P-spin Spherical Model Research Articles

Partition-function zeros of spherical spin glasses and their relevance to chaos

We investigate the partition-function zeros of the many-body interacting spherical spin glass, the so-called p-spin spherical model, with respect to the complex temperature in the thermodynamic limit. We use the replica method and extend the procedure of the replica symmetry breaking ansatz to be applicable in the complex-parameter case. We derive the phase diagrams in the complex-temperature plane and calculate the density of zeros in each phase. Near the imaginary axis away from the origin, there is a replica symmetric phase having a large density. On the other hand, we observe no density in the spin-glass phases, irrespective of the replica symmetry breaking. We speculate that this suggests the absence of the temperature chaos. To confirm this, we investigate the multiple many-body interacting case which is known to exhibit the chaos effect. The result shows that the density of zeros actually takes finite values in the spin-glass phase, even on the real axis. These observations indicate that the density of zeros is more closely connected to the chaos effect than the replica symmetry breaking.

Photo-induced magnetic solitons and the slow relaxation mechanism in diluted magnetic semiconductors

Taking into account the correlation effect among photo-induced magnetic solitons, the mechanism of the slow relaxation of the spin dynamics in diluted magnetic semiconductors has been discussed by the long-range p-spin spherical model.

Temperature shifts in the quantum spherical p-spin model of glass

Low-temperature slow dynamics, aging, and temperature shifts in a spin-glass phase in the quantum p-spin model of glass related to the quantum internal surroundings and located in external dc and ac magnetic fields have been studied analytically and numerically. It is shown that the correlation function and the real part of the susceptibility exhibit effects of aging and recovery (rejuvenation) at small changes in temperature. A brief analysis of the effects considered has been performed.

Spherical2+pspin-glass model: An analytically solvable model with a glass-to-glass transition

We present the detailed analysis of the spherical s+p spin glass model with two competing interactions: among p spins and among s spins. The most interesting case is the 2+p model with p > 3 for which a very rich phase diagram occurs, including, next to the paramagnetic and the glassy phase represented by the one step replica symmetry breaking ansatz typical of the spherical p-spin model, other two amorphous phases. Transitions between two contiguous phases can also be of different kind. The model can thus serve as mean-field representation of amorphous-amorphous transitions (or transitions between undercooled liquids of different structure). The model is analytically solvable everywhere in the phase space, even in the limit where the infinite replica symmetry breaking ansatz is required to yield a thermodynamically stable phase.

Free energy of the spherical mean field model

We compute at any temperature the free energy of the multi p-spin spherical model when only terms for p even are considered.

Low-temperature dynamic behavior of a spherical p-spin model of glass in an alternating field

Coupled dynamic Schwinger-Dyson equations for autocorrelation and linear-response functions are used for studying nonequilibrium dynamics of a p-spin model of glass. The numerical solution of these equations reveals an interesting dynamic behavior of the system in the external alternating field. The behavior is also shown to depend on the amplitude and frequency of the foregoing field and on the value of coupling between the spins.

On the nature of the low-temperature phase in discontinuous mean-field spin glasses

The low-temperature phase of discontinuous mean-field spin glasses is generally described by a one-step replica symmetry breaking (1RSB) Ansatz. The Gardner transition, i.e. a very-low-temperature phase transition to a full replica symmetry breaking (FRSB) phase, is often regarded as an inessential, and somehow exotic phenomenon. In this paper we show that the metastable states which are relevant for the out-of-equilibrium dynamics of such systems are always in a FRSB phase. The only exceptions are (to the best of our knowledge) the p-spin spherical model and the random energy model (REM). We also discuss the consequences of our results for aging dynamics and for local search algorithms in hard combinatorial problems.

Dynamical TAP approach to mean field glassy systems

The Thouless, Anderson, Palmer (TAP) approach to thermodynamics of mean field spin-glasses is generalised to dynamics. A method to compute the dynamical TAP equations is developed and applied to the p-spin spherical model. In this context we show to what extent the dynamics can be represented as an evolution in the free energy landscape. In particular the relationship between the long-time dynamics and the local properties of the free energy landscape shows up explicitly within this approach. Conversely, by an instantaneous normal modes analysis we show that the local properties of the energy landscape seen by the system during its dynamical evolution do not change qualitatively at the dynamical transition.

Quenched complexity of the mean-fieldp-spin spherical model with external magnetic field

We consider the p-spin spherical spin-glass model in the presence of an external magnetic field as a general example of a mean-field system where a one-step replica symmetry breaking (1-RSB) occurs. In this context we compute the complexity of the Thouless-Anderson-Palmer states, performing a quenched computation. We find what the general connection is between this method and the standard static 1-RSB one, formulating a clear mapping between the parameters used in the two different calculations. A dynamical analysis of the model confirms the validity of our results.

Stationary points of the Thouless-Anderson-Palmer free energy

In the context of the p-spin spherical model, we introduce a method for the computation of the number of stationary points of any nature (minima, saddles, etc.) of the TAP free energy. In doing this we clarify the ambiguities related to the approximations usually adopted in the standard calculations of the number of states in mean field spin glass models.

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.

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.

Barriers and metastable states as saddle points in the replica approach

In the context of the p-spin spherical model we show how saddle points of the n - 0 replica free energy surface can be interpreted as metastable states or barriers.