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.

Figure 1

Squares $c_N(S{)}^2$ of the coefficients of the expansion of the polar state $|{\psi }_{0,N,0}⟩$ in the eigenstates $|S,m=0⟩$ of ${\overrightarrow{S}}_{\mathrm{tot}}^2$ as a function of $S$ for $N=400$. Inset: value of the spin $S_N$ up to which one has to sum to satisfy the sum rule $\sum _S|c_N(S){|}^2=1$ to a given accuracy. It scales as $\sqrt{N}$.

Figure 2

Low-energy spectra for five sites with $N/N_s=1$ (top) and $N/N_s=2$ (bottom) with $U_2/U_0=0.005$. The energies are in unit of $U_0$ and measured from the ground state. The vertical dashed line indicates the Mott-superfluid transition according to the QMC simulations [12].

Figure 3

Ferromagnetic quadrupolar structure factor for four-, five-, and eight-site clusters with two bosons per site.

Figure 4

Charge gap as a function of the inverse number of sites for various values of $t/U_0$. It opens at $t_c/U_0\simeq 0.037$, which marks the superfluid-insulator transition.