Half-filled band dimerization, gap opening at k_F, CDW order parameter
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Gap 2Δ / t
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T_Peierls / t
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Dimerization u
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Kohn anomaly
A 1D metal with half-filled band spontaneously dimerizes: lattice distortion u → electron-phonon coupling → gap 2Δ at k=±k_F = ±π/2a. BCS-like gap equation: Δ = 2ℏω_D exp(−1/λ) where λ = g²/πv_F K is the dimensionless coupling. CDW order parameter ⟨ρ_{2k_F}⟩ ∝ Δ; phonon dispersion softens at 2k_F (Kohn anomaly: ω²(2k_F) → 0 at T_P). The condensation energy ~ (DOS)×Δ² competes with elastic cost K×u².