臺灣博碩士論文加值系統

我們考慮在二維電子氣(2DEG)在正方形晶格和垂直磁場中的拓樸性質。每個晶胞的磁通量是單位磁通量的一半。針對Hamiltonian我們用TKNN所提出( PRL 49,405(1982) )，滿足磁平移對稱的基底|n,k>做矩陣表示，並以數值方法做對角化以得到第N個藍道能階之能帶|N,k>。當位能打開後，我們關心在不同能帶之間是否有發生能帶接觸、在布里淵區中的哪一點接觸、發生接觸的兩個能帶之色散關係、Berry curvature、以及Chern number的改變。
我們發現在布里淵區的一個簡併，附近的色散關係是線性(二次式)會造成Chern number改變一(二或零)。更進一步的分析顯示靠近的能帶會由兩個基底|n1,k>,|n2,k>主導。而若|n1-n2|=1或3(2)，其色散關係會是線性(二次式)。其中二次式的色散關係，Berry curvature在對稱點k附近會形成火山分布，這是由於在對稱點，位能晶格只會耦合|n1-n2|=4的基底。
我們使用kp理論得到等效Hamiltonian，令人驚訝的，在二次式色散關係的情況，不論能隙有多小，只用接觸的能帶做基底展開是不足夠的，因為這兩個基底不會被等效Hamiltonian耦合。而等效Hamiltonian最小的維度是三。我們可以更進一步用Lowdin微擾理論，將第三個能帶的資訊放入，而得到一個二乘二的等效Hamiltonian，其能量和Berry curvature與full Hamiltonian很好地吻合。

In this work we consider the topological features in a square lattice on a 2DEG under the action of a normal magnetic field. Specifically, the magnetic flux per unit cell is fixed at one half of a flux quanta. Exact numerical diagonalization of the full Hamiltonian is performed to obtain energy band |N,k> of the N-th Landau level (LL).This calculation is facilitated by a TKNN-type (PRL 49,405(1982)) basis wavefunctions |n,k>, where the magnetic translation symmetry is built-in and the n-th eigenstate of a simple harmonic oscillator is used. As the lattice potential U is tuned on, our focus is upon the gap-closing between |N,k>, the k at which the gap-closing occurs, the dispersion
relation of the 2 gap-closing LL, the Berry curvatures, and the change in the Chern number.
The change in the Chern number for each gap-closing point in the k space is one (two) when the closing bands have linear (quadratic) energy dispersion. Further analysis
shows that the closing bands have typically two dominating components in |n1,k> and |n2,k> such that |n1-n2| equals 1 or 3 (2) for the linear (quadratic) energy dispersion
case. For the quadratic energy dispersion case, the Berry curvature takes on a volcano type protrusion encycling the high-symmetry k point (also the gap-closing point). This
is resulted from our finding that at this high-symmetry point the lattice potential couples basis wavefunctions with the condition |n1-n2|=4 only.
We use the kp method to obtain the effective Hamiltonian. To our surprise, for the quadratic energy dispersion case, using two eigenstates of the closing bands as basis is not enough no matter how small the gap between them is since these two basis won't be coupled by the effective Hamiltonian. The smallest dimension of the effective Hamiltonian is three. We further use the Lowdin perturbation to put in the coupling with third basis to get the appropriate two-by-two effective Hamiltonian which matches the energy dispersion and the Berry curvature of the full Hamiltonian well.

Abstract in Chinese i
Abstract in English ii
Acknowledgements iv
1 Introduction 1
2 Theory I: Magnetic translation symmetry and construction of basis 4
2.1 Magnetic translation operator 5
2.2 Construction of the basis 7
3 Theory II: Band structure calculation 10
3.1 Derivation of the matrix elements 10
3.2 Band structure and its symmetric discussions 16
4 Comparison with recent work in the literature 20
4.1 Formalism of abrupt single-well and smooth lattice potential problems 22
4.1.1 Abrupt single-well potential 22
4.1.2 Smooth lattice potential well 24
4.2 Spectrum comparison in different potential regimes 24
5 Topological invariants 30
5.1 Berry phase 30
5.2 Berry curvature 31
5.3 Chern number 33
6 Results and discussions 36
6.1 Where do LLs touch? 36
6.2 How do LLs touch 42
6.3 What are the effects of LLs touching? 48
7 Conclusion and future work 59
7.1 Conclusion 59
7.2 Future work 60
A Derivation of the solution of the differential equation related to the abrupt single-well potential problem 61
B Derivation of the kp method 64
C Derivation of the Lowdin perturbation theory 68

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