MPS Parameters
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Area Law & MPS
Area law: For gapped 1D systems, entanglement entropy S(l) ≤ const (independent of system size). This means an MPS with finite bond dimension χ can represent the ground state exactly (or to exponential accuracy).
Schmidt decomposition: At each bond, |Ψ⟩ = Σᵢ λᵢ |φᵢ_L⟩|φᵢ_R⟩. Entanglement entropy S = -Σλᵢ² log λᵢ². Truncating to χ largest λᵢ gives the optimal MPS approximation (Vidal 2003).
TEBD: Time-Evolving Block Decimation evolves |Ψ(t+δt)⟩ = e^{-iHδt}|Ψ(t)⟩ by applying 2-site gates and re-truncating. Entanglement grows as ~t/6 for critical systems.
χ scaling: For critical systems (CFT), need χ ~ e^{S} ~ polynomial in L. For gapped phases, χ = O(1) suffices.
Schmidt decomposition: At each bond, |Ψ⟩ = Σᵢ λᵢ |φᵢ_L⟩|φᵢ_R⟩. Entanglement entropy S = -Σλᵢ² log λᵢ². Truncating to χ largest λᵢ gives the optimal MPS approximation (Vidal 2003).
TEBD: Time-Evolving Block Decimation evolves |Ψ(t+δt)⟩ = e^{-iHδt}|Ψ(t)⟩ by applying 2-site gates and re-truncating. Entanglement grows as ~t/6 for critical systems.
χ scaling: For critical systems (CFT), need χ ~ e^{S} ~ polynomial in L. For gapped phases, χ = O(1) suffices.