Mean Field Theory
Ising model self-consistency: m = tanh(βJzm + βh) — spontaneous symmetry breaking
Parameters
Temperature T (units of Jz/k_B)
T =
0.80
T_c =
1.00
T/T_c =
0.80
External field h (units of Jz)
h =
0.00
Coordination number z
z =
4
(2D square lattice = 4)
Order parameter m*:
—
Phase:
—
Free energy minimum:
—
Self-consistency equation
: m = tanh(βJzm + βh)
β = 1/T. Solutions = intersections of y=tanh and y=m.
Critical temperature
: T_c = Jz/k_B (slope=1 at m=0)
T < T_c: two stable solutions (ferromagnet)
T > T_c: unique solution m=0 (paramagnet)
Critical exponent
: m ~ (T_c−T)^(1/2) mean-field (β_MF=1/2), vs exact Ising β=1/8 in 2D.