Exotic Order in Simple Models of Bosonic Systems

We show that simple Bose Hubbard models with unfrustrated hopping and short range two-body repulsive interactions can support stable fractionalized phases in two and higher dimensions, and in zero magnetic field. The simplicity of the constructed models advances the possibility of a controlled experimental realization and novel applications of such unconventional states.

Figure 1

Josephson junction array on a 2D bond-centered square lattice modeled by the Hamiltonian Eq. (1). Each shaded area indicates schematically cluster charging energy $U{N}_r^2$.

Figure 2

Schematic phase diagram of the boson Hubbard model Eq. (1) for a particular cut $w\equiv w_1\simeq w_2$ and $u_b\simeq u_{\psi }$ through the parameter space.

Figure 3

Diagram of the large $U$ insulating phases of the boson model Eq. (1) in two dimensions. The effective Hamiltonian Eq. (3) is equivalent to the $(2+1)\mathrm{D}$ compact QED gauge theory coupled to a charge 2 scalar. The Mott insulator is conventional or fractionalized depending on whether the effective gauge theory is confined or deconfined.

Figure 4

Large $U$ insulating phases of the 2D boson model at half filling, Eq. (6). The region of stability of the fractionalized phase is enhanced compared with Fig. 3 at integer filling. There may be several (?) nonfractionalized insulating states with broken translational invariance; here, we focus on the fractionalized state only.

Figure 5

Same as in Fig. 3 but in three dimensions. There is an additional fractionalized phase, the “Coulomb” phase, with the distinct excitations being a gapped charge $q_b/2$ chargon, gapless photon, and a gapped monopole.