An N = 8 spin-½ chain (open boundary) evolves under the mixed-field Ising Hamiltonian, exactly diagonalized every time you move the hz slider:
H = -J Σ Sz(i)Sz(i+1) - hx Σ Sx(i) - hz Σ Sz(i)
J = 1, hx = 1.05 (fixed) — h_z: yours to tune
At hz = 0 the model maps onto free fermions (transverse-field Ising) — it is integrable: it has an extensive set of conserved quantities, and individual energy eigenstates keep memory of the initial condition. At hz ≈ 0.5 integrability is broken and the level spacings become Wigner-Dyson (quantum-chaotic).
The Eigenstate Thermalization Hypothesis (Deutsch 1991, Srednicki 1994) proposes that in a chaotic system, a local observable's matrix elements in the energy eigenbasis behave as
⟨n|O|m⟩ = O_micro(Ē) δ(n,m) + e^(-S(Ē)/2) f(Ē,ω) R(n,m)
— i.e. the diagonal elements ⟨n|O|n⟩ vary smoothly with energy En alone, so a single eigenstate already "looks thermal" for local observables; off-diagonal elements are exponentially small and act as thermal noise. That is why an isolated, closed quantum system can relax to thermal-looking local expectation values under purely unitary (Schrödinger) evolution, with no bath and no wavefunction collapse.
- The chain of spheres shows ⟨Sz(i)⟩(t) after the domain-wall/Néel quench — the initial imbalance dissolves toward zero (thermal equilibrium) fast when chaotic, and only partially / with persistent oscillations when integrable.
- The scatter cloud plots ⟨n|Sz(mid)Sz(mid+1)|n⟩ against En for all 256 eigenstates — a two-site correlator, not a single-site magnetization, because a single Sz is pinned to exactly zero at hz=0 by the model's exact global spin-flip symmetry (a trivial selection rule that would masquerade as "perfect ETH"). A narrow band across the spectrum = ETH satisfied. A wide, scattered cloud = ETH violated (extra conserved quantities let eigenstates disagree at the same energy).
- ETH σ is the standard deviation of that diagonal expectation value across the bulk of the spectrum — it measurably shrinks as hz grows from 0 to ≈0.6 in this exact diagonalization, the same trend seen in the numerical ETH literature (e.g. Kim & Huse) for this exact Hamiltonian.