Flat Band Characteristics in Hexagonal Lattice Circuit QED Systems with Triple-leg Stripline Resonators*

In circuit QED systems, a flat band have previously been observed in Kagome lattices and Lieb lattices with tunable hopping strengths coupled superconducting resonators. In this study, we delve into various geometric aspects of a hexagonal optical lattice, akin to graphene, which has upper and lower flat band with Triple-leg stripline resonators(TSRs). The inherent two-fold degenerate modes of the TSR resemble a behavior of orbitals in 2D lattice systems. Our analysis reveals how the system capitalizes on destructive interference to establish flat bands, underpinned by stabilized compact localized states (CLSs). Furthermore, we explore the real-space topology corresponding to the flat bands by finding proper non-contractible loop states (NLSs). Additionally, in a zigzag-structured hexagonal lattice, we demonstrate the induction of flat edge modes at zero energy by calculating the Zak phase. We also elucidate the origin of other dispersive edge bands that arise from the singular point of the flat band.