Mean-Field Analysis of the q-Voter Model on Networks

We present a detailed investigation of the behavior of the nonlinear q-voter model for opinion dynamics. At the mean-field level we derive analytically, for any value of the number q of agents involved in the elementary update, the phase diagram, the exit probability and the consensus time at the transition point. The mean-field formalism is extended to the case that the interaction pattern is given by generic heterogeneous networks. We finally discuss the case of random regular networks and compare analytical results with simulations.     

Dynamic transitions in Domany-Kinzel cellular automata on small-world network

We study Domany-Kinzel cellular automata on small-world network. Every link on a one dimensional chain is rewired and coupled with any node with probability p. We observe that, the introduction of long-range interactions does not remove the critical character of the model and the system still exhibits a well-defined phase transition to absorbing state. In case of directed percolation (DP), we observe a very anomalous behavior as a function of size. The system shows long lived metastable states and a jump in order parameter. This jump vanishes in thermodynamic limit and we recover second-order transition. The critical exponents are not equal to the mean-field values even for large p. However, for compact directed percolation(CDP), the critical exponents reach their mean-field values even for small p.     

Effective scalar field theory for the electroweak phase transition

We investigate an effective model for the finite-temperature symmetry-restoration phase transition of the electroweak theory. It is obtained by dimensional reduction of the (3 + 1)-dimensional full theory and by subsequent integration over all static gauge degrees of freedom. The resulting theory corresponds to a 3-dimensional O(4) ferromagnet containing cubic and quartic terms of the field in its potential function. Possible nonperturbative effects of a magnetic screening mass are parametrically included in the potential. We analyse the theory using mean-field and numerical Monte Carlo (MC) simulation methods. At the value of the physical Higgs mass, m_H = 37 GeV, considered in the present investigation, we find a discontinuous symmetry-restoring phase transition. We determine the critical temperature, order parameter jump, interface tension and latent heat characteristics of the transition. The Monte Carlo results indicate a somewhat weaker first-order phase transition as compared to the mean-field treatment, demonstrating that non-perturbative fluctuations of the Higgs field are relevant. This effect is especially important for the interface tension. Any observation of hard first-order transition could result only from non-perturbative effects related to the gauge degrees of freedom.     

Mean-field renormalization group for the q-state clock spin-glass model

The mean-field renormalization group is used to study the phase diagrams of a d-dimensional q-state clock spin-glass model. We found, for q = 3 clock, the transition from paramagnet to spin glass is an isotropic spin-glass phase, but for q = 4 clock, the transition from paramagnet to spin glass is an anisotropic spin-glass phase. However, for q ⩾ 5 clock, the result of anisotropic spin-glass phase depends on the temperature and the distribution of random coupling. While the coordinate number approaches infinity, the critical temperature evaluated by the mean-field renormalization group method is equal to that by the replica method.     

Nonlocal PNJL model beyond mean field

A nonlocal chiral quark model is consistently extended beyond mean field using a strict 1/N _c expansion scheme. It is found that the 1/N _c corrections lead to a lowering of the temperature of the chiral phase transition in comparison with the mean-field result. On the other hand, near the phase transition the 1/N _c expansion breaks down and a nonperturbative scheme for the inclusion of mesonic correlations is needed in order to describe the phase transition region.     

Kinetic phase transitions and tricritical point in an Ising model with competing dynamics

The nonequilibrium phase diagram of an Ising model in which the evolution of the spins reflects diffusion as well as ordering, is determined via dynamic mean-field theory (pair approximation). There is a first-order transition (in zero field) for sufficiently large diffusion rates.     

Step-by-step first order antiferroelectric-paraelectric transition induced by frustration and electric field

We show analytically that even not too strong frustrating next neighbor interaction strongly affects first order antiferroelectric-paraelectric transition in an external electric field. We apply mean-field Landau theory. In the electric field a single phase transition at T ₀ splits into a step-by-step staircase with a series of intermediate phases. Unexpectedly enough we found that the equilibrium structures of the phases differ substantially from structures formed at low temperature both without field and in field. Polarization of intermediate structures decreases with temperature in a stepwise manner. Similar step-by-step transitions can occur also in magnetic materials with frustrating interaction.     

Noisy Hegselmann-Krause Systems: Phase Transition and the 2<Emphasis Type="Italic">R</Emphasis>-Conjecture

The classic Hegselmann-Krause (HK) model for opinion dynamics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. We introduce an order parameter to track the phase transition and resolve the corresponding phase diagram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis confirms previous simulations and predicts properties of the noisy HK model in higher dimension.     

Competition between crystalline electric field singlet and itinerant states of f electrons

A new kind of phase transition is proposed for lattice fermion systems with simplified f ² configurations at each site. The free energy of the model is computed in the mean-field approximation for both the itinerant state with the Kondo screening, and a localized state with the crystalline electric field (CEF) singlet at each site. The presence of a first-order phase transition is demonstrated in which the itinerant state changes into the localized state toward lower temperatures. In the half-filled case, the insulating state at high temperatures changes into a metallic state, in marked contrast with the Mott transition in the Hubbard model. For comparison, corresponding states are discussed for the twoimpurity Kondo system with f ¹ configuration at each site.     

Infinite Range Interaction Model of a Structural Glass

In this paper a simple mean-field model for the liquid-glass phase transition is proposed. This is the low density D-dimensional system of N particles interacting via infinite-range oscillating potential. In the framework of the replica approach it is shown that such a system exhibits the phase transition between the high-temperature liquid phase and the low-temperature glass phase. This phase transition is described in terms of the standard one-step replica symmetry breaking scheme.     

On the Interplay of Magnetic and Molecular Forces in Curie–Weiss Ferrofluid Models

We consider a mean-field continuum model of classical particles in R ^d with Ising or Heisenberg spins. The interaction has two ingredients, a ferromagnetic spin coupling and a spin-independent molecular force. We show that a feedback between these forces gives rise to a first-order phase transition with simultaneous jumps of particle density and magnetization per particle, either at the threshold of ferromagnetic order or within the ferromagnetic region. If the direct particle interaction alone already implies a phase transition, then the additional spin coupling leads to an even richer phase diagram containing triple (or higher order) points.     

Magnets with random uniaxial anisotropy: Thermodynamic properties in the large-N limit

We show that the random-axis model lends itself to a systematic large-N calculation. The model shows different behavior below and above four dimensions. The equation of state is derived and discussed in terms of “Arrott” plots. Higher-order terms in the disorder, when summed, have a crucial effect on the susceptibility which is found to be finite below four dimensions (and above four dimensions for strong disorder). A spin-glass to paramagnetic phase transition is characterized by the vanishing of the Edwards-Anderson order parameter, which differs from zero in the spin-glass phase. A cusp in the specific-heat and susceptibility is seen across the transition. The cross-over exponent and other exponents of interest are calculated. Above four dimensions a third phase appears for weak disorder and low-temperature ferromagnetic in nature. The transverse and longitudinal susceptibilities are discussed. Whereas the ferromagnetic transition is characterized by mean-field exponents, the ferromagnetic to spin-glass exponents are equal to their counterparts in the non-random system in d ? 2 dimensions. This is shown to originate from an effective random field proportional to the EA order parameter. The flow equations in the large-N limit are also discussed.     

Interpretation of the specific heat at the Jahn-Teller phase transition in DyVO4 using a “compressible” Ising model

The specific heat at the Jahn-Teller phase transition in DyVO₄ is too sharply peaked to be explained by simple mean-field theory, but is very well described by using a mean-field model in which the effective interaction increases upon ordering according to J′=J₀′(1+ζ〈σ^z〉²) where ζ is an adjustable parameter.     

Spin glass behavior and critical analysis across the magnetic phase transition in Gd2CoMnO6: dc magnetization and ac susceptibility study

We report here on critical analysis across magnetic phase transition and spin dynamics in Gd₂CoMnO₆. We found that this material behaves differently below and above the applied magnetic field of 20 kOe. The magnetic phase transition switches from nearly mean-field type to unusual class and T_c shifts towards the high temperature above 20 kOe field. The nature of the magnetic phase transition is explored by carrying out critical analysis at low as well as at high magnetic field. The critical exponents obtained at low field using Kouvel-Fisher method are β = 0.65 (2) γ = 0.90 (2), δ = 2.43 and T_c = 120 K. Apparently, these values of critical exponents appear close to mean-field model. For high field the critical exponents are β = 1.24 (2) γ = 0.64 (5), δ = 1.51 (3) and T_c = 128 K. The critical exponents show significant deviation from any universal class. This switchover in the nature of the magnetic phase transition is unique and not seen in many compounds. The formation of non-Griffiths-like clusters in this compound can be a reason for such unique behavior. Further, ac susceptibility has been measured to understand the spin dynamics in detail. The dispersion of frequency-dependent χ_ac below T_c confirms a spin glass state in this material. The observed value of τ_o and T_o indicate the slow dynamic spin which is caused by co-existence of Co/Mn spin magnetic moments. The magneto-caloric effect is also presented for Gd₂CoMnO₆ in this study. The magnetic study and critical analysis across the phase transition reveal a switchover in the nature of phase transition in this material. A non-Griffiths like cluster formation above T_c is found and dynamic susceptibility study reveals a spin glass state below T_c in Gd₂CoMnO₆.     

Aspects of the phase diagram in (P)NJL-like models

We discuss three applications of NJL- and PNJL-like models to assess aspects of the QCD phase diagram: First, we study the effect of mesonic correlations on the pressure below and above the finite temperature phase transition within a non-local PNJL model beyond the mean-field approximation. Second, we reconstruct the phase boundary of an NJL model from a Taylor expansion of the chiral susceptibility about μ=0 and compare the result with the exact phase boundary. Finally, we demonstrate the realization of the “non-standard scenario” for the critical surface in a three-flavor PNJL model with a μ-dependent determinant interaction.     

A New First-Order Phase Transition for an Extended Jaynes–Cummings–Dicke Model with a High-Finesse Optical Cavity in the BEC System<Superscript>†</Superscript>

We present a two-level atomic Bose–Einstein condensate (BEC) with dispersion, which is coupled to a high-finesse optical cavity. We call this model the extended Jaynes–Cummings–Dicke (JC-Dicke) model and introduce an effective Hamiltonian for this system. From the direct product of Heisenberg–Weyl (HW) coherent states for the field and U(2) coherent states for the matter, we obtain the potential energy surface of the system. Within the framework of the mean-field approach, we evaluate the variational energy as the expectation value of the Hamiltonian for the considered state. We investigate numerically the quantum phase transition and the Berry phase for this system. We find the influence of the atom–atom interactions on the quantum phase transition point and obtain a new phase transition occurring when the microwave amplitude changes. Furthermore, we observe that the coherent atoms not only shift the phase transition point but also affect the macroscopic excitations in the superradiant phase.     

Finite temperature neutral-ionic transition and lattice dimerization in charge-transfer complexes: QMC study

We numerically investigate the neutral-ionic (NI) phase transition on the basis of the ionic extended Hubbard model with electron-lattice coupling as well as inter-chain Coulomb interaction. Finite-temperature (T) phase transitions are examined by a quantum Monte Carlo method, within adiabatic approximation for the lattice displacement together with inter-chain mean-field treatment. The NI transition either with or without the electron-lattice coupling is of first-order at low-T and transforms into a cross-over regime with increasing T. We confirm there exist two phases in the ionic region: with and without lattice dimerization, as suggested in recent experiments. The former is ferroelectric, which is rooted in the Coulomb interactions and a spin-Peierls mechanism.     

Progress in numerical simulations of systems with a θ-vacuum like term: The two- and three-dimensional Ising model within an imaginary magnetic field

Using the approach developed in Azcoiti et al. (2003) [1], we succeeded in reconstructing the behaviour of the antiferromagnetic Ising model with imaginary magnetic field iθ for two and three dimensions in the low temperature regime. A mean-field calculation, expected to work well for high dimensions, is also carried out, and the mean-field results coincide qualitatively with those of the two- and three-dimensional Ising model. The mean-field analysis reveals also a phase structure more complex than the one expected for QCD with a topological θ-term.     

Dynamical generation of a Coulomb phase in the mean-field approach to Z(N) lattice gauge theories

The phase diagram of Z(N) lattice gauge theories is examined in the mean-field approach. It is shown how large non-perturbative fluctuations around the mean-field may naturally occur in these theories. When this happens, a massless phase which is absent in the mean-field approximation is dynamically generated. An order parameter which characterizes the Coulomb to Higgs phase transition is introduced. The predictions for the values of the critical coupling for this transition are in excellent agreement with the Monte Carlo data in four dimensions.