** Topological Pico-Electrodynamics**

One of the key achievements of the last decade has been the realization that topology may be used to categorize and find new phases of matter (Nobel Prize in Physics in 2016). The inherent spin-orbit interaction in some materials, known as topological insulators, leads to an insulating behavior for electrons in the bulk of the system, with the presence of exotic metallic surface states due to their topological order. Lossless transport channels offered by these surface states have found promising applications ranging from spintronics to quantum computation. Topological photonic features, on the other hand, have only been proposed and achieved in artificial dielectric structures. However, natural materials themselves can host electrodynamic excitations at the atomistic level. One can state that the topological phases of matter discovered so far in condensed matter systems are at the zero-frequency regime and correspond to the electrostatic observables. Unique topological electrodynamic phases due to electromagnetic excitations can be realized by entering the regime of pico-electrodynamics. In this regard, our group has made foundational discoveries in pioneering the field of topological pico-electrodynamics of matter. Our efforts bridge the field of quantum optics and condensed matter physics and spawn a new area of research in material science.

**Optical N-invariant**

Recently, we discovered that graphene with repulsive Hall viscosity is the first candidate to display a topological electromagnetic phase of matter. We have shown that the graphene monolayer supports spin-1 skyrmions in the bulk and topologically protected electromagnetic edge states at the boundary. This topological electrodynamic phase of matter is characterized by an optical N-invariant fundamentally distinct from the Chern number and Z2 invariant. The optical N-insulators form a new family of topological materials with exotic electromagnetic features such as obstructions to molecular polarizabilities [2]. We have also underlined a comprehensive classification of the topological electromagnetic phase of matter, based on the underlying crystalline symmetry.

**Non-local Topological Electrodynamic Phases of Matter**

We generalized the topological electromagnetic phases beyond the continuum approximation to the exact lattice field theory of a periodic atomic crystal [3]. To accomplish this, we put forth the concept of atomistic electrodynamic bandstructure of solids, analogous to the traditional theory of electronic band structure. For the photon propagating within a periodic atomic crystal, our theory shows that besides the Chern invariant C ∈ Z, there are also symmetry-protected topological (SPT) invariants ν ∈ ZN which are related to the cyclic point group CN of the crystal ν=CmodN. Due to the rotational symmetries of light R(2π) = +1, these SPT phases are manifestly bosonic and behave very differently from their fermionic counterparts R(2π) = −1 encountered in conventional condensed-matter systems. Remarkably, the nontrivial bosonic phases ν≠0 are determined entirely from rotational (spin-1) eigenvalues of the photon at high-symmetry points in the Brillouin zone.

**Viscous Maxwell-Chern-Simons theory for topological electromagnetic phases of matter**

Chern-Simons theories have been very successful in explaining integer and fractional quantum Hall phases of matter, topological insulators, and Weyl semimetals. We have developed a viscous Maxwell-Chern-Simons theory [3] to capture the fundamental physics of a topological electromagnetic phase of matter. We show the existence of a unique spin-1 skyrmion in the viscous Hall fluid arising from a photonic Zeeman interaction in momentum space. Our work bridges the gap between electromagnetic and condensed matter topological physics while also demonstrating the central role of photon spin-1 quantization in identifying new phases of matter.

References:

[1] T. V. Mechelen, W. Sun, and Z. Jacob, Nature Communications 12, 4729 (2021).

[2] T. Van Mechelen, S. Bharadwaj, Z. Jacob, and R.-J. Slager, Physical Review Research 4, 023011 (2022).

[3] T. V. Mechelen, W. Sun, and Z. Jacob, Physical Review B 99, 205146 (2019).

[4] T. V. Mechelen, and Z. Jacob, Optical Materials Express 9, (2019).

[5] T. V. Mechelen, and Z. Jacob, Physical Review A 98, 023842 (2018).