Invariants of disordered semimetals via the spectral localizer

Abstract

The spectral localizer consists in placing the Hamiltonian in a Dirac trap. For topological insulators its spectral asymmetry is equal to the topological invariants, providing a highly efficient tool for numerical computation. Here this technique is extended to disordered semimetals and allows to access the number of Dirac or Weyl points as well as weak invariants. These latter invariants imply the existence of surface states.

Spectral localization for semimetals and Callias operators

“…This indicates that there is a rather weak dependence of the approximate kernel on the random potential, a fact that was already observed in the numerical study [47]. Hence the spectral localizer can safely be used to detect Weyl points even in a weakly disordered system, or inversely, one can use the approximate kernel dimension to define the number of Weyl points in such a system.…”

“…This allows to access the low-lying spectrum of L κ numerically. Such a numerical illustration of Theorem 1.2 for models in d = 2 and d = 3 was provided in [47]. Numerics also indicate that the low-lying eigenvalues are actually in a window much smaller than κ 2 3 .…”

“…Even though non-generic for d > 3, we only focus on ideal semimetals (basically because d = 2 and d = 3 are the cases of greatest physical relevance). Let us provide two concrete examples which were studied numerically in [47].…”

“…The main motivation of this paper is to provide the analytical justification for an earlier numerical study on the Weyl point count in semimetals via a spectral localizer [47]. More precisely, let H be a one-particle periodic tight-binding Hamiltonian describing an ideal semimetal in odd dimension d with I Weyl points, see Definition 1.1 below for details.…”

Revealing Topology in Metals using Experimental Protocols Inspired by K-Theory

“…The supplemental materials contain detailed mathematical discussions of the properties of the spectral localizer and its spectrum, as well as a validation of the chiral symmetry of our acoustic metamaterial. It contains citations to the references [48][49][50]. * wc327@njit.edu † awcerja@sandia.gov ‡ sc945@njit.edu § prodan@yu.edu ¶ loring@math.unm.edu * * cprodan@njit.edu…”