Electronic Instabilities In Correlated Electron Systems:Functional Renormalization Group And Finite-temperature Renormalized Mean Field Theory Studies

In condensed matter physics, the electronic instabilities in correlated electron sys-tems has been a long-standing problems. We couldn’t resolve the problems exactly due to the existence of correlation effect except for some one-dimensional systems. Thus, it is important to develop effective methods to deal with the problems. In this dissertation, we develop two methods to deal with the problems, one is singular-mode functional renormalization group (SMFRG) method, the other is finite-temperature renormalized mean-field theory(TRMFT). Then we study some correlated electron systems using those two methods.On the one hand, the magnitude of energy scale in correlated electron systems usually bridges several orders. On the other hand, there are various unstable channels and complex interplays between them. Thus, it is difficult to analysis the electronic instabilities in such systems. However, the functional renormalization group method, which is a powerful method in condensed matter physics, naturally resolve those prob-lems and give a low-energy effective Hamiltonian for the system. Here, we develop the SMFRG method based on the idea of Wetterich FRG, and then study the electronic instabilities in doped graphene and competing orders in Kagome lattice at van Hove filling. Recently, topological insulators(TIs) and topological superconductors(TSCs) have attracted considerable interest in condensed matter physics. It has been proposed and discovered TIs in real materials, but there has no unambiguous evidence for TSCs, especially for those materials which are TSCs by itself. Using SMFRG method, we establish a strong tie between topological superconductivity and ferromagnetic spin fluctuations.In the strongly correlated electronic systems, one of the effects of correlation is the renormalization of the energy band of single particles. At zero temperature, un-der Gutzwiller projection, the variational theory lead to renormalized mean-field the-ory(RMFT). Most times, however, we need to know the properties of excited state, i.e. the properties of the systems at finite temperature. Here, we extend the RMFT method to finite temperature (i.e. TRMFT) by introduce the projection entropy. Then, we il-lustrate the application of the theory to the Anderson impurity problem and half-filled Hubbard model and compare the results to more elaborate techniques, such as dynam-ical mean-field theory(DMFT) and numerical renormalization group(NRG) methods. We find qualitative agreement.The dissertation is organized as follows:In chapter one, we give an introduction to correlated electron systems, renormal-ization group, and functional renormalization group.In chapter two, firstly, we derive the exact flow equation of FRG, and then, we give the process of SMFRG.In chapter three, we study the electronic instabilities in doped graphene using the SMFRG method. At 1/4 doping, the system is a chiral spin-density wave(SDW) state exhibiting the quantized anomalous Hall effect. When the doping deviates from 1/4, the d+ id Cooper pairing becomes the leading instability. Both states are topological nontrivial. Our results suggest that near 1/4 doping the graphene is either a Chern insulator or a topological superconductor.In chapter four, we study the topological superconductivity in correlated electron systems. We find the ferromagnetic fluctuations will drive two-fold degenerate triplet pairing, then any infinitesimal Rashba spin-orbital coupling(SOC) will break the de-generacy by linearly recombining of those two degenerate pairing into [(p+ip) ↑↑ ,(p - ip)↓↓] pairing state, which is time-reversal invariant and topologically nontrivial. Our result can be used as a guideline for the search for TSCs whose pairing symmetry is invariant under time reversal.In chapter five, we study the competing electronic orders at the van Hove fillings on Kagome lattice model. Because of the matrix element effect and the breaking of particle-hole symmetry, there are some exotic states on Kagome lattice at van Hove filling, and the phase diagrams of the two van Hove fillings are different. Under short-range correlations, the Kagome lattice will develop following instabilities in upper van Hove filling:ferromagnetism, intra-unit-cell antiferromagnetism, charge bond or- der(CBO) state, charge density wave(CDW), s and d+ id wave superconductivity and spin bond order(SBO1) state. At lower van Hove filling, we find ferromagnetism, or-thogonal antiferromagnetism(SDWo), CDW, s and d+id wave superconductivity and a new spin bond order(SBO2) state. Here, SDWo,SBO1,2,d+id states are topologi-cal nontrivial, and SDWo,SBO1,2 states exhibit quantum anomalous Hall effect as the chiral SDW order in graphene.In chapter six, firstly, we give the derivation of TRMFT, in which the projection entropy is the key ingredient, and compare the entropy between TRMFT and slave-boson Mean-field(SBMF) method. Then, we illustrate the application of the theory to the Anderson impurity problem and half-filled Hubbard model and compare the results to DMFT and NRG methods. We find qualitative agreement.In chapter seven, we summarize the dissertation. In the Appendix, we give the properties of symmetry and the self-adaptive momentum points which are used in SM-FRG.