Dynamical Generation of Topological Magnetic Lattices for Ultracold Atoms

We propose a scheme to dynamically synthesize a space-periodic effective magnetic field for neutral atoms by time-periodic magnetic field pulses. When atomic spin adiabatically follows the direction of the effective magnetic field, an adiabatic scalar potential together with a geometric vector potential emerges for the atomic center-of-mass motion, due to the Berry phase effect. While atoms hop between honeycomb lattice sites formed by the minima of the adiabatic potential, complex Peierls phase factors in the hopping coefficients are induced by the vector potential, and these phase factors facilitate a topological Chern insulator. With further tuning of external parameters, both a topological phase transition and topological flat bands can be achieved, highlighting realistic prospects for studying strongly correlated phenomena in this system. Our Letter presents an alternative pathway towards creating and manipulating topological states of ultracold atoms by magnetic fields.

Figure 1

A schematic illustration for synthesizing a magnetic lattice. (a) An atomic cloud is exposed to a uniform-bias magnetic field $B_0{\widehat{e}}_z$ (pointing out of the page), and subjected in sequence to four pairs of opposite magnetic pulses as shown in (b). Pulse pairs 1, 2, and 3 are gradient magnetic fields lying in the $x\text{−}y$ plane along directions separated by 120°. Pulse pair 4 executes $\mp \pi$ spin rotations along the $y$ direction. The color gradient represents the corresponding magnetic field strength. (b) The time-periodic pulse sequence with four pairs of pulses forming one complete evolution period. (c) A three-dimensional view of the effective magnetic field ${\mathbf{B}}_{\mathrm{eff}}$ [Eq. (5)], which forms a Skyrmion lattice [42] with its Wigner-Seitz unit cell shown by the hexagon [of edge length $4\pi /(3k_{\mathrm{so}})$]. The arrows are colored by the magnitude of the third component of ${\mathbf{B}}_{\mathrm{eff}}$.

Figure 2

Mapping the effective Hamiltonian [Eq. (4)] to the Haldane model. (a) The density plot of the adiabatic potential ${\epsilon }_0$ (in blue) and its associated vector potential $\mathbf{A}$ shown by field of arrows. A darker color denotes a smaller ${\epsilon }_0$, the minima of which form a honeycomb lattice with two sites per unit cell denoted, respectively, by red and green filled circles. The dashed lines denote NNN hopping paths along the directions of positive Peierls phases. (b) The flux density $n_{\varphi }$, in units of $10^{-2}{k}_{\mathrm{so}}^2$. The hexagons in (a) and (b) denote the primitive unit cell [the same as in Fig. 1], over which the net flux is unity. (c) Energy spectrum. Solid lines represent the lowest two energy bands along lines with high symmetry in the first Brillouin zone (inserted hexagon) for the effective Hamiltonian Eq. (4), with ${\omega }_0=32.3{\omega }_{\mathrm{so}}$. Dashed lines represent the fitted band structure using the Haldane model results. The edge length of the inserted hexagon is $k_{\mathrm{so}}/\sqrt{3}$. (d) A logarithmic plot of the Berry curvature for the lowest band ${\mathrm{\Omega }}_1$. The integration of ${\mathrm{\Omega }}_1$ in the first Brillouin zone gives its Chern number, $C_1=1$.

Figure 3

Illustration of the structural and topological phase transition with the tuning of the $z$ component of ${\mathbf{B}}_{\mathrm{eff}}$ in Eq. (8). (a) Top panel: the density plots of the adiabatic potentials for $\alpha =0$ (left), $\alpha =0.720$ (middle), and $\alpha =1$ (right) with the minima (dark colors) forming a simple triangular lattice, a decorated triangular lattice, and a honeycomb lattice, respectively. Bottom panel: The Chern number for the lowest energy band as a function of $\alpha$. A topological phase transition occurs at the critical point of $\alpha =0.720$. (b) The band gap between the lowest two bands and the bandwidth for the lowest band, as a function of $\alpha$. The band gap closes at $\alpha =0.720$ (marked by the dotted vertical line). The inset shows the dependence of the band gap-over-width ratio on $\alpha$. It peaks at $\alpha =0.749$ with a value of 16. (c) The lowest three bands at $\alpha =0.749$.