| Geometric aspect of condensed matter has arouse a lot of interests in recent years. The idea of Berry phase is highly appreciated in various systems. We explored the geometric features of two specific electron systems, fractional quantum Hall (FQH) states and d-wave superconducting states. For FQH states, we propose a two body operator which generates the geometric change of Laughlin state in the guiding center degrees of freedom on torus. This operator therefore generates the adiabatic evolution between Laughlin states on regular tori and the quasi-one-dimensional thin torus limit. For d-wave superconducting model, we study the local and topological features of Berry phases associated with the adiabatic transport of vortices in a lattice fermions. We find bosonic statistics for vortices in hall filling. Away from half filling, we find the complicate Berry phase to be path dependent. However, it is shown that "statistical" flux attached to the vortex are still absent. The average flux density associated with Berry curvature is tied to the average density of cooper pair in the magnetic unit cell. This is familiar from dual theories of bosonic systems, even though the underlying particles are fermions in this case. |
