Unveiling hidden millihertz quasi-periodic oscillations in 1A 0535+262
Be/X-ray binary pulsars show transient outbursts and complex timing behaviour, including millihertz quasi-periodic oscillations (QPOs), whose physical origin and energy dependence remain poorly understood. We aim to characterise the temporal evolution and energy-dependent properties of the mHz QPO during the 2020 giant outburst of 1A 0535+262. We use the multi-Lorentzian fitting framework to jointly model the power spectra and the real and imaginary parts of the cross-spectrum, incorporating simultaneous broadband X-ray observations from NICER and Insight-HXMT (0.2-120 keV). We report the first detection of weak, but significant, mHz QPOs at low X-ray energies (below 27 keV), extending their detection to a new energy regime. The centroid frequency evolves from 41 to 93 mHz, with the peak root-mean-square (rms) amplitude detected in the 50-65 keV. Throughout the outburst, the QPOs generally exhibit a hard lag between 0.12 pi rad and 0.9 pi rad. However, at the outburst peak, the higher-energy bands (above 35 keV) display a soft lag of up to -0.93 pi rad. We propose that interactions between soft seed photons and an extended outflow located outside the magnetosphere can account for the observed hard lags. Furthermore, we detect a double-peaked mHz QPO only at high energies (E above 35 keV) near peak luminosity. The two peaks maintain an approximately constant separation of 2*nu_spin and exhibit anti-correlated phase evolution. Our results indicate that the mHz QPOs in 1A 0535+262 are closely linked to the coupled evolution of a soft-photon source and a Comptonizing outflow or corona. The joint cross-spectral framework provides a complementary probe of mHz QPOs beyond traditional power-spectral analyses.