Non-Hermitian band topology in twisted bilayer graphene aligned with hexagonal boron nitride

TL;DR

This work investigates non-Hermitian topology in twisted bilayer graphene aligned with hexagonal boron nitride by introducing non-reciprocal hopping with strength $\beta$ and a top-layer mass term $M_0$ within the BM continuum framework. It analyzes both untwisted BLG and moiré tBLG, computing direct band gaps and NH Chern numbers in the moiré Brillouin zone to identify NH valley-Hall insulating phases and exceptional magic angles, with emphasis on how $\beta$ and $M_0$ modulate flat-band physics near the magic angle $\theta \approx 1.05^\circ$. The study finds that EMAs emerge where the real and imaginary bandwidths vanish, and that increasing $\beta$ generally widens the NHEMA region in the chiral limit but suppresses topological regions near the magic angle, as evidenced by $C_K=-1$ in certain parameter windows and $C_{\text{valley}}=-2$. These results offer predictions for NH topological signatures in tBLG-hBN and point to experimental platforms (photonic/metamaterial setups, cold-atom systems) where non-reciprocity can be engineered to realize NH valley-Hall insulators and related phenomena.

Abstract

Utilizing the established Bistritzer-MacDonald model for twisted bilayer graphene (tBLG), we theoretically investigate the non-Hermitian (NH) topological properties of this in the presence of non-reciprocal (NR) hopping on both layers and hexagonal boron nitride (hBN) induced mass term incorporated only on the top layer of the tBLG system. It is well known that the hBN mass term breaks the \(C_{2}\) symmetry of tBLG and gaps out the Dirac cones inducing a valley Hall insulating phase. However, when NR hopping is introduced, this system transits into a NH valley Hall insulator (NH-VHI). Our analysis reveals that, in the chiral limit, the bandwidth of the system vanishes under NH effects for a wide range of twist angles. Such range can be visibly expanded as we enhance the degree of non-Hermiticity (\(β\)). At the magic angle, we observe that enhancement of \(β\) inflates the robustness of the gapless Dirac points, requiring a progressively larger mass term to induce a gap in the NH tBLG system. Additionally, for a fixed NH parameter, we identify a range of twist angles where gap formation is significantly obstructed. To explore the topological aspects of the NH tBLG, we analyze the direct band gap in the Moiré Brillouin zone (mBZ) and compute the Chern number for the NH system. We find that the corresponding topological phase transitions are associated with corresponding direct band gap closings in the mBZ.

Figures (11)

• Figure 1: (a) Schematic diagram of the single layer graphene with NR hopping [($t + \gamma$) and ($t - \gamma$)] between the nearest neighbour lattice sites (b) Two hexagons (red and cyan) indicate layer-1 and layer-2 of the single layer graphene rotated by $+\theta/2$ and $-\theta/2$ respectively. This rotational misalignment gives rise to three distinct momenta ($\mathbf{q_1}, \mathbf{q_2}, \mathbf{q_3}$) at three equivalent Dirac points of the single layer graphene via which the two layers couple to each other. These three momenta form the mini Brillouin zone (highlighted by the green hexagon) as directed by an arrow. The black line over the mBZ manifests the high symmetry path we follow to establish the band dispersions of the twisted system.
• Figure 2: Energy spectra of untwisted NH-BLG is presented, incorporating non-reciprocity in the nearest neighbour hopping. The band structures are depicted along the corner ($K$ and $K^{\prime}$) and centre ($\Gamma$) of the hexagonal BZ. The Re(E) (upper panel) and Im(E) (lower panel) spectra have been shown for different values of the non-Hermiticity ($\beta$) and hBN-induced mass ($M_0$) terms. In panel (a), $\beta = 0$ and $M_0 = 0$ i.e., this corresponds to the Hermitian limit, in panel (b) $\beta = 0.9$ and $M_0 = 0$, two bands out of four become degenerate before valley-$K$ and after valley-$K^{\prime}$ and the band width also decreases. Finally, in panel (c) $\beta = 0.9$ and $M_0 = 1$ eV and the bands become gapped.
• Figure 3: Density plots for the band gap (in meV), considering the real part of the first valence and conduction bands, are depicted in the $k_x - k_y$ plane within the BZ. In the upper panel (i.e., panels (a) and (b)) we show the band gap for NH-BLG choosing different values of the NR hopping strength ($\beta = 0$ and $\beta = 0.9$ respectively) and observe to form a nodal line due to non-Hermiticity. In the lower panel we display the band gap for $\beta = 0$ and $\beta = 0.1$ respectively in panels (c) and (d) and observe similar nodal line for NH-tBLG near valley-$K$ when $\beta \neq 0$.
• Figure 4: The energy spectra of NH-tBLG are presented for different values of the non-reciprocal hopping strength ($\beta$) and the hBN-induced mass ($M_0$) at a twist angle of $\theta = 1.05^\circ$. The band structures are plotted along the high-symmetry path $K_m - \Gamma_m - M_m - K'_m$ of the mBZ. The red and black lines represent the energy bands corresponding to valleys $K$ and $K'$ (i.e., $\xi = \pm 1$), respectively. The real and imaginary parts of the spectra are shown as $\text{Re}(E)$ (solid lines, upper panel) and $\text{Im}(E)$ (dashed lines, lower panel). In panel (a), we set $\beta = 0$ and $M_0 = 0$, corresponding to the Hermitian model, where the characteristic tBLG flat bands appear. In panel (b), with $\beta = 0.1$ and $M_0 = 0$, the bands from the two valleys become degenerate along the high-symmetry lines, and the bandwidth decreases significantly. Panel (c) corresponds to $\beta = 0.1$ and $M_0 = 20 ,\text{meV}$, where the flat bands are separated and a gap opens. Panels (d)--(f) show the imaginary parts of the band dispersions for the same parameter values as in panels (a)--(c), respectively.
• Figure 5: The Real (represented with dotted lines) and Imaginary (denoted by solid lines) part of the bandwidth (Re($W_{0}$) and Im($W_{0}$)) are depicted with respect to the twist angle ($\theta$) of the NH-tBLG. The black and red color represent two different values of the NR-hopping strength $\beta = 0.1, 0.2$.