二維SSH模型的拓樸性質與分類

Abstract

當塊材具有拓樸性質時，其對應的邊界上會存在邊界態，這就是所謂的「塊材與邊 界對應性」。此對應可由簡單的一維模型 − SSH 模型或 extended SSH 模型來做驗證。 我們嘗試將一維 SSH 長鏈交錯編織以推廣成二維系統，並稱之為二維 SSH 模型。

我們發現透過調整二維 SSH 模型的參數，系統有可能為半金屬，弱拓樸絕緣體或 是一般的絕緣體。由於二維 SSH 模型具有時間反演對稱性，我們利用這個特性定義出 一個強拓樸量與兩個弱拓樸量，並用它們來為系統做分類。此外，我們也發現這個分 類方法等價於一個圖像化的分類方式。利用數值方法，我們驗證了二維 SSH 模型的塊 材與邊界對應性。最後，當選取特定的參數與邊界條件時，可以得出不同邊界型態的 奈米碳管的結果。

When a system carries non-vanishing topological numbers in the bulk, there will be edge states on the boundary. This is the so-called bulk-edge correspondence. It can be verified explicitly by using the 1D SSH or extended SSH models. In this thesis, we consider the 2D-SSH model which may be constructed by interweaving the 1D SSH chains into a two dimensional system.

For the 2D-SSH model, we find that it can be a semimetal, weak topological insulator or trivial insulator depending on the values of the parameters. Because the 2D-SSH model has time reversal symmetry, we may use this to define a strong topological number and two weak topological numbers. We can classify the system by using these topological numbers. Furthermore, we also show that this is equivalent to a graphical way to classify the system. We use numerical calculation to verify the bulk-edge correspondence for the 2D-SSH model. Finally, we can reproduce the results of various carbon nanotubes by choosing specific parameters and boundary conditions.