An Improved Heterogeneous Mean-Field Theory for the Ising Model on Complex Networks

Heterogeneous mean-field theory is commonly used methodology to study dynamical processes on complex networks,such as epidemic spreading and phase transitions in spin models.In this paper,we propose an improved heterogeneous mean-field theory for studying the Ising model on complex networks.Our method shows a more accurate prediction in the critical temperature of the Ising model than the previous heterogeneous mean-field theory.The theoretical results are validated by extensive Monte Carlo simulations in various types of networks.     

On the Mean-Field Spherical Model

Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci/ of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N. The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor σ-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.     

Asymptotics of the Mean-Field Heisenberg Model

We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramér- and Sanov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Stein’s method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.     

Nambu-Jona-Lasinio Model Beyond the Mean-Field Approximation

1. Department of Technical Physics, Peking University, Beijing 100871, China;
2. Department of Physics, Peking University, Beijing 100871, China     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

Slave-Fermion Mean-Field Theory of Heisenberg Model

A nearly half-filled two-dimensional Heisenberg model is investigated. A slave-fermion method with fermions as the charge carriers and bosons as the spin carriers is proposed. The ground state shows antiferromagnetic long range order at T = 0. The spin-spin correlation and static susceptibility are also obtained.     

Slave-Fermion Mean-Field Theory of Heisenberg Model

A nearly half-filled two-dimensional Heisenberg model is investigated. A slave-fermion method with fermions as the charge carriers and bosons as the spin carriers is proposed. The ground state shows antiferromagnetic long range order at T = 0. The spin-spin correlation and static susceptibility are also obtained.     

Consensus,Polarization and Hysteresis in the Three-State Noisy q-Voter Model with Bounded Confidence

In this work, we address the question of the role of the influence of group size on the emergence of various collective social phenomena, such as consensus, polarization and social hysteresis. To answer this question, we study the three-state noisy q-voter model with bounded confidence, in which agents can be in one of three states: two extremes (leftist and rightist) and centrist. We study the model on a complete graph within the mean-field approach and show that, depending on the size q of the influence group, saddle-node bifurcation cascades of different length appear and different collective phenomena are possible. In particular, for all values of q>1, social hysteresis is observed. Furthermore, for small values of q∈(1,4), disagreement, polarization and domination of centrists (a consensus understood as the general agreement, not unanimity) can be achieved but not the domination of extremists. The latter is possible only for larger groups of influence. Finally, by comparing our model to others, we discuss how a small change in the rules at the microscopic level can dramatically change the macroscopic behavior of the model.     

Infrared Bound and Mean-Field Behaviour in the Quantum Ising Model

We prove an infrared bound for the transverse field Ising model. This bound is stronger than the previously known infrared bound for the model, and allows us to investigate mean-field behaviour. As an application we show that the critical exponent γ for the susceptibility attains its mean-field value γ = 1 in dimension at least 4 (positive temperature), respectively 3 (ground state), with logarithmic corrections in the boundary cases.     

Double-Chain Mean-Field Renormalizat ion Group Approach for the Ising Model

A double-chain mean-field renormalization group approach for the Ising model is formulated. With the approach, transition temperatures and critical exponents for several numbers of nearest-neighbor chains are obtained.     

Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs

A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the d-dimensional Euclidean space with \(d\ge 2\). Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clustered random graphs. This random network model can be thought of as a corrected mean-field model for the interaction graph between the attempted spheres. Using functional limit theorems, we characterize the fraction of accepted spheres and its fluctuations.     

Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling

We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two-dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.     

The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n ^3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n ^3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β _c  = 1. In particular, we find a scaling window of order around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ ² n → ∞ with n, the mixing-time has order (n/δ) log(δ ² n), and exhibits cutoff with constant and window size n/δ. In the critical window, β = 1± δ, where δ ² n is O(1), there is no cutoff, and the mixing-time has order n ^3/2. At low temperature, β = 1 + δ for δ > 0 with δ ² n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order . Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.     

Small-Scale Analysis of Hydrodynamical Helicity Suppression in the Mean-Field Dynamo-Model

Journal of Experimental and Theoretical Physics - We compare the classical mean-field dynamo model proposed by Steenbeck, Krause, and Rädler to describe the generation of large-scale magnetic...     

Shape Coexistence in Neutron-Deficient At Isotopes in Relativistic Mean-Field Model

The potential energy surfaces are calculated for neutron-deficient At isotopes from A - 190 to 207 in an axially deformed relativistic mean-field approach, using a quadratic constraint scheme for the first time. We find several minima in the potential energy surface for each nucleus, shape-coexistence, and quadratic deform are discussed.     

Self-Organized Criticality Analysis of Earthquake Model Based on Heterogeneous Networks

The original Olami-Feder-Christensen (OFC) model, which displays a robust power-law behavior, is a quasistatic two-dimensional version of the Burridge--Knopoff spring-block model of earthquakes. In this paper, we introduce a modified OFC model based on heterogeneous network, improving the redistribution rule of the original model. It can be seen as a generalization of the original OFC model. We numerically investigate the influence of theparameters θ and β, which respectively control the intensity of the evolutivemechanism of the topological growth and the inner selection dynamicsin our networks, and find that there are two distinct phases in theparameter space (θ, β). Meanwhile, we study the influence of the control parameter a either. Increasing a, the earthquake behavior of the model transfers from local to global.