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Topological Invariant and Topological Phases

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Topological States on Interfaces Protected by Symmetry

Part of the book series: Springer Theses ((Springer Theses))

Abstract

In this chapter, we review several topics for an introduction to topological insulators and the edge states resulting from the bulk-boundary correspondence. In condensed matter physics, researches in terms of topology have been intensively done after the discovery of the quantum Hall effect (von Klitzing et al., Phys Rev Lett 49:494, 1980, [1]). Systems are characterized topologically and topological classification is done in terms of symmetry. We first introduce the quantum Hall effect to explain a topological phase with time-reversal symmetry breaking. Next, we discuss topological insulators (Hasan and Kane, Rev Mod Phys 82:3045, 2010, [2]) as a natural extension of the quantum Hall system. The topological aspect of the quantum Hall effect teaches us the background of the quantized conductivity, and these basic notions can be applied to topological insulators. Finally, we introduce another kind of a topological material, which has flat-band states localized at the edge at the surface.

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Notes

  1. 1.

    We assume that the Berry curvature does not depend on time \(t\), because the Hall current is the lowest order in the time-dependent term, when we assume that the time dependence of the system is weak.

  2. 2.

    For a vector \(\mathrm {x}=x_{1}\mathbf {e}_{1}+x_{2}\mathbf {e_{2}}+x_{3}\mathbf {e_{3}}\) where \(\mathbf {e}_{i=1,2,3}\) are the basis vectors, the wedge product \(\wedge \) is defined as

    figure a
  3. 3.

    When the Landau level is filled, the electron density per Landau level is given as \(\frac{N_{0}}{L_xL_y}=\frac{eB}{h}=\frac{1}{2\pi \ell ^{2}}\).

  4. 4.

    In 1D the Z\(_2\) topological number cannot be defined. For example, by a gauge transformation \(|u\rangle \rightarrow \mathrm {e}^{ik/(2\varGamma )}|u\rangle \) where \(\varGamma \) is a half of the reciprocal lattice vector, the Z\(_2\) topological number changes as \((-1)^{v}\rightarrow -(-1)^{v}\).

  5. 5.

    We assume the tuning affect only the four bands around the Fermi energy.

  6. 6.

    For example \(\psi _{1}\) consists of s-orbitals, and \(\psi _{2}\) consists of p-orbitals.

  7. 7.

    The Hamiltonian (Eq. 2.68) is transformed into \(v(\varvec{\sigma }\cdot \mathbf {k})\) by the unitary transformation, which is known as the massless Dirac Hamiltonian. Therefore we call Eq. (2.68) as the Dirac Hamiltonian.

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  1. Department of Applied Physics, The University of Tokyo, Tokyo, Japan

    Ryuji Takahashi

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  1. Ryuji Takahashi
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Correspondence to Ryuji Takahashi .

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Takahashi, R. (2015). Topological Invariant and Topological Phases. In: Topological States on Interfaces Protected by Symmetry. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55534-6_2

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