  In a recent study [Phys. Rev. X 10, 021042 (2020)], we showed using
large-scale density matrix renormalization group (DMRG) simulations on infinite
cylinders that the triangular lattice Hubbard model has a chiral spin liquid
phase. In this work, we introduce hopping anisotropy in the model, making one
of the three distinct bonds on the lattice stronger or weaker compared with the
other two. We implement the anisotropy in two inequivalent ways, one which
respects the mirror symmetry of the cylinder and one which breaks this
symmetry. In the full range of anisotropy, from the square lattice to weakly
coupled one-dimensional chains, we find a variety of phases. Near the isotropic
limit we find the three phases identified in our previous work: metal, chiral
spin liquid, and 120$^\circ$ spiral order; we note that a recent paper suggests
the apparently metallic phase may actually be a Luther-Emery liquid, which
would also be in agreement with our results. When one bond is weakened by a
relatively small amount, the ground state quickly becomes the square lattice
N\'{e}el order. When one bond is strengthened, the story is much less clear,
with the phases that we find depending on the orientation of the anisotropy and
on the cylinder circumference. While our work is to our knowledge the first
DMRG study of the anisotropic triangular lattice Hubbard model, the overall
phase diagram we find is broadly consistent with that found previously using
other methods, such as variational Monte Carlo and dynamical mean field theory.
