具有手性對稱性及具有反射對稱性的(1+1)維電荷守恆系統的費米子對稱保護拓樸相

Abstract

本文旨在為受兩種不同對稱性保護的(1+1)維費米子對稱保護拓樸相提供物理解釋，其分別是U(1)和手性對稱性及U(1)和反射對稱性。 第一章提供了一些與對稱保護拓樸相相關知識的簡要回顧。除了對稱保護拓樸相的基本定義及分類它們的方法，我們還介紹了如何透過配邊理論建構多體拓樸不變量及體邊對應的概念。 在第二和第三章中，我們提出了一種能夠協助人們物理地理解這兩種(1+1)維費米子對稱保護拓樸相的方法，主要是透過建立它們對應的多體拓樸不變量及可分解系統之間的關係。其中，可分解系統是指系統在週期性邊界條件下可以被分解成某個子系統的多個副本。此外，我們還指出了有限尺寸效應在多體系統和自由費米子系統之間的不同。 第四章主要在探討平移對稱性如何改變這兩種 (1+1) 維費米子對稱保護拓樸相。同時，我們也為在考慮平移對稱性後產生的拓樸相建立對應的拓樸不變量。

This thesis aims to provide a physical interpretation of the (1+1)d fermionic symmetry-protected topological (SPT) phases protected by U(1) and chiral (U(1)+S) symmetry and protected by U(1) and reflection (U(1) + R) symmetry. The first chapter contains some basic knowledge related to the SPT phases. Beyond the concept of SPT, we introduce the classification schemes of SPT phases, how to construct many-body topological invariants through the cobordism theory, and the idea of bulk-boundary correspondence. In the second and third chapters, as an approach to physically understanding the (1+1)d fermionic SPT phases protected by U(1) + S and U(1) + R, we construct the relations between their corresponding many-body topological invariants and the decomposable systems, defined as systems that can be decomposed into n copies of a subsystem under the consideration of periodic boundary conditions. Additionally, we indicate that the finite-size effect on many-body systems differs from that on free fermion systems. The fourth chapter focuses on how translation symmetry changes two SPT phases studied in the previous two chapters. We also construct a set of topological invariants to describe the SPT phases that involve translation symmetry.