The existence of negative absolute temperatures in Axelrod’s social influence model

We introduce the concept of temperature as an order parameter in the standard Axelrod’s social influence model. It is defined as the relation between suitably defined entropy and energy functions, T=(∂S/∂E)⁻¹. We show that at the critical point, where the order/disorder transition occurs, this absolute temperature changes in sign. At this point, which corresponds to the transition homogeneous/heterogeneous culture, the entropy of the system shows a maximum. We discuss the relationship between the temperature and other properties of the model in terms of cultural traits.     

Monte Carlo analysis of critical properties of the two-dimensional randomly site-diluted Ising model via Wang-Landau algorithm

The influence of random site dilution on the critical properties of the two-dimensional Ising model on a square lattice was explored by Monte Carlo simulations with the Wang-Landau sampling. The lattice linear size was L=20-120 and the concentration of diluted sites q=0.1,0.2,0.3. Its pure version displays a second-order phase transition with a vanishing specific heat critical exponent α, thus, the Harris criterion is inconclusive, in that disorder is a relevant or irrelevant perturbation for the critical behaviour of the pure system. The main effort was focused on the specific heat and magnetic susceptibility. We have also looked at the probability distribution of susceptibility, pseudocritical temperatures and specific heat for assessing self-averaging. The study was carried out in appropriate restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the correlation length exponent ν was found to be greater than one, whereas the ratio of the critical exponents (α/ν) is negative and (γ/ν) retains its pure Ising model value supporting weak universality.     

Third and fourth order phase transitions: Exact solution for the Ising model on the Cayley tree

In this work we present the first exact solution of a system of interacting particles with phase transitions of order higher than two. The presented analytical derivation shows that the Ising model on the Cayley tree exhibits a line of third order phase transition points, between temperatures and , and a line of fourth order phase transitions between T_BP and ∞, where k_B is the Boltzmann constant, and J is the nearest-neighbor interaction parameter.     

The strange man in random networks of automata

We have performed computer simulations of Kauffman’s automata on several graphs, such as the regular square lattice and invasion percolation clusters, in order to investigate phase transitions, radial distributions of the mean total damage (dynamical exponent) and propagation speeds of the damage when one adds a damaging agent, nicknamed “strange man”. Despite the increase in the damaging efficiency, we have not observed any appreciable change of the transition threshold to chaos neither for the short-range nor for the small-world case on the square lattices when the strange man is added, in comparison to when small initial damages are inserted in the system. Particularly, we have checked the damage spreading when some connections are removed on the square lattice and when one considers special invasion percolation clusters (high boundary-saturation clusters). It is seen that the propagation speed in these systems is quite sensible to the degree of dilution on the square lattice and to the degree of saturation on invasion percolation clusters.     

Phase transitions in self-dual generalizations of the Baxter–Wu model

We study two types of generalized Baxter–Wu models, by means of transfer-matrix and Monte Carlo techniques. The first generalization allows for different couplings in the up- and down-triangles, and the second generalization is to a q-state spin model with three-spin interactions. Both generalizations lead to self-dual models, so that the probable locations of the phase transitions follow. Our numerical analysis confirms that phase transitions occur at the self-dual points. For both generalizations of the Baxter–Wu model, the phase transitions appear to be discontinuous.     

Percolation properties of the antiferromagnetic Blume-Capel model in the presence of a magnetic field

The problem of order-order and order-disorder transitions in the system described by the 2D antiferromagnetic Blume-Capel model in the presence of a magnetic field is studied by the Wang and Landau flat-histogram simulation method and by the classical Monte Carlo. Anomalous thermodynamic characteristics in low temperatures indicate different type orderings in finite temperatures. The existence of pure antiferromagnetic phases as well as mixed state is shown by detailed phenomenological analysis of the system. The border lines on the phase diagram between various orderings are determined by the complementary microscopic study of the percolation problem for c(2×2) elementary structures of antiferromagnetic ordered phases. This new approach has also shown a full agreement between the percolation threshold for the cluster of mixed phase and the critical temperature of the ordered system.     

Phase diagram of the two-dimensional Ising model with random competing interactions

An Ising model with ferromagnetic nearest-neighbor interactions J₁ (J₁>0) and random next-nearest-neighbor interactions [+J₂ with probability p and −J₂ with probability (1−p); J₂>0] is studied within the framework of an effective-field theory based on the differential-operator technique. The order parameters are calculated, considering finite clusters with n=1,2, and 4 spins, using the standard approximation of neglecting correlations. A phase diagram is obtained in the plane temperature versus p, for the particular case J₁=J₂, showing both superantiferromagnetic (low p) and ferromagnetic (higher values of p) orderings at low temperatures.     

Exact results of a dimerized S=1 Ising chain with single-ion anisotropy

The dimerized spin-1 Ising chain with both longitude and transverse single-ion anisotropies D_z and D_x is solved exactly by means of a mapping to the spin- Ising chain with the alternating transverse fields and the Jordan-Wigner transformation. The analytical expressions of the quasi-particles’ spectra Λ_k, the minimal energy gap Δ₀ for exciting a fermion quasi-particle, the minimal energy gap Δ_h for exciting a hole, and the ground-state energy E_g are obtained. The phase diagram of the ground state is also given. The results show that the system exhibits a series of quantum phase transitions depending on the dimerization strength of the crystal fields, while the quantum critical points are determined exactly.     

Phase diagram of the two-dimensional ferromagnetic three-color Ashkin-Teller model

In the present work we study the critical properties of the ferromagnetic three-color Ashkin-Teller model (3AT) by means of a Migdal-Kadanoff renormalization group approach on a diamond-like hierarchical lattice. The analysis of the fixed points and flux diagram of the recursion relations is used to determine the corresponding phase diagram (including its symmetry properties) and critical exponents. Our numerical results show the presence of four universality classes, three of them are associated to the Potts model with q=2, 4 and 6 states. Finally, a connection between our findings and some known results from the literature is presented.     

Cluster evaporation transition in finite system

Using Wang-Landau entropic sampling we study the Ising model in the framework of microcanonical ensemble (fixed magnetization). We are working for lattice size up to 1500×1500 in two dimensions and 100×100×100 in three dimensions. As we approach the coexistence curve from inside, varying temperature and keeping the magnetization constant, a first-order phase transition takes place for a temperature near the coexistence curve if the lattice size is large enough. We analyze various features of this transition as well as the scaling behavior of characteristic quantities and we compare our numerical results with existing theories.     

Synchronization of Local Oscillations in a Spatial Rock-Scissors-Paper Game Model

We study a spatial rock-scissors-paper model in a square lattice and a quenched small-world network. The system exhibits a global oscillation in the quenched small-world network, but the oscillation disappears in the square lattice. We find that there is a local oscillation in the square lattice the same as in the quenched small-world 1 network. We define σ = 1/N ∑i（di-〈di〉^2 （where di is the density of a kind of species and （di） is the average value） as the variance of the oscillation amplitude in a certain local patch. It is found that σ decays in a power law with an increase of the local patch size R in the square lattice σ ∝ R^-σ, but it remains constant with an increase of the patch size in the quenched small-world network. We can speculate that in the square lattice, superposition between the local oscillations in different patches leads to global stabilization, while in the quenched small-world network, long-range interactions can synchronize the local oscillations, and their coherence results in the global oscillation.     

Stability conditions for fermionic Ising spin-glass models in the presence of a transverse field

The stability of a spin-glass (SG) phase is analyzed in detail for a fermionic Ising SG (FISG) model in the presence of a magnetic transverse field Γ. The fermionic path integral formalism, replica method and static approach have been used to obtain the thermodynamic potential within one step replica symmetry breaking ansatz. The replica symmetry (RS) results show that the SG phase is always unstable against the replicon. Moreover, the two other eigenvalues λ_± of the Hessian matrix (related to the diagonal elements of the replica matrix) can indicate an additional instability to the SG phase, which enhances when Γ is increased. Therefore, this result suggests that the study of the replicon cannot be enough to guarantee the RS stability in the present quantum FISG model, especially near the quantum critical point. In particular, the FISG model allows changing the occupation number of sites, so one can get a first order transition when the chemical potential exceeds a certain value. In this region, the replicon and the λ_± indicate instability problems for the SG solution close to all ranges of a first order boundary.     

Absorbing phase transition in contact process on fractal lattices

We investigate the critical behavior of nonequilibrium phase transition from an active phase to an absorbing state on two selected fractal lattices, i.e., on a checkerboard fractal and on a Sierpinski carpet. The checkerboard fractal is finitely ramified with many dead ends, while the Sierpinski carpet is infinitely ramified. We measure various critical exponents of the contact process with a diffusion-reaction scheme A→AA and A→0, characterized by a spreading with a rate λ and an annihilation with a rate μ, and the results are confirmed by a crossover scaling and a finite-size scaling. The exponents, compared with the ?-expansion results assuming , being the fractal dimension of the underlying fractal lattice, exhibit significant deviations from the analytical results for both the checkerboard fractal and the Sierpinski carpet. On the other hand, the exponents on a checkerboard fractal agree well with the interpolated results of the regular lattice for the fractional dimensionality, while those on a Sierpinski carpet show marked deviations.     

Evolution of ethnocentrism on undirected and directed Barabási-Albert networks

Using Monte Carlo simulations, we study the evolution of contingent cooperation and ethnocentrism in the one-shot game. Interactions and reproduction among computational agents are simulated on undirected and directed Barabási-Albert (BA) networks. We first replicate the Hammond-Axelrod model of in-group favoritism on a square lattice and then generalize this model on undirected and directed BA networks for both asexual and sexual reproduction cases. Our simulations demonstrate that irrespective of the mode of reproduction, the ethnocentric strategy becomes common even though cooperation is individually costly and mechanisms such as reciprocity or conformity are absent. Moreover, our results indicate that the spread of favoritism towards similar others highly depends on the network topology and the associated heterogeneity of the studied population.     

Partition function zeros of the antiferromagnetic Ising model on triangular lattice in the complex temperature plane for nonzero magnetic field

The grand partition functions Z(T,B)$Z(T,B)$ of the Ising model on L×L$L\times L$ triangular lattices with fully periodic boundary conditions, as a function of temperature T and magnetic field B  , are evaluated exactly for L<12$L<12$ (using microcanonical transfer matrix) and approximately for L?12$L?12$ (using Wang–Landau Monte Carlo algorithm). From Z(T,B)$Z(T,B)$, the distributions of the partition function zeros of the triangular-lattice Ising model in the complex temperature plane for real B≠0$B\ne 0$ are obtained and discussed for the first time. The critical points aN(x)$a_N(x)$ and the thermal scaling exponents yt(x)$y_t(x)$ of the triangular-lattice Ising antiferromagnet, for various values of x=e−2βB$x=e^{-2\beta B}$, are estimated using the partition function zeros.     

Critical behavior of sound attenuation in a metamagnetic Ising system

In this study, the sound attenuation coefficient of a spin- metamagnetic Ising system is calculated by the method of thermodynamics of irreversible processes. The behavior of sound attenuation near the phase transition temperatures is analyzed according to various values of phenomenological rate coefficients (γij). For all γm and γs values it is found that sound attenuation peaks occur below TN(H) and depend on frequency ω and the value of the off-diagonal rate coefficient γ. On the other hand, the critical behavior of the sound attenuation in the hydrodynamic regime is obtained analytically via the critical exponents. Moreover, the behavior of the sound attenuation as a function of frequency is also investigated and ω² dependence is observed for the attenuation coefficient. These results are in a good agreement with ultrasonic investigations of magnetic systems.     

Phase diagram of the Ising antiferromagnet with nearest-neighbor and next-nearest-neighbor interactions on a square lattice

The phase diagram of the Ising model in the presence of nearest-neighbor (J₁) and next-nearest-neighbor (J₂) interactions on a square lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by effective-field theory in finite cluster (we have chosen N=4 spins). We have proposed a functional for the free energy (similar to Landau expansion) to obtain the phase diagram in the (T,α) space (α=J₂/J₁), where the transition line from the superantiferromagnetic (SAF) to the paramagnetic (P) phase is of first-order in the range 1/2<α<0.95 in contrast to previous study of CVM (Cluster Variational Method) that predict first-order transition for α=1.0. Our results for α=1.0 are in accordance with MC (Monte Carlo) simulations, that predict a second-order transition.     

Universal properties of population dynamics with fluctuating resources

Starting from the well-known field theory for directed percolation (DP), we describe an evolving population, near extinction, in an environment with its own nontrivial spatio-temporal dynamics. Here, we consider the special case where the environment follows a simple relaxational (Model A) dynamics. Two new operators emerge, with upper critical dimension of four, which couple the two theories in a nontrivial way. While the Wilson-Fisher fixed point remains completely unaffected, a mismatch of time scales destabilizes the usual DP fixed point, suggesting a crossover to a first-order transition from the active (surviving) to the inactive (extinct) state.