Short version of my question is this : what is the nature of the phase transition in the Kitaev honeycomb model ?
Kitaev honeycomb model undergoes a phase transition from a gapped to a gapless-single-mode phase when the coupling constants are tuned. This way of describing the phases and the transition between them is different from the way classical thermal phase transitions are described, where one talks about order parameters, diverging correlation lengths and scale invariant critical theory (if it is a continuous transition), ergodicity breaking etc. Are there similar notions in the case of the Kitaev model ?
In the case of another phase transition in a quantum system, namely the transverse field Ising model (TFIM), I can define an order parameter (spin expectation value in a suitable direction). Since there is this order parameter, and a relevant tuning parameter (say the transverse field strength) I can think of quantum fluctuations of the order parameter, scaling of various correlations under tuning of the parameter etc in a manner similar to the classical transitions.