Differential Topology of Semimetals

Mathai, Varghese; Thiang, Guo Chuan

doi:10.1007/s00220-017-2965-z

Cited by 43 publications

(68 citation statements)

References 96 publications

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“…Let us note that this is a generalization of the Poincaré-Hopf theorem [53][54][55], which relates zeros of a tangent vector field to the Euler characteristic of the manifold, to rank two real Bloch bundles (i.e., two real Bloch states).…”

Section: B Winding Number and The Euler Classmentioning

confidence: 99%

Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle

2019

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We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two dimensional real fermions in systems with space-time inversion IST symmetry. The Euler class is an integer topological invariant classifying real two band systems. We show that a two-band system with a nonzero Euler class cannot have an IST -symmetric Wannier representation. Moreover, a two-band system with the Euler class e2 has band crossing points whose total winding number is equal to −2e2. Thus the conventional Nielsen-Ninomiya theorem fails in systems with a nonzero Euler class. We propose that the topological phase transition between two insulators carrying distinct Euler classes can be described in terms of the pair creation and annihilation of vortices accompanied by winding number changes across Dirac strings. When the number of bands is bigger than two, there is a Z2 topological invariant classifying the band topology, that is, the second Stiefel Whitney class (w2). Two bands with an even (odd) Euler class turn into a system with w2 = 0 (w2 = 1) when additional trivial bands are added. Although the nontrivial second Stiefel-Whitney class remains robust against adding trivial bands, it does not impose a Wannier obstruction when the number of bands is bigger than two. However, when the resulting multi-band system with the nontrivial second Stiefel-Whitney class is supplemented by additional chiral symmetry, a nontrivial second-order topology and the associated corner charges are guaranteed. * These authors contributed equally to this work. † bjyang@snu.ac.kr the U v (1) valley and the space-time inversion C 2z T symmetries, where C 2z denotes a two-fold rotation about the z-axis and T is time-reversal symmetry [29,30]. In the presence of U v (1) and C 2z T symmetries, the four nearly flat bands are decoupled into two independent valley-filtered two-band systems, and each two-band system possesses Dirac points at K and K . The fact that both the valley charge conservation and C 2z T symmetries are not the exact symmetry of the TBG indicates that the symmetry of the low energy physics is larger than the exact lattice symmetry [29].Interestingly, by putting together all the emergent symmetries including U v (1) and C 2z T symmetries, Po et al. have found an obstruction to constructing well-localized Wannier functions describing the four nearly flat bands in TBG [29,30]. Moreover, it has been shown that the obstruction originates from the fact that the two Dirac points in each valleyfiltered two-band system have the same winding number, which is generally not allowed in 2D periodic systems due to the Nielsen-Ninomiya theorem [41]. In addition, based on the observation that the winding number is defined only for a two-band model in each valley, it is conjectured that the Wannier obstruction is fragile [29,3...

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Section: B Winding Number and The Euler Classmentioning

confidence: 99%

Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle

2019

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“…There have been a lot of efforts to classify such bulk gapless topological phases. 30,35,36,69 Whereas the bulk gapless phases resemble to gapless boundary and defect modes in TCIs and TCSCs, their theoretical treatment is different from that of the latter: While the topological structure of the latter can be examined by a bulk Hamiltonian flattened in the entire BZ, that of the former cannot be, since the information on the band touching structure is obviously lost by the flattening. Therefore, one needs a different approach to characterize gapless topological phases in the K-theory formulation.…”

Section: A Formulation By K-theorymentioning

confidence: 99%

Topological crystalline materials: General formulation, module structure, and wallpaper groups

2017

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We formulate topological crystalline materials on the basis of the twisted equivariant K-theory. Basic ideas of the twisted equivariant K-theory are explained with application to topological phases protected by crystalline symmetries in mind, and systematic methods of topological classification for crystalline materials are presented. Our formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states, as well as bulk gapless topological materials such as Weyl and Dirac semimetals, and nodal superconductors. As an application of our formulation, we present a complete classification of topological crystalline surface states, in the absence of time-reversal invariance. The classification works for gapless surface states of three-dimensional insulators, as well as full gapped two-dimensional insulators. Such surface states and two-dimensional insulators are classified in a unified way by 17 wallpaper groups, together with the presence or the absence of (sublattice) chiral symmetry. We identify the topological numbers and their representations under the wallpaper group operation. We also exemplify the usefulness of our formulation in the classification of bulk gapless phases. We present a new class of Weyl semimetals and Weyl superconductors that are topologically protected by inversion symmetry. CONTENTS

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“…and measuring the differential integrated rate [33] ∆Γ int = ∞ 0 dω Γ + (ω) − Γ − (ω) = (8πE 2 )g µν (q 0 ), where Γ ± refer to the excitation rates resulting from the drives q ± ν (t) in Eq. (35). Such a protocol could be applied to the general threelevel Hamiltonian in Eq.…”

Section: Measuring a Tensor Monopole In A Three-level Systemmentioning

confidence: 99%

“…[33] requires driving a set of parameters {q µ (t), q ν (t)} periodically in time, which can be realized in this setting by modulating the real and imaginary parts of the Rabi amplitudes Ω 12 and Ω 23 in a proper manner; see Eqs. (33) and (35). Measuring the resulting excitation rates Γ(ω), for a proper range of drive frequencies ω, would then provide the different components of the quantum metric tensor, and hence, reveal the tensor monopole associated with this 4D model (see main text).…”

Section: Measuring a Tensor Monopole In A Three-level Systemmentioning

confidence: 99%

Revealing Tensor Monopoles through Quantum-Metric Measurements

Palumbo

Goldman

2018

Phys. Rev. Lett.

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Monopoles are intriguing topological objects, which play a central role in gauge theories and topological states of matter. While conventional monopoles are found in odd-dimensional flat spaces, such as the Dirac monopole in three dimensions and the non-Abelian Yang monopole in five dimensions, more exotic objects were predicted to exist in even dimensions. This is the case of "tensor monopoles", which are associated with generalized (tensor) gauge fields, and which can be defined in four dimensional flat spaces. In this work, we investigate the possibility of creating and measuring such a tensor monopole, by introducing a realistic three-band model defined over a four-dimensional parameter space. Our probing method is based on the observation that the topological charge of this tensor monopole, which we relate to a generalized Berry curvature, can be directly extracted from the quantum metric. We propose a realistic three-level atomic system, where tensor monopoles could be generated and revealed through quantum-metric measurements. arXiv:1805.01247v2 [cond-mat.mes-hall]

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Differential Topology of Semimetals

Cited by 43 publications

References 96 publications

Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle

Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle

Topological crystalline materials: General formulation, module structure, and wallpaper groups

Revealing Tensor Monopoles through Quantum-Metric Measurements

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