The system is chaotic. But there's no classical system to compare it to.
Quantum chaos is traditionally diagnosed by comparison: a quantum system is “chaotic” if its classical analogue is chaotic. Wigner-Dyson level statistics, eigenstate thermalization, sensitivity to perturbations — all these signatures were first understood through the quantum-classical correspondence. But some quantum systems have no classical analogue (arXiv:2603.20540).
The PXP model: a spin-1/2 chain with a kinetic constraint — a spin can flip only if both neighbors are down. This constraint has no classical equivalent. There's no phase space, no Lyapunov exponents, no semiclassical limit to take. Yet the system shows signatures of chaos: Wigner-Dyson level spacing for large system sizes, eigenstate thermalization (most eigenstates look thermal), and ballistic energy propagation in quench dynamics.
But also signatures of non-chaos: rare spectral violations of ETH (quantum many-body scars — special eigenstates that resist thermalization), semi-Poisson statistics at small sizes transitioning to Wigner-Dyson only at large sizes, and non-Gaussian eigenvector component distributions (a previously unstudied diagnostic).
The structural insight: chaos in constrained quantum systems is not binary. The PXP model is neither fully chaotic (the scars break ergodicity) nor fully integrable (the majority of states thermalize). It occupies a middle ground — weakly ergodicity-breaking — that has no classical analogue because the constraint itself is purely quantum. The diagnostics of chaos work (level statistics, ETH) but the results are mixed because the system is genuinely intermediate. Chaos without a classical shadow.