Functional renormalization group (FRG) is a relatively new theoretical machinery for correlated fermion and boson systems, and has found great success in applications in correlated electrons systems of interest in this talk. In general, FRG provides the flow (versus a decreasing infrared cutoff) of the exact one-particle-irreducible (1PI) vertices, or the effective interactions seen by quasiparticles, and hence provides unbiased predictions on the various instabilities of the normal state in a correlated electron system. However, the power of FRG relies very sensitively on the implementation scheme in practice. In this talk, the previous implementation of FRG, called patch-FRG, is introduced and criticized in terms of the violation of momentum conservation and vertex hermiticity therein. Then a new implementation is developed, based on the expansion of the four-point vertices as scattering matrices between fermion bilinears in the three Mandelsdam channels. On physical basis, the fermion bilinears can be truncated in the internal spatial distance between the fermions within a bilinear, in such a way that all potential singular scattering modes can be captured during the RG flow. This scheme, called singular-mode functional renormalization group (SM-FRG), is asymptotically exact and avoids all difficulties in patch-FRG, providing immediate and more reliable informations on the competing orders. Finally, examples of the applications of SM-FRG are discussed in light of unconventional electronic orders, some of which are topological in nature, in some correlated electron systems.

Prof. Qiang-Hua Wang recieved his PhD in Nanjing University, 1993, and work in the university thereafter. In the intervening stages, he worked as a postdoc in the University of Honkong (1995-1997), and as a visiting professor in the University of California, Berkeley (2000-2002). He became a full professor in 2002, Nanjing University. He was supported by the Natural Science Foundation as an Outstanding Youngest in 2003, and was honored by the Ministry of Education the Changjiang Scholar Professorship in 2006. His main research interest is to unravel the physis, the mechanism and properties of unconventional and/or topological states of matter in particular, in strongly correlated electron systems, such as the cuprates, iron-based superconductors, and other correlated materials.