Parameters
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Physics
Heisenberg model: H = JΣᵢ(SˣᵢSˣᵢ₊₁+SʸᵢSʸᵢ₊₁+ΔSᶻᵢSᶻᵢ₊₁). Exactly solvable by Bethe ansatz (1931) but ED gives full spectrum.
Sectors: Conserved: total Sᶻ, momentum k. We diagonalize Sz=0 sector (half-filling).
Level statistics: Integrable (Bethe): Poisson distribution of level spacings. Chaotic/thermal: Wigner-Dyson (GOE). Adding disorder: Poisson → MBL transition.
Structure factor S(k): S(k)=Σⱼe^{ikj}⟨S⁰·Sⱼ⟩. AFM (J>0): peak at k=π. FM: peak at k=0.
Bethe ansatz: Ground state energy E₀/N → -ln(2)+¼ = -0.4431... as N→∞.
Sectors: Conserved: total Sᶻ, momentum k. We diagonalize Sz=0 sector (half-filling).
Level statistics: Integrable (Bethe): Poisson distribution of level spacings. Chaotic/thermal: Wigner-Dyson (GOE). Adding disorder: Poisson → MBL transition.
Structure factor S(k): S(k)=Σⱼe^{ikj}⟨S⁰·Sⱼ⟩. AFM (J>0): peak at k=π. FM: peak at k=0.
Bethe ansatz: Ground state energy E₀/N → -ln(2)+¼ = -0.4431... as N→∞.