  The anomalous surface states of symmetry protected topological (SPT) phases
are usually thought to be only possible in conjunction with the higher
dimensional topological bulk. However, it has recently been realized that a
class of anomalous SPT surface states can be realized in the same dimension if
symmetries are allowed to act in a nonlocal fashion. An example is the
particle-hole symmetric half filled Landau level, which effectively realizes
the anomalous surface state of a 3D chiral Topological Insulator (class AIII).
A dual description in terms of Dirac composite fermions has also been
discussed. Here we explore generalizations of these constructions to
multicomponent quantum Hall states. Our results include a duality mapping of
the bilayer case to composite bosons with Kramers degeneracy and the
possibility of a particle hole symmetric integer quantum Hall state when the
number of components is a multiple of eight. Next, we make a further extension
by half filling other classes of topological bands and imposing particle hole
symmetry. When applied to time-reversal invariant topological insulators we
realize a different chiral class (CII) topological surface state. Notably,
half-filling a 3D TI band allows for the realization of the surface of the
otherwise inaccessible 4D topological insulator, which supports an anomalous 3D
Dirac cone. Surface topological orders equivalent to the 3D Dirac cone (from
the global anomaly standpoint) are constructed and connections to Witten's
SU(2) anomaly are made. These observations may also be useful for numerical
simulations of topological surface states and of Dirac fermions without fermion
doubling.
