Merging of Dirac points and Floquet topological transitions in ac-driven graphene

We investigate the effect of an in-plane ac electric field coupled to electrons in the honeycomb lattice and show that it can be used to manipulate the Dirac points of the electronic structure. We find that the position of the Dirac points can be controlled by the amplitude and the polarization of the field for high-frequency drivings, providing a new platform to achieve their merging, a topological transition which has not been observed yet in electronic systems. Importantly, for lower frequencies we find that the multiphoton absorptions and emissions processes yield the creation of additional pairs of Dirac points. This provides an additional method to achieve the merging transition by just tuning the frequency of the driving. Our approach, based on Floquet formalism, is neither restricted to specific choice of amplitude or polarization of the field, nor to a low-energy approximation for the Hamiltonian.

Figure 1

Honeycomb lattice irradiated by an ac electric field with arbitrary polarization. The inset shows the merging of the Dirac points induced by the external field.

Figure 2

Semimetallic (SM)/insulating (I) phase diagram ($\phi ,a{E}_y/\omega$) in the high-frequency regime for fixed $A_x/A_y=\sqrt{3}$, with $a$ the lattice spacing. NI (ZI) denotes a normal (Zak) insulating phase. The different phases, which are represented in different colors, are separated by four distinct merging lines corresponding to the four $M$${}_i$ points of the first Brillouin zone (shown on the right) where a PDP can be created/annihilated.

Figure 3

(a) Semimetallic (SM)/insulating (I) phase diagram in the high-frequency regime for $\phi =\pi /2$, where only three phases appear, due to the overlap between the $M$${}_1$ and $M$${}_2$ merging lines. (c) This overlap gives rise to localization lines separating two semimetallic phases. At the transition between these two semimetallic phases, the topological charges assigned to each Dirac point change sign [(b) and (d)]. This topological transition goes together with a change of the value of the Zak phase, which is 1 (0) in the SM${}_3$ (SM${}_0$) phase along the path represented by a full (dashed) arrow.

Figure 4

Sketch of the first uncoupled sidebands $\alpha =0,\pm 1$ as a function of a parameter (for instance, the field amplitude), for different driving frequencies. The colored area highlight the range where the sidebands are inverted.

Figure 5

Quasienergy spectra for $A_y=3.6$, $A_x=0$, and $\phi =0$ with (top left) $\omega =10t$, (top right) $\omega =2.5t$, (bottom left) $\omega =2.1t$, and (bottom right) $\omega =1.5t$.

Figure 6

Phase diagram as a function of the external ac electric field parameters $\phi$ and $E_y$ for the condition $A_x=\sqrt{3}{A}_y$. The different colors label the four inequivalent $M_j$ points for the merging. The color code is the same as the one in Fig. 2.

Figure 7

Phase diagram as a function of the external ac electric field parameters $E_x$ and $E_y$ for the condition $\phi =\pi /2$. The different colors label the four inequivalent $M_j$ points for the merging. Note that in this case the merging lines $M_1$ and $M_2$ overlap, giving rise to the existence of two and four simultaneous crossings. The color code is the same as the one in Fig. 2.

Figure 8

Merging lines as a function of the external ac electric field parameters $E_x$ and $E_y$ for the condition $\phi =0$. The different colors label the four inequivalent $M_j$ points for the merging. The color code is the same as the one in Fig. 2.