| My thesis work consists of several applications of the symmetry principle in condensed matter and cold atomic systems.; First, we study the relative spin-orbit symmetry breaking effect in the Fermi surface instability of the Landau-Pomeranchuk type in the Fa1 channel. It provides a new mechanism to generate spin-orbit coupling arising from many-body interactions, in contrast to the conventional mechanism which originates from the single-body relativistic effect. This mechanism is essentially a p-wave generalization of ferromagnetism, and also a particle-hole channel analogy of the superfluid 3He-B phase.; Second, we rigorously prove a hidden and generic SO(5) symmetry in spin-3/2 systems with contact interactions, which may be realized in cold atomic systems in optical traps or lattices with candidate atoms such as 132Cs, 9Be, 135Ba, 137Ba. This symmetry sets up a framework to unify many seemingly unrelated properties of Fermi liquid theory, Cooper pairing structures in such systems.; Third, we study the relation between time-reversal symmetry and the sign problem in fermionic quantum Monte-Carlo (QMC) simulations. We rigorously prove a general criterion dubbed T-invariant decomposition for the absence of this problem in the auxiliary field QMC method. We apply this criterion to the two-dimensional staggered current phase, which has received extensive attention in high Tc superconductivity. Its existence is conclusively demonstrated for the first time in a bi-layer model. |
