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Interacting Topological Crystalline Insulators

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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

We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and theorized in antiperovskite materials \(A_3BX\) with \(A=\) (Sr, La, Ca), \(B=\) (Sn, Pb) and \(X=\) (O, N, C). They host multivalley Dirac fermion surface states. Without interactions, such TCIs are classified by the mirror Chern number both for two and three dimensions. We find that electron interactions reduce the integer classification of noninteracting TCIs to a finite group \(\mathbb {Z}_4\) in two dimensions and \(\mathbb {Z}_8\) in three dimensions. The classification of the two-dimensional case is obtained by analyzing the one-dimensional edge modes using the bosonization method. For the classification of the three-dimensional case, the argument exploits the nonlocal nature of mirror symmetry and an explicit construction of surface states shows a reduction of the classification. Our construction builds on interacting edge states of \(U(1)\times Z_2\) symmetry-protected topological phases of fermions in two dimensions, which we classify. It reveals a deep connection between 3D topological phases protected by spatial symmetries and 2D topological phases protected by internal symmetries.

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Notes

  1. 1.

    For any 2D system including multilayers, one can choose single-particle basis states that are either even or odd under the reflection \(z\rightarrow -z\). In this basis, the mirror symmetry takes the explicit form of a \(\mathbb {Z}_2\) internal symmetry.

  2. 2.

    We consider the operator of the form \(e^{i\Phi _{\varvec{L}}(x)}\), which describes a local excitation in the sense that it can be written by the electron creation and annihilation operators. The commutation relation for \(e^{i\Phi _{\varvec{L}}(x)}\) is given by

    $$\begin{aligned} e^{i\Phi _{\varvec{L}_a}(x)} e^{i\Phi _{\varvec{L}_b}(x')}&= e^{i\Phi _{\varvec{L}_b}(x')} e^{i\Phi _{\varvec{L}_a}(x)} e^{-[\Phi _{\varvec{L}_a}(x), \Phi _{\varvec{L}_b}(x')]} \nonumber \\&= e^{i\Phi _{\varvec{L}_b}(x')} e^{i\Phi _{\varvec{L}_a}(x)} e^{\pi i \varvec{L}_a^T K \varvec{L}_b \text {sgn} (x-x')}. \end{aligned}$$

    Under the commutativity condition (5.14), the operators \(e^{i\Phi _{\varvec{L}_a}(x)}\) commute each other, and hence a pair of nonchiral edge modes is decoupled from others. Therefore, if we have n linearly-independent integer-valued vectors \(\{ \varvec{L}_a \}\), the edge modes are decoupled into n pairs of nonchiral modes, and a gap-opening scattering process can be assigned to each decoupled pair.

  3. 3.

    This redefinition is possible when the two flavors can be regarded as equivalent, i.e., the velocity of different edge modes are chosen to be the same. We note that relaxing the condition does not affect any of the results.

  4. 4.

    x is actually defined on a lattice \(x=aj\), where a is a lattice constant and j is an integer, and finally we take the continuum limit. Here we take the lattice constant \(a=1\) for simplicity.

  5. 5.

    An argument of similar spirit has been made in Ref. [5] to prove that in the presence of time-reversal symmetry, TCI surface states cannot be localized under disorder.

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  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Hiroki Isobe

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Isobe, H. (2017). Interacting Topological Crystalline Insulators. In: Theoretical Study on Correlation Effects in Topological Matter. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-3743-6_5

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