We study electron-electron and electron-phonon mediated pairing in the Holstein extended Hubbard model on the Kagome lattice near the van Hove fillings, and investigate their combined effects on electron pairing states. We find that their combination can promote exotic pairings in a crossover region, where the filling is close to a van Hove singularity. In particular, at the $p$-type van Hove filling the $E_{1u}$ ($p$-wave) and $B_{2u}$ ($f_{y^3-3yx^2}$-wave) pairings become leading, and at the $m$-type van Hove filling the $E_{1u}$ and $A_{2g}$ ($i$-wave) pairings get promoted. Moreover, we show that the electron-phonon interaction acquires a significant momentum dependence, due to the sublattice texture of the Fermi surfaces, which can promote non $s$-wave pairing. We present a detailed analysis of these pairing propensities and discuss implications for the vanadium-based kagome superconductors AV$_3$Sb$_5$.

Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev-Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.

Within a homologous series of cuprate superconductors, variations in the stacking of CuO$_2$ layers influence the collective charge dynamics through the long-range Coulomb interactions. We use O $K$-edge resonant inelastic x-ray scattering to reveal plasmon excitations in the optimally-doped trilayer Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$. The observed plasmon exhibits nearly $q_z$-independent dispersion and a large excitation gap of approximately 300 meV. This mode is primarily ascribed to the $\omega_{-}$ mode, where the charge density on the outer CuO$_2$ sheets oscillates out of phase while the density in the inner sheet remains unaltered at $q_z=0$. The intensity of the acoustic $\omega_3$ mode is relatively weak and becomes vanishingly small near $(q_x, q_y)=(0, 0)$. This result highlights a qualitative change in the eigenmode of the dominant low-energy plasmon with the number of CuO$_2$ layers.