Line of Dirac Nodes in Hyperhoneycomb Lattices

Abstract

We propose a family of structures 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.Introduction.− In honeycomb lattices, the existence of the Dirac point results from the planar trigonal connectivity of the sites and its sub-lattice symmetry [1]. Less well known are "Dirac loops", three dimensional (3D) closed lines of Dirac nodes in momentum space, on which the energy vanishes linearly with the perpendicular components of momentum [2]. To date there are no experimental observations of Dirac loops, and they were predicted to exist only in topological superconductors [3] and 3D Dirac semimetals [4] in which the parameters such as interactions and magnetic field are finely tuned [2].Theoretically, graphene is not the only possible lattice realization with planar trigonally connected atoms [5]. It is therefore natural to ask if there are variations on the honeycomb geometry that might produce exotic Fermi surfaces with Dirac-like excitations and topologically non-trivial states. In this Letter, we propose a family of trigonally connected 3D lattices that admit simple tight-binding Hamiltonians having Dirac loops, without requiring any tuning or spin-orbit coupling. Some of these structures lie in the family of harmonic honeycomb lattices, which have been studied in the context of the Kitaev model [7][8][9][10][11], and experimentally realized in honeycomb iridates [12]. The simplest example is the hyper-honeycomb lattice, shown in Fig. 1a.We derive the low energy Hamiltonian of this family of systems, and analyze the quantization of the conductivity and possible surface states. Even though these systems are 3D semimetals, their Fermi surface is multiply connected, with the shape of a torus, and highly anisotropic. When a magnetic field with toroidal geometry is applied, we find that the Hall conductivity is quantized in 3D at sufficiently large field. Additional spin-orbit coupling effects can create topologically protected surface states in these crystals. We claim that in the presence of spin-orbit coupling, these structures conceptually correspond to a new family of strong 3D topological insulators [13,14]. We finally discuss the experimental feasibility of realizing those structures as new allotropic forms of carbon.Tight-binding lattice.− Our discussion starts with the simplest structure, the hyper-honeycomb lattice (see Fig.

Topological nodal line semimetals

“…In Ref. [49], the Landau levels are calculated in a lattice model describing a 3D honeycomb lattice that has nodal rings in the BZ. A key distinction between this work and Ref.…”

Topological semimetals with helicoid surface states

“…(20) shows that at θ = 0, π (where k b = π), the two degenerate states have the same eigenvalue of R 2 . Therefore the following expression δ 0 = n=1,...,Nocc/2 e ik0/2 γ 2n (0) e −ik0/2 γ 2n (π) .…”

“…The nontrivial topology gives rise to anomalous bulk properties of topological semimetals such as the chiral anomaly [3][4][5]. Several classes of topological semimetals have been theoretically proposed so far, including Weyl, [2,[6][7][8][9][10][11][12][13], Dirac [14][15][16][17] and nodal line semimetals [7,[17][18][19][20][21][22][23][24][25][26][27][28][29][30], some among which have been experimentally observed [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47].Surface states of topological semimetals have attracted much attention. On the surface of a Weyl semimetal, the Fermi surface consists of open ar...…”

CaTe: a new topological node-line and Dirac semimetal

“…[39][40][41][42] Unlike DSM and WSM whose band crossing points distribute at separate k points in the BZ, for a NLS the crossing points around the Fermi level form a closed loop. Several compounds have been proposed as NLSs, including Mackay-Terrones crystals, 25 Bernal graphite, 43 hyperhoneycomb lattices, 44 and antiperovskite Cu 3 PdN 26,27 and Cu 3 NZn. 27 In the absence of spin-orbit coupling (SOC), time reversal symmetry together with inversion or mirror symmetry will guarantee the occurrence of node line in 3D BZ for systems with band inversion.…”