Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model.

We present a thorough numerical analysis of the relaxational dynamics of the Sherrington-Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes N. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition from a spin glass to a ferromagnetic phase. We show that finite-size effects induce the appearance of a distinctive slow regime in the relaxation dynamics, the extension of which depends on the size of the system and also on the strength of the non-disordered perturbation. The long time dynamics are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues of the spike random matrices in the different regimes, sub-critical, critical, and super-critical, confirming some known results and anticipating others, even in the less studied critical regime. We also numerically characterize the finite-size statistics of the gap, which we hope may encourage analytical work which is lacking. Finally, we compute the finite-size scaling of the long time relaxation of the energy, showing the existence of power laws with exponents that depend on the strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap.

Barriers, trapping times, and overlaps between local minima in the dynamics of the disordered Ising p-spin model.

We study the low-temperature out-of-equilibrium Monte Carlo dynamics of the disordered Ising p-spin Model with p=3 and a small number of spin variables. We focus on sequences of configurations that are stable against single spin flips obtained by instantaneous gradient descent from persistent ones. We analyze the statistics of energy gaps, energy barriers, and trapping times on subsequences such that the overlap between consecutive configurations does not overcome a threshold. We compare our results to the predictions of various trap models finding the best agreement with the step model when the p-spin configurations are constrained to be uncorrelated.

Three-state model with competing antiferromagnetic and pairing interactions.

Motivated by the rich phase diagram of the high-temperature superconductors, we introduce a pseudospin model with three state variables which can be interpreted as two states (spin ±1/2) particles and holes. The Hamiltonian has a term which favors antiferromagnetism and an additional competing interaction which favors bonding between pairs of antiparallel spins mediated by holes. For low concentration of holes the dominant interaction between particles has antiferromagnetic character, leading to an antiferromagnetic phase in the temperature-hole concentration phase diagram, qualitatively similar to the antiferromagnetic phase of doped Mott insulators. For growing concentration of holes antiferromagnetic order is weakened and a phase with a different kind of order mediated by holes appears. This last phase has the form of a dome in the T-hole concentration plane. The whole phase diagram resembles those of some families of high-T_{c} superconductors. We compute the phase diagram in the mean-field approximation and characterize the different phase transitions through Monte Carlo simulations.

Index statistical properties of sparse random graphs.

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability P(N)(K,λ) that a large N×N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of P(N)(K,λ), from which the typical sample-to-sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N≫1 for |λ|>0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a nonmonotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as lnN, with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.

Nematic phase in the J(1)-J(2) square-lattice Ising model in an external field.

The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.

Nature of long-range order in stripe-forming systems with long-range repulsive interactions.

We study two dimensional stripe forming systems with competing repulsive interactions decaying as r(-α). We derive an effective Hamiltonian with a short-range part and a generalized dipolar interaction which depends on the exponent α. An approximate map of this model to a known XY model with dipolar interactions allows us to conclude that, for α<2 long-range orientational order of stripes can exist in two dimensions, and establish the universality class of the models. When α≥2 no long-range order is possible, but a phase transition in the Kosterlitz-Thouless universality class is still present. These two different critical scenarios should be observed in experimentally relevant two dimensional systems like electronic liquids (α=1) and dipolar magnetic films (α=3). Results from Langevin simulations of Coulomb and dipolar systems give support to the theoretical results.

Energy landscape of the finite-size mean-field 2-spin spherical model and topology trivialization.

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite-size 2-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-Hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.

Structural signatures of (two) characteristic dynamical temperatures in lithium metasilicate.

We report on the dynamic and structural characterization of lithium metasilicate Li2SiO3, a network-forming ionic glass, by means of molecular dynamics simulations. The system is characterized by a network of SiO4 tetrahedra disrupted by Li ions which diffuse through the network. Measures of mean square displacement and the diffusion constant of Si and O atoms allow us to identify the mode-coupling temperature, Tc ≈ 1500 K. At a much lower temperature, a change in the slope of the specific volume versus temperature singles out the glass transition at Tg ≈ 1000 K, the temperature below which the system goes out of equilibrium. We find signatures of both dynamical temperatures in structural order parameters related to the orientation of the tetrahedra. At lower temperatures we find that a set of order parameters which measure the relative orientation of neighbouring tetrahedra cease to increase and stay constant below Tc. Nevertheless, the bond orientational order parameter, which in this system measures local tetrahedral order, is found to continue growing below Tc until Tg, below which it remains constant. Although these structural signatures of the two dynamical temperatures do not imply any real thermodynamic transition in terms of the order parameters, they do give insight into the relaxation processes that occur between Tc and Tg, in particular they allow us to characterize the nature of the crossover happening around Tc.

Nematic phase in two-dimensional frustrated systems with power-law decaying interactions.

We address the problem of orientational order in frustrated interaction systems as a function of the relative range of the competing interactions. We study a spin model Hamiltonian with short-range ferromagnetic interaction competing with an antiferromagnetic component that decays as a power law of the distance between spins, 1/r(α). These systems may develop a nematic phase between the isotropic disordered and stripe phases. We evaluate the nematic order parameter using a self-consistent mean-field calculation. Our main result indicates that the nematic phase exists, at mean-field level, provided 0<α<4. We analytically compute the nematic critical temperature and show that it increases with the range of the interaction, reaching its maximum near α~0.5. We also compute a coarse-grained effective Hamiltonian for long wavelength fluctuations. For 0<α<4 the inverse susceptibility develops a set of continuous minima at wave vectors |k[over arrow]|=k(0)(α) which dictate the long-distance physics of the system. For α→4, k(0)→0, making the competition between interactions ineffective for greater values of α.

Glassy behavior of two-dimensional stripe-forming systems.

We study two-dimensional frustrated but nondisordered systems applying a replica approach to a stripe-forming model with competing interactions. The phenomenology of the model is representative of several well-known systems, like high-Tc superconductors and ultrathin ferromagnetic films, which have been the subject of intense research. We establish the existence of a glass transition to a nonergodic regime accompanied by an exponential number of long-lived metastable states, responsible for slow dynamics and nonequilibrium effects.

Energy landscape of the finite-size spherical three-spin glass model.

We study the three-spin spherical model with mean-field interactions and Gaussian random couplings. For moderate system sizes of up to 20 spins, we obtain all stationary points of the energy landscape by means of the numerical polynomial homotopy continuation method. On the basis of these stationary points, we analyze the complexity and other quantities related to the glass transition of the model and compare these finite-system quantities to their exact counterparts in the thermodynamic limit.

Nematic phase in stripe-forming systems within the self-consistent screening approximation.

We show that in order to describe the isotropic-nematic transition in stripe-forming systems with isotropic competing interactions of the Brazovskii class it is necessary to consider the next to leading order in a 1/N approximation for the effective Hamiltonian. This can be conveniently accomplished within the self-consistent screening approximation. We solve the relevant equations and show that the self-energy in this approximation is able to generate the essential wave vector dependence to account for the anisotropic character of a two-point correlation function characteristic of a nematic phase.

Coarse-grained models of stripe forming systems: phase diagrams, anomalies, and scaling hypothesis.

Two coarse-grained models which capture some universal characteristics of stripe forming systems are studied. At high temperatures, the structure factors of both models attain their maxima on a circle in reciprocal space, as a consequence of generic isotropic competing interactions. Although this is known to lead to some universal properties, we show that the phase diagrams have important differences, which are a consequence of the particular k dependence of the fluctuation spectrum in each model. The phase diagrams are computed in a mean field approximation and also after inclusion of small fluctuations, which are shown to modify drastically the mean field behavior. Observables like the modulation length and magnetization profiles are computed for the whole temperature range accessible to both models and some important differences in behavior are observed. A stripe compression modulus is computed, showing an anomalous behavior with temperature as recently reported in related models. Also, a recently proposed scaling hypothesis for modulated systems is tested and found to be valid for both models studied.

Interplay between coarsening and nucleation in an Ising model with dipolar interactions.

We study the dynamical behavior of a square lattice Ising model with exchange and dipolar interactions by means of Monte Carlo simulations. After a sudden quench to low temperatures, we find that the system may undergo a coarsening process where stripe phases with different orientations compete, or alternatively it can relax initially to a metastable nematic phase and then decay to the equilibrium stripe phase through nucleation. We measure the distribution of equilibration times for both processes and compute their relative probability of occurrence as a function of temperature and system size. This peculiar relaxation mechanism is due to the strong metastability of the nematic phase, which goes deep into the low-temperature stripe phase. We also measure quasiequilibrium autocorrelations in a wide range of temperatures. They show a distinct decay to a plateau that we identify as due to a finite fraction of frozen spins in the nematic phase. We find indications that the plateau is a finite-size effect. Relaxation times as a function of temperature in the metastable region show super-Arrhenius behavior, suggesting a possible glassy behavior of the system at low temperatures.

Competing interactions, the renormalization group, and the isotropic-nematic phase transition.

We discuss 2D systems with Ising symmetry and competing interactions at different scales. In the framework of the renormalization group, we study the effect of relevant quartic interactions. In addition to the usual constant interaction term, we analyze the effect of quadrupole interactions in the self-consistent Hartree approximation. We show that in the case of a repulsive quadrupole interaction, there is a first-order phase transition to a stripe phase in agreement with the well-known Brazovskii result. However, in the case of attractive quadrupole interactions there is an isotropic-nematic second-order transition with higher critical temperature.

Fickian crossover and length scales from two point functions in supercooled liquids.

Particle motion of a Lennard-Jones supercooled liquid near the glass transition is studied by molecular dynamics simulations. We analyze the wave vector dependence of relaxation times in the incoherent self-scattering function and show that at least three different regimes can be identified and its scaling properties determined. The transition from one regime to another happens at characteristic length scales. The length scale associated with the onset of Fickian diffusion corresponds to the maximum size of heterogeneities in the system, and the characteristic time scale is several times larger than the alpha relaxation time. A second crossover length scale is observed, which corresponds to the typical time and length of heterogeneities, in agreement with results from four point functions. The different regimes can be traced back to the behavior of the van Hove distribution of displacements, which shows a characteristic exponential regime in the heterogeneous region before the crossover to Gaussian diffusion and should be observable in experiments. Our results show that it is possible to obtain characteristic length scales of heterogeneities through the computation of two point functions at different times.

Topology, phase transitions, and the spherical model.

The topological hypothesis states that phase transitions should be related to changes in the topology of configuration space. The necessity of such changes has already been demonstrated. We characterize exactly the topology of the configuration space of the short range Berlin-Kac spherical model, for spins lying in hypercubic lattices of dimension d. We find a continuum of changes in the topology and also a finite number of discontinuities in some topological functions. We show, however, that these discontinuities do not coincide with the phase transitions which happen for d > or = 3, and conversely, that no topological discontinuity can be associated with them. This is the first short range, confining potential for which the existence of special topological changes are shown not to be sufficient to infer the occurrence of a phase transition.

Quasistationary trajectories of the mean-field XY Hamiltonian model: a topological perspective.

We employ a topological approach to investigate the nature of quasistationary states of the mean-field XY Hamiltonian model. We focus on the quasistationary states reached when the system is initially prepared in a fully magnetized configuration. By means of numerical simulations and analytical considerations, we show that, along the quasistationary trajectories, the system evolves in a manifold of critical points of the potential energy function. Although these critical points are maxima, the large number of directions with marginal stability may be responsible for the slow relaxation dynamics and the trapping of the system in such trajectories.

Topological hypothesis on phase transitions: the simplest case.

We critically analyze the possibility of finding signatures of a phase transition by looking exclusively at static quantities of statistical systems, like, e.g., the topology of potential energy submanifolds (PES's). This topological hypothesis has been successfully tested in a few statistical models but up to now there has been no rigorous proof of its general validity. We make a new test of it analyzing the, probably, simplest example of a nontrivial system undergoing a continuous phase transition: the completely connected version of the spherical model. Going through the topological properties of its PES it is shown that, as expected, the phase transition is correlated with a change in their topology. Nevertheless, this change, as reflected in the behavior of a particular topological invariant, the Euler characteristic, is small, at variance with the strong singularity observed in other systems. Furthermore, it is shown that in the presence of an external field, when the phase transition is destroyed, a similar topology change in the submanifolds is still observed at the maximum value of the potential energy manifold, a level which nevertheless is thermodynamically inaccessible. This suggests that static properties of the PES's are not enough in order to decide whether a phase transition will take place; some input from dynamics seems necessary.