## 4 Mott Insulator and Superconductivity  
The superconductivity in TBG with the magic angle was firstly reported by [PERSON] et al. in 2018 [12], and confirmed by many other researchers soon after [14; 42; 43]. The superconducting behavior of TBG is largely similar to the doped Mott insulator, similar to unconventional high T\({}_{\rm C}\) superconductor. Although it is well known that the monolayer graphene is a zero gap semiconductor with the highest mobility [44; 45; 46] among all the materials, [PERSON] et al. surprisingly discovered the insulating behavior in the TBG in 2016 [26]. They had successfully fabricated dual-gate hall bar devices (Figure 3a) with two TBG of different twist angles. The one with a twist angle \(>3^{\circ}\) shows a normal V-shape dip in the conductivity-carrier density curve, as shown in Figure 3b, which indicates the normal behavior of conductivity increases with carrier density around the zero density (charge neutral) point. However, in the small angle sample \(\theta=(2.0\pm 0.5)\)\({}^{\circ}\), they observe two insulating states, which are symmetric on both sides of the charge neutrality point. Such insulating states are suppressed when the  
Figure 2: Van Hove singularities on the electronic structure of TBG. **(a)** Emergence of van Hove singularities (VHS) as a consequence of the rotation in reciprocal space. **(b)** Scanning tunneling microscopy (STM) image of the boundary of the moiér pattern. **(c)** Corresponding spatial dependence of tunneling spectra along a line connecting point M2 to G in **(b)**. **(d)** Angle dependence of the Fermi velocity renormalization. Inset, Dirac cones of twisted layers. **(e)** VHS separation as a function of rotation angle \(\theta\). **(f)**\(\upmu\)-ARPES spectra of TBG. Red arrows indicate the minigap band topology arising from interlayer coupling. Panels **(a,e)** adapted with permission from Ref. [40], Copyright American Physical Society 2012; Panels **(b,c)** adapted with permission from Ref. [3], Copyright Nature Publishing Group 2010; Panel **d** adapted with permission from Ref. [4], Copyright American Physical Society 2011 and Panel **(f)** adapted with permission from Ref. [30], Copyright Nature Publishing Group 2016.  
temperature increases. Interestingly, the total density of the states required to fill up the insulating gaps, \(n=7.5\times 10^{12}\) cm\({}^{-2}\), is 4 times of the mini Brillouin zone area, indicating two fold valley degeneracy and two fold spin degeneracy. Therefore, they could derive the unit cell area of the moire superlattice to be \(4/n=53.3\) nm\({}^{2}\), which correspond to the twist angle of \(\theta=1.8^{\circ}\), matching the target value of \(\theta=(2.0\pm 0.5)^{\circ}\). The insulating behavior could be well explained by the flat band in magic angle TBG, as introduced in Section 1. Basically, when the Fermi level crosses the flat band, the dynamic energy of electron is limited by the width of the flat band. The Bloch electron in flat band would have a very large effective mass, and thus the Coulomb interaction between electrons would dominate their behavior, i.e., Mott insulator at low temperature. Additionally, due to the combination of valley degeneracy and spin degeneracy, the flat band is fourfold degeneracy. In 2018, [PERSON] et al. [13] verified the Mott insulating states of TBG by measuring devices with twist angles carefully controlled to \(1.1^{\circ}\), close to the first magic angle. The full filling condition is four electrons per moire unit cell, corresponding to a carrier density \(n_{\text{s}}\) (in this work \(n_{\text{s}}=2.7\times 10^{12}\) cm\({}^{-2}\)). It was found that the insulating states not only appear at the charge neutral point (n = 0) and the  
Figure 3: Mott insulator and superconductivity verified by the transport measurement in magic angle TBG. (**a**) Schematic of a typical TBG device and the four-probe measurement. (**b**) Comparison of the conductivity of two TBG devices.