Optical N-invariant of graphene's topological viscous Hall fluid

Abstract

Over the past three decades, graphene has become the prototypical platform for discovering topological phases of matter. Both the Chern [Formula: see text] and quantum spin Hall [Formula: see text] insulators were first predicted in graphene, which led to a veritable explosion of research in topological materials. We introduce a new topological classification of two-dimensional matter - the optical N-phases [Formula: see text]. This topological quantum number is connected to polarization transport and captured solely by the spatiotemporal dispersion of the susceptibility tensor χ. We verify N ≠ 0 in graphene with the underlying physical mechanism being repulsive Hall viscosity. An experimental probe, evanescent magneto-optic Kerr effect (e-MOKE) spectroscopy, is proposed to explore the N-invariant. We also develop topological circulators by exploiting gapless edge plasmons that are immune to back-scattering and navigate sharp defects with impunity. Our work indicates that graphene with repulsive Hall viscosity is the first candidate material for a topological electromagnetic phase of matter.

Fig. 1. Topological phases of graphene.

a The Chern phase $C\in \mathbb{Z}$ arises from complex next-nearest-neighbor (NNN) hopping and is related to charge transport. b The quantum spin Hall phase $\upsilon \in {\mathbb{Z}}_2$, also known as the 2D topological insulator, is due to spin–orbit coupling and leads to nontrivial spin transport. c The optical phase $N\in \mathbb{Z}$ we put forth in this paper is a consequence of repulsive Hall viscosity and connected to polarization transport. These three phases can be identified as the Chern insulator, quantum spin Hall insulator and viscous Hall insulator, respectively.

Fig. 2. Visualization of the 2 + 1D momentum space in the continuum limit.

a Due to regularization from viscosity, the q space is equivalent to the sphere ${\mathbb{R}}^2\simeq S^2$ as all paths at q = ∞ are compactified into a single point. Likewise, the f-sum rule ensures imaginary contours in Ω space are compactified to a circle $\mathbb{R}\simeq S^1$. The product of these two spaces S² × S¹ is a 2 + 1D manifold without boundary. b Contour in the complex frequency plane used to evaluate the optical N-invariant. Integration is performed vertically Ω ∈ (ω − i∞, ω + i∞) over all Matsubara frequencies. ω = ℜ(Ω) is the photon energy that lies within the electronic band gap 0 < ω < ∣ω_c∣. c Optical invariant N(ω) as a function of the photon energy ω. The N-invariant is calculated for damped, low-loss, and quantum Hall fluids in the nontrivial repulsive regime ω_cν_H > 0. In the dissipationless limit Γ = 0, the optical invariant is quantized to ∣N∣ = 2 within the entire band gap.

Fig. 3. Experimental e-MOKE setup of topological viscous Hall fluid.

a Overview of e-MOKE spectroscopy for direct measurement of the optical N-invariant through B-field repulsion. b Evolution of the reflected polarization ellipse ${\overrightarrow{\mathcal{E}}}_{\mathrm{r}}$ due to an incident $\widehat{s}$ polarized wave ${\overrightarrow{\mathcal{E}}}_0$ at the e-MOKE interface. The frequency of incident light is ω/2π = 286 THz. The ellipticity gradually changes for various incident angles (θ) but abruptly switches handedness due to skyrmionic vorticity. This cyclotron null coincides with an incident angle $\tan {\theta }_{\mathrm{H}}=\sqrt{{\left(2\pi n_{-}{D}_{\mathrm{H}}/\lambda \right)}^2-1}$ and only occurs in the nontrivial phase N = 2. c, d Magnitude Θ and phase Φ of the Kerr rotation plotted against the in-plane momentum q. e The optical N-invariant encodes the angular momentum texture of spin-1 Néel-type skyrmions and is identified with the skyrmion winding number Δj_z = N. This is an experimentally measurable signature of the N-invariant. In the nontrivial phase N = 2, the spin flips direction which is indicated by the arrows at the high-symmetry points q = 0 and q = ∞.

Fig. 4. Boundary layer of the topological viscous Hall fluid.

a Depiction of the additional boundary conditions (ABCs) on the current density J in magnetohydrodynamic systems. The boundary scattering velocity v_b dictates the amount of slip at the interface, with v_b = 0 and v_b = ∞ being the extreme cases of stress-free and no-slip BCs respectively. b Traditional magneto-optics does not consider nonlocal phenomenon and the associated ABCs on J. This theory cannot describe the physics in the hydrodynamic boundary layer. c Bulk and edge dispersion of the dissipationless Γ = 0 nontrivial N = 2 viscous Hall fluid. No edge states exist in the trivial regime N = 0. Magenta and black lines are the BMP and topological EMP dispersion respectively, ω_b(q_y) and ω_e(q_y). Gray denotes the continuous bulk spectrum. The simulations include the Coulomb interaction and we have imposed arbitrary boundary slip conditions. Topological EMPs exist for all values of the scattering velocity v_b. d Numerical simulation of the topological circulator on dielectric substrate. The center to edge length is d = 20r_c = 548 nm. A dipole source is placed at the boundary and oscillates periodically in the band gap at ω/2π = ∣ω_c∣/4π = 2.26 THz for a total of 1.7 ps. The color plot shows the normalized charge density fluctuation δρ/ρ₀ where red and blue indicate positive and negative values respectively. The topological edge state navigates sharp defects with zero back-scattering and is immune to boundary effects.