Zero-Temperature Quantum Criticality

The transverse-field Ising model H = -J Σσ_z σ_z - Γ Σσ_x has a quantum critical point at Γ/J = 1. Quantum fluctuations, not thermal ones, drive this T=0 phase transition.

Phase Diagram
Spectral Gap
Correlations

Parameters

H = −J Σ σ_z^i σ_z^{i+1}
− Γ Σ σ_x^i
QCP at Γ/J = 1 (T=0)
Δ ∝ |Γ/J − 1|^(zν)

Observables (MF)

Order param ⟨σ_z⟩---
Spectral gap Δ/J---
Corr. length ξ---
Phase---
Critical exponent zν1 (Ising)
Quantum criticality facts:
QCP: Γ_c/J = 1 (exact, 1D)
Universality: same as 2D classical
(quantum-classical mapping)
Non-Fermi liquid near QCP
ω/T scaling at criticality