Naive questions on the ground states of Kitaev model

Question

Up to now, I found myself still does not have a deep understanding of the honeycomb Kitaev model, and I got some naive questions about the ground states (GSs) of the this model:

(1) Consider a simple case that $J_x=J_y=0$, then the model reduces to $$H=J_z\sum_{z\text{ }\text{links}}S_i^zS_j^z.$$ It's obvious that $H$ has highly degenerate GSs (degeneracy$=2^N$, where $N$ is the number of unit cells), and in each GS configuration, every two spins connected by a $z$ link are aligned parallel (if $J_z<0$) or antiparallel (if $J_z>0$). Thus, each of these GSs corresponds to a spin configuration and is not a spin liquid (SL).

On the other hand, the Majorana exact solution tells us that the GS of $H$ is a gapped SL.

So, does the exact solution just single out one of the highly degenerate GSs, which happens to be a SL (some certain manner of superposition of the above $2^N$ spin configurations) ??

(2) Following question (1), if the exact solution does single out just one of the highly degenerate GSs of $H$, and how could this be?

According to Kitaev's discussion on P.19 in his paper, the GS is achieved by the vortex-free field configuration, which follows from a theorem proved by Lieb.

But shouldn't an exact solution gives all the $2^N$ GSs ? Why only single out one which strongly contradicts the obviously correct GS degeneracy$=2^N$ ? Is there something wrong with applying the Lieb's theorem here ? I'm confused....

This post imported from StackExchange Physics at 2014-04-13 11:24 (UCT), posted by SE-user K-boy