In the former Phys.SE post Can gapped state and gapless state be adiabatically connected to each other?, I saw different answers from Norbert Schuch and Xiaogang Wen.
I am confused by the question: Can we define if a STATE is gapped or gapless without mentioning any Hamiltonian?
In Norbert Schuch's example on the Toric code, he gave an example that a state can be the ground state of a gapped Hamiltonian and of another gapless Hamiltonian as well. So it seems we have to specify the Hamiltonian when we talk about gapped or gapless state.
In Wen's answer talking about the definition of gapped and gapless state (here), he mentioned
I wonder, if some one had consider the definition of gapped many-body system very carefully, he/she might discovered the notion on topological order mathematically.
Here it seems that Prof. Wen suggest, that the gapped or gapless property is intrinsic to the state itself (as in his papers to explain phases of states as equivalent sets w.r.t. finite depth quantum circuits).
So my question:
If Schuch is right, then there is no absolute definition of the phase of a state since it's Hamiltonian dependent.
If Wen is right, then the phase of a state will not depend on the Hamiltonian, or there exists a kind of mapping between the state and the Hamiltonian so that there is no ambiguity on the gapped or gapless property of a state.