P-spin Glass Model Research Articles

Gauge theory for quantum XYZ spin glasses

Nishimori’s gauge theory is extended to the quantum XYZ p-spin glass model in finite dimensions. This enables us to obtain useful correlation equalities, which show also that Duhamel correlation functions at an arbitrary temperature are bounded by those in the corresponding classical model on the Nishimori line. These bounds give that the spontaneous magnetization vanishes in any low temperature even if the model enters the Z2 -symmetry broken spin glass phase. This theory explains well-known fact from experiments and numerical calculations that the magnetic susceptibility does not diverge in the spin glass transition. The new gauge theory together with the known phase diagram of the Edwards–Anderson model can specify the spin glass region in the coupling constant space of the quantum Heisenberg XYZ spin glass model.

Probing chaos in the spherical p-spin glass model

We study the dynamics of a quantum p-spin glass model starting from initial states defined in microcanonical shells, in a classical regime. We compute different chaos estimators, such as the Lyapunov exponent and the Kolmogorov-Sinai entropy, and find a marked maximum as a function of the energy of the initial state. By studying the relaxation dynamics and the properties of the energy landscape we show that the maximal chaos emerges in correspondence with the fastest spin relaxation and the maximum complexity, thus suggesting a qualitative picture where chaos emerges as the trajectories are scattered over the exponentially many saddles of the underlying landscape. We also observe hints of ergodicity breaking at low energies, indicated by the correlation function and a maximum of the fidelity susceptibility.

Quantum-to-Classical Crossover in Many-Body Chaos and Scrambling from Relaxation in a Glass.

Chaotic quantum systems with Lyapunov exponent λ_{L} obey an upper bound λ_{L}≤2πk_{B}T/ℏ at temperature T, implying a divergence of the bound in the classical limit ℏ→0. Following this trend, does a quantum system necessarily become "more chaotic" when quantum fluctuations are reduced? Moreover, how do symmetry breaking and associated nontrivial dynamics influence the interplay of quantum mechanics and chaos? We explore these questions by computing λ_{L}(ℏ,T) in the quantum spherical p-spin glass model, where ℏ can be continuously varied. We find that quantum fluctuations, in general, make paramagnetic phase less and the replica symmetry-broken spin glass phase more chaotic. We show that the approach to the classical limit could be nontrivial, with nonmonotonic dependence of λ_{L} on ℏ close to the dynamical glass transition temperature T_{d}. Our results in the classical limit (ℏ→0) naturally describe chaos in supercooled liquid in structural glasses. We find a maximum in λ_{L}(T) substantially above T_{d}, concomitant with the crossover from simple to slow glassy relaxation. We further show that λ_{L}∼T^{α}, with the exponent α varying between 2 and 1 from quantum to classical limit, at low temperatures in the spin glass phase.

The quantum p-spin glass model: a user manual for holographers

We study a large-N bosonic quantum mechanical sigma-model with a spherical target space subject to disordered interactions, more colloquially known as the p-spin spherical model. Replica symmetry is broken at low temperatures and for sufficiently weak quantum fluctuations, which drives the system into a spin glass (SG) phase. The first half of this paper is dedicated to a discussion of this model’s thermodynamics, with particular emphasis on the marginally stable SG. This phase exhibits an emergent conformal symmetry in the strong coupling regime, which dictates its thermodynamic properties. It is associated with an extensive number of nearby states in the free energy landscape. We discuss in detail an elegant approximate solution to the SG equations, which interpolates between the conformal regime and an ultraviolet-complete short distance solution. In the second half of this paper we explore the real-time dynamics of the model and uncover quantum chaos as measured by out-of-time-order four-point functions, both numerically and analytically. We find exponential Lyapunov growth, which intricately depends on the model’s couplings and becomes strongest in the quantum critical regime. We emphasize that the SG phase also exhibits quantum chaos, albeit with parametrically smaller Lyapunov exponent than in the replica symmetric phase. An analytical calculation in the marginal SG phase suggests that this Lyapunov exponent vanishes in a particular infinite coupling limit. We comment on the potential meaning of these observations from the perspective of holography.

Large deviations of glassy effective potentials

The theory of glassy fluctuations can be formulated in terms of disordered effective potentials. While the properties of the average potentials are well understood, the study of the fluctuations has been so far quite limited. Close to the MCT transition, fluctuations induced by the dynamical heterogeneities in supercooled liquids can be described by a cubic field theory in presence of a random field term. In this paper, we set up the general problem of the large deviations going beyond the assumption of the vicinity to T MCT and analyze it in the paradigmatic case of spherical (p-spin) glass models. This tool can be applied to study the probability of the observation of dynamic trajectories with memory of the initial condition in regimes where, typically, the correlation C(t, 0) decays to zero at long times, at finite T and at T = 0.

On Spin Distributions for Generic p-Spin Models

We provide an alternative formula for spin distributions of generic p-spin glass models. As a main application of this expression, we write spin statistics as solutions of partial differential equations and we show that the generic p-spin models satisfy multiscale Thouless-Anderson-Palmer equations as originally predicted in the work of Mezard-Virasoro.

A method that reveals the multi-level ultrametric tree hidden in p -spin-glass-like systems

.In the study of disordered models like spin glasses the key object of interest is the rugged energy hypersurface defined in configuration space. The statistical mechanics calculation of the Gibbs–Boltzmann partition function gives the information necessary to understand the equilibrium behavior of the system as a function of the temperature but is not enough if we are interested in the more general aspects of the hypersurface: it does not give us, for instance, the different degrees of ruggedness at different scales. In the context of the replica symmetry breaking (RSB) approach we discuss here a rather simple extension that can provide a much more detailed picture. The attractiveness of the method relies on the fact that it is conceptually transparent and the additional calculations are rather straightforward. We think that this approach reveals an ultrametric organisation with many levels in models like p-spin glasses when we include saddle points. In this first paper we present detailed calculations for the spherical p-spin glass model where we discover that the corresponding decreasing Parisi function q(x) codes this hidden ultrametric organisation.

Continuous and discontinuous transitions in generalized p-spin glass models

We investigate the generalized p-spin models that contain arbitrary diagonal operators with no reflection symmetry. We derive general equations that give an opportunity to uncover the behavior of the system near the glass transition at different (continuous) p. The quadrupole glass with J = 1 is considered as an illustrative example. It is shown that the crossover from continuous to discontinuous glass transition to the one-step replica breaking solution (1RSB) takes place at p = 3.3 for this model. For p < 2 + Δp, where Δp = 0.5 is a finite value, the stable 1RSB solution disappears. This behavior is very different from that of the p-spin Ising glass model.

Calculation of the 1RSB transition temperature of spin glass models on regular random graphs under the replica symmetric ansatz

We study p-spin glass models on regular random graphs. By analyzing the Franz–Parisi potential with a two-body cavity field approximation under the replica symmetric ansatz, we obtain a good approximation of the 1RSB transition temperature for p = 3. Our calculation method is much easier than the 1RSB cavity method because the result is obtained by solving self-consistent equations with Newton’s method.

Universality in p-spin glasses with correlated disorder

We introduce a new method, based on the recently developed random tensor theory, for studying the p-spin glass model with non-Gaussian, correlated disorder. Using a suitable generalization of Gurau’s theorem on the universality of the large N limit of the p-unitary ensemble of random tensors, we exhibit an infinite family of such non-Gaussian distributions which leads to the same low temperature phase as the Gaussian distribution. While this result is easy to show (and well known) for uncorrelated disorder, its robustness with respect to strong quenched correlations is surprising. We show in detail how the critical temperature is renormalized by these correlations. We close with speculation on possible applications of random tensor theory to finite-range spin glass models.

Following states in temperature in the spherical s + p-spin glass model

In many mean-field glassy systems, the low-temperature Gibbs measure is dominated by exponentially many metastable states. We analyze the evolution of the metastable states as temperature changes adiabatically in the solvable case of the spherical s + p-spin glass model, extending the work of Barrat et al (1997 J. Phys. A: Math. Gen. 30 5593). We confirm the presence of level crossings, bifurcations, and temperature chaos. For the states that are at equilibrium close to the so-called dynamical temperature Td, we find, however, that the following state method (and the dynamical solution of the model as well) is intrinsically limited by the vanishing of solutions with non-zero overlap at low temperature.

Singular perturbation near mode-coupling transition

We study the simplest mode-coupling equation which describes the time-correlation function of the spherical p-spin glass model. We formulate a systematic perturbation theory near the mode-coupling transition point by introducing multiple time scales. In this formulation, the invariance with respect to the dilatation of time in a late stage yields an arbitrary constant in a leading-order expression of the solution. The value of this constant is determined by a solvability condition associated with a linear singular equation for perturbative corrections in the late stage. The solution thus constructed provides exactly the α-relaxation time.

Antiferromagnetic spherical spin-glass model

We study the thermodynamic properties and the phase diagrams of a multi-spin antiferromagnetic spherical spin-glass model using the replica method. It is a two-sublattice version of the ferromagnetic spherical p-spin glass model. We consider both the replica-symmetric and the one-step replica-symmetry-breaking solutions, the latter being the most general solution for this model. We find paramagnetic, spin-glass, antiferromagnetic and mixed or glassy antiferromagnetic phases. The phase transitions are always of second order in the thermodynamic sense, but the spin-glass order parameter may undergo a discontinuous change.

A quantum spherical p-spin glass model: Low-temperature behavior

Low-temperature slow dynamics, aging, and temperature and field shifts are studied analytically and numerically for a quantum p-spin (p = 3) glass model interacting with internal quantum environment and external magnetic fields. It is demonstrated that the correlation function and the real part of susceptibility are subject to aging and restoration (rejuvenation) under little changes of temperature and field. The examined effects are briefly analyzed.

Rigorous Inequalities Between Length and Time Scales in Glassy Systems

Glassy systems are characterized by an extremely sluggish dynamics without any simple sign of long range order. It is a debated question whether a correct description of such phenomenon requires the emergence of a large correlation length. We prove rigorous bounds between length and time scales implying the growth of a properly defined length when the relaxation time increases. Our results are valid in a rather general setting, which covers finite-dimensional and mean field systems. As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin glass models on random regular graphs. We present the first proof that a model of this type undergoes a purely dynamical phase transition not accompanied by any thermodynamic singularity.

Interface Free Energies inp-Spin Glass Models

The replica method is used to calculate the interface free energy associated with the change from periodic to antiperiodic boundary conditions in finite-dimensional p-spin glass models in the phase which at mean-field level has one-step replica symmetry breaking (1RSB). In any finite dimension the interface free energy is exponentially small for a large system. This result implies that, in finite dimensions, the 1RSB state does not exist, as it is destroyed by thermal excitation of arbitrarily large droplets. The implications of this for the theory of structural glasses are discussed.

Exact solutions for diluted spin glasses and optimization problems.

We study the low temperature properties of p-spin glass models with finite connectivity and of some optimization problems. Using a one-step functional replica symmetry breaking ansatz we can solve exactly the saddle-point equations for graphs with uniform connectivity. The resulting ground state energy is in perfect agreement with numerical simulations. For fluctuating connectivity graphs, the same ansatz can be used in a variational way: For p-spin models (known as p-XOR-SAT in computer science) it provides the exact configurational entropy together with the dynamical and static critical connectivities (for p = 3, gamma(d) = 0.818, and gamma(s) = 0.918), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds ( gamma(var)(c) = 4.396 and gamma(var)(c) = 2.149).

Multispin Ising spin glasses with ferromagnetic interactions

We consider the thermodynamics of an infinite-range Ising p-spin glass model with an additional r-spin ferromagnetic interaction. For r=2 there is a continuous transition to a ferromagnetic phase, while for r>2 the transition is first order. We find both glassy and non-glassy ferromagnetic phases, with replica symmetry breaking of both the one step and full varieties. We obtain new results for the case where r=p>2, demonstrating the existence of a non-glassy ferromagnetic phase, of significance to error-correcting codes.

Dynamics of structural models with a long-range interaction: glassy versus nonglassy behavior

By making use of the Langevin dynamics and its generating functional (GF) formulation, the influence of the long-range nature of the interaction on the tendency of the glass formation is systematically investigated. In doing so, two types of models are considered: (i) the nondisordered model with a pure repulsive type of interaction, and (ii) the model with a randomly distributed strength of interaction (a quenched disordered model). The long-ranged potential of interaction is scaled with a number of particles N in such a way as to enable for the GF the saddle-point treatment as well as the systematic 1/N expansion around it. We show that the nondisordered model has no glass transition, which is in line with the mean-field limit of the mode-coupling theory (MCT) predictions. On the other hand, the model with a long-range interaction that has a quenched disorder leads to MC equations which are generic for the p-spin glass model and polymeric manifold in a random media.

Barriers in the p-spin interacting spin-glass model: the dynamical approach

We investigate the barriers separating metastable states in the spherical p-spin glass model using the instanton method. We show that the problem of finding the barrier heights can be reduced to the causal two-real-replica dynamics. We find the probability for the system to escape one of the highest energy metastable states and the energy barrier corresponding to this process.