Fractal geometry of spin-glass models

Stability and diversity are two key properties that living entities share with spin glasses, where they are manifested through the breaking of the phase space into many valleys or local minima connected by saddle points. The topology of the phase space can be conveniently condensed into a tree structure, akin to the biological phylogenetic trees, whose tips are the local minima and internal nodes are the lowest-energy saddles connecting those minima. For the infinite-range Ising spin glass with p-spin interactions, we show that the average size-frequency distribution of saddles obeys a power law $\sim w^{-D}$, where w=w(s) is the number of minima that can be connected through saddle s, and D is the fractal dimension of the phase space

An investigation of the hidden structure of states in a mean field spin glass model

We study the geometrical structure of the states in the low temperature phase of a mean field model for generalized spin glasses, the p-spin spherical model. This structure cannot be revealed by the standard methods, mainly due to the presence of an exponentially high number of states, each one having a vanishing weight in the thermodynamic limit. Performing a purely entropic computation, based on the TAP equations for this model, we define a constrained complexity which gives the overlap distribution of the states. We find that this distribution is continuous, non-random and highly dependent on the energy range of the considered states. Furthermore, we show which is the geometrical shape of the threshold landscape, giving some insight into the role played by threshold states in the dynamical behaviour of the system.Comment: 18 pages, 8 PostScript figures, plain Te

How a spin-glass remembers. Memory and rejuvenation from intermittency data: an analysis of temperature shifts

The memory and rejuvenation aspects of intermittent heat transport are explored theoretically and by numerical simulation for Ising spin glasses with short-ranged interactions. The theoretical part develops a picture of non-equilibrium glassy dynamics recently introduced by the authors. Invoking the concept of marginal stability, this theory links irreversible `intermittent' events, or `quakes' to thermal fluctuations of record magnitude. The pivotal idea is that the largest energy barrier $b(t_w,T)$ surmounted prior to $t_w$ by thermal fluctuations at temperature $T$ determines the rate $r_q \propto 1/t_w$ of the intermittent events occurring near $t_w$. The idea leads to a rate of intermittent events after a negative temperature shift given by $r_q \propto 1/t_w^{eff}$, where the `effective age' $t_w^{eff} \geq t_w$ has an algebraic dependence on $t_w$, whose exponent contains the temperatures before and after the shift. The analytical expression is verified by numerical simulations. Marginal stability suggests that a positive temperature shift $T \to T'$ could erase the memory of the barrier $b(t_w,T)$. The simulations show that the barrier $b(t_w,T') \geq b(t_w,T)$ controls the intermittent dynamics, whose rate is hence $r_q \propto 1/t_w$. Additional `rejuvenation' effects are also identified in the intermittency data for shifts of both signs.Comment: Revised introduction and discussion. Final version to appear in Journal of Statistical Mechanics: Theory and Experimen

Phase transition and landscape statistics of the number partitioning problem

The phase transition in the number partitioning problem (NPP), i.e., the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution, is investigated by examining the energy landscape of this classic optimization problem. This is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure, termed a barrier tree, the leaves and internal nodes of which represent, respectively, the minima and the lowest energy saddles connecting those minima. Here we apply several measures of shape (balance and symmetry) as well as of branch lengths (barrier heights) to the barrier trees that result from the landscape of the NPP, aiming at identifying traces of the easy/hard transition. We find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights. Only the {\it difficulty} measure, given by the maximum value of the ratio between the barrier height and the energy surplus of local minima, succeeded in detecting traces of the phase transition in the tree. In adddition, we show that the barrier trees associated with the NPP are very similar to random trees, contrasting dramatically with trees associated with the $p$ spin-glass and random energy models. We also examine critically a recent conjecture on the equivalence between the NPP and a truncated random energy model

Spin glass overlap barriers in three and four dimensions

For the Edwards-Anderson Ising spin-glass model in three and four dimensions (3d and 4d) we have performed high statistics Monte Carlo calculations of those free-energy barriers $F^q_B$ which are visible in the probability density $P_J(q)$ of the Parisi overlap parameter $q$. The calculations rely on the recently introduced multi-overlap algorithm. In both dimensions, within the limits of lattice sizes investigated, these barriers are found to be non-self-averaging and the same is true for the autocorrelation times of our algorithm. Further, we present evidence that barriers hidden in $q$ dominate the canonical autocorrelation times.Comment: 20 pages, Latex, 12 Postscript figures, revised version to appear in Phys. Rev.

Metastable States in Spin Glasses and Disordered Ferromagnets

We study analytically M-spin-flip stable states in disordered short-ranged Ising models (spin glasses and ferromagnets) in all dimensions and for all M. Our approach is primarily dynamical and is based on the convergence of a zero-temperature dynamical process with flips of lattice animals up to size M and starting from a deep quench, to a metastable limit. The results (rigorous and nonrigorous, in infinite and finite volumes) concern many aspects of metastable states: their numbers, basins of attraction, energy densities, overlaps, remanent magnetizations and relations to thermodynamic states. For example, we show that their overlap distribution is a delta-function at zero. We also define a dynamics for M=infinity, which provides a potential tool for investigating ground state structure.Comment: 34 pages (LaTeX); to appear in Physical Review

Energy barriers in SK spin-glass model

The height of barriers separating metastable states is studied for the infinite range Ising spin-glass model. A barrier configuration corresponds to a saddle point {mi, i = 1, ..., N, — 1 ≤ mi ≤ 1} of an energy surface that smoothly interpolates the energy in the hypercube. For low energy barriers, which are relevant for the dynamics, the number of independent descending directions from the saddle point is finite when N → ∞. It is found that these descending directions are contained in the linear subspace of the hypercube [—1,1]N generated by directions for which Σ j Jij mj = 0. Numerical estimates were performed by two distinct algorithms which lead to consistent results. Assuming barrier heights growing with system size N as Nα we find α = 0.34 ± 0.08.Nous étudions les hauteurs de barrière séparant des états métastables pour le modèle de verre de spin avec symétrie d'Ising et portée infinie. Une configuration de barrière correspond à un col de {mi, i = 1, ..., N, - 1 ≤ mi ≤ 1} de la surface d'énergie qui interpole de façon régulière l'énergie dans l'hypercube. Pour des barrières d'énergie faibles, qui sont importantes pour la dynamique, le nombre de directions descendantes au col est fini dans la limite N → ∞. Nous trouvons que ces directions sont contenues dans un sous-espace linéaire de l'hypercube [—1,1]N engendré par les directions pour lesquelles Σj Jij mj = 0. Nous avons fait des simulations numeriques par deux algorithmes différents. Les résultats sont cohérents. Si nous supposons que les hauteurs de barrière croissent avec la taille N du système comme Nα, nous trouvons α = 0,34 ± 0,08