Topological transition induced by selective random defects on a honeycomb lattice

TL;DR

This work investigates how selective random defects mediate the spectral and topological evolution of electrons on a honeycomb lattice as it interpolates to the 1/6-depleted Bishamon-kikko (BK) lattice, using the Haldane model with a staggered potential $M_i$, nearest-neighbor hopping $t_1$, next-nearest-neighbor hopping $t_2$, and a complex phase $\phi$. By defining a defect ratio $r \in [0,1]$, the authors interpolate between the pristine honeycomb ($r=0$) and the BK lattice ($r=1$) and analyze spectral gaps, edge states, and topological invariants. They identify two regimes: (i) a smooth connection where the topological character (e.g., Chern numbers) remains intact across $0\le r \le 1$, and (ii) a defect-induced topological transition at particular parameter sets where a gap closes around $E\approx 0.3$ and topological markers jump near $r \approx 0.7$. To interpret the transition, they introduce an effective model on the honeycomb lattice that mimics defects as hopping-amplitude modulations $\alpha_{ij}$, finding qualitative agreement and a proposed mapping $\alpha \simeq 1 - r$ with the transition occurring near $\alpha \approx 0.3$. The results demonstrate lattice engineering as a viable approach to tailor spectral and topological properties across materials, with potential realizations in vdW materials, graphene-based systems, photonic crystals, and MOFs.

Abstract

We investigate how the spectral and topological properties of electron systems evolve on a lattice that interpolates between the honeycomb and its 1/6-depleted structures through the introduction of selective random defects. We find that in certain parameter regimes, the topological properties of the two lattice systems are smoothly connected, whereas in other regimes, selective random defects induce a topological transition. Analysis based on an effective model reveals that the effect of selective random defects can be understood as a modulation of hopping amplitudes. Our results highlight the potential for designing and controlling the spectral and even topological properties of electronic systems across a wide range of material platforms.

Figures (9)

• Figure 1: Schematic pictures of the hierarchy of two dimensional lattices derived from a triangular lattice. 1/3 and 1/4 periodic site depletions applied to a triangular lattice lead to honeycomb and kagome lattices, respectively, which can host remarkable electronic band structures such as Dirac cones and flat bands. Similarly, 1/6 periodic site depletion from the honeycomb lattice yields the Bishamon-kikko lattice, which can exhibit both Dirac cones and flat bands in the electronic band structure. These examples illustrate that various periodic depletions applied to a single parent lattice can generate a wide variety of derived lattices with rich electronic properties.
• Figure 2: Schematic picture of the Haldane model on the honeycomb lattice with random depletion from selected sites, connecting to the BK lattice. The defect ratio $r$ is defined as the fraction of defect sites among the yellow-marked ones [Eq. \ref{['eq:def-r']}]. The figure represents an example with the defect ratio $r=3/6=0.5$. The pink arrows indicate the direction of complex next-neighbor hopping with a phase $\phi$ in Eq. \ref{['eq:Hamiltonian1']}. The dashed translucent pink bonds and arrows represent hoppings that are modulated by the parameter $\alpha$ in an effective model on the honeycomb lattice introduced in Sec. \ref{['sec3-3']}
• Figure 3: A real space distribution of (a) the local Chern marker and (b) the crosshair marker on a lattice under the OBC with the defect ratio $r=0.5$. The Hamiltonian parameters are $M/t_2=-3$, $\phi=\pi/2$, and $t_2=0.3$. The intersection of the two blue dashed lines in (b) represents the position of the crosshair. To represent the bulk contributions, the LCM is averaged over the sites inside the green circle in (a), and the crosshair marker is summed over the same region in (b).
• Figure 4: Spectral properties of the Haldane model in Eq. \ref{['eq:Hamiltonian1']} at $M=0$, $t_2=0.3$, and $\phi=\pi/2$. (a), (b) Band structures along the high symmetric lines in the Brillouin zones for the model on (a) the honeycomb lattice ($r=0$) and (b) the BK lattice ($r=1$). The Chern numbers $C$ associated with each band are also shown. The inset represents the first Brillouin zone for the honeycomb (outer black hexagon) and BK (inner green hexagon) lattices. (c) DOS under the PBC averaged over 60 selective random defect configurations and (d) energy spectrum under PBC and OBC as functions of the defect ratio $r$.
• Figure 5: Spectral properties of the Haldane model at $M/t_2=-3, t_2=0.3$, and $\phi=\pi/2$. Band structures and Chern numbers $C$ of the Haldane model on (a) the honeycomb and (b) the BK lattices. (c) DOS under the PBC and (d) energy spectrum under PBC and OBC as functions of the defect ratio $r$. (e) DOS at defect ratio $0.5, 0.6, 0.7, 0.8,$ and $0.9$ under the PBC. The solid lines represent the DOS averaged over 60 independent selective random defect configurations, and the shades indicate the standard deviation around each average. The white dashed line in (c) and red dashed lines in (d) represent the chemical potential for calculating the topological invariants in Fig. \ref{['fig:marker']}.