Transport properties of nonequilibrium systems under the application of light: Photoinduced quantum Hall insulators without Landau levels

In this paper, we study transport properties of nonequilibrium systems under the application of light in many-terminal measurements, using the Floquet picture. We propose and demonstrate that the quantum transport properties can be controlled in materials such as graphene and topological insulators, via the application of light. Remarkably, under the application of off-resonant light, topological transport properties can be induced; these systems exhibit quantum Hall effects in the absence of a magnetic field with a near quantization of the Hall conductance, realizing so-called quantum Hall systems without Landau levels first proposed by Haldane.

Figure 1

The many-terminal measurements of dc current for graphene under the application of light. Graphene is attached to multiple leads labeled by $\left\{\alpha \right\}$, and the leads are connected to reservoirs at chemical potentials, $\left\{{\mu }_{\alpha }\right\}$. Off-resonant, circularly polarized light is applied to the graphene. In the absence of impurities and interactions, electrons coming from leads coherently propagate under the application of light, asborb or emit photons, and leak out into leads. Current measurements between each lead determine longitudinal and Hall conductances.

Figure 2

(a) The modification of the Hamiltonian due to the virtual photon process can be intuitively understood as the sum of two second-order processes where electrons absorb and then emit a photon and electrons first emit and absorb a photon. (b) The illustration of the structure of $H_{\mathrm{eff}}$ in real space for graphene under the application of right circularly polarized light in Eq. (3). The commutator $[H_1,H_1]$ is the second-neighbor hopping with phase $\varphi =\pi /2$. Thus, the tight-binding model under the application of light realizes Haldane model proposed in Ref.11.

Figure 3

(a) The spectrum of graphene for a single spin near one of the Dirac points for an infinite system. Here we plot the energy spectrum of a static system corresponding to $\mathcal{A}=0$ (top figure) and the spectrum of $H_{\mathrm{eff}}$ for the system under the application of light with light intensity $\mathcal{A}=0.3$ (bottom figure). A finite gap $\Delta$ opens at the Dirac point. (b) The spectrum of $H_{\mathrm{eff}}$ for a single spin near one of the Dirac points for infinite system in the $x$ direction and 150 sites in the $y$ direction for $\mathcal{A}=0.3$. The frequency of light is chosen to be off-resonant with $\Omega =7.5J$. Here we chose the armchair edges, and in this case the Dirac points are at $k_x=0$. $H_{\mathrm{eff}}$ shows the existence of gapless chiral edge states for each spin originating from nonzero Chern numbers of the bands, which are colored as blue and green, corresponding to the edge states in the upper and lower edges, respectively. The propagation of chiral edge states for a single spin is illustrated in the top figure.

Figure 4

Illustration of the response conduction current in the Floquet Landauer formula given in Eq. (5). The transmission of the electrons can now happen with $n$ photon absorption/emissions. The total conductance is simply given by the sum of these contributions for each $n$.

Figure 5

(a) Measurements of Faraday/Kerr rotations in three dimensional topological insulator. Circularly polarized light with a large frequency is used to open a gap at the Dirac point and linearly polarized light with small frequency within the gap is used to measure the Faraday/Kerr rotations. The polarization angle of the light is denoted by blue arrows in the figure. (b) The induction of magnetic monopole inside a three dimensional topological insulator through the application of light. Electron charge is placed near the surface, and circularly polarized light is applied on the surface to break the time-reversal symmetry and open the effective gap. The magnetic monopole is induced as a mirror image of the electron charge.

Figure 6

Illustration of the configuration considered in Sec. 3b3 to demonstrate the insulating behavior for gapped effective Hamiltonian $H_{\mathrm{eff}}$. $N_L$ is the number of leads that are attached to the infinite plane of graphene as “left” leads and $N_R$ is the number of leads that are attached as “right” leads. The chemical potentials of the reservoirs connected to left leads are assumed to be at $V/2$ and those of the reservoirs connected to the right leads are at $-V/2$ with $\left|V\right|\ll J$

Figure 7

The Floquet Dyson equation. The propagator goes from right to left. Double lines represent the full propagator $\widehat{\mathcal{G}}(n,\omega )$ and single lines are a bare propagator $\widehat{\mathcal{G}}(n,\omega )=\frac{1}{\omega +n\Omega -H_0}$ which does not include the effect of photon absorptions.