Dynamical competition between quantum Hall and quantum spin Hall effects

In this paper, we investigate the occurrence of quantum phase transitions in topological systems out of equilibrium. More specifically, we consider graphene with a sizable spin-orbit coupling, irradiated by circularly polarized light. In the absence of light, the spin-orbit coupling drives a quantum spin Hall phase where edge currents with opposite spins counterpropagate. On the other hand, the light generates a time-dependent vector potential, which leads to a hopping parameter with staggered time-dependent phases around the benzene ring. The model is a dynamical version of the Haldane model, which considers a static staggered flux with zero total flux through each plaquette. Since the light breaks time-reversal symmetry, a quantum Hall (QH) phase protected by an integer topological invariant arises. An important difference with the static QH phase is the existence of counterpropagating edge states at different momenta, which are made possible by zero- and two-photon resonances. By numerically solving the complete problem, with spin-orbit coupling and light, and investigating different values of the driving frequency $\omega$, we show that the spectrum exhibits nontrivial gaps not only at zero energy but also at $\omega /2$. This additional gap is created by photon resonances between the valence and conduction band of graphene, and the symmetry of the spectrum forces it to lie at $\omega /2$. By increasing the intensity of the irradiation, the topological state in the zero energy gap undergoes a dynamical phase transition from a quantum spin Hall to a quantum Hall phase, whereas the gap around $\omega /2$ remains in the quantum Hall regime.

Figure 1

A plaquette of the honeycomb lattice is shown. Sites of the $A$ ($B$) sublattice are depicted as red (blue) dots. The arrows denote counterclockwise hopping along the lattice bonds by an electron. Next to each arrow the hopping phase $\varphi$ is given. Important is that all the hopping phases have the same functional form, but due to a different orientation of the bonds they are evaluated at times differing by a multiple $\delta =T/6$.

Figure 2

The spectrum of the Floquet Hamiltonian defined in Eqs. (1) and (5), for a graphene cylinder with zigzag edges; note that the entire spectrum is spin degenerate. Even though there are only two energy bands in graphene, the periodicity of the spectrum causes the existence of two inequivalent band gaps. The inlays show the gapless edge states with a magnification of 2 for greater clarity. Black bands are localized in the bulk, and red and blue bands are localized on the upper and lower edges of the system, respectively. Parameter values are (a) $V=J,\omega =3J$ and (b) $V=J,\omega =2.2J$. Since the band gaps open at multiples of $\omega /2$ for all values of $\omega$, the quasienergies $\varepsilon$ are given in units of $\omega /2$, which takes a different value in (a) and (b).

Figure 3

The spectrum of the Floquet Hamiltonian defined in Eqs. (1) and (6) for $\omega =3J$ and $\lambda =0.06J$, for a graphene cylinder with zigzag edges. The inlays show the gapless edge states with a magnification of 2, for greater clarity. Black bands are localized in the bulk, and red and blue bands are localized on the upper and lower edges of the system, respectively. (a) For $V=0$, the known dispersion for the QSH effect is observed, apart from a periodic continuation in $\omega$, caused by artificially considering the Floquet spectrum despite a time independence. Since the spectrum is spin degenerate, we indicated the spin direction for each edge state in the inlay. (b) For $V=J$, in the presence of both ISO coupling and circularly polarized light, the spin degeneracy in both edge and bulk states is lifted. In the gap at $\varepsilon =0$, the QSH still dominates. (c) For $V=1.3J$ and $\varepsilon =0$, the QH and QSH effects annihilate each other for spin down, and enhance each other for spin up. This causes one of the states on each edge to become dispersionless and thus also localized. (d) For V=$2J$, the irradiation dominates; aside from a lifting of the spin degeneracy, the graph resembles the one in Fig. 2, where QH states are generated.