P-spin Interaction Research Articles

Triadic interaction in the background of a pairwise spin-glass.

Developing an equilibrium solution for a pairwise spin-glass with a quenched random infinite range shows a continuous phase transition. Models with p-spin interactions have been studied and the exact solution was provided that shows a continuous phase transition for p=2 and a first-order one for p>2. Although the p-spin interactions were studied individually without considering lower-order interactions, is it always feasible to ignore the lower ones? Here, we are interested in finding an analytical solution for considering a triadic interaction as a perturbation in the background of a pairwise interaction in the Sherrington-Kirkpatrick spin-glass model. Our results indicate a sudden phase transition as a consequence of considering triadic interactions that signal a switch from a continuous to an explosive phase transition.

Complexity as information in spin-glass Gibbs states and metastates: Upper bounds at nonzero temperature and long-range models.

In classical finite-range spin systems, especially those with disorder such as spin glasses, a low-temperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly as the logarithm of this number, assuming the question is meaningful in a finite system. As nontrivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] Höller and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all T≥0. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are found for each of these complexities for general finite-range (i.e., short- or long-range, in a sense we define) mixed p-spin interactions of discrete or continuous spins (such as m-vector models), but only for T>0. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.

Circumventing spin-glass traps by microcanonical spontaneous symmetry breaking.

The planted p-spin interaction model is a paradigm of random-graph systems possessing both a ferromagnetic phase and a disordered phase with the latter splitting into many spin-glass states at low temperatures. Conventional simulated annealing dynamics is easily blocked by these low-energy spin-glass states. Here we demonstrate that actually this planted system is exponentially dominated by a microcanonical polarized phase at intermediate energy densities. There is a discontinuous microcanonical spontaneous symmetry breaking transition from the paramagnetic phase to the microcanonical polarized phase. This transition can serve as a mechanism to avoid all the spin-glass traps, and it is accelerated by the restart strategy of microcanonical random walk. We also propose an unsupervised learning problem on microcanonically sampled configurations for inferring the planted ground state.

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.

Dynamical mean-field theory and aging dynamics

Dynamical mean-field theory (DMFT) replaces the many-body dynamical problem with one for a single degree of freedom in a thermal bath whose features are determined self-consistently. By focusing on models with soft disordered p-spin interactions, we show how to incorporate the mean-field theory of aging within DMFT. We study cases with only one slow time-scale, corresponding statically to the one-step replica symmetry breaking phase, and cases with an infinite number of slow time-scales, corresponding statically to the full replica symmetry breaking (FRSB) phase. For the former, we show that the effective temperature of the slow degrees of freedom is fixed by requiring critical dynamical behavior on short time-scales, i.e. marginality. For the latter, we find that aging on an infinite number of slow time-scales is governed by a stochastic equation where the clock for dynamical evolution is fixed by the change of the effective temperature, hence obtaining a dynamical derivation of the stochastic equation at the basis of the FRSB phase. Our results extend the realm of the mean-field theory of aging to all situations where DMFT holds.

Fluctuations of the free energy in the mixed p-spin models with external field

We show that the free energy in the mixed p-spin models of spin glasses does not superconcentrate in the presence of external field, which means that its variance is of the order suggested by the Poincare inequality. This complements the result of Chatterjee who showed that the free energy superconcentrates when there is no external field. For models without odd p-spin interactions for $$p\geqslant 3$$ , we prove the central limit theorem for the free energy at any temperature and give an explicit formula for the limiting variance. Although we only deal with the case of Ising spins, all our results can be extended to the spherical models as well.

The Aizenman-Sims-Starr scheme and Parisi formula for mixed $p$-spin spherical models

The Parisi formula for the free energy in the spherical models with mixed even p-spin interactions was proven in Michel Talagrand. In this paper we study the general mixed p-spin spherical models including p-spin interactions for odd p. We establish the Aizenman Sims-Starr scheme and from this together with many well-known results and Dmitry Panchenko's recent proof on the Parisi ultrametricity conjecture, we prove the Parisi formula.

Replica analysis of the generalized p-spin interaction glass model

We investigate the stability of replica symmetry breaking solutions in generalized p-spin models. It is shown that the kind of the transition to the one-step replica symmetry breaking state depends not only on the presence or absence of the reflection symmetry of the generalized ‘spin’-operators but on the number of interacting operators and their individual characteristics.

Rigorous construction of the pure states for certain spin glass models

For a large class of Gaussian Hamiltonians, that includes the Sherrington–Kirkpatrick model, the p-spin interaction model for even p, and many others, we prove at each temperature (under a very mild and essentially necessary condition) that generically the “pure states” predicted by physics do exist. The configuration space decomposes in an essentially unique manner in a sequence of subsets on which the overlap takes essentially its maximum value. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Small-world hypergraphs on a bond-disordered Bethe lattice

We study the thermodynamic properties of spin systems with bond-disorder on small-world hypergraphs, obtained by superimposing a one-dimensional Ising chain onto a random Bethe graph with p-spin interactions. Using transfer-matrix techniques, we derive fixed-point equations describing the relevant order parameters and the free energy, both in the replica symmetric and one step replica symmetry breaking approximation. We determine the static and dynamic ferromagnetic transition and the spinglass transition within replica symmetry for all temperatures, and demonstrate corrections to these results when one step replica symmetry breaking is taken into account. The results obtained are in agreement with Monte-Carlo simulations.

The magnetization at high temperature for a p-spin interaction model with external field

This paper is devoted to a detailed and rigorous study of the magnetization at high temperature for a $p$-spin interaction model with external field, generalizing the Sherrington–Kirkpatrick model. In particular, we prove that $\langle \sigma_i \ra

Phase diagrams of soluble multi-spin glass models

We include p-spin interactions in a spherical version of a soluble mean-field spin-glass model proposed by van Hemmen. Due to the simplicity of the solutions, which do not require the use of the replica trick, we are able to carry out a detailed investigation of a number of special situations. For p ⩾ 3 , there appear first-order transitions between the paramagnetic and the ordered phases. In the presence of additional ferromagnetic interactions, we show that there is no stable mixed phase, with both ferromagnetic and spin-glass properties.

Fluctuations in a p-spin interaction model

We investigate the behavior of the free energy of the p-spin interaction variant of the SK-model for high temperatures. We admit arbitrary distributions of the interactions given that their distribution is symmetric around the origin and some exponential moment is finite. We show that there is a critical temperature β ˜ depending on p such that for β < β ˜ the free energy of the p-spin interaction model has normally distributed fluctuations.

Some remarks on p-spin interaction models in a random field

We present some calculations for the thermodynamic behavior of mean-field ferromagnetic p-spin interaction models in the presence of quenched random fields. For both Ising and spherical spin variables, we use the law of large numbers, without recourse to the replica trick, to obtain a free-energy functional in terms of the order parameter and an extra non-ordering density. In the spherical limit, we show that the transition is continuous for p=2, but turns into first order for p⩾3, regardless of the probability distribution of the random fields. In the Ising case, for p=2, we recover previously known results. The free-energy functional obtained in this treatment can be used as a starting point for a dynamical study of these models.

MAGNETIZATION OF THE PURE STATES IN THE p-SPIN INTERACTION MODEL

We study the magnetization of the pure states in (a suitable version of) the p-spin interaction model at low temperature. We give a rigorous proof of the phenomenon, discovered in 1 that (very roughly speaking), at a given level of accuracy, the set of sites decomposes in a finite number of cells on which the magnetization of any different pure states are uncorrelated.

On Guerra's broken replica-symmetry bound

Consider a random Hamiltonian H N( σ → ) for σ → ∈Σ N={0,1} N . We assume that the family (H N( σ → )) is jointly Gaussian centered and that for σ → 1, σ → 2∈Σ N, N −1EH N( σ → 1)H N( σ → 2) = ξ( N −1∑ i⩽ N σ 1 i σ 2 i ) for a certain function ξ on R . F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian H N( σ → )+h∑ i⩽Nσ i is bounded below by the free energy of the Parisi solution provided that ξ is convex on R . We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on R + . This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

The generalized Parisi formula

For a large class of Gaussian Hamiltonians, we use Guerra's interpolation scheme to show that at any temperature the free energy of the corresponding spin-glass system is asymptotically given by Parisi's formula. This class includes the Sherrington–Kirkpatrick model, the p-spin interaction model for even p, and many others. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Self-generated disorder: from spin glasses to the glassy homopolymer globule

We have investigated the interrelation between spin glasses and structural glasses. Spin glasses in this case are p-spin interaction spin glasses (at p>2) or Potts glasses which contain quenched disorder, whereas the structural glasses are here exemplified by the homopolymeric globule, which can be viewed as a liquid of connected units on nanoscales. It is argued that the homopolymeric globule problem can be mapped onto a disorder field theoretical model whose effective Hamiltonian resembles the corresponding one for the spin glass model. In this sense the disorder in the globule is self-generated (in contrast to spin glasses) and can be related with competing interactions (virial coefficients of different signs) and the chain connectivity. The work is aimed at giving a quantitative description of this analogy. We have investigated in the mean-field (MF) approximation the phase diagram of the homopolymeric globule where the transition line from the liquid to glassy globule is treated in terms of the replica symmetry breaking paradigm. The configurational entropy temperature dependence is also discussed.

A p-spin interaction Ashkin–Teller spin-glass model

A p-spin interaction Ashkin–Teller spin glass, with three independent Gaussian probability distributions for the exchange interactions, is studied by means of the replica method. A simple phase diagram is obtained within the replica-symmetric approximation, presenting an instability of the paramagnetic solution at low temperatures. The replica-symmetry-breaking procedure is implemented and a rich phase diagram is obtained; besides the paramagnetic phase, three distinct spin-glass phases appear. Three first-order critical frontiers are found and they all meet at a triple point; among such lines, two of them present discontinuities in the order parameters, but no latent heat, whereas the other one exhibits both discontinuities in the order parameters and a finite latent heat.

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 $<\psi(w) > \sim w^{-D}$, 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.