Charge Pumping of Interacting Fermion Atoms in the Synthetic Dimension

Recently it has been theoretically proposed and experimentally demonstrated that a spin-orbit coupled multicomponent gas in a 1D lattice can be viewed as a spinless gas in a synthetic 2D lattice with a magnetic flux. In this Letter we consider interaction effects in such a Fermi gas, and propose these effects can be easily detected in a charge pumping experiment. Using $1/3$ filling of the lowest 2D band as an example, in the strongly interacting regime, we show that the charge pumping value gradually approaches a universal fractional value for large spin components and low filling of the 1D lattice, indicating a fractional quantum Hall-type behavior, while the charge pumping value is zero if the 1D lattice filling is commensurate, indicating a Mott insulator behavior. The charge-density-wave order is also discussed.

Figure 1

Illustration of two different physical pictures of this system and the idea of charge pumping. (a) Two Raman beams along $\widehat{x}$ are applied to a multicomponent quantum gas in optical lattices. The Raman beams couple different spin states and generate a coupling between spin and momentum $k_x$. (b) Different spin components are viewed as another dimension, and the system is mapped into a 2D spinless particle in a magnetic field. By applying an electric field along the physical dimension $\widehat{x}$, it generates a charge pumping along the synthetic dimension, which can be detected by measuring spin populations.

Figure 2

Charge pumping for different ${\nu }_{1\mathrm{D}}$ and $W$. The numbers in the boxes are charge pumping $Q$ after insertion of one flux [15]. Blue triangles are points calculated by ED and red circles are points calculated by DMRG, with $\nu$ fixed at $1/3$, $\mathrm{\Omega }=t$ and $U=6t$. (a)–(c) Three marked cases where spectral flow will be shown in Fig. 3.

Figure 3

Left column: Spectral flow under the insertion of flux (i.e., changing periodic boundary condition $\theta$ from zero to $6\pi$.) Right column: Charge polarization $Y$ as a function of $\theta$. (a)–(c) correspond to different $W$ and ${\nu }_{1\mathrm{D}}$ as marked in Fig. 2. (a) $W=9$, ${\nu }_{1\mathrm{D}}=1$ ($N=L=5$), $q=3$; (b) $W=5$, ${\nu }_{1\mathrm{D}}=5/6$ ($N=5$, $L=6$), $q=2$, and (c) $W=14$, ${\nu }_{1\mathrm{D}}=2/3$ ($N=4$, $L=6$), $q=7$. All of these cases are calculated by ED. $U=6t$ and $\mathrm{\Omega }=t$.

Figure 4

Charge pumping $Q$ as a function of Raman coupling strength $\mathrm{\Omega }/t$. The dashed line is $1/3$. Here $U=6t$ and other parameters are shown in the legend.

Figure 5

Energy diagram with periodic boundary condition in the synthetic dimension. (a) Energy levels for different momenta $K_y+W{K}_x$. (b) Energy of the lowest three states as a function of $\theta$. Here $W=4$, ${\nu }_{1\mathrm{D}}=1/3$ ($N=4$, $L=12$), $q=4$. This case is also calculated by ED. $U=6t$ and $\mathrm{\Omega }=t$.

Figure 6

Fourier transformation of density $\rho (q_x)$ for different values of $W$ and ${\nu }_{1\mathrm{D}}$. $U=6t$, $\mathrm{\Omega }=t$ and other parameters are shown in the legend.