We present a new class of structured optical media - composite materials that support guided modes which originate from the recently discovered electromagnetic "ghost" waves. Such "ghost crystals" show a broad range of novel optical phenomena - from the exceptional points to the linear-optical equivalent of the exchange interaction in correlated fermionic systems. In particular, the formation of the propagating waves in optical "ghost crystals" is related to the spin interaction in the Ising model, offering full control on the exchange dynamics and leading to a realization of the ferromagnetic – antiferromagnetic phase transition in a linear-optical system.
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