Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases

Abstract

Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$$^*$$ ∗ -algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $$K$$ K -theory (or $$KR$$ KR -theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $$KR$$ KR -theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants.

Toeplitz Extensions in Noncommutative Topology and Mathematical Physics

“…The algebra C(S 1 ) admits a convenient representation on the Hilbert space L 2 (S 1 ) of square-integrable functions on S 1 . This Hilbert space is isomorphic to the Hilbert space of sequences 2 (Z), and the isomorphism is implemented by the discrete Fourier transform…”

“…Crossed product C * -algebras are also used to describe disordered systems. The recent paper [1] describes the bulk-boundary correspondence for disordered freefermion topological phases in terms of Van Daele K-theory for graded C * -algebras [49,50]. The relevant observable algebra is the crossed product of the algebra of continuous functions on a compact disorder space by the action of a lattice.…”

Classification of topological invariants related to corner states

“…(1) * 2n , and thus, C (a) τ = C (a) holds. When u = 1, we can take a = 1 and C (1) = I (1) in this case. Combined with this, the proof is completed once we have checked that ∂ For u = 1, we take a = 1 and C (1) = I (6) holds.…”

“…Bellissard-van Elst-Schulz-Baldes studied quantum Hall effects by means of noncommutative geometry [10,11], and Kellendonk-Richter-Schulz-Baldes went on to prove the bulk-boundary correspondence by using index theory for Toeplitz operators [40]. The study of topological insulators, especially regarding its classification and the bulk-boundary correspondence for each of the ten Altland-Zirnbauer classes by using K -theory and index theory, has been much developed [1,16,17,24,28,30,39,40,45,49,58,65,65]. In [33], three-dimensional (3-D) class A bulk periodic systems are studied on one piece of a lattice cut by two specific hyperplanes, which is a model for systems with corners.…”

Bulk-boundary-defect correspondence at disclinations in rotation-symmetric topological insulators and superconductors

“…In particular, a d-dimensional topological crystalline phase of order n hosts (d − n−1)-dimensional anomalous states at hinges or corners of the corresponding dimension. This correspondence between bulk topology and boundary anomaly is a fundamental aspect of topological insulators and superconductors [12,25,[28][29][30][31][32][33].…”