Renormalization Group: Applications in Statistical Physics I-II

Landau expansion; mean-field theory
Expand free energy (density) in terms of order parameter (scalar
field) φ near a continuous (second-order) phase transition at T c :
f (φ) = r 2 φ2 + u 4! φ4 + . . . − h φ ,
f
r > 0
r = a(T − T c ), u > 0; conjugate field
h breaks Z(2) symmetry φ → −φ.
f ′ (φ) = 0 ⇒ equation of state:
φ - 0 φ+
r = 0
r < 0
φ
h(T , φ) = r(T ) φ + u 6 φ3 ;
φ
stability: f ′′ (φ) = r + u 2 φ2 > 0.
◮ Critical isotherm at T = T c :
h(T c , φ) = u 6 φ3 .
0
h > 0
T c
h < 0
T
◮ Spontaneous order parameter for
r < 0: φ ± = ±(6|r|/u) 1/2 .