Floquet Engineering of Multifold Fermions

We next consider shining light on a material with three fold crossing. If a system is subjected to an external time-dependent periodic perturbation such that $H(t+T)=H(t)$, then the Floquet approximation states that in the high frequency ($\omega$) limit, an effective Hamiltonian describing the system can be written as Cayssol et al. (2013); Oka and Kitamura (2019)

where $\omega=2\pi/T$ and $H_{m}(\textbf{k})=\frac{1}{T}\int_{0}^{T}H(\textbf{k},t)e^{-im\omega t}dt$. In cases where the Hamiltonian $H(t)$ can be separated into a time-independent and a time-dependent part such that $H(\textbf{k},t)=H_{0}(\textbf{k})+V(\textbf{k},t)$, the expression simplifies to

If light is applied on our three fold fermion system, we can use Floquet theory to study the resulting effective band structure if the frequency of light, $\omega$, is sufficiently large such that $\omega\gg V$ (see Appendix A). We denote the vector potential of applied light by $\boldsymbol{A}(t)$. For non-cavity modes, light is plane-polarised, allowing us to choose $A_{z}=0$. Thus with a phase difference $\theta$ between the $x$ and $y$ components, we have $\boldsymbol{A}=[A_{x}\cos(\omega t),A_{y}\cos(\omega t+\theta),0]$. The effect of this vector potential on our Hamiltonian can be incorporated through the standard Peierls substitution, i.e., $\textbf{k}\to\textbf{k}+\boldsymbol{A}$. This gives, $H_{3f}(\textbf{k},t)=H_{3f}(\textbf{k})+V(t)$, where,

Now, for large $\omega$, we can use the Floquet approximation given by Equation 3 to study the effective photon-dressed static Hamiltonian $H_{floq}$. After integrating over one time period, we obtain the form of the Fourier components as

This gives the light induced term as

where we have defined the prefactor $\gamma$, having the same dimensions as $k$, as

Note that because of the $\sin\theta$ factor, $\gamma$ and hence the light-induced term vanishes for linearly polarized light (i.e. for $\theta=0$), while taking the maximum value for $\theta=\pi/2$ (elliptically polarized light). We can understand this as follows. Elliptical polarization provides the chirality needed to break time-reversal symmetry. In contrast, a linear polarization, made of equal superposition of clockwise and anticlockwise circular polarizations, cannot break time-reversal symmetry and hence does not lead to observable changes.

Our effective Floquet Hamiltonian after applying light thus reads

For certain cases this Hamiltonian can be mapped back to the original low energy model, $H_{3f}$ for three fold fermions. For $\phi=\frac{\pi}{6}$ (mod $\pi/3$), we can write $H_{floq}(\textbf{k})=H_{3f}(\tilde{\textbf{k}})$, where $\tilde{\textbf{k}}=(k_{x},k_{y},k_{z}-\gamma\sin 3\phi)$. The system in presence of light is thus exactly like the unperturbed system but with the 3-BTP now shifted along the $k_{z}$ axis. We now find the new eigenvalues

which are degenerate at $\textbf{k}=(0,0,\gamma\sin 3\phi)$. In contrast, however, for $\phi=0$ (mod $\pi/3$), we observe that a band gap opens up at $\textbf{k}=0$ (see Figure 1). For other intermediate values of $\phi$, we find that a band gap opens up along with a shift in the 3-BTP. For $\phi\in(0,\frac{\pi}{3})$, it shifts to the right, while for $\phi\in(\frac{\pi}{3},\frac{2\pi}{3})$, it shifts to the left, and this alternating pattern continues in a periodic manner.

Next, we find the analytical expressions for the light-induced band gap ($\Delta_{g}$) and location of the 3-BTP after the shift ($k_{z}^{0}$). The eigenvalues of $H_{floq}$ satisfy the following equation

The exact form of the eigenvalues for any general $\phi$ can be derived (given in Appendix B), using which we obtain

These expressions succinctly summarize the trends described earlier (see Figure 2). For $\phi=0$ (mod $\pi/3$) we obtain the largest band gap opening $\Delta_{g}=2\gamma$ at $\textbf{k}=0$, while for $\phi=\pi/6$ (mod $\pi/3$), we find the maximum shift of the 3-BTP to $k_{z}=\gamma$. The $\cos 3\phi$, $\sin 3\phi$ nature of variation of the band gap and shift respectively also captures the inherent $\pi/3$ periodicity in the response of the system. We note that the results recently reported in the literature Firoz Islam and Zyuzin (2019) for the special case of $\phi=\pi/2$ are consistent with ours.

Interestingly, we observe the maximum shift of the 3-BTP along the $k_{z}$-axis, with the three fold degeneracy being maintained, for cases where the original band structure was fully centrosymmetric around the degeneracy. Analogously, we observe the maximum band gap opening up for cases where the original band structure showed the maximal tilt along the $(1,1,1)$ direction. This matches Weyl semimetal studies where it has been shown that lifting the degeneracy in a fully centrosymmetric Weyl semimetal is difficult Wu et al. (2012). It has also been shown that tilted Weyl semimetals have a better response to light, and they can support significant photocurrents while centrosymmetric Weyls cannot Chan et al. (2017).

Effect of higher order terms: For our low energy model under illumination, all multi-photon terms $H_{m}$ for $m\neq 0,1,-1$ are zero. The only non-zero terms in the Floquet-Magnus expansion thus have the form: $[H_{0},H_{\pm 1}]/\omega$, $[H_{\pm 1},[H_{0},H_{\mp 1}]]/\omega^{2}$ and so on. All the properties discussed above hold true even when these higher-order terms are considered.

We discovered that depending on the value of $\phi$, the three fold degeneracy can either be lifted or shifted upon illumination with elliptically polarized light. To understand the underlying reason, it is insightful to look at the symmetries of our system, and see if they change after applying light. Multifold degeneracies in space group 199 are characterized by three unitary symmetries which protect the threefold degeneracy: $\{C_{2x}|\bar{\frac{1}{2}}\frac{1}{2}0\}$, $\{C_{2y}|0\frac{1}{2}\bar{\frac{1}{2}}\}$ and $\{C^{-1}_{3,111}|101\}$. They can be chosen to be Bradlyn et al. (2016)

A Hamiltonian obeying these symmetries is constrained to satisfy the following conditions

where, the matrices $D(G_{i})$ have the representation

It is easy to verify that our original Hamiltonian $H_{3f}$ satisfies Equation 13. Next, we investigate whether these symmetries are preserved once light is applied. We find that Equation 13 is satisfied by $H_{floq}(\tilde{\textbf{k}})$ with $\tilde{\textbf{k}}=(k_{x},k_{y},k_{z}-\gamma\sin 3\phi)$, whenever $\phi=n\frac{\pi}{6}$ (mod $\pi/3$). This implies that for these values of $\phi$, the three-fold degeneracy shifts but remains symmetry protected even after illumination. On the other hand, for all other values of $\phi$, $H_{floq}(\textbf{k})$ fails to satisfy the symmetry conditions. The degeneracy is no longer symmetry protected and can be lifted by light leading to opening up of a band gap. Note that because both circular and elliptically polarised light break all three lattice symmetries, the results are similar for both these choices of polarisation. Thus, we can understand our results based on these symmetry arguments.

Having discussed the tuning of the band structure with light, we next illustrate these changes using the k-resolved density of states (DOS) which can be directly measured in angle resolved photoemission (ARPES) experiments. The DOS at an energy $E$ given by $\rho(E,\textbf{k})$, can be calculated using Bruus and Flensberg (2004)

where the energy dependent Green’s function $G(E,\textbf{k})$ is given by

where $\eta\to 0^{+}$ is a positive infinitesimal. Contour plots for $\rho(E,\textbf{k})$ obtained are presented in Figure 3. For the unperturbed system, we observe that the DOS for $E$ close to the Fermi energy is large near $\textbf{k}=0$ due to the degeneracy, but decreases gradually as we move away from it. (Note that exactly at $E=0$, the DOS would be very large due to the contributions from the flat band.) When light is applied, the high density point representing the three fold degeneracy shifts along $k_{z}$ for $\phi=\pi/6$ (mod $\pi/3$). While, for other $\phi$ values, the high DOS near the center vanishes as the degeneracy is no longer present and a band gap opens up. The dependence of $\Delta_{g}$ and $k_{z}^{0}$ on the intensity of applied light can also be clearly seen from our results.

The signatures of light-induced changes to three fold fermions could be observed in several experimental probes. We propose measurement of the anomalous Hall effect as a direct way to verify the light-induced changes. As we discussed before, for $\phi=\pi/6$ (mod $\pi/3$), we can write $H_{floq}(\textbf{k})=H_{3f}(\textbf{k}-\delta\textbf{k})$ where $\delta\textbf{k}=(0,0,\gamma\sin 3\phi)$. A band gap does not open up at these values of $\phi$, and the 3-BTP continues to behave like a source or sink of Berry curvature with monopole charge $\pm 2$. Moreover, since the flat band is topologically trivial with Chern number zero, for these particular values of $\phi$, we can calculate the change in the anomalous Hall conductivity, $\Delta\sigma_{xy}$, at half filling using the relation Chan et al. (2016a)

Here $n$ is the Chern number of the lower band and $\nu_{z}$ is the shift of the 3-BTP along the $k_{z}$ direction after applying light. Here, we have restored factors of $\hbar$, $v_{f}$ and $e$ for ease of estimation. Thus, for our case we get,

As the monopole charge of our three fold fermion is twice that of a Weyl fermion, the resulting anomalous Hall signature also comes out to be double of its Weyl counterpart Firoz Islam and Zyuzin (2019). We will next discuss the values of different parameters required for experimental verification of our proposal and show that they lie well within current reach.

Now, we estimate values for the change in Hall voltage and band gap likely to be measured in experiments. If our sample with transverse dimensions $l_{x}\crossproduct l_{y}$ and thickness $l_{z}=d$ is illuminated with a high frequency laser of power $P$ shining normally along $z$ direction, the penetration depth for light would be given by $\delta(\omega)=\frac{n(\omega)c\epsilon_{0}}{\real\sigma_{xx}(\omega)}$, where $n$ is the refractive index of the material, $\sigma_{xx}$ is the longitudinal Hall conductivity, $c$ is the speed of light and $\epsilon_{0}$ is the permittivity of free space. When a current $I_{x}$ is applied along $x$-direction, we can thus measure a Hall voltage $V_{y}$ along the perpendicular direction, given by Chan et al. (2016a); Yan and Wang (2016)

As $\sigma_{xx}\gg\sigma_{xy}$, we can ignore the $\sigma_{xy}$ term in the denominator. The change in the Hall conductivity due to light will then read

For circularly polarised light, $\sin\theta=1$ and $A_{x}=A_{y}=A$. Using $E\sim A\omega$, we thus get

where $R$ is the reflectivity of the sample. This yields

We choose the typical sample parameters as $l_{x}=l_{y}=100$ $\mu$m, $d=100$ nm, $\sigma_{xx}\sim 10^{6}$ $\Omega^{-1}$m^-1, $\real\sigma_{xx}(\omega)\sim 10^{5}$ $\Omega^{-1}$m^-1 and $v_{f}\sim 5\crossproduct 10^{5}$ m/s and consider shining a mid-infrared laser pulse with $\omega=30$ THz and $P=1$ W on our sample. With $R=0.8$ and $I_{x}=1$ A, we finally obtain $\Delta V_{y}=290$ nV, which can be easily detected in experiment.

As the band gap is directly proportional to the intensity of applied light $I\sim P/l_{x}l_{y}$, using a laser with the same power $P=1$ W but smaller spot size would lead to larger effect on the band structure. With $l_{x}=l_{y}=1$ $\mu$m, we get $\Delta_{g}=2\hbar v_{f}\gamma\approx 25$ meV, which can again be readily measured in current ARPES experiments.

Heating in Floquet systems is a major concern for experimental realisation. However, there are some known ways of evading the problem Harper et al. (2020). Opening a gap above the conduction and below the valence bands McIver et al. (2020) is a possible way of preventing heating. Additionally, Floquet phases can be supported in prethermal regimes, where heating takes a parametrically long time Kuwahara et al. (2016); Weidinger and Knap (2017); Abanin et al. (2017). Finally, driven systems can be actively cooled by coupling to a bath. Under suitable conditions, Floquet phases can be stabilized in such systems Iadecola et al. (2015); Dehghani et al. (2015); Seetharam et al. (2015, 2018).

Next, we examine the properties of the tight binding model upon illumination. When we shine light on our lattice system, we can again use Floquet theory to study the resulting effective band structures. From our analysis of the low energy model, we know that linearly polarised light cannot change the band structure while elliptically polarised light affects it the most. We thus choose $\theta=\pi/2$, giving the vector potential of light as $\boldsymbol{A}=(A_{x}\cos\omega t,-A_{y}\sin\omega t,0)$. After some calculations, we find the lowest order light induced term to be

with $\zeta$ defined as

where $J_{1}$ is the Bessel function of the first kind. The photon-dressed effective Hamiltonian reads $H_{F}=H_{tb}+\frac{[V_{+1},V_{-1}]}{\omega}$. Plots of the eigenvalues of $H_{F}$ are presented in Figures 4(b) and 4(c). From the band structure, we observe that a band gap opens up and/or the two 3-BTPs shift along the $k_{z}$ axis depending on the value of $\phi$, similar to the case of the low energy model. The pattern exactly matches the one shown in Figure 2, in agreement with our low energy effective model analysis. For $\phi\in(0,\frac{\pi}{3})$, the two 3-BTPs shift inward towards $k_{z}=0$, while for $\phi\in(\frac{\pi}{3},\frac{2\pi}{3})$, they shift outwards, and the pattern continues in a periodic fashion.

As our system has a non-trivial topology, we expect to observe topologically protected surface states representing the bulk-boundary correspondence. To investigate these surface states and the effect of light on them we consider our generalised tight binding model under open boundary conditions. We make our lattice finite along the $x$-direction, while keeping it periodic along the $y$ and $z$ directions. The numerical results for the band structure are shown in Figure 5(a). Apart from the bulk bands, we find that surface states connecting the two 3-BTPs appear in this slab geometry. As shown in Figure 5(b), their corresponding wavefunctions are highly localized near the surface for $k_{z}$ close to zero but start penetrating into the bulk as the surface states mix with the bulk bands. Both the upper and the lower bands contribute two such states, in agreement with the bulk Chern number for the two bands being $\pm 2$.

After applying light, we observe that the surface states remain intact [see Figures 5(c) and 5(d)], indicating that light does not change the Chern number of the bands. The structure of the surface states remains preserved for $\phi=\pi/6$ (mod $\pi/3$), however, they now connect the shifted 3-BTPs. For other values of $\phi$ where a gap opens up, the surface states do not traverse the bulk gap. The wavefunction for these surface states, however, does not show any significant change with respect to surface localization and decay upon illumination with light.

We now move on to calculate the change in the anomalous Hall signature for our lattice model for $\phi=\pi/2$ and compare it to our low energy effective model results. Analogous to a Weyl semimetal Armitage et al. (2018); Bernevig and Hughes (2013), our three fold semimetal can be thought of as being constructed by stacking planes of anomalous Hall insulators with Chern numbers $\pm 2$ along $k_{z}$ Nandy et al. (2019). We show two such anomalous Hall insulator planes in Figure 6. For $k_{z}\in(-\pi/2,\pi/2)$, each plane possesses chiral surface states traversing the gap which show a quantized Hall conductance. However, for planes beyond $\pi/2$, no such surface states are present. Thus, at half filling the semimetal would show a large anomalous Hall conductivity coming from contributions from each of the planes between the two 3-BTPs implying $\sigma_{xy}=2\times\frac{e^{2}}{h}\times\frac{K_{0}}{2\pi}$ where $K_{0}$ is the separation between the two 3-BTPs. This gives $\sigma_{xy}=\frac{e^{2}}{h}$.

When light is applied, $K_{0}$ changes, leading to a change in the anomalous Hall signature. To get an analytical form for the 3-BTP shift, we consider the mass term in $H_{tb}$ at $k_{x}=k_{y}=0$ which reads $\cos k_{z}+\zeta$. For small $\zeta$, the mass term vanishes for $k_{z}=\pm(\pi/2+\delta k_{z})$. From the condition, $-\zeta=\cos(\pi/2+\delta k_{z})$, where $\cos(\pi/2+\delta k_{z})\approx-\delta k_{z}^{2}/2$, we thus obtain $\delta k_{z}=\zeta$. This tells us that the Hall signature changes by

The contribution from each node $\sim\frac{e^{2}\zeta}{\pi h}$ is similar to the low energy model results. Such a change in the anomalous Hall conductivity of the three fold fermion material upon illumination could be measured directly to test our predictions.