Step-dependent quantum walks on odd and even cyclic graphs produce gapped and gapless topological phases, rotationally symmetric flat bands in 4n-site graphs, and robust edge states at phase interfaces.
Hsieh et al., A topological Dirac insulator in a quan- tum spin Hall phase, Nature (London) 452, 970 (2008)
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Quantum walks reveal topological flat bands, robust edge states and topological phase transitions in cyclic graphs
Step-dependent quantum walks on odd and even cyclic graphs produce gapped and gapless topological phases, rotationally symmetric flat bands in 4n-site graphs, and robust edge states at phase interfaces.