Disorder effects in two-dimensional flat-band system with next-nearest-neighbor hopping

TL;DR

This study analyzes localization and topological phase transitions in a 2D Lieb lattice hosting a flat band, focusing on how complex next-nearest-neighbor hopping t' and intrinsic spin-orbit coupling λ open gaps and generate topologically protected edge states. A transfer-matrix formalism for the Lieb lattice with complex NNN hopping is developed to compute localization lengths via Lyapunov exponents, complemented by real-space Chern-number calculations. Results show that nonzero λ opens gaps with robust edge states that counteract flat-band localization, while t' introduces mobility; under uncorrelated disorder localization strengthens, but symmetric correlated disorder can stabilize edge states and even produce inverse Anderson localization. The findings provide a unified framework connecting flat-band physics, topology, and disorder, with implications for robust topological transport in flat-band materials and avenues for future work on non-Hermitian disorder.

Abstract

For two-dimensional Lieb lattice, while intrinsic spin-orbit coupling is responsible for opening the gap that exhibits the quantum spin Hall effect, topological phase transitions are driven by a real next-nearest-neighbor (NNN) hopping. In this work, we utilize the transfer matrix method to study the flat-band localization mechanism in the presence of complex NNN hoppings. We demonstrate that the geometric localization in flat bands can be alleviated by topological edge states under weak disorder. Furthermore, correlated disorders are shown to induce inverse Anderson transition with the topological edge states persisting under strong disorder, a robustness confirmed by Chern number calculations, which identifies the root cause of this phenomenon. These findings establish a unified platform for investigating topological phase transitions, flat bands, and disorder effects.

Figures (7)

• Figure 1: (a)2D Lieb lattice. One unit cell contains three sublattice $\{A,B,C\}$ (yellow area). The location of unit cell is labeled by $(m, n)$ in real space. The NN hopping $t$ is set to 1 for simplicity and the NNN hopping between B and C sublattices $t'$ is indicated using dashed red line. The square unit cell has lattice constant $a \equiv 1$. (b) Energy gap (gap2) between top and middle branches as a function of parameter $t'$ and $\lambda$. The two sets of parameters are marked as $P_1(t'=0.1, \lambda=0.8)$ and $P_2(t'=0.9, \lambda=0.8)$.
• Figure 2: Energy Dispersions along the high symmetry lines in the first Brillouin zone. (a) for $P_1$ with $t' = 0.1, \lambda=0.8$. (b) for $P_2$ with $t' = 0.9, \lambda=0.8$.(c) fix parameter $\lambda=0$ and vary $t' = 0.1, 0.3, \cdots, 0.9$. and (d) fix parameter $\lambda=0.8$ and vary $t' = 0.1, 0.3, \cdots, 0.9$. $\mathcal{C}$ denotes the Chern number of corresponding energy branch. $\Gamma(0,0), M(\pi,\pi), X(\pi,0)$ are high symmetry points.
• Figure 3: Reduced localization length $\Lambda(N_y)$ as a function of disorder strength $W$ with width $N_y = 22$. Panel (a) and (b) depict the results for a fixed parameter $\lambda=0.8$ and $\lambda=0.0$ respectively, with parameter $t'$ varied across the interval $[0.1, 1]$. The inset shows the flatness ratio $\gamma$ versus $t'$ with fixed $\lambda = 0.8$. Panel (c) and (d) depict the results for symmetric and anti-symmetric disorder configurations respectively, i.e., $\varepsilon_{m,n}^B = \pm \varepsilon_{m,n}^C \in [-\frac{W}{2}, \frac{W}{2}], \varepsilon_{m,n}^A=0$.
• Figure 4: (a)(b) Reduced localization length $\Lambda(N_y)$ as a function of energy $E$ for $P_1$ and $P_2$ cases, respectively. Several disorder strengths are considered. Averaged $\langle \rm{IPR} \rangle$ values for eigenstates with energy $E$ for (c) $P_1$ and (d) $P_2$.
• Figure 5: The energy spectrum for different disorder strength with $20\times20$ unit cells. (a) for $P_1$ and (b) for $P_2$ case. The inset depicts the averaged two energy gap $\langle \Delta E \rangle$ versus disorder strength $W$. (c)(d) Chern number as a function of electron Fermi energy $E$ for two phases under several different disorder strengths. The result is averaged over 200 disorder realizations.