在凝聚態系統中尋找和模擬馬約拉納費米子是凝聚態物理中的熱門課題之一，在通常的分類表中，馬約拉納邊緣態的存在是基於對均勻的超導序參數做分類。但實際上，超導序參數在邊界附近會耗盡，在本文中，我們研究了超導序參數在邊界耗盡的輪廓如何影響馬約拉納邊緣態。特別是，我們檢驗了在有限蜂窩晶格上 d +id 超導序參數自洽解所導致的邊緣態。藉由考慮在蜂窩晶格上的 t-J 模型，和 Kane-Mele 模型的表徵跳躍項，我們解出超導序參數的自洽方程。我們發現除了 d+id 手性馬約拉納邊緣態，額外的高能區的螺旋狀邊緣態與馬約拉納態共存，這是因為在邊界附近超導序參數的耗盡。這些螺旋狀邊緣態起源於常態的拓撲結構，即使進入超導狀態，仍然在耗盡的邊界區域持續維持。最後，我們證明了當化學勢變為極大的負值，進入 BEC 極限時，手性馬約拉納邊緣態消失，這些螺旋狀邊緣態仍然存在，表明該系統在通常 BEC 極限下是費米子-玻色子混合物。

Searching and simulating the Majorana fermions in condensed matter systems is one of the popular topics in condensed matter physics. In the usual classification scheme, the existence of Majorana edge states is based on the classification of uniform superconducting order parameter. In reality, the superconducting order depletes near the boundary. Here in this thesis, we investigate how the depleted boundary profile of the superconducting order parameter affect Majorana edge states. In particular, we examine edge states resulting from the presence of self-consistent d + id superconducting order parameter on a finite honeycomb lattice. By considering the t-J model on a honeycomb lattice with the Kane-Mele model characterizing the hopping terms, we solve the self-consistent equations for the superconducting order parameter. We find that in addition to the d + id chiral Majorana edge states, extra helical-like edge states coexist with the Majorana states in high energies regime due to the depletion of the superconducting order near the boundary. These helical-like edge states originate from the topological structure of the normal state and persist into the superconducting state in the depleted boundary region. Finally, we show that in the BEC limit when the chemical potential goes to large negative values, while the chiral Majorana edge states diminish, these helical-like edge states survives, indicating that the system is a fermion-boson mixture in the usual BEC limit.

摘要
目錄
圖片目錄

1 Introduction----------------------------------1
2 Model and Method------------------------------4
2.1 Hubbard model-----------------------------4
2.2 Effective 1D model and zigzag edge---------6
2.3 Self-consistent equation------------------11
3 Result----------------------------------------14
3.1 Uniform situation-------------------------14
3.2 Boundary situation------------------------19
4 Summary---------------------------------------28

附錄
A Gutzwiller’s approximation--------------------29
B The mean field of spin-spin interaction--------30
C The uniform case’s phase transition point-----31
Reference --------------------------------------33

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