Extend the prethermal decomposition to long-range entangled metastable states

Develop an analogue of Theorem H=H0+V for long-range entangled metastable states by constructing, for any such metastable state in a local lattice system, a decomposition H=H0+V where the state is an exact eigenstate of H0, H0 and V are quasi-local, and V is small in an appropriate local norm. Establish precise locality and norm bounds comparable to those obtained in Theorem H=H0+V for short-range entangled states.

Background

A key structural result of the paper (Theorem H=H0+V) shows that any short-range entangled metastable state is an exact eigenstate of a nearby Hamiltonian H0, with the difference V small in local norm, enabling rigorous prethermalization bounds.

The authors partially extend phenomenology to long-range entangled states under additional assumptions but explicitly lack an analogue of the structural decomposition theorem in that setting, leaving open whether an appropriate H0+V representation exists.