本研究探討了兩體SSH模型的局域化、邊緣態以及拓樸特性。首先我們引入單體
的SSH模型來說明本研究的一些重要概念：邊緣態、局域化、對稱性和bulk-boundary correspondence。我們直接對實空間的矩陣進行數值對角化來計算邊緣態；至於局域化的研究，我們使用了inverse participation ratio (IPR)。IPR不只可以在連續體中區分束縛態，還可以用來研究邊緣態的局域化性質。我們定義了么正和反么正對稱性，並且說明其在凝態系統中的重要性。我們特別專注於平移、共轉譯、反轉以及手徵對稱性，這些特性之後能讓我們定義出交互作用模型的拓樸性質。

接著我們利用絕熱定理以及貝里相位建立拓樸。我們定義了拓樸不變量和札克相位，並討論其在反對稱下的量子化。我們更進一步地將此拓樸不變量從平移對稱系統延伸到可能是非厄米特、雙正交，具共轉譯對稱性的交互作用系統。我們呈現了一種利用威爾森迴圈算出此不變量數值的方法，另外，也呈現了捲繞數、簡單SSH模型的拓樸不變量和bulk-boundary correspondence。

最後，我們研究兩體SSH模型。我們證明了此系統具有反轉和共轉譯對稱性，但是並沒有手徵對稱性。此外，我們用正則變換，將內部自由度映射到準自旋自由度，並利用質心座標和相對座標，將波函數分解。如此一來我們便可以套用由共轉譯對稱導出的類布洛赫定理，去計算系統的左和右布洛赫週期函數，進而定義出札克相位。最後，我們得到D1和D2二聚體的兩邊緣態，我們可以利用IPR和數值計算出的顯式波函數來分辨兩者。這個邊緣態為幾何結構、邊界以及系統的交互作用交織出的結果。此外，我們定義出的札克相位是用來描述三個離散能帶的拓樸結構，導出非普通拓樸，因為π和0這兩個相位不只和ω/ν有關，它們和交互作用的長度跟bulk boundarycorrespondence的破壞也有關係。

In this work, we study the localization, edge states and topology of the two-body SSH model. First, we introduce the single-particle SSH model to illustrate some important concepts for the development of the work: Edge states, localization, symmetries and the bulk-boundary correspondence. To compute the edge states, we directly diagonalize numerically the real space representation matrix of the model. For the study of the localization, we use the inverse participation ratio (IPR), which can not only be used to distinguish bound states even in the continua, but also to investigate the localization properties of edge states. We define unitary and anti-unitary symmetries and explain their importance in condensed matter systems; in particular, we focus in translational, cotranslational inversion and chiral symmetries, which later allow us to define the topological properties
of the interacting model.

Next, we establish the topology by the means of the adiabatic theorem and the Berry phase. With these, we define our topological invariant, the Zak phase, and discuss its quantization under inversion symmetry. Moreover, we extend this topological invariant from translational symmetric systems, to interacting systems with cotranslational
symmetry which may be non-Hermitian, but biorthogonal. Furthermore, a way to numerically compute this invariant through the Wilson-loop approach is provided. The winding number, as the topological invariant for the simple SSH model, and the bulkboundary correspondence are presented as well. Finally, we examine the two-body SSH model. We show that this system presents inversion and cotranslational symmetries, but no chiral symmetry. In addition, we utilize a canonical transformation to map the internal degrees of freedom into pseudo-spin ones, together with the center-of-mass and relative coordinates, to separate the wave function such that
we can apply a Bloch-like theorem, derived from the cotranslational symmetry, to calculate the left and right Bloch periodic functions of the system, and hence, define the Zak phase.

Finally, we obtain two edge states in both dimerizations D1 and D2, this are distinguished with the IPR and the explicit form of the wave functions computed numerically; these edge states come as a result between the interplay between the geometry, the boundaries
and the interaction of the system. On the other hand, the defined Zak phase is used to characterize the topological structure of three of the discrete bands, yielding non-trivial topology as the values of π and 0 depend not only on the ratio ω/ν but also on the interaction strength, and breaking the bulk boundary correspondence.

1 Introduction 1
2 The Su-Schrieffer-Heeger model 3
2.1 The SSH Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 The bulk Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 The bulk-momentum space Hamiltonian . . . . . . . . . . . . . . 5
2.3 Edge states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Localization of the edge states . . . . . . . . . . . . . . . . . . . . 8
2.4 Symmetries in the SSH model . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1 Unitary symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.2 Non-unitary symmetries . . . . . . . . . . . . . . . . . . . . . . . 12
3 Topological properties in one-dimension 15
3.1 The adiabatic theorem and the Berry phase . . . . . . . . . . . . . . . . 15
3.1.1 Gauge invariance of the Berry phase . . . . . . . . . . . . . . . . 16
3.2 Topological invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 The Zak phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.2 Quantization of the Zak phase . . . . . . . . . . . . . . . . . . . . 17
3.2.3 Wilson-loop approach . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.4 The winding number and bulk-boundary correspondence in the SSH
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Topology in interacting systems . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 Cotranslational symmetry . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Biorthogonal Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.1 The Zak phase for biorthogonal systems . . . . . . . . . . . . . . 25
4 The two-body SSH model 27
4.1 The interacting Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Symmetries in the two-body SSH model . . . . . . . . . . . . . . . . . . 29
4.3 Bulk momentum-space Hamiltonian . . . . . . . . . . . . . . . . . . . . . 30
4.3.1 Canonical transformation: Pseudo-spin basis . . . . . . . . . . . . 30
4.3.2 Center-of-mass and relative coordinates . . . . . . . . . . . . . . . 32
4.4 Energy and bound states in the bulk Hamiltonian . . . . . . . . . . . . . 35
4.5 Energy and edge states in D1 and D2 . . . . . . . . . . . . . . . . . . . . 38
4.6 Topology of the two-body SSH model . . . . . . . . . . . . . . . . . . . . 42
5 Conclusions and outlook 46
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Appendix 48
A Numerical implementation of the two-body SSH matrix in the Fock
basis 48
B Pseudo-spin and center-of-mass coordinates’ Schrödinger equation 50
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