Quantum Critical Point — Transverse-Field Ising Model

The 1D transverse-field Ising model H = −J∑σᵢᶻσᵢ₊₁ᶻ − h∑σᵢˣ has an exact quantum phase transition at h/J = 1. At the QCP, the correlation length diverges, the gap closes, and the system is described by a (1+1)D conformal field theory with central charge c = 1/2.

Transverse field h/J: 1.0

Chain length N: 40

Temperature T/J: 0.1

Exact dispersion:
ε(k) = 2J√((h/J−cos k)² + sin²k)
= 2√(1 + (h/J)² − 2(h/J)cos k)

Gap: Δ = 2|h/J − 1|
Corr. length: ξ ~ |h/J−1|⁻¹

CFT at QCP: c=1/2, η=1/4
C(r) ~ r^{-1/4}