Solutions of the Ising model mean field theory: trivial vs nontrivial solutions


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6 months ago by

Having explained the Ising model mean field theory and Heisenberg model of magnetic ions in another post, we can attempt to digest possible solutions of the Ising model.
As it is presented in the Bruus's textbook (electronic version), page 74:
The mean-field Hamiltonian of the Ising model is 

$H_{MF}=-2\sum_i\vec{m}\cdot\vec{S_i}+mN<\vec{S_z}>$HMF=2im·Si +mN<Sz> ,

where $<\vec{S_z}>$<Sz> is the mean field parameter -- so the magnetization $\vec{m}$m.  
By minimizing the free energy of the mean field Hamiltonian, one obtains the following self-consistent equation  in terms of the mean field parameter  $\vec{m}$m,

$\alpha=tanh\left(\alpha b\right)$α=tanh(αb) , with  $\alpha=\frac{m}{nJ_0}$α=mnJ0 and  $b=nJ_0\beta$b=nJ0β .
The self-consistent equation can be expanded for small values of the  $\alpha$α, which means  $J_0$J0 takes high values or there is a strong exchange interaction.
$\alpha\approx b\alpha-\frac{1}{3}\left(b\alpha\right)^3$αbα13 (bα)3
The solutions are presented on page 75 of the textbook; https://imgur.com/6jHGyqs
We can see that there is no solution for  $b<1$b<1, or better to say; there is no ferromagnetic phase at high temperatures. This can easily be seen by looking at  \( b < 1  \rightarrow  nJ_0 < k_B T \) . This condition can be satisfied for the antiferromagnet case \( J_0 < 0\), but not necessarily for the ferromagnetic case \( J_0 >0 \). In the other words, if the thermal energy \( k_B T  \) of each degree of freedom--here it's spin degree of freedom--exceeds from the spin exchange energy \( nJ_0 \) coming from the neighboring spins, the parallel configuration of spins would not be the ground state of the system.
Using such condition \( (b<1) \) for the existence of a solution, we can define a critical temperature \( T_c \) which shows that there is a solution below it and no solution above.
\( b=1 \rightarrow nJ_0\frac{1}{k_BT_c}=1 \rightarrow T_c = \frac{nJ_0}{k_B} \)

Now we come to the questions:

1) By looking at the phase diagram, the link is given above, one can see that the mean field parameter \( m \) has finite values below \( T_c\), which displays having a ferromagnetically ordered phase in that region and a vanishing mean field parameter above \( T_c\)  representing the disordered phase. Now one ask that where the antiferromagnetic solutions are located?.
We know that in an antiferromagnetic phase the magnetization and so the mean field parameter \( m \) is zero. Then how could one distinguish the disordered phase from the antiferromagnetic phase at all?

Comment: the previous question can also be revisited when one tries to classify different possible solutions of the Ising model mean field theory as follows,
\( m \neq 0 : \) ferromagnetic phase  (Nontrivial case),
\(m = 0 : \) antiferromagnetic phase (Nontrivial case),
No solution: disordered phase with \( m=0 \) (trivial case).

One can see that in principle, this mean field theory is not keen enough to physically distinguish second and third case since the mean field parameter is vanishing for both cases!

#The Art of Mean-Field Theory

1 Answer


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5 months ago by
the order parameter in the AF case is the staggered magnetization, \( m_i \, (-1)^i \). With that, F and AF solutions for \( T_c \) are the same, at the MF level :)
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