Avalanches in mean-field models and the Barkhausen noise in spin-glasses (original) (raw)
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Equilibrium avalanches in spin glasses
Physical Review B, 2012
We study the distribution of equilibrium avalanches (shocks) in Ising spin glasses which occur at zero temperature upon small changes in the magnetic field. For the infinite-range Sherrington-Kirkpatrick model we present a detailed derivation of the density ρ(∆M) of the magnetization jumps ∆M. It is obtained by introducing a multi-component generalization of the Parisi-Duplantier equation, which allows us to compute all cumulants of the magnetization. We find that ρ(∆M) ∼ ∆M −τ with an avalanche exponent τ = 1 for the SK model, originating from the marginal stability (criticality) of the model. It holds for jumps of size 1 ∆M < N 1/2 being provoked by changes of the external field by δH = O(N −1/2) where N is the total number of spins. Our general formula also suggests that the density of overlap q between initial and final state in an avalanche is ρ(q) ∼ 1/(1 − q). These results show interesting similarities with numerical simulations for the out-ofequilibrium dynamics of the SK model. For finite-range models, using droplet arguments, we obtain the prediction τ = (d f +θ)/dm where d f , dm and θ are the fractal dimension, magnetization exponent and energy exponent of a droplet, respectively. This formula is expected to apply to other glassy disordered systems, such as the random-field model and pinned interfaces. We make suggestions for further numerical investigations, as well as experimental studies of the Barkhausen noise in spin glasses.
Static chaos and scaling behavior in the spin-glass phase
Physical review. B, Condensed matter, 1994
We discuss the problem of static chaos in spin glasses. In the case of magnetic field perturbations, we propose a scaling theory for the spin-glass phase. Using the mean-field approach we argue that some pure states are suppressed by the magnetic field and their free energy cost is determined by the finite-temperature fixed point exponents. In this framework, numerical results suggest that mean-field chaos exponents are probably exact in finite dimensions. If we use the droplet approach, numerical results suggest that the zero-temperature fixed point exponent θ is very close to d−3 2. In both approaches d = 3 is the lower critical dimension in agreement with recent numerical simulations.
Spinodals with Disorder: from Avalanches in Random Magnets to Glassy Dynamics
Motivated by the connection between the dynamical transition predicted by the mean-field theory of glass-forming liquids and the spinodal of an Ising model in a quenched random field (RFIM) beyond mean-field, we revisit the phenomenon of spinodals in the presence of quenched disorder and develop a complete theory for it. By working at zero temperature in the quasi-statically driven RFIM, thermal fluctuations are eliminated and one can give a rigorous content to the notion of spinodal. We show that the spinodal transition is due to the depinning and the subsequent expansion of rare droplets. We work out the critical behavior, which, in any finite dimension, is very different from the mean-field one: the characteristic length diverges exponentially and the thermodynamic quantities display very mild non-analyticities much like in a Griffith phenomenon. On the basis of our results we assess the physical content and the status of the dynamical transition predicted by the mean-field theory of glassy dynamics.
Fluctuations and thermodynamic variables in mean field spin glass models
1992
Two rigorous results on the Sherrington-Kirkpatrick mean field model for spin glasses are presented. Elementary methods, based on properties of fluctuations, with respect to the external quenched noise, of the thermodynamic variables and order parameters, are applied. The first result gives the uniform convergence of the quenched average of the free energy in the thermodynamic limit to its annealed approximation, in the high temperature regime, including the assumed critical point. The second result shows that the free energy can be expressed through a functional order parameter, of the type introduced by Parisi in the frame of the replica symmetry breaking method. The functional order parameter is implicitly given in terms of fluctuations of thermodynamic variables.
On the mean-field spin glass transition
European Physical Journal B, 2008
In this paper we analyze two main prototypes of disordered mean-field systems, namely the Sherrington-Kirkpatrick (SK) and the Viana-Bray (VB) models, to show that, in the framework of the cavity method, the transition from the annealed regime to a broken replica symmetry phase can be thought of as the failure of the saturability property (detailed explained along the paper) of the overlap fluctuations which act as the order parameters of the theory. We show furthermore how this coincides with the lacking of the commutativity of the infinite volume limit with respect to a, suitably chosen, vanishing perturbing field inducing the transition as prescribed by standard statistical mechanics. This is another step towards a complete theory of disordered systems. As a well known consequence it turns out that the annealed and the replica symmetric regions must coincide, implying that the averaged overlap is zero in this phase. Within our framework the finding of the values of the critical point for the SK and line for the VB becomes available straightforwardly and the method is of a large generality and applicable to several other mean field models
We review recent theoretical progress on glassy dynamics, with special emphasis on the importance and universality of the aging regime, which is relevant to many experimental situations. The three main subjects which we address are: (i) Phenomenological models of aging (coarsening, trap models), (ii) Analytical results for the low-temperature dynamics of mean-field models (corresponding to the mode-coupling equations); and (iii) Simple non-disordered models with glassy dynamics. We discuss the interrelation between these approaches, and also with previous work in the field. Several open problems are underlined -in particular the precise relation between mean-field like (or mode-coupling) descriptions and finite dimensional problems.
Critical dynamics of spin-glasses
Physical Review B, 1984
We have studied the critical behavior of a purely relaxational dynamic model for Ising spins with quenched random exchange. In zero magnetic field we find a nontrivial stable fixed point below six dimensions in agreement with previous static calculations. It is shown, to lowest order in a=6d, that van Hove theory correctly predicts the dynamic exponent z=2(2g). I. INTROjDUCTION The critical behavior as well as the low-temperature phase of magnetic systems with quenched random exchange continues to attract intensive research. ' Most theoretical work has concentrated on a spin-glass (SG) model introduced by Edwards and Anderson (EA), or rather its long-range version as proposed by Sherrington and Kirkpatrick, to provide a realization of the meanfield (MF) liinit of the short-range model. Whereas the physical interpretation of the MF results for the lowtemperature phase are still controversial, the predictions of MF theory for T) T, are generally accepted. Within MF theory the transition is second order, signaled by a divergence of the static uniform order-parameter susceptibility and a divergence of the relaxation rate of the dynamic spin autocorrelation. The first attempt to go beyond MF theory was undertaken by Harris, t.ubensky, and Chen. These authors studied the static critical behavior in zero field using the renormalization group (RG). A nontrivial fixed point was found in an expansion around the upper critical dimensionality and critical exponents were calculated in an expansion in @=6d. Subsequently, Bray and Roberts' extended the work of Ref. 4 to finite fields in order to study the critical behavior along the de Almeida-Thouless line. Surprisingly they were unable to locate a stable fixed point below six dimensioris. Another surprising result was obtained by Chen and Lubensky, who considered a model with competing ferromagnetic and antiferromagnetic ordering. They found complex exponents for XY' and Heisenberg spins at the ferromagneticspin-glass multicritical point. Neither of these results appears to be understood. All of these calculations made use of the replica trick. Even though it is generally believed that above T, no problems arise with replica symmetry breaking, it is not yet clear whether this is true only within MF theory or also in the more general case, that is, when fluctuations are important.
Dynamic scaling in spin glasses
Physical Review B, 2003
We present new Neutron Spin Echo (NSE) results and a revisited analysis of historical data on spin glasses, which reveal a pure power-law time decay of the spin autocorrelation function s(Q, t) = S(Q, t)/S(Q) at the glass temperature Tg, each power law exponent being in excellent agreement with that calculated from dynamic and static critical exponents deduced from macroscopic susceptibility measurements made on a quite different time scale. It is the first time that this scaling relation involving exponents of different physical quantities determined by completely independent experimental methods is stringently verified experimentally in a spin glass. As spin glasses are a subgroup of the vast family of glassy systems also comprising structural glasses and other non-crystalline systems the observed strict critical scaling behaviour is important. Above the phase transition the strikingly non-exponential relaxation, best fitted by the Ogielski (power-law times stretched exponential) function, appears as an intrinsic, homogeneous feature of spin glasses.
Static chaos in spin glasses: the case of quenched disorder perturbations
Journal of Physics A: Mathematical and General, 1995
We study the chaotic nature of spin glasses against perturbations of the realization of the quenched disorder. This type of perturbation modifies the energy landscape of the system without adding extensive energy. We exactly solve the mean-field case, which displays a very similar chaos to that observed under magnetic field perturbations, and discuss the possible extension of these results to the case of short-ranged models. It appears that dimension four plays the role of a specific critical dimension where mean-field theory is valid. We present numerical simulation results which support our main conclusions.