  Phonon polaritons in natural anisotropic crystals hold great promise for
infrared nano-optics. However, the direct electrical control of these
polaritons is difficult, preventing the development of active polaritonic
devices. Here we propose the heterostructures of graphene on a biaxial crystal
({\alpha}-phase molybdenum trioxide) slab and theoretically study the
hybridized plasmon-phonon polaritons with dependence on the Fermi level of
graphene from three aspects: dispersion relationships, iso-frequency contours,
and the quantum spin Hall effects. We demonstrate the distinct wavelength
tunability of the plasmon-phonon polaritons modes and the optical topologic
transitions from open (hyperbolic) to closed (bow-tie-like) iso-frequency
contours as the increase of the Fermi level of graphene. Furthermore, we
observe the tunable quantum spin Hall effects of the plasmon-phonon polaritons,
manifesting propagation direction switching by the Fermi level tuning of the
graphene. Our findings open opportunities for novel electrically tunable
polaritonic devices and programmable quantum optical networks.
