Low dimensional ordering on a lattice model

A simple d-dimensional lattice model is proposed, incorporating some degree of frustration and thus capable of describing some aspects of molecular orientation in covalently bound molecular solids. For d = 2 the model is shown to be equivalent to the standard two-dimensional Ising model, while for d > 2 it describes a peculiar transition from an isotropic high temperature phase to a low-dimensional anisotropic low temperature state. A general mean field analysis is presented and compared to some exact limit properties.     

Analysis of the migdal transformation for models with Heisenberg spins on a d-dimensional lattice

We show for classical Heisenberg spins, with a general nearest neighbour interaction, that in the Migdal approximation the only low-temperature phase transitions are Ising ones (ferror antiferromagnetic). For d=2 neither the pure Heisenberg model nor the Lebwohl-Lasher model show a phase transition at a finite temperature. For d>2 transitions do exist at intermediate temperature and the complete flow diagram together with a two-parameter phase diagram is obtained numerically for d=3. Apart from critical temperatures and thermal exponents, also the magnetic exponents (for both Heisenberg and XY spins) are calculated. The latter are in very good agreement with exact results.     

Upper and lower bounds to a correlation function for an Ising model with random interactions

We consider the general Ising model with random interactions Jp. We assume that the probability densities of the random interactions are statistically independent and that the averages of the absolute values of the random interactions, |Jp|, are finite. We then show that a correlation function for the regular Ising model with interactions |Jp| and the same quantity with opposite sign are an upper and a lower bound to the corresponding averaged correlation function of the random Ising model under consideration.     

Vanishing Critical Magnetization in the Quantum Ising Model

Adapting the recent argument of Aizenman, Duminil-Copin and Sidoravicius for the classical Ising model, it is shown here that the magnetization in the transverse-field Ising model vanishes at the critical point. The proof applies to the ground state in dimension d ≥ 2 and to positive-temperature states in dimension d ≥ 3, and relies on graphical representations as well as an infrared bound.     

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.     

Convergence to Equilibrium of Random Ising Models in the Griffiths Phase

We consider Glauber-type dynamics for disordered Ising spin systems with nearest neighbor pair interactions in the Griffiths phase. We prove that in a nontrivial portion of the Griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1. This condition has, in turn, been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential, and that the average over the disorder of the time-autocorrelation function goes to equilibrium faster than exp[–k(log t) ^d/(d–1)]. We then show that for the diluted Ising model these upper bounds are optimal.     

Dilute antiferromagnets: Imry-Ma argument,hierarchical model,and equivalence to random field Ising models

We discuss some aspects of the problem of the equivalence of dilute antiferromagnets and random field Ising models. We first investigate for dilute antiferromagnets the validity of the arguments of Imry and Ma. It turns out that they are applicable, but some care is required concerning the role played by the so-called internal Peierls contours. Next we consider a hierarchical version of a dilute antiferromagnetic Ising model in the presence of a uniform magnetic field and show that a renormalization group transformation maps it exactly into a hierarchical version of the random field Ising model, thus proving their equivalence as far as the critical behavior is concerned. In particular this implies that phase transition with spontaneous magnetization occurs only for dimensiond>2. Finally we show that in the absence of internal Peierls contours both models, in their hierarchical versions, exhibit phase transition already in dimensiond=2.     

Stabilization of a steady state in oscillators coupled by a digital delayed connection

We propose the mapping of polynomial of degree 2S constructed as a linear combination of powers of spin-S (for simplicity, we called as spin-S polynomial) onto spin-crossover state. The spin-S polynomial in general can be projected onto non-symmetric degenerated spin up (high-spin) and spin down (low-spin) momenta. The total number of mapping for each general spin-S is given by 2(2^2S ? 1). As an application of this mapping, we consider a general non-bilinear spin-S Ising model which can be transformed onto spin-crossover described by Wajnflasz model. Using a further transformation we obtain the partition function of the effective spin-1/2 Ising model, making a suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising model with a fixed external magnetic field, which in general cannot be solved exactly. However, for a particular case of non-bilinear spin-S Ising model could become equivalent to an exactly solvable Ising model. The transformed Ising model exhibits a residual entropy, then it should be understood also as a frustrated spin model, due to competing parameters coupling of the non-bilinear spin-S Ising model.     

Connection between the KdV equation and the two-dimensional Ising model

We construct the one-parameter family of solutions to d²w/dzsu2 = zw + 2w³ that tend to zero for z → +∞ by specializing an equation previously solved in connection with the two-dimensional Ising model. These solutions are intimately related to the KdV equation.     

Monte Carlo study of the Yukawa coupled two-spin Ising model

We consider a particular four state spin system composed of two Ising spins (s_x, σ_x) with independent hopping parameters κ₁ κ₂, coupled by a bilinear Yukawa term, ys_xσ_x. The Yukawa term is solely responsible for breaking the global Z₂ × Z₂ symmetry down to Z2. This model is intended as an illustration of general coupled Higgs system where scalars can arise both as composite and elementary excitations. For the Ising example in 2d, we give convincing numerical evidence of the universality of the two-spin system with the one-spin Ising model, by Monte Carlo simulations and finite size scaling analysis. We also show that as we approach the phase transition, universality arises by having a single correlation length that diverges.     

Large q expansions for q-state gauge-matter Potts models in lagrangian form

We consider the lagrangian form of a q-state generalization of Ising gauge theories with matter fields in d = 3 and 4 dimensions. The theory is exactly soluble in the limit q → ∞ and corrections are easily calculable in power series in $\text{1}\text{q}{}^{\mathrm{<mtext>1</mtext><mtext>d</mtext>}}$. Extrapolating the series for the free energies and latent heats by the method of Padé approximants, we have constructed the phase diagrams for all values of q. Our results agree well with known results for pure spin systems and, for the case q = 2, with Ising Monte Carlo data.     

On the Absence of Ferromagnetism in Typical 2D Ferromagnets

We consider the Ising systems in d dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions that decay with power s of the distance. The physical context of such models is discussed; primarily this is d = 2 and s = 3 where, at long distances, genuine magnetic interactions between genuine magnetic dipoles are of this form. We prove that when the power of decay lies above d and does not exceed d + 1, then for all temperatures the spontaneous magnetization is zero. In contrast, we also show that for powers exceeding d + 1 (with d ≥ 2) magnetic order can occur.     

Statistical mechanical methods in particle structure analysis of lattice field theories

We illustrate on simple examples a new method to analyze the particle structure of lattice field theories. We prove that the two-point function in Ising and rotator models has an Ornstein-Zernike correction at high temperature. We extend this to Ising models at low temperatures if the lattice dimensiond3. We prove that the energy-energy correlation function at high temperatures (for Ising orN=2 rotators) decays according to mean field theory (i.e. with the square of the Ornstein-Zernike correction) ifd4. We also study some surface models mimicking the strong-coupling expansion of the glueball correlation function. In the latter model, besides Ornstein-Zernike decay, we establish the presence of two nearly degenerate bound states.     

Damage spreading and critical exponents for “model A” Ising dynamics

Using damage spreading and heat bath dynamics, we study the Ising model in 2 and 3 dimensions with non-conservative dynamics. Our algorithm differs in some important points from previous ones, which makes it rather efficient. We give estimates for the exponent z which seem to be the most precise published so far (2.172 ± 0.006 for d = 2, 2.032 ± 0.004 for d = 3). We also give precise estimates of the exponent θ′ introduced by Janssen et al. (Z. Phys. B 73 (1989) 539) and of analogous but in principle independent exponents. We find surprisingly that some of the latter agree with θ′, and give an explanation for this.     

Mean-Field Behavior for Long- and Finite Range Ising Model,Percolation and Self-Avoiding Walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d>2(α ∧2) for self-avoiding walk and the Ising model, and d>3(α ∧2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886–1944, 2005).      

Energy dissipation in the Nagel-Schreckenberg model with open boundary condition

In this paper, we numerically investigate energy dissipation caused by traffic in the Nagel-Schreckenberg (NaSch) model with open boundary conditions (OBC). Boundary results in excess energy dissipation. The effects of the stochastic boundary conditions on energy dissipation are discussed. The behaviors of energy dissipation in different traffic phase are distinct. As an order parameter, energy dissipation rate E _d characterizes the phase transition behaviors well. It is shown that there is no true free-flow state in nondeterministic NaSch model with OBC. We refer to this non-true free-flow state as quasi-free-flow (QFF) phase in which there are interactions between vehicles caused by stochastic braking but no backward moving jam exists. In the maximum current phase, E _d is minimal thus the social payoff is maximal. Energy dissipation profiles in QFF, jammed and maximum current phase are presented. Theoretical analyses are in good agreement with numerical results for the case v _max = 1.     

Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D

We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)-model reduces in the limit N → 0 to self-avoiding polymers, the N-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension D → 1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent ν of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to this model. By comparison of low- and high-temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent ν in d = 3. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order.     

Reentrant phases in the ± J spin glass

We consider an enlarged phase space of the ±J spin glass which includes the dilute Ising model and the frustrated system. The three orthogonal axes in this space are: (i) The fraction of ferro- to antiferro-magnetic bonds, p; (ii) the ratio of the strengths of the antiferro- to ferromagnetic interacions, q; and (iii) the temperature, T. Within this phase space we observe extended regions of the low-temperature spin-glass phase which is characterized by a unique distribution of the local-order parameter. We observe reentrant phase transitions: for fixed p and q with varying T the distribution of the local order parameter shows paramagnetic, ferromagnetic and then spin-glass phases; for fixed p and T and varying q the distribution shows ferromagnetic to paramagnetic and then spin-glass phases.     

Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model

We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T ^*=0.9527(1), p ^*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T ^*, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T ^*. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν?η=d?2=0.