Elettra | APE-LE Measuring electron’s quantum geometry Following the seminar works of Lev Landau (1908-1968), conventional phase transitions in solids have long been understood within the framework of symmetry breaking. For instance, the transition from a paramagnetic to a ferromagnetic state is characterized by the breaking of spin-rotational symmetry, while the transition to superconducting state is marked by the gauge symmetry breaking. In the late 20th century, however, a paradigm shift emerged through the groundbreaking work of Michael Kosterlitz, David Thouless, and Duncan Haldane, who introduced a new class of phase transitions governed not by symmetry breaking, but by changes in electronic topology of solids. This new framework – often called topological phase transitions – earned them the Nobel Prize in Physics in 2016. Since then, understanding the topological properties of wavefunctions at the microscopic level and exploring their profound impact on macroscopic physical phenomena has become a central subject of modern condensed matter physics. Mathematically, topology is a global invariant property that is defined by the integration of local geometric quantities over a given parameter space. In condensed matter physics, the topology in solids is determined by the integrating the quantum geometry of electronic wavefunctions across momentum space. In this regard, a central quantity for describing topology and quantum geometry in solids is the so-called Quantum Geometric Tensor (QGT). The real and imaginary parts of the QGT are the quantum metric (Fig. 1 a) and the Berry curvature (Fig. 1 b), which describes the distance and phase information in the wavefunction space. Together, the quantum metric and Berry curvature encode the complete geometric information of electrons in solids, and underpin a wide range of exotic quantum phenomena, such as topologically-protected charge transport that can be harnessed for nextgeneration transistors, nonlinear Hall effects that have potential application in wireless energy transfer and rectification, and even unconventional superconductivity with critical temperature exceeding conventional limits. Despite its central importance, however, a direct experimental method to measure the quantum geometry of electrons in solids has long remained elusive. Roughly speaking, measuring the QGT of electrons is analogous to obtaining a complete understanding of electrons’ behavior in solids in wavefunctionlevel, which is an inherently challenging task. As a result, previous studies on quantum geometric properties in solids have almost always relied on theoretical calculation of the QGT, rather than its experimental measurements. To overcome this challenge, we have recently developed a novel strategy to measure the quantum geometry of electrons in solids using Angle-Resolved Photoemission Spectroscopy (ARPES) with both polarization- and spinresolution. While ARPES is best known as a direct tool for measuring the band structure of solids (a scalar property), the ARPES signal as a function of light polarization and electron spin also encodes wavefunction information (a vector property). This direct sensitivity to wavefunction properties suggests that carefully designed photoemission experiments and analysis can uncover the underlying quantum geometry of electronic wavefunctions in solids. We applied this idea to a model kagome lattice material, CoSn. Owing to its unique lattice symmetry, kagome lattice materials are known to host exotic electronic states, such as Dirac fermions (massless electrons that behave like light, m = 0) and flat band electrons (electrons with infinite mass, m = ∞) with intrinsic topology. By performing spin- and polarization-resolved ARPES measurements at the state-of-theart spin-ARPES facility at the APE-LE beamline of Elettra, we successfully reconstructed the quantum geometry and topology of these special electrons in the kagome lattice. Importantly, unlike previous approaches that were restricted to specific model systems, our methodology is completely general and broadly applicable to a wide class of quantum materials, including twodimensional semiconductors and topological metals. The range of exotic quantum matter phenomena governed by quantum geometry in solids is rapidly expanding. For example, while earlier studies primarily focused on the role of the imaginary part of the QGT (Berry curvature), more recent studies have demonstrated that the real part of the QGT (quantum metric) also gives rise to an equally important set of unconventional phenomena, such as anomalous Landau levels, flat band superfluidity, and exitonic Lamb shifts. Given this growing importance of quantum geometry in solids, we anticipate that our work – offering a general framework to quantify the QGT – will serve as an essential foundation for advanced understanding of quantumgeometric phenomena in solids and for guiding the discovery of new quantumgeometric materials. Elettra Highlights 2024–2025 14
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