Artificial topological models based on a one-dimensional spin-dependent optical lattice

Topological matter is a popular topic in both condensed matter and cold-atom research. In the past decades, a variety of models have been identified with fascinating topological features. Some, but not all, of the models can be found in materials. As a fully controllable system, cold atoms trapped in optical lattices provide an ideal platform to simulate and realize these topological models. Here we present a proposal for synthesizing topological models in cold atoms based on a one-dimensional spin-dependent optical lattice potential. In our system, features such as staggered tunneling, staggered Zeeman field, nearest-neighbor interaction, beyond-near-neighbor tunneling, etc. can be readily realized. They underlie the emergence of various topological phases. Our proposal can be realized with current technology and hence has potential applications in quantum simulation of topological matter.

Figure 1

Optical lattice configuration: the empty and solid circles represent the spin $\uparrow \text{and}\text{spin}\downarrow$ atoms, respectively. The black solid lines describe the conventional tunneling with the amplitude $t$. The blue and red solid wavy lines describe the rf-induced tunneling with the amplitude $J_1$ and $J_2$, respectively. The intersublattice tunneling marked by the gray dashed line is considered to be negligible.

Figure 2

(a) Energy spectrum of the generalized SSH model. The blue solid line shows energy of the edge states. (b) Intersublattice-tunneling amplitudes $J_1$ (blue solid line) and $J_2$ (red dashed line) as functions of the phase $\varphi$. (c) The spatial distributions of two degenerate edge states at $\varphi =0.2\pi$. In our calculation, we set 100 sites for each sublattice. The trap depth is set as $V_L=9.0E_R$ with the intrasublattice-tunneling amplitude $t=0.0242E_R$. The rf field strength $\mathrm{\Omega }=2.0E_R$.

Figure 3

Energy spectrum of the lattice model for (a) $\varphi =\pi /5$ and (b) $\varphi =0$. We set 100 sites for each sublattice. Other parameters are the same as in Fig. 2, and the corresponding values of $J_{1,2}$ at a given $\varphi$ can be inferred from Fig. 2.

Figure 4

Site index mapping from the 1D spin-1/2 model (top panel) to the 1D spinless chain model (bottom panel).

Figure 5

(a) Three paths of the quantum pump progress: (i) blue solid line, $\mathrm{\Delta }=0$; (ii) red dashed line, $\mathrm{\Delta }=0.5E_R$; (iii) green dash-dotted line, $\mathrm{\Delta }=0.5cos\left(2\varphi \right)E_R$. The dependence of $J_1-J_2$ on $\varphi$ has been given in Eq. (3). The initial states of each pump trajectory are marked by the empty diamonds. After $\varphi$ evolves a $\pi$ period from $-\pi /4$ to $3\pi /4$, the system returns to each initial state. The arrows describe the direction along each pump trajectory in the $(J_1-J_2)-\mathrm{\Delta }$ plane. (b) Evolution of the averaged Berry curvature $\langle {\mathrm{\Omega }}_{-}\left(\varphi \right)\rangle$ along the three trajectories in (a), respectively. The Berry curvature of the other branch is given by $\langle {\mathrm{\Omega }}_{+}\left(\varphi \right)\rangle =-\langle {\mathrm{\Omega }}_{-}\left(\varphi \right)\rangle$.

Figure 6

Effects of unequal numbers of sublattice sites on the topological properties. We set $L=100$. (a) Optical lattice configuration for different $\varphi$: the empty or solid circles represent the spin $\uparrow \text{or}\downarrow$ atoms. (b) Energy spectrum of the system as a function of $\varphi$. The system is topologically nontrivial with a single nondegenerate edge mode as long as $\varphi \ne 0$, i.e., $J_1\ne J_2$. (c) The spatial wave function of the edge mode at $\varphi =\mp 0.1\pi$ for the upper or lower panel. Here the edge mode is localized at opposite ends of the chain for the sign of $\varphi$. Other parameters are the same as in Fig. 2.

Figure 7

(a) Optical lattice configuration: Raman-assisted tunneling in each linear tilted sublattice with an energy offset hosts a spatial dependent phase. Gray triangles along the chain denote the tilted directions. (b) Hofstadter butterfly: energy spectrum $E_{\eta }$ versus the particular flux $\theta$. $E_{\eta }$ is in units of $t$ and we set $J=t$.