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Register a new userBulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the Z2-invariant in topological insulators
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We study the bulk-edge correspondence in topological insulators by taking Fu-Kane spin pumping model as an example. We show that the Kane-Mele invariant in this model is Z2 invariant modulo the spectral flow of a single-parameter family of 1+1-dimensional Dirac operators with a global boundary condition induced by the Kramers degeneracy of the system. This spectral flow is defined as an integer which counts the difference between the number of eigenvalues of the Dirac operator family that flow from negative to non-negative and the number of eigenvalues that flow from non-negative to negative. Since the bulk states of the insulator are completely gapped and the ground state is assumed being no more degenerate except the Kramers, they do not contribute to the spectral flow and only edge states contribute to. The parity of the number of the Kramers pairs of gapless edge states is exactly the same as that of the spectral flow. This reveals the origin of the edge-bulk correspondence, i.e., why the edge states can be used to characterize the topological insulators. Furthermore, the spectral flow is related to the reduced eta-invariant and thus counts both the discrete ground state degeneracy and the continuous gapless excitations, which distinguishes the topological insulator from the conventional band insulator even if the edge states open a gap due to a strong interaction between edge modes. We emphasize that these results are also valid even for a weak disordered and/or weak interacting system. The higher spectral flow to categorize the higher-dimensional topological insulators are expected.
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We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Kreiger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras.
We present an exact solution of a modifed Dirac equation for topological insulator in the presence of a hole or vacancy to demonstrate that vacancies may induce bound states in the band gap of topological insulators. They arise due to the Z_2 classification of time-reversal invariant insulators, thus are also topologically-protected like the edge states in the quantum spin Hall effect and the surface states in three-dimensional topological insulators. Coexistence of the in-gap bound states and the edge or surface states in topological insulators suggests that imperfections may affect transport properties of topological insulators via additional bound states near the system boundary.
We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the $eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion, we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $eta$ invariant comes entirely from the massless edge-localized modes.
The bulk-edge correspondence (BEC) refers to a one-to-one relation between the bulk and edge properties ubiquitous in topologically nontrivial systems. Depending on the setup, BEC manifests in different forms and govern the spectral and transport properties of topological insulators and semimetals. Although the topological pump is theoretically old, BEC in the pump has been established just recently [1] motivated by the state-of-the-art experiments using cold atoms [2,3]. The center of mass (CM) of a system with boundaries shows a sequence of quantized jumps in the adiabatic limit associated with the edge states. Although the bulk is adiabatic, the edge is inevitably non-adiabatic in the experimental setup or in any numerical simulations. Still the pumped charge is quantized and carried by the bulk. Its quantization is guaranteed by a compensation between the bulk and edges. We show that in the presence of disorder the pumped charge continues to be quantized despite the appearance of non-quantized jumps.
The bulk-boundary correspondence is a generic feature of topological states of matter, reflecting the intrinsic relation between topological bulk and boundary states. For example, robust edge states propagate along the edges and corner states gather at corners in the two-dimensional first-order and second-order topological insulators, respectively. Here, we report two kinds of topological states hosting anomalous bulk-boundary correspondence in the extended two-dimensional dimerized lattice with staggered flux threading. At 1/2-filling, we observe isolated corner states with no fractional charge as well as metallic near-edge states in the C = 2 Chern insulator states. At 1/4-filling, we find a C = 0 topologically nontrivial state, where the robust edge states are well localized along edges but bypass corners. These robust topological insulating states significantly differ from both conventional Chern insulators and usual high-order topological insulators.
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