## 5 Ferromagnetism and Quantum Anomalous Hall Effect  
The ferromagnetic state in TBG has been predicted by [PERSON] et al. [62] in 2017 and antiferromagnetism was suggested as well [63]. Basically, fine tuning of the band filling would lift the degeneracy, and thus induce ferromagnetism in the system. For example, the spin symmetry and valley symmetry could be partially or completely broken in the 1/4, 1/2 and 3/4 filling. [PERSON] et al. reported that the transport measurement shows hysteretic behavior with respect to an applied out-of-plane magnetic field B below 3.9 K, near the 3/4 filling of the moire band in a TBG Hall bar device [64], as shown in Figure 5a. Considering the absence of transition metal and other heavy elements, the large amorous hall signal in TBG is surprising. This suggests spin and valley symmetries in the correlated insulated states are spontaneously broken. [PERSON] et al. also found the ferromagnetic state near the 3/4 filling in TBG devices [16]. The Hall resistivity is hysteretic, with a critical field of several tens of mT. At low temperature, lateral resistance R\({}_{yx}\) quickly saturated at quantum resistivity h/e\({}^{2}\) along with a low longitudinal resistivity \(R_{xx}<1\) k\(\Omega\), as shown in Figure 5b, which reveals a topological insulating state with Chern number = 1, similarly to the result by [PERSON] et al. [14]. These results again indicate a quantum anomalous Hall (QAH) state by spontaneously broken time-reversal symmetry of the moire band. Additionally, they both found the quantum anomalous Hall state can be switched on/off by a low direct current [16; 64], as shown in Figure 5c,d, providing a possibility for functional devices.  
This unusual magnetism in TBG, revealed by the anomalous Hall effect, cannot be explained by a trivial reason like adsorbed impurities or defects. Due to the absence of the \(d\) electron or \(f\) electron, the unusual magnetism at the 1/4 filling can be attributed to a field stabilized orbital magnetic state [14; 65]. Additionally, the certainly potential ferromagnetic topological insulator, found at the 3/4 filling, is explained by spin or valley symmetry broken from the flat band. A series of theoretical works have attempted to find the reason of the electrically controllable magnetism [58; 66; 67; 68; 69] and the origin of the Chern number in TBG by calculating the topology of the flat band [34; 35; 37]. To better understand the magnetism experimentally, these spin or valley polarized states, which are around the AA region as suggested by the theory, need to be verified by performing more detailed local probe characterization.  
Figure 4: Tunable superconducting and insulate state in TBG. (**a**) Color plot of longitudinal resistance versus carrier density and temperature. SC: Superconducting state; CS: correlated state; BI: band insulator. (**b**) Longitudinal resistance plotted versus magnetic field at various temperatures. (**c**) Longitudinal resistance versus magnetic field and filling factor up to \(\pm\)12. (**d**) Longitudinal resistance versus back gate voltage at various temperatures. Inset: an optical image of the TBG device with a scale bar of 10 \(\upmu\)m. Panel (**a,b**) adapted with permission from Ref. [14], Copyright Nature Publishing Group 2019. Panel (**c,d**) adapted with permission from Ref. [42], Copyright AAAS 2019.