We demonstrate that the fidelity between a pure state trivial insulator and the mixed state density matrix of a Chern insulator under decoherence can be mapped to a variety of two-dimensional conformal field theories (CFT); more specifically, the quantity $\mathcal{ Z} = \text{ tr}\{ \hat{ \rho}^D_c \hat{ \rho}_\Omega \}$ is mapped to the partition function of the desired CFT, where $\hat{ \rho}^D_c$ and $\hat{ \rho}_\Omega$ are respectively the density matrices of the decohered Chern insulator and a pure state trivial insulator. For a pure state Chern insulator with Chern number $2N$, the fidelity $\mathcal{ Z}$ is mapped to the partition function of the $\text{ U}(2N)_1$ CFT; under weak decoherence, the Chern insulator density matrix can experience certain instability, and the "partition function" $\mathcal{ Z}$ can flow to other interacting CFTs with smaller central charges. The R\'{ e}nyi relative entropy $\mathcal{ F} = - \log \text{ tr}\{ \hat{ \rho}^D_c \hat{ \rho}_\Omega \}$ is mapped to the free energy of the CFT, and we demonstrate that the central charge of the CFT can be extracted from the finite size scaling of $\mathcal{ F}$, analogous to the well-known finite size scaling of $2d$ CFT.
Categories: cond-mat.str-el hep-th quant-ph