The Four Exits

An integrable system is exactly solvable — its dynamics are organized by conserved quantities that prevent chaos. Deform the system slightly, and integrability can break. The standard picture has two outcomes: the deformation preserves integrability, or it destroys it. Binary.

There are four exits, not two.

The first two are expected. Some deformations maintain exact solvability (the deformed XXZ spin chain remains integrable under certain parameter shifts). Others immediately introduce chaos (generic perturbations break all conserved quantities at once).

The third exit is stranger: deformations that are integrable only when all orders of the deformation parameter are included. Truncate the perturbation series at any finite order, and the system is chaotic. Only the full, infinite series is integrable. These appear in holographic models, where the long-range deformations generated by gauge/gravity duality are exactly this type — order-by-order chaos, all-orders order.

The fourth exit is the mirror image: deformations that are perturbatively integrable to some finite order but cannot be extended to an integrable model at all orders. The system appears solvable when you don’t look too closely. It passes every low-order test for integrability and fails at a specific higher order. The integrability is a mirage that vanishes at finite depth.

The volume scaling at which chaos appears differs for each type. The third and fourth types exhibit intermediate scaling between the strong and weak integrability breaking that characterize the first two types. Four distinct routes to chaos, each with its own signature.

Some systems are solvable only in their totality. Any finite approximation is chaotic. And some systems appear solvable at every order of approximation but are chaotic in truth.