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Flatband generators

초록/요약

Flatbands (FBs) are dispersionless energy bands in the single-particle spectrum of a translational invariant tight-binding network. The FBs occur due to destruc- tive interference, resulting in macroscopically degenerate eigenstates living in a finite number of unit cells, which are called compact localized states (CLSs). Such macroscopic degeneracy is in general highly sensitive to perturbations, such that even slight perturbation lifts the degeneracy and leads to various in- teresting physical phenomena. In this thesis, we develop an approach to identify and construct FB Hamilto- nians in 1D, 2D Hermitian, and 1D non-Hermitian systems. First, we introduce a systematic classification of FB lattices by their CLS properties, and propose a scheme to generate tight-binding Hamiltonians having FBs with given CLS properties—a FB generator. Applying this FB generator to a 1D system, we identify all possible FB Hamiltonians of 1D lattices with arbitrary numbers of bands and CLS sizes. Extending the 1D approach, we establish a FB generator for 2D FB Hamiltonians that have CLSs occupying a maximum of four unit cells in a 2*2 plaquette. Employing this approach in the non-Hermitiaon regime, we realize a FB generator for a 1D non-Hermitian lattice with two bands. Ultimately, we apply our methods to propose a tight-binding model that explains the spectral properties of a microwave photonic crystal. Our results and methods in this thesis further our understanding of the prop- erties of FB lattices and their CLSs, provide more flexibility to design FB lattices in experiments, and open new avenues for future research.

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목차


1 Motivation and outline 1
2 Introduction: Flatbands in discrete systems 7
2.1 Tight-bindingmodelfordiscretesystems . . . . . . . . . . . 8
2.1.1 Tight-bindingmodel .................. 9
2.1.2 Bloch theorem and band structure . . . . . . . . . . . 12
2.2 Flatbands, compact localized states, and macroscopic degeneracy 15
2.3 Flatband construction methods ................. 19
2.3.1 Geometrical methods.................. 19
2.3.2 Flatband generation from local unitary transformations 22
2.3.3 Flatband construction from symmetry . . . . . . . . . 25
2.4 Applications and experimental realizations of flatband systems 29
2.4.1 Perturbation as a playmaker .............. 30
2.4.2 Experimental realizations and applications . . . . . . 33
2.5 Summary ............................ 41
3 Classification and construction of flatbands by CLS . . . . . . 43
3.1 Classification of flatband networks by compact localized states (CLSs).............................. 44
3.2 Properties of CLSs ....................... 46
3.2.1 Relation between CLSs and Bloch wave functions . . 46
3.2.2 Irreducibility condition of CLSs . . . . . . . . . . . . 47
3.2.3 Completeness of CLSs................. 52
3.2.4 Relation between CLS completeness and band touching 54
3.3 CLS existence conditions and the flatband generator . . . . . . 55
3.3.1 Block matrix representation of the tight-binding model 56
3.3.2 CLS existence conditions and flatband tester . . . . . 59
3.3.3 Flatband Generator................... 62
3.4 Summary ............................ 64
4 Flatband generator in one dimension . . . . . . . . . . . . 65
4.1 Introduction........................... 66
4.2 Two-band problem ....................... 67
4.2.1 U=1case ........................ 69
4.2.2 U=2case ........................ 70
4.3 Arbitrary number of bands ................... 74
4.3.1 Thegenerator...................... 76
4.3.2 Solutions ........................ 77
4.3.3 Chiral symmetry .................... 82
4.3.4 Network constraints .................. 86
4.4 Summary ............................ 87
5 Flatband generator in two dimensions . . . . . . . . . . . . 89
5.1 The eigenvalue problem..................... 90
5.2 Classification of CLSs in 2D .................. 91
5.3 The flatband generator ..................... 95
5.4 Nearest neighbor hoppings ................... 97
5.5 Next nearest neighbor hoppings ................ 100
5.6 Summary ............................ 103
6 Non-Hermitian flatband generator 105
6.1 Overview of non-Hermitian physics and flatbands in non-Hermitian systems ...... 106
6.2 Non-HermitianHamiltonian.................. 107
6.3 The generator.......................... 108
6.4 Results ............................. 110
6.4.1 Completely flat case.................. 110
6.4.2 Partially flat case.................... 113
6.4.3 Modulus-flat case.................... 114
6.5 Summary............................ 115
7 Flatband in a microwave photonic crystal .......... 117
7.1 Photonic crystals and Dirac billiards . . . . . . . . . . . . . . 118
7.2 Tight-binding model for a microwave photonic crystal . . . . 120
7.3 The honome lattice ....................... 123
7.4 Methods............................. 125
7.4.1 Positioning van Hove singularities . . . . . . . . . . . 126
7.4.2 Deviations from experiment .............. 127
7.4.3 The algorithm...................... 128
7.5 Results.............................. 130
7.5.1 Fitting with three hoppings............... 131
7.5.2 Fitting with five hoppings ............... 132
7.6 Summary ............................ 134
8 Conclusions and outlook .......... 137
References ....... 143
Appendices 168
A Supplementary materials for CLS properties ....... 169
A.1 Linear dependence and reducibility of CLSs in 1D (U=3 case) ..... 169
B Supplementary materials for the flatband generator in 1D ...... 173
B.1 On the linear independence of CLSs . . . . . . . . . . . . . . 173
B.2 Generator and band structure for two-band U = 2 FB networks 174
B.2.1 Real H1......................... 175
B.2.2 Complex H1 ...................... 176
B.2.3 Degenerate H0 ..................... 177
B.2.4 FB energy equals one of the eigenvalues of H0: Reduc- tiontoU=1...................... 178
B.3 Generalized sawtooth chain................... 178
B.4 Inverse eigenvalue problem: A toy example and the solution of the U=2 CLS ......................... 181
B.4.1 Toy example ...................... 181
B.4.2 U=2 case ........................ 182
B.4.3 Bipartite lattices and chiral symmetry . . . . . . . . . 185
B.5 Resolving the non-linear constraints . . . . . . . . . . . . . . 186
B.5.1 U=2 case ........................ 187
B.5.2 U=3 case........................ 190
B.6 Examples for FB generators .................. 191
B.6.1 nu=3,U=2 case................... 191
B.6.2 nu=3,U=3 case ................... 194
B.7 Network constraints....................... 196
B.7.1 U=2 case ........................ 196
B.7.2 U=3case........................ 200
C Supplementary materials for the flatband generator in 2D 203
C.1 Two-bandproblem ....................... 203
C.2 More than two bands with nearest neighbor hoppings . . . . . 205
C.2.1 U=(2,1,0) case ..................... 205
C.2.2 U=(2,2,1) case ..................... 206
C.2.3 Special case for U=(2,2,1) with nearest neighbor hoppings .......................... 212
C.2.4 U=(2,2,0) case with three bands . . . . . . . . . . . . 214
C.3 Next nearest neighbor hoppings ................ 221
C.3.1 U=(2,2,1) case ..................... 221
C.3.2 U=(2,1,0) case ..................... 225
D Supplementary materials for the non-Hermitian FB generator 229
D.1 CLS-based generator ...................... 229
D.1.1 U=1case ....................... 231
D.1.2 U=2 case ........................ 234
D.1.3 U=3 case ........................ 235
D.1.4 Inverse eigenvalue method for CLS approach . . . . . 236
D.2 Solving completely flatbands.................. 237
D.2.1 Both bands are completely flat . . . . . . . . . . . . . 237
D.2.2 Oneband is completely flat............... 239
D.3 Solving partially flatbands ................... 241
D.3.1 Real parts of both bands are flat . . . . . . . . . . . . 242
D.3.2 Real part of one band is flat .............. 246
D.3.3 Imaginary parts of both bands are flat . . . . . . . . . 249
D.3.4 Imaginary part of one band is flat . . . . . . . . . . . 252
D.3.5 Modulus of a band is flat................ 253

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