Xing, Ruiqi2019-08-202019-08-202019-05http://hdl.handle.net/11299/206217University of Minnesota Ph.D. dissertation.May 2019. Major: Physics. Advisor: Andrey Chubukov. 1 computer file (PDF); viii, 159 pages.This dissertation focuses on the theoretical understanding of the interesting phases observed in one kind of the high temperature superconductors, iron-based superconductors. In the introduction chapter, I introduce what superconductivity and high temperature superconductors is and the motivation to study them; then I list out some of the basic, important, and relevant experiment results of iron-based superconductors, such as their lattice structures, phase diagrams, superconductivity, magnetic orders and charge orders observed; after that, I give a brief review of motivation, high level summary, and importance of each work being presented in later chapters. I finish the introduction with outlines and an educational introduction for people not that familiar with this area. The following chapters consist of three topics. First, we attack the issue of methodology of studying the FeSC materials. The approach we use in this entire thesis is itinerant-scenario approach. Our analytical calculation finds out the orders developed for a FeSc model agree with non-biased(though has its own limitations) quantum Monte Carlo calculations. Both of these methods find $s^{++}$ superconductivity and anti-ferro orbital orders as the leading orders. Secondly, we use parquet renormalization group theory to study a 4-pocket and a 5-pocket model for iron-based superconductors to shed more light upon the competing instabilities in these materials. We find amazingly simple behaviors in these complex models. These results can explain the interplay between superconductivity, charge order and magnetism in different kinds of iron-based superconductors. Thirdly, we study the details about the charge order(orbital order) discussed in previous chapters. In FeSe, the orbital order is in d-wave form, i.e., the sign of the orbital order is different between hole and electron pockets. We reproduce this sign difference by including vertex renormalization in d-wave orbital channel. Lastly, the conclusion follows.eniron-baseditinerant scenarionematicparquetrenormalization groupsuperconductivityThesis or Dissertation