Magic continuum in a twisted bilayer square lattice with staggered flux

We derive the general continuum model for a bilayer system of staggered-flux square lattices, with arbitrary elastic deformation in each layer. Applying this general continuum model to the case where the two layers are rigidly rotated relative to each other by a small angle, we obtain the band structure of the twisted bilayer staggered-flux square lattice. We show that this band structure exhibits a magic continuum in the sense that an exponential reduction of the Dirac velocity and bandwidths occurs in a large parameter regime. We show that the continuum model of the twisted bilayer system effectively describes a massless Dirac fermion in a spatially modulating magnetic field, whose renormalized Dirac velocity can be exactly calculated. We further give an intuitive argument for the emergence of flattened bands near half filling in the magic continuum and provide an estimation of the large number of associated nearly zero-energy states. We also show that the entire band structure of the twisted bilayer system is free of band gaps due to symmetry constraints.

Figure 1

The square lattice with staggered flux is shown. The blue/black dots correspond to the $A$/$B$ sublattice, respectively. The hopping amplitudes from sublattices $A$ to $B$ through solid black links are given by $(it+\mathrm{\Delta })$, while those through the dashed links are given by $(it-\mathrm{\Delta })$. The neighboring plaquettes thus host staggered flux $\pm \mathrm{\Phi }$. To obtain the spectrum, we consider a four-site unit cell with the four sites within a unit cell labeled as shown above.

Figure 2

Spectrum of the monolayer staggered-flux square lattice with $t=\mathrm{\Delta }=1$. There are two degenerate Dirac cones for each spin located at $\mathit{K}=(\pi /2,\pi /2)$ in the reduced Brillouin zone.

Figure 3

The bilayer staggered-flux square lattice with a uniform relative deformation $(u_x,u_y)=(1,0)$ is depicted. The blue, green, orange, and pink vertices correspond, respectively, to the lattices site indexed by $i=1,2,3,4$ following Eq. (3). In the horizontal plane, the unit cell of the bilayer system still covers two lattice spacings in both the $x$ and $y$ directions. There are four vertical plaquettes in each unit cell labeled $F$, $B$, $L$ and $R$. There are four vertical links in each unit cell colored blue, green, orange, and pink. The interlayer tunneling only occurs along vertical links. See the main text for the tunneling strength along each link. The interlayer tunneling corresponds to having magnetic fluxes $2\mathrm{\Phi }+2\varphi -\pi$, $-2\mathrm{\Phi }-2\varphi +\pi$, $\pi -2\mathrm{\Phi }+2\varphi$, $-\pi +2\mathrm{\Phi }-2\varphi$ through the $F$, $B$, $L$, and $R$ plaquettes, respectively.

Figure 4

The shifted Dirac cones at half filling for a uniform deformation $\mathit{u}=\widehat{\mathit{x}}$. The two colors correspond to the two valleys ${\mu }^3=\pm 1$.

Figure 5

Top panel: Band structures in the chiral limit $w_s=10$, $w_a=0$. The momentum $q_1$ is conserved while $q_2$ takes value in the moiré Brillouin zone $q_2\in (-1/2,1/2)$. Bottom panel: Zoomed-in view of the flattened bands (with the coupling constants $w_a$ and $w_s$) near half filling.

Figure 6

Band structures along the moiré Brillouin zone trajectory $\mathrm{\Gamma }\to X\to M\to \mathrm{\Gamma }$ for various $w_s$'s with a fixed $w_a=0.5$ is plotted. For fixed spin and valley indices, the spectrum is twofold degenerate [corresponding the quantum number $s=\pm 1$ in Eq. (24)]. All bands are connected by Dirac cones at the $\mathrm{\Gamma }$ and the $M$ points in the moiré Brillouin zone. The ten bands (with each doubly degenerate) which are nearest to the zero energy are colored. As $w_s$ increases, the spectrum is compressed toward the zero energy and the bands near zero-energy are flattened. Note that the energy range of the plot with $w_s=3$ in the right bottom panel is about 200 times smaller than the other plots.

Figure 7

The renormalized Dirac velocity $v^{*}$ as a function of $w_s$ in the chiral limit $w_a=0$. There is a magic continuum for $w_s\gtrsim 1$. The dots are the numerically computed Dirac velocity. The smooth line is the modified Bessel function $I_0{\left(4w_s\right)}^{-1}$, which is the analytical expression of the reduced Dirac velocity $v^{*}$ we obtained in Eq. (31).

Figure 8

Number of moiré bands with nearly zero energies scales linearly with $|w_1|+|w_2|$ in the regime with $|w_1|,|w_2|\gtrsim 1$. We fix $|w_1|-|w_2|=1$ in the plot and count all the bands (obtained from numerical calculations) that completely lie within the energy window $E\in (-0.05,0.05)$. Note that bands only partially lying inside the energy window are not included in the counting. The steplike feature is due to the fact that the number of moiré bands counted has to be an integer. The distance between steps is always two due to the fact that the spectrum is symmetric in energy about the $E=0$ axis. The black straight line is obtained from connecting the middle points of each step.