1. Mean Field Approximation
Defining mean field (molecular field or effective field) & mean field Hamiltonian
𝑆 𝑖 = 𝑆 𝑖 + 𝑆 𝑖 − 𝑆 𝑖
𝑆 𝑖 𝑆 𝑗 = 𝑆 𝑖 + 𝑆 𝑖 − 𝑆 𝑖 𝑆 𝑗 + 𝑆 𝑗 − 𝑆 𝑗
𝑆 𝑗 = 𝑆 𝑗 + 𝑆 𝑗 − 𝑆 𝑗
𝑆 𝑖 𝑆 𝑗 = 𝑆 𝑖 𝑆 𝑗 + 𝑆 𝑖 𝑆 𝑗 − 𝑆 𝑗 + 𝑆 𝑗 𝑆 𝑖 − 𝑆 𝑖 + 𝑆 𝑖 − 𝑆 𝑖 𝑆 𝑗 − 𝑆 𝑗
term quadratic in fluctuations
𝑆 𝑖 𝑆 𝑗 ≈ 𝑆 𝑖 𝑆 𝑗 + 𝑆 𝑖 𝑆 𝑗 − 𝑆 𝑗 + 𝑆 𝑗 𝑆 𝑖 − 𝑆 𝑖

2. Mean field Hamiltonian
𝐻=− 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖 𝑆 𝑗 −ℎ 𝑖 𝑆 𝑖
𝐻 𝑚𝑓 =− 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖 𝑆 𝑗 + 𝑆 𝑖 𝑆 𝑗 − 𝑆 𝑗 + 𝑆 𝑗 𝑆 𝑖 − 𝑆 𝑖 −ℎ 𝑖 𝑆 𝑖
Note I dropped here the factor ½ introduced previously
to keep the notation simple.
How is Hmf simplifying the problem?
𝑆 𝑗 = 𝑆 = S av
independent of summation index
𝐻 𝑚𝑓 =− 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖 𝑆 𝑗 + 𝑆 𝑖 𝑆 𝑗 − 𝑆 𝑗 + 𝑆 𝑗 𝑆 𝑖 − 𝑆 𝑖 −ℎ 𝑖 𝑆 𝑖 =− 𝑆 2 𝑖,𝑗 𝐽 𝑖𝑗 + 𝑆 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑗 − 𝑆 2 𝑖,𝑗 𝐽 𝑖𝑗 + 𝑆 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖 − 𝑆 2 𝑖,𝑗 𝐽 𝑖𝑗 −ℎ 𝑖 𝑆 𝑖
=− 𝑆 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑗 + 𝑆 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖 − 𝑆 2 𝑖,𝑗 𝐽 𝑖𝑗 −ℎ 𝑖 𝑆 𝑖
𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑗 = 𝑗 𝑆 𝑗 𝑖 𝐽 𝑖𝑗 = 𝑖 𝑆 𝑖 𝑗 𝐽 𝑖𝑗 = 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖
using

3. with z= # of nearest neighbors
=− 𝑆 𝑖 2 𝑗 𝐽 𝑖𝑗 𝑆 𝑗 −ℎ 𝑖 𝑆 𝑖 + 𝐸 0
𝐸 0 = 𝑆 2 𝑖,𝑗 𝐽 𝑖𝑗
where
independent of i,j
𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑗 = 𝑖,𝑗 𝐽 𝑖𝑗 𝑆 𝑖
𝐻 𝑚𝑓 =− 𝑆 𝑖 2 𝑗 𝐽 𝑖𝑗 𝑆 𝑖 −ℎ 𝑖 𝑆 𝑖 + 𝐸 0
𝜂 𝑖 =ℎ+ 𝑆 𝑗 2 𝐽 𝑖𝑗
with the mean field
For nearest neighbor interaction
identical for all nearest neighbors
such that, e.g., 2J12=J
𝑗 2 𝐽 𝑖𝑗 =𝑧𝐽
with z= # of nearest neighbors
𝜂=ℎ+ 𝑆 𝑧𝐽
𝐻 𝑚𝑓 =− 𝑖 𝜂 𝑖 𝑆 𝑖 + 𝐸 0 =−𝜂 𝑖 𝑆 𝑖 + 𝐸 0

4. is formally identical to non-interacting spins in a field
𝐻 𝑚𝑓 =−𝜂 𝑖 𝑆 𝑖 + 𝐸 0
Thermodynamics of
is formally identical to non-interacting spins in a field
(paramagnetism)
For Ising spins with
𝑆 𝑖 ±1
𝑆 𝑖 = 𝑆 = 𝑆 𝑖 =−1 +1 𝑆 𝑖 𝑒 𝛽𝜂 𝑆 𝑖 𝑆 𝑖 =−1 +1 𝑒 𝛽𝜂
= − 𝑒 −𝛽𝜂 + 𝑒 𝛽𝜂 𝑒 −𝛽𝜂 + 𝑒 𝛽𝜂 = tanh 𝛽𝜂
𝑆 𝑎𝑣 = tanh 𝛽 ℎ+ 𝑆 𝑧𝐽
Zero field solution (h=0)
𝑆 = tanh 𝛽 𝑆 𝑧𝐽

5. Let’s explore
𝑦= tanh 𝑎𝑦
on variation of parameter a
a=0.7
a=0.5
a=1.0
a=1.2
Y=0.66
Y=-0.66

6. For a<1 <S>=0 only solution
For a>1 two solution for non-zero <S>
It can be shown that the non-trivial solutions minimize the
free energy
a=1 defines the critical temperature TC
𝑎=𝑧𝐽𝛽=1
𝑇 𝐶 = 𝑧𝐽 𝑘 𝐵
in mean field approximation

7. Discussion of the thermodynamics
T near the critical temperature TC
𝑆 ->0
If we allow for a small applied magnetic field h such that h≪ 𝑆 𝑎𝑣 𝑧𝐽
we can explore magnetization and susceptibility near TC
tanh 𝑥 = 𝑒 𝑥 − 𝑒 −𝑥 𝑒 𝑥 + 𝑒 −𝑥 ≈ 1+𝑥 𝑥 𝑥 3 −1+𝑥− 1 2 𝑥 𝑥 3 1+𝑥 𝑥 𝑥 3 +1−𝑥 𝑥 2 − 1 6 𝑥 3
𝑆 = tanh 𝛽 ℎ+ 𝑆 𝑧𝐽
x≈0
= 2𝑥 𝑥 𝑥 2 =𝑥 𝑥 𝑥 2 =𝑥 𝑥 𝑥 ≈𝑥 𝑥 2 1− 𝑥 2 2
tanh 𝑥≈𝑥− 𝑥 …
=𝑥 𝑥 2 − 𝑥 2 2 − 𝑥 4 ≈𝑥 𝑥 2 − 𝑥 =𝑥 1− 1 3 𝑥 2
tanh 𝛽 ℎ+ 𝑆 𝑧𝐽 ≈𝛽 ℎ+ 𝑆 𝑧𝐽 − 𝛽 𝑆 𝑧𝐽 3 3
𝑆 =𝛽ℎ+ 𝑆 T C T − 𝑆 T C T 3
For T>TC we can neglect the cubic term

8. 𝑆 =𝛽ℎ+ 𝑆 𝑆 T C T
𝜒∝ 𝜕 𝑆 𝜕ℎ
with
𝜒 1− T C T = const k B T ≈ const k B T C
𝜕 𝑆 𝜕ℎ =𝛽+ 𝜕 𝑆 𝜕ℎ T C T
𝜒= const k B T +𝜒 T C T
𝜒∝ 𝑇 𝐶 𝑇− 𝑇 𝐶
T-dependence of <S> (∝ magnetization) at h=0 near TC
𝑆 =0 𝑓𝑜𝑟 𝑇> 𝑇 𝐶
𝑆 = 𝑆 T C T − 𝑆 T C T 3
1= T C T − 𝑆 T C T 3
𝑇< 𝑇 𝐶 :

9. 1= T C T − 𝑆 T C T 3
3 𝑇 T 𝐶 3 =3 𝑇 2 T C 2 − 𝑆 2
𝑆 2 =3 𝑇 2 T C 2 1− 𝑇 𝑇 𝐶 ≈3 1− 𝑇 𝑇 𝐶
𝑆 ∝ 𝑇 𝐶 −𝑇
𝑆 = tanh 𝑆 𝑇 𝐶 𝑇