Metastable States In Spin Glasses Research Articles

Distribution of metastable states of spin glasses

The complex behavior of systems like spin glasses, proteins or neural networks is typically explained in terms of a rugged energy or fitness landscape. Within the high-dimensional conformation space of these systems one finds features like barriers, saddle points, and metastable states whose number – at least in the case of spin glasses – grows exponentially with the size of the system. We propose a novel Monte Carlo sampling algorithm that employs an ensemble of short Markovian chains in order to visit all metastable states with equal probability. We apply this algorithm in order to measure the number of metastable states for the two-dimensional and the three-dimensional Edwards-Anderson model and compare with theoretical predictions.

Metastable states of spin glasses on random thin graphs

In this paper we calculate the mean number of metastable states for spin glasses on so called random thin graphs with couplings taken from a symmetric binary distribution $\pm J$. Thin graphs are graphs where the local connectivity of each site is fixed to some value $c$. As in totally connected mean field models we find that the number of metastable states increases exponentially with the system size. Furthermore we find that the average number of metastable states decreases as $c$ in agreement with previous studies showing that finite connectivity corrections of order $1/c$ increase the number of metastable states with respect to the totally connected mean field limit. We also prove that the average number of metastable states in the limit $c\to\infty$ is finite and converges to the average number of metastable states in the Sherrington-Kirkpatrick model. An annealed calculation for the number of metastable states $N_{MS}(E)$ of energy $E$ is also carried out giving a lower bound on the ground state energy of these spin glasses. For small $c$ one may obtain analytic expressions for $<N_{MS}(E)>$.

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 sigma(t), a zero-temperature dynamical process with flips of lattice animals up to size M and starting from a deep quench, to a metastable limit sigma(infinity). 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.

Structure of metastable states in spin glasses by means of a three replica potential

We introduce a three replica potential useful to examine the structure of metastables states above the static transition temperature, in the spherical p-spin model. Studying the minima of the potential we are able to find which is the distance between the nearest equilibrium and local equilibrium states, obtaining in this way information on the dynamics of the system. Furthermore, the analysis of the potential at the dynamical transition temperature suggests that equilibrium states are not randomly distributed in the phase space.

Recipes for Metastable States in Spin Glasses

In this paper we develop a method introduced by one of us to study metastable states in spin glasses. We consider a `potential function' defined as the free energy of a system at a given temperature $T$ constrained to have a fixed overlap with a reference configuration of equilibrium at temperature $T'$. We apply the method to the spherical p-spin glass and to some generalization of this model in the range of temperatures between the dynamic and the static transition. The analysis suggests a correspondence among local minima of the potential and metastable states. This correspondence is confirmed studying the relaxation dynamics at temperature $T$ of a system starting from an initial configuration equilibrated at a different temperature $T'$.

Correlations between metastable states in spin glasses.

The distribution of local minima in the energy landscape of the infinite-ranged Ising spin glass is studied by calculation of the average number of pairs of metastable states with fixed overlap and energies. The average number forms an upper bound on the typical number. It is known that below a critical energy metastable states are globally (strongly) correlated. Above the critical energy the upper bound on pairs of metastable states suggests that although metastable states are only locally (weakly) correlated, an interesting local structure of the energy landscape may be present.

Barrier heights versus temperature in spin glasses

Recent experimental results on the relaxation of the remanent magnetization (after various aging procedures) have been used to study the temperature dependence of the barrier heights between the metastable states of spin glasses. It is suggested that, in the range 0.8< T/ T g<1, the temperature vation of the barriers only dependes on T-T ∗ where T ∗ characterizes each particular barrier Δ ∗. As the temperature is lowered, the height of Δ ∗ increases and diverges at T ∗ . The emerging picture is consistent with that of a contiuous series of micro-phase transitions below T g, as suggested from theoretical arguments.

Renormalisation group theory of spin glasses

The renormalisation group scheme for the thermodynamic theory of metastable states in spin glasses is proposed. On the basis of the hierarchical organisation of states in spin glasses, it is possible to calculate an effective distribution functional of spin-spin interactions with increasing scale in a phase space. For the Sherrington-Kirkpatrick model (1975), its explicit form has been found both above and below the phase transition point. Below the phase transition point the system is shown to be 'trapped' in a finite region of the phase space which corresponds to one of the ancestor states of the hierarchical tree.

Infinite set of exponents describing physics on fractal networks

The generalised resistance between connected points a distance L apart on fractal networks of nonlinear (V approximately Ialpha ) resistors scales as Lzeta (/alpha ). It is shown that zeta ( alpha ) for alpha =- infinity , -1, 0-, 0+, 1 and infinity , describes physically relevant geometrical properties and d zeta /d alpha <or=0. For percolating clusters approximants are given for zeta for - infinity < alpha < infinity in 2-6 dimensions. For alpha <0 a family of solutions to Kirchhoff's equations exists, reminiscent of metastable states in spin glasses.

On the interplay of replicas and metastable states in spin glasses

For the Ising Sherrington-Kirkpatrick spin glass (1975), the author studies the constraints imposed on the distributions P(q,(Jij)), P(U,M,(Jij)) for the overlap q, energy U and magnetisation M of the underlying metastable states, by the general nature of current replica symmetry breaking schemes. He confirms that whilst P(q,(Jij)) shows significant sample to sample variations the thermodynamic energy and field cooled magnetisation are well behaved. For both the Edwards-Anderson order parameter and the Minimum overlap the case is less clear.

Distributions of the metastable energy levels and the internal magnetic fields in spin glasses

Energy distributions in the metastable states of spin glasses are studied for two-component spin systems in the limit of infinitely long-ranged exchange interactions. To determine how fundamental properties change by spin symmetry, the Edwards-Anderson model is extended to the vector Potts model, each spin of which can take only n directions in a plane. The authors' theory includes both n=2 (Ising model) and n to infinity limit (XY model). The upper limit of the average ground-state energy per spin is evaluated to be -0.594 J for the XY model. The internal magnetic field distribution P(h) is also studied. For n>or=3, its small h behaviour is found to be quite different from that of the Ising model.

Metastable states in spin glasses with short-ranged interactions

A formalism is developed for calculating the density NS(E) of metastable (i.e. one spin-flip stable) states of energy E for Ising spin glasses. Results are presented, for nearest-neighbour interactions on hypercubic lattices to O(1/z), where z is the coordination number, for E>or=Ec, a critical energy below which the metastable states become correlated. The critical behaviour associated with the transition at Ec is non-classical in less than six dimensions, with critical exponents identical to those of the thermal spin glass transition. Results are also presented for an asymmetric distribution of exchange interactions.

Broken replica symmetry and metastable states in spin glasses

The average number (NS(f)) of uncorrelated metastable states of free energy kBTf per spin in the Sherrington-Kirkpatrick model of an Ising spin glass is shown to equal the extremum with respect to n of exp(nf N)(Zn), where (Zn) is the bond-averaged nth power of the partition function calculated within the two-group replica symmetry breaking scheme.

Metastable states in spin glasses

The number of solutions of the equations of Thouless et al. (1977) is obtained as a function of temperature. The density of solutions with a given free energy is calculated for free energies greater than a (temperature-dependent) critical value.