Chern-number spectrum of ultracold fermions in optical lattices tuned independently via artificial magnetic, Zeeman, and spin-orbit fields

We discuss the Chern number spectrum of ultracold fermions in square optical lattices as a function of artificial magnetic, Zeeman, and spin-orbit fields that can be tuned independently. We show the existence of topological quantum phase transitions induced by Zeeman and spin-orbit fields, where the total number and chirality of edge states change for fixed magnetic flux ratio, thus leading to topological-insulator phases which are different from those found at zero Zeeman and spin-orbit fields. We construct phase diagrams of chemical potential versus Zeeman field or spin-orbit coupling and characterize all insulating phases by their topological invariants. Lastly, we obtain a staircase structure in the filling factor versus chemical potential for various Zeeman and spin-orbit fields, showing the existence of incompressible states at rational filling factors derived from a generalized Diophantine equation.

Figure 1

Flux ratio $\alpha =\mathrm{\Phi }/{\mathrm{\Phi }}_0$ vs energy $E_n(k_y=0)/t$ for various values of spin-orbit coupling parameter $k_{T}a$ and Zeeman field $h_x/t$. The parameters are (a) $k_{T}a=0$ and $h_x/t=0$, (b) $k_{T}a=\pi /4$ and $h_x/t=0$, (c) $k_{T}a=0$ and $h_x/t=1$, and (d) $k_{T}a=\pi /4$ and $h_x/t=1$. The dotted lines in the white areas correspond to midgap edge states.

Figure 2

Eigenvalues $E_n\left(k_y\right)/t$ of the spin-dependent Harper's matrix vs $k_{y}a$ for magnetic flux ratio $\alpha =1/3$. The parameters are (a) $k_{T}a=0$ and $h_x/t=0$, (b) $k_{T}a=\pi /4$ and $h_x/t=0$, (c) $k_{T}a=0$ and $h_x/t=1$, and (d) $k_{T}a=\pi /4$ and $h_x/t=1$. The vertical dashed lines located at $k_{y}a=\pm \pi /3$ indicate the boundaries of the magnetic Brillouin zone. The bulk bands have periodicity $2\pi /3a$, and the edge bands have periodicity $2\pi /a$ along the $k_y$ direction.

Figure 3

Phase diagrams of chemical potential $\mu /t$ vs Zeeman field $h_x/t$ and the associated Chern-number spectrum are shown for spin-orbit coupling parameters (a) $k_{T}a=0$, (b) $k_{T}a=\pi /4$, (c) $k_{T}a=\pi /2$, and (d) $k_{T}a=3\pi /4$. The white regions correspond to conducting phases, and the nonwhite regions of different colors and textures correspond to insulating phases as indicated in the legend key provided. The values of the Chern numbers for each colored region are +2 (blue with negative slope lines), +1 (red with positive slope lines), 0 (gray, black, beige with horizontal lines), $-1$ (gold with crossed lines), and $-2$ (green with vertical lines). The gray regions are topologically trivial with no edge states, the black regions are topologically trivial with nonchiral edge states, and the beige regions with horizontal lines are topologically nontrivial, having two edge states with opposite chirality characteristic of quantum spin-Hall phases.

Figure 4

Chemical potential $\mu /t$ vs spin-orbit parameter $k_{T}a$ for flux ratio $\alpha =1/3$ and Zeeman fields (a) $h_x/t=0$, (b) $h_x/t=1$, (c) $h_x/t=2$, and (d) $h_x/t=3$. The color code for the Chern numbers is the same as in Fig. 3, as indicated in the legend.

Figure 5

Eigenvalues $E_n\left(k_y\right)/t$ of the spin-dependent Harper's matrix vs $k_y{a}_y$ for magnetic flux ratio $\alpha =1/3$, with $t_y=t$ and $t_x=t/2$. The parameters are (a) $k_T{a}_x=0$ and $h_x/t=0$, (b) $k_T{a}_x=\pi /4$ and $h_x/t=0$, (c) $k_T{a}_x=0$ and $h_x/t=1$, and (d) $k_T{a}_x=\pi /4$ and $h_x/t=1$. The vertical dashed lines located at $k_y{a}_y=\pm \pi /3$ indicate the boundaries of the magnetic Brillouin zone. The bulk bands have periodicity $2\pi /3a_y$, and the edge bands have periodicity $2\pi /a_y$ along the $k_y$ direction.

Figure 6

Eigenvalues $E_n\left(k_y\right)/t$ of the spin-dependent Harper's matrix vs $k_y{a}_y$ for magnetic flux ratio $\alpha =1/3$, with $t_y=t$ and $t_x=2t$. The parameters are (a) $k_T{a}_x=0$ and $h_x/t=0$, (b) $k_T{a}_x=\pi /4$ and $h_x/t=0$, (c) $k_T{a}_x=0$ and $h_x/t=1$, and (d) $k_T{a}_x=\pi /4$ and $h_x/t=1$. The vertical dashed lines located at $k_y{a}_y=\pm \pi /3$ indicate the boundaries of the magnetic Brillouin zone. The bulk bands have periodicity $2\pi /3a_y$, and the edge bands have periodicity $2\pi /a_y$ along the $k_y$ direction.

Figure 7

Phase diagrams of chemical potential $\mu /t$ vs Zeeman field $h_x/t$ for $t_y=t$ and $t_x=t/2$. The associated Chern-number spectrum is shown for spin-orbit coupling parameters (a) $k_T{a}_x=0$, (b) $k_T{a}_x=\pi /4$, (c) $k_T{a}_x=\pi /2$, and (d) $k_T{a}_x=3\pi /4$. The white regions correspond to conducting phases, and the nonwhite regions of different colors and textures correspond to insulating phases as indicated in the legend key provided. The values of the Chern numbers for each colored region are +2 (blue with negative slope lines), +1 (red with positive slope lines), 0 (gray, black, beige with horizontal lines), $-1$ (gold with crossed lines), and $-2$ (green with vertical lines). The gray regions are topologically trivial with no edge states, the black regions are topologically trivial with nonchiral edge states, and the beige regions with horizontal lines are topologically nontrivial, having two edge states with opposite chirality characteristic of quantum spin-Hall phases.

Figure 8

Phase diagrams of chemical potential $\mu /t$ vs Zeeman field $h_x/t$ for $t_y=t$ and $t_x=2t$. The associated Chern-number spectrum is shown for spin-orbit coupling parameters (a) $k_T{a}_x=0$, (b) $k_T{a}_x=\pi /4$, (c) $k_T{a}_x=\pi /2$, and (d) $k_T{a}_x=3\pi /4$. The white regions correspond to conducting phases, and the nonwhite regions of different colors and textures correspond to insulating phases as indicated in the legend key provided. The values of the Chern numbers for each colored region are +2 (blue with negative slope lines), +1 (red with positive slope lines), 0 (gray, black, beige with horizontal lines), $-1$ (gold with crossed lines), and $-2$ (green with vertical lines). The gray regions are topologically trivial with no edge states, the black regions are topologically trivial with nonchiral edge states, and the beige regions with horizontal lines are topologically nontrivial, having two edge states with opposite chirality characteristic of quantum spin-Hall phases.