Topological edge states in the Su-Schrieffer-Heeger model

Daichi Obana, Feng Liu, and Katsunori Wakabayashi

Phys. Rev. B 100, 075437 – Published 29 August 2019

Abstract

The Su-Schrieffer-Heeger (SSH) model on a two-dimensional square lattice exhibits a topological phase transition which is related to the Zak phase determined by bulk band topology. The strong modulation of electron hopping causes nontrivial charge polarization even in the presence of inversion symmetry. The energy band structures and topological edge states have been calculated numerically in previous studies. Here, however, the full energy spectrum and explicit form of wave functions for two-dimensional bulk and one-dimensional ribbon geometries of the SSH model are analytically derived using the wave mechanics approach. Explicit analytic representations of wave functions provide the information of parity for each subband, localization length, and critical point of the topological phase transition in the SSH ribbon. IThe dimensional crossover of the topological transition point for the SSH model from one to two dimensions is also shown.

Received 13 June 2019

DOI:https://doi.org/10.1103/PhysRevB.100.075437

Physics Subject Headings (PhySH)

Edge states Electronic structure Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Daichi Obana^1,*, Feng Liu^1,†, and Katsunori Wakabayashi ^1,2,‡

¹Department of Nanotechnology for Sustainable Energy, School of Science and Technology, Kwansei Gakuin University, Gakuen 2-1, Sanda, Hyogo 669-1337, Japan
²National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan

^*1227banao@kwansei.ac.jp
^†ruserzzz@gmail.com
^‡waka@kwansei.ac.jp

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Vol. 100, Iss. 7 — 15 August 2019

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Images

Figure 1
(a) Schematic of the SSH model on a square lattice. Thick and thin bonds represent the intra- and intercell electron hoppings, respectively. The primitive translation vectors are $a_{1} = (a, 0)$ and $a_{2} = (0, a)$ . (b) Corresponding first BZ. The reciprocal lattice vectors are $b_{1} = (\frac{2 π}{a}, 0)$ and $b_{2} = (0, \frac{2 π}{a})$ .
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Figure 2
Energy band structure of the 2D SSH model for (a) $γ^{'} / γ = 1 / 3.0$ ( $γ^{'} / γ = 3.0$ ) and (b) $γ^{'} / γ = 1$ . The signs ( $\pm$ ) in the plot of the energy band structure represent the parity of the wave function. It should be noted that band inversion occurs at the $X (Y)$ point.
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Figure 3
Density plot of the phase $ϕ_{l} (k_{l})$ for (a) trivial and (b) nontrivial phases. In the trivial phase, no phase jumps occurs. In the nontrivial phase, however, the phase jumps by $2 π$ occur along the lines of $k_{x} = \pm π$ and $k_{y} = \pm π$ , respectively.
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Figure 4
(a) SSH ribbon model on a square lattice. $N_{x}$ is the ribbon width. The $\times$ marks indicate the missing atoms for the open boundary condition. Energy band structure of $N_{x} = 20$ for (b) $γ^{'} / γ = 1 / 3.0$ and (c) $γ^{'} / γ = 3.0$ . In the nontrivial case, edge states appear (red lines).
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Figure 5
Schematic chart of functions $v$ (magenta line) and $w$ (cyan line) for $N_{x} = 5$ in (a) the trivial $[| γ^{'} / γ | \leq {(γ^{'} / γ)}_{c}]$ and (b) nontrivial $[| γ^{'} / γ | > {(γ^{'} / γ)}_{c}]$ phases, respectively. White circles at $k_{x} = 0$ and $π$ are unphysical solutions. The black circle in the region of $0 < k_{x} < π$ indicates the real-value solution of the transverse wave number $k_{x}$ . It should be noted that there are $N_{x}$ solutions for the trivial phase but only $N_{x} - 1$ solutions for the nontrivial phase. The insets show the dependence of $w$ on the longitudinal wave number $k_{y}$ near $k_{x} = π$ . The relation between the obtained complex transverse wave number $k_{x} + i η$ and the corresponding energy band structure for the 1D SSH model with $N_{x} = 5$ for (c) the trivial and (d) nontrivial phases. In the top panels of (c) and (d), solutions of the complex transverse wave number $k_{x} + i η$ in the BZ are plotted. Both $k_{x}$ and $η$ do not have $k_{y}$ dependence, and no imaginary part exists in the trivial phase. The bottom panels of (c) and (d) show the corresponding energy band structures. Real-value solutions $k_{x}$ lead to energy dispersion of bulk states, denoted by thin black lines; however, the imaginary solution $η$ leads to energy dispersion of TESs, denoted by thin magenta lines.
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Figure 6
(a) Dependence of the localization length $1 / η$ of TES on $γ^{'} / γ$ . The solid line is calculated from analytic wave functions of Eq. (25). The dashed line is the fitting for the large coupling limit. The fitting curve is $1 / η = 1.327 {(γ^{'} / γ - 1)}^{- 0.481}$ . (b) Ribbon width $N_{x}$ dependence of the critical value ${(γ^{'} / γ)}_{c}$ , where the topological phase transition occurs. In the 2D limit of $N_{x} \to \infty, {(γ^{'} / γ)}_{c}$ converges to 1. Black circles are ${(γ^{'} / γ)}_{c}$ obtained from Eq. (24). The dashed line is the fitting for the limit of large $N_{x}$ . The fitting curve is ${(γ^{'} / γ)}_{c} = 1.165 {(N_{x} - 1)}^{- 0.034}$ .
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