Mean field theory for Ising model and Weiss approximation
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For this exercise we use the Weiss Molecular Field approximation for the Ising model in 3 dimensions. Here the interaction between neighbouring spins is replaced by an interaction of the averaged field over all the spins: the Weiss Molecular Field, which we name m.
The Hamiltonian is given by H = −Jzm ∑si - B ∑si, i = 1,,...,N, si = ±1
where z is the number of nearest neighbors, e.g for z = 6 we would have a cube lattice. In addition we have the self-consistency equation <si > = m
a) Calculate the canonical partition function from the Hamiltonian. Derive <si> from it. Prove that the corresponding self-consistency equation for m is given by
m = tanh (βB + βJzm)
b) Let B = 0. Using a graph, determine the solution for m in dependence on zβJ. Discuss the stability of the solution(s). Argue that a phase transition between one phase with and one phase without spontaneous magnetization takes place and determine its critical temperature Tc as a function of J.
c) Let B = 0. If the temperature is near the critical point then there is a small magnetization. Develop the self-consistency equation as a Taylor Series in m to the third order term and derive from this an expression for m(T,J)
d) The susceptibility is defined by χT = N (∂m/∂B)T. Calculate the susceptibility for B = 0. For temperatures slightly under the critical temperature (T < Tc),
we have χT is proportional to (T − Tc)^(−γ), where γ is a critical exponent. Determine γ for the three-dimensional Ising model in the molecular field approximation.
e) Explain why the Molecular Field Theory cannot be applied in 1 dimension
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A detailed solution is given. A mean field theory for Ising model and Weiss approximation is determined.
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documentclass[a4paper]{article}
usepackage{amsmath,amssymb,graphicx}
newcommand{haak}[1]{!left(#1right)}
newcommand{rhaak}[1]{!left [#1right]}
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newcommand{ahaak}[1]{!left{#1right}}
newcommand{gem}[1]{leftlangle #1rightrangle}
newcommand{gemc}[2]{leftlangleleftlangleleft. #1right | #2
rightranglerightrangle}
newcommand{geml}[1]{leftlangle #1right.}
newcommand{gemr}[1]{left. #1rightrangle}
newcommand{haakl}[1]{left(#1right.}
newcommand{haakr}[1]{left.#1right)}
newcommand{rhaakl}[1]{left[#1right.}
newcommand{rhaakr}[1]{left.#1right]}
newcommand{lhaakl}[1]{left |#1right.}
newcommand{lhaakr}[1]{left.#1right |}
newcommand{ket}[1]{lhaakl{gemr{#1}}}
newcommand{bra}[1]{lhaakr{geml{#1}}}
newcommand{brak}[2]{gem{#1lhaakl{#2}}}
newcommand{braket}[3]{gem{#1lhaak{#2}#3}}
newcommand{floor}[1]{leftlfloor #1rightrfloor}
newcommand{half}{frac{1}{2}}
newcommand{kwart}{frac{1}{4}}
renewcommand{imath}{text{i}}
begin{document}
title{Weiss Molecular Field approximation}
date{}
author{}
maketitle
Let's calculate the partition function:
begin{equation}label{part}
Z = sum_{ahaak{s_{i}}}exphaak{-beta H} = sum_{ahaak{s_{i}}}prod_{i=1}^{N}exprhaak{beta s_{i}haak{j z m + B}} = ahaak{2coshrhaak{betahaak{j z m + ...
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