Corner modes of the breathing kagome lattice: Origin and robustness

We study the non-trivial phase of the two-dimensional breathing kagome lattice, displaying both edge and corner modes. The corner localized modes of a two-dimensional flake were initially identified as a signature of a higher-order topological phase but later shown to be trivial for perturbations that were thought to protect them. Using various theoretical and simulation techniques, we confirm that it does not display higher-order topology the corner modes are of trivial nature. Nevertheless, they might be protected. First, we show a set of perturbations within a tight-binding model that can move the corner modes away from zero energy, also repeat some perturbations that were used to show that the modes are trivial. In addition, we analyze the protection of the corner modes in more detail and find that only perturbations respecting the sublattice or generalized chiral and crystalline symmetries, and the lattice connectivity, pin the corner modes to zero energy robustly. A destructive interference model corroborates the results. Finally, we analyze a muffin-tin model for the bulk breathing kagome lattice. Using topological and symmetry markers, such as Wilson loops and Topological Quantum Chemistry, we identify the two breathing phases as adiabatically disconnected different obstructed atomic limits.

Figure 1

The breathing kagome model. (a) Lattice model with three sublattice sites A, B, and C, and hopping parameters $t_a$ and $t_b$. The black triangles represent the $1a$ Wyckoff position, the blue triangles represent the $1b$ Wyckoff position, and the empty triangles represent the $1c$ Wyckoff position. The lattice sites fall on the $3d$ Wyckoff positions, represented with green circles. (b) Band structure of the kagome model for the breathing (solid lines) and nonbreathing (dashed lines) phases. Notice that the nonbreathing phase is gapless at the $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points. The corresponding values of $t_a=0.38t_b$ and $t_b=0.075\text{eV}$ are obtained from Ref. [21] by fitting the bands calculated within the tight-binding approach with the ones derived using the muffin-tin method. This corresponds to $\delta \approx -0.45$ and $t_0\approx 52$ meV. (c) Spectrum of a finite-size lattice and (d) zoom-in for low-energy scale. For $\delta <0$, threefold-degenerated energy eigenvalues can be found pinned at zero. (e), (f), (g) Exponential decay of the wave function for $\delta =-0.5$ in the bulk, edge, and corner, respectively. In the last three panels, the size of the dots is set for convenience proportional to ${\left|\psi \right|}^{0.2}$.

Figure 2

Set of perturbations that have been studied to detect the protection mechanisms of the corner modes. Each panel display the close-up of the spectrum plus the localization in real space of the wave function, thus, a visualization of the local density of states. (a) Random disorder in all lattice sites; (b) perturbation breaking generalized chiral symmetry, thus connecting lattice sites of the same species; (c) perturbation connecting second-order nearest neighbors; (d) perturbation connecting third-order nearest neighbors; (e) perturbation connecting fourth-order nearest neighbors; (f) long-range local perturbation with different sign; (g) long-range local perturbation with same sign; (h) and (i) influence of the size on the average, in absolute value, of the corner modes on each of the perturbations. Panel (h) has a logarithmic scale due to exponential localization.

Figure 3

Destructive interference models. (a) In the case of the 1D chain, the probability is finite only in the A sublattice site and decays as ${(-t_a/t_b)}^{2n}$, where $n$ is the unit-cell index. (b) The breathing kagome rhombus. Due to destructive interference, there is a wave function that has zero amplitude on the B and C sublattices and a finite amplitude on the A, where the probability decays as ${(-t_a/t_b)}^{\left(2m^{\prime }\right)}$ along $m$ (and analogously along $m^{\prime }$), whereas it decays as ${(-t_a/t_b)}^{2(m+m^{\prime })}$ along $m+m^{\prime }$. In all the panels, we have set $t_2=2t_1$.

Figure 4

Set of hopping terms that lift (a) or preserve (b) the corner modes pinned to zero energy. Although they are only shown on the lower-left corner of the flake, these hopping terms can be established between any $m$ and $m^{\prime }$ layers.

Figure 5

Summary of the results obtained for the muffin-tin calculation of the canonical/breathing kagome lattice. The overall legend affects the three cases, while each column has its own legend for the Wyckoff positions, which are called differently due to the group/subgroup relation. Potential wells defining the unit cells for the nonbreathing (a) and breathing phases (c), (e). The Wyckoff positions have been represented with an element with the same symmetries as in the point group associated with each Wyckoff position: hexagon, $C_{6v}$, triangle, $C_{3v}$, and ellipse, $C_{2v}$. In the case of the $3d$ Wyckoff position, we have used a circle for simplicity due to the reduced symmetry of this Wyckoff position (the point group associated is $C_m$, which includes just mirror and identity). Upper corner of a finite-size sample of kagome lattice; nonbreathing (b) and breathing phases (d), (f). Panels (g) to (i) show the corresponding band structures plus the irrep assignment at each high-symmetry point. Finally, panels (j) to (n) show the Wilson spectra obtained for the first three bands of each configuration.

Figure 6

Steps for realizing the muffin-tin potential for the kagome lattice in the canonical phase (left panel) and the two breathing phases (right panel) using CO molecules.

Figure 7

Wave functions for the first and second bands of the trivial (left) and nontrivial (right) configurations of the muffin-tin setup.