Conformal field theories generated by Chern insulators under decoherence or measurement
Kaixiang Su, Nayan Myerson-Jain, Cenke Xu
Abstract
We demonstrate that the fidelity between a pure state trivial insulator and the mixed state density matrix of a Chern insulator generated from decoherence or measurement can be mapped to a variety of two-dimensional conformal field theories (CFTs); more specifically, the quantity $\mathcal{Z}=\mathrm{tr}{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}}$ is mapped to the partition function of the desired CFT, where ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}$ are respectively the density matrix of a pure state trivial insulator and the mixed state density matrix generated from the Chern insulator. For a pure state Chern insulator with Chern number $2N$, the fidelity $\mathcal{Z}$ is mapped to the partition function of the $\mathrm{U}{(2N)}_{1}$ CFT; under decoherence or measurement, the Chern insulator density matrix can experience a certain instability, and the ``partition function'' $\mathcal{Z}$ can flow to other interacting CFTs with smaller central charges. The R\'enyi relative entropy $\mathcal{F}=\ensuremath{-}ln\mathrm{tr}{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}}$ 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.