Replica Formalism Research Articles

Dynamics of randomly branched polymers: configuration averages and solvable models.

Treating the relaxation dynamics of an ensemble of random hyperbranched macromolecules in dilute solution represents a challenge even in the framework of Rouse-type approaches, which focus on generalized Gaussian structures (GGSs). The problem is that one has to average over a large class of realizations of molecular structures, and that each molecule undergoes its own dynamics. We show that a replica formalism allows to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum. Interestingly, for a specific probability distribution of the spring strengths of the GGSs, the integral equation takes a particularly simple form. Given that several dynamical observables, such as the mechanical moduli G'(omega) and G"(omega), as well as the averaged monomer displacement <Y(t)> are relatively simple functions of the eigenvalues, we can use the obtained spectra to compute the corresponding averaged dynamical forms. Comparing the results obtained from this approach and from extensive diagonalizations of hyperbranched GGSs we find a very good agreement.

Infinite-range Ising spin glass with a transverse field under the static approximation

In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of ${C}_{H}/T$ near the glass transition temperature ${T}_{g},$ associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.

An approximation scheme for the density of states of the Laplacian on random graphs

In a number of recent papers, the spectral properties of the Laplacian on randomly connected graphs have been studied within the replica formalism. In this letter we show how the replica formalism can be unravelled and we find an approximation for calculating the density of states which substantially simplifies the numerical resolution of the equations obtained, whilst giving results in excellent accord with the exact solution.

Adsorption of a fluid in an aerogel: Integral equation approach

We present a theoretical study of the phase diagram and the structure of a fluid adsorbed in high-porosity aerogels by means of an integral-equation approach combined with the replica formalism. To simulate a realistic gel environment, we use an aerogel structure factor obtained from an off-lattice diffusion-limited cluster–cluster aggregation process. The predictions of the theory are in qualitative agreement with the experimental results, showing a substantial narrowing of the gas–liquid coexistence curve (compared to that of the bulk fluid), associated with weak changes in the critical density and temperature. The influence of the aerogel structure (nontrivial short-range correlations due to connectedness, long-range fractal behavior of the silica strands) is shown to be important at low fluid densities.

Learning to coordinate in a complex and nonstationary world.

We study analytically and by computer simulations a complex system of adaptive agents with finite memory. Borrowing the framework of the minority game and using the replica formalism we show the existence of an equilibrium phase transition as a function of the ratio between the memory lambda and the learning rates Gamma of the agents. We show that, starting from a random configuration, a dynamic phase transition also exists, which prevents agents from reaching optimal coordination. Furthermore, in a nonstationary environment, we show by numerical simulations that the phase transition becomes discontinuous.

Imaginary-time replica formalism study of a quantum sphericalp-spin-glass model

We study a quantum extension of the spherical $p$-spin-glass model using the imaginary-time replica formalism. We solve the model numerically and we discuss two analytical approximation schemes that capture most of the features of the solution. The phase diagram and the physical properties of the system are determined in two ways: by imposing the usual conditions of thermodynamic equilibrium and by using the condition of marginal stability. In both cases, the phase diagram consists of two qualitatively different regions. If the transition temperature is higher than a critical value $T^{\star}$, quantum effects are qualitatively irrelevant and the phase transition is {\it second} order, as in the classical case. However, when quantum fluctuations depress the transition temperature below $T^{\star}$, the transition becomes {\it first order}. The susceptibility is discontinuous and shows hysteresis across the first order line, a behavior reminiscent of that observed in the dipolar Ising spin-glass LiHo$_x$Y$_{1-x}$F$_4$ in an external transverse magnetic field. We discuss in detail the thermodynamics and the stationary dynamics of both states. The spectrum of magnetic excitations of the equilibrium spin-glass state is gaped, leading to an exponentially small specific heat at low temperatures. That of the marginally stable state is gapless and its specific heat varies linearly with temperature, as generally observed in glasses at low temperature. We show that the properties of the marginally stable state are closely related to those obtained in studies of the real-time dynamics of the system weakly coupled to a quantum thermal bath. Finally, we discuss a possible application of our results to the problem of polymers in random media.

Statistical physics and practical training of soft-committee machines

Equilibrium states of large layered neural networks with differentiable activation function and a single, linear output unit are investigated using the replica formalism. The quenched free energy of a student network with a very large number of hidden units learning a rule of perfectly matching complexity is calculated analytically. The system undergoes a first order phase transition from unspecialized to specialized student configurations at a critical size of the training set. Computer simulations of learning by stochastic gradient descent from a fixed training set demonstrate that the equilibrium results describe quantitatively the plateau states which occur in practical training procedures at sufficiently small but finite learning rates.

Frustrated Blume-Emery-Griffiths model

A generalised integer S Ising spin glass model is analysed using the replica formalism. The bilinear couplings are assumed to have a Gaussian distribution with ferromagnetic mean . Incorporation of a quadrupolar interaction term and a chemical potential leads to a richer phase diagram with transitions of first and second order. The first order transition may be interpreted as a phase separation, and contrary to what has been argued previously, it persists in the presence of disorder. Finally, the stability of the replica symmetric solution with respect to fluctuations in replica space is analysed, and the transition lines are obtained both analytically and numerically.

Dynamics versus replicas in the random field Ising model

In a previous article we have shown, within the replica formalism, that the conventional picture of the random field Ising model breaks down, due to the effect of singularities in the interactions between fields involving several replicas below dimension eight. In the zero-replica limit, several coupling constants have thus to be considered, instead of just one. As a result we found that there is no stable fixed point in the vicinity of dimension six. It is natural to reconsider the problem in a dynamical framework, which does not require replicas, although the equilibrium properties should be recovered in the large time limit. Singularities in the zero-replica limit are a priori not visible in a dynamical picture. In this note we show that in fact new interactions are also generated in the stochastic approach. Similarly these interactions are found to be singular below dimension eight. These critical singularities require the introduction of a time origin t 0 at which initial data are given. The dynamical properties are thus dependent upon the waiting time. It is shown here that one can indeed find a complete correspondence between the equilibrium singularities in the limit at n = 0, and the singularities in the dynamics when the initial time t 0 goes to minus infinity, with n replaced by −1/ t 0. There is thus complete coherence between the two approaches.

Phase diagram of randomly polymerized membrane

Using a replica formalism, a generalization of a recent mean field model corresponding to the observed wrinkling transition in randomly polymerized membranes is presented. In this model we study the effects of global fluctuations of the surface normals to the flat membrane, which can be introduced by a random local field. In absence of these global fluctuations, we show that, the model exhibits both continuous and discontinuous transitions between flat and wrinkled phases, contrary to what has been predicted by Bensimon et al. and Attal et al. Phase diagrams both in replica symmetry and in breaking of replica symmetry in sense of Almeida and Thouless are given. We have also investigated the effects of global fluctuations on the replica symmetry phase diagram. We show that, the wrinkled phase is favored and the flat phase is unstable. For large global fluctuations, the transition between wrinkled and flat phases becomes first order.

Generalizing with perceptrons in the case of structured phase- and pattern-spaces

We investigate the influence of different kinds of structures on the learning behaviour of a perceptron performing a classification task defined by a teacher rule. The underlying pattern distribution is permitted to have spatial correlations. The prior distribution for the teacher coupling vectors itself is assumed to be nonuniform. Thus, classification tasks of quite different difficulty are included. As learning algorithms we discuss Hebbian, Gibbs, and Bayesian learning with different priors, using methods from statistics and the replica formalism. We find that the Hebb rule is quite sensitive to the structure of the actual learning problem, failing asymptotically in most cases. In contrast, the behaviour of the more sophisticated methods of Gibbs and Bayes learning is influenced by the spatial correlations only in an intermediate regime of , where specifies the size of the training set. In view of the Bayesian case, we show how enhanced prior knowledge improves the performance.

Multiscale analysis of hierarchical landscapes

The Statistical Mechanics calculation of the Gibbs-Boltzmann partition function does not give all the necessary information for a complete analysis of a rugged energy hypersurface. We show instead, that the partition function of a conveniently defined system composed of a number of replicas constrained to lie at specific distances provides such a description. We further sketch how the usual Replica formalism can be used to calculate these more complicated functions. As a by-product we show that in this way unstable solutions in the usual Replica method acquire a new meaning.

Multifractality and percolation in the coupling space of perceptrons

The coupling space of perceptrons with continuous as well as with binary weights gets partitioned into a disordered multifractal by a set of p=\ensuremath{\gamma}N random input patterns. The multifractal spectrum f(\ensuremath{\alpha}) can be calculated analytically using the replica formalism. The storage capacity and the generalization behavior of the perceptron are shown to be related to properties of f(\ensuremath{\alpha}) which are correctly described within the replica symmetric Ansatz. Replica symmetry breaking is interpreted geometrically as a transition from percolating to nonpercolating cells. The existence of empty cells gives rise to singularities in the multifractal spectrum. The analytical results for binary couplings are corroborated by numerical studies.

Vapnik-Chervonenkis entropy of the spherical perceptron

Perceptron learning of randomly labeled patterns is analyzed using a Gibbs distribution on the set of realizable labelings of the patterns. The entropy of this distribution is an extension of the Vapnik-Chervonenkis (VC) entropy, reducing to it exactly in the limit of infinite temperature. The close relationship between the VC and Gardner entropies can be seen within the replica formalism.

Replica Symmetry Breaking and the Kuhn-Tucker Cavity Method in Simple and Multilayer Perceptrons

Within a Kuhn-Tucker cavity method introduced in a former paper, we study optimal stability learning for situations, where in the replica formalism the replica symmetry may be broken, namely (i) the case of a simple perceptron above the critical loading, and (ii) the case of two-layer AND-perceptrons, if one learns with maximal stability. We find that the deviation of our cavity solution from the replica symmetric one in these cases is a clear indication of the necessity of replica symmetry breaking. In any case the cavity solution tends to underestimate the storage capabilities of the networks.

A Field Theory for Finite-Dimensional Site-Disordered Spin Systems.

We present a new field theoretic approach for finite-dimensional site-disordered spin systems by introducing the notion of grand canonical disorder, where the number of spins in the system is random but quenched. We perform the simplest nontrivial analysis of this field theory by using the variational replica formalism. We explicitly discuss a three-dimensional RKKY-like system where we find a spin glass phase with continuous replica symmetry breaking.

Some non-perturbative calculations on spin glasses

Models of spin glasses are studied with a phase transition discontinuous in the Parisi order parameter. It is assumed that the leading order corrections to the thermodynamic limit of the high temperature free energy are due to the existence of a metastable saddle point in the replica formalism. An ansatz is made on the form of the metastable point and its contribution to the free energy is calculated. The Random Energy Model is considered along with the p-spin and the p-state Potts Models in their p < infinity expansion.

Equilibrium properties of a quadrupolar glass.

We introduce a semimicroscopic discrete-state model appropriate to the orientational glass phase in mixed alkali halide cyanides with 〈111〉 equilibrium orientations of the ${\mathrm{CN}}^{\mathrm{\ensuremath{-}}}$ ions, such as K(CN${)}_{\mathit{x}}$${\mathrm{Br}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$. The order-parameter fields are defined as symmetry adapted combinations of the occupation number operators along the cubic body diagonals, which transform according to the ${\mathit{T}}_{2\mathit{g}}$ representation of the cubic group. These interact via an infinite-range random interaction in the presence of quenched local random strains. We then use the replica formalism to derive a replica-symmetric solution for the components of the orientational-glass order parameter, the linear susceptibilities, and the elastic compliances. The high-temperature orientational-glass phase is characterized by an isotropic order-parameter matrix with only the diagonal elements ${\mathit{q}}_{\mathrm{\ensuremath{\mu}}}$ being nonzero. At high temperatures, the behavior of the order parameter q=${\mathrm{\ensuremath{\Sigma}}}_{\mathrm{\ensuremath{\mu}}}$${\mathit{q}}_{\mathrm{\ensuremath{\mu}}}$/3 is similar to that of an Ising spin glass, however, at intermediate and low temperatures the two models differ significantly. We also derive the instability line ${\mathit{T}}_{\mathit{f}}$(\ensuremath{\Delta}) separating the replica-symmetric isotropic phase from the low-temperature anisotropic orientational glass phase, which is characterized by broken replica symmetry. In contrast to the random-bond--random-field model of an Ising spin glass, the instability temperature increases with random-field variance, implying that in quadrupolar glasses replica-symmetry breaking may be relevant already at relatively high temperatures. Finally, an expression for the distribution of local strains related to the NMR line shape is derived. It is also shown that the quadrupolar glass order parameter can be determined by NMR.

Partially annealed neural networks

We consider the Hopfield model of neural networks in which the patterns, as well as the spins, are dynamical variables. The characteristic time scales of the dynamics of the spins and the patterns are assumed to be widely separated such that the spins completely equilibrate at the time scale at which the elementary changes in the patterns take place. We study the situation in which each type of variable thermalizes at different temperatures, respectively, T and T'. In this case, such a system is described in terms of the traditional replica formalism in which the number of replicas n=T/T' is still the finite parameter. The complete phase diagram of the model in the space of the parameters T, alpha and n is obtained. If the parameter n is negative, the model is argued to present some similarities with the unlearning training algorithm. In this case a substantial increase in size of the retrieval phase in the plane (T, alpha ) is found.

Statistical mechanics of flux lines in oxide superconductors

The structure of the unpinned vortex liquid in strongly anisotropic layered high- T c oxide superconductors in a magnetic field B perpendicular to the layer plane is investigated by using appropriate classical liquid-state theories. The direct correlation function so obtained is used as input for a parameter-free, first-principles calculation of the liquid-to-Abrikosov-lattice transition boundary T f( B) within the framework of the density functional formalism. This transition boundary is found to have a shape and location similar to that observed in experiments on strongly anisotropic materials, e.g. BiSrCaCuO. The effects of random pinning disorder on the equilibrium density correlations of the vortex liquid and its freezing transition are investigated by combining the methods mentioned above with the replica formalism.