How to define the 'gapped ' and 'gapless' states?

Question

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.

Answer 1

Although it's true, as Norbert Schuch pointed out, that the same state can be the ground state of Hamiltonians with and without gap, in general this behavior seems rather fine-tuned. I would expect that, for a generic perturbation to the gapless Hamiltonians he discussed, a gap would open up. For this reason, physicists tend to ignore this subtlety and treat gapped/gapless as a property of the state. In particular:

• A ground state of gapped Hamiltonian must have correlations which decay exponentially with distance (this has been proved rigorously by Hastings and Koma).

• Empirically, it appears that the ground state of a gapless Hamiltonian generically has correlations with decay as a power-law with distance. (Though this may fail at some fine-tuned points, as Norbert Schuch's examples demonstrate).

However, if one wants to make the notion of a "gapped state" totally rigorous without reference to its parent Hamiltonian, one could make the following definition:

A state $|\psi\rangle$ is gapped if there exists a gapped Hamiltonian $H$ such that $|\psi\rangle$ is its ground state.

This obviously does not rule out the existence of another Hamiltonian $H'$ which is gapless and has the same ground state. However, the existence of a gapped parent Hamiltonian already ensures that $|\psi\rangle$ has sufficiently nice properties (e.g. exponentially decaying correlations, area law for entanglement entropy) that its topological order can be defined (you don't ever need to refer to the Hamiltonian $H$ -- its only role is to ensure that the state has these nice properties).