Unified optical response of two-dimensional Lieb-to-kagome lattices

In this study we present a thorough theoretical and numerical investigation of the unified optical conductivity response in two-dimensional (2D) lattice systems that transition between the Lieb and kagome lattices through a tunable shear angle, $\ensuremath{\phi}$. This continuous transformation enables us to analyze how optical properties evolve with lattice geometry. Our findings reveal that distinct interband transitions, either Dirac to Dirac and flat to Dirac, dominate at various structural stages, with the real longitudinal optical conductivity potentially reaching twice that of graphene (${\ensuremath{\sigma}}_{0}=\frac{{e}^{2}}{4\ensuremath{\hbar}}$) at a very low frequency. We establish that shear-induced structural changes lead to significant band hybridization while maintaining a low-frequency optical conductivity quantized at $2{\ensuremath{\sigma}}_{0}$. Sharp resonances arise in the finite-frequency regime due to strain-induced van Hove singularities. We also derive both real and imaginary components of optical conductivity, uncovering polarization-dependent plasmonic excitations, including TM-polarized modes in the low-frequency range (1--10 THz) and TE-polarized modes across a wider high-frequency window (65--240 THz) than seen in graphene and other 2D materials. These findings not only reinforce the physical relevance of flat bands and valley degeneracy but also highlight a feature of intraband conductivity under impurity-induced transitions. We explore intraband transitions in a distorted flat band during the Lieb-to-kagome lattice transformation, enabling a tunable low-frequency optical response. Our study combines analytical methods and numerical techniques, providing a platform to optimize the optical responses of tunable 2D lattices.