Helical Topological Edge States in a Quadrupole Phase

A topological electric quadrupole is a recently proposed concept that extends the theory of electric polarization of crystals to higher orders. Such a quadrupole phase supports topological states localized on both edges and corners. In this work, we show that in a quadrupole phase of a honeycomb lattice, topological helical edge states and pseudospin-polarized corner states appear by making use of a pseudospin degree of freedom related to point group symmetry. Furthermore, we argue that a general condition for the emergence of helical edge states in a (pseudo)spinful quadrupole phase is the existence of either mirror or time-reversal symmetry. Our results offer a way of generating topological helical edge states without spin-orbital couplings.

Figure 1

(a) Schematic of the model that is characterized by two hopping parameters $\gamma$ and ${\gamma }^{\prime }$. Within a unit cell, there are six atomic orbitals $|j⟩$, $j=1,\dots ,6$. (b) Reciprocal lattice vectors ${\mathbf{b}}_{1,2}$ and the first Brillouin zone. (c) Energy bands spectrum for $|\gamma |>|{\gamma }^{\prime }|$ and $|\gamma |<|{\gamma }^{\prime }|$. “$\pm$” indicates the eigenvalue of $\pi$ rotation of wave functions at $\mathrm{\Gamma }$ and $M$ points. $M$ refers to either $M_1$ or $M_2$ in the 1st Brillouin zone. The first three energy bands in the case of $|\gamma |<|\gamma {|}^{\prime }$ are colored as red, blue, and green, respectively, for clarifying the dipole moment of each band.

Figure 2

Schematic of pseudospins made up by linear combination of atomic orbitals. Pseudo spin-up and -down transform to each by either time-reversal operation or mirror reflection. Inset: combination factors indicated in color.

Figure 3

(a) Schematic of a ribbon supporting topological helical edge states. The ribbon is periodic along the $y$ direction. There are two mirror planes denoted as $M_x$ and $M_y$. (b) Energy spectrum of the ribbon for $\gamma =-1.0$ and ${\gamma }^{\prime }=-2.0$. The wave number $k$ refers to direction ${\mathbf{b}}_1$. Within the bulk energy gap, a pair of spin-polarized bands consisting of edge states appear. (c) Helical edge states. Because of mirror symmetry (time reversal symmetry), edge states of the same energy must propagate oppositely—as indicated by the cross and dot—with opposite spin polarization on the same edge.

Figure 4

(a) Schematic of a sample comprised of $10\times 10$ unit cells supporting the corner states. We call the edge along the $x$ direction zigzag, while that along the $y$ direction armchair. (b) Spin decomposition of the corner states with positive energies, blue (red) for the spin-up (-down) component. Zigzag corner states are spin-polarized up while armchair corner states are spin-polarized down. Insets: charge density maps for the corner states.

Figure 5

(a) Pumping spectrum of a spinful quadrupole for a half period. The spectrum consists of three types of states: the bulk states (black), edge states (blue), and corner states (red). (b) By replacing $\theta$ with a quasi wave number $k_z$ along the third direction, a class of 3D topological insulators characterized by spin polarized hinge states—blue for spin-up and red for spin-down—emerge.