**Atomistic Topological Electrodynamics**

*Zubin Jacob, School of Electrical and Computer Engineering, **Purdue University, Indiana, U.S.A.*

*zjacob@purdue.edu**, **www.electrodynamics.org*

**Abstract.—**Over the last decade, the concept of Dirac matter has emerged to the forefront of condensed matter physics. Prominent examples include the physics of the Dirac point in Graphene, Weyl points in topological semi-metals (TaAs) and edge states in topological insulators (Bi_{2}Te_{3}). These phases of matter are a playground for studying effects related to topology and spin-1/2 Dirac Hamiltonians. We note, however, that they are defined only with respect to the properties of the electron wavefunction (eg: electronic bandstructure). This opens intriguing questions 1) Can there exist new topological phases of matter where the topology is tied to spin-1 Maxwellian electrodynamics? 2) Are there new “atomistic electrodynamic” topological invariants beyond the Chern number and Z_{2} invariant that classify topological electronic phases of matter? These questions will be the focus of the talk.

To answer these open questions, we will introduce two theoretical tools (a) Photon wavefunctions inside matter and (b) Atomistic Maxwell Hamiltonians. We develop a viscous Maxwell-Chern-Simons theory to rigorously define effective photon mass and photon spin inside matter. Exploring the geometry of Maxwellian electrodynamics in matter, we introduce a new topological classification of two-dimensional matter – the optical *N*-phases. This topological quantum number (optical N-invariant) is connected to polarization transport and captured solely by the spatiotemporal dispersion of the atomistic susceptibility tensor *χ*. We discover that *N* ≠ 0 in graphene with the underlying physical mechanism being repulsive Hall viscosity. We propose two experimental signatures to isolate the new optical N-invariant (i) On the edge, the GHz frequency electromagnetic band-gap is closed by viscous hydrodynamic edge states which are unidirectional and topologically robust. (ii) In the bulk, a spin-1 skyrmion causes magnetic field expulsion reminiscent of the Meissner effect but for electrodynamics at finite frequency and momentum. Our work indicates that graphene with repulsive Hall viscosity is the first candidate material for a topological electromagnetic phase of matter. Introduction to the formalism and experimental proposals for atomistic topological electrodynamics can be found in [1-4].

*[1] **Nat. Commun.** 12, 4729 (2021) *

*[2] Physical Review B 102 (15), 155425 (2020) *

*[3] Optical Materials Express 9 (1), 95-111 (2019) *

*[4] Physical Review A 98 (2), 023842 (2018)*

Host: Giuseppe Strangi