  A number of strongly correlated heavy fermion compounds, such as samarium
(Sm), ytterbium (Yb), plutonium (Pu) hexaboride, are predicted to become
topological insulators at low temperatures. These systems support massless
Dirac fermions near certain (three) points of the surface Brillouin zone,
hereafter referred to as the valleys. In strong perpendicular magnetic fields,
the conical Dirac dispersions of these surface states quench onto three sets of
Landau levels and we predict various possible hierarchies of incompressible
quantum Hall states on the surface of hexaborides. In addition, we address the
effects of strong electron-electron interaction within the surface zeroth
Landau levels. Specifically, we show that depending on the relative strength of
the long-range (Coulomb-type) and the finite-range (Hubbard-type) interactions
the ground state can display either a valley-polarized or a valley-coherent
distribution of electronic density. We also show that the transition between
two valley-polarized states is always discontinuous, while that between a
valley-polarized and a valley-coherent phase is continuous. The Zeeman
splitting and/or an applied uniaxial strain on the surface can drive the system
through various quantum phase transitions and place it in different
broken-symmetry phases. Application of uniaxial strain is also shown to
considerably modify the precise sequence of quantum Hall states. We also
highlight the role of topology in determining the broken symmetry phases,
disorder on the surface of topological hexaborides in strong magnetic fields.
