Consider a quantum system described by a gapped Hamiltonian $H_0$ with degenerate ground states (GS), adding a perturbation term $V$ to $H_0$, then the low-energy physics can be described by an effective Hamiltonian $H_{eff}$ acting within the GS subspace of $H_0$, where $H_{eff}$ can be calculated from degenerate perturbation theory.
What if the GS degeneracy of $H_0$ is a topological degeneracy ?
I learned that the topological degeneracy is robust against ANY local perturbations. Does this imply that: if the above $H_0$ describes a topologically ordered system defined on a torus with a topological GS degeneracy, and $V$ represents ANY local perturbations, then would the resulting effective Hamiltonian $H_{eff}$ be always trivial (i.e., $H_{eff}=$constant number) ??
And in the opposite case, if we find $H_{eff}$ (corresponding to ANY local perturbations $V$) is a constant number at any finite order from a degenerate perturbation theory calculation, then does this imply that $H_0$ describes a topologically ordered system??
Of course, for a finite system, $H_{eff}$ is usually NOT a constant number. All the above we talk about is in the thermodynamic limit.
Thanks a lot.