Non-linear transport in multifold semimetals

TL;DR

This work addresses how bulk nonlinear transport in 3D multifold semimetals reflects their quantum geometry. It develops a complete DC intrinsic transport framework up to third order in the electric field within Boltzmann theory, incorporating Drude and purely geometric contributions such as Berry curvature, Berry curvature dipole, and Berry connection polarization, and expresses results as Fermi-surface integrals. Symmetry analysis for magnetic point groups is used to constrain the tensor structure and guide interpretation, with explicit case studies for space groups 213 and 199 demonstrating how nonlinear responses differ between valleys and nodal charges. The findings pave the way for nonlinear valley-tronics in multifold systems and offer a route to disentangle geometric from dispersive contributions in experiments, with prospects for magnetic variants and AC physics.

Abstract

Transport measurements are a powerful way to probe the electronic structure of quantum materials, but the information they contain is often convoluted. Yet, in particular for simple low-energy fermiologies, and by combining linear and non-linear responses, definite conclusion can be drawn -- such as, for instance, in the case of the circular photogalvanic effect in Weyl semimetals. Here, we derive the complete DC intrinsic transport response functions up to third order in the applied electric field within Boltzmann theory that hold combined information about quantum geometry and band dispersion. We discuss the responses for multifold fermions at high-symmetry momenta in time-reversal symmetric crystals as well as their reduction by symmetry constraints. We exemplify in detail the cases of space group 213 and space group 199, which realize different multifold fermions, and show under which conditions these low-energy excitations can be differentially addressed through their bulk nonlinear responses, enabling nonlinear valley-tronics.

Figures (3)

• Figure 1: $\textbf{k}$-resolved Drude conductivity $\sigma^{(1),\text{D}}_{xx}$ on the Fermi pocket centered at $P$. When summed other the entire Fermi surface, this gives $a_1(=a_2)$ in Eq. \ref{['jEgeneral']}.
• Figure 2: $\textbf{k}$-resolved contributions to the second order conductivity on the Fermi pocket centered at $P$, coming from the Drude conductivity $\sigma^{(2),\text{D}}_{\alpha \beta \gamma}$, the Berry curvature dipole conductivity $\sigma^{(2),\text{BCD}}_{\alpha \beta \gamma}$ and the intrinsic Berry connection polarization conductivity $\sigma^{(2),\text{BCP}}_{\alpha \beta \gamma}$. When summed other the entire Fermi surface, these give $b_1$ in Eq. \ref{['jEgeneral']}. The only contributions which can sum to a finite value are $\sigma_{x[yz]}$ type where $[...]$ denotes index symmetrization. Note that the BCD contribution sums to zero.
• Figure 3: $\textbf{k}$-resolved contributions to the third order conductivity on the Fermi pocket centered at $P$, coming from the Drude conductivity $\sigma^{(3),\text{D}}_{\alpha \beta \gamma \eta}$, the Berry curvature quadrupole conductivity $\sigma^{(3),\text{BCQ}}_{\alpha \beta \gamma \eta}$, the Berry connection polarizability dipole $\sigma^{(3),\text{BCPD}}_{\alpha \beta \gamma \eta}$ and the intrinsic second Berry connection polarization conductivity $\sigma^{(3),\text{SBCP}}_{\alpha \beta \gamma \eta}$. When summed other the entire Fermi surface, these give $b_1$ in Eq. \ref{['jEgeneral']}. The only contributions which can sum to a finite value are of type $\sigma_{xxxx}$ and $\sigma_{x[xyy]}$ where $[...]$ denotes index symmetrization. Note that the BCQ contributions sum to zero, as does the SBCP contribution to $\sigma_{xxxx}$ (which are not plotted here).