Strong and weak second-order topological insulators with hexagonal symmetry and \ensuremath{\mathbb{Z}}${}_{3}$ index

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Strong and weak second-order topological insulators with hexagonal symmetry and ℤ $_{3}$ index

Motohiko Ezawa

Phys. Rev. B 97, 241402(R) – Published 4 June 2018

Abstract

We propose second-order topological insulators (SOTIs) whose lattice structure has a hexagonal symmetry $C_{6}$ . We start with a three-dimensional weak topological insulator constructed on a stacked triangular lattice, which has only side topological surface states. We then introduce an additional mass term which gaps out the side surface states but preserves the hinge states. The resultant system is a three-dimensional SOTI. The bulk topological quantum number is shown to be the $Z_{3}$ index protected by inversion time-reversal symmetry $I T$ and rotoinversion symmetry $I C_{6}$ . We obtain three phases: trivial, strong, and weak SOTI phases. We argue the origin of these two types of SOTIs. A hexagonal prism is a typical structure respecting these symmetries, where six topological hinge states emerge at the side. The building block is a hexagon in two dimensions, where topological corner states emerge at the six corners in the SOTI phase. Strong and weak SOTIs are obtained when the interlayer hopping interaction is strong and weak, respectively.

Received 12 March 2018

DOI:https://doi.org/10.1103/PhysRevB.97.241402

Physics Subject Headings (PhySH)

Geometric & topological phases Topological insulators Topological phase transition Topological phases of matter

Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Motohiko Ezawa

Department of Applied Physics, University of Tokyo, Hongo 7-3-1, Tokyo 113-8656, Japan

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Vol. 97, Iss. 24 — 15 June 2018

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Figure 1
(a), (b) A hexagonal prism is a typical structure respecting rotoinversion symmetry ${\bar{C}}_{6}$ . The real-space plot of the local density $ρ_{i}$ of the zero-energy states for a hexagonal prism in the case of (a) a weak TI with $Δ = 0$ and (b) a topological hinge insulator with $Δ = 0.7$ . The amplitude is represented by the radius of spheres. The length of one side of the hexagon is $L = 8$ , and the height of the prism is $H = 11$ . (c), (d) A hexagon is a typical structure respecting rotoinversion symmetry ${\bar{C}}_{6}$ . The real-space plot of $ρ_{i}$ of the zero-energy states for a hexagon in the case of (c) a TI and (d) a topological corner insulator. A weak (strong) topological hinge insulator is obtained by stacking topological corner insulators when the interlayer interaction $t_{z}$ is weak (strong). We have set $t = 1, t_{z} = 2, λ = λ_{z} = 1$ , and $m = 3$ .
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Figure 2
(a)–(c) Band structures of the top surface states in a thin film for $Δ = 0$ , where the gaps are open. (d)–(f) Band structures of the side surface states in a thin film for $Δ \neq 0$ . Two and one Dirac cones are observed in (d) and (e), respectively, showing the system is in a weak TI phase. A gap exists in (f), showing the system is in a trivial phase. (g)–(i) The figures corresponding to (d)–(f) for $Δ \neq 0$ , showing the weak TI phase is transformed into the SOTI phase. (j)–(l) Band structures of a hexagonal prism, where hinge states are shown in red. They are remnants of the Dirac cones observed in (d) and (e). The gap closes at two or one points in the weak or strong SOTI as in (j) or (k). Parameters $(Δ, m, t_{z})$ are displayed in the figures. The other parameters are $t = 1 = λ = λ_{z} = 1$ . (m) Topological phase diagram of the 3D model. The horizontal axis is $m / t$ , while the vertical axis is $t_{z} / t$ . Numbers in red represent the symmetry indicator $κ_{6}$ . The topological quantum number $ν_{3D}$ is defined by $ν_{3D} = {mod}_{3} κ_{6}$ . Thus, there are two SOTI phases shown by $κ_{6} = \pm 1$ , corresponding to (j) and (k). It does not depend on $λ, λ_{z}$ and $Δ$ . The horizontal axis of (j)–(l) is $k_{z}$ in the range of $(- π, π)$ .
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Figure 3
Band structures of the 2D model (a)–(c) with $Δ = 0$ and (d)–(f) with $Δ \neq 0$ in a nanoribbon geometry for typical points in the phase diagram. (g)–(i) Eigenenergy of the 2D model in a hexagonal nanodisk geometry for typical points in the phase diagram. (h) Six degenerate zero-energy states emerge in the SOTI phase. Parameters $(Δ, m)$ are displayed in the figures. The horizontal axis of (a)–(f) is $k_{x}$ in the range of $(- π, π)$ .
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Physical Review B

covering condensed matter and materials physics

Strong and weak second-order topological insulators with hexagonal symmetry and ℤ $_{3}$ index

Motohiko Ezawa

Phys. Rev. B 97, 241402(R) – Published 4 June 2018

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Physical Review B

covering condensed matter and materials physics

Strong and weak second-order topological insulators with hexagonal symmetry and ℤ3 index

Motohiko Ezawa

Phys. Rev. B 97, 241402(R) – Published 4 June 2018

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Strong and weak second-order topological insulators with hexagonal symmetry and ℤ $_{3}$ index