ETH violation in PXP model — Néel state revivals, entanglement entropy, thermalization contrast
Quantum Revival — Néel State Fidelity |⟨ψ(0)|ψ(t)⟩|²
8
0.00
PXP model Hamiltonian: H = Σᵢ Pᵢ₋₁ σˣᵢ Pᵢ₊₁ (P = projector onto |0⟩, blocking adjacent excitations — Rydberg blockade).
Scar states: |Z₂⟩ = |↑↓↑↓...⟩ shows anomalously large fidelity oscillations → ETH violation. Period ≈ π/Ω_scar.
Entanglement Entropy Growth
Scar initial state: S(t) oscillates, grows slowly — area law behavior.
Thermal state: S(t) ~ t (linear growth to volume law).
ETH prediction: all states at same E should thermalize identically.
Energy Eigenstate Expectation Values (ETH Test)
Eigenstate Thermalization Hypothesis: ⟨n|O|n⟩ = f(E) smooth function of energy.
Scar states (red) are outliers — they have anomalous expectation values amid otherwise thermal spectrum.
Floquet Spectrum — Level Statistics
Thermal (GOE): Wigner-Dyson level spacing distribution P(s) ~ s·e^(-πs²/4) — level repulsion.
Integrable/scar states show Poisson P(s) ~ e^(-s) — no repulsion.