  Magnetic order typically emerges due to the short-range exchange interaction
between the constituent electronic spins. Recent discoveries have found a
crucial role for spin-phonon coupling in various phenomena from optical
ultrafast magnetization switching to dynamical control of the magnetic state.
Here, we demonstrate theoretically the emergence of a biquadratic long-range
interaction between spins mediated by their coupling to phonons hybridized with
vacuum photons into polaritons. The resulting ordered state enabled by the
exchange of virtual polaritons between spins is reminiscent of
superconductivity mediated by the exchange of virtual phonons. The biquadratic
nature of the spin-spin interaction promotes ordering without favoring ferro-
or antiferromagnetism. It further makes the phase transition to magnetic order
a first-order transition, unlike in conventional magnets. Consequently, a large
magnetization develops abruptly on lowering the temperature which \aknew{could}
enable magnetic memories admitting ultralow-power thermally-assisted writing
while maintaining a high data stability. The role of photons in the phenomenon
further enables an in-situ static control over the magnetism. These unique
features make our predicted spin-spin interaction and magnetism highly
unconventional paving the way for novel scientific and technological
opportunities.
