We propose a model which includes a nearest-neighbor intrinsic spin-orbit coupling and a trimerized Hamiltonian in the Kagomé lattice and promises to host the transition from the quantum spin Hall insulator to the normal insulator. In addition, we design an experimental scheme to simulate and detect this transition in the ultracold atom system. The lattice intrinsic spin-orbit coupling is generated via the laser-induced-gauge-field method. Furthermore, we establish the connection between the spin Chern number and the spin-atomic density which enables us to detect the quantum spin Hall insulator directly by the standard density-profile technique used in the atomic systems.

We obtain a rigorous solution of universal Bose gases near resonance and offer an answer to one of the long-standing challenges of quantum gases at large scattering lengths, where the standard dilute theory breaks down. The solution was obtained by using an ǫ expansion near four spatial dimension. In dimension d = 4 − ǫ, the chemical potential of Bose gases near resonance is shown to approach the universal value ǫ 2 4−ǫ ǫF 2 3 (1 + 0.474ǫ − i1.217ǫ + · · · ), where ǫF is the Fermi energy defined for a Fermi gas of density n, and the condensation fraction is equal to 2 3 (1 + 0.0877ǫ + · · · ). We also discuss the implications on ultra-cold gases in physical dimensions.