Critical behaviour of the hard-core lattice gas on the honeycomb lattice

The hard triangle lattice-gas model (lattice-gas on the honeycomb lattice with first neighbour exclusion) is studied by the phenomenological renormalization method. The critical activity is found to be z = 7.85 and the critical exponents suggest that this model belongs to the 2-D Ising universality class.     

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.     

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.     

The Ising model on random lattices in arbitrary dimensions

We study analytically the Ising model coupled to random lattices in dimension three and higher. The family of random lattices we use is generated by the large N limit of a colored tensor model generalizing the two-matrix model for Ising spins on random surfaces. We show that, in the continuum limit, the spin system does not exhibit a phase transition at finite temperature, in agreement with numerical investigations. Furthermore we outline a general method to study critical behavior in colored tensor models.     

A mechanism of long-range order induced by random fields: Effective anisotropy created by defects

A microscopic mechanism of the long-range order in two-dimensional space induced by random local fields of crystal defects has been found. The impurity-induced effective anisotropy has been shown to arise in the system due to anisotropic distribution of impurity-induced random local field directions in the n-dimensional space of vector order parameter with the O(n) symmetry. The expression for the effective anisotropy constant has been obtained. A weak anisotropy of the “easy axis” type transforms the X–Y model and the Heisenberg model to the class of Ising models, and brings into long-range order existence in the system.     

Renormalization Group Maps for Ising Models in Lattice-Gas Variables

Real-space renormalization group maps, e.g., the majority rule transformation, map Ising-type models to Ising-type models on a coarser lattice. We show that each coefficient in the renormalized Hamiltonian in the lattice-gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice-gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice-gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.     

A freezing transition in the lattice-gas model

In the lattice-gas model, condensation of a gas to its liquid phase is identified as the long range ordering transition in the equivalent Ising model. The same model has a second percolation transition at a lower temperature, which is identified here as the freezing transition of the liquid to its solid phase. It predicts that the mass diffusion in a liquid should decay near the freezing point T_F ≈ (T ? T_F)^t?β, where t is the conductivity exponent and β is the percolation probability exponent.     

A remark on different norms and analyticity for many-particle interactions

We compare a recent result of Dobrushin and Martirosyan with previous results by Gallavotti and Miracle-Sole and by Israel and point out that the analytic behavior at high temperatures for many-particle interactions is different depending on whether the interactions are weighted with a lattice-gas or Ising norm or, on the other hand, with the supremum norm.     

Spin anisotropy and slow dynamics in spin glasses

We report on an extensive study of the influence of spin anisotropy on spin glass aging dynamics. New temperature cycle experiments allow us to compare quantitatively the memory effect in four Heisenberg spin glasses with various degrees of random anisotropy and one Ising spin glass. The sharpness of the memory effect appears to decrease continuously with the spin anisotropy. Besides, the spin glass coherence length is determined by magnetic field change experiments for the first time in the Ising sample. For three representative samples, from Heisenberg to Ising spin glasses, we can consistently account for both sets of experiments (temperature cycle and magnetic field change) using a single expression for the growth of the coherence length with time.     

Variational principle for regular and random Ising models on the cactus tree or on the usual lattice in the “cactus approximation”

The distribution functions and the free energy are expressed in terms of the effective fields for the regular and random Ising models of an arbitrary spin S on the generalized cactus tree. The same expressions apply to systems on the usual lattice in the “cactus approximation” in the cluster variation method. For an ensemble of random Ising models of an arbitrary spin S on the generalized cactus tree, the equation for the probability distribution function of the effective fields is set up and the averaged free energy is expressed in terms of the probability distribution. The same expressions apply to the system on the usual lattice in the “cactus approximation”. We discuss the quantities on the usual lattice when the system or the ensemble of random systems has the translational symmetry. Variational properties of the free energy for a system and of the averaged free energy for an ensemble of random systems are noted. The “cactus approximations” are applicable to the Heisenberg model as well as to the Ising model of an arbitrary spin, and to ensembles of random systems of these models.     

Statistical Entropy of a Lattice-Gas Model: Multiparticle Correlation Expansion

A formula expressing the statistical entropy of a lattice-gas model as a multiparticle correlation expansion is derived in the grand-canonical and in the canonical ensembles. The differences from the analogous expansion in the continuum case are elucidated. The Ising model in one dimension is discussed as a case study.     

Hysteresis and noise from electronic nematicity in high-temperature superconductors

An electron nematic is a translationally invariant state which spontaneously breaks the discrete rotational symmetry of a host crystal. In a clean square lattice, the electron nematic has two preferred orientations, while dopant disorder favors one or the other orientations locally. In this way, the electron nematic in a host crystal maps to the random field Ising model. Since the electron nematic has anisotropic conductivity, we associate each Ising configuration with a resistor network and use what is known about the random field Ising model to predict new ways to test for local electronic nematic order (nematicity) using noise and hysteresis. In particular, we have uncovered a remarkably robust linear relation between the orientational order and the resistance anisotropy which holds over a wide range of circumstances.     

T=0 phase diagram and nature of domains in ultrathin ferromagnetic films with perpendicular anisotropy

We present the complete zero temperature phase diagram of a model for ultrathin films with perpendicular anisotropy. The whole parameter space of relevant coupling constants is studied in first order anisotropy approximation. Because the ground state is known to be formed by perpendicular stripes separated by Bloch walls, a standard variational approach is used, complemented with specially designed Monte Carlo simulations. We can distinguish four regimes according to the different nature of striped domains: a high anisotropy Ising regime with sharp domain walls, a saturated stripe regime with thicker walls inside which an in-plane component of the magnetization develops, a narrow canted-like regime, characterized by a sinusoidal variation of both the in-plane and the out of plane magnetization components, which upon further decrease of the anisotropy leads to an in-plane ferromagnetic state via a spin reorientation transition (SRT). The nature of domains and walls are described in some detail together with the variation of domain width with anisotropy, for any value of exchange and dipolar interactions. Our results, although strictly valid at T=0, can be valuable for interpreting data on the evolution of domain width at finite temperature, a still largely open problem.     

An adaptation of the lattice gas to the water problem. II. Second-order approximation

The adaptation of the lattice-gas model to embody features possessed by water is further explored. On the basis of Martin's functional derivative formulation of Ising problems, a perturbation scheme is developed which allows calculation of the free energy to any desired order in the interaction potential at fixed density. The free energy correct to second order in the interaction strength is utilized here for calculation of other thermodynamic properties of the model. With reasonable choices of values of the interaction parameters these thermodynamic properties of the model can be brought into agreement with those of real water.     

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.     

Critical behaviour in random field gauge theory

We study the thermodynamic behaviour of spin and gauge systems in the presence of a quenched external random field. In particular, we show that forZ(2) andSU (2) gauge theory in two space dimensions, the random field destroys the ordered phase and thus leads to a shift in the lower critical dimension, just as found for the corresponding Ising model.     

Entropy and multi-particle correlations in two-dimensional lattice gases

We analyse the statistical entropy of two-dimensional lattice-gas models in terms of the contributions which arise from space correlations of increasing order. The “residual multiparticle entropy”, defined as the contribution to the excess entropy that is associated with correlations involving more than two particles, is calculated for the Ising and Coulomb lattice gases. The thermodynamic behaviour of the residual multiparticle entropy is then discussed in relation to the phase diagram of the model and the existence of underlying signatures of order-disorder phase transitions is also investigated. Received 31 December 1998 and Received in final form 8 March 1999     

Symmetric vertex models on planar random graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ³) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.