Line of Dirac Nodes in Hyper-Honeycomb Lattices (1408.5522v3)

Abstract: We propose a family of free fermion lattice models that have "Dirac loops", closed lines of Dirac nodes in momentum space, on which the density of states vanishes linearly with energy. Those lattices all possess the planar trigonal connectivity present in graphene, but are three dimensional. We show that their highly anisotropic and multiply-connected Fermi surface leads to quantized Hall conductivities in three dimensions for magnetic fields with toroidal geometry. In the presence of spin-orbit coupling, we show that those structures have topological surface states. We discuss the feasibility of realizing the structures as new allotropes of carbon.

Line of Dirac Nodes in Hyper-Honeycomb Lattices

The paper "Line of Dirac Nodes in Hyper-Honeycomb Lattices" presents a theoretical proposal for a family of three-dimensional structures characterized by the presence of "Dirac loops"—closed lines of Dirac nodes in momentum space where the density of states vanishes linearly with energy. These structures extend the planar trigonal connectivity found in graphene into three dimensions, offering new fodder for exploration in condensed matter physics and materials science, especially in the context of topologically non-trivial states and exotic Fermi surfaces.

The authors, Kieran Mullen, Bruno Uchoa, and Daniel T. Glatzhofer, employ a tight-binding model to elucidate the properties of these hyper-honeycomb lattices. A key finding of this paper is that these lattices naturally host Dirac loops without the need for fine-tuning parameters or the presence of spin-orbit coupling. This property significantly simplifies the realizability of such structures compared to previous theoretical models predicting Dirac loops in topological superconductors and 3D Dirac semimetals, which require finely tuned experimental conditions.

Summary and Analysis

1. Lattice Structure and Hamiltonian: The paper introduces a variety of hyper-honeycomb lattices, including harmonic honeycomb lattices denoted as $\mathcal{H}$-n, with $n$ specifying different structural arrangements. The simplest, $\mathcal{H}$-0, contains a four-atom unit cell that exemplifies the three-dimensional extension of graphene's connectivity. The authors develop a tight-binding Hamiltonian which, when expanded around the nodal line, reveals a Dirac-like form.
2. Dirac Loops and Energy Spectrum: The existence of Dirac loops is a central feature, with these loops forming lines of nodes at zero energy in the 3D Brillouin zone. The nodal line results in a torus-shaped Fermi surface with distinct anisotropic characteristics. The projected low-energy Hamiltonian describes quasiparticles with linear dispersions normal to the loop, emblematic of Dirac-like excitations.
3. Transport Properties and Quantum Hall Effect: A notable outcome of these Dirac loops is the potential for a quantized three-dimensional Hall effect under a toroidal magnetic field geometry. The Hall conductivity, in this configuration, is quantized, indicating possibilities for observing 3D quantum Hall effects at feasible magnetic field strengths. The transport calculations reveal a non-universal conductivity independent of scattering rate, driven by the complex topological nature of the nodal loops.
4. Topological Surface States: When considering spin-orbit interactions, the authors propose that these structures can host topological surface states, akin to those found in 3D topological insulators. These surface states are protected by the topological characteristics of the bulk, adding another layer of rich physics to these materials.
5. Implications for Material Realization: The paper briefly discusses potential pathways for realizing these theoretical structures as new carbon allotropes, suggesting methods such as layer-by-layer synthesis using oriented polymer chains. This synthesis could unveil new classes of carbon materials with both fundamental and technological implications, particularly if enhanced by chemical doping to increase the spin-orbit coupling effects.

Future Directions

The theoretical predictions in this work necessitate experimental verification, particularly the synthesis of hyper-honeycomb carbon allotropes and the observation of Dirac loops. Future research could focus on exploring the impact of external fields or mechanical strain on these structures, potentially uncovering novel quantum states or enhancing specific desirable properties such as quantum conductance or magnetic response.

In summary, this paper provides a comprehensive exploration of hyper-honeycomb lattices, underscoring their significance as potential hosts for novel topological phenomena in three dimensions. The interplay between structure, topology, and electronic properties presents intriguing avenues for further investigations, paving the way for advancements in fields ranging from quantum materials to nanotechnology.