  Graphite under high magnetic field exhibits consecutive metal-insulator (MI)
transitions as well as re-entrant insulator-metal (IM) transition in the
quasi-quantum limit at low temperature. In this paper, we identify the low-$T$
insulating phases as excitonic insulators with spin nematic orderings. We first
point out that graphite under the relevant field regime is in the charge
neutrality region, where electron and hole densities compensate each other.
Based on this observation, we introduce interacting electron models with
electron pocket(s) and hole pocket(s) and enumerate possible umklapp scattering
processes allowed under the charge neutrality. Employing effective boson
theories for the electron models and renormalization group (RG) analyses for
the boson theories, we show that there exist critical interaction strengths
above which the umklapp processes become relevant and the system enter
excitonic insulator phases with long-range order of spin superconducting phase
fields ("spin nematic excitonic insulator"). We argue that, when a pair of
electron and hole pockets get smaller in size, a quantum fluctuation of the
spin superconducting phase becomes larger and destabilizes the excitonic
insulator phases, resulting in the re-entrant IM transitions. We also show that
an odd-parity excitonic pairing between the electron and hole pockets
reconstruct surface chiral Fermi arc states of electron and hole into a
2-dimensional helical surface state with a gapless Dirac cone. We discuss
field- and temperature-dependences of in-plane resistance by surface transports
via these surface states.
