Entanglement Entropy & Area Law

Chain length N: 40

Cut position ℓ: 20

Coupling Δ: 1.00

Phase: Critical

S(ℓ) von Neumann

S_max (Page)

Central charge c

c/3 · ln(ℓ_eff)

Area Law: For gapped 1D systems, S(ℓ) = const (independent of ℓ). At critical points (gapless, CFT), S(ℓ) = (c/3)·ln[sin(πℓ/N)·N/π] + const — logarithmic growth governed by central charge c. The c/3 prefactor is universal; c=1 for free boson (XX chain), c=1/2 for Ising.