Plasmonics is a mature subfield of optoelectronics where light-matter interactions and propagating collective charge density excitations in a conductor are used to confine and steer electromagnetic energy in nanoscale devices. Progress in conventional metal-based plasmonics, however, has been hampered by substantial losses. Indeed, when electromagnetic fields are confined through the use of e.g. noble-metal plasmons, losses tend to be high and greatly limit the propagation distance of these collective modes. Substantial efforts have been recently made to increase the lifetime of these modes at room temperature, without decreasing the associated confining power. For example, one can utilize high-quality graphene sheets encapsulated in hexagonal boron nitride , where graphene plasmons scatter essentially only against the acoustic phonons of the two-dimensional (2D) carbon lattice, which are weakly coupled to the electronic degrees of freedom. Another possible pathway is to use plasmons in topologically-non-trivial materials. In the particular case of crystals displaying broken time-reversal symmetry (BTRS), the existence of unidirectional propagating modes akin to the ultra-long-lived  topological  edge magnetoplasmons that occur in 2D electron systems in the quantum Hall regime is expected. Technologically, it would be extremely useful to use materials where BTRS occurs without the aid of an external magnetic field. Natural candidates among topological materials with BTRS are recently discovered Weyl semimetals (WSMs) [4-7]. These are semimetals with protected linear band crossings in the Brillouin zone, which act as power-law-decaying sources of Berry curvature. Some of these compounds do display intrinsic BTRS  and, at the same time, have intriguing topological surface states called “Fermi arcs” (FAs).
In this talk, I will present a fully quantum-mechanical theory of WSM FA plasmons . The present derivation focuses on the simplest microscopic model Hamiltonian of a (type-I) WSM with BTRS [4-7] and is based on linear response theory  and the random phase approximation (RPA) . We focus on the electrostatic regime, where the plasmon wave number is much larger than the photon one, enabling great concentration of electromagnetic energy. I will discuss how quantum non-local effects are crucial to understand WSM FA plasmon physics. Since the FA wavefunctions are in strong spatial overlap with a bulk of gapless excitations, FA plasmons are susceptible to Landau damping even at zero temperature and deep in the long-wavelength limit. Our theory fully quantifies this intrinsic dissipation mechanism, which is dominated by processes whereby FA plasmons decay by emitting electron-hole pairs in the bulk, and puts strict theoretical bounds on the observability of certain angular portions of the highly-anisotropic FA plasmon dispersion.
Finally, if time allows, I will also discuss recent progress in understanding quantum non-local effects in graphene plasmonics . In this case, we have used a combination of graphene plasmons and engineered dielectric-metallic environments, to probe the local shape of density correlations in the graphene electron liquid. Near-field imaging experiments in the Terahertz (THz) spectral range have revealed a parameter-free match with the full theoretical quantum description of the massless Dirac electron gas, in which we have identified three types of quantum effects as keys to understanding the response of graphene to short-ranged THz electric fields. The first type is of single-particle nature and is related to shape deformations of the Fermi surface during a plasmon oscillation. The second and third types are a many-body effect controlled by the inertia and compressibility of the interacting electron liquid in graphene. Our work paves the way for accessing the full non-local conductivity tensor of electron liquids in 2D materials and surface states of WSMs and topological insulators.
 A. Woessner, M.B. Lundeberg, Y. Gao, A. Principi, P. Alonso-Gonz\’alez, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F.H.L. Koppens, Nature Mater. 14, 421 (2015).
 N. Kumada, P. Roulleau, B. Roche, M. Hashisaka, H. Hibino, I. Petković, and D.C. Glattli, Phys. Rev. Lett. 113, 266601 (2014).
 D. Jin, L. Lu, Z. Wang, C. Fang, J.D. Joannopoulos, M. Soljačić, L. Fu, and N.X. Fang, Nature Commun. 7, 13486 (2016).
 P. Hosur and X. Qi, C.R. Physique 14, 857 (2013).
 M.Z. Hasan, S.-Y. Xu, I. Belopolski, and S.-M. Huang, Annu. Rev. Condens. Matter Phys. 8, 289 (2017).
 B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. 8, 337 (2017).
 A.A. Burkov, Annu. Rev. Condens. Matter Phys. 9, 359 (2018).
 C. Shekhar, A.K. Nayak, S. Singh, N. Kumar, S.-C. Wu, Y. Zhang, A.C. Komarek, E. Kampert, Y. Skourski, J. Wosnitza, W. Schnelle, A. McCollam, U. Zeitler, J. Kubler, S.S.P. Parkin, B. Yan, and C. Felser, arXiv:1604.01641.
 G.M. Andolina, F.M.D. Pellegrino, F.H.L. Koppens, and M. Polini, arXiv:1706.06200.
 D. Pines and P. Noziéres, The Theory of Quantum Liquids (W.A. Benjamin, Inc., New York, 1966).
 M.B. Lundeberg, Y. Gao, R. Asgari, C. Tan, B. Van Duppen, M. Autore, P. Alonso-Gonzalez, A. Woessner, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, and F.H.L. Koppens, Science 357, 187 (2017).
Work supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 696656 “Graphene Core1”.