  We consider linear periodic chains of exciton-polariton condensates formed by
pumping polaritons non-resonantly into a linear network. To the leading order,
such a sequence of condensates establishes relative phases as to minimize a
classical one-dimensional XY Hamiltonian with nearest and next to nearest
neighbors. We show that the low energy states of polaritonic linear chains
demonstrate various classical regimes: ferromagnetic, antiferromagnetic and
frustrated spiral phases, where quantum or thermal fluctuations are expected to
give rise to a spin liquid state. At the same time nonlinear interactions at
higher pumping intensities bring about phase chaos and novel exotic phases.
