Topological defects and gapless modes in insulators and superconductors

Abstract

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 H(k,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. 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 H͑k , r͒ 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. Disciplines Physical Sciences and Mathematics | Physics

Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band insulators

“…In contrast, in a weak phase characterized by the weak topological index M, protected helical modes are obtained only when the product M · b(mod 2π ) is nontrivial [3,[17][18][19]. This is consistent with the fact that double pairs of modes originating from two K inv not related by symmetry may be gapped out.…”

“…In two dimensions (2D), the role of these lattice defects has been recently elucidated in TBIs [10,11], as well as in topological superconductors [12,13] and interacting topological states [14][15][16]. In particular, it has been shown that these lattice defects in two-dimensional TBIs act as probes of distinct topological states through binding of the localized zero-energy modes [10].Although early on it was identified that in three-dimensional TBIs dislocation lines support propagating helical modes [17], the precise role of dislocations has not been explored thoroughly [18][19][20]. In particular, the relation between the lattice symmetry and the electronic topology, as well as the characterization of these topological states through the response of the dislocation lines, still needs to be addressed.…”

“…Although early on it was identified that in three-dimensional TBIs dislocation lines support propagating helical modes [17], the precise role of dislocations has not been explored thoroughly [18][19][20]. In particular, the relation between the lattice symmetry and the electronic topology, as well as the characterization of these topological states through the response of the dislocation lines, still needs to be addressed.…”

Ripple-modulated electronic structure of a 3D topological insulator

“…An intriguing consequence of our observations is that in a magnetic field, such periodic chemical potential variations can play host to 1D chiral metallic modes. Analogous to edge states in an integer quantum Hall system, 1D modes are predicted at the spatial boundaries between two regions with successive half-integer fillings 2,6 . These modes are interesting as practical realizations of 1D quantum wire states and can potentially be used to achieve dissipationless transport in 1D (Fig.…”

“…T he helical Dirac fermions that exist on the surface of topological insulators (TIs) are condensed matter analogues of relativistic fermions extensively studied in high-energy physics [1][2][3][4][5][6] . Many theoretical scenarios in high-energy physics such as Majorana fermions and axion electrodynamics are also anticipated to manifest experimental signatures in TIs 2, 7-9 .…”