  A remarkable discovery in recent years is that there exist various kinds of
topological insulators and superconductors characterized by a periodic table
according to the system symmetry and dimensionality. To physically realize
these peculiar phases and study their properties, a critical step is to
construct experimentally relevant Hamiltonians which support these topological
phases. We propose a general and systematic method based on the quaternion
algebra to construct the tight binding Hamiltonians for all the
three-dimensional topological phases in the periodic table characterized by
arbitrary integer topological invariants, which include the spin-singlet and
the spin-triplet topological superconductors, the Hopf and the chiral
topological insulators as particular examples. For each class, we calculate the
corresponding topological invariants through both geometric analysis and
numerical simulation.
