Topological Defects and Gapless Modes in Insulators and Superconductors

By J. C. Teo, C. L. Kane. 2010

598 citations recorded by the source corpus.

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians Hk,r that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/663f1263734cf815dbd87ac66a582e3f5f2a6c69

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Weyl and Dirac semimetals in three-dimensional solids (2017, background): “Then, following (Teo and Kane, 2010), who considered space dependent band structures H(k, r) to account for topological defects, the space coordinates can be treated as ‘negative’ dimensions.”
Topological Semimetals from First Principles (2018, background): “The Z2 invariant of Dirac nodal line semimetals given by the Berry phase formula can also be understood as the topological invariant characterizing one-parameter families of Hamiltonians in class AI (124).”
Topological photonics (2018, background): “…complete classification of noninteracting fermionic topological phases in any dimension based on the time-reversal, particle-hole, and chiral symmetries is known in the literature (Kitaev, Lebedev, and Feigelman, 2009; Schnyder et al., 2009; Ryu et al., 2010; Teo and Kane, 2010; Chiu et al., 2016).”
Topological insulators and superconductors (2010, background): “More generically, there is a one-to-one correspondence between topological insulators and robust gapless thories in one lower dimension(Freedmanet al., 2010; Kitaev, 2009; Teo and Kane, 2010)”
Topological Phases of Non-Hermitian Systems (2018, methodology): “Our guiding principles also enable a systematic application of the K theory [118], a technique widely used in classifying Hermitian topological systems [44,46,51], to the non-Hermitian AZ classes, leading to a complete classification in all spatial dimensions.”
Higher-order non-Hermitian skin effect (2020, background): “This can be understood in view of the topological invariant in momentum space and the adiabatic parameter by Teo and Kane [116].”
Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases (2018, methodology): “isomorphism to a real tenfold-way class, then using Teo and Kane’s dimension-raising isomorphisms κ and ρ [52], and then using the inverse of the isomorphism (B7).”
Search for Majorana Fermions in Superconductors (2011, background): “As the gap goes through zero, Majorana fermions emerge as zero-modes bound to magnetic or electrostatic defects [61, 62].”
Topological superconductors: a review (2016, background): “Consequently, the classification of the gapless modes on defects reduces to that of the δ-dimensional insulators/superconductors [59].”