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Theoretical Study on Correlation Effects in Topological Matter

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

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Abstract

To begin with the thesis, which focuses on how electron interactions play in topological matters, we first look at the stage; we review the basic idea of topological electronic states and their material realizations, after giving the scope and the outline of the thesis. Starting from the celebrated quantum Hall effect, we introduce the basic notion of the topology of electron wave functions. This evolves to the important discovery in the recent decade of topological insulators. A characteristic feature of topological insulators is the gapless linear dispersion on the surface. It is described by the Weyl or Dirac equation, originated in the high-energy physics. Not only on a surface, such linear energy dispersion can emerge in a bulk; those materials are dubbed as Weyl or Dirac semimetals, whose band crossings are protected by the topological characters of the electron wave function. The protection of the topological properties is dependent on the symmetry of systems. We give the classification of topological electronic states for noninteracting cases, and also state on the realization of topological states driven by electron correlations.

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Notes

  1. 1.

    An axion insulator is an analog of a time-reversal invariant strong TI, but it is protected solely by inversion symmetry. Since inversion symmetry is absent on the surface, protected surface modes are not guaranteed in general [46]. However we can observe the quantized magnetoelectric effect \( \varvec{P} = (\theta e^2/2\pi h) \varvec{B} \) with \(\theta = \pi \).

  2. 2.

    There are a single time-reversal operator \(\mathcal {T}\) and a single particle-hole operator \(\mathcal {C}\). Assuming two particle-hole operators \(\mathcal {C}_1\) and \(\mathcal {C}_2\), their product \(\mathcal {C}_1 \cdot \mathcal {C}_2\) is a unitary operator that commutes with the Hamiltonian.

  3. 3.

    In case (i), a gapless topological system in d dimensions can be interpreted as topologically stable surface of a \((d+1)\)D TI, and the symmetry partner is embedded on the opposite side of the \((d+1)\)D TI. Now we have the \(d_\text {FS}\)-D Fermi surface in the \((d+1)\)D system, and hence its topological phase is characterized by the codimension of the Fermi surface \(d-d_\text {FS}+1\). A \(\mathbb {Z}_2\) topological number is protected only when \(d_\text {FS}=0\) [133]. On the other hand, for case (ii), the Fermi surface is regarded as defect in the Brillouin zone. We consider a closed object or sphere that surrounds the Fermi surface. Note that it wraps only a single Fermi surface and it separates the two Fermi surfaces related by symmetries. Therefore, the topological phase is determined by the dimension of the sphere \(d-d_\text {FS}-1\). In case (ii), a \(\mathbb {Z}_2\) topological number does not protect Fermi surfaces, but may support surface states at TRIMs of the surface Brillouin zone [133].

  4. 4.

    There is another definition for SRE states, which are defined by gapped, non-degenerate bulk spectrum [145]. By this definition, SRE phases include SPT phases as a subset.

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    Hiroki Isobe

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Isobe, H. (2017). Introduction. In: Theoretical Study on Correlation Effects in Topological Matter. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-3743-6_1

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