  Cold atoms tailored by an optical lattice have become a fascinating arena for
simulating quantum physics. In this area, one important and challenging problem
is creating effective spin-orbit coupling (SOC), especially for fashioning a
cold atomic gas into a topological phase, for which prevailing approaches
mainly rely on the Raman coupling between the atomic internal states and a
laser field. Herein, a strategy for realizing effective SOC is proposed by
exploiting the geometric effects in the effective-mass theory, without
resorting to internal atomic states. It is shown that the geometry of Bloch
states can have nontrivial effects on the wave-mechanical states under external
fields, leading to effective SOC and an effective Darwin term, which have been
neglected in the standard effective-mass approximation. It is demonstrated that
these relativisticlike effects can be employed to introduce effective SOC in a
two-dimensional optical superlattice, and induce a nontrivial topological
phase.
