PDF Publication Title:
Text from PDF Page: 035
the pull-backs are given by L∗1Ω = 0, L∗2Ω = dθ/π. For f(t) = (R,Reit), it follows that μa f−4,0 =0, μa f0,4 =μa f−4,4 =1. π π π π Notes on topological insulators The spectral flow of a family of self-adjoint elliptic operators, in practice Dirac operators, Dt = D(L1(t),L2(t)),a ≤ t ≤ b is defined as usual by the net number of eigenvalues changing sign (+1 for increasing, −1 for decreasing) from a to b, denoted by sf(Dt). A second integer-valued analytical Maslov index is defined by the spectral flow of the family of Dirac operators associated with the pair f(t) = (L1(t), L2(t)), μa(f) := sf(D(L1(t),L2(t))), a ≤ t ≤ b. (4.8) Recall that D(R,Reiθ) has eigenvalues πl + θ for −π ≤ θ ≤ π. When θ approaches 0 from the left, the eta invariant has the limit lim η(D(L1,L2))= lim −1−2θ =−1. θ→0− θ→0− π 44 Meanwhile, we also have the right limit lim η(D(L1,L2))= lim 1−2θ =1. crossing 0, i.e. sf(D(L1(t),L2(t))) = 1[η(D(L1,L2),0+)−η(D(L1,L2),0−)] = 1. 2 The analytical Maslov index unifies the spectral flow and the eta invariant in the symplectic setting, that is, the spectral flow of a family of Dirac operators can be computed as the variation of eta invariants. To end this subsection, we define the edge Z2 index [10] by the Maslov index μ modulo two, i where the Fermi level is fixed as R = L(0) and one chiral edge state is assumed to be piecewisely modeled by a finite collection of 1d Lagrangian subspaces L(θi) ∈ Λ(1). The equivalence with the Kane–Mele invariant can be seen by expressing the edge Z2 index as the winding number of a closed unitary path, for more details see [10]. 4.2. Topological Z2 index In this subsection, we will discuss how to compute the Z2 invariant of a 3d topolog- ical insulator by the topological Z2 index. One starts out with a 4d time reversal invariant model and applies dimensional reduction to obtain a 3d Chern–Simons θ→0+ θ→0− π Immediately, the spectral flow is easily obtained from the jump of the eta invariant ν ≡ μ(R, L(θi)) (mod 2) (4.9) 1630003-35 Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
PDF Search Title:
Notes on topological insulatorsOriginal File Name Searched:
RMP.pdfDIY PDF Search: Google It | Yahoo | Bing
Sulfur Deposition on Carbon Nanofibers using Supercritical CO2 Sulfur Deposition on Carbon Nanofibers using Supercritical CO2. Gamma sulfur also known as mother of pearl sulfur and nacreous sulfur... More Info
CO2 Organic Rankine Cycle Experimenter Platform The supercritical CO2 phase change system is both a heat pump and organic rankine cycle which can be used for those purposes and as a supercritical extractor for advanced subcritical and supercritical extraction technology. Uses include producing nanoparticles, precious metal CO2 extraction, lithium battery recycling, and other applications... More Info
CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com (Standard Web Page)