Concise tables of James numbers and some homotopy of classical Lie groups and associated homogeneous spaces

Lundell, Albert T.

doi:10.1007/bfb0087515

Cited by 20 publications

(29 citation statements)

References 34 publications

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“…Here, n, > 0 indicate the number of occupied and unoccupied bands (n = whenever P or C is present). The next block of the table lists the large n, limit homotopy groups of MCL which exhibit the Bott periodicity π d+8 (MCL) = π d (MCL) [90] with the exception of π0(MCL) which counts the number of connected components of MCL. Note that in some cases we write 2Z instead of Z as the naturally formulated topological invariant takes even values, and we type 0 for a trivial (one-element) group.…”

Section: CLmentioning

confidence: 99%

“…I and II that for D = 3 in the large n, limit, doubly charged nodal lines appear in AZ+I classes AI and CI, and doubly charged nodal surfaces exist in classes BDI and D. More care is required if one studies few-band models not reaching the large n, limit of Ref. [90]. In that case, the homotopy groups may differ from those listed in Tab.…”

Section: Topological Chargesmentioning

confidence: 99%

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Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems

2017

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Weyl points in three spatial dimensions are characterized by a Z-valued charge -the Chern number -which makes them stable against a wide range of perturbations. A set of Weyl points can mutually annihilate only if their net charge vanishes, a property we refer to as robustness. While nodal loops are usually not robust in this sense, it has recently been shown using homotopy arguments that in the centrosymmetric extension of the AI symmetry class they nevertheless develop a Z2 charge analogous to the Chern number. Nodal loops carrying a non-trivial value of this Z2 charge are robust, i.e. they can be gapped out only by a pairwise annihilation and not on their own. As this is an additional charge independent of the Berry π-phase flowing along the band degeneracy, such nodal loops are, in fact, doubly charged. In this manuscript, we generalize the homotopy discussion to the centrosymmetric extensions of all Atland-Zirnbauer classes. We develop a taylored mathematical framework dubbed the AZ+I classification and show that in three spatial dimensions such robust and multiply charged nodes appear in four of such centrosymmetric extensions, namely AZ+I classes CI and AI lead to doubly charged nodal lines, while D and BDI support doubly charged nodal surfaces. We remark that no further crystalline symmetries apart from the spatial inversion are necessary for their stability. We provide a description of the corresponding topological charges, and develop simple tight-binding models of various semimetallic and superconducting phases that exhibit these nodes. We also indicate how the concept of robust and multiply charged nodes generalizes to other spatial dimensions.arXiv:1705.07126v2 [cond-mat.mes-hall]

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Section: CLmentioning

confidence: 99%

Section: Topological Chargesmentioning

confidence: 99%

Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems

2017

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“…The relevant homotopy groups were computed in [20]. See also [21] for a concise summary of the results used here, and in following sections. From the long exact sequence in homotopy (3.5) we have that…”

Section: Computations In Eight Dimensionsmentioning

confidence: 99%

8d gauge anomalies and the topological Green-Schwarz mechanism

García-Etxebarria

Hayashi

Ohmori

et al. 2017

J. High Energ. Phys.

115

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String theory provides us with 8d supersymmetric gauge theory with gauge algebras su(N ), so(2N ), sp(N ), e 6 , e 7 and e 8 , but no construction for so(2N +1), f 4 and g 2 is known. In this paper, we show that the theories for f 4 and so(2N +1) have a global gauge anomaly associated to π d=8 , while g 2 does not have it. We argue that the anomaly associated to π d in d-dimensional gauge theories cannot be canceled by topological degrees of freedom in general. We also show that the theories for sp(N ) have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the GreenSchwarz mechanism.

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“…Space M may allow non-trivial embeddings of S p that cannot be continuously deformed to a point (such as S 1 winding around a cylinder). This property is formalized by homotopy groups π p (M) [90][91][92][93][94].…”

Section: Local-in-k Symmetries Restrict Admissible Gappedmentioning

confidence: 99%

Conversion Rules for Weyl Points and Nodal Lines in Topological Media

2018

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According to a widely held paradigm, a pair of Weyl points with opposite chirality mutually annihilate when brought together. In contrast, we show that such a process is strictly forbidden for Weyl points related by a mirror symmetry, provided that an effective two-band description exists in terms of orbitals with opposite mirror eigenvalue. Instead, such a pair of Weyl points convert into a nodal loop inside a symmetric plane upon the collision. Similar constraints are identified for systems with multiple mirrors, facilitating previously unreported nodal-line and nodal-chain semimetals that exhibit both Fermi-arc and drumhead surface states. We further find that Weyl points in systems symmetric under a π rotation composed with time reversal are characterized by an additional integer charge that we call helicity. A pair of Weyl points with opposite chirality can annihilate only if their helicities also cancel out. We base our predictions on topological crystalline invariants derived from relative homotopy theory, and we test our predictions on simple tight-binding models. The outlined homotopy description can be directly generalized to systems with multiple bands and other choices of symmetry.

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Concise tables of James numbers and some homotopy of classical Lie groups and associated homogeneous spaces

Cited by 20 publications

References 34 publications

Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems

Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems

8d gauge anomalies and the topological Green-Schwarz mechanism

Conversion Rules for Weyl Points and Nodal Lines in Topological Media

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