Quantum geometry in hexagonal circuit QED lattice with triple leg stripline resonators

In this study, we investigate various geometric aspects of a photonic hexagonal lattice made of triple-leg stripline resonators (TSRs) in a circuit QED system. The inherent two-fold degenerate spatial modes of the TSR act as two distinct orbitals in our 2D lattice system. Remarkably the energy spectra of the system exhibits the dispersive quadratic band-touching to the top and bottom flat bands. Our analysis reveals how the system harnesses destructive interference to establish flat bands via stabilized compact localized states (CLSs). We further 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 topological flat edge modes at zero energy by analyzing the Zak phase. We also elucidate the quantum geometric origin of other dispersive edge bands that arise from the singular point of the flat band.