Z$_{2}$ topological classification in BEC-BCS crossover phenomena

Recently, the BEC-BCS crossover is experimentally realized in ultracold atomic Fermi gases--- the Bose-Einstein condensation of real space molecules in a strongly attractive system and the BCS superfluidity in a weakly coupled case. They are not clearly distinguished by the standard order parameters. They are just separated as a crossover, that is, the two ground states are adiabatically connected even in the thermodynamic limit. Introducing a topological order parameter as the Z$_{2}$ Berry phase with a local U(1) twist[1], we have discussed the BCS hamiltonian in a real space for the chiral symmetric case. This local Z$_{2}$ Berry phase distinguishes the BEC-BCS crossover as a local quantum phase transition, that is the phases are separated by closing of the energy gap under the local twist, ~although the gap of the translational invariant system is always open. Physically it characterizes the paired electron is itinerant or localized. This comes from the bulk-edge correspondence in the BEC-BCS crossover as is well established in the Quantum Hall effects. [1] ~Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006) .