BibSonomy publications for /user/statphys23/nestedWed Jun 20 10:16:09 CEST 2007Genova, ItalyAbstract Book of the XXIII IUPAP International Conference on Statistical Physics9-13 JulyOne-Dimensional $\delta$-Function Fermi Gases with Imbalanced Spin Populations2007systems gas topic-1 yang nested gaudin equation statphys23 one-dimensional integrable integral The possibility to access the predicted crossover between Bardeen-Cooper-Schrieffer (BCS) superfluidity of momentum pairs and Bose-Einstein condensation (BEC) of molecules has been exploited in the ultra cold atom systems with tunable inter-atomic interaction via Feshbach resonance. More recently, experimental studies of an atomic Fermi gas with unequal numbers of two components have attracted much attention of physicists related to many fundamental problems in quantum physics. In this presentation, we would like to report an exact analysis of one-dimensional integrable two-component Fermi gas with arbitrary spin polarization at zero temperature. Rigorously speaking, neither the BCS state nor the BEC state occurs in one-dimension. However, the quasi-one dimensional experimental settings are possible and realized in laboratory, and for integrable models we have a theoretical advantage of analyzing the physical properties by using the Bethe ansatz method. We discuss a one-dimensional integrable systems of $N$ spin-$1/2$ Fermi gas with either attractive or repulsive $\delta$-function interaction at zero temperature. The ground state with arbitrary spin polarization (i.e., for all $0\le S\le N/2$, where $S$ is the total spin) is characterized by the sets of integral equations; the Gaudin integral equation for the attractive case and the Yang integral equation for the repulsive case. They determine the distribution of quasi-momenta and that of spin rapidities. We analytically investigate these integral equations with arbitrary spin polarization. The first few terms of the asymptotic expansions of the distribution functions are calculated explicitly for both strong and weak coupling cases. In the weakly attractive case, BCS-like paired fermions and unpaired separate in the quasi-momentum space. This property remains with a weakly repulsive interaction. On the other hand, in the strongly attractive case, bound pairs of fermions behave like hard core bosons. We further study some physical quantities, such as the ground state energy and the chemical potentials, as functions of the coupling constant and the polarization. These expressions are valid for all values of $S$. The method of solution and the detail of the results are found in [1] and [2]. \\ References\\ 1) T. Iida and M. Wadati, J. Phys. Soc. Jpn. \textbf{74} 1724 (2005). \\ \phantom{[1]}\ M. Wadati and T. Iida, Phys. Lett. A \textbf{360} 423 (2007). 2) T. Iida and M. Wadati, J. Low. Temp. Phys., in press. \\ \phantom{[2]}\ T. Iida and M. Wadati, in preparation.