Emulating topological currents arising from a dipolar parity anomaly in two-dimensional optical lattices

Dipolar parity anomaly can be induced by spatiotemporally weak-dependent energy-momentum separation of paired Dirac points in two-dimensional Dirac semimetals. Here we reveal topological currents arising from this kind of anomaly. A corresponding lattice model is proposed to emulate the topological currents by using two-component ultracold atoms in a two-dimensional optical Raman lattice. In our scheme, the topological currents can be generated by varying on-site coupling between the two atomic components in time and tuned via the laser fields. Moreover, we show that the topological particle currents can directly be detected from measuring the drift of the center of mass of the atomic gases.

Figure 1

(a) A sketch of the dipole momentum $b_{\mu }=(b_0,\mathbf{b})$ of paired Dirac points in energy-momentum space for $b_1=0$, where $b_{\mu }$ is a function of space and time. The Dirac points ${\mathbf{k}}_D^{+}$ and ${\mathbf{k}}_D^{-}$ are, respectively, in blue and red Dirac cones, with $\pm$ denoting the two opposite parity signs. (b) A sketch of the topological current induced by the dipolar parity anomaly ${\mathbf{J}}_D$ from a weak time-dependent $\mathbf{b}$ perturbation.

Figure 2

(a) A sketch of Raman lattice potential $V^R\left(x,y\right)$ as a function of $y$ and $x$ (in units of $a_x$ and let $V_{0,x}^R=V_{0,y}^R$). The green rings region denotes the Raman lattice site [minimal Raman lattice potential $V_{\mathrm{Min}}^R\left(x,y\right)$] and the red rings denote the site with potential $V^R\left(x,y\right)=-V_{\mathrm{Min}}^R\left(x,y\right)<V_{\mathrm{Max}}^R\left(x,y\right)$. Moreover, the orange ovals denote the optical lattice site $(x_i,y_j)$, where the $i$ and $j$ is the shorthand notation of $x_i$ and $y_j$, respectively. The orange and light-blue arrows denote the laser beams, which are used to construct the optical lattice and Raman lattice, respectively. From $V^R\left(x,y\right)$, we easily find $V^R\left(x,y=y_j\right)=-V^R\left(x+a_x,y=y_j\right)$ as well as $V^R\left(x,y\approx y_j\right)\approx -V^R\left(x+a_x,y\approx y_j\right)$ and $V^R\left(x,y\right)=V^R\left(x,y+a_y\right)$. Two typical topological band structures (having one or two pairs of Dirac points) are showed in (b) and (c), where the Dirac points are denoted by $k_{D,1/2}^{\pm }$ and the $\pm$ denote the different chiralities. The parameters are (b),(c) ${\delta }_t=0.32$ and (b),[(c)] $\lambda \left(\tau \right)=1.2 [\lambda \left(\tau \right)=0.48]$, corresponding to the point of blue (red) lines with $\tau \sim 17.7\text{or}49.1$ in Fig. 3.

Figure 3

The topological current ${\mathbf{J}}_D$ (in units of $t/a_y$) for the slowly periodically driving case with $\lambda =0.2$ in (a) and (b). We set $\omega \ll 1$ and $\lambda \left(\tau \right)-{\lambda }_0/2\ll 1$ for $\forall \tau$. ${\lambda }_0=2.44,1.0$ are denoted by blue and red dashed lines, respectively. In (a), the red and blue lines denote ${\mathbf{J}}_D$ for two pairs and a single pair of Dirac points, respectively, with $\omega ={\omega }_{max}=0.1$ and ${\delta }_t=0.32$. In (b), ${\mathbf{J}}_D$ as a function ${\delta }_t$ with $\tau =10.0$ is shown. The red line denotes the case with two pairs of Dirac points, while the blue line represents the case undergoing a topological phase transition from the trivial state to the nontrivial state (a pair of Dirac points); the discontinuity of the topological currents ${\mathbf{J}}_D$ reflects this topological phase transition. The inset is an enlarged figure of ${\mathbf{J}}_D$.

Figure 4

The drift of the center of mass $\mathrm{\Delta }{x}_c=x_c\left(\tau \right)-x_c(\tau =0)$ (in units of $a_x$) as a function of $\tau$ in the $x$ direction are shown by taking $\rho =0.01{\left(a_y{a}_x\right)}^{-1}$. The parameters for blue and red dashed lines are the same as those in Fig. 3. Here the parameters for the red transparent thick line are the same as the others color lines except ${\delta }_t$, where ${\delta }_t$ is 0 (0.32) for the red transparent thick line (the others lines). It is clear that there is no drift of the center of mass $\mathrm{\Delta }{x}_c$ for ${\delta }_t=0$, even though there are two pairs of Dirac points (due to the current cancellation from the two pairs). Thus this nonvanishing $\mathrm{\Delta }{x}_c$ feature may be taken as an experimental signature for topological currents of dipolar parity anomaly.

Figure 5

(a)–(c) The cartoon pictures of the spin-flip hopping along the $x$ and $y$ directions, the radio-frequency field induced on-site spin-flip hopping, and the normal spin-independent hopping, respectively, where $x_j$ ($y_j$) denotes the lattice site along the $x$ ($y$) direction and $a_x$ ($a_y$) denotes the lattice length along the $x$ ($y$) direction. $t_{0,x}$ is the amplitude of normal spin-independent hopping along the $x$ direction (we do not show the $y$ direction), and $\mathrm{\Delta }$ is the detuning of the $|\uparrow \rangle$ state and $|\downarrow \rangle$.

Figure 6

(a) The solid lines denote $t_x=t_y$ and the dashed lines denote the ${\delta }_t=0.32t_x$, which have been chosen to calculate the topological current $J_{\mathrm{D}}$ and the corresponding drifting of center mass $x_{\mathrm{c}}$; the orange, red, black, blue, and magenta lines denote the values of trapping potential ${\tilde{V}}_{0,x}^L=4,5,6,7,8$, respectively. (b) ${\delta }_t/t_x$ as a function of $m$ and $l$ with ${\tilde{V}}_{0,x}^L=5$.

Figure 7

$t_x$ as a function of trapping potential ${\tilde{V}}_{0,x}^L$ and Raman potential ${\tilde{V}}_{0,x}^R$ takes the unit of $\mathrm{kHz}$ for (a) ${}^{87}\mathrm{Rb}$ and (b) ${}^{23}\mathrm{Na}$. In these regions, the maximum drifting of the center of mass $\mathrm{\Delta }{x}_c$, which is presented in Fig. 4, will be observed within the coherent time ${\tau }_{\mathrm{coh}}$. The trapping laser wavelength is $\lambda =790.1\mathrm{nm}$ and recoil energy is $E_R^x={\left(k_x\right)}^2{\hslash }^2/\left(2M\right)$ ($E_R^x=\hslash \times 2\pi \times 3.68\mathrm{kHz}$ for ${}^{87}\mathrm{Rb}$ and $E_R^x=\hslash \times 2\pi \times 13.9\mathrm{kHz}$ for ${}^{87}\mathrm{Rb}$), where $k_x=2\pi /\lambda$ is the recoil momentum [66].

Figure 8

The sketch of the solution of Eqs. (C2) and (C3).