Analytical solutions for Ising models on high dimensional lattices

Abstract

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions d>=3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with results of computer simulations. For d=5, 6, 7, our theoretical estimates match the experiments both qualitatively and quantitatively. For d=3, 4, our method is sufficiently accurate for calculation of the critical temperatures, however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (d=3), this contradicts to the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (d = 4) the character of the singularity is under current discussion. For the dimensions d=1, 2 the m-vicinity method is not applicable.

arXiv comment22 pages, 7 figures

References in corpus (10)

• An efficient, multiple range random walk algorithm to calculate the density of states
• Colloquium: Statistical mechanics of money, wealth, and income
• Pushing the Limits of Monte Carlo Simulations for the 3d Ising Model
• Critical universality and hyperscaling revisited for Ising models of general spin using extended high-temperature series
• Critical behaviour of the Ising model on the 4-dimensional lattice
• Phase transition of four-dimensional Ising model with higher-order tensor renormalization group
• The discontinuity of the specific heat for the 5D Ising model
• The surprising convergence of the Monte Carlo renormalization group for the d=3 Ising Model
• Applicability of n-vicinity method for calculation of free energy of Ising model
• Weighted Patterns as a Tool for Improving the Hopfield Model

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