The topology of mobility-gapped insulators
- PDF / 392,997 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 48 Downloads / 151 Views
The topology of mobility-gapped insulators Jacob Shapiro1 Received: 27 November 2018 / Revised: 15 December 2019 / Accepted: 8 July 2020 © Springer Nature B.V. 2020
Abstract Studying deterministic operators, we define a topology on the space of mobilitygapped insulators such that topological invariants are continuous maps into discrete spaces, we prove that this is indeed the case for the integer quantum Hall effect, and lastly we show why our “insulator” condition makes sense from the point of view of the localization theory using the fractional moments method. Keywords Topological insulators · Strong disorder · Integer quantum Hall effect · Random Schrödinger operators · Mobility gap Mathematics Subject Classification 82C10 · 37B25 · 37H05
1 Introduction Topological insulators [18] are usually studied in physics by assuming translation invariance, which allows for a topological description of Hamiltonians in terms of continuous maps from the Brillouin torus Td → X where X is some smooth manifold which depends on the symmetry class under consideration (for example, for a system in class A (no symmetry) gapped after n levels, X = Gr n (C∞ ), the Grassmannian manifold). Such a description is extremely convenient because one may immediately apply classical results from algebraic topology, for example, that the set of homotopy classes [T2 → Gr n (C∞ )] is isomorphic to Z. This approach led to many classification results [26,27], which culminated in Kitaev’s periodic table of topological insulators [21], all the while ignoring the fact that the systems to be analyzed are actually not translation invariant, and in fact certain physical features of the phenomena demand strong disorder. Indeed, the plateaus of the integer quantum Hall effect (IQHE henceforth) are explained only when assuming the Fermi energy lies in a region of
B 1
Jacob Shapiro [email protected] Mathematics Department, Columbia University, New York, NY 10027, USA
123
J. Shapiro
localized states (the mobility gap regime) which cannot appear in a translation-invariant system. Hence, a physically more realistic description calls for understanding disordered systems in which Bloch decomposition cannot be used. This has been done for the IQHE in [6] by applying ideas from non-commutative geometry and later generalized to the Kitaev table in [7,22] (and references therein). One problem with the application of non-commutative geometry is that it still required the Fermi energy to be placed in a spectral gap, which is why [6] goes beyond the C*-algebra generated by continuous functions of the Hamiltonian by defining a so-called non-commutative Sobolev spaces. Such an approach still uses crucially the translation invariance of the system; in contrast to the studies in physics, however, translation is used in terms of the probability distributions defining a random model. That is, a whole statistical ensemble of models is considered simultaneously, proofs use the covariance property, and statements are made either almost surely or about disorder av
Data Loading...
