  Orbital degrees of freedom play an important role for understanding the
emergence of unconventional quantum phases. Ultracold atomic gases in optical
lattices provide a wonderful platform to simulate orbital physics. In this
work, we consider spinless fermionic atoms loaded into $p$-orbital bands of a
two-dimensional frustrated triangular lattice. The system can be described by
an extended Fermi-Hubbard model, which is numerically solved by using the
orbital version of real-space dynamical mean-field theory. Low-temperature
phase diagrams are obtained, which contain stripe-, ferro- and para-orbital
ordered quantum phases, due to the interplay of anisotropic hoppings and
geometrical frustration. In order to understand the underlying mechanics of
competing orbital orders, we derive an effective orbital-exchange model, which
yields consistent explanation with our main numerical results.
