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

Journal Article · Fri Jul 15 00:00:00 EDT 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2010.12.013· OSTI ID:21583306

Loring, Terry A ^[1]

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1. Department of Physics, Duke University, Durham, NC 27708 (United States)

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.

• OSTI ID: 21583306
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Journal Issue: 7 Vol. 326; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
ALGORITHMS
ANGULAR MOMENTUM
CHIRAL SYMMETRY
DIMENSIONLESS NUMBERS
HAMILTONIANS
MATHEMATICAL LOGIC
MATHEMATICAL OPERATORS
MATHEMATICS
MATRICES
ORDER PARAMETERS
PARTICLE PROPERTIES
PHASE TRANSFORMATIONS
QUANTUM OPERATORS
SCATTERING
SPIN
SYMMETRY
THREE-DIMENSIONAL CALCULATIONS
TOPOLOGY
TWO-DIMENSIONAL CALCULATIONS