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

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