Topological phases in ultracold polar-molecule quantum magnets

We show how to use polar molecules in an optical lattice to engineer quantum spin models with arbitrary spin $S\ge 1/2$ and with interactions featuring a direction-dependent spin anisotropy. This is achieved by encoding the effective spin degrees of freedom in microwave-dressed rotational states of the molecules and by coupling the spins through dipolar interactions. We demonstrate how one of the experimentally most accessible anisotropies stabilizes symmetry protected topological phases in spin ladders. Using the numerically exact density matrix renormalization group method, we find that these interacting phases—previously studied only in the nearest-neighbor case—survive in the presence of long-range dipolar interactions. We also show how to use our approach to realize the bilinear-biquadratic spin-1 and the Kitaev honeycomb models. Experimental detection schemes and imperfections are discussed.

Figure 1

A lattice of polar molecules in the $XY$ plane is subjected to a dc electric field along $\widehat{z}$. We define the $xyz$ coordinate system as the rotation of the XYZ coordinate system around $\widehat{Z}$ by ${\Phi }_0$ and then around $\widehat{y}$ by ${\Theta }_0$. A vector $\mathbf{R}$ with polar coordinates $(R,\Phi )$ in the $XY$ plane has spherical coordinates $(R,\theta ,\varphi )$ in the $xyz$ coordinate system.

Figure 2

(a) The level scheme and resonant microwave coupling used to realize the Hamiltonian, Eq. (2). The dressed states we choose are $\left\{\right|\uparrow \rangle ,|\downarrow \rangle \}=\left\{\right|0,0\rangle ,\left({\Omega }_{-}\right|1,-1\rangle -{\Omega }_{+}|1,1\rangle )/\sqrt{{\Omega }_{-}^2+{\Omega }_{+}^2}\}$. (b) Microwave-dressed rotational levels for the SU(2)-symmetric spin-1 model. The dressed states are $|1\rangle$ (linear combination of states indicated by triangles), $|0\rangle$ (ovals), and $|-1\rangle$ (the rest). The diagram is schematic: The real system is anharmonic and levels $|N,M\rangle$ with the same $N$ are nondegenerate (unless the levels have the same $\left|M\right|$).

Figure 3

Spin ladder of Ref. 50 and its phase diagram showing SPT phases in the presence of long-range interactions. (a) The nearest-neighbor model. (b) Phase diagram in the presence of long-range interactions. The shaded area indicates points achievable with the simple configuration of Fig. 2a. The shaded area does not extend past the limits of the vertical axis, while it extends infinitely far along the horizontal axis. (c) Entanglement splitting ${\Delta }_{\mathrm{ES}}$ (open boxes, left axis) and energy gap (solid boxes, right axis) of the SPT phases along ${\lambda }_z=-0.5$ cuts (bold, red lines) shown in (b).