2D Ising Model (Triangle Lattice)

Xintian Wu, Shenghong Sun

Wolfhard Janke E, Mohammad W. Katoot, Ramon Villanova

A. I. Bugrij, O. Lisovyy

H. J. Hilhorst

Abstract We consider infinite or periodic 2D triangular Ising lattices with arbitrary positive or negative nearest-neighbor couplings K i ( r ⃗ ) , where r ⃗ and i indicate the bond position and orientation, respectively. Iterative application of the star–triangle transformation to an initial lattice T ( 0 ) with a set of couplings { K i ( 0 ) ( r ⃗ ) } generates a sequence of lattices T ( n ) , for n = 1, 2, …, with couplings { K i ( n ) ( r ⃗ ) } . When T ( 0 ) includes sufficiently strongly frustrated plaquettes, complex couplings will appear. We show that, nevertheless, the variables 1 / sinh 2 K i ( n

Stjepan Pavić

Ising model is a description given for a set of magnetic mooments on a grid. This thesis presents exact analytical solutions for one-dimensional Ising model using the matrix transfer method and two-dimensional model using combinatorial method. Further on it shows Baxter’s star-triangle transformations and a c++ code for Monte-Carlo simulation over Metropolis algorithm showing the dynamics of a 2D square lattice for different temperatures. The goal is to show mathematical method for solving initially easy to setup, but very hard to solve problems that have direct impact on physics and describe feromagnetic/paramagnetic phase transtition.

George Allen Baker Jr.