  Fermions in two-dimensions (2D) when subject to anisotropic spin-dependent
hopping can potentially give rise to unusual paired states in {\it unpolarized}
mixtures that can behave as non-Fermi liquids. One possibility is a fully
paired state with a gap for fermion excitations in which the Cooper pairs
remain uncondensed. Such a "Cooper-pair Bose-metal" phase would be expected to
have a singular Bose-surface in momentum space. As demonstrated in the context
of 2D bosons hopping with a frustrating ring-exchange interaction, an analogous
Bose-metal phase has a set of quasi-1D descendent states when put on a ladder
geometry. Here we present a density matrix renormalization group (DMRG) study
of the attractive Hubbard model with spin-dependent hopping on a two-leg ladder
geometry. In our setup, one spin species moves preferentially along the leg
direction, while the other does so along the rung direction. We find compelling
evidence for the existence of a novel Cooper-pair Bose-metal phase in a region
of the phase diagram at intermediate coupling. We further explore the phase
diagram of this model as a function of hopping anisotropy, density, and
interaction strength, finding a conventional superfluid phase, as well as a
phase of paired Cooper pairs with d-wave symmetry, similar to the one found in
models of hard-core bosons with ring-exchange. We argue that simulating this
model with cold Fermi gases on spin dependent optical lattices is a promising
direction for realizing exotic quantum states.
