Quantum Hall and Light Responses in a 2D Topological Semimetal

TL;DR

This work addresses a graphene-based 2D topological semimetal with a protected zero-energy edge mode by linking its bulk quantum Hall response to a resonant circularly polarized light response at the Dirac points. Using a four-band Haldane-like model ${\mathcal H}({\bf k})=\bm{d}_{\bf k}\cdot\bm{\sigma}\otimes \mathbb{I} + r\mathbb{I}\otimes \mathbf{s}_x$ in the nodal-ring regime, the authors compute the Hall conductivity via a Kubo/Berry-curvature framework and show it interpolates as $\sigma^{xy}=(e^2/h)(\cos\theta_c-1)$, with a quantized $\mathbb{Z}$ invariant for the lowest band and a $\mathbb{Z}_2$ marker for the partially filled bands. Circularly polarized light resolved at the Dirac points reproduces these topological markers, yielding a local Berry-curvature picture $F_{p_x p_y}=\zeta\frac{(\hbar v_F)^2}{2|{\bf d}|^2}\cos\theta_{\bf p}$ and, for $\theta_c=\frac{\pi}{2}$, a half-quantized (half-Skyrmion) response linked to the equatorial plane on the sphere. The bulk-edge correspondence then yields a topological half-metal: one spin channel supports a quantized edge conductance while the other is metallic, corresponding to a $\frac{1}{2}-\frac{1}{2}$ conductance along $z$, with experimental fingerprints in spin-polarized ARPES and edge transport and a link to bilayer half-invariant physics.

Abstract

We have recently identified a protected topological semimetal in graphene which presents a zero-energy edge mode robust to disorder and interactions. Here, we address the characteristics of this semimetal and show that the $\mathbb{Z}$ topological invariant of the Hall conductivity associated to the lowest energy band can be equivalently measured from the resonant response to circularly polarized light resolved at the Dirac points. The (non-quantized) conductivity responses of the intermediate energy bands, including the Fermi surface, also give rise to a $\mathbb{Z}_2$ invariant. We emphasize on the bulk-edge correspondence as a protected topological half metal, i.e. one spin-population polarized in the plane is in the insulating phase related to the robust edge mode while the other is in the metallic regime. The quantized transport at the edges is equivalent to a $\frac{1}{2}-\frac{1}{2}$ conductance for spin polarizations along $z$ direction. We also build a parallel between the topological Hall response and a pair of half numbers (half Skyrmions) through the light response locally resolved in momentum space and on the sphere.

Figures (2)

• Figure 1: Color Online: (Left) Two-dimensional band structure of the system for a chosen path in the Brillouin zone with the spin polarizations associated to the energy bands shown in distinct colors. (Right) Energy bands showing the structure of the edge modes on a cylinder geometry. The structure of these edge modes crossing the Fermi energy reveals the $\mathbb{Z}_2$ and $\mathbb{Z}$ invariants in Eq. (\ref{['quantumnumber']}) obtained from the analysis of the quantum Hall responses at half-filling. In momentum space, the band structure with the edge modes can be observed with spin-resolved Angle Resolved Photoemission Spectroscopy (ARPES). In real space, the system shows one robust edge mode in blue (distinct from the bulk modes in red) crossing the Fermi energy. The different responses to circularly polarized light at the $K$ Dirac point will also reveal the quantized quantum Hall conductivity of the lowest-energy band. The parameters chosen for all panels are $M = 3\sqrt{3}t/4$, $r = t/\sqrt{3}$, $t_2 = t/3$ with $a=1$.
• Figure 2: Color online: One-dimensional energy bands of the topological semimetal showing the two underlying band structures of opposite spin polarization, with the spin polarization shown in the third direction. The parameters chosen for all panels are $M = 3\sqrt{3}t/4$, $r = t/\sqrt{3}$, $t_2 = t/3$ with $a=1$.