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Register a new userLocal minima in disordered mean-field ferromagnets
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We consider the complexity of random ferromagnetic landscapes on the hypercube ${pm 1}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph $mathcal G(N,p)$. Previous results had shown that, with high probability as $Ntoinfty$, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as $Ntoinfty$. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.
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We investigate the properties of local minima of a recently introduced spin glass model of soft spins subjected to an anharmonic quartic local potential which serves as a model of low temperature molecular or soft glasses. We track the long time gradient descent dynamics in the glassy phase through dynamical mean field theory and show that spins are separated in two groups depending on their local stiffness. For spins having local stiffness that is right above its smallest possible value, the local fields distribution displays a depletion around the origin while those having a stiffness right below its largest possible value have a regular local fields distribution. We rationalize these findings through the replica method and show that the finite temperature phase transition to the glass phase is of continuous (full) replica-symmetry-breaking (RSB) type at low temperatures, down to zero temperature. Furthermore, marginal stability of the zero temperature fullRSB solution implies a linear pseudogap in the density of cavity fields for the spins with a local effective stiffness that is below a certain threshold. This generates a hole around the origin in the corresponding local field distribution. Those spins are natural candidates to model two level systems (TLS). The behavior of the cavity fields distribution for spins having stiffness close to the threshold one determines the tail of the low frequency density of states which is gapless. Therefore the corresponding spins are the natural candidates to model quasi localized modes (QLM) in glasses.
We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of ``admissible transitions. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.
We generalize the strategy, we recently introduced to prove the existence of the thermodynamic limit for the Sherrington-Kirkpatrick and p-spin models, to a wider class of mean field spin glass systems, including models with multi-component and non-Ising type spins, mean field spin glasses with an additional Curie-Weiss interaction, and systems consisting of several replicas of the spin glass model, where replicas are coupled with terms depending on the mutual overlaps.
We apply the Kovacs experimental protocol to classical and quantum p-spin models. We show that these models have memory effects as those observed experimentally in super-cooled polymer melts. We discuss our results in connection to other classical models that capture memory effects. We propose that a similar protocol applied to quantum glassy systems might be useful to understand their dynamics.
We study a recently introduced and exactly solvable mean-field model for the density of vibrational states $mathcal{D}(omega)$ of a structurally disordered system. The model is formulated as a collection of disordered anharmonic oscillators, with random stiffness $kappa$ drawn from a distribution $p(kappa)$, subjected to a constant field $h$ and interacting bilinearly with a coupling of strength $J$. We investigate the vibrational properties of its ground state at zero temperature. When $p(kappa)$ is gapped, the emergent $mathcal{D}(omega)$ is also gapped, for small $J$. Upon increasing $J$, the gap vanishes on a critical line in the $(h,J)$ phase diagram, whereupon replica symmetry is broken. At small $h$, the form of this pseudogap is quadratic, $mathcal{D}(omega)simomega^2$, and its modes are delocalized, as expected from previously investigated mean-field spin glass models. However, we determine that for large enough $h$, a quartic pseudogap $mathcal{D}(omega)simomega^4$, populated by localized modes, emerges, the two regimes being separated by a special point on the critical line. We thus uncover that mean-field disordered systems can generically display both a quadratic-delocalized and a quartic-localized spectrum at the glass transition.
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