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Register a new userEvidences Against Temperature Chaos in Mean Field and Realistic Spin Glasses
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A. Billoire
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We discuss temperature chaos in mean field and realistic 3D spin glasses. Our numerical simulations show no trace of a temperature chaotic behavior for the system sizes considered. We discuss the experimental and theoretical implications of these findings.
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Optimizing a high-dimensional non-convex function is, in general, computationally hard and many problems of this type are hard to solve even approximately. Complexity theory characterizes the optimal approximation ratios achievable in polynomial time in the worst case. On the other hand, when the objective function is random, worst case approximation ratios are overly pessimistic. Mean field spin glasses are canonical families of random energy functions over the discrete hypercube ${-1,+1}^N$. The near-optima of these energy landscapes are organized according to an ultrametric tree-like structure, which enjoys a high degree of universality. Recently, a precise connection has begun to emerge between this ultrametric structure and the optimal approximation ratio achievable in polynomial time in the typical case. A new approximate message passing (AMP) algorithm has been proposed that leverages this connection. The asymptotic behavior of this algorithm has been analyzed, conditional on the nature of the solution of a certain variational problem. In this paper we describe the first implementation of this algorithm and the first numerical solution of the associated variational problem. We test our approach on two prototypical mean-field spin glasses: the Sherrington-Kirkpatrick (SK) model, and the $3$-spin Ising spin glass. We observe that the algorithm works well already at moderate sizes ($Ngtrsim 1000$) and its behavior is consistent with theoretical expectations. For the SK model it asymptotically achieves arbitrarily good approximations of the global optimum. For the $3$-spin model, it achieves a constant approximation ratio that is predicted by the theory, and it appears to beat the `threshold energy achieved by Glauber dynamics. Finally, we observe numerically that the intermediate states generated by the algorithm have the properties of ancestor states in the ultrametric tree.
Mean-field spin glasses are families of random energy functions (Hamiltonians) on high-dimensional product spaces. In this paper we consider the case of Ising mixed $p$-spin models, namely Hamiltonians $H_N:Sigma_Nto {mathbb R}$ on the Hamming hypercube $Sigma_N = {pm 1}^N$, which are defined by the property that ${H_N({boldsymbol sigma})}_{{boldsymbol sigma}in Sigma_N}$ is a centered Gaussian process with covariance ${mathbb E}{H_N({boldsymbol sigma}_1)H_N({boldsymbol sigma}_2)}$ depending only on the scalar product $langle {boldsymbol sigma}_1,{boldsymbol sigma}_2rangle$. The asymptotic value of the optimum $max_{{boldsymbol sigma}in Sigma_N}H_N({boldsymbol sigma})$ was characterized in terms of a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here we ask whether a near optimal configuration ${boldsymbol sigma}$ can be computed in polynomial time. We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of $H_N$, and characterize the typical energy value it achieves. When the $p$-spin model $H_N$ satisfies a certain no-overlap gap assumption, for any $varepsilon>0$, the algorithm outputs ${boldsymbol sigma}inSigma_N$ such that $H_N({boldsymbol sigma})ge (1-varepsilon)max_{{boldsymbol sigma}} H_N({boldsymbol sigma})$, with high probability. The number of iterations is bounded in $N$ and depends uniquely on $varepsilon$. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.
We construct and analyze a family of $M$-component vectorial spin systems which exhibit glass transitions and jamming within supercooled paramagnetic states without quenched disorder. Our system is defined on lattices with connectivity $c=alpha M$ and becomes exactly solvable in the limit of large number of components $M to infty$. We consider generic $p$-body interactions between the vectorial Ising/continuous spins with linear/non-linear potentials. The existence of self-generated randomness is demonstrated by showing that the random energy model is recovered from a $M$-component ferromagnetic $p$-spin Ising model in $M to infty$ and $p to infty$ limit. In our systems the quenched disorder, if present, and the self-generated disorder act additively. Our theory provides a unified mean-field theoretical framework for glass transitions of rotational degree of freedoms such as orientation of molecules in glass forming liquids, color angles in continuous coloring of graphs and vector spins of geometrically frustrated magnets. The rotational glass transitions accompany various types of replica symmetry breaking. In the case of repulsive hardcore interactions in the spin space, continuous the criticality of the jamming or SAT/UNSTAT transition becomes the same as that of hardspheres.
Starting from the second law of thermodynamics applied to an isolated system consisting of the system surrounded by an extremely large medium, we formulate a general non-equilibrium thermodynamic description of the system when it is out of equilibrium. We then apply it to study the structural relaxation in glasses and establish the phenomenology behind the concept of the fictive temperature and of the empirical Tool-Narayanaswamy equation on firmer theoretical foundation.
We investigate numerically the time dependence of window overlaps in a three-dimensional Ising spin glass below its transition temperature after a rapid quench. Using an efficient GPU implementation, we are able to study large systems up to lateral length $L=128$ and up to long times of $t=10^8$ sweeps. We find that the data scales according to the ratio of the window size $W$ to the non-equilibrium coherence length $xi(t)$. We also show a substantial change in behavior if the system is run for long enough that it globally equilibrates, i.e. $xi(t) approx L/2$, where $L$ is the lattice size. This indicates that the local behavior of a spin glass depends on the spin configurations (and presumably also the bonds) far away. We compare with similar simulations for the Ising ferromagnet. Based on these results, we speculate on a connection between the non-equilibrium dynamics discussed here and averages computed theoretically using the metastate.
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