Floquet topological semimetal with a helical nodal line in 2+1 dimensions

Topological semimetals with a nodal line is a class of topological matter extending the concept of topological matter beyond topological insulators and Weyl/Dirac semimetals. Here we show theoretically that a Floquet topological semimetal with a helical nodal line can be generated in 2+1 dimensions by irradiating graphene or the surface of a topological insulator with circularly polarized light. The helical nodal line is the nodal line running across the Brillouin zone with helical winding. Specifically, it is shown that the dynamics of irradiated graphene is described by the time Stark Hamiltonian, which can host a Floquet topological insulator and a weakly driven Floquet topological semimetal with a helical nodal line in the high and low frequency limits, respectively. One of the most striking features of the Floquet topological semimetal at low frequency is that the Berry phase accumulated along the time direction, also known as the Zak phase, has a topological discontinuity of $\pi$ across the projected helical nodal line. It is predicted that such a topological discontinuity of the Berry phase manifests itself as the topological discontinuity of the Floquet states. At intermediate frequency, this topological discontinuity can create an interesting change of patterns in the quasienergy dispersion of the Floquet states.

Figure 1

Schematic diagram showing the mapping of irradiated graphene from the real (a) to the extended momentum (b) space. Here, ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ denote the primitive lattice vectors of graphene in the real space, while ${\widehat{\mathbf{b}}}_1$ and ${\widehat{\mathbf{b}}}_2$ denote the unit reciprocal lattice vectors in the extended momentum space. The dynamics of irradiated graphene is described by the time Stark Hamiltonian in the extended momentum space, where an effective dc electric field with the strength equal to the radiation frequency $\mathrm{\Omega }$ is applied along the axis of the helical nodal line aligned with the time, i.e., $\kappa =-\mathrm{\Omega }t$ direction. Note that the helical nodal line is the trajectory of each Dirac node encircling either $K$ or $K^{\prime }$ as a function of time. The radius of the helical nodal line is given by $A=e{E}_0/\mathrm{\Omega }$ with $E_0$ being the radiation electric field strength. The Zak phase accumulated along the $\kappa$ direction acquires a relative $\pi$ shift inside the projected helical nodal line due to the same reason why the Su-Schrieffer-Heeger model becomes topological. At low frequency, the $\pi$ shift of the Zak phase gives rise to the topological discontinuity across the projected helical nodal line in the momentum spectrum of the Floquet states.

Figure 2

Zero-energy momentum spectrum of the Floquet states in irradiated graphene at low frequency. Here, the radiation frequency is set to be $\mathrm{\Omega }/\tau =0.05$ with $\tau$ being the hopping parameter, and the radius of the helical nodal line is set to be $A=0.8$. (a) Zak phase accumulated along the time direction, showing the $\pi$ shift inside the projected helical nodal line. (b) Zero-energy momentum spectrum of the full Floquet states obtained by exactly solving either the Floquet or the time Stark Hamiltonian. (c) Closeup of the zero-energy momentum spectrum of the time Wannier-Stark ladder eigenstates obtained by solving the Abelian approximation of the time Stark Hamiltonian. (d) Closeup of the zero-energy momentum spectrum of the full Floquet states in the same region as (c), denoted by the green box in (b).

Figure 3

Quasienergy dispersion of the Floquet states in irradiated graphene. Here, the radiation frequency and the radius of the helical nodal line are set to be the same corresponding values as Fig. 2. (a) Quasienergy dispersion of the time Wannier-Stark ladder eigenstates showing the mismatch of the quasienergy eigenvalues and hence the topological discontinuity at the projected helical nodal line. Note that the inside of the projected helical nodal line is denoted as a yellow region. (b) Quasienergy dispersion of the full Floquet states showing the opening of the quasienergy gap near the projected helical nodal line. The quasienergy gap is very well captured by the analytical formula ${\mathcal{E}}_{\mathrm{gap}}$ in Eq. (24), the result of which is plotted as the shaded (grey) regions bounded by the thin black lines in the figure. Note that the thin black lines denote the plus and minus halves of ${\mathcal{E}}_{\mathrm{gap}}$. Color indicates the Abelian weight measuring how much the Floquet states belongs to the conduction (red) or valence (blue) band. By construction, the time Wannier-Stark ladder (WSL) eigenstates belong to either one of the two bands exclusively. The green lines in the insets show the scanned momentum path, along which the quasienergy dispersion is computed.

Figure 4

Evolution of the Floquet states in irradiated graphene as a function of the radius of the helical nodal line. (a)–(e) Zero-energy momentum spectrum of the time Wannier-Stark ladder eigenstates. (f)–(j) Zero-energy momentum spectrum of the full Floquet states. Here, the radius of the helical nodal line $A$ is changed from 0.4 [(a), (f)] to 0.6 [(b), (g)] to 1.0 [(c), (h)] to 1.2 [(d), (i)] to 1.6 [(e), (j)], while the radiation frequency $\mathrm{\Omega }/\tau$ is fixed as 0.05. Note that the projected helical nodal lines encircling different Dirac nodes are just about to touch at (d) and (i) and pass through each other at (e) and (j).

Figure 5

Quasienergy dispersion of the Floquet states in the continuum limit at an intermediate frequency with three representative ratios to the radius of the helical nodal line. Specifically, $\mathrm{\Omega }/\tau =16A/\pi$ (a)–(d), $8A/\pi$ (e)–(h), and $A$ (i)–(l) with $\mathrm{\Omega }/\tau =0.2$. (a), (e), (i) Quasienergy dispersion of the time Wannier-Stark ladder eigenstates. (b), (f), (j) Quasienergy dispersion of the full Floquet states. (c), (g), (k) Local density of Floquet states via the Gaussian probe weight with the standard deviation $\sigma$ being $1/2$. (d), (h), (l) Local density of Floquet states via the Gaussian probe weight with the standard deviation $\sigma$ being $3/2$. As one can see, at $\mathrm{\Omega }/\tau =A$ (i)–(l), the quasienergy dispersion of the Floquet states cannot be understood in terms of the simple overlapping Floquet copies of the Dirac dispersion.