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We study a one-dimensional atomic lattice gas in which Rydberg atoms areexcited by a laser and whose external dynamics is frozen. We identify aparameter regime in which the Hamiltonian is well-approximated by a spinHamiltonian with quasi-local many-body interactions which possesses an exactanalytic ground state solution. This state is a superposition of all states ofthe system that are compatible with an interaction induced constraint weightedby a fugacity. We perform a detailed analysis of this state which exhibits across-over between a paramagnetic phase with short-ranged correlations and acrystal. This study also leads us to a class of spin models with many-bodyinteractions that permit an analytic ground state solution.

Many-body spin interactions and the ground state of a dense Rydberg lattice gas