$XYZ$ Quantum Heisenberg Models with $p$-Orbital Bosons

We demonstrate how the spin-$1/2$ $XYZ$ quantum Heisenberg model can be realized with bosonic atoms loaded in the $p$ band of an optical lattice in the Mott regime. The combination of Bose statistics and the symmetry of the $p$-orbital wave functions leads to a nonintegrable Heisenberg model with antiferromagnetic couplings. Moreover, the sign and relative strength of the couplings characterizing the model are shown to be experimentally tunable. We display the rich phase diagram in the one-dimensional case and discuss finite size effects relevant for trapped systems. Finally, experimental issues related to preparation, manipulation, detection, and imperfections are considered.

Figure 1

(a) Schematic phase diagram of the $XYZ$ chain. (b) Finite size “phase diagram” obtained by exact diagonalization of 18 spins. The finite size phase diagram comprises an incomplete devil’s staircase of SDW between the PP and AFM phases. The anisotropy parameter is $\gamma =0.2$ in (b).

Figure 2

Different types of models are achieved by varying the relative tunneling strength and the relative orbital squeezing. The three different parameter regions are (I) antiferromagnetic couplings in all spin components with $\Delta >J(1+|\gamma |)$, (II) ferromagnetic or antiferromagnetic couplings in the $z$ component and antiferromagnetic in the $y$ component with $J(1+|\gamma |)>|\Delta |$, and (III) same as in (II) but with $|\Delta |>J(1+|\gamma |)$. The inset shows one example of the spin parameters $J_{xx}=(1+\gamma )$, $J_{yy}=(1-\gamma )$, and $J_{zz}=\Delta /J$ for $t_y/t_x=-0.1$.