Fig. 4: Transition properties of the thermal dissipation within the of the conductive moiré superlattice.

a–c present the effective band structures of the first Brillouin zones, respectively, at the twisted angles of 36.87°, 22.62°, and 30°. The line colors indicate the effective bands at different frequencies. Significant degeneracies occur with non-flat bands in the commensurable superlattice at 36.87°and 22.62°, while flat bands appear in the incommensurable superlattice at 30°. d–f exhibit thermal profiles under the incident temperature wave through a pulsing point source (150 Hz), whose positions are indicated by orange stars. Rapid delocalization and localization of the temperature fields are respectively observed in the commensurable and incommensurable superlattices. g plots the phase diagram of the localized degrees (integral form factor) of the passing-through temperature wave under twisted angles and the ratio of conductivity (\({p}_{c}\)). The white-dashed lines indicate some representative Pythagorean angles. h, i present the threshold frequencies for motivating the transitions between thermal delocalization and localization at a specific non-Pythagorean angle. The error bars denote the standard deviations among measurements. Notably, such a threshold also exists in commensurable superlattices at Pythagorean angles. However, only field delocalization is significant due to the quite small integral form factor.