Floquet band structure of a semi-Dirac system

In this work we use Floquet-Bloch theory to study the influence of circularly and linearly polarized light on two-dimensional band structures with semi-Dirac band touching points, taking the anisotropic nearest neighbor hopping model on the honeycomb lattice as an example. We find that circularly polarized light opens a gap and induces a band inversion to create a finite Chern number in the two-band model. By contrast, linearly polarized light can either open up a gap (polarized in the quadratically dispersing direction) or split the semi-Dirac band touching point into two Dirac points (polarized in the linearly dispersing direction) by an amount that depends on the amplitude of the light. Motivated by recent pump-probe experiments, we investigated the nonequilibrium spectral properties and momentum-dependent spin texture of our model in the Floquet state following a quench in the absence of phonons, and in the presence of phonon dissipation that leads to a steady state independently of the pump protocol. Finally, we make connections to optical measurements by computing the frequency dependence of the longitudinal and transverse optical conductivity for this two-band model. We analyze the various contributions from interband transitions and different Floquet modes. Our results suggest strategies for optically controlling band structures and experimentally measuring topological Floquet systems.

Figure 1

The honeycomb lattice and the first Brillouin zone. (a) The honeycomb lattice with primitive lattice vectors $a_1=a(\frac{3}{2},\frac{\sqrt{3}}{2}),a_2=a(\frac{3}{2},-\frac{\sqrt{3}}{2})$. A, B sublattices are colored as red and blue, respectively. (b) Reciprocal lattice vectors $b_1=2\pi /3a\left(1,\sqrt{3}\right),b_1=2\pi /3a\left(1,-\sqrt{3}\right)$ and the first Brillouin zone with high-symmetry points are marked. We note that ${\mathrm{\Gamma }}^{\prime }=b_1+b_2$ is equivalent to $\mathrm{\Gamma }$ for the convenience of plotting band structures along $\mathrm{\Gamma }\to M\to {\mathrm{\Gamma }}^{\prime }$.

Figure 2

The band energies from Eq. (4) with $t^{\prime }=2t$ and the semi-Dirac band touching point at $\mathbf{M}=(\frac{2\pi }{3},0)$ for (a) energy dispersions viewed along the $k_y$ axis, (b) dispersion along $\mathrm{\Gamma }\to K\to M\to K^{\prime }\to \mathrm{\Gamma }$, (c) energy dispersions viewed along the $k_x$ axis, and (d) dispersion along $\mathrm{\Gamma }\to M\to {\mathrm{\Gamma }}^{\prime }\to K\to \mathrm{\Gamma }$.

Figure 3

The Floquet band structures of the semi-Dirac honeycomb lattice model embedded in a normally incident polarized light for (a) circularly polarized light with $A=1.5,\mathrm{\Omega }=5t$; (b) circularly polarized light with $A=1.5,\mathrm{\Omega }=10t$; (c) circularly polarized light with $A=2.4,\mathrm{\Omega }=5t$; (d) linearly polarized light with $A=1.5,\mathrm{\Omega }=5t$ along the $x$ direction; (e) linearly polarized light with $A=1.5,\mathrm{\Omega }=5t$ along the $y$ direction. The semi-Dirac point linearly disperses in $k_x$ and has a quadratic dispersion along $k_y$.

Figure 4

Chern number as a function of the laser amplitude for different laser frequencies. Upper: $\mathrm{\Omega }=5.0$; middle: $\mathrm{\Omega }=7.5$; lower: $\mathrm{\Omega }=10.0$.

Figure 5

The spectral density of the semi-Dirac lattice model under the circularly polarized light. (a), (b) ARPES spectrum $i{\sum }_{\sigma }{g}_{\sigma \sigma }^{<}(k,\omega )$ for (a) $A=1.5,\mathrm{\Omega }=5t$ and (b) $A=2.4,\mathrm{\Omega }=5t$ along high-symmetry lines. Upper panel: Without phonons and for a quench. Lower panel: Steady state with phonons at $T=0.01\mathrm{\Omega }$. (c), (d) Time-averaged pseudospin density $P_z\left(k_x,k_y\right)$ for (c) $A=1.5,\mathrm{\Omega }=5t$ and (d) $A=2.4,\mathrm{\Omega }=5t$ in the first BZ with $\mathbf{k}·{\mathbf{a}}_1$ and $\mathbf{k}·{\mathbf{a}}_2$ as $x$ and $y$ axes. Left panel: Without phonons and for a quench. Right panel: Steady state with phonons at $T=0.01\mathrm{\Omega }$. (e), (f) Time-averaged pseudospin density $P_z\left(k_x,k_y\right)$ for (e) $A=1.5,\mathrm{\Omega }=5t$ and (f) $A=2.4,\mathrm{\Omega }=5t$ along the high-symmetry line. In both (e) and (f), the pseudospin polarization textures show a discontinuity at the $K$ point following a quench.

Figure 6

ARPES spectrum $i{\sum }_{\sigma }{g}_{\sigma \sigma }^{<}(k,\omega )$ for the lattice model embedded in the linearly polarized light with $A=1.5,\mathrm{\Omega }=5t$ along the high-symmetry lines with (a), (b) $x$ polarization and (c), (d) $y$ polarization. Upper panel: Without phonons and for a quench. Lower panel: Steady state with phonons at $T=0.01\mathrm{\Omega }$.

Figure 7

The longitudinal and transverse optical conductivity $\text{Re}\left[\sigma \right]$ under a circularly polarized laser field as a function of the frequency of the probe light, in units of $e^2/h$. The driving laser frequency, laser amplitude, and the Chern number are (a) $\mathrm{\Omega }=5t,A=1.5,C=1$; (b) $\mathrm{\Omega }=5t,A=2.4,C=2$; (c) $\mathrm{\Omega }=10t,A=1.5,C=1$. Top panel: Ideal electron distribution with ${\rho }_{kd}-{\rho }_{ku}=1$. Middle panel: Distribution following a quench. Bottom panel: Steady state distribution with phonons at $T=0.01\mathrm{\Omega }$.

Figure 8

The longitudinal and transverse optical conductivity $\text{Re}\left[\sigma \right]$ decomposed into different Floquet modes in the ideal case ${\rho }_{kd}-{\rho }_{ku}$ under a circularly polarized laser field as a function of the frequency of the probe light, in units of $e^2/h$ for $\mathrm{\Omega }=5t,A=1.5,C=1$. Top panel: ${\sigma }_{xx}^m$. Middle panel: ${\sigma }_{yy}^m$. Bottom panel: ${\sigma }_{xy}^m$.

Figure 9

The longitudinal optical conductivity $\text{Re}\left[\sigma \right]$ under a linearly polarized laser field and its decompositions into $\text{Re}\left[{\sigma }^m\right]$ as a function of the frequency of the probe light with $\mathrm{\Omega }=5t,A=1.5$. (a) $\text{Re}\left[\sigma \right]$ for $x$ polarization, (b) $\text{Re}\left[{\sigma }^m\right]$ for $x$ polarization, (c) $\text{Re}\left[\sigma \right]$ for $y$ polarization, (d) $\text{Re}\left[{\sigma }^m\right]$ for $y$ polarization. Top panel: Ideal electron distribution with ${\rho }_{kd}-{\rho }_{ku}=1$. Middle panel: Distribution with quench. Bottom panel: Distribution with phonons at $T=0.01\mathrm{\Omega }$.