Finite-Size Scaling Analysis of a Three-Dimensional Blume--Capel Model in the Presence of External Magnetic Field

The Blume-Capel model in the presence of external magnetic field H has been simulated using a cellular automaton algorithm improved from the Creutz cellular automaton in three-dimension lattice. The field critical exponent 5 is estimated using the power law relations and the finite size scaling functions for the magnetization and the susceptibility in the range -0.1≤ h = H/J ≤0. The estimated value of the field critical exponent 5 is in good agreement with the universal value （δ = 5） in three dimensions. The simulations are carried out on a simple cubic lattice under periodic boundary conditions.     

A higher symmetry in the sol-gel transition at the consolute point

By methods of renormalized field theory we show that generically the multicritical behavior of the sol-gel transition at the consolute point is dominated by a fixed point symmetry which is higher than the symmetry of the original Hamiltonian. In general, this leads to Ising-like behavior of all fluctuations. We show in particular that the Fisher exponent _P of the percolation field coincides with the corresponding exponent _I of Ising fields. We perform a preliminary stability analysis which indicates that the higher symmetry is not broken in the physical 3-dimensional case.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

On the critical temperature of non-periodic Ising models on hexagonal lattices

The critical temperature of layered Ising models on triangular and honeycomb lattices are calculated in simple, explicit form for arbitrary distribution of the couplings.     

Behavior of the antiferromagnetic phase transition near the fermion condensation quantum phase transition in YbRh2Si2

Low-temperature specific-heat measurements on YbRh₂Si₂ at the second order antiferromagnetic (AF) phase transition reveal a sharp peak at TN=72 mK. The corresponding critical exponent α turns out to be α=0.38, which differs significantly from that obtained within the framework of the fluctuation theory of second order phase transitions based on the scale invariance, where α?0.1. We show that under the application of magnetic field the curve of the second order AF phase transitions passes into a curve of the first order ones at the tricritical point leading to a violation of the critical universality of the fluctuation theory. This change of the phase transition is generated by the fermion condensation quantum phase transition. Near the tricritical point the Landau theory of second order phase transitions is applicable and gives α?1/2. We demonstrate that this value of α is in good agreement with the specific-heat measurements.     

Green function method study of the anisotropic ferromagnetic Heisenberg model on a square lattice

We study the phase diagram of the anisotropic ferromagnetic Heisenberg model on a square lattice. We use the double-time Green’s function method within the Callen decoupling approximation. The dependence of the Curie temperature T_c on the spin S and on the anisotropy parameter Δ (Δ=0 and 1 correspond to the isotropic Heisenberg and Ising model, respectively) is obtained explicitly. Our results are in agreement with results obtained from other theoretical approaches.     

A worm algorithm for the fully-packed loop model

We present a Markov-chain Monte Carlo algorithm of worm   type that correctly simulates the fully-packed loop model with n=1$n=1$ on the honeycomb lattice, and we prove that it is ergodic and has uniform stationary distribution. The honeycomb-lattice fully-packed loop model with n=1$n=1$ is equivalent to the zero-temperature triangular-lattice antiferromagnetic Ising model, which is fully frustrated and notoriously difficult to simulate. We test this worm algorithm numerically and estimate the dynamic exponent zexp=0.515(8)$z_{\exp }=0.515(8)$. We also measure several static quantities of interest, including loop-length and face-size moments. It appears numerically that the face-size moments are governed by the magnetic dimension for percolation.     

Finite size effects for the Ising model on random graphs with varying dilution

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with N nodes and N^γ edges, with 1<γ≤2. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of γ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter γ, using two different approaches: a replica based finite N expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics.     

Periodic states, local effects and coexistence in the BML traffic jam model

The Biham-Middleton-Levine (BML) model is simple lattice model of traffic flow, self-organization and jamming. Rather than a sharp phase transition between free-flow and jammed, it was recently shown that there is a region where stable intermediate states exist, with details dependent on the aspect ratio of the underlying lattice. Here we investigate square aspect ratios, focusing on the region where random, disordered intermediate (DI) states and conventional global jam (GJ) states coexist, and show that DI states dominate for some densities and timescales. Moreover, we show that periodic intermediate (PI) states can also coexist. PI states converge to periodic limit cycles with short recurrence times and were previously conjectured to arise from idiosyncrasies of relatively prime aspect ratios. The observed coexistence of DI, PI and GJ states shows that global parameters, density together with aspect ratio, are not sufficient to determine the full jamming outcome. We investigate additional features that lead towards jamming and show that a strategic perturbation of a few selected bits can change the nature of the flow, nucleating a global jam.     

Phase diagram of the three states Potts model with next nearest neighbour interactions on the Bethe lattice

We have found an exact phase diagram of the Potts model with competing nearest neighbor and next nearest neighbor interactions on the Bethe lattice of order two. The diagram consists of five phases: ferromagnetic, paramagnetic, modulated, antiphase and paramodulated, all meeting at the multicritical point . We report on a new phase which we denote as paramodulated, found at low temperatures and characterized by zero average magnetization lying inside the modulated phase. Such a phase, inherent in the Potts model has no analogues in the Ising setting.     

Percolation on patchwise heterogeneous surfaces under equilibrium conditions

The site percolation problem on square lattices whose sites are grouped in two types of energetically different patches is studied. Several lattices formed by collections of either randomly or orderly localized and no overlapped patches of different sizes are generated. The system is characterized by two parameters, namely, the size of each patch, l, and the energy difference between the two kind of sites, ΔE. Particles are adsorbed at equilibrium on the lattice. The critical coverage is determined by means of Monte Carlo simulations and finite-size scaling analysis. The percolative behavior of the system as a function of the parameters characterizing the heterogeneity of the energetic surface topography is presented and discussed.     

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.     

The paramagnetic properties of one-dimensional spin-1 single-ion anisotropic ferromagnet

One-dimensional single-ion anisotropic ferromagnet with spin-1 is investigated by means of Green's function treatment in this paper. The model Hamiltonian includes a Heisenberg ferromagnetic term, an external magnetic field, and a second-order single-ion anisotropy. The magnetic properties of the system are treated by the random phase approximation for the exchange interaction term and the Anderson-Callen approximation for the anisotropy term. Our paramagnetic results are in agreement with the other theoretical results.     

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.     

NMR and magnetic studies on 7 K anomaly in Tl3H(SO4)2

Measurements of the spin-lattice relaxation time, NMR absorption line and magnetization have been carried out on the Tl₃H(SO₄)₂ crystal below 50 K. The anomaly at around 7 K was: (1) the spin-lattice relaxation times of ¹H and ²⁰⁵Tl nuclei increase steeply with decreasing temperature below 7 K, (2) the NMR absorption lines below 7 K shift to the high-magnetic field side in comparison with that above 7 K, and (3) the ¹H NMR line width exhibits a drastic increase of the line width with decreasing temperature below 7 K. These results indicate that the magnetic dipole fluctuation of the proton changes at 7 K. On the other hand, there are no remarkable anomalies of magnetic susceptibility at around 7 K. From these results it is deduced that the anomaly at around 7 K is caused by the change in quantum mechanical process of the proton from proton tunneling to zero-point vibration of hydrogen in the hydrogen bond with the decrease of temperature.     

Front-type solutions of fractional Allen-Cahn equation

Super-diffusive front dynamics have been analysed via a fractional analogue of the Allen-Cahn equation. One-dimensional kink shape and such characteristics as slope at origin and domain wall dynamics have been computed numerically and satisfactorily approximated by variational techniques for a set of anomaly exponents 1<γ<2. The dynamics of a two-dimensional curved front has been considered. Also, the time dependence of coarsening rates during the various evolution stages was analysed in one and two spatial dimensions.     

Investigation of the Potts Model on Triangular Lattices by the Second Renormalization of Tensor Network States

We employ the second renormalization group method of tensor-network states to investigate thermodynamic properties of the ferromagnetic and antiferromagnetic Potts model on triangular lattices. From the temperature dependence of the internal energy and the specific heat, both the critical temperatures and critical exponents are evaluated. For the q = 3 antiferromagnetic Potts model, the critical temperature is found to be Tc = 0.627163±0.000003, which is at least one order of magnitude more accurate than that obtained by other methods.     

Electromagnetic Coulomb gas with vector charges and “elastic” potentials: Renormalization group equations

We present a detailed derivation of the renormalization group equations for two-dimensional electromagnetic Coulomb gases whose charges lie on a triangular lattice (magnetic charges) and its dual (electric charges). The interactions between the charges involve both angular couplings and a new electromagnetic potential. This motivates the denomination of “elastic” Coulomb gas. Such elastic Coulomb gases arise naturally in the study of the continuous melting transition of two-dimensional solids coupled to a substrate, either commensurate or with quenched disorder.     

Magnetic phase diagram of the Ca1−xMnxO systems

The magnetic properties of the Ca_1−xMn_xO systems in the range 0?x?1 have been studied by mean field theory and high-temperature series expansions (HTSEs). By using the first theory, we have evaluated the nearest neighbour and the next-neighbour super-exchange interaction J₁(x) and J₂(x) respectively, in the range 0.45?x?1. The corresponding classical exchange energy for magnetic structure is obtained for the Ca_1−xMn_xO systems. The HTSEs combined with the Padé approximants (PA) method is applied to the Ca_1−xMn_xO systems; we have obtained the magnetic phase diagrams (TN or TSG versus dilution x) in the range 0?x?1. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) are deduced in the range 0?x?1.