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Register a new userThe Percolation Signature of the Spin Glass Transition
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Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering -- both in short-range (EA) and infinite-range (SK) models -- within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the $pm J$ EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of {it two} percolating clusters of {it unequal} densities.
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Understanding the physics of glass formation remains one of the major unsolved challenges of condensed matter science. As a material solidifies into a glass, it exhibits a spectacular slowdown of the dynamics upon cooling or compression, but at the same time undergoes only minute structural changes. Among the numerous theories put forward to rationalize this complex behavior, Mode-Coupling Theory (MCT) stands out as the only framework that provides a fully first-principles-based description of glass phenomenology. This review outlines the key physical ingredients of MCT, its predictions, successes, and failures, as well as recent improvements of the theory. We also discuss the extension and application of MCT to the emerging field of non-equilibrium active soft matter
We study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e. the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. We find that the replica symmetric (RS) solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at $T = 0$. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e. magnetic field).
It has been shown recently that predictions from Mode-Coupling Theory for the glass transition of hard-spheres become increasingly bad when dimensionality increases, whereas replica theory predicts a correct scaling. Nevertheless if one focuses on the regime around the dynamical transition in three dimensions, Mode-Coupling results are far more convincing than replica theory predictions. It seems thus necessary to reconcile the two theoretic approaches in order to obtain a theory that interpolates between low-dimensional, Mode-Coupling results, and mean-field results from replica theory. Even though quantitative results for the dynamical transition issued from replica theory are not accurate in low dimensions, two different approximation schemes --small cage expansion and replicated Hyper-Netted-Chain (RHNC)-- provide the correct qualitative picture for the transition, namely a discontinuous jump of a static order parameter from zero to a finite value. The purpose of this work is to develop a systematic expansion around the RHNC result in powers of the static order parameter, and to calculate the first correction in this expansion. Interestingly, this correction involves the static three-body correlations of the liquid. More importantly, we separately demonstrate that higher order terms in the expansion are quantitatively relevant at the transition, and that the usual mode-coupling kernel, involving two-body direct correlation functions of the liquid, cannot be recovered from static computations.
In this paper we introduce a new algorithm to study some NP-complete problems. This algorithm is a Markov Chain Monte Carlo (MCMC) inspired by the cavity method developed in the study of spin glass. We will focus on the maximum clique problem and we will compare this new algorithm with several standard algorithms on some DIMACS benchmark graphs and on random graphs. The performances of the new algorithm are quite surprising. Our effort in this paper is to be clear as well to those readers who are not in the field.
We perform Monte-Carlo simulations to study the Bernoulli ($p$) bond percolation on the enhanced binary tree which belongs to the class of nonamenable graphs with one end. Our numerical results show that the system has two different percolation thresholds $p_{c1}$ and $p_{c2}$. All the points in the intermediate phase $(p_{c1} < p < p_{c2})$ are critical and there exist infinitely many infinite clusters in the intermediate phase. In this phase the corresponding fractal exponent continuously increases with $p$ from zero to unity.
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