  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.
