Experimental evidence for non-Abelian gauge potentials in twisted graphene bilayers

Non-Abelian gauge potentials are quite relevant in subatomic physics, but they are relatively rare in a condensed matter context. Here we report the experimental evidence for non-Abelian gauge potentials in twisted graphene bilayers by scanning tunneling microscopy and spectroscopy. At a magic twisted angle, $\theta \approx {\left(1.11\pm 0.05\right)}^{\circ }$, a pronounced sharp peak, which arises from the nondispersive flat bands at the charge neutrality point, is observed in the tunneling density of states due to the action of the non-Abelian gauge fields. Moreover, we observe confined electronic states in the twisted bilayer, as manifested by regularly spaced tunneling peaks with energy spacing $\delta E\approx v_{\mathrm{F}}/D\approx 70\mathrm{meV}$ (here $v_{\mathrm{F}}$ is the Fermi velocity of graphene and $D$ is the period of the moiré patterns). This indicates that the non-Abelian gauge potentials in twisted graphene bilayers confine low-energy electrons into a triangular array of quantum dots following the modulation of the moiré patterns. Our results also directly demonstrate that the Fermi velocity in twisted bilayers can be tuned from about ${10}^6\mathrm{m}/\mathrm{s}$ to zero by simply reducing the twisted angle of about $2^{\circ }$.

Figure 1

STM topographs of twisted graphene bilayers. (a)–(d) STM topographic images, $40\mathrm{nm}\times 40\mathrm{nm}$, of twisted bilayers with the twisted angle $\theta ={\left(0.88\pm 0.03\right)}^{\circ }$ and the period of the moiré patterns $D=(16.0\pm 0.5)\mathrm{nm}$ (a), $\theta ={\left(1.11\pm 0.05\right)}^{\circ }$ and $D=(12.6\pm 0.6)\mathrm{nm}$ (b), $\theta ={\left(1.88\pm 0.08\right)}^{\circ }$ and $D=(7.5\pm 0.3)\mathrm{nm}$ (c), and $\theta ={\left(3.1\pm 0.1\right)}^{\circ }$ and $D=(4.5\pm 0.3)\mathrm{nm}$ (d) on graphite surface. The error bar of the twisted angle is estimated according to the error bar in determining the period of the moiré patterns. (e) Atomic resolution STM image of a twisted bilayer with $\theta ={\left(3.8\pm 0.1\right)}^{\circ }$ and $D=\left(3.7\pm 0.1\right)\mathrm{nm}$. It shows different stacking orders depending on positions of the moiré structure. In the AA-stacking region, we can observe honeycomb lattice, whereas in the AB-stacking region, only triangular lattice can be observed. (f) Range of STM measured corrugation of different moiré patterns as a function of sample voltages.

Figure 2

STS spectra and $dI/dV$ maps of twisted graphene bilayers. (a)–(d) Spatial evolution of $dI/dV$ spectra along the dashed lines in STM images of the insets for the twisted bilayers with $\theta ={\left(0.88\pm 0.03\right)}^{\circ }$ (a), $\theta ={\left(1.11\pm 0.05\right)}^{\circ }$ (b), $\theta ={\left(1.88\pm 0.08\right)}^{\circ }$ (c), and $\theta ={\left(3.1\pm 0.1\right)}^{\circ }$ (d). The VHSs of the twisted bilayers are marked by arrows in the spectra. For the $1.{88}^{\circ }$ and $3.1^{\circ }$ bilayers, the two peaks with energy spacing of 80 and $350\mathrm{meV}$, respectively, are the two low-energy VHSs. For the $1.{11}^{\circ }$ and $0.{88}^{\circ }$ samples, a pronounced peak mainly localized in the AA-stacking regions is observed at the charge neutrality point of the spectra, indicating the emergence of nondispersive flat bands. The FWHMs of the peaks for the $1.{11}^{\circ }$ and $0.{88}^{\circ }$ bilayers are 18 and $30\mathrm{meV}$, respectively. (e) $25\mathrm{nm}\times 25\mathrm{nm}$ STM topography image of the $1.{11}^{\circ }$ moiré patterns. (f) and (g) $dI/dV$ maps of the same region in (e) at two bias voltages. The map recorded at 18 mV, the energy of the flat bands, shows localized states following the modulation of the moiré patterns, whereas the map recorded at 500 mV, the energy far from the flat bands, exhibits extended states. (h) $10\mathrm{nm}\times 10\mathrm{nm}$ STM topography image of the $1.{88}^{\circ }$ moiré patterns. (i) and (j) $dI/dV$ maps of the same region in (h) at two bias voltages, 100 mV (i) and 400 mV (j). It also displays localization of the DOS in the $dI/dV$ map recorded at 100 mV, the position of one of the VHSs in (c). The charge density becomes nearly homogeneous for energies away from the VHSs.

Figure 3

Electronic band structures and quantum confinement in twisted bilayers. (a) Low-energy subband of twisted bilayers with different twisted angles. Here $\left|\mathrm{\Delta }\mathit{K}\right|=2\left|\mathit{K}\right|sin(\theta /2)$ is the relative shift of the Dirac points of the adjacent bilayers with $K$ the reciprocal-lattice vector. The energy separation of the two saddle points decreases with decreasing the rotation angle and the low-energy nondispersive flat bands appear in the $1.{11}^{\circ }$ bilayer. For clarity, only the lowest subbands are shown. (b) Left panel: A typical differential conductance spectrum recorded in the $1.{11}^{\circ }$ bilayer. Besides the pronounced peak at the charge neutrality point, several regularly spaced resonances appear in the spectrum, as marked by the red crosses. Right panel: Energy positions of the resonance peaks in the left panel as a function of the positive integers. The energy separation of these peaks is about $70\mathrm{meV}$.

Figure 4

STS spectra of twisted graphene bilayers in different magnetic fields. (a) $dI/dV$ spectra taken at a bright dot in the $1.{11}^{\circ }$ bilayer with different magnetic fields. (b) Tunneling spectra of the $3.1^{\circ }$ moiré pattern measured under various magnetic fields. For clarity, the curves are offset in the $Y$ axis and the Landau level indices of graphene monolayer are marked. (c) FWHM of the peaks at the charge neutrality points of the $0.{88}^{\circ }$ and $1.{11}^{\circ }$ moiré patterns in different magnetic fields. (d) Landau level peak energies for different magnetic fields obtained in (b) are shown to be linear with $\mathrm{sgn}\left(n\right){\left(\left|n\right|B\right)}^{1/2}$, as expected for massless Dirac fermions. The solid line is a linear fitting of the data with the slope yielding a Fermi velocity of $v_F=\left(0.82\pm 0.01\right)\times {10}^6\mathrm{m}/\mathrm{s}$.