Competing Orders: Superconductor vs Antiferromagnet

Landau Free Energy with Two Order Parameters — Phase Diagram vs Doping & Temperature
Phase diagram (doping p vs temperature T)
Free energy landscape (Δ_SC vs Δ_AFM)
p = 0.100
T = 0.050 t
g_SC = 0.80
g_AFM = 1.00
γ = +0.50
Landau free energy:
F = a_SC Δ² + b_SC Δ⁴
+ a_AFM m² + b_AFM m⁴
+ γ Δ² m²
Coefficients:
a_SC = α_SC(T − T_SC(p))
a_AFM = α_AFM(T − T_AFM(p))
γ > 0: competing
γ < 0: cooperating
Phases:
SC only: Δ≠0, m=0
AFM only: m≠0, Δ=0
Coexist: both ≠ 0
In underdoped cuprates and iron-based superconductors, superconductivity (Δ) and antiferromagnetism (m) compete for the same carriers. The cross-coupling γ determines whether they coexist or phase-separate. For γ > √(b_SC · b_AFM), phases are mutually exclusive (first-order transition).