Topological And Physical Properties Of Topological Crystalline Insulators And Non-hermitian Systems

Landau phase transition theory embodies the importance of symmetry in the study of phases in condensed matter physics,while the discovery of topological order and topological insulators introduces the concept of topology from mathematics into condensed matter physics.In the study of topological insulators and topological superconductors,researchers found that topology is protected by the intrinsic symmetry of the system.However,topological crystalline insulators are different from traditional topological insulators,in addition to the non-spatial intrinsic symmetry,the crystalline symmetry of the system also protects the topological properties of the system.This additional symmetry increases the research difficulty and brings richer physical phenomena.On the other hand,traditional condensed matter physical systems have Hermitian Hamiltonians,but open systems have non-Hermitian effective Hamiltonians.The study of the non-Hermitian system found that different from the Hermitian system,the non-Hermitian system has a unique point-gap topology,which brings a series of interesting physics including the non-Hermitian skin effect and the exponentially growing Green’s function.In this thesis,I will introduce topological crystalline insulators and non-Hermitian systems with the nontrivial point-gap topology.The structure of this thesis is as follows:The first chapter is the introduction.We briefly review the development from the integer quantum Hall effect to topological insulators,and briefly summarize the main characteristics,important research methods and problems of topological crystalline insulators.In addition,starting from the problem of the bulk-boundary correspondence,I introduce the non-Hermitian skin effect and recall the motivation of the non-Bloch band theory in non-Hermitian systems.In chapter 2,I introduce the basic knowledge of the topological crystalline insulators,and present important methods to study the topological and physical properties of topological crystalline insulators,which are Wilson loops and symmetry indicators.After that,I introduce high-order topological insulators described by nested Wilson loops and symmetry indicators,respectively.Finally,I introduce the general theory of the classification of topological crystalline insulators.Chapter 3 discusses my work on two dimensional twofold rotation symmetric and time-reversal invariant systems:Proposing symmetry protected topological invariants based on lifted Wilson loops.Specifically,in this system,the symmetry indicators is trivial,but the topological classification results predict a new Z2 topological invariant.We propose a new homotopy invariant to characterize this topological invariant,and mathematically prove that it is the invariant predicted by the topological classification.Our method is based on the Wilson loop method,but different from the usual Wilson loop method,our method lifts the Wilson loop and time-reversal symmetry to its universal covering group and studies the topology class of the Wilson loop after lifting.We conclude that the invariant predicted by topological classification originates from the disconnectedness of the time-reversed fixed points set on the universal covering group.We correct the view in previous literature that in any system of 4 occupied bands,the winding of the Wilson loop is only protected by the Fu-Kane-Mele invariant.In contrast,we conclude that in a phase with zero Fu-Kane-Mele invariant and nonzero homotopy invariant,the gapless Wilson loop spectrum is protected by symmetries,implying that this topological phase cannot be continuously deformed into an atomic insulator.Finally,we find two examples with different topological phases but the same Wilson loop spectrum,illustrating the failure of the conventional Wilson loop method.Chapter 4 discusses my work on Green’s functions in non-Hermitian systems:obtaining the exact formula of the end-to-end Green’s function and deriving the speed at which the bulk region Green’s function tends to the generalized Brillouin zone integral formula.I first briefly review the non-Bloch band theory and the generalized Brillouin zone and introduce the work of characterizing the Green’s function under open boundary conditions by the generalized Brillouin zone integral formula.After that,I introduce the exact formula of the end-to-end Green’s functions.In addition,we verify that the Green’s function integral formula agrees well with Green’s function in the bulk region.Furthermore,we illustrate that as the system scale increases,the Green’s function at the bulk region approaches the generalized Brillouin zone integral formula no slower than exponential decay.Finally,we establish the correspondence between the signal amplification and the non-Hermitian skin effect.In chapter 5,I make a brief summary and outlook.