Nondiagonal disorder enhanced topological properties of graphene with laser irradiation

Laser irradiation, as a versatile tool to tune the topological properties of electronic systems, is under intensive study. Experimentally, a laser irradiation induced anomalous Hall effect in graphene has been observed [J. W. McIver et al., Nat. Phys. 16, 38 (2020)]. Disorder is ubiquitous in real materials, and it has been shown that diagonal disorders, i.e., on-site disorder, can enhance the topological properties of time-periodically driven quantum materials [P. Titum et al., Phys. Rev. Lett. 114, 056801 (2015)]. Here, we investigate circularly polarized laser irradiated graphene with nondiagonal disorders, i.e., disordered tunneling, and find that disorder can induce nontrivial topological properties, characterized by a Bott index and the real-space Chern number. Moreover, we show that one can turn on the laser irradiation nonadiabatically to drive the disordered graphene into a nontrivial topological phase. It is a scheme which is especially interesting for experimental implementations.

Figure 1

The phase diagram characterized by the Bott index of pristine graphene irradiated by a circularly polarized laser. $A_0=1.6$. A periodical boundary condition is adopted. The red dots indicate the parameters $(M,\mathrm{\Omega })$ for the disordered system we have investigated in Figs. 2 and 3.

Figure 2

The Bott index ${\overline{C}}_b(-\frac{\mathrm{\Omega }}{2},0)$ for the Hamiltonian (2) as a function of disordered tunneling strength for a lattice of $24\times 24$ with different staggered potentials $M$. The driving frequency $\mathrm{\Omega }=6$. $A_0=1.6$. A periodical boundary condition is adopted. In this and the following calculations, the energy unit is $J{\mathcal{J}}_0\left(A_0\right)$ with ${\mathcal{J}}_0\left(A_0\right)$ as the zeroth-order Bessel function of the first kind. 960 realizations are calculated.

Figure 3

The Bott index ${\overline{C}}_b(-\frac{\mathrm{\Omega }}{2},0)$ for the Hamiltonian (2) with disordered tunneling for a lattice of $24\times 24$ for different driving frequencies. A periodical boundary condition is adopted. The staggered potential $M=0.8$. $A_0=1.6$. 960 realizations of disorder are calculated.

Figure 4

A graphene lattice of $25\times 24$ with open boundary conditions in (a) where sites in the rectangular box are chosen as bulk sites and sites in the 12th and 13th rows are indicated by zigzag lines, and (b)–(d) are Chern markers on sites in the box of (a) for different disorder strengths. The driving frequency $\mathrm{\Omega }=6$. $M=0.6.A_0=1.6$. 960 realizations of disorder are calculated.

Figure 5

The Chern marker ${\mathcal{C}}_i$ for a lattice of $25\times 24$ for different nondiagonal disorder strengths. The blue and red lines are Chern markers for the 12th and 13th rows of lattice sites. An open boundary condition is adopted. The driving frequency $\mathrm{\Omega }=6$. $M=0.6.A_0=1.6$. 960 realizations of disorder are calculated.

Figure 6

With a quenched ramp up of the laser irradiation, the time evolution of the Bott index ${\overline{C}}_b(-\frac{\mathrm{\Omega }}{2},0)$ in (a), and level spacing ratio (LSR) in (b) for the Hamiltonian (2) with disordered tunneling from an initial state with a trivial topological phase for a lattice of $18\times 18$ for different disorder strengths. A periodical boundary condition is adopted. $\mathrm{\Omega }=6$. The staggered potential $M=0.7$. $A_0=1.6$. $N_0=400, N=150$. 256 realizations of disorder are calculated.

Figure 7

With a linear ramp up of the laser irradiation by setting $\mathit{A}=A_0\frac{t}{\tau }(cos\left(\mathrm{\Omega }t\right),sin\left(\mathrm{\Omega }t\right))$, the time evolution of the Bott index ${\overline{C}}_b(-\frac{\mathrm{\Omega }}{2},0)$ in (a), and level spacing ratio (LSR) in (b) for the Hamiltonian (2) with disordered tunneling from an initial state with trivial topology for a lattice of $18\times 18$ for different disorder strengths. $\tau =200T$. The other parameters are the same as in Fig. 6.