Can we write the effective field theory for the toric code model?

Question

If not, then why not? If yes, then what is the effective field theory?

Answer 1

No.

Effective field theory only describes system near critical point. Toric code model is far from critical point. Thus "No".

Toric code model realizes a $Z_2$-topological order. When a state with a $Z_2$-topological order is close to a continuous phase transition, then the state will have an effective field theory description. But which effective field theory will depend on which continuous phase transition the state is close to.