Higher-order topological knots in non-Hermitian lattices Restricted; Files Only

Abstract

In two dimensions, Hermitian lattices with non-zero Chern numbers and non-Hermitian

lattices with a higher-order skin e↵ect (HOSE) bypass the constraints of the Nielsen–Ninomiya

“no-go” theorem at their one-dimensional boundaries. This allows the realization of

topologically-protected one-dimensional edges with nonreciprocal dynamics. However,

unlike the edge states of Chern insulators, the nonreciprocal edges of HOSE phases

exist only at certain edges of the two-dimensional lattice, not all, leading to cornerlocalized

states. In this work, we investigate the topological connections between

these two systems and uncover novel non-Hermitian topological phases possessing

“higher-order topological knots” (HOTKs). These phases arise from multiband topology

protected by crystalline symmetries and host point-gap-protected nonreciprocal edge

states that circulate the entire boundary of the two-dimensional lattice. We show that

phase transitions typically separate HOTK phases from “Complex Chern insulator”

phases –non-Hermitian lattices with nonzero Chern numbers protected by imaginary

line gaps in the presence of time-reversal symmetry.

Table of Contents

1 Introduction 1

1.1 Symmetry, topology, and condensed matter physics . . . . . . . . . . 1

1.2 Rise of non-Hermiticity and topology . . . . . . . . . . . . . . . . . . 6

2 Background 13

2.1 Complex Chern insulators under time-reversal symmetry . . . . . . . 13

2.2 Higher-order skin e↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Deforming Chern insulators into HOSE phases . . . . . . . . . . . . . 19

3 Approach 23

3.1 Classification of Higher-order topological knots . . . . . . . . . . . . . 23

3.1.1 Classification of Cn-symmetric NH Hamiltonians in class AI . 24

4 Analysis 28

4.1 Model Hamiltonians with HOTK phases . . . . . . . . . . . . . . . . 28

5 Conclusion 42

A Appendix 45

A.1 Winding number under TRS and TRS† . . . . . . . . . . . . . . . . . 45

A.2 Z2 quantization of the Berry phase under TRS . . . . . . . . . . . . . 46

A.3 Details on the deformation of complex Chern insulators into HOSE

phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

A.4 Construction of the topological classification of Cn-symmetric NH Hamiltonians

in class AI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A.4.1 Time-Reversal symmetry . . . . . . . . . . . . . . . . . . . . . 54

A.4.2 Rotation symmetry . . . . . . . . . . . . . . . . . . . . . . . . 55

A.4.3 Constraints due to rotation . . . . . . . . . . . . . . . . . . . 57

A.4.4 Constraints due to TRS . . . . . . . . . . . . . . . . . . . . . 59

A.4.5 chi^(n) indices for Cn-symmetric NH crystals . . . . . . . . . . . 61

A.5 chi^(2) index of the minimal model for a HOSE phase . . . . . . . . . . 63

A.6 Spectra of HOTK phases with C3 and C6 symmetries . . . . . . . . . 64

A.7 A C6-symmetric real-line-gap Chern insulator . . . . . . . . . . . . . 64

Bibliography 67

About this Honors Thesis

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School	Emory College
Department	Physics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Physics, Condensed Matter
Stichwort	Topological phases of matters Topological insulator Non-Hermitian
Committee Chair / Thesis Advisor	Benalcazar, Wladimir, Emory University
Committee Members	Santos, Luiz, Emory University Srivastava, Ajit, Emory University

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