Bulk photovoltaic effects in the Haldane model

TL;DR

This work analyzes the bulk photovoltaic effect (BPVE) in the two-dimensional Haldane model under mirror-time ($\mathcal{MT}$) symmetry. By deriving symmetry-constrained forms for the second-order photoconductivity and computing the associated quantum geometric quantities, the study shows that linearly polarized light can drive orthogonal shift and injection currents, while circularly polarized light is forbidden by $C_{3z}$ symmetry. The injection current is tied to the quantum metric and does not change sign across a topological phase transition, whereas the shift current is linked to the symplectic connection and can flip sign due to band inversion. Zone-averaged responses and vortex structures in the symplectic connection reveal distinct signatures of topological vs trivial phases, offering a route to experimentally probe quantum geometry and symmetry breaking in BPVE, e.g., in ultracold-atom realizations of the Haldane model.

Abstract

The bulk photovoltaic effect (BPVE) refers to the direct current generation in a noncentrosymmetric material under illumination and can be applied to solar energy technology. BPVE includes injection and shift currents, led by the change of velocity and displacement of wave packet during optical transitions, respectively. We derive the constraints on the conductivity tensors imposed by mirror-time ($\mathcal{MT}$) symmetry for two-dimensional systems. For the Haldane model, we show that linearly polarized light can induce shift and injection currents. In contrast, circularly polarized light can not induce shift or injection currents, as constrained by the three-fold rotation symmetry. Additionally, due to the presence of $\mathcal{MT}$ symmetry, a separation of responses is shown in the Haldane model. Under linearly polarized light, shift current, allowed by time-reversal symmetry, flows perpendicularly to the injection current, allowed by $\mathcal{MT}$ symmetry. Across the topological phase transition, the injection current does not change sign since the group velocity's sign remains unchanged. On the contrary, shift current shows a sign flip, as a result of band inversion. Furthermore, we calculate quantum geometry, including quantum metric and symplectic connection, to demonstrate the microscopic quantum origin of the BPVE. We found that the vector field of symplectic connection in the Brillouin zone possesses vortices in the topological phase, but not in the trivial phase.

Figures (12)

• Figure 1: Schematic of response separation in the Haldane model. For a $y$-polarized plane electromagnetic wave incident normally upon the $x$-$y$ plane, the $\mathcal{T}$-allowed shift current flows along $x$-direction, and the $\mathcal{M}_y\mathcal{T}$-allowed injection current flows along $y$-direction.
• Figure 2:
• Figure 3: Energy dispersion along $\bm{K'-\Gamma-M-K}$ for $\phi=0$ (a) and $\phi=-\pi/2$ (b). The numerical values indicate the energy gaps at $\bm{K',M,K}$ in the figure.
• Figure 4: (a) The injection conductivity as a function of photon energy for the Haldane model with $M/t_1=0.4$, $\phi=-\pi/2$ ($\mathbf{C}=-1$) and $\mu=0$. The vertical dashed lines, ordered from left to right, indicate the photon energies corresponding to the $\bm{K' ,M,K}$ energy gaps, as annotated in Fig. \ref{['fig:bands']}. (b) The momentum resolved quantum metric $g^{yy}$ for the conductivity in (a). (c) The momentum resolved quantum metric $g^{yy}$ for $M/t_1=0.4$, $\phi=0$ ($\mathbf{C}=0$) and $\mu=0$.
• Figure 5: The $xxx$ component of the shift conductivity for (a) $\phi=0$ and $\mu=-0.5$ and (b) $\phi=-\pi/2$ and $\mu=0$. $M/t_1=0.4$ for both panels. The color-coded vertical dashed lines correspond to the energy gaps annotated in Fig. \ref{['fig:bands']}.