Landau level quantization and almost flat modes in three-dimensional semimetals with nodal ring spectra

We investigate Landau level structures of semimetals with nodal ring dispersions. When the magnetic field is applied parallel to the plane in which the ring lies, there exist almost nondispersive Landau levels at the Fermi level ($E_F=0$) as a function of the momentum along the field direction inside the ring. We show that the Landau levels at each momentum along the field direction can be described by the Hamiltonian for the graphene bilayer with fictitious interlayer couplings under a tilted magnetic field. Near the center of the ring where the in-terlayer coupling is negligible, we have Dirac Landau levels which explain the appearance of the zero modes. Although the interlayer hopping amplitudes become finite at higher momenta, the splitting of zero modes is exponentially small and they remain almost flat due to the finite artificial in-plane component of the magnetic field. The emergence of the density of states peak at the Fermi level would be a hallmark of the ring dispersion.

Figure 1

(a) Low energy band structures on the $k_b=-\pi$ plane where the nodal ring (black ellipse) resides. (b) The nodal chain appears when $2t_2=t_4$. Two coupled doubly degenerate Dirac cones reside at the points where the nodal lines intersect the plane for given $q_c$. In (c) and (d), we compare energy spectra around the $U$ point calculated from the full tight binding Hamiltonian (solid) and the continuum model ${\mathcal{H}}^U$ (dashed).

Figure 2

(a), (b) Landau levels of the NRS as functions of $q_c$ for different TB parameters when the magnetic field ($B=1$ T) is along the $c$ direction. Blue solid lines are numerical results while red dashed curves are analytic ones obtained from the fictitious graphene bilayer model near $q_c=0$. (c) Central Landau levels ($N=1,2$) near $q_c=r_c$ ($*$) which are fitted by a formula $\varepsilon \sim \pm {(1-a{q}_c^2)}^{\alpha }{q}_c^{\beta }exp\left(b{q}_c^2\right)$ for various magnetic fields (red dashed lines). $e_0=1\text{eV}$. (d) $b$ in the exponent of the fitting function and its fitting by $b_0{l}_{\mathrm{B}}^2$. (e), (f) Landau levels as functions of the Landau level index and magnetic field for some momenta.