Synergistic control of transverse and longitudinal spin excitations in the Lieb lattice via electron–phonon coupling and magnetic fields

We investigate the dynamical and static spin structure factors of the Lieb lattice in the Holstein model under a perpendicular magnetic field. Using the Green’s function formalism combined with the random phase approximation, we examine how electron–phonon coupling and Zeeman splitting jointly influence the magnetic excitations in this flat-band system. The Lieb lattice, which hosts a dispersive remnant of the ideal flat band coexisting with Dirac cones, provides a versatile platform for exploring the synergistic effects of Holstein coupling, next-nearest-neighbour hopping, sublattice energy asymmetry, and external magnetic fields. Our numerical results show that both the Holstein coupling strength and the perpendicular magnetic field act as effective external control parameters. These parameters systematically modify the positions, intensities, and spectral weights of the resonance peaks in the dynamical spin structure factors in both transverse and longitudinal channels. Furthermore, the static spin structure factors exhibit non-monotonic dependence on temperature and magnetic field, with enhanced spin correlations appearing at moderate coupling and field strengths. These findings indicate that the Lieb lattice constitutes a tunable platform in which electron–phonon interaction and Zeeman field can be utilized to engineer collective magnetic excitations. The identified control mechanisms provide valuable insights that may guide the design of reconfigurable magnonic waveguides, logic gates, and hybrid quantum devices based on artificial quantum lattices. The results also offer direct theoretical benchmarks for inelastic neutron scattering experiments in ultracold atomic gases, photonic lattices, and superconducting circuit platforms.