Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory
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Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory Eli Fonseca1 · Jacob Shapiro1 Kohtaro Yamakawa1
· Ahmed Sheta1 · Angela Wang1 ·
Received: 26 January 2020 / Accepted: 16 April 2020 / © Springer Nature B.V. 2020
Abstract We study spinful non-interacting electrons moving in two-dimensional materials which exhibit a spectral gap about the Fermi energy as well as time-reversal invariance. Using Fredholm theory we revisit the (known) bulk topological invariant, define a new one for the edge, and show their equivalence (the bulk-edge correspondence) via homotopy. Keywords Topological insulators · Fredholm theory · Z2 invariants · Time-reversal Mathematics Subject Classification (2010) 47A53 · 35Q99
1 Introduction Insulators in two space dimensions obeying fermionic time-reversal symmetry [1, Class AII] have two distinct topological phases [2–4]. The fact that there exists a non-trivial phase for such systems had not been immediately clear since the integer quantum Hall effect (IQHE) [5] is always trivial in the presence of time-reversal symmetry. In pioneering studies on Hall fluids in the early 1990s Fr¨ohlich and Studer [6] discovered that despite time-reversal symmetry, such systems may exhibit non-trivial effects; they used Chern-Simons effective field theory. Much later, this non-triviality was rediscovered in [7–9], now from the perspective of single-particle translation invariant Hamiltonians, and experimental investigations [10–14] followed. Physically, the topological non-triviality of such systems is associated with an unpaired state at the boundary of the sample. Via the discovery of [15] there came an association with the field of algebraic topology. Mathematically, in the presence of translation invariance for bulk systems, Jacob Shapiro
[email protected] 1
Mathematics Department, Columbia University, New York, NY 10027, USA
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bulk insulators have associated with them a C-vector-bundle over Td , the Brillouin zone. The topological classes of such vector bundles are studied via the well-known Chern characteristic classes [16]. In the case d = 2 one obtains an isomorphism of all classes with Z via the Chern number. In the presence of time-reversal symmetry, as mentioned already, the Chern number is always zero. However, time-reversal which squares to −1 defines a quaternionic structure on such vector bundles where now the Pontryagin classes may be non-trivial [17, 18]. The breakthrough study of [19] allowed one to do away with the assumption of translation invariance, which was crucial to explain the IQHE; one uses K-theory of C∗ -algebras as the main algebraic tool, and index theorems relating physical quantities to indices of Fredholm operators guarantee topological properties. A major theme in the study of topological insulators is the bulk-edge correspondence: the fact that topological invariants computed for an infinite bulk system agree with those computed from that system truncated to the half-infinite
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