Non-vanishing boundary effects and quasi-first order phase transitions in high dimensional Ising models

In order to gain a better understanding of the Ising model in higher dimensions we have made a comparative study of how the boundary, open versus cyclic, of a d-dimensional simple lattice, for d=1,...,5, affects the behaviour of the specific heat C and its microcanonical relative, the entropy derivative -dS/dU. In dimensions 4 and 5 the boundary has a strong effect on the critical region of the model and for cyclic boundaries in dimension 5 we find that the model displays a quasi first order phase transition with a bimodal energy distribution. The latent heat decreases with increasing systems size but for all system sizes used in earlier papers the effect is clearly visible once a wide enough range of values for K is considered. Relations to recent rigorous results for high dimensional percolation and previous debates on simulation of Ising models and gauge fields are discussed.Comment: 12 pages, 27 figure

On the p,qp,q-binomial distribution and the Ising model

A completely new approach to the Ising model in 1 to 5 dimensions is developed. We employ $p,q$-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case $p=q$. We take our investigation to the simple $d$-dimensional lattices for $d=1,2,3,4,5$ and fit $p,q$-binomial distributions to our data, some of which are exact but most are sampled. For $d=1$ and $d=5$ the magnetisation distributions are remarkably well-fitted by $p,q$-binomial distributions. For $d=4$ we are only slightly less successful, while for $d=2,3$ we see some deviations (with exceptions!) between the $p,q$-binomial and the Ising distribution. We begin the paper by giving results on the behaviour of the $p,q$-distribution and its moment growth exponents given a certain parameterization of $p,q$. Since the moment exponents are known for the Ising model (or at least approximately for $d=3$) we can predict how $p,q$ should behave and compare this to our measured $p,q$. The results speak in favour of the $p,q$-binomial distribution's correctness regarding their general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution is not settled though.Comment: 51 pages, 23 figures, submitted to PRB on Oct 23 200

The Ising universality class in dimension three: Corrections to scaling

International audienceSimulation data are analyzed for four 3D spin- 1∕2 Ising models: on the FCC lattice, the BCC lattice, the SC lattice and the Diamond lattice. The observables studied are the susceptibility, the reduced second moment correlation length, and the normalized Binder cumulant. From measurements covering the entire paramagnetic temperature regime the corrections to scaling are estimated. We conclude that a correction term having an exponent which is consistent within the statistics with the bootstrap value of the universal subleading thermal confluent correction exponent, θ2∼2.454(3) , is almost always present with a significant amplitude. In all four models, for the normalized Binder cumulant the leading confluent correction term has zero amplitude. This implies that the universal ratio of leading confluent correction amplitudes aχ4∕aχ=2 in the 3D Ising universality class

Ising spin glasses in dimension five

International audienceIsing spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two