Tensor network state approach to quantum topological phase transitions and their criticalities of $\mathbb{Z}_2$ topologically ordered states

We construct a general wave function with the topological order by introducing the $\mathbb{Z}_{2}$ gauge degrees of freedom, characterizing both the toric code state and double semion state. Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter $\lambda $ and identify three different quantum critical points (QCPs) at $\lambda =0$, $\pm 1.73$. The first one separates the toric code phase and double semion phase, while later two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry breaking phase, respectively. When mapping to the exactly solved statistical models, the norm of the tensor network wave function is transformed into the partition function of the eight-vertex model. Actually such a quantum-classical mapping can not reveal the rich structures of low-energy excitations at these three QCPs. So we further demonstrate that the full eigenvalue spectra of the transfer operators with/without the flux insertions can describe the complete quantum criticalities, which are characterized by the two-dimensional compactified free boson conformal field theories (CFTs) with the compactified radii $R=\sqrt{6}$ for the QCPs at $\lambda =\pm\sqrt{3}$ and $R=\sqrt{8/3}$ for the QCP at $\lambda =0$. For the QCP at $\lambda =0$, there are no anyon condensation, and the emerged matrix product operator symmetries result in a rich structure of the low-energy excitations, distinct from those of both toric code and double semion phases. Finally, we discuss the possible relation between our conformal quantum criticalities and the general (2+1) spatial-time dimensional CFTs for quantum topological phase transitions.
Comment: 17 pages, 16 figures, including three appendices, revised version