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

Abstract Based on the interconvertibility between the Lieb and Kagome lattices, parameterized by θ ∈ [ π / 2 , 2 π / 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 of θ induces a transition in the point group from D 4 h (Lieb) to D 6 h (Kagome), passing through D 2 h , altering the band symmetries and leading to the lifting of degeneracies. In the intermediate stages, band crossings along the M → – Γ → and M → – 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 the X → and M → 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.