Topological insulators and C*algebras: Theory and numerical practice
Abstract
Research Highlights: > We classify topological insulators using C* algebras. > We present new Ktheory invariants. > We develop efficient numerical algorithms based on this technique. > We observe unexpected quantum phase transitions using our algorithm.  Abstract: We apply ideas from C*algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed Ktheory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show nonexistence of localized Wannier functions for these systems. We use this approach to calculate the index for timereversal invariant systems with spinorbit scattering in three dimensions, on sizes up to 12{sup 3}, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which canmore »
 Authors:

 Department of Physics, Duke University, Durham, NC 27708 (United States)
 Publication Date:
 OSTI Identifier:
 21583306
 Resource Type:
 Journal Article
 Journal Name:
 Annals of Physics (New York)
 Additional Journal Information:
 Journal Volume: 326; Journal Issue: 7; Other Information: DOI: 10.1016/j.aop.2010.12.013; PII: S00034916(10)002277; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 00034916
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; ALGORITHMS; CHIRAL SYMMETRY; HAMILTONIANS; MATRICES; ORDER PARAMETERS; PHASE TRANSFORMATIONS; SCATTERING; SPIN; THREEDIMENSIONAL CALCULATIONS; TOPOLOGY; TWODIMENSIONAL CALCULATIONS; ANGULAR MOMENTUM; DIMENSIONLESS NUMBERS; MATHEMATICAL LOGIC; MATHEMATICAL OPERATORS; MATHEMATICS; PARTICLE PROPERTIES; QUANTUM OPERATORS; SYMMETRY
Citation Formats
Hastings, Matthew B., Email: mahastin@microsoft.com, Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, Loring, Terry A, Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, and Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131. Topological insulators and C*algebras: Theory and numerical practice. United States: N. p., 2011.
Web. doi:10.1016/j.aop.2010.12.013.
Hastings, Matthew B., Email: mahastin@microsoft.com, Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, Loring, Terry A, Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, & Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131. Topological insulators and C*algebras: Theory and numerical practice. United States. doi:10.1016/j.aop.2010.12.013.
Hastings, Matthew B., Email: mahastin@microsoft.com, Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, Loring, Terry A, Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, and Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131. Fri .
"Topological insulators and C*algebras: Theory and numerical practice". United States. doi:10.1016/j.aop.2010.12.013.
@article{osti_21583306,
title = {Topological insulators and C*algebras: Theory and numerical practice},
author = {Hastings, Matthew B., Email: mahastin@microsoft.com and Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106 and Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131 and Loring, Terry A and Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106 and Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131},
abstractNote = {Research Highlights: > We classify topological insulators using C* algebras. > We present new Ktheory invariants. > We develop efficient numerical algorithms based on this technique. > We observe unexpected quantum phase transitions using our algorithm.  Abstract: We apply ideas from C*algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed Ktheory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show nonexistence of localized Wannier functions for these systems. We use this approach to calculate the index for timereversal invariant systems with spinorbit scattering in three dimensions, on sizes up to 12{sup 3}, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an 'order parameter' for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C*algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.},
doi = {10.1016/j.aop.2010.12.013},
journal = {Annals of Physics (New York)},
issn = {00034916},
number = 7,
volume = 326,
place = {United States},
year = {2011},
month = {7}
}