UC San Diego

The study of topological states has become a major research focus of contemporary physics. This thesis consists of several investigations on novel states of matter with non-trivial topological properties in both condensed matter and ultra-cold atom systems. We have systematically generalized Landau levels (LL) from the two dimensions (2D) to 3D and above for both non-relativistic and relativistic fermions. LLs with the full 3D rotation symmetry and flat energy spectra are constructed by coupling fermions to the SU(2) Aharanov-Casher potential. Fermion spins are coupled to orbital motions with a helicity structure, and time-reversal symmetry is maintained. The lowest LL wavefunctions exhibit the quaternionic analyticity as a generalization of the complex analyticity of the 2D cases. Each LL contributes one branch of gapless helical Dirac modes to the surface spectra. The elegant analytic properties together with the flat energy spectra are expected to facilitate future studies of high dimensional fractional topological states. LLs in the Landau-like gauge are also constructed in high dimensions, which exhibit spatial separations of 2D helical Dirac modes with opposite helicities or 3D Weyl modes with opposite chiralities. As a square root problem of the non-relativistic cases, LLs of Dirac electrons are constructed in 3D and above. The zeroth LL states are a branch of half-fermion Jackiw-Rebbi modes. We have also found parity breaking LL quantization in harmonic traps in the presence of strong spin-orbit(SO) couplings. We have studied topological properties in fermion systems with magnetic dipolar interactions. Different from electric dipoles which are classic vectors, atomic magnetic dipoles are quantum-like. Magnetic dipolar interactions are isotropic under simultaneous SO rotations. This feature gives rise to a novel p-wave spin triplet pairing symmetry with the total angular momentum J=1 of the Cooper pair. Such a state is fundamentally different from both ³He A and B-phases and exhibits Weyl type Bogoliubov excitations. In Fermi liquid theory, there exists a novel SO coupled zero-sound-like collective mode exhibiting non- trivial spin configurations over the Fermi surface