## 6 Local Probe Measurement of TBG  
Although the transport measurement already revealed many fascinating properties of TBG systems, one should note that the published results sometime are puzzling or contradicting to each other, for example, different determination to the insulating states at a certain filling to the band [13; 42; 64]; disagreed positions of the superconducting states in the phase diagram [12; 14] and different critical magnetic field or critical temperature values [12; 56]. These contradictions are mainly attributed to the unavoidable variation in device fabrication, local inhomogeneity or twist angle disorder. To avoid these deviations in the measurement that are carried out on a relatively large area, a local probe measurement that could zoom in to the individual moire unit cell would be very helpful. Due to the atomic resolution STM measurements on TBG, [PERSON] et al. [70] observed the one-dimensional conducting channels locating at the edges of the domain wall between the AB and BA regions; [PERSON] et al. [71] observed a double-line network of enhanced density of states on TBG with the moire unit cell lattice constant of about 50 nm, as shown in Figure 6a. The density of the state is enhanced only when the gap was opened by applying a positive electric field, suggesting topologically protected helical edge states on the network. On the TBG samples with varying twist angles from 0.79\({}^{\circ}\) to 3.48\({}^{\circ}\), [PERSON] et al. [19] detected the VHS features in STS spectra. They also found that the half-width of each individual VHS is minimized at the magic angle, as shown in Figure 6b, which means the electron correlations are maximized. Figure 5: Ferromagnetism and quantum anomalous Hall effect (QAHE) in TBG. (**a**) Magnetic field dependence of the longitudinal resistance (upper panel) and Hall resistance (lower panel) with \(n/n_{s}=0.746\) at 30 mK. (**b**) Hall resistance as a function of magnetic field at various temperatures with \(n/n_{s}\approx\) 3/4. (**c**) Differential Hall resistance tuned by a DC bias with \(n/n_{s}=0.746\) at 2.1 K. (**d**) Nonvolatile electrical writing and reading of a magnetic bit at \(T=6.5\) K and \(B=0\). Panel (**a**,**c**) adapted with permission from Ref. [64], Copyright AAAS 2019. Panel (**c**,**d**) adapted with permission from Ref. [16], Copyright AAAS 2020. The influences of the electron-electron correlation interaction to the local electronic structure and local charge ordering are also very suitable for scanning probe measurement. By taking STS on the AA region of the moire unit cell in TBG with the magic angle, [PERSON] et al. [20] studied the flat band peak position under different filling conditions. It was found that in the fully filled or empty condition, the shift of the flat band peak is proportional to the density of states (DOS) at the Fermi level, as shown in Figure 6d, indicating that the electron correlation is weak; while in the partial filling condition, an abrupt distortion of the spectra that cannot be explained by the mean-field model has been observed, indicating that a strong correlation effect is indeed playing a crucial role. Such a conclusion is also supported by the STS measurement on a similar system by [PERSON] et al. [17]. In addition, [PERSON] et al. observed an enhanced flat band peak splitting, which was attributed to exchange interactions. Furthermore, the strong correlation effect could also induce a stripe charge order, as reported by [PERSON] et al. [18], as shown in Figure 6c. Besides the scanning tunneling microscope, the nanoscale on-tip scanning superconducting quantum interference device (SQUID-on-tip) [73] is a powerful tool to study the local magnetic properties at high spatial resolution. Recently, [PERSON] et al. [72] investigated the influence of local \(\theta\) variation on the Landau levels in the quantum Hall states in magic angle TBG, at an ultra-high precision of 0.002\({}^{\circ}\) with SQUID-on-tip, as shown in Figure 6e.