The Markov property method applied to Ising model calculations

George Allen Baker Jr.

George Allen Baker Jr.

S. I. Kubarev

Hiroshi Takano, Masuo Suzuki

A systematic real-space dynamic renormalization group method is proposed and it is applied to the kinetic Ising model on a triangular lattice. The coarse grained master equation is constructed by using the memory function formalism and the Markov approximation. The high temperature series renormalization group method proposed by Betts, Cuthiell and Plischke is extended for the present purpose to make a perturbation expansion in a self-consistent way with respect to all orders of interactions. Terms nonlocal in space appear in the coarse grained time-evolution operator in our approach, because of the memory effect accompanied with the coarse graining procedure. It is shown within the second order calculation that these nonlocal terms are irrelevant around the nontrivial fixed point. The values of the static exponent ν and dynamic exponent z are given by 1/ν = 0.99 and z = 2.23 from the present second order calculation. These results are quite reasonable.

M. L. GUERRA, Jose Daniel Muñoz

This work measures the time to equilibrium for the multicanonical method on the 2D-Ising system by using a new criterion, proposed here, to find the time to equilibrium, t eq , of any sampling procedure based on a Markov process. Our new procedure gives the same results that the usual one, based on the magnetization, for the canonical Metropolis sampling on a 2D-Ising model at several temperatures. For the multicanonical method we found a power-law relationship with the system size, L, of t eq =0.27(15) L2.80(13) and with the number of energy levels to explore, k E , of [Formula: see text], in perfect agreement with the above result. In addition, a kind of critical slowing down was observed around the critical energy. Our new procedure is completely general, and can be applied to any sampling method based on Markov processes.

Chris J. Hamer, A.C. Irving

The cluster expansion methods of Nickel are applied to calculate high-temperature series for the vacuum energy and specific heat, the susceptibility, and the mass gap in the (2+1)D Ising model. Critical points and critical exponents are estimated for the square and triangular lattices. The results demonstrate universality with the 3D Ising model, within errors. Exact linked cluster expansions are formulated for the quantities above, and their convergence and scaling properties are investigated.