Magnetic-field-induced deformation and spin-resolved flat-band localization in a Lieb lattice

We investigate the magnetic-field-induced deformation of electronic bands in the Lieb lattice within a tight-binding framework, focusing on the interplay between Aharonov-Bohm (AB) caging and Zeeman splitting. By incorporating Peierls phases and Zeeman effects, we demonstrate that magnetic flux can modulate the degeneracy and bandwidth of flat bands, leading to spin-resolved localization phenomena. Specifically, at a perpendicular flux of φ = π, AB caging localizes particles into flat-band states, while Zeeman coupling lifts spin degeneracy and induces spin-selective energy shifts. This stabilization of flat bands facilitates tunable, spin-selective transport and magnetic responses. Our findings position the Lieb lattice as a versatile platform for quantum simulation and spin-resolved control, with potential applications in synthetic systems such as ultracold atoms, photonic lattices, and superconducting circuits. These results offer insights into the controlled engineering of spin-resolved localization and flat-band physics, providing a pathway for future experiments and applications in quantum materials.