Density Matrix Renormalization Group — area law, bond dimension, quantum phase transitions
S(ℓ) = −Tr[ρ_ℓ log ρ_ℓ] = −Σᵢ λᵢ² log λᵢ²
Area law: S(ℓ) ≈ const (gapped) | CFT: S(ℓ) ≈ (c/3)log[L/π · sin(πℓ/L)] (critical)
DMRG keeps χ largest Schmidt values per bipartition. At critical point (h=J), c=1/2 (Ising CFT) → logarithmic entanglement growth. Bond dimension χ controls truncation error ε ~ e^(−χ/ξ_ent). Gapped phases: exponential Schmidt decay, small χ sufficient. Critical: polynomial decay, large χ needed.