  It has been known that encoding Boltzmann weights of a classical spin model
in amplitudes of a many-body wave function can provide quantum models whose
phase structure is characterized by using classical phase transitions. In
particular, such correspondence can lead to find new quantum phases
corresponding to well-known classical phases. Here, we investigate this problem
for Kosterlitz-Thouless (KT) phase in the d-state clock model where we find a
corresponding quantum model constructed by applying a local invertible
transformation on a d-level version of Kitaev's Toric code. In particular, we
show the ground state fidelity in such quantum model is mapped to the heat
capacity of the clock model. Accordingly, we identify an extended topological
phase transition in our model in a sense that, for $d \geq 5$, a KT-like
quantum phase emerges between a $Z_d$ topological phase and a trivial phase.
Then, using a mapping to the correlation function in the clock model, we
introduce a non-local (string) observable for the quantum model which
exponentially decays in terms of distance between two endpoints of the
corresponding string in the $Z_d$ topological phase while it shows a power law
behavior in the KT-like phase. Finally, using well-known transition
temperatures for d-state clock model we give evidences to show that while
stability of both $Z_d$ topological phase and the KT-like phase increases by
increasing d, the KT-like phase is even more stable than the $Z_d$ topological
phase for large d.
