Quantum Hall and Light Responses in a 2D Topological Semimetal

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.

💡 Research Summary

This paper presents a comprehensive theoretical analysis of a protected topological semimetal phase realizable in graphene, focusing on the intricate connections between quantum Hall conductivity, optical response, and bulk-edge correspondence.

The authors begin by introducing the model Hamiltonian, which consists of two copies of the Haldane model (described by the vector d_k · σ) for the two spin projections along the z-axis, coupled by an in-plane Zeeman term (r I ⊗ s_x). This setup breaks spin degeneracy, resulting in four energy bands labeled by their spin polarization along the x-direction: ↓x,-, ↓x,+, ↑x,-, and ↑x,+. Within a specific parameter range (3√3 t₂ - M < r < 3√3 t₂ + M), the two middle bands (↓x,+ and ↑x,-) overlap, forming a closed one-dimensional Fermi surface known as a nodal ring, thereby realizing a semimetallic state.

The core of the analysis investigates the quantum Hall response of this semimetal at half-filling (Fermi level μ=0). The study first establishes that the standard Kubo formula remains well-defined despite the band crossings, as the relevant velocity operator matrix elements between the overlapping bands vanish. The total Hall conductivity is expressed as the sum of Berry curvature integrals over occupied states. This sum decomposes into three distinct contributions: 1) the Chern number (C) of the fully occupied lowest band (↓x,-), integrated over the entire Brillouin Zone (BZ); 2) the Berry flux (Č) of the partially occupied band ↑x,-, integrated over the region outside the nodal ring; and 3) the Berry flux (Ĉ) of the partially occupied band ↓x,+, integrated over the region inside the nodal ring.

Using a mapping from the BZ to a sphere S² via the d_k vector, these integrals are computed analytically. The Chern number C is found to be quantized to the integer -1 (for 3√3 t₂ > M), serving as a robust Z topological invariant directly linked to the presence of a zero-energy edge mode observed in the band structure on a cylinder geometry. The other two contributions, Ĉ and Č, are not quantized and vary continuously with the parameter ‘r’, reflecting the metallic nature of the Fermi surface. However, the structure of these terms suggests the possibility of defining an additional Z₂ invariant related to the intermediate bands and the Fermi surface.

A key finding is the demonstration of equivalence between this Z invariant and the material’s resonant response to circularly polarized light. Specifically, the optical response resolved at the Dirac points (K and K’) provides a direct measure of the quantized Chern number of the lowest band. This proposes a non-contact method for probing topological order.

Furthermore, the paper builds a geometric analogy by showing that, when analyzed locally in momentum space and on the sphere S², the topological Hall response can be interpreted as a pair of “half numbers” or “half-Skyrmions.” This decomposition is linked to the optical response from the equatorial region of the sphere under linear polarization.

Finally, the work explores the bulk-edge correspondence. The analysis of edge states reveals that the system behaves as a “topological half-metal.” When the chemical potential is tuned, one spin population (polarized along -x) resides in an insulating phase with a quantized edge conductance, while the other spin population (polarized along +x) exhibits metallic bulk behavior. This quantized edge transport can be equivalently interpreted as a conductance of (1/2 - 1/2) for spin polarizations along the z-direction, drawing a parallel to bilayer topological systems and highlighting a novel topological metallic phase where quantized transport and metallicity coexist.

In summary, this research provides a unified framework for understanding a new class of 2D topological semimetals, connecting quantized bulk invariants, characteristic optical responses, and protected edge state transport, thereby enriching the landscape of topological quantum matter.