  An interplay between electron correlation and reduced dimensionality due to
the Landau quantization gives rise to exotic electronic phases in
three-dimensional semimetals under high magnetic field. Using an unbiased
theoretical method, we clarify for the first time comprehensive ground-state
phase diagrams of a three-dimensional semimetal with a pair of electron and
hole pockets in the quantum limit. For the electron interaction, we consider
either screened Coulomb repulsive interaction or an attractive
electron-electron interaction mediated by a screened electron-phonon coupling,
where a screening length is generally given by a dimensionless constant times
magnetic length $l$. By solving the parquet RG equation numerically and
employing a mean-field argument, we construct comprehensive ground-state phase
diagrams of the semimetal in the quantum limit for these two cases, as a
function of the Fermi wave length and the screening length (both normalized by
$l$). In the repulsive interaction case, the ground state is either excitonic
insulator (EI) in strong screening regime or Ising-type spin density wave in
weak screening regime. In the attractive interaction case, the ground state is
either EI that breaks the translational symmetries (strong screening regime),
topological EI, charge Wigner crystal (intermediate screening regime), plain
charge density wave or possible non-Fermi liquid (weak screening regime). We
show that the topological EI supports a single copy of massless Dirac fermion
at its side surface, and thereby exhibit a $\sqrt{H_{\perp}}$-type surface
Shubnikov-de Haas (SdH) oscillation in in-plane surface transports as a
function of a canted magnetic field $H_{\perp}$. Armed with these theoretical
knowledge, we discuss implications of recent transport experiments on graphite
under the high field.
