Possible 3-dimensional Topological Excitonic Insulator in a Semimetal under Magnetic Field

Motivated by a recent graphite experiment under high magnetic field, we introduce an interacting electron model with a pair of electron pocket and hole pocket under magnetic field. We use a fermionic renormalization group (RG) method and study a parquet RG equation of the model for various types of effective electron-electron interactions. We found that, for a screened Coulomb interaction, the one-loop level RG equation exhibit an instability of the excitonic pairing, while, for a short-ranged repulsive interaction case, the excitonic instability and a charge-density wave instability appear simultaneously. We further argue that, depending on the sign of the interaction, the excitonic insulator phase can be either a symmetry-protected topological excitonic insulator or topologically trivial excitonic insulator. In the former case, an odd-parity excitonic pairing in the bulk reconstruct a chiral Fermi arc state of electron type and that of hole type into a helical surface state with gapless Dirac cone. The reconstructed surface state may provide a simple explanation for a mysterious in-plane transport behaviour observed in the graphite experiment under high magnetic field.