# Winding of electrons in the quantum world

What makes an orange different from a coffee mug?

Let’s take two objects and transform one into another without making or removing holes. If such transformation is possible, those objects share the same *topology*. Therefore, *topologically-speaking*, a donut is identical to a coffee mug (1 hole), but it is different from a tangerine or a tennis ball (zero holes). This geometrical concept has been used also by physicists to describe the electronic structure of solid state systems. A metal is a material where the electrons of the valence and conduction bands overlap without discontinuities (no gaps/holes). Instead, a semiconductor or an insulator have an energy gap separating the valence from the conducting band.

However, there are systems, so-called topologically non-trivial, which falls in between the standard classification of metals and insulators. These systems have properties where both electrons and spins are strongly entangled, and give rise to the appearance of novel phenomena, highly desired for spintronics applications. The energy needed to flip one spin is significantly lower than the energy required to put one charge in motion. By generating a sequence of spin-flips, information can be transferred without moving any particles. Thus, scattering is suppressed. This enables the opportunity for faster, more sensitive, more efficient, more complex and non-volatile processes.

The discovery of topological materials marks one of the biggest milestones in condensed matter physics because they can be tailored to host currents which are robust against disorder and various external conditions. One of the main parameters related to topology and which describes "*how the electrons wind*" around the space where they live is called *Berry-curvature* and could deepen our understanding on topological quantum matter drastically. We have measured for the first time this quantum parameter, opening a potential pathway to understanding, studying, and potentially controlling the topology of quantum materials.

To do this, the APE Low-Energy beamline at the Elettra synchrotron was used, thanks to the availability of circularly polarized synchrotron radiation and the electron spin-detector present at the end-station. By shining the light on the sample with circular polarization we were able to access directly the angular and orbital momentum of the electrons, while by filtering the photoemitted electrons according to their spins we mapped how the spins are oriented. Effectively, this methodology can be used to track what is known as “spin-Berry curvature”.

**Figure 1**: (a) Calculations of the spin-Berry curvature for a simplistic kagome lattice performed by using density functional theory (DFT). The blue and red colours represent the minimum and maximum value of the spin-Berry curvature. (b) Same as (a) performed for the real material TbV_{6}Sn_{6}. (c) Real measurements performed for spin up (red) and spin down (blue) species with circular left and (d) circular right light. As a result of the measurements, the e) circular dichroism resolved in spin is plotted (for both species) along with the integrated value (gray line), which is significantly smaller.

In **Fig.1**, the main finding of the study is reported. Density functional theory (DFT) calculations (**Fig.1 a-b**) can be used, first of all, to explain the phenomenology that was experimentally discovered: As shown in the simplified version of **Fig. 1a**, the kagome system used has a Dirac cone similar to the one reported for graphene. Together with this cone, a flat band is also present. These two features are a peculiarity in common to all kagome metals. The minimum of the Dirac cone (at the Γ point) is separated from the flat band by an energy gap. The latter is predicted to be topological. The same scenario is shown in **Fig. 1b** in a realistic picture. The experiment will be therefore aimed at demonstrating the topological nature of that gap. This was done in the present work: by reversing the polarization chirality of the light, the spin character is flipped too (**Fig.1 c-d**) and this results in a circular dichroism (**Fig.1 e**) with nearly 100% signal. The latter is the ultimate proof of the topological nature of the states which form the gap.

As a proof of principle, we have placed the incoming synchrotron light within a mirror plane of the crystal (this allows to understand the geometrical contributions to the matrix elements) and measured in the centre of the Brillouin zone, where the matrix elements are zero. This experimental geometry is crucial to the present experiment: by doing this, in fact, it was possible to remove the contribution given by the geometrical matrix elements, to access directly a ‘pure’ signal from the Berry curvature, for the first time.

We have performed this study on other systems of the kagome family and have shown that the topological character is ubiquitous to all of them, as it was previously suggested by the theoretical calculations. It is important to stress that this study has implications for understanding the relationship between band curvature and topology. The importance of the Berry-curvature in the description of topology is a concept dear to the cold atoms community. The idea of this branch of physics is to "freeze" a few atoms down to very low temperatures (from which the name cold-atoms) to build artificial lattices that mimic clean and simple versions of particular structures, i.e., honeycomb, hexagonal, kagome, Lieb. Cold-atoms experiments have shown an immense potential. We demonstrated how to measure the Berry curvature by detecting with circularly polarized light the *"winding of the eigenstates across the Brillouin zone*". However, the detection of the Berry-curvature in real materials is more challenging than in optical lattices: for a multi-orbital system, where very often electronic bands are fully or partially spin-degenerate, observing the spin-projection is required to get a signal different from zero. The present study is the first on a real material and sheds light on one of the experimentally most elusive quantities tightly connected to the concept of topology in quantum matter.

**This research was conducted by the following research team:**

Domenico Di Sante^{1-2}, Chiara Bigi^{3}, Philipp Eck^{4}, Stefan Enzner^{4}, Armando Consiglio^{4}, Ganesh Pokharel^{5}, Pietro Carrara^{6-7}, Pasquale Orgiani^{8}, Vincent Polewczyk^{8}, Jun Fujii^{7}, Phil D. C. King^{3}, Ivana Vobornik^{7}, Giorgio Rossi^{6-7}, Ilija Zeljkovic^{9}, Stephen D. Wilson^{5}, Ronny Thomale^{4}, Giorgio Sangiovanni^{4}, Giancarlo Panaccione^{7}, Federico Mazzola^{7-10}

^{1} Department of Physics and Astronomy, University of Bologna, Bologna, Italy

^{2} Center for Computational Quantum Physics, Flatiron Institute, New York, NY, USA

^{3} School of Physics and Astronomy, University of St Andrews, St Andrews, UK

^{4} Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, Würzburg, Germany.

^{5} Materials Department, University of California Santa Barbara, Santa Barbara, CA, USA

^{6} Dipartimento di Fisica, Università degli Studi di Milano, Milano, Italy

^{7} Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche, Trieste, Italy

^{8} CNR-IOM TASC Laboratory, Trieste, Italy

^{9} Department of Physics, Boston College, MA, USA

^{10} Department of Molecular Sciences and Nanosystems, Ca’ Foscari University of Venice, Venice, Italy

### Reference

D. Di Sante, C. Bigi, P. Eck, S. Enzner, A, Consiglio, G. Pokharel, P. Carrara, P. Orgiani, V. Polewczyk, J. Fujii, P.D.C. King, I. Vobornik, G. Rossi, I. Zeljkovic, S.D. Wilson, R. Thomale, G. Sangiovanni, G. Panaccione & F. Mazzola , *“Flat band separation and robust spin Berry curvature in bilayer kagome metals”*, Nat. Phys. (2023). DOI:10.1038/s41567-023-02053-z.