  Since the latter half of the 20th century, the use of metal in optics has
become a promising plasmonics field for controlling light at a deep
subwavelength scale. Surface plasmon polaritons localized on metal surfaces are
crucial in plasmonics. However, despite the long history of plasmonics, the
underlying mechanism producing the surface waves is not fully understood. This
study unveils the hidden symmetry protection that ensures the existence of
degenerated electric zero modes. These zero modes are identified as physical
origins of surface plasmon polaritons, and similar zero modes can be directly
excited at a temporal boundary. In real space, the zero modes possess
vector-field rotation related to surface impedance. Focusing on the surface
impedance, we prove the bulk-edge correspondence, which guarantees the
existence of surface plasmon polaritons even with nonuniformity. Lastly, we
extract the underlying physics in the topological transition between metal and
dielectric material using a minimal circuit model with duality. The transition
is considered the crossover between electric and magnetic zero modes.
