  We study the effective dipole-dipole interactions in ultracold quantum gases
on optical lattices as a function of asymmetry in confinement along the
principal axes of the lattice. In particular, we study the matrix elements of
the dipole-dipole interaction in the basis of lowest band Wannier functions
which serve as a set of low-energy states for many-body physics on the lattice.
We demonstrate that the effective interaction between dipoles in an optical
lattice is non-algebraic in the inter-particle separation at short to medium
distance on the lattice scale and has a long-range power-law tail, in contrast
to the pure power-law behavior of the dipole-dipole interaction in free space.
The modifications to the free-space interaction can be sizable; we identify
differences of up to 36% from the free-space interaction at the
nearest-neighbor distance in quasi-1D arrangements. The interaction difference
depends essentially on asymmetry in confinement, due to the d-wave anisotropy
of the dipole-dipole interaction. Our results do not depend on statistics,
applying to both dipolar Bose-Einstein condensates and degenerate Fermi gases.
Using matrix product state simulations, we demonstrate that use of the correct
lattice dipolar interaction leads to significant deviations from many-body
predictions using the free-space interaction. Our results are relevant to up
and coming experiments with ultracold heteronuclear molecules, Rydberg atoms,
and strongly magnetic atoms in optical lattices.
