Conformal Field Theories generated by Chern Insulators under Quantum Decoherence

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 $\mathcal{Z}=\text{tr}\{{\widehat{\rho }}_c^D{\widehat{\rho }}_{\mathrm{\Omega }}\}$ is mapped to the partition function of the desired CFT, where ${\widehat{\rho }}_c^D$ and ${\widehat{\rho }}_{\mathrm{\Omega }}$ 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 Rényi relative entropy $\mathcal{F}=-\log \text{tr}\{{\widehat{\rho }}_c^D{\widehat{\rho }}_{\mathrm{\Omega }}\}$ 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.