Topologically Stratified Energy Minimizers in a Product Abelian Field Theory

We study a recently developed product Abelian gauge field theory by Tong and Wong hosting magnetic impurities. We first obtain a necessary and sufficient condition for the existence of a unique solution realizing such impurities in the form of multiple vortices. We next reformulate the theory into an extended model that allows the coexistence of vortices and anti-vortices. The two Abelian gauge fields in the model induce two species of magnetic vortex-lines resulting from $N_s$ vortices and $P_s$ anti-vortices ($s=1,2$) realized as the zeros and poles of two complex-valued Higgs fields, respectively. An existence theorem is established for the governing equations over a compact Riemann surface $S$ which states that a solution with prescribed $N_1, N_2$ vortices and $P_1,P_2$ anti-vortices of two designated species exists if and only if the inequalities $$\left|N_1+N_2-(P_1+P_2)\right|<\frac{|S|}{\pi},\quad \left|N_1+2N_2-(P_1+2P_2)\right|<\frac{|S|}{\pi},$$ hold simultaneously, which give bounds for the `differences' of the vortex and anti-vortex numbers in terms of the total surface area of $S$. The minimum energy of these solutions is shown to assume the explicit value $$E= 4\pi (N_1+N_2+P_1+P_2),$$ given in terms of several topological invariants, measuring the total tension of the vortex-lines.Comment: 22 page

BCS-BEC Crossover in Mix-dimensional Fermi Gases

We investigate a mix-dimensional Fermi-Fermi mixture in which one species is confined in two-dimensional(2D) space while the other is free in three-dimensional space(3D). We determine the superfluid transition temperature $T_{c}$ for the entire BCS-BEC crossover including the important effects of noncondensed pairs. We find that the transition temperature reduces while the imbalance of mass is increased or lattice constant ($dz$) is reduced. In population imbalance case, the stability of superfluid is sharply destroyed by increasing the polarization.Comment: 7 pages, 5 figure