Quantum Critical Point — Hertz-Millis

Zero-Temperature Phase Transitions

A quantum critical point (QCP) occurs at T=0 when a continuous phase transition is driven by quantum fluctuations (tuned by pressure, field, or doping) rather than thermal ones.

The QCP radiates a "quantum critical fan" in (g,T) space where non-Fermi liquid behavior dominates. Above it: resistivity ρ~T (linear), unlike Fermi liquid ρ~T².

Hertz-Millis theory: integrating out fermions gives a bosonic action with dynamic exponent z. For ferromagnetic QCP: z=3; antiferromagnetic: z=2. The effective dimension d+z determines criticality.

S ~ ∫(q,ω) |φ|²(r + q² + |ω|/q^{z-2})
ξ ~ |g−g_c|^{-ν}, τ ~ ξ^z
C/T ~ log(1/T) near AF-QCP
ρ ~ T^{(d+z-2)/(z)} non-FL

Tuning parameter g/g_c 1.0 Dynamic exponent z 3 Show: Phase diagram

Quantum Critical Point — Hertz-Millis Theory

Zero-Temperature Phase Transitions