Topological Flat Bands from Dipolar Spin Systems

We propose and analyze a physical system that naturally admits two-dimensional topological nearly flat bands. Our approach utilizes an array of three-level dipoles (effective $S=1$ spins) driven by inhomogeneous electromagnetic fields. The dipolar interactions produce arbitrary uniform background gauge fields for an effective collection of conserved hard-core bosons, namely, the dressed spin flips. These gauge fields result in topological band structures, whose band gap can be larger than the corresponding bandwidth. Exact diagonalization of the full interacting Hamiltonian at half-filling reveals the existence of superfluid, crystalline, and supersolid phases. An experimental realization using either ultracold polar molecules or spins in the solid state is considered.

Figure 1

Schematic representation of a 2D dipolar droplet. The grey droplet represents a 2D array of interacting tilted dipoles. The dipoles are tilted by a static field in the $\widehat{z}$ direction, oriented at ${\Theta }_0$, ${\Phi }_0$ relative to the lattice basis $\{X,Y,Z\}$. ${\mathbf{R}}_{ij}$ is a vector connecting dipoles in the $XY$ plane.

Figure 2

(a) Depicts the on-site level structure and the two-photon driving scheme. These levels could, for example, be adiabatically connected to the $J=1$ manifold of a rigid rotor as one turns on a dc electric field [see Eq. (6)]. The resonance frequency of the dressing lasers is detuned by $\Delta$, while their Rabi frequencies are ${\Omega }_{-}(r)$ and ${\Omega }_{+}(r)$. We consider $|{\Omega }_{\pm }|\ll \Delta$ to operate in the far-detuned limit. In the case of polar molecules, $\delta$ is the electric-field induced splitting within the $J=1$ manifold, which we require to be larger than the typical dipolar interaction strength. (b) Square lattice with a single tilted dipole per vertex. We index columns of the lattice by $\ell$ and plaquettes by $p_{\ell }$. For a particle traversing the edge of a single plaquette, there are two contributions $t_{\ell }$ and $t_{\ell }^{\prime }$ to $W(p_{\ell })$; each contribution occurs twice as represented by the red and blue colored arrows. A simple periodic gradient of $\beta$ enables uniform $\pi /N$ flux per plaquette.

Figure 3

(a) Schematic representation of the two-site unit cell lattice with $\beta ={\beta }_1$, ${\beta }_2$. The dotted box outlines a single unit cell. There is a flux $\Psi$, $-\Psi$ which alternates in neighboring square plaquettes. The direct lattice vectors $g_1$ and $g_2$ are depicted as purple arrows. While all hops are present with amplitude decaying as $1/R^3$, only nearest-neighbor (solid) and next-nearest-neighbor (dashed) hops are shown. (b) The topology of bulk bands as a function of complex ${\beta }_2$ for ${\beta }_1\in \mathbb{R}$. The Chern number is $c=\frac{1}{4\pi }\int d{k}_{x}d{k}_y({\partial }_{k_x}\widehat{d}\times {\partial }_{k_y}\widehat{d})·\widehat{d}$, where $H(k)=\overrightarrow{d}(k)·\overrightarrow{\sigma }+f(k)$.

Figure 4

Phase transitions in topological flat bands of 2D driven dipoles. (a) Band structure for $({\Theta }_0,{\Phi }_0)=(0.46,0.42)$, ${\beta }_1=3.6e^{2.69i}$, and ${\beta }_2=5.8e^{5.63i}$. We have verified that the Chern number does not change upon adding in dipolar interactions up to order $1/27R_0$. Significantly flatter band structures with a flatness ratio $>10$ can be obtained for slightly generalized configurations involving a tripod level structure and optical superlattice [42]. (b) Structure factor $S(R,0)=⟨n(R)n(0)⟩$ for filling $\nu =1/2$ in KMS [30] and (c) SSS regime; size of circles indicates weight. (d) Spectral gap density plot as a function of varying microwave drive for parameters: $({\Theta }_0,{\Phi }_0)=(0.66,\pi /4)$, ${\beta }_1=-2.82e^{i{\varphi }_1}$, ${\beta }_2=-4.84e^{-i{\varphi }_2}$, and $(d^0-d^1{)}^2/d_{01}^2\approx 2.8$. The transition from the SF, which has a unique finite-size ground state, to the degenerate SSS shows as a collapse of this gap. (e) Spectral flow in the ground state momentum sector of the SSS under twisting of the boson boundary condition in the ${\widehat{g}}_1$ and (f) ${\widehat{g}}_2$ directions. For the $N_s=24$ lattice with 6 bosons, momentum sectors return to themselves after $2\pi$ in ${\theta }_1$ and after $4\pi$ in ${\theta }_2$.