很多凝態相位的成因一般都跟費米面的拓墣特性相關像是超導相、自
旋密度波、電荷密度波等，因此當能帶和晶格動量無關時(平坦能帶)，在
交互作用力下會產生怎樣的相位就不是那麼清楚了，一個有趣的例子是，
最近在魔角雙層石墨烯的平坦能帶上發現超導相，在此論文，我們探索
在Lieb晶格的平坦能帶上會有什麼凝態相位，藉由重整化均場理論，我們
在Lieb晶格上分析t-j模型，我們分別在實空間和動量空間用自洽方程去計
算系統允許的相位，並且獲得一至的數值結果，我們也給出了系統的整
體相圖，我們的結果指出由於Lieb晶格是一個不平衡的二分晶格(bipartite
lattice)，使得在低參雜區域亞鐵磁相的升起並且分別和兩種有著不同配對
對稱性的均勻超導相共存，當參雜增加SDW和CDW在系統升起並且依舊
和亞鐵磁相共存，這些凝態相位對凝態相位如何被有平坦能帶的系統容納
提供了有用的線索。

Phases in condensed matters, such as superconductivity (SC), spin density wave (SDW), charge density wave (CDW) and etc, normally arises from the instability of the Fermi surface and are tied up with the topology of Fermi surface. It is therefore not clear when the energy band involved is dispersionless, i.e., a flat band, how different phases may arise in the presence of interaction. An interesting example is the surprising SC that is recently found at flat-bands of a magic-angle twisted bilayer graphene. In this thesis, we explore how condensed matter phases arise in a flat band in a Lieb lattice. By resorting to the renormalized mean field theory (RMFT), we analyze the t-J model on the Lieb lattice. We compute allowed phases self-consistently in the real space as well as in the momentum space and found consistent numerical results from these two spaces. The global phase diagram is constructed. Our results indicate that due to being an imbalanced bipartite for the Lieb lattice, two uniform SC phases with different pairing symmetries that are in coexistence with the ferrimagnetic order arise at the low doping regime. When doping increases, SDW and CDW emerge and coexist with the ferrimagnetic phase. These condensed matter phases provide useful hints on how condensed matter phases are adapted to systems with flat bands.

1 Introduction 1
2 Model and RMFT 3
2.1 t-j model in Lieb lattice . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 RMFT and diagonalization in real space . . . . . . . . . . . . . . . 4
3 RMFT in momentum space 9
3.1 Analyze HMF in momentum space . . . . . . . . . . . . . . . . . . . 9
3.2 Pairing symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Numerical result 16
4.1 Result and phase diagram . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 The energy band of quasiparticle of SC phase . . . . . . . . . . . . 28
5 Summary 31
Appendix 32
A derivation of HMF 32
B Other diagrams 34
Reference 39

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