DMRG — Density Matrix Renormalization Group

DMRG (White 1992) finds the ground state of 1D quantum chains by iteratively truncating the Hilbert space using the entanglement spectrum. Keep the χ most entangled Schmidt states; the discarded weight ε tells you the error. Energy converges exponentially with bond dimension χ.

Parameters

Bond dimension χ: 12 System size N: 20 Coupling ratio J₂/J₁: 0.0

Truncation error ε: —

Ground energy/site: —

Entanglement S: —

ρ_A = Tr_B|ψ⟩⟨ψ|
Keep χ largest eigenvalues
ε = 1 - Σ_{i≤χ} λᵢ
E_0(χ) → E_exact as χ→∞

DMRG is essentially MPS variational optimization. For gapped 1D systems, area law guarantees χ = O(1) is sufficient. Critical systems (J₂/J₁ = 0.5 Majumdar-Ghosh) need larger χ due to logarithmic growth. White's key insight: truncate by entanglement, not energy.