## 7 Twisted Multilayer Graphene  
Apart from the gating as a major tool to tune the correlated states in TBG, the number of layers is also a knob for tuning. Actually, correlation effects also exist in the multilayer graphene with or without the twist angle, for example, the twisted double-bilayer graphene (TDBG) [74; 75; 21; 76; 77] formed between twisted AB stacking bilayer graphene. It is well known that the AB stacking bilayer graphene  
Figure 6: Local probe measurements of TBG. (**a**) STM density of states (DOS) mapping of the moiré pattern with period —50 nm. Inset: scheme of the band structure. (**b**) Experimental conduction and valence VHS half-widths and tight-binding half-widths versus twist angle. (**c**) Superimposed STM DOS map showing the stripe net charge order. (**d**) Evolution of the TBG point spectrum with back-gate voltage. Right panel: schematic of the TBG band structure. (**e**) Schematic of SQUID on tip scanning over magic angle TBG. Panel (**a**) adapted with permission from Ref. [71], Copyright American Physical Society 2018. Panel (**b**) adapted with permission from Ref. [19], Panel (**c**) adapted with permission from Ref. [18] and Panel (**d**) adapted with permission from Ref. [17], Copyright Nature Publishing Group 2019. Panels (**e**) adapted with permission from Ref. [72], Copyright Nature Publishing Group 2020.  
has parabolic bands touching at low energies close to the K point, and the gap opening at the charge neutral point could be induced under electric displacement fields, as shown in Figure 7a. Theory suggests that strong electron-electron interaction may occur in such a TDBG system at the 1/2 and1/4 filling of the flat band [78]. Indeed, a number of experimental works show evidence that the correlated insulate states exist in TDBG. [PERSON] et al. recently investigated the response of the correlated states to magnetic fields, perpendicular or in-plane [75]. At \(B_{\perp}\) = 8 T, the correlated insulate state at 1/2 \(n_{s}\) shifts to a bigger D region in the n-D map, and the 1/4 \(n_{s}\) insulate state shifts to a lower D region, as shown in Figure 7b. At the paralleled magnetic field, no such strong shift is observed at all fractional filling. In the evolution of the thermally activated gap of the 1/2 \(n_{s}\) state, they find a g-factor of \(g_{\parallel}\approx\) 1.5 for the in-plane magnetic field, closer to g = 2 that corresponds to electron spin contribution, rather than \(g_{\perp}\approx\) 3.5 for the out of plane magnetic field. These results suggest that the correlated states are spin-polarized. [PERSON] et al. obtained an effective g factor of 2.12 by fitting the thermal activation gap according to the Zeeman effect in in-plane magnetic field [74], as shown in Figure 7c, which indicates spin polarization at half-filling. [PERSON] et al. also reported the ferromagnetic order and superconductivity at 1/2 filling in TDBG [77]. Additionally, they surprisingly observed that the superconductivity is enhanced in the low in-plane magnetic field region, revealed by a 50% increase of critical temperature as \(B_{\parallel}\) increases from 0 to about 0.3 T, again suggesting unusual spin-polarized electron pairing [77]. On an ABC stacking trilayer graphene with the 1/4 or 1/2 filling of the flat band, [PERSON] et al. observed the Mott-insulating state [79], superconductivity [80], ferromagnetism and Chern insulator [15], as shown in Figure 8a-c. By applying the vertical field, they observed splitting of the low energy band and suppressing of the high energy flat band, indicating correlation due to  
Figure 7: The correlated insulated and potential spin-polarized state in TBG. (**a**) Schematics of a moiré unit cell in TBG and the band structure with and without tuning by the displaced field. (**b**) Longitudinal resistance \(R_{xx}\) plot versus the displaced field and carrier density for the TDBG device in magnetic fields of \(B_{\perp}\)= 8 T. The dashed blue line denoted the original center of the insulated state with the 1/4 and 1/2 filling at \(B=0\).