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.     

Conformal invariance of the Ising model in three dimensions

We investigate a critical Ising-like model in the curved geometry S2 x R1 obtained by a conformal mapping of the infinite 3D space R3. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as X(h) = 0.5178(12) and X(t) = 1.423(19), respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.     

Ising model in the high density limit

It is proved that the free energy per spin in the thermodynamic limit of an Ising model on a lattice with coordination numberz approaches the classical Curie-Weiss free energy in the limitz→∞. The infinite spacial dimension limit of nearest neighbour lattice models is a special case of this result.     

Self-avoiding random loops versus Ising model in three dimensions

On the basis of a recently developed lattice model for ensembles of random loops which contains a parameter interpolating between self-avoiding and Ising-like loops, we calculate the average length and the length fluctuations of self-avoiding loops in three dimensions, using analytic as well as Monte Carlo methods, and compare the results with the Ising case. Applying finite-size scaling techniques, we show that the critical behavior of self-avoiding random loops is consistent with universality predictions based on the Ising model.     

A note on differentiability of the cluster density for independent percolation in high dimensions

The cluster density function of independent percolation in ad-dimensional lattice is considered. For eachn, it is shown that(p) has finitenth leftderivative at critical probabilityp _c ifd is sufficiently large. This result agrees with the Bethe lattice approximation, where thenth one-sided derivative of(p) is bounded atp _c for alln.     

A note on unparticle in lower dimensions

Using the gauge-invariant but path-dependent variables formalism, we examine the effect of the space–time dimensionality on a physical observable in the unparticle scenario. We explicitly show that long-range forces between particles mediated by unparticles are still present whenever we go over into lower dimensions.     

Ising droplets in five dimensions

Monte Carlo simulations indicate that the size distribution of Coniglio-Klien droplets in the five-dimensional nearest-neighbor Ising model corresponds to mean field Ising exponents.     

Multiscale entanglement renormalization ansatz in two dimensions: quantum Ising model

We propose a symmetric version of the multiscale entanglement renormalization ansatz in two spatial dimensions (2D) and use this ansatz to find an unknown ground state of a 2D quantum system. Results in the simple 2D quantum Ising model on the 8x8 square lattice are found to be very accurate even with the smallest nontrivial truncation parameter.     

Improved Migdal recursion formula for the Ising model in two dimensions on a triangular lattice

The ε perturbation around the Migdal transformation proposed by Martinelli and Parisi has been computed up to second order for the Ising model in two dimensions on a triangular lattice. Perturbation around the value ε = 2 has also been introduced to evaluate higher orders in the series. The condition of null derivatives around ε = 1 for the critical values is analyzed. The critical temperature and the critical index ν are obtained with a satisfactory accuracy of a few percent.     

A note on Edwards' hypothesis for zero-temperature Ising dynamics

We give a simple criterion for testing a behavior à la Edwards in certain zero-temperature, ferromagnetic spin-flip dynamics and use it to show that the limiting distributions of those dynamics do not coincide with the uniform distribution over the blocked configurations of the dynamics. We provide explicit examples in dimension one and higher.     

A spin-one Ising model on the Bethe lattice

We study the statistical mechanics of spin-one Ising models on the Bethe lattice assuming that the spins interact by dipolar and quadrupolar interactions. An exact calculation of the properties of the system is performed on the basis of the general formulation of Morita. An exact expression for the Curie temperature is derived and the results are found to be in agreement with those of Obokata and Oguchi who utilized a generalized Bethe approximation to a spin-one Ising system. The nature of variation of the Curie temperature with respect to the change of quadrupolar exchange is discussed for various coordination numbers and the results agree qualitatively with the earlier works. The temperature variation of both dipolar and quadrupolar moments is studied.     

A note on the summation of the virial expansion in two dimensions

The infinite subset of Cayley trees in the cluster expansion is considered and the expressions for two-dimensional pressure and adsorption isotherm are obtained.     

Ising lattice gauge theory in three dimensions

By raising the transfer matrix to a finite power the partition function for a finite lattice Z(2) gauge model is obtained exactly. The zeros of the resultant polynomial are found and some plaquette-plaquette expectation values are extracted. An exponential fit for the inverse correlation length matches onto both strong- and weak-coupling results but breaks down close to the second-order phase transition point.Similar calculations for the three-dimensional Ising model are also discussed.     

A note on supersymmetry breaking in 1+1 dimensions

The spontaneous breakdown of supersymmetry for a single chiral superfield in 1+1 dimensions is investigated and a number of puzzling features are resolved by noting that the loop expansion is not always valid. It is found that both broken and unbroken supersymmetry are stable with respect to radiative corrections.     

Interface profile of the Ising ferromagnet in two dimensions

The interface profile of the two-dimensional Ising ferromagnet is obtained for all temperatures in the thermodynamic limit. The width of the interface depends on its length as (length)^1/2.On leave from Oxford University. Partially supported by NSF grant no. MPS 75 20638.     

A study of metastability in the Ising model

We consider a two-dimensional Ising ferromagnet with (+) boundary conditions and negative external field, where a Markovian time evolution is assumed. We construct, suitably restricting the allowed configurations att=0, a non equilibrium state with positive magnetization such that:

1. only one phase is present,
2. the relaxation time for unit volume is finite and can be made very large.

These results are obtained following a general method for describing metastable states proposed by Lebowitz and Penrose and exploiting the analysis of the Ising-spin-configurations in terms of contours given by Minlos and Sinai.