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Cold atoms coupled to photonic crystals constitute an exciting platform forexploring quantum many-body physics. Here we investigate the strong couplingbetween atomic internal ("spin") degrees of freedom and motion, which arisesfrom spin-dependent forces associated with the exchange of guided photons. Weshow that this system can realize a remarkable and extreme limit of quantumspin-orbital systems, where both the direct spin exchange between neighboringsites and the kinetic energy of the orbital motion vanish. We find that thispreviously unexplored system has a rich phase diagram of emergent orders,including spatially dimerized spin-entangled pairs, a fluid of compositeparticles comprised of joint spin-phonon excitations, phonon-induced N\'eelordering, and a fractional magnetization plateau associated with trimerformation.

Designing exotic many-body states of atomic spin and motion in photonic crystals