N/A

In this thesis, we investigate the interface between condensed matter physics and cold atom. On the one hand, topological phases in condensed matter physics have attracted significant attention due to their fundamental physics implications and potential applications. For example, information encoded in topological state is robust to local perturbations and topological quantum computing with Majorana zero modes. On the other hand, the high experimental controllability of cold atoms in optical lattices makes them ideal candidates for simulation Hamiltonians of condensed matter system.

We propose that topological states can be prepared and observed in a bilayer model, which constructed by a bilayer optical lattice with spin polarized fermionic atoms are loaded. Our self-consistent mean-field calculation shows that the gapped topological (p-wave) superfluids in each layer are coupled together by the s-wave pairing in an intermediate inter-layer distance with a spontaneously modulated phases between these two order parameters. We systematically investigate the topological properties of this bilayer model. One of state we interest in is a gapless paired topological superfluid. This state belong to DIII gapless topological superconductor since it has the time-reversal symmetry and particle-hole symmetry. Our work provides a way for future investigation of non-Abelian statistics of Majorana zero modes in the many- body physics with time-reversal symmetry.

Besides, we develop the bilayer optical lattice into a mulilayer system to explore the phenomenon of dimerization. The ground state is obtained by minimizing the ground state energy. By controlling the strength of the Rydberg-dressed interaction, we can indeed find the ground state with the dimerized pairing. We also find the ground state with coexisted interlayer order parameter and in-plane order parameter in unconventional pairing phases.

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