Index of Dirac operators and classification of topological insulators

Ertem, Ümit

doi:10.1088/2399-6528/aa8ab7

Cited by 5 publications

(15 citation statements)

References 29 publications

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“…The central part of this paper is concerned with the construction of a concrete example of this general formalism describing the parity anomaly in odd spacetime dimensions. As the parity anomaly is related to an index in one higher dimension [NS83, AGDPM85,Wit16a], this suggests that quantum field theories with parity anomaly should take values in an extended field theory constructed from index theory; this naturally fits in with the classification of topological insulators and superconductors using index theory and K-theory, see [Ert17] for a recent exposition of this. We build such a theory using the index theory for manifolds with corners developed in [LM02,Loy04], which extends the well-known Atiyah-Patodi-Singer index theorem [APS75] to manifolds with corners of codimension 2.…”

Section: Summary Of Resultsmentioning

confidence: 89%

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

Müller

Szabo

2018

Commun. Math. Phys.

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We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general background fields in a stack, and together with the theory of symmetric monoidal bicategories we use it to provide the concrete forms of invertible extended quantum field theories which capture anomalies in both the path integral and Hamiltonian frameworks. Specialising this situation by using the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity anomaly. We compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space, which is defined in a parity-symmetric way by suitably augmenting the standard chiral fermionic Fock spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that naturally appear in the index theorem. We describe the significance of our constructions for the bulkboundary correspondence in a large class of time-reversal invariant gauge-gravity symmetryprotected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3 + 1 dimensions. • Dai-Freed theories describing chiral anomalies in odd dimensions d have been sketched in [Mon15]. They are an extended version of the field theories constructed in [DF95].• Wess-Zumino theories describing the anomaly in self-dual field theories have been constructed in [Mon15].• The theory describing the anomaly of the worldvolume theory of M5-branes has been constructed as an unextended quantum field theory in [Mon17].

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Section: Summary Of Resultsmentioning

confidence: 89%

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

Müller

Szabo

2018

Commun. Math. Phys.

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“…In fact, these topological invariants correspond to the indices of Dirac operators corresponding to the Dirac Hamiltonians of topological materials [5].…”

Section: Topological Materialsmentioning

confidence: 99%

“…There are two types of topological insulators and superconductors that are Z-insulators and Z 2 -insulators and the topological invariants corresponding to these classes take values in Z and Z 2 groups which are described by Chern numbers or winding numbers and Kane-Mele invariants or Chern-Simons invariants, respectively [3,4]. In fact, all these topological invariants correspond to the indices of Dirac operators defined from the Dirac Hamiltonians of topological materials [5]. Classification topological insulators and superconductors can be maintained by using the Bott periodicity of K-theory groups and Clifford algebras which results to a periodic table of topological insulators and superconductors for different symmetry classes and dimensions [6,7].…”

Section: Introductionmentioning

confidence: 99%

“…Classification topological insulators and superconductors can be maintained by using the Bott periodicity of K-theory groups and Clifford algebras which results to a periodic table of topological insulators and superconductors for different symmetry classes and dimensions [6,7]. The periodic table of topological insulators and superconductors can also be obtained from the Clifford chessboard of Clifford algebras and the index theorems of Dirac operators [5].…”

Section: Introductionmentioning

confidence: 99%

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Weyl semimetals and spin$^c$ cobordism

Ertem¹

2020

Preprint

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Classification of topological insulators and superconductors is manifested in terms of spin cobordism groups for lower dimensions. It is discussed that the periodic table of topological insulators is a result of the possible choices of spin structures on Brillouin zones of relevant topological materials. This framework is extended to the case of Weyl semimetals. It is shown that the classification of Weyl semimetals can be managed in terms of spin c cobordism groups via the extension of spin structures to spin c structures. Topological invariants of Weyl semimetals are connected to the topological invariants of spin c cobordism groups and Fermi arcs of Weyl semimetals are interpreted in terms of the choices of spin c structures.

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“…Kane-Mele invariants for semimetals have also been discussed in [71]. A variety of mathematical approaches to the Kane-Mele invariant are known: they range from algebraic topology perspectives [27,59] over index theory [19,54] and C * -algebras [56,57] to geometric approaches [31][32][33]. An overview relating various different treatments of the Kane-Mele invariant is provided by [46].…”

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confidence: 99%

Topological insulators and the Kane–Mele invariant: Obstruction and localization theory

Bunk

Szabo

2019

Rev. Math. Phys.

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We present homotopy theoretic and geometric interpretations of the Kane–Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the sewing matrix up to non-equivariant homotopy. We prove a Mayer–Vietoris Theorem for manifolds with [Formula: see text]-actions which intertwines Real and [Formula: see text]-equivariant de Rham cohomology groups, and apply it to derive a new localization formula for the Kane–Mele invariant. This provides a unified cohomological explanation for the equivalence between the discrete Pfaffian formula and the known local geometric computations of the index for periodic lattice systems. We build on the relation between the Kane–Mele invariant and the theory of bundle gerbes with [Formula: see text]-actions to obtain geometric refinements of this obstruction and localization technique. In the preliminary part we review the Freed–Moore theory of band insulators on Galilean spacetimes with emphasis on geometric constructions, and present a bottom-up approach to time-reversal symmetric topological phases.

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Index of Dirac operators and classification of topological insulators

Cited by 5 publications

References 29 publications

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

Weyl semimetals and spin$^c$ cobordism

Topological insulators and the Kane–Mele invariant: Obstruction and localization theory

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