Symmetry analysis of the mono- and bilayer Lieb-Kagome lattices

Based on the interconvertibility between the Lieb and Kagome lattices, parameterized byθ∈[π/2,2π/3], we investigate the electronic properties of monolayer and bilayer Lieb-Kagome systems using a group-theoretical approach to classify and identify degeneracies at high-symmetry points via irreducible representations. In the monolayer case, the variation ofθinduces a transition in the point group fromD4h(Lieb) toD6h(Kagome), passing throughD2h, altering the band symmetries and leading to the lifting of degeneracies. In the intermediate stages, band crossings along theM→-Γ→andM→-K→paths are guaranteed by compatibility relations between representations. In the bilayer system, AA stacking preserves the monolayer symmetries, while AB stacking induces an effectively non-symmorphic structure, with glide operations protecting double degeneracies at theX→andM→points. Using operator algebra and Herring's method, we show that such degeneracies remain robust throughout the entire morphological transition. Beyond the electronic classification, we extend the group-theoretical framework to derive the selection rules for interband optical transitions and to determine the vibrational normal modes, distinguishing those that are Raman-active and infrared-active. This unified and explicit symmetry-based approach not only consolidates previous analyses of Lieb and Kagome lattices but also provides a general framework that connects crystalline symmetry with measurable optical and vibrational responses along the Lieb-Kagome morph evolution.