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We investigate the structure of the spectrum of antiferromagnetically coupledspin-1 bosons on a square lattice using degenerate perturbation theory andexact diagonalizations of finite clusters. We show that the superfluid phasedevelops an Anderson tower of states typical of nematic long-range order withbroken SU(2) symmetry.We further show that this order persists into the Mottinsulating phase down to zero hopping for one boson per site, and down to acritical hopping for two bosons per site, in agreement with mean-field andQuantum Monte Carlo results. The connection with the transition between afragmented condensate and a polar one in a single trap is briefly discussed.

Anderson tower of states and nematic order of spin-1 bosonic atoms on a 2D lattice