DMRG & Matrix Product States

DMRG (White 1992) finds the ground state of 1D Hamiltonians by variationally optimizing a Matrix Product State |ψ⟩ = Σ A¹_{s₁}A²_{s₂}…Aᴺ_{sN}|s₁…sN⟩. The key parameter is the bond dimension χ: it controls how much entanglement the ansatz can capture. For gapped phases, entanglement satisfies an area law S ~ const, so finite χ is exact. At critical points, S ~ (c/6)log(N) (CFT), requiring χ ~ poly(N). Disorder localizes the system (many-body localization), reducing entanglement even at criticality.