Topological phase transitions and quantum Hall effect in the graphene family

Monolayer staggered materials of the graphene family present intrinsic spin-orbit coupling and can be driven through several topological phase transitions using external circularly polarized lasers and static electric or magnetic fields. We show how topological features arising from photoinduced phase transitions and the magnetic-field-induced quantum Hall effect coexist in these materials and simultaneously impact their Hall conductivity through their corresponding charge Chern numbers. We also show that the spectral response of the longitudinal conductivity contains signatures of the various phase-transition boundaries, that the transverse conductivity encodes information about the topology of the band structure, and that both present resonant peaks which can be unequivocally associated with one of the four inequivalent Dirac cones present in these materials. This complex optoelectronic response can be probed with straightforward Faraday rotation experiments, allowing the study of the crossroads between quantum Hall physics, spintronics, and valleytronics.

Figure 1

Schematics of the system considered: a staggered monolayer of the graphene family exposed to out-of-plane static electric and magnetic fields together with a normally incident circularly polarized laser. Optoelectronic properties of the system can be probed with Faraday rotation measurements, where incident linearly polarized light becomes elliptically polarized and undergoes a rotation of the polarization plane after transmission through the monolayer.

Figure 2

(a) Static Hall conductivity ${\sigma }_{xy}$ plotted in the $(E_z,\mathrm{\Lambda })$ plane at zero magnetic field for a neutral and dissipationless monolayer. The charge Chern number at the phase-transition boundaries (lines and points) is given by the average of the charge Chern numbers of the adjacent regions. The phase diagram remains unchanged in the presence of a static magnetic field provided $\mu =0$ (see text). (b) Relevant transitions for the case $\left|\mu \right|<{\varepsilon }_1$. For illustration purposes only, we consider a cone for which ${\varepsilon }_0>0$ and choose $\mu <{\varepsilon }_0$. (c) Phase diagram for the static ${\sigma }_{xy}$ for a dissipationless but doped monolayer with $\mu /{\lambda }_{\text{SO}}=0.5$. The magnetic field intensity chosen $\left(E_B/\left|\mu \right|=100\right)$ is to ensure $\left|\mu \right|<{\varepsilon }_1$ everywhere in the phase plane and for all Dirac cones. Under these conditions, the plot is independent of the particular value of the magnetic field. The chemical potential has a role similar to $\mathrm{\Lambda }$ and therefore shifts the phase diagram vertically.

Figure 3

(a) Relevant transitions for the case $\left|\mu \right|>{\varepsilon }_1$, such that ${\varepsilon }_N<\mu <{\varepsilon }_{N+1}$. The interband transitions are shown with black arrows, and they include the lone transition $-N\to N+1$ as well as the pair of transitions $-l\to l+1$ and $-(l+1)\to l$ for all $l\ge N+1$. Also shown is the intraband transition $N\to N+1$, depicted with a gold arrow. (b) Static Hall conductivity ${\sigma }_{xy}$ plotted in the $(E_z,\mathrm{\Lambda })$ plane for $\mu /{\lambda }_{\text{SO}}=0.5,E_B/{\lambda }_{\text{SO}}=0.34$, and a dissipationless monolayer. $N=1$ regions (between adjacent solid and dashed lines) and $N=2$ regions (between adjacent dashed lines) appear near the unshifted phase boundaries, where the Dirac mass is sufficiently small. The shifted boundaries seen in Fig. 2 are still present.

Figure 4

(a) Static Hall conductivity ${\sigma }_{xy}$ plotted in the $(E_z,\mu )$ plane for $E_B/{\lambda }_{SO}=1.5,\mathrm{\Lambda }=0$, and a dissipationless monolayer. The horizontal strip in between the dashed lines corresponds to $\left|\mu \right|<E_B$. (b) Hall conductivity as a function of doping displaying the interplay between photoinduced phase transitions and the quantum Hall effect for $\mathrm{\Lambda }/{\lambda }_{SO}=0$ (blue), 1/2 (red), and 1 (black). We choose $E_z=0$, so that Dirac cones are degenerate in the valley index. The pair of labels in each plateau shows the last filled Landau level for spin up and down. For cases with $\left|\mu \right|<|{\varepsilon }_0|$ for all four cones, plateaus are not labeled. When this condition is fulfilled for just two cones, the corresponding plateau has a single label for the last filled Landau level with spin up (see the $1\uparrow$ black plateau in the top left corner).

Figure 5

(a) The central panel shows the real part of ${\sigma }_{xx}\left(\omega \right)$ plotted versus frequency for various values of $\mathrm{\Lambda }/{\lambda }_{SO}$ for the case $E_z=0$, which is valley degenerate. The arrows correspond to spin-up and -down cones. The left subpanels show the allowed transitions for spin up for the corresponding $\mathrm{\Lambda }/{\lambda }_{SO}$ values, while the right subpanels show the allowed transitions for spin down for the two selected extreme values of $\mathrm{\Lambda }/{\lambda }_{SO}$. (b)–(d) Same as the central panel in (a) but for nonzero electrostatic field, $e\ell E_z/{\lambda }_{SO}=0.25,0.5$, and 0.75, respectively. Parameters are $\mu /{\lambda }_{\text{SO}}=0.5,E_B/{\lambda }_{\text{SO}}=5$ (so that $\left|\mu \right|<{\varepsilon }_1$ always), and $\hslash \mathrm{\Gamma }/{\lambda }_{SO}=0.02$. For clarity, all curves are vertically shifted by 1.5 with respect to each other.

Figure 6

Real part of ${\sigma }_{xy}\left(\omega \right)$ plotted versus frequency for various values of $\mathrm{\Lambda }/{\lambda }_{SO}$ for $e\ell E_z/{\lambda }_{SO}$ equal to (a) 0, (b) 0.25, (c) 0.5, and (d) 0.75. Color scheme, vertical shifting of curves, and parameters are the same as in Fig. 5.