Mean-field Spin Glass Research Articles

An emergent autonomous flow for mean-field spin glasses

We study the dynamics of symmetric and asymmetric spin-glass models of size N. The analysis is in terms of the double empirical process: this contains both the spins, and the field felt by each spin, at a particular time (without any knowledge of the correlation history). It is demonstrated that in the large N limit, the dynamics of the double empirical process becomes deterministic and autonomous over finite time intervals. This does not contradict the well-known fact that SK spin-glass dynamics is non-Markovian (in the large N limit) because the empirical process has a topology that does not discern correlations in individual spins at different times. In the large N limit, the evolution of the density of the double empirical process approaches a nonlocal autonomous PDE operator \(\Phi _t\). Because the emergent dynamics is autonomous, in future work one will be able to apply PDE techniques to analyze bifurcations in \(\Phi _t\). Preliminary numerical results for the SK Glauber dynamics suggest that the ‘glassy dynamical phase transition’ occurs when a stable fixed point of the flow operator \(\Phi _t\) destabilizes.

Low-frequency vibrational spectrum of mean-field disordered systems

We study a recently introduced and exactly solvable mean-field model for the density of vibrational states $\mathcal{D}(\ensuremath{\omega})$ of a structurally disordered system. The model is formulated as a collection of disordered anharmonic oscillators, with random stiffness $\ensuremath{\kappa}$ drawn from a distribution $p(\ensuremath{\kappa})$, 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 $p(\ensuremath{\kappa})$ is gapped, the emergent $\mathcal{D}(\ensuremath{\omega})$ 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, $\mathcal{D}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2}$, 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 $\mathcal{D}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{4}$, 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.

Uniqueness of Ground State in the Edwards–Anderson Spin Glass Model

It is proven rigorously that the ground state in the Edwards-Anderson spin glass model is unique in any dimension for almost all continuous random exchange interactions under a condition that a single spin breaks the global ${\mathbb Z}_2$ symmetry. This theorem implies that replica symmetry breaking does not occur at zero temperature. The site- and bond-overlap are concentrated at their maximal values. It is argued that behaviors of short range spin glass models are much different from those of mean field spin glass models near zero temperature. Errata have been attached at the final page.

Gradient descent dynamics in the mixed p-spin spherical model: finite-size simulations and comparison with mean-field integration

We perform numerical simulations of a long-range spherical spin glass with two and three body interaction terms. We study the gradient descent dynamics and the inherent structures found after a quench from initial conditions well thermalized at temperature T in. In very large systems, the dynamics perfectly agrees with the integration of the mean-field dynamical equations. In particular, we confirm the existence of an onset initial temperature, within the liquid phase, below which the energy of the inherent structures undoubtedly depends on T in. This behavior is in contrast with that of pure models, where there is a ‘threshold energy’ that attracts all the initial configurations in the liquid. Our results strengthen the analogy between mean-field spin glass models and supercooled liquids.

Out-of-equilibrium phase diagram of the quantum random energy model

In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's Random Energy Model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body localized phase at low energy, a fully ergodic phase at high energy, and a multifractal "bad metal" phase at intermediate energy. In the latter, eigenfunctions occupy a diverging volume, yet an exponentially vanishing fraction of the total Hilbert space. We discuss the limitations of our approximations and the relationship with previous studies.

The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space

AbstractThe Parisi formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We showthat this quantity can be recast as the solution of a Hamilton–Jacobi equation in the Wasserstein space of probability measures on the positive half-line.

Analysis of landscape hierarchy during coarsening and aging in Ising spin glasses

We use record dynamics (RD), a coarse-grained description of the ubiquitous relaxation phenomenology known as "aging", as a diagnostic tool to find universal features that distinguish between the energy landscapes of Ising spin models and the ferromagnet. According to RD, a non-equilibrium system after a quench relies on fluctuations that randomly generate a sequence of irreversible record-sized events (quakes or avalanches) that allow the system to escape ever-higher barriers of meta-stable states within a complex, hierarchical energy landscape. Once these record events allow the system to overcome such barriers, the system relaxes by tumbling into the following meta-stable state that is marginally more stable. Within this framework, a clear distinction can be drawn between the coarsening dynamics of an Ising ferromagnet and the aging of the spin glass, which are often put in the same category. To that end, we interpolate between the spin glass and ferromagnet by varying the admixture $p$ of ferromagnetic over anti-ferromagnetic bonds from the glassy state (at 50% each) to wherever clear ferromagnetic behavior emerges. The accumulation of record events grows logarithmic with time in the glassy regime, with a sharp transition at a specific admixture into the ferromagnetic regime where such activations saturate quickly. We show this effect both for the Edwards-Anderson model on a cubic lattice as well as the Sherrington-Kirkpatrick (mean-field) spin glass. While this transition coincides with a previously observed zero-temperature equilibrium transition in the former, that transition has not yet been described for the latter.

On the Almeida-Thouless transition line in the Sherrington-Kirkpatrick model with centered Gaussian external field

We study the phase transition of the free energy in the Sherrington-Kirkpatrick mean-field spin glass model with centered Gaussian external field. We show that the corresponding Almeida-Thouless line is the correct transition curve that distinguishes between the replica symmetric and replica symmetry breaking solutions in the Parisi formula.

The Multi-species Mean-Field Spin-Glass on the Nishimori Line

In this paper we study a multi-species disordered model on the Nishimori line. The typical properties of this line, a set of identities and inequalities among correlation functions, allow us to prove the replica symmetry i.e. the concentration of the order parameter. When the interaction structure is elliptic we rigorously compute the exact solution of the model in terms of a finite dimensional variational principle and we study its properties.

Replica symmetry breaking in neural networks: a few steps toward rigorous results

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e. a pairwise mean-field neural-network whose couplings are drawn according to Hebb’s learning rule): this is accomplished by grafting Guerra’s telescopic averages on the transport equation technique, recently developed by some of the authors. As an overture, we apply the technique to solve the Sherrington-Kirkpatrick model with i.i.d. Gaussian couplings centered at J0 and with finite variance J; the mean J0 provides a ferromagnetic contribution to be detected in a noisy environment tuned by J, hence making this model a natural test-case to be investigated before addressing the Hopfield model. For both the models, an explicit expression of their quenched free energy in terms of their natural order parameters is obtained at the Kth step (K arbitrary, but finite) of replica-symmetry-breaking. In particular, for the Hopfield model, by assuming that the overlaps respect Parisi’s decomposition (following the ziqqurat ansatz) and that the Mattis magnetization is self-averaging, we recover previous results obtained via replica-trick by Amit, Crisanti and Gutfreund (1RSB) and by Steffan and Kühn (2RSB).

The quantum random energy model as a limit of p-spin interactions

We consider the free energy of a mean-field quantum spin glass described by a [Formula: see text]-spin interaction and a transversal magnetic field. Recent rigorous results for the case [Formula: see text], i.e. the quantum random energy model (QREM), are reviewed. We show that the free energy of the [Formula: see text]-spin model converges in a joint thermodynamic and [Formula: see text] limit to the free energy of the QREM.

Remarks on the Interpolation Method

We discuss a generalization of the classic condition of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the L^2 metric structure of the Gaussian random variables. As an example of application we deduce the existence of the thermodynamic limit for a GREM model with infinite branches for which the classic conditions of validity fail. We underline the dependence of the density of quenched free energy just on the metric structure and discuss the models from a metric viewpoint.

Strong ergodicity breaking in aging of mean-field spin glasses

Out-of-equilibrium relaxation processes show aging if they become slower as time passes. Aging processes are ubiquitous and play a fundamental role in the physics of glasses and spin glasses and in other applications (e.g., in algorithms minimizing complex cost/loss functions). The theory of aging in the out-of-equilibrium dynamics of mean-field spin glass models has achieved a fundamental role, thanks to the asymptotic analytic solution found by Cugliandolo and Kurchan. However, this solution is based on assumptions (e.g., the weak ergodicity breaking hypothesis) which have never been put under a strong test until now. In the present work, we present the results of an extraordinary large set of numerical simulations of the prototypical mean-field spin glass models, namely the Sherrington-Kirkpatrick and the Viana-Bray models. Thanks to a very intensive use of graphics processing units (GPUs), we have been able to run the latter model for more than [Formula: see text] spin updates and thus safely extrapolate the numerical data both in the thermodynamical limit and in the large times limit. The measurements of the two-times correlation functions in isothermal aging after a quench from a random initial configuration to a temperature [Formula: see text] provides clear evidence that, at large times, such correlations do not decay to zero as expected by assuming weak ergodicity breaking. We conclude that strong ergodicity breaking takes place in mean-field spin glasses aging dynamics which, asymptotically, takes place in a confined configurational space. Theoretical models for the aging dynamics need to be revised accordingly.

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond diluted to an overall density p, such that the average degree diverges as ∼pN with the system size N. Ground state energies are obtained with high accuracy for random instances over a wide range of fixed p. Since this is an NP-hard combinatorial problem, we employ the extremal optimization heuristic to that end. We find that the exponent describing the finite-size corrections ω varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond dilution would change the T=0 universality class of a statistical model. For p→1, the familiar result of ω(p=1)≈2/3 for the SK model is obtained.

RANDOM COMBINATORIAL OPTIMIZATION PROBLEMS: MEAN FIELD AND FINITE-DIMENSIONAL RESULTS

This PhD thesis is organized as follows. In the first two chapters I will review some basic notions of statistical physics of disordered systems, such as random graph theory, the mean-field approximation, spin glasses and combinatorial optimization. The replica method will also be introduced and applied to the Sherrington-Kirkpatrick model, one of the simplest mean-field models of spin-glasses. The second part of the thesis deals with mean-field combinatorial optimization problems. The attention will be focused on the study of finite-size corrections of random integer matching problems (chapter 3) and fractional ones (chapter 4). In chapter 5 I will discuss a very general relation connecting multi-overlaps and the moments of the cavity magnetization distribution. In the third part we consider random Euclidean optimization problems. I will start solving the traveling-salesman-problem (TSP) in one dimension both in its bipartite and monopartite version (chapter 6). In chapter 7 I will discuss the possible optimal solutions of the 2-factor problem. In chapter 8 I will solve the bipartite TSP in two dimensions, in the limit of large number of points. Chapter 9 contains some conclusions.

Phase transition in the spiked random tensor with Rademacher prior

We consider the problem of detecting a deformation from a symmetric Gaussian random $p$-tensor $(p\geq3)$ with a rank-one spike sampled from the Rademacher prior. Recently, in Lesieur et al. (Barbier, Krzakala, Macris, Miolane and Zdeborová (2017)), it was proved that there exists a critical threshold $\beta_{p}$ so that when the signal-to-noise ratio exceeds $\beta_{p}$, one can distinguish the spiked and unspiked tensors and weakly recover the prior via the minimal mean-square-error method. On the other side, Perry, Wein and Bandeira (Perry, Wein and Bandeira (2017)) proved that there exists a $\beta_{p}'<\beta_{p}$ such that any statistical hypothesis test cannot distinguish these two tensors, in the sense that their total variation distance asymptotically vanishes, when the signa-to-noise ratio is less than $\beta_{p}'$. In this work, we show that $\beta_{p}$ is indeed the critical threshold that strictly separates the distinguishability and indistinguishability between the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure $p$-spin model with Ising spin, arising initially from the field of spin glasses. In particular, we identify the signal-to-noise criticality $\beta_{p}$ as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure $p$-spin mean-field spin glass model.

Asymptotic Behavior of Memristive Circuits.

The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such a Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only.

Dynamics of mean field spin glasses on short and long timescales

We survey some recent rigorous results regarding the dynamics of spin glasses. We first survey recent results with Gheissari and Ben Arous regarding spectral gaps for these models. In particular, we briefly present the extension of the large deviations based approach of Ben Arous and Jagannath [Commun. Math. Phys. 361, 1–52 (2018)] to the setting of spherical spin glasses, unifying the treatment of the Ising and spherical models. We then turn to the new bounding flows method introduced by Ben Arous et al. [“Bounding flows for spherical spin glass dynamics,” e-print arXiv:1808.00929] regarding the nonactivated dynamics. We end with a report on progress on activated dynamics.

Gardner physics in amorphous solids and beyond.

One of the most remarkable predictions to emerge out of the exact infinite-dimensional solution of the glass problem is the Gardner transition. Although this transition was first theoretically proposed a generation ago for certain mean-field spin glass models, its materials relevance was only realized when a systematic effort to relate glass formation and jamming was undertaken. A number of nontrivial physical signatures associated with the Gardner transition have since been considered in various areas, from models of structural glasses to constraint satisfaction problems. This perspective surveys these recent advances and discusses the novel research opportunities that arise from them.

The TAP–Plefka Variational Principle for the Spherical SK Model

We reinterpret the Thouless-Anderson-Palmer approach to mean field spin glass models as a variational principle in the spirit of the Gibbs variational principle and the Bragg-Williams approximation. We prove this TAP-Plefka variational principle rigorously in the case of the spherical Sherrington-Kirkpatrick model.