Topology has become a critical factor in the field of modern condensed matter physics and beyond. It explains the way solid materials may possess two distinct and seemingly conflicting characteristics. An example of this is topological insulators, materials whose bulk acts as an insulator, and can still conduct electricity at their surfaces and edges.

Over the past several decades, the idea of topology has revolutionized the understanding of electronic structure and the overall properties of materials. Additionally, it has opened doors to technological advancements by facilitating the integration of topological materials into electronic applications.

At the same time, topology is quite tricky to measure, often requiring combinations of multiple experimental techniques such as photoemission and transport measurements. A method known as high harmonic spectroscopy has recently emerged as a key technique to observe the topology of a material. In this approach a material is irradiated by intense laser light.

The interactions between electrons in the material and the laser result in the emission of a broadband optical spectrum – which contains clues about the topological phase of the solid. With the help of theoretical calculations, those clues can be extracted in order to measure the material topology.

However, theoreticians at the Max Planck Institute for the Structure and Dynamics of Matter in Hamburg, Germany, now report in *Physical Review X* that they found no evidence of any universal topological signatures after performing the first *ab initio* investigation of high harmonic generation from topological insulators.

Focusing on a quantum spin Hall insulator in a monolayer of Bismuth atoms, and a quantum anomalous Hall insulator in a single monolayer of Na_{3}Bi, the researchers questioned the underlying assumptions of topological high harmonic spectroscopy: That topological information is imprinted on the emitted spectra and can be subsequently extracted.

“We specifically set out to avoid common approximations and simplified models,” explains lead author Ofer Neufeld. “In this vast and thorough analysis, we could not identify any universal topological signatures, hinting that it is unlikely such signatures exist. Even if at first glance some features seemed to strongly correlate with a topological property, whenever we dug into their origin it was never topological.”

Instead, the non-topological aspects of the system dominated its response, suggesting that topology may play a more minor role than previously thought. “For instance, a solid can react differently to laser light that is left or right elliptically polarized,” Nicolas Tancogne-Dejean, the paper’s second author, explains. “Initially it might seem that that typical response originates in the topology. However, on closer examination, this effect turns out to stem from the crystal structure, rather than the topological structure.”

The team’s findings raise important questions about the potential use of topology for applications in highly nonlinear optics. On a more positive note, the MPSD theoreticians stress that they do not rule out the existence of topological signatures in high harmonic generation altogether. However, they argue that other non-topological aspects of the material usually dominate the resulting spectra, such as the band structure, lattice symmetry, and the chemical nature of the participating orbitals.

“We hope that our study will not only provide a ‘cautionary tale’ to warn others of potentially misleading topological fingerprints, but more importantly, that it will motivate the community to come up with more complex and robust ideas for how to measure topology through nonlinear optics,” Neufeld concludes.

Reference: “Are There Universal Signatures of Topological Phases in High-Harmonic Generation? Probably Not.” by Ofer Neufeld, Nicolas Tancogne-Dejean, Hannes Hübener, Umberto De Giovannini and Angel Rubio, 28 July 2023, *Physical Review X*.

DOI: 10.1103/PhysRevX.13.031011

Even if at first glance some features seemed to strongly correlate with a topological property, whenever we dug into their origin it was never topological. According to topological vortex gravitational field theory, symmetry only manifests between topological vortices and their twin anti vortices, It is difficult for us to observe any universal topological signatures in the high-dimensional space-time matter formed by the interaction of topological vortices.