Gauge invariance of light-matter interactions in first-principle tight-binding models

We study the different ways of introducing light-matter interaction in first-principle tight-binding (TB) models. The standard way of describing optical properties is the velocity gauge, defined by linear coupling to the vector potential. In finite systems a transformation to represent the electromagnetic radiation by the electric field instead is possible, albeit subtleties arise in periodic systems. The resulting dipole gauge is a multi-orbital generalization of the Peierls substitution. In this work we investigate the accuracy for both pathways, with particular emphasis on gauge invariance, for TB models constructed from maximally localized Wannier functions. This approach accurately captures the light-matter interaction close to the Fermi level. Focusing on paradigmatic two-dimensional materials, we construct first-principle models and calculate the response to electromagnetic fields in linear response and for strong excitations. Benchmarks against fully converged first-principle calculations allow for ascertaining the accuracy of the TB models. We find that the dipole gauge provides a more accurate description than the velocity gauge in all cases. The main deficiency of the velocity gauge is an imperfect cancellation of paramagnetic and diamagnetic current. Formulating a corresponding sum rule however provides a way to explicitly enforce this cancellation. This procedure corrects the TB models in the velocity gauge, yielding excellent agreement with dipole gauge and thus restoring gauge invariance.

Figure 1

Calculated band structures along typical paths in the respective Brillouin zone for the four considered systems. The energy scale is chosen relative to the Fermi energy $E_F$ (red dashed line). The inset for FeSe illustrates the geometry and chosen unit cell.

Figure 2

Longitudinal optical conductivity of the considered 2D systems obtained from time propagation of the TB Hamiltonian in dipole (TB-DG) and velocity gauge (TB-VG), respectively. We propagated until a maximum time of $T_{\max }=8000$ a.u. and used $256\times 256$ sampling of the Brillouin zone, ensuring convergences. For the calculation of the conductivity with DFT we used a $200\times 200$ grid of Brillouin zone. We checked the convergence of the spectra in the considered frequency range with respect to the number of included bands.

Figure 3

Total Berry curvature of SnC (left) and ${\mathrm{WSe}}_2$ (right panel) along a characteristic path in the Brillouin zone with the TB-DG model [Eqs. (29) and (30)], TB-VG model [Eqs. (28) and (6)], and directly from the Bloch states (DFT). The turquoise line corresponds to the dipole contribution (30).

Figure 4

Current induced by a few cycle pulse (electric field shown in top panels) in the case of weak (middle) and strong driving (bottom panels). Calculations were performed with a $48\times 48$ sampling of the Brillouin zone for all cases. For better readability, $J_x$ has been multiplied by the factor ${10}^3$.

Figure 5

Current induced by a few cycle pulse as in Fig. 4 for longer times $t\gg \tau$ where $\mathbf{A}\left(t\right)\approx 0$. The color coding is consistent with Fig. 4.

Figure 6

Optical conductivity as in Fig. 2 (using the same parameters), but for the reduced TB models.