在這篇論文中我們探討了安德森晶格中的半金屬相。在過去的研究文獻中我們可以得知外爾半金屬相一般發生在考慮系統的價帶電子以及傳導帶電子之間的自旋軌道耦合後，破壞了系統本身的時間繁衍對稱性或空間繁衍對稱性。特別的是，我們發現外爾半金屬相總是出現在兩個絕緣相之間。藉由調控系統的溫度或是自旋軌道耦合的強度，安德森晶格可以在外爾半金屬相以及絕緣體相之間相變。確切的說，在一個考慮價帶電子以及傳導帶電子之間有自旋軌道耦合的安德森晶格裡存在外加的交互作用場破壞了時間繁衍對稱性，一個狄拉克半金屬點將分裂出兩個分別帶有手徵電荷+1及-1的外爾半金屬點。另一方面來說，當我們考慮了兩個傳導帶電子之間也具有自旋軌道耦合時，安德森晶格的空間繁衍對稱性將被破壞。我們的研究發現以這種方式所產生的外爾半金屬點將分別帶有手徵電荷+2以及-2。當我們更近一步的考慮系統的旋轉對稱性被躍遷強度不再各項同性破壞時，外爾半金屬相出現在不同的拓樸絕緣相之間。同樣地，在外爾半金屬相以及拓樸絕緣相之間的相變可以藉由調控溫度來達成。最後，我們發現手徵電荷為+2以及-2的外爾半金屬相可以藉由考慮Rashba效應調控為手徵電荷為+1以及-1的外爾半金屬相。我們的研究指出安德森晶格藉由引入適當的自旋軌道耦合提供了一個可以用來了解拓樸絕緣相以及外爾半金屬相的可調性系統。

In this thesis, we investigate semi-metallic phases in an Anderson lattice. It is shown that by including spin-orbit interactions both in the conduction electron and in the coupling between the conduction electron and more localized f electron, Weyl semi-metallic phases generally emerges upon either time-reversal symmetry broken or inversion symmetry broken. In particular, we find that Weyl semi- metallic phases always emerge between insulating phases and can be accessed by tuning either temperature or spin-orbit interaction so that the Anderson lattice goes through two insulating phases. Specifically, for an Anderson lattice with the spin-orbit coupling between the conduction electron and f electron, we find that in an exchange field, two Weyl points of chirality charge ±1 split from a Dirac point at a time-reversal momentum. The Weyl semi-metallic phases can be accessed by tuning temperature, resulting in a Fermionic critical point at a finite temperature. On the other hand, when the spin-orbit interaction is included in the conduction electron, the inversion symmetry of the Anderson lattice is bro- ken. We find that Weyl semi-metallic phases emerges with Weyl points carrying chirality charges of ±2. Furthermore, when the rotation symmetry of the system is broken with anisotropic hopping amplitudes, Weyl semi-metallic phase emerges between topological insulating phases and can be thus accessed by changing temperatures. Finally, we find that Weyl semi-metallic phases with chirality charges of ±2 can be tuned into Weyl semi-metallic phases with chirality charges of ±1 through the inclusion of the Rashba spin-orbit interaction. Our analyses indicate that Anderson lattices with appropriate spin-orbit interactions provide a tunable platform for realizing both insulating and semi-metallic topological phases.

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