Effective Conformal Field Theory generated from Pure and Dephased Chern insulator

We demonstrate that the fidelity between two states with different Chern numbers $\mathcal Z = \mathrm{tr} \{ \rho \rho' \} $ serves as a generating theory for an effective conformal field theory (CFT) at the $(2+0)d$ temporal interface. $\rho$ can be chosen as a pure trivial insulator, and $\rho'$ can be taken as a pure or dephased Chern insulator density matrix. More specifically, we obtain the following results: (1) through evaluation of the effective central charge, stiffness, and correlation function (the ``strange correlators"), we demonstrate that the fidelity between a trivial insulator and an insulator with Chern number $C=1$ maps to a CFT with effective central charge $c_{\rm eff} = 1$; while the fidelity between two Chern insulators with Chern numbers $C = \pm 1$ maps to a CFT with $c_{\rm eff} = 2$. (2) The density matrix of the Chern insulator becomes a quantum spin Hall insulator in the doubled Hilbert space, and the dephasing acts as interaction between the two spin species. (3) In the limit of infinite dephasing the Chern insulator becomes a superconductor in the doubled Hilbert space, featuring the ``strong-weak" U(1) spontaneous symmetry breaking. The analysis based on Laughlin wave function and previous studies of projected wave function of quantum spin Hall insulator suggest this is a power-law superconductor. (4) With increasing strength of dephasing, the amplitude of single particle strange correlator is suppressed, while the Cooper pair strange correlator is enhanced, consistent with the trend of emerging superconductivity.