Kosterlitz-Thouless phase and Zd topological quantum phase

It has been known that encoding Boltzmann weights of a classical spin model in amplitudes of a manybody wave function can provide quantum models whose phase structure is characterized by using classical phase transitions. In particular, such correspondence can lead to finding new quantum phases corresponding to well-known classical phases. Here, we investigate this problem for the 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 a quantum model is mapped to the heat capacity of the clock model. Accordingly, we identify an extended topological phase transition in our model in the sense that, for d >= 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 nonlocal (string) observable for the quantum model which exponentially decays in terms of distance between two end points 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 the d-state clock model, we give evidence to show that while the stability of both the 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.