在特定自旋模型，由自旋波激發產生的拓樸磁振子邊界態已經在戈薇晶格，蜂窩狀行晶格和正方形晶格被發現。然而，這些邊界態只出現在波色子能帶高能量區間，不像是靠近零能量的狄拉克模態。為了發現這有種模態的模型，我們嘗試數個二維模型。在這篇論文裡，我們分析了三個模型:"類P波"施溫格波色子模型、XYZ模型和玻色霍夫施塔特-哈伯模型。最後一個模型我們可能可以找到"類狄拉克模態"磁振子邊界態。
在第2和第3章，我們先介紹如何對角化一個BdG哈密頓算子和定義玻色系統的貝里曲率。因為波色子要滿足對易關係，所以不再是么正變換而是稱為反么正變換。為了對角化，我們必須要做波戈留波夫轉換就可以得到一組能量解，接著我們就可以找到相對應的本徵態和他們的規一化關係。貝里曲率的定義是本徵態維分的內積，在波色子系統，我們必須要加入規一化條件在貝里曲率的定義裡，然後我們用哈密頓算自維分取代本徵態維分，我們找到一個簡單精準的方法在數值上算出貝理曲率。此外，我們發現在能帶是正能量半部和負能量半部的貝里曲率差一個負號。
從最簡單可比被分離成自旋向上和自旋向下部分的配對的施溫格波色子模型，必需要加入化學能來避免虛數能量。我們在晶格上切邊界並觀察邊界能帶，我們得到的結果是沒有任何邊界態存在。
在第5章，我們討論XYZ模型在蜂窩狀反鐵磁晶格上，並加上異向性的反對稱交換作用。我們假設反對稱交換作用是次近鄰作用並且波色子在2種次晶格中是相同方向跳躍，這或許可以產生一個不為零的掌性效應在邊界上，所以我們討論在蜂窩狀晶格的鋸齒邊的結果。然而我們發現存在在鋸齒邊界的邊界態並不強健，在晶格上掌性效應依然是零，原因是粒子和洞的掌性是相反的。
在第6章，我們討論玻色霍夫施塔特-哈伯模型在正方形晶格上。這原來是一種費米子模型加上外加磁場，所以會有一個規範場在哈密頓算子裡並改變準粒子的動量。這邊我們使用相同物理圖像在波色子上並且加上哈伯作用變成一個新模型稱為玻色霍夫施塔特-哈伯模型。非零的貝里曲率表示有拓樸邊界態，所以我們沿著晶格的一個邊界發現到有在高能量的邊界態但是在中間零能量附近卻沒有。接著我們調整化學能來平移能帶，貝里曲率也會跟著平移而不改變值，所以我們可以藉由調整化學能讓邊界態出現在零能量附近的區間。
在第7章，基於玻色霍夫施塔特-哈伯模型，我們使用霍爾斯坦-普里馬科夫轉換找到一個對應的自旋模型，我們改變一些耦合參數和外加場並找出基態，我們可以得到這個一般自旋模型的相圖。

Topological magnon edge states in spin-wave excitation of certain spin models were found at kagome, honey-comb lattice and square lattice. However, instead of being near zero energy like gapless Dirac modes, these bosonic edge states appear only in high energy level of positive bosonic energy band. To create or find
such kind of model, we try several 2D Hamiltonians. In this thesis, we analyze three models: "P-wave like" Schwinger boson model, XYZ model and "bosonic" Hofstadter-Hubbard model that we may possibly find "Dirac mode-like" magnon edge states.
In chapter 2 and 3, we first introduce the diagonalization of a BdG Hamiltonian and redefine the Berry curvature in bosonic system. Since boson satisfies commutation relation, this is no longer an unitary transformation and it is called para-unitary. To diagonalize, we need to do Bogoliubov transformation and eigenenergy will appear in pair due to BdG Hamiltonian. We can then find the correspondent eigenstates and find the normalization condition of these states. The definition of Berry curvature is the inner product of states' derivative. We will add normalization condition of eigenstates into Berry curvature definition to define the Berry curvature in bosonic system. Then, we replace states' derivative to Hamiltonian derivative. We find an alternative way to simply and precisely calculate Berry curvature in numerical. Furthermore, we find that the sign of Berry curvature is opposite in positive and negative bulk energy bands.
Start from the simplest Schwinger boson model that can be decoupled into spin up and down part with pairing, we should add a chemical potential to avoid imaginary energy. We set open boundary on lattice to study edge spectrum. Our demonstration shows that there is no edge states.
In chapter 5, we use XYZ model in AFM honeycomb lattice with an anisotropic DM interaction. We assume this DM interaction is next-nearest neighbour(NNN) interaction and bosons hop the same direction in A, B sublattice. This may emerge a non-zero chiral effect at the edge of lattice. We set open boundary at zigzag
edge of honeycomb. However, we find that the existing of these edge states is not robust. The net chiral here is still zero due to the opposite chiral of particle and holes.
In chapter 6, we use "bosonic" Hofstadter-Hubbard model in the suare lattice. Hofstadter model is a model that originally describes fermion systems with applying an uniform magnetic field. There will be a gauge appearing in Hamiltonian and change the momentum of quasi-particles. Here, we use the same physical
image but insteading to bosons' interaction and add Hubbard interaction to become a new model called "bosonic"Hofstadter-Hubbard model (BHH) or bosonic Hubbard model. The non-zero Berry curvature implies topological edge states. We set open boundary at one edge. We can find that there are edges states in the high energy regime but no appearing in the middle part. Then, we tune chemical to shift energy bands. The Berry curvature will not change as bands shifting. Therefore, we can tune chemical potential to let these edge states appearing near zero energy region.
In chapter 7, based on "bosonic" Hofstadter-Hubbard model, we use Holstein-Primakoff transformation to find a correspondent spin model. Then, we change the coupling constant and external field and calculate its ground state. We get the phase diagram of this general spin model.

1 INTRODUCTION 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 DIAGONALIZE BOSONIC BdG HAMILTONIAN 4
2.1 Bosonic BdG Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 4
3 CHERN INTEGERS IN BOSONIC BdG SYSTEMS 7
3.1 Topological Chern Number of Bosonic Bands . . . . . . . . . . . . . 7
3.2 Alternative Numerical Method to Calculate Berry Curvature . . . . 8
3.3 Characters of Berry Curvature in Negative Energy Band . . . . . . 10
4 "P-WAVE" PAIR SCHWINGER BOSON MODEL 11
4.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1.1 Eigenenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Edge Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 XYZ MODEL 16
5.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Schwinger Boson Approach in Anti-ferromagnetic System . . . . . . 17
5.3 Self-Consistent Equations . . . . . . . . . . . . . . . . . . . . . . . 21
5.4 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.4.1 Berry curvature,
= 0 . . . . . . . . . . . . . . . . . . . . 23
5.4.2 Berry curvature,
6= 0 . . . . . . . . . . . . . . . . . . . . 23
6 "BOSONIC" HOFSTADTER-HUBBARD MODEL 25
6.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.1.1 mean eld approximation . . . . . . . . . . . . . . . . . . . 27
6.2 BdG Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3 Self-Consistent Equation and  Determined . . . . . . . . . . . . . 29
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.4.1 Berry Curvatures . . . . . . . . . . . . . . . . . . . . . . . . 30
6.4.2 Edge State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.4.3 Edge State Near E=0 . . . . . . . . . . . . . . . . . . . . . . 35
7 MAP to GENERAL SPIN MODEL 38
7.1 Map to Spin System . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.2 General Spin System . . . . . . . . . . . . . . . . . . . . . . . . . . 40
8 CONCLUSION and DISCUSSION 43
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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