Robustness of chiral edge modes in fractal-like lattices below two dimensions: A case study

Fischer, Sonja; Hooft, Michal van; Meijden, Twan van der; Smith, C. Morais; Fritz, Lars; Fremling, Mikael

doi:10.1103/physrevresearch.3.043103

Cited by 15 publications

(10 citation statements)

References 43 publications

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“…4(a), at the transition point between the spectral insulator and topological random fractal phase, the energy gap closes and remains closed in the topological and Anderson insulator phases. Despite being gapless, and in contrast to the studies of deterministic fractals 50,55,57 , we uncover unambiguous and robust quantization of conductance in the topological random fractal phase. The formation of the quantized plateau as a function of finite t 2 is demonstrated in Fig.…”

Section: Resultscontrasting

confidence: 64%

“…In this vein, the possibility of topological states in fractals has stirred a new research direction. Despite reported signatures of topology in a number of fractal lattices, many aspect of these systems remains unclear or controversial [50][51][52][53][54][55][56][57][58][59][60][61][62] . The existence of the spectral gap, possibility of supporting quantized responses and anomalous dependence on system details, such as the coordination number and connectivity of lattice sites, remain under debate.…”

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confidence: 99%

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Topological random fractals

et al. 2022

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The search for novel topological quantum states has recently moved beyond naturally occurring crystalline materials to complex and engineered systems. In this work we generalize the notion of topological electronic states to random lattices in non-integer dimensions. By considering a class D tight-binding model on critical clusters resulting from a two-dimensional site percolation process, we demonstrate that these topological random fractals exhibit the hallmarks of topological insulators. Specifically, our large-scale numerical studies reveal that topological random fractals display a robust mobility gap, support quantized conductance and represent a well-defined thermodynamic phase of matter. The finite-size scaling analysis further suggests that the critical properties are not consistent with the expectations of class D systems in two dimensions, hinting to the nontrivial relationship between fractal and integer-dimensional topological states. Our results establish topological random fractals as the most complex systems known to support nontrivial band topology with their distinct unique properties.

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Section: Resultscontrasting

confidence: 64%

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confidence: 99%

Topological random fractals

et al. 2022

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“…The latter experiments were done on SGF structures grafted either mainly by the femtosecond laser writing techniques, or designed with evanescently coupled helical waveguides, for example. Parallel theoretical proposals and analyses of topological phase transitions and edge states in fractal space [32][33][34][35][36] came up, exploring the unique geometry of an SGF. This domain of research has opened up the possibility of looking back at the well understood physics and geometry of the SGF and its variants [37][38][39][40] from a new perspective.…”

Section: Introductionmentioning

confidence: 99%

Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

Biswas¹,

Chakrabarti²

2023

Preprint

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Compact localized single particle eigenstates on a deterministic fractal substrate, modelled by a triangular Sierpinski gasket of arbitrarily large size, are unravelled and examined analytically. We prescribe an exact real space renormalization group (RSRG) decimation scheme within a tight binding formalism to discern these states, and argue that the number of such states can be infinite if the fractal substrate is enlarged to its thermodynamic limit. Interestingly, these localized states turn out to populate the non-dispersive, flat bands in a periodic array of Sierpinski gasket motifs, however large they may be. Our results match and corroborate the recently observed compact localized, flat band states engineered on two dimensional photonic waveguide networks with a fractal geometry, and provide a whole subset of them, which, in principle, should be observable in fractal photonic lattice experiments.

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“…Recently, interest has developed with respect to investigating topological phases of matter on fractal lattices [11][12][13][14][15][16][17][18][19][20], such as the Sierpinski triangle. This is, in part, motivated by experimental developments, e.g., within molecular [21] and electronic [22] fractals.…”

Section: Introductionmentioning

confidence: 99%

“…Topological phases of non-interacting systems, including integer quantum Hall phases, have been found on fractal lattices [11,12,14,15,17,19,20]. Recently, fractional quantum Hall models have also been constructed * These authors contributed equally to this manuscript.…”

Section: Introductionmentioning

confidence: 99%

Laughlin topology on fractal lattices without area law entanglement

Li,

Jha,

Nielsen

2022

Preprint

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Laughlin states have recently been constructed on fractal lattices, and the charge and braiding statistics of the quasiholes were used to confirm that these states have Laughlin type topology. Here, we investigate density, correlation, and entanglement properties of the states on a fractal lattice derived from a Sierpinski triangle with the purpose of identifying similarities and differences compared to two-dimensional systems and with the purpose of investigating whether various probes of topology work for fractal lattices. Similarly to two-dimensional systems, we find that the connected particle-particle correlation function decays roughly exponentially with the distance between the lattice sites measured in the two-dimensional plane, but the values also depend on the local environment. Contrary to two-dimensional systems, we find that the entanglement entropy does not follow the area law if one defines the area to be the number of nearest neighbor bonds that cross the edge of the selected subsystem. Considering bipartitions with two bonds crossing the edge, we find a close to logarithmic scaling of the entanglement entropy with the number of sites in the subsystem. This also means that the topological entanglement entropy cannot be extracted using the Kitaev-Preskill or the Levin-Wen methods. Studying the entanglement spectrum for different bipartitions, we find that the number of states below the entanglement gap is robust and the same as for Laughlin states on two-dimensional lattices.

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Robustness of chiral edge modes in fractal-like lattices below two dimensions: A case study

Cited by 15 publications

References 43 publications

Topological random fractals

Topological random fractals

Designer quantum states on a fractal substrate: compact localization, flat bands and the edge modes

Laughlin topology on fractal lattices without area law entanglement

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