Intrinsic nonlinear conductivities induced by the quantum metric

Three mechanisms are known to contribute to second-order nonlinear current: Extrinsic nonlinear Drude, Berry curvature dipole, and intrinsic Berry connection polarizability. Here, we predict an intrinsic contribution to the current related to the quantum metric, a quantum geometric property of the electronic wave function. This contribution manifests in systems that simultaneously break the time-reversal and the inversion symmetry. Interestingly, this contribution is dissipative and contributes to both longitudinal and transverse response. The quantum metric-induced NL current dominates transport in parity-time reversal symmetric systems near the band edges, something we show explicitly for topological antiferromagnets.

Figure 1

A schematic of all four different second-order NL transport responses in the dc limit. The two contributions in the top row depend on the scattering time. The Drude conductivity arises from the second-order correction to the distribution function and the band gradient velocity. In contrast, the anomalous Hall conductivity arises from the first-order correction to the distribution function and the anomalous Hall velocity. The two intrinsic contributions are shown in the bottom row. The left panel represents the nonlinear BCP Hall conductivity. The right panel shows the NL BCP dissipative conductivity.

Figure 2

(a) Schematic of the dispersion of the tilted massive Dirac model. (b), (c) The momentum space distribution of the BCPH and BCPD components of the dipoles. They are in units of ${\mathrm{eV}}^{-1}{\text{Å}}^{-3}$. (d) Variation of the nonzero BCPH contributions with chemical potential, $\mu$. (e) The BCP-induced NL dissipative conductivities. The various parameters for the Hamiltonian are chosen to be $\mathrm{\Delta }=0.1\mathrm{eV}, v_t=0.1\mathrm{eV}$ Å, and $v_F=1\mathrm{eV}$ Å. We have considered temperature $T=50$ K.

Figure 3

(a) The energy gap between the conduction and the valence band in units of eV. Note the gapped Dirac points near $(k_x,k_y)=(\pm 1,0.5)\pi$ and $(k_x,k_y)=(0,0.8)\pi$. Near $(\pm 1,0.5)\pi$, the BCP Hall dipole is shown in (b), and the BCP longitudinal dipole is shown in (c). They are in units of ${\mathrm{eV}}^{-1}$ ${\text{Å}}^{-3}$. d) The chemical potential dependence of the NL Hall conductivity in which the contributions are induced by the BCP Hall dipole. (e) The chemical potential dependence of the longitudinal nonlinear conductivity induced by the BCP longitudinal dipole. We have used the Hamiltonian parameters $t=0.08$ eV and $\tilde{t}=1$ eV. The other parameters are ${\alpha }_{\mathrm{R}}=0.8, {\alpha }_{\mathrm{D}}=0$, and ${\mathit{h}}_{\mathrm{AFM}}=(0.85,0,0)$ eV. For the conductivity calculation we have considered temperature $T=50$ K.