Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices

Gomi, Kiyonori; Thiang, Guo Chuan

doi:10.1007/s11005-018-1129-1

Cited by 8 publications

(21 citation statements)

References 50 publications

(97 reference statements)

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“…The linear space of boundary zero modes should therefore host a graded representation of the boundary symmetries. This expectation was verified in [22] through a new mod 2 index theorem (valued in K graded • (C * r (p11g))) which counts the "glide zero modes" that appear along a boundary of a 2D topological insulator with glide reflection symmetry. More generally, the framework of graded groups and graded equivariant twistings of K-theory allows us to formulate appropriate "super-indices" for exotic topological boundary modes arising in crystalline topological phases.…”

mentioning

confidence: 85%

“…A second "partial Fourier transform" producesT 2 on the right-hand-side of Eq. (22), showing that T G factorises into partial T-dualities. In these cases, the factorisation is essentially a combination of the circle bundle T-dualities in Propositions 5 .3, 5.4, 5.7.…”

Section: D Crystallographic T-dualitiesmentioning

confidence: 99%

“…In this section, we apply crystallographic T-duality, T G of Eq. (22), to the 17 wallpaper groups. When the point group G is Z 2 , T 2 G fibres as a 'Real' or Z 2 -equivariant circle bundle over another circle (or even both), so that we can T-dualise only the fibre circle while keeping the base circle fixed.…”

Section: D Crystallographic T-dualitiesmentioning

confidence: 99%

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Crystallographic T-duality

Gomi

Thiang

2019

Journal of Geometry and Physics

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We introduce the notion of crystallographic T-duality, inspired by the appearance of K-theory with graded equivariant twists in the study of topological crystalline materials. Besides giving a range of new topological T-dualities, it also unifies many previously known dualities, motivates generalisations of the Baum-Connes conjecture to graded groups, provides a powerful tool for computing topological phase classification groups, and facilitates the understanding of crystallographic bulk-boundary correspondences in physics.

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confidence: 85%

Section: D Crystallographic T-dualitiesmentioning

confidence: 99%

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Crystallographic T-duality

Gomi

Thiang

2019

Journal of Geometry and Physics

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“…Let us briefly note that while any crystallographic group G ⊂ R d is a Delone set and our choice of boundary is quite general, the factorisation and bulk-boundary result in Theorem 3.5 is too coarse to detect boundary indices derived from the crystalline structure as in [38].…”

Section: The Bulk-boundary Extensionmentioning

confidence: 99%

Index Theory and Topological Phases of Aperiodic Lattices

Bourne

Mesland

2019

Ann. Henri Poincaré

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We examine the noncommutative index theory associated to the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials, and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

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“…As in the Su-Schrieffer-Heeger model [17], the bulk topological invariant of class AIII in 1D is the integer winding number over the 1D Brillouin zone. Its nonzero value guarantees topologically protected dangling edge modes [18,19] even in the presence of disorder [20,21]. For dipolar spin waves, each subsystem gives winding number ±1 which remains constant as the slice is varied, unless a vortex line is crossed, forcing a discontinuous jump by 2.…”

Section: Mssws As a Realizationmentioning

confidence: 99%

Topological Characterization of Classical Waves: The Topological Origin of Magnetostatic Surface Spin Waves

et al. 2019

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We propose a topological characterization of Hamiltonians describing classical waves. Applying it to the magnetostatic surface spin waves that are important in spintronics applications, we settle the speculation over their topological origin. For a class of classical systems that includes spin waves driven by dipole-dipole interactions, we show that the topology is characterized by vortex lines in the Brillouin zone in such a way that the symplectic structure of Hamiltonian mechanics plays an essential role. We define winding numbers around these vortex lines and identify them to be the bulk topological invariants for a class of semimetals. Exploiting the bulk-edge correspondence appropriately reformulated for these classical waves, we predict that surface modes appear but not in a gap of the bulk frequency spectrum. This feature, consistent with the magnetostatic surface spin waves, indicates a broader realm of topological phases of matter beyond spectrally gapped ones.The principle of bulk-edge correspondence is a cornerstone in the field of topological phases of matter [1]: at the boundary of a system whose bulk frequency spectrum is topologically nontrivial, there should appear localized edge modes with eigenfrequencies in a gap of the bulk spectrum. This principle underlies the unconventional stability of chiral edge states in quantum Hall insulators [2] and Dirac surface states of topological insulators [3], and has more recently led to predictions of edge modes in various classical systems [4-6]. The bulk system topology is usually characterized by a topological invariant defined for Hamiltonians describing spatially unbounded systems with specified symmetry operations. It dictates the existence and number of topologically protected edge modes. The corresponding hallmark of these edge states is their robustness against symmetrypreserving perturbations.The insensitiveness of edge states to material parameters strikes a chord in the field of magnetism. Since their discovery in 1960 [7], ferromagnetic spin waves known as "magnetostatic surface waves" (MSSWs) have been a subject of various experimental and theoretical studies. These edge modes owe their intrinsic chiral structure to dipole-dipole interactions. MSSWs propagate perpendicular to the ordered magnetization regardless of the sample geometry, be it a slab [8] or a sphere [9]. They are known to be anomalously robust against back scatterings [10,11], hinting towards a topological origin. The chirality and robustness render them interesting for many fundamental studies, e.g. for non-reciprocal transport of spin [12] and heat [13]. Today, in the context of magnon spintronics [14], MSSWs are almost exclusively used in studies of spin-wave transport in microstructures since they offer the largest decay length of all available modes and are easily excited by the commonly used inductive microwave antennas. It is therefore of fundamental interest whether MSSWs are indeed topologically protected or not.In this Letter, we show that the bulk Hamiltonian of spin ...

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Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices

Cited by 8 publications

References 50 publications

Crystallographic T-duality

Crystallographic T-duality

Index Theory and Topological Phases of Aperiodic Lattices

Topological Characterization of Classical Waves: The Topological Origin of Magnetostatic Surface Spin Waves

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