Degenerate perturbation theory applied to topological degeneracy? 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.
 A: The effective Hamiltonian need not be a constant. In fact it's usually a sum of terms which connect different degenerate ground states by topologically non-trivial actions.
For example in the Toric code ($d=2$ square lattice of length $L$ with periodic boundary conditions...) the effective Hamiltonian is going to be a sum of the 4 string/logical operators which correspond to winding the electric or magnetic excitations around the torus. The $2^2$ ground states are labelled by the presence or absence of closed strings, so these operators connect different ones.
The statement the degeneracy is robust is that the level splitting between different ground states is $\delta E \approx e^{-L}$; you need to go to order $O(L)$ in perturbation theory.
Now this is assuming the perturbation is local, like something that flips a couple spins around a site. If you allow non-local terms to enter you could just imagine adding something which may flip all the spins needed for a string operator at once which ruins this conclusion of robustness.
