One-step Replica Symmetry Breaking Research Articles

Interplay between spin-glass clusters and geometrical frustration.

The presence of spin-glass (SG) order in highly geometrically frustrated systems is analyzed in a cluster SG model. The model considers infinite-range disordered interactions among cluster magnetic moments and the J(1)-J(2) model couplings between Ising spins of the same cluster. This model can introduce two sources of frustration: one coming from the disordered interactions and another coming from the J(1)-J(2) intracluster interactions (intrinsic frustration). The framework of one-step replica symmetry breaking is adopted to obtain a one-cluster problem that is exactly solved. As a main result we create phase diagrams of the temperature T versus intensity of the disorder J, where the paramagnetic-SG phase transition occurs at T(f) when T decreases for high-J values. For low-J values, the SG order is absent for antiferromagnetic clusters without intrinsic frustration. However, the SG order can be observed within the intracluster intrinsic frustration regime even for lower intensity of disorder. In particular, the results indicate that the presence of small clusters in geometrically frustrated antiferromagnetic systems can help stabilize the SG order within a weak disorder.

Extremal Optimization for Ground States of the Sherrington-Kirkpatrick Spin Glass with Levy Bonds

Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds J distributed according to a Levy distribution P(J) / 1/|J| 1+α ; |J| > 1; are investigated for a range of powers α. We determine ground state energy density variation with α and their finite-size corrections. We find that the energies attain universally the Parisi-energy of the SK as long as the second moment of P (J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ω SK = = 2/3 already well above α = 2, at which point it reaches a minimum.

Monte Carlo simulations of the three-dimensionalXYspin glass focusing on chiral and spin order

The ordering of the three-dimensional isotropic {\it XY} spin glass with the nearest-neighbor random Gaussian coupling is studied by extensive Monte Carlo simulations. To investigate the ordering of the spin and the chirality, we compute several independent physical quantities including the glass order parameter, the Binder parameter, the correlation-length ratio, the overlap distribution and the non-self-averageness parameter, {\it etc}, for both the spin-glass (SG) and the chiral-glass (CG) degrees of freedom. Evidence of the spin-chirality decoupling, {\it i.e.}, the CG and the SG order occurring at two separated temperatures, $0<T_{SG}<T_{CG}$, is obtained from the glass order parameter, which is fully corroborated by the Binder parameter. By contrast, the CG correlation-length ratio yields a rather pathological and inconsistent result in the range of sizes we studied, which may originate from the finite-size effect associated with a significant short-length drop-off of the spatial CG correlations. Finite-size-scaling analysis yields the CG exponents $\nu_{CG}=1.36^{+0.15}_{-0.37}$ and $\eta_{CG}=0.26^{+0.29}_{-0.26}$, and the SG exponents $\nu_{SG}=1.22^{+0.26}_{-0.06}$ and $\eta_{SG}=-0.54^{+0.24}_{-0.52}$. The obtained exponents are close to those of the Heisenberg SG, but are largely different from those of the Ising SG. The chiral overlap distribution and the chiral Binder parameter exhibit the feature of a continuous one-step replica-symmetry breaking (1RSB), consistently with the previous reports. Such a 1RSB feature is again in common with that of the Heisenberg SG, but is different from the Ising one, which may be the cause of the difference in the CG critical properties from the Ising SG ones despite of a common $Z_2$ symmetry.

Spin-glass freezing in Kondo-lattice compounds in the presence of a random and a transverse magnetic field

The present work studies the effects of a random magnetic field on the competition among Kondo effect, spin glass (SG) phase and ferromagnetic (FE) order in disordered cerium systems such as CeNi1−xCux. A Kondo lattice model is used with an intersite disordered interaction Jij between localized Ising spins in the presence of a transverse magnetic field Γ and a longitudinal random magnetic field. The disorders introduced by Jij and the random field follow Gaussian distributions. They are treated within the one-step replica symmetry breaking procedure. The results suggest that the presence of a random field affects strongly the FE and SG phases. It can disrupt the FE order completely and reduces the SG phase region. It can also introduce an independent spin phase that can coexist within the Kondo regime at high Kondo interaction Jk. On the other hand, the Γ field introduces quantum spin fluctuations able to destroy the magnetic order by decreasing the critical temperature towards a quantum critical point. In addition, the Kondo states can only occur at higher Jk values as Γ increases.

A theoretical study of the spin glass-Kondo-magnetic disordered alloys in the presence of a random field

We study here the influence of a random applied magnetic field on the competition between the Kondo effect, the spin glass phase and a ferromagnetic order in disordered cerium systems such as CeNi1−xCux. The model used here takes an intrasite Kondo coupling and an intersite random coupling; both the intersite random coupling and the random magnetic field are described within the Sherrington-Kirkpatrick model and the one-step replica symmetry breaking procedure is also used here. We present phase diagrams giving Temperature versus the Kondo exchange parameter and the random magnetic field makes decrease particularly the importance of the spin glass and ferromagnetic phases.

Inverse freezing in a cluster Ising spin-glass model with antiferromagnetic interactions

Inverse freezing is analyzed in a cluster spin-glass (SG) model that considers infinite-range disordered interactions between magnetic moments of different clusters (intercluster interaction) and short-range antiferromagnetic coupling J(1) between Ising spins of the same cluster (intracluster interaction). The intercluster disorder J is treated within a mean-field theory by using a framework of one-step replica symmetry breaking. The effective model obtained by this treatment is computed by means of an exact diagonalization method. With the results we build phase diagrams of temperature T/J versus J(1)/J for several sizes of clusters n(s) (number of spins in the cluster). The phase diagrams show a second-order transition from the paramagnetic phase to the SG order at the freezing temperature T(f) when J(1)/J is small. The increase in J(1)/J can then destroy the SG phase. It decreases T(f)/J and introduces a first-order transition. In addition, inverse freezing can arise at a certain range of J(1)/J and large enough n(s). Therefore, the nontrivial frustration generated by disorder and short-range antiferromagnetic coupling can introduce inverse freezing spontaneously.

Pure Glass in Finite Dimensions

Pure glass is defined as a thermodynamic phase in which typical equilibrium particle configurations have macroscopic overlaps with one of some special irregular configurations. By employing 128 types of artificial molecules, a pure glass model is constructed in the cubic lattice.

Next nearest neighbour Ising models on random graphs

This paper develops results for the next nearest neighbour Ising model on random graphs. Besides being an essential ingredient in classic models for frustrated systems, second neighbour interactions arise naturally in several applications, such as the colour diversity problem and graphical games. We demonstrate ensembles of random graphs, including regular connectivity graphs, that have a periodic variation of free energy, with either the ratio of nearest to next nearest couplings, or the mean number of nearest neighbours. When the coupling ratio is integer then paramagnetic phases can be found at zero temperature. This is shown to be related to the locked or unlocked nature of the interactions. For anti-ferromagnetic couplings, spin glass phases are demonstrated at low temperature. The interaction structure is formulated as a factor graph, the solution on a tree is developed. The replica symmetric and energetic one-step replica symmetry breaking solution is developed using the cavity method. We calculate within these frameworks the phase diagram and demonstrate the existence of dynamical transitions at zero temperature for cases of anti-ferromagnetic coupling on regular and inhomogeneous random graphs.

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.

Vector Precoding for Gaussian MIMO Broadcast Channels: Impact of Replica Symmetry Breaking

The “replica method” of statistical physics is employed for the large-system analysis of vector precoding for the Gaussian multiple-input multiple-output broadcast channel. The transmitter comprises a linear front-end combined with nonlinear precoding, minimizing transmit energy by means of input alphabet relaxation. For the common discrete lattice-based relaxation, the problem violates replica symmetry and a replica symmetry breaking (RSB) ansatz is taken. The limiting empirical distribution of the precoder's output and the limiting transmit energy are derived for one-step RSB. Particularizing to a “zero-forcing” (ZF) linear front-end, a decoupling result is derived. For discrete lattice-based relaxations, the impact of RSB is demonstrated for the transmit energy. The spectral efficiencies of the aforementioned precoding methods are compared to linear ZF and Tomlinson-Harashima precoding (THP). Focusing on quaternary phase shift-keying (QPSK), significant performance gains of both lattice and convex relaxations are revealed for medium to high signal-to-noise ratios (SNRs) when compared to linear ZF precoding. THP is shown to be outperformed as well. Comparing certain lattice-based relaxations for QPSK against a convex counterpart, the latter is found to be superior for low and high SNRs but slightly inferior for medium SNRs in terms of spectral efficiency.

Ground states of the Sherrington–Kirkpatrick spin glass with Levy bonds

Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds J distributed according to a Levy distribution P(J) ∝ 1/|J|1+α, |J| > 1, are investigated for a range of powers α. The results are compared with those for the Sherrington–Kirkpatrick (SK) model, where bonds are Gaussian distributed. In particular, we determine the variation of the ground-state energy densities with α, their finite-size corrections, measure their fluctuations, and analyze the local field distribution. We find that the energies themselves at infinite system size attain universally the Parisi-energy of the SK as long as the second moment of P(J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ω SK = 2/3 already well above α = 2, at which point it reaches a minimum. This result is justified with a speculative calculation of a random energy model with Levy bonds. The exponent ρ that describes the variations of the ground-state energy fluctuations with system size decays monotonically from its SK value for decreasing α and appears to vanish at α = 1.

Random, thermodynamic and inverse first-order transitions in the Blume–Capel spinglass

A class of model systems undergoing a glass transition with inversion of the fluid and glassyphase in temperature is investigated in order to qualitatively characterize the so-calledinverse freezing phenomenon occurring in some complex glassy polymeric systems such asmethylcellulose. The leading model we analyze is the spherical mean-field approximation of a spin-1 modelwith p-body quenched disordered interaction. Depending on temperature and chemical potentialthe system is found in a paramagnetic or in a glassy phase and the transition between thesephases can be of a different nature. In the given conditions inverse freezing occurs. Asp = 2 theglassy phase is replica-symmetric and the transition is always continuous in the phase diagram. Forp > 2 the exact solution for the glassy phase is obtained by the one-step replica symmetrybreaking ansatz. Different scenarios arise for both the dynamic and the thermodynamictransitions. These include (i) the usual random first-order transition (Kauzmann-like)preceded by a dynamic transition, typical of mean-field glasses, (ii) a thermodynamicfirst-order transition with phase coexistence and latent heat and (iii) a regime of inversionof static and dynamic transition lines. In the latter case a thermodynamic stable glassyphase, with zero configurational entropy, is dynamically accessible from the paramagneticphase. Crossover between different transition regimes is analyzed by means of replicasymmetry breaking theory and a detailed study of the complexity and of the stability ofthe static solution is performed throughout the space of external thermodynamicparameters.

Replica analysis of the generalized p-spin interaction glass model

We investigate the stability of replica symmetry breaking solutions in generalized p-spin models. It is shown that the kind of the transition to the one-step replica symmetry breaking state depends not only on the presence or absence of the reflection symmetry of the generalized ‘spin’-operators but on the number of interacting operators and their individual characteristics.

Inverse freezing in the cluster fermionic spin glass model with a transverse field

The inverse freezing (IF) transition is studied with a cluster fermionicIsing spin glass model in the presence of a transverse magnetic fieldΓ. The model considers a disordered infinite-range interaction among magneticmoments of the clusters with a short-range ferromagnetic intracluster couplingJ0. The intercluster disorder is treated within a mean-field theory by using theframework of one-step replica symmetry breaking and the static approximation. Thistreatment results in an effective model that is computed by means of an exactdiagonalization method. In particular, in this fermionic problem, the chemical potentialμ andthe Γ introduce cluster charge fluctuations and quantum spin flipping,respectively. The results indicate an IF transition for a certain range ofμ. When the short-range interaction increases, the IF is still found. However, the intensity ofJ0 can affectthe charge fluctuation, introducing a second-order IF transition. On the other hand, the enhancementof Γ destroys the inverse freezing.

Learning from Stochastic Rules by Spherical Perceptrons under Finite Temperature –Optimal Temperature and Asymptotic Learning Curve–

In the problem of learning under external disturbance, there is a possibility that the existence of some tolerance or flexibility in the system weakens the effect of noise and helps the system to perform more efficiently. In a previous letter, we gave one example of such phenomena in learning from stochastic rules by spherical perceptrons adopting the Gibbs algorithm using statistical mechanical methods. By the replica method, we showed that, in the output noise model, there exists an optimal temperature at which the generalization error takes its minimum for the stable replica symmetric (RS) solution. On the other hand, for other types of noise including input noise, it was shown that no such temperature exists up to the one-step replica symmetry breaking (1RSB) solution. That is, it was shown that for the asymptotic region of a large number of training sets, the RS solution becomes unstable, and the asymptotic behavior is determined by the 1RSB solution, The asymptotic expressions for learning curves we...

Replica symmetry breaking, complexity and spin representation in the generalized random energy model

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 the 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 the nature of the higher step RSB, we generalize the notion of complexity developed for the one-step RSB to the higher step and demonstrate how the GREM is characterized by the generalized complexity. In addition, we propose a novel mean-field spin-glass model with a hierarchical structure, which is equivalent to the GREM at a certain limit. We also show that the same hierarchical structure can be implemented to other mean-field spin models than the GREM. Such models with hierarchy exhibit phase transitions of multiple steps in common.

Cluster fermionic Ising spin glass model with a transverse field

The fermionic Ising spin glass (SG) model in the presence of a transverse magnetic field Γ is studied within a cluster mean field theory. The model considers an infinite-range interaction among magnetic moments of clusters with a short-range ferromagnetic intracluster coupling J 0 . The spin operators are written as a bilinear combination of fermionic operators. In this quantum SG model, the intercluster disorder is treated by using a framework of one-step replica symmetry breaking (RSB) within the static approximation. The effective intracluster interaction is then computed by means of an exact diagonalization method. Results for several values of cluster size n s , Γ and J 0 are presented. For instance, the specific heat can show a broad maximum at a temperature T ∗ above the freezing temperature T f , which is characterized by the intercluster RSB. The difference between T ∗ and T f is enhanced by Γ , which suggests that the quantum effects can increase the ratio T ∗ / T f . Phase diagrams ( T versus Γ ) show that the critical temperature T f ( Γ ) decreases for any values of n s and J 0 when Γ increases until it reaches a quantum critical point at some value of Γ c .

Thermodynamics of the Lévy spin glass

We investigate the Lévy glass, a mean-field spin-glass model with power-law distributed couplings characterized by a divergent second moment. By combining extensively many small couplings with a spare random backbone of strong bonds the model is intermediate between the Sherrington-Kirkpatrick and the Viana-Bray models. A truncated version where couplings smaller than some threshold ε are neglected can be studied within the cavity method developed for spin glasses on locally treelike random graphs. By performing the limit ε→0 in a well-defined way we calculate the thermodynamic functions within replica symmetry and determine the de Almeida-Thouless line in the presence of an external magnetic field. Contrary to previous findings we show that there is no replica-symmetric spin-glass phase. Moreover we determine the leading corrections to the ground-state energy within one-step replica symmetry breaking. The effects due to the breaking of replica symmetry appear to be small in accordance with the intuitive picture that a few strong bonds per spin reduce the degree of frustration in the system.

An extended ensemble Monte Carlo study of a lattice glass model

The thermodynamic transition to a glass phase from a liquid phase is studied in a simple lattice glass model both on a random graph and in two dimensions by employing an extended ensemble Monte Carlo method, which enables us to explore its equilibrium behavior in a high density regime. The results for the random-graph case support the scenario of an equilibrium glass transition as predicted from one-step replica symmetry breaking schemes via the cavity method. Numerical evidence for the glass transition is also found for the system in two dimensions. The equation of state measured indicates a discontinuous change in the compressibility at the transition point and the overlap distribution function has a sharp double peak structure in the glassy phase. They suggest that this singularity originates from a one-step replica symmetry breaking.

Optimal location of sources in transportation networks

We consider the problem of optimizing the locations of source nodes in transportationnetworks. A reduction of the fraction of surplus nodes induces a glassy transition. Incontrast to most constraint satisfaction problems involving discrete variables, ourproblem involves continuous variables which lead to cavity fields in the form offunctions. The one-step replica symmetry breaking (1RSB) solution involves solving astable distribution of functionals, which is in general infeasible. In this paper, weobtain small closed sets of functional cavity fields and demonstrate how functionalrecursions are converted to simple recursions of probabilities which make the 1RSBsolution feasible. The physical results in the replica symmetric (RS) and the 1RSBframeworks are thus derived and the stabilities of the RS and 1RSB solutions areexamined.