases the first two moments of the distribution are chosen to be J_0=-1, Delta J=+1, so that no ferromagnetic ordering of the Potts spins can occur. We find that for all temperatures investigated the spin glass susceptibility remains finite, the spin glass order parameter remains zero, and that the specific heat has only a smooth Schottky-like peak. These r…
We study a 7-state Potts glass model in three dimensions with first, second, and third neighbor interactions with a bimodal distribution of couplings by Monte Carlo simulations. Our results show the existence of a spin-glass transition at a finite temperature T_c, a discontinuous jump of an order parameter at T_c without latent heat, and a non-trivial structure of the order-parameter distribution below T_c. They are compatible with a one-step replica symmetry breaking.
The fermionic Ising spin glass models in a transverse field are investigated in a Grassmann path integral formalism. The Parisi's scheme of one-step replica symmetry breaking (RSB) is used within the static ansatz. This formalism has already been applied in a theory in which $m$ (Parisi's block-size parameter) is taken as a constant (PRB 72, 014447 (2005)). Now, it is extended to consider $m$ as a variational parameter. In this case, the results show that RSB is present when $T\to0$, in which the system is driven by quantum fluctuations.
In a recent letter Marinari et al [Phys. Rev. Lett. 81, 1698 (1998)] introduced a new method to study spin glass transitions and argued that by probing replica symmetry (RS) as opposed to time reversal symmetry (TRS), their method unambiguously shows that replica symmetry breaking (RSB) occurs in short-range spin glasses. In this comment we show that while the new method is indeed useful for studying transitions in systems where TRS is absent (such as the p-spin model studied by them), the conclusion that it shows the existence of RSB in short-range spin glasses is wrong.
Similarities between fragile glasses and spin glasses (SG) suggest the study of frustrated spin model to understand the complex dynamics of glasses above the glass transition. We consider a frustrated spin model with Ising spins and s-state Potts spins both with and without disorder. We study the two models by Monte Carlo simulations in two dimensions. The Potts spins mimic orientational degrees of freedom and the coupled frustrated Ising spins take into account for frustrating effects like the geometrical hindrance. We show that in this model dynamical transitions and crossovers are related t…
We expand asymptotically mean-field solutions of the $p<4$ Potts glass with various levels of replica-symmetry breaking below the transition temperature to the glassy phase. We find that the ordered phase is degenerate and solutions with one hierarchy of spin replicas and with the full continuous replica-symmetry breaking coexist for $p> p^{*} \approx 2.82$. The latter emerges immediately with the instability of the replica-symmetric one. Apart from these two solutions there exists also a succession of unstable states converging to the solution with the continuous replica-symmetry breaking tha…
We report on the image reconstruction (IR) problem by making use of the random chiral q-state Potts model, whose Hamiltonian possesses the same gauge invariance as the usual Ising spin glass model. We show that the pixel representation by means of the Potts variables is suitable for the gray-scale level image which can not be represented by the Ising model. We find that the IR quality is highly improved by the presence of a glassy term, besides the usual ferromagnetic term under random external fields, as very recently pointed out by Nishimori and Wong. We give the exact solution of the infini…
We investigate the critical properties of the four-state commutative random permutation glassy Potts model in three and four dimensions by means of Monte Carlo simulation and of a finite size scaling analysis. Thanks to the use of a field programmable gate array we have been able to thermalize a large number of samples of systems with large volume. This has allowed us to observe a spin-glass ordered phase in d=4 and to study the critical properties of the transition. In d=3, our results are consistent with the presence of a Kosterlitz-Thouless transition, but we cannot exclude transient effect…
We investigate the phase structure of the random-field Ising model with a bimodal random field distribution. Our aim is to test for the possibility of an equilibrium spin-glass phase, and for replica symmetry breaking (RSB) within such a phase. We study a low-temperature region where the spin-glass phase is thought to occur, but which has received little numerical study to date. We use the exchange Monte-Carlo technique to acquire equilibrium information about the model, in particular the $P(q)$ distribution and the spectrum of eigenvalues of the spin-spin correlation matrix (which tests for t…
The fixed point structure of the 2D 3-state random-bond Potts model with a bimodal ($\pm$J) distribution of couplings is for the first time fully determined using numerical renormalization group techniques. Apart from the pure and T=0 critical fixed points, two other non-trivial fixed points are found. One is the critical fixed point for the random-bond, but unfrustrated, ferromagnet. The other is a bicritical fixed point analogous to the bicritical Nishimori fixed point found in the random-bond frustrated Ising model. Estimates of the associated critical exponents are given for the various fi…
We investigate by means of Monte Carlo simulations the fully connected p-state Potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the, exactly known, behavior of the system in the thermodynamic limit. Using p=10 we are able to study the equilibrium dynamics for system sizes as large as N=2560. We find that the static quantities, such as the energy, the entropy, the spin glass susceptibility as well as the distribution of the order parameter P(q) show very strong finite size effects. From P(q) we calculate the forth order …
