Magnetisation and Mean Field Theory in the Ising Model

Abstract

In this set of notes, a complete, pedagogical tutorial for applying mean field theory to the two-dimensional Ising model is presented. Beginning with the motivation and basis for mean field theory, we formally derive the Bogoliubov inequality and discuss mean field theory itself. We proceed with the use of mean field theory to determine a magnetisation function, and the results of the derivation are interpreted graphically, physically, and mathematically. We give a new interpretation of the self-consistency condition in terms of intersecting surfaces and constrained solution sets. We also include some more general comments on the thermodynamics of the phase transition. We end by evaluating symmetry considerations in magnetisation, and some more subtle features of the Ising model. Together, a self-contained overview of the mean field Ising model is given, with some novel presentation of important results.

LaOMS2 (M = Ti, V, and Cr): novel crystal spin valves without contact

“…The Curie temperatures ( T c ) are measured to be 184, 275 and 74 K for LaOTiS 2 , LaOVS 2 , and LaOCrS 2 , respectively, within the mean field approximation. 37…”

“…The La 2 O 2 layer is semiconductive, which can be observed from PDOSs (Fig. 2c, d 37 To further investigate the interaction between two MS 2 layers, spin-orbit coupling (SOC) was considered. The energy difference between the inter-AFM and FM states of LaOMS 2 is still near zero (B0.2 meV per unit cell).…”

Comparative study of galaxy clustering using halo approximation and mean-field theory in the light of modified theories of gravity

“…Also, the purpose of using the molecular field [34] or mean-field theory is to reduce the many-body problem (here, the statistical many-body problem) to an effective one-body problem by replacing all gravitational interaction with an effective interaction. The MFT has been used to study various phenomena like phase transitions [17], Bogoliubov inequality [35,36] and the Ising model (magnetization) [37]. MFT applies to the case of high dimensionality, for the system of extended nature of particles or long-range interaction, including by Hamiltonian.…”