## 4 Mott Insulator and Superconductivity
(**c**) Measured conductance of the TBG device D1 at T = 0.3 K. Inset, the carries density locations of half-filling states in the four different devices. (**d**) Longitudinal resistance measured in two devices respectively. Inset: an optical image of device M1. (**e**) Longitudinal resistance versus temperature and carries density for device M2. Panel (**a**,**d**,**e**) adapted with permission from Ref. [12], Copyright Nature Publishing Group 2018. Panel (**b**) adapted with permission from Ref. [26], Copyright American Physical Society 2016. Panel (**c**) adapted with permission from Ref. [13], Copyright Nature Publishing Group 2018.  
full filling condition (\(n=n_{s}\)), but also at the half filling condition (\(n=\pm n_{s}\)/2), as shown in Figure 3c. Additionally, by measuring the Shubnikov-de Haas oscillation frequency \(f_{SdH}\), they deduced that the degeneracy was to be 4 at charge neutrality and 2 at the half-filling states, and suggested the halved degeneracy of the Fermi pockets is related to the spin-charge separation. These results are unambiguous proof of Mott-like correlated behavior at half-filling. In a back to back paper on the same journal, [PERSON] et al. reported unconventional superconductivity in the same material system, as shown in Figure 3d, which was realized by back gate tuned electrostatic doping into the TBG slightly away from the correlated insulating state [12]. They observed tunable zero-resistance states, i.e., superconductivity, in two TBG devices with critical temperature of 1.7 K and 0.5 K respectively. Both devices show a maximum critical field of about 70 mT. The temperature and magnetic field-carrier density phase diagram demonstrate two dome-shaped superconducting phase regions approximate to both sides of the half-filling Mott-insulator region, as in Figure 3e. Such behavior of magic angle TBG is very similar to that of cuprates superconductors. These works immediately inspired enormous attention to the TBG systems, both experimental and theoretical [42; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59]. Due to the fourfold degeneracy of the moire flat band, one can define a filling factor \(\
u=n/n_{0}=4n/n_{s}\) (\(n_{0}\) is the carrier density for one electron per moire unit cell). It is expected that insulating states appears when \(\
u\) is the integer and superconducting exists between them. Indeed, [PERSON] et al. reported that on a more accurate (\(\theta\) = 1.1 \(\pm\) 0.02\({}^{\circ}\)) magic angle TBG device, the insulating state appears at the integer filling factor positions, as shown in Figure 4a [14]. The temperature-carrier density phase diagram shows superconductivity up to 3 K around \(\
u\approx-2\). Surprisingly, there are also three new superconducting domes at much lower temperatures, close to the \(\
u\) = 0 and \(\
u\) = \(\pm\)1 insulating states. Moreover, at \(\
u\) = \(-1\), when the perpendicular magnetic field is higher than 3.6 T, the insulating state exhibits a sharp hysteretic resistance and the longitudinal resistance saturated at about 13 k\(\Omega\) (Figure 4b). These details imply orbital magnetism and a non-zero Chern number induced by a possible field-driven phase transition at low temperature. [PERSON] et al. [42] reported an insulating state at a filling factor as high as 12 in the TBG with a twist angle of 0.93\({}^{\circ}\), as shown in Figure 4c,d, which is surprising since at such a high filling condition the Fermi level has reached the dispersing bands. [PERSON] et al. [56] induced superconductivity in TBG with twist angle \(\theta\) = 1.27\({}^{\circ}\), larger than the first magic angle, by varying the interlayer spacing with hydrostatic pressure [60]. These superconducting (SC) domes are almost next to the correlated insulate states region in the B-T phase diagram, similar to the effect of the doping of the Mott insulator and implying the strong correlation effect between the electrons on the flat band in special filling. However, some theoretical works attributed the superconductivity in TBG to phonon driven [48; 51].