  Three-dimensional (3D) topological insulators in general need to be protected
by certain kinds of symmetries other than the presumed $U(1)$ charge
conservation. A peculiar exception is the Hopf insulators which are 3D
topological insulators characterized by an integer Hopf index. To demonstrate
the existence and physical relevance of the Hopf insulators, we construct a
class of tight-binding model Hamiltonians which realize all kinds of Hopf
insulators with arbitrary integer Hopf index. These Hopf insulator phases have
topologically protected surface states and we numerically demonstrate the
robustness of these topologically protected states under general random
perturbations without any symmetry other than the $U(1)$ charge conservation
that is implicit in all kinds of topological insulators.
