Exotic paired states with anisotropic spin-dependent Fermi surfaces

We propose a model for realizing exotic paired states in cold atomic Fermi gases. By using a {\it spin dependent} optical lattice it is possible to engineer spatially anisotropic Fermi surfaces for each hyperfine species, that are rotated 90 degrees with respect to one another. We consider a balanced population of the fermions with an attractive interaction. We explore the BCS mean field phase diagram as a function of the anisotropy, density, and interaction strength, and find the existence of an unusual paired superfluid state with coexisting pockets of gapless unpaired carriers. This state is a relative of the Sarma or breached pair states in polarized mixtures, but in our case the Fermi gas is unpolarized. We also propose the possible existence of an exotic paired ``Cooper-pair Bose-Metal" (CPBM) phase, which has a gap for single fermion excitations but gapless and uncondensed ``Cooper pair" excitations residing on a ``Bose-surface" in momentum space. We extend our study of the model to a ladder geometry by using the density matrix renormalization group method, and we unveil a phase diagram with paired states that reveal a striking resemblance to the physics of hard-core bosons with a frustrating ring-exchange term.