Kagome metals $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K}$,Rb,Cs) exhibit an exotic charge order (CO), involving three order parameters, with broken translation and time-reversal symmetries compatible with the presence of orbital currents. The properties of this phase are still intensely debated, and it is unclear if the origin of the CO is mainly due to electron-electron or electron-phonon interactions. Most of the experimental studies confirm the nematicity of this state, a feature that might be enhanced by electronic correlations. However, it is still unclear whether the nematic CO becomes stable at a temperature equal to ($T_{\text{nem}}=T_{\text{C}}$) or lower than ($T_{\text{nem}}<T_{\text{C}}$) that of the CO itself. Here, we systematically characterize several CO configurations, some proposed for the new member of the family ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$, by combining phenomenological Ginzburg-Landau theories, valid irrespective of the specific ordering mechanism, with mean-field analysis. We find a few configurations for the CO that are in agreement with most of the experimental findings to date and that are described by different Ginzburg-Landau potentials. We propose to use resonant ultrasound spectroscopy to experimentally characterize the order parameters of the CO, such as the number of their components and their relative amplitude, and we provide an analysis of the corresponding elastic tensors. This might help to explain which mean-field configuration found in our study is the most representative for describing the CO state of kagome metals, and it can provide information regarding the nematicity onset temperature $T_{\text{nem}}$ with respect to $T_{\text{C}}$.

Kagome materials have emerged as a setting for emergent electronic phenomena that encompass different aspects of symmetry and topology. It is debated whether the XV₆Sn₆ kagome family (where X is a rare-earth element), a recently discovered family of bilayer kagome metals, hosts a topologically non-trivial ground state resulting from the opening of spin–orbit coupling gaps. These states would carry a finite spin Berry curvature, and topological surface states. Here we investigate the spin and electronic structure of the XV₆Sn₆ kagome family. We obtain evidence for a finite spin Berry curvature contribution at the centre of the Brillouin zone, where the nearly flat band detaches from the dispersing Dirac band because of spin–orbit coupling. In addition, the spin Berry curvature is further investigated in the charge density wave regime of ScV₆Sn₆ and it is found to be robust against the onset of the temperature-driven ordered phase. Utilizing the sensitivity of angle-resolved photoemission spectroscopy to the spin and orbital angular momentum, our work unveils the spin Berry curvature of topological kagome metals and helps to define its spectroscopic fingerprint.

In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.

From first-principles calculations, we investigate the structural and electronic properties of the kagome metals ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ $(A=\mathrm{Cs}, \mathrm{K}, \mathrm{Rb})$ under isotropic and anisotropic pressure. Charge-ordering patterns are found to be unanimously suppressed, while there is a significant rearrangement of $p$-type and $m$-type Van Hove point energies with respect to the Fermi level. Already for moderate tensile strain along the V plane and compressive strain normal to the V layer, we find that a Van Hove point can be shifted to the Fermi energy. Such a mechanism provides an invaluable tuning knob to alter the correlation profile in the kagome metal, and suggests itself for further experimental investigation. It might allow us to reconcile possible multidome superconductivity in kagome metals not only from phonons but also from the viewpoint of unconventional pairing.

Long-range electronic ordering descending from a metallic parent state constitutes a rich playground to study the interplay of structural and electronic degrees of freedom. In this framework, kagome metals are in the most interesting regime where both phonon and electronically mediated couplings are significant. Several of these systems undergo a charge density wave transition. However, to date, the origin and the main driving force behind this charge order is elusive. Here, we use the kagome metal ScV₆Sn₆as a platform to investigate this problem, since it features both a kagome-derived nested Fermi surface and van-Hove singularities near the Fermi level, and a charge-ordered phase that strongly affects its physical properties. By combining time-resolved reflectivity, first principles calculations and photo-emission experiments, we identify the structural degrees of freedom to play a fundamental role in the stabilization of charge order, indicating that ScV₆Sn₆features an instance of charge order predominantly originating from phonons.

Optical spectroscopy of ultimately thin materials has significantly enhanced our understanding of collective excitations in low-dimensional semiconductors. This is particularly reflected by the rich physics of excitons in atomically thin crystals which uniquely arises from the interplay of strong Coulomb correlation, spin-orbit coupling (SOC), and lattice geometry. Here we extend the field by reporting the observation of room temperature excitons in a material of non-trivial global topology. We study the fundamental optical excitation spectrum of a single layer of bismuth atoms epitaxially grown on a SiC substrate (hereafter bismuthene or Bi/SiC) which has been established as a large-gap, two-dimensional (2D) quantum spin Hall (QSH) insulator. Strongly developed optical resonances are observed to emerge around the direct gap at the K and K’ points of the Brillouin zone, indicating the formation of bound excitons with considerable oscillator strength. These experimental findings are corroborated, concerning both the character of the excitonic resonances as well as their energy scale, by ab-initio GW and Bethe-Salpeter equation calculations, confirming strong Coulomb interaction effects in these optical excitations. Our observations provide evidence of excitons in a 2D QSH insulator at room temperature, with excitonic and topological physics deriving from the very same electronic structure.