Effect of magnetic field and light on energy levels of (1+3+1) chirally twisted multilayer graphene system

We study the Hofstadter butterfly spectrum in (1+3+1) chirally twisted multilayer graphene (CTMLG) subject to perpendicular magnetic field and light with different polarizations. We focus on the interplay between twist angles and light-induced effects. In equilibrium, we examine symmetric ($\theta_1 = \theta_2$) and asymmetric ($\theta_1 \neq \theta_2$) configurations. Our results show that asymmetric configurations cause distinct effects in the electronic energy spectrum. However, the unique symmetry of the system ensures that the spectra remain identical when the twist angles are interchanged. This highlights the role of interlayer coupling in shaping the electronic structure of CTMLG. We then explored the effects of external periodic perturbations, such as circularly polarized light (CPL) and waveguide-generated linearly polarized light (WGL). CPL breaks chiral symmetry, creating a gap that distorts the Hofstadter spectrum. These distortions are more pronounced for asymmetric twist configurations. In contrast, WGL preserves chiral symmetry and has a tunable, non-monotonic effect on the bandwidth. This makes WGL a reliable tool for engineering electronic properties. These results demonstrate how (1+3+1)-CTMLG combines the effects of light-matter interactions with moir\'e physics. This allows accurate control of the electronic properties and fractal spectra by adjusting external fields and twist angles.