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

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

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

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

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

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

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+ 1 2 𝑥 2 + 1 6 𝑥 3 −1+𝑥− 1 2 𝑥 2 + 1 6 𝑥 3 1+𝑥+ 1 2 𝑥 2 + 1 6 𝑥 3 +1−𝑥+ 1 2 𝑥 2 − 1 6 𝑥 3 𝑆 = tanh 𝛽 ℎ+ 𝑆 𝑧𝐽 x≈0 = 2𝑥+ 1 3 𝑥 3 2+ 𝑥 2 =𝑥 2+ 1 3 𝑥 2 2+ 𝑥 2 =𝑥 1+ 1 6 𝑥 2 1+ 𝑥 2 2 ≈𝑥 1+ 1 6 𝑥 2 1− 𝑥 2 2 tanh 𝑥≈𝑥− 𝑥 3 3 +… =𝑥 1+ 1 6 𝑥 2 − 𝑥 2 2 − 1 12 𝑥 4 ≈𝑥 1+ 1 6 𝑥 2 − 𝑥 2 2 =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 − 1 3 𝑆 T C T 3 For T>TC we can neglect the cubic term

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 − 1 3 𝑆 T C T 3 1= T C T − 1 3 𝑆 2 T C T 3 𝑇< 𝑇 𝐶 :

1= T C T − 1 3 𝑆 2 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