Occupation probabilities and current densities of bulk and edge states of a Floquet topological insulator

Results are presented for the occupation probabilities and current densities of bulk and edge states of half-filled graphene in a cylindrical geometry and irradiated by a circularly polarized laser. It is assumed that the system is closed and that the laser has been switched on as a quench. Laser parameters corresponding to some representative topological phases are studied: one where the Chern number of the Floquet bands equals the number of chiral edge modes, a second where anomalous edge states appear in the Floquet Brillouin zone boundaries, and a third where the Chern number is zero, yet topological edge states appear at the center and boundaries of the Floquet Brillouin zone. Qualitative differences are found for the high-frequency off-resonant and low-frequency on-resonant laser with edge states arising due to resonant processes occupied with a high effective temperature on the one hand, while edge states arising due to off-resonant processes occupied with a low effective temperature on the other. For an ideal half-filled system where only one of the bands in the Floquet Brillouin zone is occupied and the other empty, particle-hole and inversion symmetry of the Floquet Hamiltonian implies zero current density. However the laser switch-on protocol breaks the inversion symmetry, resulting in a net cylindrical sheet of current density at steady state. Due to the underlying chirality of the system, this current density profile is associated with a net charge imbalance between the top and bottom of the cylinders.

Figure 1

Sketch of the four topological phases studied, and labeled by $P_{1,2,3,4}$. Laser frequency $\mathrm{\Omega }$ is in units of the hopping strength $t_h$ with $6t_h$ being the bandwidth of graphene. Laser amplitude $A_0$ is in units of the lattice spacing. Solid lines are edge states from the center of the FBZ, while dashed lines are edge states from the boundary of the FBZ. The arrows indicate the chirality of the edge states.

Figure 2

Spectrum and occupation probabilities due to a quench for the case $P_1$, where $A_{0}a=0.5,\mathrm{\Omega }=10t_h$, and the Chern number is $C=1$. The system supports a pair of chiral edge modes at the center of the FBZ. The area of the circles is proportional to the occupation probability $O_{\alpha }\left(k_x\right)$.

Figure 3

Spectrum and occupation probabilities due to a quench for the case $P_2$, where $A_{0}a=1.5,\mathrm{\Omega }=5t_h$, and the Chern number is $C=1$. The system supports a pair of chiral edge modes at the center of the FBZ. The area of the circles is proportional to the occupation probability $O_{\alpha }\left(k_x\right)$.

Figure 4

Spectrum and occupation probabilities due to a quench for the case $P_3$, where $A_{0}a=0.5,\mathrm{\Omega }=5t_h$, and the Chern number is $C=3$. The system supports a pair of chiral edge modes at the center of the FBZ, and two pairs of chiral edge modes on the Floquet zone boundaries (see Figs. 1 and 8). The area of the circles is proportional to the occupation probability $O_{\alpha }\left(k_x\right)$.

Figure 5

Spectrum and occupation probabilities due to a quench for the case $P_4$, where $A_{0}a=10,\mathrm{\Omega }=0.5t_h$, and the Chern number is $C=0$. The system supports two pairs of chiral edge modes at the center of the FBZ, and two pairs of chiral edge modes on the Floquet zone boundaries (see Figs. 1 and 8). The area of the circles is proportional to the occupation probability $O_{\alpha }\left(k_x\right)$.

Figure 6

Quench occupation probabilities of the Floquet quasienergy levels averaged over the momenta $k_x$. Clockwise from top left, phases $P_1,P_2,P_4$, and $P_3$ (see Fig. 1). $P_4$ has an almost infinite effective temperature, while $P_1$ has an almost zero effective temperature.

Figure 7

Upper panel: Case $P_1$, where $A_{0}a=0.5,\mathrm{\Omega }=10t_h,C=1$. Lower panel: Case $P_2$, where $A_{0}a=1.5,\mathrm{\Omega }=5t_h,C=1$. Both correspond to strip width $N_y=40$. Current densities of three exact Floquet eigenstates: one from the Floquet band edge $(\alpha =1$), another from the center of the Floquet band $(\alpha =N_y/4=10$), and a third from the edge state located at the center of the FBZ $(\alpha =N_y/2=20$). The current densities from the two bulk states $\alpha =1,10$ are much smaller than that from the edge states, and are also of the opposite chirality from that of the edge state.

Figure 8

Upper panel: Case $P_3$, where $A_{0}a=0.5,\mathrm{\Omega }=5t_h,C=3$ with strip width $N_y=200$. Lower panel: Case $P_4$, where $A_{0}a=10,\mathrm{\Omega }=0.5t_h,C=0$ with strip width $N_y=100$. For each panel, current densities of three exact Floquet eigenstates, one state from the Floquet band edge $(\alpha =1$), another from the Floquet band center $[\alpha =N_y/4=50\left(P_3\right),25\left(P_4\right)]$, and a third from the edge state(s) located at the center of the FBZ $[\alpha =N_y/2=100\left(P_3\right),50\left(P_4\right)]$ are shown. The current density from the bulk state $\alpha =N_y/4$ is much smaller than that from the edge states $(\alpha =1,N_y/2$). The current density over only half the strip has been plotted as for the other half the current is antisymmetric to the first half.

Figure 9

Current density flowing in the $\widehat{x}$ direction and plotted as a function of the $y\equiv n_y$, at long times after the laser quench for (clockwise from top left) $P_1\equiv \left[A_{0}a=0.5,\mathrm{\Omega }=10t_h,C=1\right],P_2\equiv \left[A_{0}a=1.5,\mathrm{\Omega }=5t_h,C=1\right],P_4\equiv \left[A_{0}a=10,\mathrm{\Omega }=0.5t_h,C=0\right]$, and $P_3\equiv \left[A_{0}a=0.5,\mathrm{\Omega }=5t_h,C=3\right]$. The quench current is symmetric in $n_y$; hence for the lower two panels, the current density is plotted over only half the strip. For a half-filled Floquet band, the current density is zero due to particle-hole symmetry.