Effects of periodically kicked Dirac mass term in the Chern insulators

Floquet engineering offers a unique approach to generate nonequilibrium topological phases in which the unbounded nature of a quasienergy band allows two kinds of topological edge modes, one of them traversing the 0 gap and another one traversing the $\pi$ gap. Characterizing of these two modes is the main topic of Floquet topological insulators, where they are usually characterized by different topological invariants. However, in this paper, for a specific protocol of Floquet engineering where the Dirac mass term of the Chern insulator is periodically kicked, its topological phases are characterized with the Floquet Chern number $C_F$. Specifically, in our illustrative example, the periodically kicked Qi-Wu-Zhang model, there are six different topological phases in total, denoted as $C_F=\{-1_0,-2,-1_{\pi },1_{\pi },2,1_0\}$. Topological phases with larger topological number are observed, i.e., $C_F=\pm 2$, where the chiral edge modes traversing the 0 gap and those traversing $\pi$ gap have the opposite chirality. The mechanism of its topology is revealed by studying the corresponding low-energy effective Dirac Hamiltonian, and the phase boundaries between different topological phases are explicitly found. Additionally, we investigate the orders of phase transitions between different topological phases by studying the von Neumann entropy of the Floquet steady state (FSS), where the FSS corresponds to a stationary state of the Floquet system that is coupled to a Markovian environment. The hallmark of a periodically kicked Dirac mass term is uncovered in this paper, which may inspire further explorations of the physical effects of periodically kicked Dirac mass terms in other systems.

Figure 1

(a) Chern number $C_F$ as a function of $u$ for $u_0=0.1\pi$ and $u_0=0.2\pi$ in the PK-QWZ model. The quasibands of the PK-QWZ model for $u=0.1\pi$ (b), $u=0.3\pi$ (c), $u=0.6\pi$ (d), $u=\pi$ (e), $u=1.3\pi$ (f), and $u=1.6\pi$ (g) are separate, where $u_0=0.2\pi , L_y=20$. The boundary condition along the $x$ direction is periodic. For the topological phases with $C_F=\pm 1$, there are two chiral edge modes, either traversing the 0 gap (marked in red) or traversing the $\pi$ gap (marked in green). For the topological phases with $C_F=\pm 2$, there are four chiral edge modes. They are separated into two pairs, one pair traverses the 0 gap and the other traverses the $\pi$ gap.

Figure 2

Quasienergy spectrum of the PK-QWZ model for $C_F=2$ (a), where the boundary condition along the $x$ direction is periodic, $L_y=20, u=1.3\pi$, and $u_0=0.2\pi$. Also presented are the chiral edge modes traversing the 0 gap (b) and the chiral edge modes traversing the $\pi$ gap (c), where ${\left|\psi \left(y\right)\right|}^2{=|\psi \left(y\right)}_{\uparrow }{|}^2+{\left|\psi {\left(y\right)}_{\downarrow }\right|}^2$. For the right-handed edge modes, those traversing the 0 gap are localized at $y=1$ (b, red solid line), while those traversing the $\pi$ gap are localized at $y=L_y$ (c, green solid line). And the left-handed chiral edge modes (dashed line) are the other way around.

Figure 3

Graphical representation of chiral edge modes in PK-QWZ model for $C_F=2$.

Figure 4

Phase diagram of the PK-QWZ model: the magenta dashed lines represent the phase boundaries between the phase with $C_F=-2$ and those with $C_F=-1_s$, the red dashed lines are the phase boundaries between the phase with $C_F=-1_{\pi }$ and those with $C_F=1_{\pi }$, green dashed lines mark the phase boundaries between the phase with $C_F=2$ and those with $C_F=1_s$, and black dashed line is the critical point of the static QWZ model, where $u_0=2$, and where $s=0\text{or}\pi$.

Figure 5

Contour plot of $n_z(k_x,k_y)$ for different values of $u$, where $u_0=0.2\pi$. The presented cases are for $u=0.1\pi$ (a), $u=0.3\pi$ (b), $u=0.6\pi$ (c), $u=\pi$ (d), $u=1.3\pi$ (e), and $u=1.6\pi$ (f) separately. In these plots the gray areas represent regions where $n^z<0$. The phases $C_F=\pm 1$ are for the case that the sign of $n^z$ changes only once in the BZ (a), (c), (d), (f), while the phases with $C_F=\pm 2$ are for the case that the sign of $n^z$ changes twice in the BZ (b), (e).

Figure 6

Pictorial illustration of the PK-QWZ model weakly coupled to the Markovian environment, where the occupation of particles with up-spin is boosted while the particles with down-spin are dissipated away. The geometry of system is a torus where the boundary conditions are periodic both for the $x$ and $y$ directions.

Figure 7

Average von Neumann entropy $\overline{S}$ vs $u_0$ (a) of FSS in the toruslike PK-QWZ model, and the first-order (b), second-order (c), and third-order (d) derivatives of $\overline{S}$ with respect to $u$, where $u_0=0.2\pi , \gamma =0.002$, and $L_x=L_y=30$. It worth noting that $\overline{S}$ is periodic in $u$ with a period of $\pi$.

Figure 8

Average von Neumann entropy $\overline{S}$ vs $u_0$ (a) of FSS in the toruslike PK-QWZ model, and the first-order (b), second-order (c), and third-order (d) derivatives of $\overline{S}$ with respect to $u_0$, where $u=0.2\pi , \gamma =0.002$, and $L_x=L_y=30$. Moreover, $\overline{S}$ decreases as $u_0$ increases, and well-defined phases only exist when $u_0<2\pi -2-u$ because the derivative of $\overline{S}$ with respect to $u_0$ becomes chaotic beyond this limit.