Extended Data Fig. 6: Complete electronic structure of liquid metals.

a,b Band dispersion (a) and probability density (b) of wavefunctions obtained by the theoretical model^46,47,48,49 that was initially developed for liquid metals but can be generally applied to any non-crystalline system in the presence of the short-range order. There are two branches in terms of partial wave analysis⁴⁶: One is the p-wave or d-wave states at resonance scattering predicted by Anderson and McMillan⁴⁷ to show the back-bending band dispersion (due to the real part of Δk as shown by the black curve) and the pseudogap (due to the imaginary part of Δk as shown by the grey area), as shown in a. This is due to the formation of quasi-bound states (QBS), as shown by the black curve in b, within the scattering potential (dotted black line). The other is s-wave states⁴⁹, for which resonance scattering is forbidden by the absence of a potential barrier as shown by the red dotted line in b. The presence of the unbound states represented by the red curve in b was predicted by Schwartz and Ehrenreich⁴⁸ to be related to another aperiodic (damped oscillatory) branch in band dispersion that extends towards the zone boundary^30,31 as shown by the red curve in a. The grey region surrounding the red curve shows the imaginary part of Δk that accounts for the small increase in the peak width of EDCs indicated by the red arrow in Extended Data Fig. 1c.