General scattering and electronic states in a quantum-wire network of moiré systems

We investigate electronic states in a two-dimensional network consisting of interacting quantum wires, a model adopted for twisted bilayer systems. We construct general operators which describe various scattering processes in the system. In a twisted bilayer structure, the moiré periodicity allows for generalized umklapp scatterings, leading to a class of correlated states at certain fractional fillings. We identify scattering processes which can lead to an insulating gapped bulk with gapless chiral edge modes at fractional fillings, resembling the quantum anomalous Hall effect recently observed in twisted bilayer graphene. Finally, we demonstrate that the description can be useful in predicting spectroscopic and transport features to detect and characterize the chiral edge modes in the moiré-induced correlated states.

Figure 1

Moiré pattern and quantum-wire network of the TBG. When two graphene monolayers (orange) are stacked with a misalignment, there appears a moiré pattern with the wavelength $\lambda =a_0/[2sin(\theta /2)]$, monolayer lattice constant $a_0$, and the angle $\theta$ between the layers. The moiré pattern results in three sets of parallel quantum wires, plotted in distinct colors and labeled by $j$, with the interwire distance $d=\sqrt{3}\lambda /2$.

Figure 2

Quantum-wire network in a moiré structure. Left: For each wire, we define the local coordinate $x$ and fermion fields ${\psi }_{\ell m\sigma }$, which experience a periodic potential $V\left(x\right)$ generated by the moiré structure. Right: Each array consists of parallel wires with the chemical potential $\mu$ and Fermi wave vector $k_F$, where we linearize the energy dispersion and bosonize the fields with Eq. (1).

Figure 3

Examples for moiré umklapp scatterings at $\nu =P/4$. (a) $O_{\mathrm{i}}$, characterized by Eq. (3) with $(s_{R0\sigma },s_{L0\sigma })=(2,-2)$. (b) $O_{\mathrm{ii}}$, with $(s_{R0\sigma },s_{L0\sigma },s_{Rn\sigma },s_{Ln\sigma })=(1,-1,1,-1)$. (c) $O_{\mathrm{iii}}$, with $(s_{R0\sigma },s_{L0\sigma },s_{Rn\sigma },s_{Ln\sigma })=(1,-1,1,-1)$. (d) $O_{\mathrm{iv}}$, with $(s_{R0\sigma },s_{L0\sigma },s_{Rn\sigma },s_{Ln\sigma })=(2,0,0,-2)$. Here we illustrate processes that are invariant upon changing the spin sign; see Table S1 for more general cases [105].

Figure 4

QPC setups for systems in a moiré correlated state with a gapped bulk (orange) and gapless chiral edge modes (green). The setup (a) allows for tunnel current $I_{\mathrm{t}}$. The setup (b) leads to backscattering current $I_{\mathrm{b}}$ and conductance correction $\delta G$.