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Title: Topological insulators and C*-algebras: Theory and numerical practice

Abstract

Research Highlights: > We classify topological insulators using C* algebras. > We present new K-theory 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 K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit 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 » 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.« less

Authors:
 [1]
  1. 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: S0003-4916(10)00227-7; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0003-4916
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; THREE-DIMENSIONAL CALCULATIONS; TOPOLOGY; TWO-DIMENSIONAL CALCULATIONS; ANGULAR MOMENTUM; DIMENSIONLESS NUMBERS; MATHEMATICAL LOGIC; MATHEMATICAL OPERATORS; MATHEMATICS; PARTICLE PROPERTIES; QUANTUM OPERATORS; SYMMETRY

Citation Formats

Hastings, Matthew B., E-mail: 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., E-mail: 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., E-mail: 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., E-mail: 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 K-theory 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 K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit 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 = {0003-4916},
number = 7,
volume = 326,
place = {United States},
year = {2011},
month = {7}
}