Wave-packet propagation in a finite topological insulator and the spectral localizer index

Michala, Jonathan; Pierson, Alexander; Loring, Terry A.; Watson, Alexander B.

doi:10.2140/involve.2021.14.209

Cited by 9 publications

(5 citation statements)

References 29 publications

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“…However, the persistence of the edge conductance in this regime suggests that localised initial data with spectral measure concentrated in the bulk gap will still propagate along the edge coherently. This remarkable behaviour has been observed numerically (see, for example, [3,80]), and even experimentally across various model systems [38,86,99,105]. Such initial data, which we again refer to as edge wave-packets, can be obtained by multiplying approximate edge states, i.e., approximate eigenfunctions of H E,d with energies in the bulk gap, by a smooth decaying function such as a Gaussian.…”

Section: Edge States Edge Conductance and Edge Wave-packetsmentioning

confidence: 64%

“…The edge currents of TIs are mediated by electronic states localised at edges known as edge states. The robustness of the edge conductance of a TI can be seen at the level of localised wave-packets formed from these edge states, which snake around corners and defects of the edge, even in the presence of disorder [3,11,17,19,41,46,59,61,66,75,80,81,85,93]. The behaviour of these wave-packets has spurred interest in building photonic and acoustic devices which mimic topological insulators for wave-guiding applications [38,72,86,99,100,105].…”

Section: Introductionmentioning

confidence: 99%

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Computing spectral properties of topological insulators without artificial truncation or supercell approximation

Colbrook¹,

Horning²,

Thicke³

et al. 2021

Preprint

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Topological insulators (TIs) are renowned for their remarkable electronic properties: quantised bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations of these physical properties generally rely on artificial periodicity (the supercell approximation), or unphysical boundary conditions (artificial truncation). In this work, we build on recently developed methods for computing spectral properties of infinite-dimensional operators. We apply these techniques to develop efficient and accurate computational tools for computing the physical properties of TIs. These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinitedimensional operator directly, even in the presence of material defects and disorder. Our methods permit computation of spectra, approximate eigenstates, spectral measures, spectral projections, transport properties, and conductances. Numerical examples are given for the Haldane model, and the techniques can be extended similarly to other TIs in two and three dimensions.

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Section: Edge States Edge Conductance and Edge Wave-packetsmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

Colbrook¹,

Horning²,

Thicke³

et al. 2021

Preprint

Self Cite

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show abstract

“…This fact is problematic from the experimental point of view but also for numerical purposes to actually compute edge indices on a finite sample. In the last decades, several strategy have been followed to compute a numerical estimate of bulk or edge indices in finite samples [6,25,26,21,22,32,24].…”

Section: Introductionmentioning

confidence: 99%

Estimating bulk and edge topological indices in finite open chiral chains

Jezequel,

Tauber,

Delplace

2022

Preprint

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We develop a formalism to extend, simultaneously, the usual definition of bulk and edge indices from topological insulators to the case of a finite sample with open boundary conditions, and provide a physical interpretation of these quantities. We then show that they converge exponentially fast to an integer value when we increase the system size, and also that bulk and edge quantities coincide at finite size. The theorem applies to any non-homogeneous system such as disordered or defect configurations. We focus on one-dimensional chains with chiral symmetry, such as the Su-Schrieffer-Heeger model, but the proof actually only requires the Hamiltonian to be short-range and with a spectral gap in the bulk. The definition of bulk and edge indices relies on a finite-size version of the switch-function formalism where the Fermi projector is smoothed in energy using a carefully chosen regularization parameter.

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“…Whereas in an eigenvalue problem, (X j − λ j )v = 0, λ j is something that is calculated using a known operator (i.e., λ j is a dependent variable), in a composite operator λ is better thought of as an input (i.e., as a set of independent variables). This is because, no matter how close the localizer is to being singular, its spectrum contains valuable information on the durability of edge modes [34] or on the number of Dirac or Weyl points [40].…”

Section: Introductionmentioning

confidence: 99%