Simulating Twistronics without a Twist

Rotational misalignment or twisting of two monolayers of graphene strongly influences its electronic properties. Structurally, twisting leads to large periodic supercell structures, which in turn can support intriguing strongly correlated behavior. Here, we propose a highly tunable scheme to synthetically emulate twisted bilayer systems with ultracold atoms trapped in an optical lattice. In our scheme, neither a physical bilayer nor twist is directly realized. Instead, two synthetic layers are produced exploiting coherently coupled internal atomic states, and a supercell structure is generated via a spatially dependent Raman coupling. To illustrate this concept, we focus on a synthetic square bilayer lattice and show that it leads to tunable quasiflatbands and Dirac cone spectra under certain magic supercell periodicities. The appearance of these features are explained using a perturbative analysis. Our proposal can be implemented using available state-of-the-art experimental techniques, and opens the route toward the controlled study of strongly correlated flatband accompanied by hybridization physics akin to magic angle bilayer graphene in cold atom quantum simulators.

Figure 1

Synthetic bilayer structure with a supercell. (a) Real-space potential of the synthetic bilayer. Each plane corresponds to one spin state $m=\pm 1/2$ (orange, green), which experiences a square lattice potential (tunneling $t$) and is connected to the other layer by a spatially dependent and complex coupling $\mathrm{\Omega }(x,y)$ (vertical red lines of variable width). A top view of the lattice indicating the unit cell of the system containing $2\times 8$ sites for $l_x=l_y=4d$ is shown (black line). (b) Sketch of the first Brillouin zone, indicating the position of the high-symmetry points, and three dimensional view of the energy spectrum in the vicinity of $E=-{\mathrm{\Omega }}_0(1-\alpha )$ for ${\mathrm{\Omega }}_0\alpha /h=20t$ with $\alpha =0.2$ and $\gamma =0$. It has two quasiflatbands intersecting a Dirac point. Note that a simple square lattice supports neither flatbands nor Dirac cones. (c) Proposed experimental realization. Top: Two retroreflected optical lattice beams (green) create the square lattice. Two Raman beams of opening angle $\theta$ (red) produce complex synthetic tunneling between the two layers. One “modulation laser” with a spatially varying intensity distribution (blue) modulates the amplitude of the Raman coupling. Bottom: laser beams involved in the synthetic bilayer coupling scheme. The single-photon detuning of the Raman beams (red arrows) is spatially modulated with respect to its initial value ${\mathrm{\Delta }}_0/2\pi \sim 75\mathrm{MHz}$ using a laser beam blue detuned with respect to the ${{}^{3}P}_1\to {{}^{3}S}_1$ transition (blue arrow). It produces a light shift of maximal amplitude $2\delta /2\pi \sim 30\mathrm{MHz}$.

Figure 2

Magic configuration band structure and DOS. Band structures around energy $-{\mathrm{\Omega }}_0(1-\alpha )$ and DOS (in arbitrary units) corresponding to $\mathrm{\Theta }(4,4)$ supercell along the paths passing through the high-symmetry points $\mathrm{\Gamma },\mathbf{X},\mathbf{M},\mathrm{\Gamma },{\mathbf{X}}^{\prime },\mathbf{M}$. Panels (a) and (b) correspond to ${\mathrm{\Omega }}_0\alpha /h=2t$ and $20t$, respectively, with $\alpha =0.2$ and $\gamma =0.8$. In the evolution from panel (a) to (b), the six central bands in panel (a), denoted with colors from orange to maroon, remain close in energy (as part of one single band, see perturbative analysis in [49]) while the remaining two bands in panel (a) (in cyan and brown) separate in energy and do not appear in panel (b).

Figure 3

Gap opening and Dirac cone widening due to the artificial flux $\gamma$. Four bands [Dirac cones (purple and red) and quasiflatbands (green and blue)] are shown in the vicinity of $E/t=-{\mathrm{\Omega }}_0(1-\alpha )=-80$, for the $\mathrm{\Theta }(4,4)$ system. The parameters and color scale of the bands are identical to those of Fig. 2, except for the magnetic flux that corresponds to (a) $\gamma =0$ and (b) $\gamma =0.8$. The energy surfaces are rotated for visibility, and only a small region centered at the $\mathrm{\Gamma }=(0,0)$ point in the Brillouin zone [yellow area in (c), corresponding to $\sim 4\%$ of the complete Brillouin zone] is depicted.