Emergence of Non-Hermitian Magic Angles and Topological Phase Transitions in Twisted Bilayer $α$-$T_3$ Lattices

Abstract

We investigate the flat-band properties and topological phase transitions in a non-Hermitian twisted bilayer $α-T_3$ lattice. Here, non-Hermiticity is introduced via Hatano-Nelson-type asymmetric hopping, while an aligned hexagonal boron nitride substrate provides a staggered sublattice mass to the system. We find that the introduction of non-reciprocal hopping splits the conventional single magic angle into three distinct non-Hermitian magic angles (NHMAs). Unlike the exceptional magic angles driven by spectral singularities, these NHMAs host perfectly isolated flat bands where the real and imaginary parts of the bandwidth simultaneously vanish. By mapping the complex eigenspectrum across the moiré Brillouin zone, we show that the scattered energy eigenvalues coalesce into well-defined, closed loop-like structures as the non-Hermitian parameter strength increases, indicating emergence of a nontrivial point-gap topology and hence the non-Hermitian skin effect. Furthermore, we characterize the topological phases by computing the direct band gap and the biorthogonal Chern number. While the system exhibits a transition to a higher topological phase at weak non-Hermiticity, we demonstrate that stronger non-Hermiticity drives the gap-closing boundaries to merge and their topological charges to mutually annihilate. This convergence results in a trivial gap closing and a complete suppression of the intermediate topological phase, confirming that non-Hermiticity fundamentally plays a crucial role with regard to destabilizing the robust topological features of this moiré system.

Figures (8)

• Figure 1: (a) Schematic representation of the $\alpha$–$T_3$ lattice, showing the three sublattices A, B, and C. The non‑Hermitian effect is introduced through asymmetric hopping amplitudes, $t(1+\beta)$ and $t(1-\beta)$, between the A and B sites. Here, $t_1$ represents the next nearest‑neighbor hopping.(b) Brillouin zones of the top layer (blue) and bottom layer (red), rotated by $-\theta/2$ and $+\theta/2$, respectively, relative to the unrotated Brillouin zone (black dotted hexagon). The vectors $\mathbf{q}b$, $\mathbf{q}{tr}$, and $\mathbf{q}_{tl}$ indicate the momentum‑transfer vectors connecting the shifted Dirac points of the two layers.(c) The corresponding mBZ highlighting the high‑symmetry points ($\Gamma_m$, $M_m$, $K_m$, $K'_m$) and their coordinates. The arrows indicate the high‑symmetry momentum path used to calculate the energy bands.
• Figure 2: (a) Real‑part bandwidth ($\Delta E_{\text{Re}}$) of the isolated band near $E_0 = -17$ meV as a function of the twist angle $\theta$ for selected non‑Hermitian strengths $\beta = 0.1$ (top panel), $0.3$ (middle panel), and $0.5$ (bottom panel) in the dice limit ($\alpha = 1$). The color scale encodes the corresponding imaginary‑part bandwidth ($\Delta E_{\mathrm{Im}}$). The NHMAs $\theta_1$, $\theta_2$, and $\theta_3$, at which $\Delta E_{\mathrm{Re}}$ and $\Delta E_{\mathrm{Im}}$ vanish within numerical precision, are marked by red arrows. These points demonstrate the splitting of the flat‑band condition into multiple NHMAs. The inset in the top panel shows a zoomed‑in view near the NHMAs for $\beta = 0.1$. (b) Phase diagram of the real‑part bandwidth $\Delta E_{\text{Re}}$ in the $(\alpha, \theta)$ plane for a fixed non‑Hermitian parameter $\beta = 0.3$. The bright yellow regions indicate perfectly flat bands ($\Delta E_{\text{Re}} \to 0$), whereas darker regions represent highly dispersive bands. The phase diagram highlights the transition from three distinct NHMAs near the dice limit ($\alpha \to 1$) to a broad, twist‑angle‑independent localized flat band in the graphene limit ($\alpha \to 0$). (c) Phase diagram of $\Delta E_{\text{Re}}$ in the $(M, \theta)$ plane for $\beta = 0.3$ and $\alpha = 1$. The vertical yellow strips indicate that the locations of the three NHMAs are robust and entirely independent of the staggered mass term $M$ induced by the aligned substrate.
• Figure 3: Complex energy bands evaluated along the high-symmetry path of the mBZ for two representative NHMAs at $(\beta=0.1, \theta=1.12^\circ)$ and $(\beta=0.5, \theta=2.01^\circ)$. The top panels show the full real-part spectrum, featuring two isolated bands near $E \approx \pm 17$ meV. The middle and bottom panels display a zoomed-in views of the real and imaginary parts of the isolated band near $E_0=-17$ meV, respectively.
• Figure 4: Complex energy spectrum $(\mathrm{Re}\,E, \mathrm{Im}\,E)$ evaluated over a dense grid in the mBZ for the isolated band near $E_0 = -17$ meV. The panels show the evolution of the eigenvalues for increasing non-Hermitian parameter $\beta$ along the third magic-angle $\theta_3$. As the strength of the non-Hermitian parameter increases, the eigenvalues transform from scattered points to sharply defined, interconnected loop-like structures in the complex plane. Despite the visual prominence of these loops, their actual span remains microscopically small (up to $\sim10^{-5}$ meV), indicating that the flat band remains robustly pinned at $E_0 \approx -17$ meV. The formation of these closed spectral loops under periodic boundary conditions mathematically signifies a nontrivial point-gap topology, which is fundamentally tied to the NHSE.
• Figure 5: Complex spectral distribution near $E_0 \simeq -17$ meV for a fixed non-Hermiticity strength $\beta=0.5$ evaluated at the three distinct NHMAs: $\theta_1=0.52^\circ$, $\theta_2=0.67^\circ$, and $\theta_3=2.01^\circ$. Although all three angles host perfectly flat bands, the geometric shape and energy span of the corresponding spectral loops vary significantly.