Direct prediction of corner state configurations from edge winding numbers in two- and three-dimensional chiral-symmetric lattice systems

Linhu Li, Muhammad Umer, and Jiangbin Gong

Phys. Rev. B 98, 205422 – Published 26 November 2018

Abstract

Higher-order topological phases feature topologically protected boundary states in lower dimensions. Specifically, the zero-dimensional corner states are protected by the $d th$ -order topology of a $d$ -dimension system. In this work, we propose to predict different configurations of corner states from winding numbers defined for one-dimensional edges of the system. We first demonstrate the winding number characterization with a generalized two-dimensional square lattice belonging to the BDI symmetry class. In addition to the second-order topological insulating phase, the system may also be a nodal point semimetal or a weak topological insulator with topologically protected one-dimensional edge states coexisting with the corner states at zero energy. A three-dimensional cubic lattice with richer configurations of corner states is also studied. We further discuss several experimental implementations of our models with photonic lattices or electric circuits.

Received 6 September 2018

DOI:https://doi.org/10.1103/PhysRevB.98.205422

Physics Subject Headings (PhySH)

Symmetry protected topological states Topological insulators Topological phase transition Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Linhu Li, Muhammad Umer, and Jiangbin Gong^*

Department of Physics, National University of Singapore, Singapore 117551, Republic of Singapore

^*phygj@nus.edu.sg

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Vol. 98, Iss. 20 — 15 November 2018

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Images

Figure 1
Sketches of the 2D lattice of Hamiltonian (1). (a) The general lattice structure. (b)–(d) Examples of different situations of the hopping amplitudes. Longer distances between two sites indicate weaker hopping amplitudes. The winding numbers of them are (b) $ν_{1} = ν_{2} = ν_{3} = ν_{4} = 1$ , (c) $ν_{1} = 0$ and $ν_{2} = ν_{3} = ν_{4} = 1$ , and (d) $ν_{1} = ν_{3} = 0$ and $ν_{2} = ν_{4} = 1$ .
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Figure 2
Spectra and the collective distribution of zero modes of the 2D square lattice, the parameters of which are given by Eq. (15). The top panels show the spectra with (a) $δ_{1} = 0.5$ , $δ_{2} = 0.6$ , $δ_{3} = 0.7$ , and $δ_{4} = 0.8$ ; (b) $δ_{1} = - 0.5$ , $δ_{2} = 0.6$ , $δ_{3} = 0.7$ , and $δ_{4} = 0.8$ ; (c) $δ_{1} = - 0.5$ , $δ_{2} = 0.6$ , $δ_{3} = - 0.7$ , and $δ_{4} = 0.8$ . The collective distribution of zero modes with the same parameters are shown in (d)–(f), respectively.
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Figure 3
A phase diagram of the system with $δ_{2} = δ_{4} = 1$ , and the spectra under OBC along $y$ direction in several different parameter regimes. (a) The phase diagram with different phases labeled on it. Different colors show the behavior of corner states, and the solid lines are the boundaries between different phases regarding the bulk and edge states. The dash lines show the boundaries between second-order topological insulators with different types of corner states. The parameters for the spectra are (b) $δ_{1} = 0.5$ , $δ_{3} = - 1.5$ ; (c) $δ_{1} = - 1.5$ , $δ_{3} = - 1.5$ ; (d) $δ_{1} = 0.5$ , $δ_{3} = 0.5$ ; (e) $δ_{1} = 0$ , $δ_{3} = 0.5$ .
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Figure 4
A sketch of the 3D lattice. Solid lines are for positive hoppings, dash lines for negative ones, following the choices in Refs. [1, 4]. In this sketch, we only illustrate a unit cell and the intracell hoppings, while in the main text the corresponding intercell hoppings are denoted as $t_{i}^{'}$ with $i = 1, 2, \dots, 12$ , respectively.
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Figure 5
Different configurations of type-1 corner states in the 3D cubic lattice. The numbers of type-1 of corner states are indicated in the figures. The insert of each panel shows a sketch of the configuration of corner states, with red circles and lines indicate the corners with type-1 corner states and the edges with winding numbers $ν_{i} = 1$ , and gray ones for the rest. The left figure in each panel shows the central part of the spectrum under OBC, which has a gap near zero energy and some in-gap corner states in panels (b), (h), (i) (j), (l), and (m); the right one shows the collective distribution of zero modes ( $E < 10^{- 4}$ in numerical calculations) along the edges, the amplitude of which is indicated by the shade, the darker ones are for larger amplitude. The bulk distributions are omitted for a clearer view. The parameters are chosen as $δ_{i} = 0.8$ for $ν_{i} = 1$ , and $δ_{i} = - 0.8$ for $ν_{i} = 0$ . The system are chosen to have $5 \times 5 \times 5$ unit cells.
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Figure 6
Observation of corner states in a 2D photonic lattice. (a) A sketch of the arranged waveguides. The red arrow indicates the injecting light, and the right panel shows an example of the outgoing intensity distribution. (b)–(d) the distribution $ρ_{fin} (m, n)$ of the final state, with initial states chosen as soliton states in different corners, indicated by the red arrows. The parameters are $δ_{1} = - 0.5$ , $δ_{2} = 0.6$ , $δ_{3} = - 0.7$ , and $δ_{4} = 0.8$ , with the evolving time $T = 100$ .
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