Constructing the Critical Curve for a Symmetric Two-Layer Ising Model

TL;DR

This paper develops a transfer matrix numerical method to accurately determine the critical temperature of a symmetric two-layer Ising model with varying inter-layer coupling, filling a gap in existing data.

Contribution

It introduces a new numerical approach to calculate the critical temperature for weak and strong inter-layer couplings in the two-layer Ising model.

Findings

Critical temperature as a function of J/K for weak and strong couplings.

Shift exponent Phi values of 1.74 and 0.5 for different intra-layer interactions.

Accurate reduced internal energy calculations for symmetric ferromagnetic case.

Abstract

A numerical method based on the transfer matrix method is developed to calculate the critical temperature of two-layer Ising ferromagnet with a weak inter-layer coupling. The reduced internal energy per site has been accurately calculated for symmetric ferromagnetic case, with the nearest neighbor coupling K1 = K2 = K (where K1 and K2 are the nearest neighbor interaction in the first and second layers, respectively) with inter layer coupling J. The critical temperature as a function of the inter-layer coupling J/K << 1, is obtained for very weak inter-layer interactions,J/K < 0.1 . Also a different function is given for the case of the strong inter-layer interactions (J/K > 1). The importance of these relations is due to the fact that there is no well tabulated data for the critical points versus J/K. We find the value of the shift exponent Phi = Gama is 1.74 for the system with the same intra-layer interaction and 0.5 for the system with different intra-layer interactions.