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: We have focused on novel quantum phases and emergent symmetries with ultra-cold atoms that are not easily accessible in usual solid state systems. For bosons pumped into high orbital bands, the concept of unconventional BEC (UBEC) is proposed in analogous to unconventional superconductivity based on symmetry properties. A novel statistical model dubbed the 4-coloring model is proposed to describe the heavy frustration of the UBEC in the diamond lattice, which leads to a 3D critical phase with dipolar correlations. For orbital fermions, an exact theorem is proved for the existence of an itinerant ferromagnetic state. For alkaline-earth fermions, the large SU(2N) symmetry greatly enhances quantum spin fluctuations, which give rises to novel quantum magnetism in the ground state and thermodynamic properties. For the research of synthetic gauge field, we derived the 3D Landau levels with the full 3D rotation symmetry and flat spectra. Their wavefunctions exhibit the elegant quaternionic analyticity, which greatly facilitates the further study of interacting topological states in 3D. We have also identified a novel p-wave Weyl type Cooper pairing instability in magnetic dipolar fermion systems, which is fundamentally different from both He-3 A and B phases.

Unconventional States of Matter with Cold Atoms and Dipolar Molecules