Step Replica Symmetry Breaking Research Articles

Pseudogaps and charge band in the Parisi solution of insulating and superconducting electronic spin glasses at arbitrary fillings

We report progress in understanding the fermionic Ising spin glass with arbitrary filling. A crossover from a magnetically disordered single band phase via two intermediate bands just below the freezing temperature to a 3-band structure at still lower temperatures - beyond an almost random field instability - is shown to emerge in the magnetic phase. An attempt is made to explain the exact solution in terms of a quantum Parisi phase. A central nonmagnetic band is found and seen to become sharply separated at T = 0 by gaps from upper and lower magnetic bands. The gap sizes tend towards zero as the number of replica symmetry breaking steps increases towards infinity. In an extended model, the competition between local pairing superconductivity and spin glass order is discussed.

On Ward-Takahashi identities for the Parisi spin glass

The introduction of ``small permutations'' allows us to derive Ward-Takahashi identities for the spin-glass, in the Parisi limit of an infinite number of steps of replica symmetry breaking. The first identities express the emergence of a band of Goldstone modes. The next identities relate components of (the Replica Fourier Transformed) 3-point function to overlap derivatives of the 2-point function (inverse propagator). A jump in this last function is exhibited, when its two overlaps are crossing each other, in the special simpler case where one of the cross-overlaps is maximal.

Glass transition and effective potential in the hypernetted chain approximation

We study the glassy transition for simple liquids in the hypernetted chain (HNC) approximation by means of an effective potential recently introduced. Integrating the HNC equations for hard spheres, we find a transition scenario analogous to that of the long range disordered systems with ``one step replica symmetry breaking''. Our result agree qualitatively with Monte Carlo simulations of three dimensional hard spheres.

Universality classes for extreme-value statistics

The low temperature physics of disordered systems is governed by the statistics of extremely low energy states. It is thus rather important to discuss the possible universality classes for extreme value statistics. We compare the usual probabilistic classification to the results of the replica approach. We show in detail that one class of independent variables corresponds exactly to the so-called one step replica symmetry breaking solution in the replica language. This universality class holds if the correlations are sufficiently weak. We discuss the relation between the statistics of extremes and the problem of Burgers turbulence in decay.

Variational Approach to Interfaces in Random Media: Negative Variances and Replica Symmetry Breaking

A Gaussian variational approximation is often used to study interfaces in random media. By considering the 1+1 dimensional directed polymer in a random medium, it is shown here that the variational Ansatz typically leads to a negative variance of the free energy. The situation improves by taking into account more and more steps of replica symmetry breaking. For infinite order breaking the variance is zero (i.e. subextensive). This situation is reminiscent of the negative entropies in mean field spin glass models, which were also eliminated by considering infinite order replica symmetry breaking.

Off-Equilibrium Fluctuation-Dissipation Relation in Fragile Glasses

In this Letter we investigate the approach to equilibrium in fragile glasses by doing numerical simulations of binary mixtures. We study the diffusion of particles and the response to an external driving force. We find evidence for the validity of the Cugliandolo-Kurchan off-equilibrium fluctuation-dissipation relation. The results agree with the predictions of one step replica symmetry breaking and the dependence of the breakpoint parameter on temperature coincides with that found in simple generalized spin glass models.

Statistical mechanics of the multi-constraint continuous knapsack problem

We apply the replica analysis established by Gardner to the multi-constraint continuous knapsack problem,which is one of the linear programming problems and a most fundamental problem in the field of operations research (OR). For a large problem size, we analyse the space of solution and its volume, and estimate the optimal number of items to go into the knapsack as a function of the number of constraints. We study the stability of the replica symmetric (RS) solution and find that the RS calculation cannot estimate the optimal number of items in knapsack correctly if many constraints are required.In order to obtain a consistent solution in the RS region,we try the zero entropy approximation for this continuous solution space and get a stable solution within the RS ansatz.On the other hand, in replica symmetry breaking (RSB) region, the one step RSB solution is found by Parisi's scheme. It turns out that this problem is closely related to the problem of optimal storage capacity and of generalization by maximum stability rule of a spherical perceptron.

Stability of the Mézard-Parisi Solution for Random Manifolds

The eigenvalues of the Hessian associated with random manifolds are constructed for the general case of $R$ steps of replica symmetry breaking. For the Parisi limit $R\to\infty$ (continuum replica symmetry breaking) which is relevant for the manifold dimension $D<2$, they are shown to be non negative.

A Puzzle on Fluctuations of Weights in Spin Glasses

In certain mean field models for spin glasses there occurs a one step replica symmetry breaking pattern. As an example of general $1/N$-corrections in such systems, the fluctuations in the internal energy are calculated. For this specific quantity the outcome is known from the specific heat. It is shown that the correct result can be derived by assuming that the both the height and the location of the breakpoint fluctuate. This effect enlarges the commonly considered space of allowed fluctuations of the order parameter in loop calculations for short range spin glasses of this type. The phenomenon does not occur in spin glasses with infinite order replica symmetry breaking.

Replica symmetry breaking in the random replicant model

We study the statistical mechanics of a model describing the coevolution of species interacting in a random way. We find that at high competition replica symmetry is broken. We solve the model in the approximation of one step replica symmetry breaking and we compare our findings with accurate numerical simulations.

Replica-symmetry breaking in the critical behaviour of the random ferromagnet

We study the critical properties of the weakly disordered $p$-component random Heisenberg ferromagnet. It is shown that if the specific heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions $D=4-\e$, which are usually considered as describing the disorder-induced universal critical behavior, are {\it unstable} with respect to replica symmetry breaking (RSB) potentials as found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have the structure known as the 1 step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.

Block diagonalizing ultrametric matrices

The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica symmetry breaking. The residual symmetry, remaining after the breaking of permutation symmetry between replicas, allows us to bring all ultrametric matrices to a block diagonal form by a common similarity transformation. A large number of these blocks are, in fact, of size 1 times 1, i.e. in a vast sector the transformation actually diagonalizes the matrix. In the other sectors we end up with blocks of size (R+1) times (R+1) where R is the number of replica symmetry breaking steps. These blocks cannot be further reduced without giving more information, in addition to ultrametric symmetry, about the matrix. Similar results for the inverse of a generic ultrametric matrix are also derived.

Replica symmetry breaking in attractor neural network models

The phenomenon of replica symmetry breaking is investigated for the retrieval phases of Hopfield-type network models. The basic calculation is done for the generalized version of the standard model introduced by Horner [1] and by Perez-Vicente and Amit [2] which can exhibit low mean levels of neural activity. For a mean activity $\bar a =1/2$ the Hopfield model is recovered. In this case, surprisingly enough, we cannot confirm the well known one step replica symmetry breaking (1RSB) result for the storage capacity which was presented by Crisanti, Amit and Gutfreund [3] ($\alpha_c^{\hbox{\mf 1RSB}}\simeq 0.144$). Rather, we find that 1RSB- and 2RSB-Ans\"atze yield only slightly increased capacities as compared to the replica symmetric value ($\alpha_c^{\hbox{\mf 1RSB}}\simeq 0.138\,186$ and $\alpha_c^{\hbox{\mf 2RSB}}\simeq 0.138\,187 $ compared to $\alpha_c^{\hbox{\mf RS}}\simeq 0.137\,905$), significantly smaller also than the value $\alpha_c^{\hbox{\mf sim}} = 0.145\pm 0.009$ reported from simulation studies. These values still lie within the recently discovered reentrant phase [4]. We conjecture that in the infinite Parisi-scheme the reentrant behaviour disappears as is the case in the SK-spin-glass model (Parisi--Toulouse-hypothesis). The same qualitative results are obtained in the low activity range.

Symmetry breaking in nonmonotonic neural networks

The optimal performance of a nonmonotonic neural network is studied by the replica method. In analogy to what happens in multi-layered networks, the authors show that replica symmetry breaking (RSB) is required. The distribution of the patterns stabilities, the correlations in the distribution of the internal representation and the optimal capacity per synapse ( alpha c approximately=4.8) are computed with one step of RSB.

Dynamics of manifolds in random media: the selfconsistent Hartree approximation

We discuss a dynamical description of fluctuating manifolds in random media, using an approximation which becomes exact in the «spherical limit» N→∞, N being the dimension of the transversal fluctuations. The system behaviour is studied in detail for the case of quenched disorder with short range correlations. In the static limit, we recover solutions known from a replica calculation with one step replica symmetry breaking. In the dynamical formulation it turns out that this replica symmetry breaking is related to broken ergodicity for temperatures below a critical point. While the static behaviour in the low temperature phase superficially seems to be not very different from that of the high temperature phase, the dynamic treatment shows that the manifolds get dynamically localized below the critical temperature. The value we find for this «freezing temperature» is higher than the corresponding result from replica theory. Introducing an additional external quadratic potential, we find that the character of the phase transition changes with increasing coupling strength μ. There is a tricitical point μ * , for μ μ * it becomes a second order one. We give the phase diagram for this generalized model and discuss the dynamical behaviour in the different regimes

Replica symmetry breaking of the Ising spin-glass with finite connectivity

The authors consider the Ising spin-glass on a lattice with finite connectivity (=M+1). Using the recent method of large connectivity expansions, the free energies at finite temperatures are evaluated numerically at first-step replica symmetry breaking (RSB). The 1/M expansion at finite temperature diverges as T0 as in the replica symmetric case but is well behaved for a larger range. The 1/M expansion at zero temperature is also calculated at higher steps of RSB. An expression for the free-energy expansion is derived for an arbitrary step of RSB up to order 1/M. Explicit numerical values at second-step RSB are obtained. From the results, they speculate that the divergence in the finite-temperature expansion might disappear for the exact infinite-step RSB solution. They also compare their results with the simulation results of graph bipartitioning.