Anomalous geometric transport signatures of topological Euler class

TL;DR

This paper explores how the topological Euler class influences nonlinear electrical responses in two-dimensional topological materials, offering methods to detect and reconstruct this invariant through bulk transport measurements.

Contribution

It introduces a novel framework linking the Euler invariant to nonlinear Hall responses and provides analytic and numerical tools for its detection in materials.

Findings

Reconstruction of the Euler invariant via nonlinear electric field coupling.

Identification of anomalous bulk dynamics related to higher-order nonlinear Hall conductivities.

Proposal of a triple-contact setup to measure the non-Abelian Euler class in ballistic conductance.

Abstract

We investigate Riemannian quantum-geometric structures in semiclassical transport features of two-dimensional multigap topological phases. In particular, we study nonlinear Hall-like bulk electric current responses and, accordingly, semiclassical equations of motion induced by the presence of a topological Euler invariant. We provide analytic understanding of these quantities by phrasing them in terms of momentum-space geodesics and geodesic deviation equations and further corroborate these insights with numerical solutions. Within this framework, we moreover uncover anomalous bulk dynamics associated with the second- and third-order nonlinear Hall conductivities induced by a patch Euler invariant. As a main finding, our results show how one can reconstruct the Euler invariant by coupling to electric fields at nonlinear order and from the gradients of the electric fields. Furthermore, we comment on the possibility of deducing the nontrivial non-Abelian Euler class invariant specifically in second-order nonlinear ballistic conductance measurements within a triple-contact setup, which was recently proposed to probe the Euler characteristics of more general Fermi surfaces. Generally, our results provide a route for deducing the topology in real materials that exhibit the Euler invariant by analyzing bulk electrical currents.