Floquet topological transitions in extended Kane-Mele models with disorder

In this work we use Floquet theory to theoretically study the influence of circularly polarized light on disordered two-dimensional models exhibiting topological transitions. We find circularly polarized light can induce a topological transition in extended Kane-Mele models that include additional hopping terms and on-site disorder. The topological transitions are understood from the Floquet-Bloch band structure of the clean system at high-symmetry points in the first Brillouin zone. The light modifies the equilibrium band structure of the clean system in such a way that the smallest gap in the Brillouin zone can be shifted from the $M$ points to the $K$ $\left(K^{\prime }\right)$ points, the $\mathrm{\Gamma }$ point, or even other lower-symmetry points. The movement of the minimal gap point through the Brillouin zone as a function of laser parameters is explained in the high-frequency regime through the Magnus expansion. In the disordered model, we compute the Bott index to reveal topological phases and transitions. The disorder can induce transitions from topologically nontrivial states to trivial states or vice versa, both examples of Floquet topological Anderson transitions. As a result of the movement of the minimal gap point through the Brillouin zone as a function of laser parameters, the nature of the topological phases and transitions is laser parameter dependent, a contrasting behavior to the Kane-Mele model.

Figure 1

(a) Honeycomb lattice with two sublattices in one unit cell (shaded area), labeled A (open circles) and B (solid circles). Three nearest-neighbor unit vectors are ${\delta }_1=(-\sqrt{3}/2,-1/2)a$, ${\delta }_2=(\sqrt{3}/2,-1/2)a$, and ${\delta }_3=(0,1)a$, with $a$ being the nearest-neighbor distance. Lattice translational vectors are labeled ${\mathbf{a}}_1={\delta }_3-{\delta }_1=(\sqrt{3}/2,3/2)a,{\mathbf{a}}_2={\delta }_3-{\delta }_2=(-\sqrt{3}/2,3/2)a$. The blue dashed lines represent the imaginary second-neighbor hopping (spin-orbit coupling), and the arrow directions represent positive signs. (b) First Brillouin zone of the underlying triangular Bravais lattice with reciprocal lattice vectors ${\mathbf{b}}_1=(\sqrt{3},1)2\pi /\left(3a\right)$ and ${\mathbf{b}}_2=(-\sqrt{3},1)2\pi /\left(3a\right)$. High-symmetry points are $\mathbf{K}=(-2\pi /\sqrt{3},2\pi )/\left(3a\right)$, ${\mathbf{K}}^{\prime }=(-4\pi /\sqrt{3},0)/\left(3a\right)$, and time-reversal-invariant momenta are ${\mathbf{M}}_{1,2}=(\pm \sqrt{3}\pi ,\pi )/\left(3a\right)$, ${\mathbf{M}}_3=(0,2\pi )/\left(3a\right)$. All solid circles are equivalent to $\mathbf{K}$, and all open circles are equivalent to ${\mathbf{K}}^{\prime }$. The Floquet quasiband structure is plotted along the momentum path $K^{\prime }\text{−}\mathrm{\Gamma }\text{−}{M}_3\text{−}K\text{−}{M}_2\text{−}{K}^{\prime }$ in the first Brillouin zone.

Figure 2

The spin Chern number as a function of laser amplitude $A_0$ for the generalized Kane-Mele model [Eq. (1)] with (a) $t_3=0.0$, (c) $t_3=0.20$, and (e) $t_3=0.40$ and the dimerized Kane-Mele model [Eq. (2)] with (b) $t_d=1.5$, (d) $t_d=2.0$, and (f) $t_d=2.50$. The remaining parameters are nearest-neighbor hopping $t_1=1.0$, spin-orbit coupling ${\lambda }_{\mathrm{soc}}=0.3$, and laser frequency $\mathrm{\Omega }=10.0$. All the calculations are done with 2500 $\mathbf{k}$ points in the first Brillouin zone and nine Floquet copies.

Figure 3

The Floquet-Bloch band structure of the Kane-Mele model [Eq. (1) with $t_3=0.0$] with parameters $t_1=1.0,{\lambda }_{\mathrm{soc}}=0.3$, and $\mathrm{\Omega }=10.0$ for (a1) $A_0=0.0$, (a2) $A_0=0.4$, (a3) $A_0=0.8$, (a4) $A_0=1.2$, (a5) $A_0=1.6$, and (a6) $A_0=2.0$. The Floquet-Bloch band structure of the generalized Kane-Mele model [Eq. (1)] with parameters $t_1=1.0$, ${\lambda }_{\mathrm{soc}}=0.3$, $t_3=0.4$, and $\mathrm{\Omega }=10.0$ for (b1) $A_0=0.0$, (b2) $A_0=0.4$, (b3) $A_0=0.8$, (b4) $A_0=1.2$, (b5) $A_0=1.6$, and (b6) $A_0=2.0$. The Floquet-Bloch band structure of the dimerized Kane-Mele model [Eq. (2)] with parameters $t_1=1.0$, ${\lambda }_{\mathrm{soc}}=0.3$, $t_d=2.4$, and $\mathrm{\Omega }=10.0$ for (c1) $A_0=0.0$, (c2) $A_0=0.4$, (c3) $A_0=0.8$, (c4) $A_0=1.2$, (c5) $A_0=1.6$, and (c6) $A_0=2.0$. Note the solid lines denote the Floquet band structure with laser intensity $A_0$ given above; the dashed lines show the data for $A_0-0.2$ for comparison.

Figure 4

The energy gaps at high-symmetry points are plotted with symbols. The dashed line indicates the energy gap in the theoretical infinite-frequency limit. (a) Generalized Kane-Mele model with $t_3=0.0$. (b) Generalized Kane-Mele model with $t_3=0.4$. (c) Dimerized Kane-Mele model with $t_d=2.5$. (d) The zeroth-order Bessel function of the first kind used in the infinite-frequency limit.

Figure 5

Top: The phase diagram in the plane of laser intensity $A_0$ and on-site disorder $U_{\mathrm{dis}}$ for both GKM with (a) $t_3=0.0$, (b) $t_3=0.2$, and (c) $t_3=0.4$ and DKM with (d) $t_d=1.5$, (e) $t_d=2.0$, and (f) $t_d=2.5$. Note the color scale for (c) is different from the others. The phase diagram can serve as a visual guide. Detailed data with the Bott index as a function of disorder strength are plotted in the lower panels. Middle: Generalized Kane-Mele model with (a1) $t_3=0.0$, (b1) $t_3=0.2$, and (c1) $t_3=0.4$. Bottom: dimerized Kane-Mele model with (d1) $t_d=1.5$, (e1) $t_d=2.0$, and (f1) $t_d=2.5$. The remaining model parameters are fixed at $t_1=1.0,{\lambda }_{\mathrm{soc}}=0.3,\mathrm{\Omega }=10.0$.

Figure 6

The Chern number as a function of on-site disorder $U_{\mathrm{dis}}$ for (a) the GKM model with $t_3=0.0$, (b) the GKM model with $t_3=0.2$, and (c) the DKM model with $t_d=1.5$. The remaining model parameters are fixed at with $t_1=1.0,{\lambda }_{\mathrm{soc}}=0.3,\mathrm{\Omega }=5.0$. The Floquet-Bloch band structure in the clean limit (absence of disorder) is plotted for different laser intensities $A_0=0.0,0.4,0.8,1.2,1.6,2.0$ in (a1)–(a6) for the GKM model with $t_3=0.0$, (b1)–(b6) for the GKM model with $t_3=0.2$, and (c1)–(c6) for the DKM model with $t_d=1.5$.

Figure 7

(a) The Bott index as a function of on-site disorder $U_{\mathrm{dis}}$ for the GKM model with $t_3=0.4$ and laser intensity $A_0=0.2$. (b) The Bott index as a function of on-site disorder $U_{\mathrm{dis}}$ for the DKM model with $t_d=2.5$ and laser intensity $A_0=0.8$. The remaining model parameters are fixed at $t_1=1.0,{\lambda }_{\mathrm{soc}}=0.3,\mathrm{\Omega }=10.0$. Different cluster sizes are chosen to illustrate the finite-size effect. The total number of lattice sites is $N\times N\times 2$, where the 2 comes from the number of atoms in one unit cell.