Computing Edge States without Hard Truncation

Thicke, Kyle; Watson, Alexander B.; Lu, Jianfeng

doi:10.1137/19m1282696

Cited by 11 publications

(8 citation statements)

References 53 publications

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“…This is an example of the general phenomenon known as 'spectral pollution', where eigenvalues of finite discretisations/truncations can cluster in gaps between the essential spectrum of infinite self-adjoint operators [37,73,83] as the truncation size increases. In the context of TIs, which necessarily have eigenstates localised at edges, spectral pollution arising from the new edges created by the truncation is inevitable, see [101].…”

Section: Rectangular As Opposed To Square Truncationsmentioning

confidence: 99%

“…It is common in the physics literature to accept the additional spectrum arising from the truncation away from the edge and simply treat the system as having a second edge. However, the spectrum of the truncated Hamiltonian is often clearly different from that of the semi-infinite operator: see, for example, Lee-Thorp [71], in particular Figure 26.7, and compare with Figure 5.3 of [101].…”

Section: Rectangular As Opposed To Square Truncationsmentioning

confidence: 99%

“…In any case, it cannot be used in the direction transverse to an edge. At the same time, artificial truncation leads immediately to spectral pollution and spurious edge states [101].…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, we adopt a "solve-then-discretise" approach. The "solve-then-discretise" paradigm has also recently been applied to other spectral problems [64,101,107], extensions of classical methods such as the QL and QR algorithms [30,106], Krylov methods [52], and the computation of resonances [8,9]. We are concerned with the following four types of spectral computations for infinite-dimensional operators:…”

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: Rectangular As Opposed To Square Truncationsmentioning

confidence: 99%

Section: Rectangular As Opposed To Square Truncationsmentioning

confidence: 99%

“…In any case, it cannot be used in the direction transverse to an edge. At the same time, artificial truncation leads immediately to spectral pollution and spurious edge states [101].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…For an analysis of the propagation of wave packets along curved interfaces in a Dirac model, see [7]. Eigenvalues and edge states for discrete and continuous models are also computed numerically in [44] using a method that avoids artificial Dirichlet boundary conditions by utilizing the resolvent of the Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Approximations of interface topological invariants

Solomon¹,

Bal²

2021

Preprint

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This paper concerns continuous models for two-dimensional topological insulators and superconductors. Such systems are characterized by asymmetric transport along a 1-dimensional curve representing the interface between two insulating materials. The asymmetric transport is quantified by an interface conductivity. Our first objective is to prove that the conductivity is quantized and stable with respect to a large class of perturbations, and to relate it to an integral involving the symbol of the system's Hamiltonian; this is a bulk-interface correspondence.Second, we define a modified interface conductivity in a periodic setting, and show that it is asymptotically robust despite lacking the topological nature of its infinite domain analogue. This periodization allows for numerical evaluations of the conductivity. We demonstrate on a class of differential systems that the periodic interface conductivity is a good approximation for the infinite domain conductivity.To illustrate the theoretical results, we present numerical simulations that approximate the (infinite domain) conductivity with high accuracy, and show that it is approximately quantized even in the presence of perturbations.

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