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We study signatures of magnetic quantum oscillations in three-dimensional nodal line semimetals at zero temperature. The extended nature of the degenerate bands can result in a Fermi surface geometry with topological genus one, as well as a Fermi surface of electron and hole pockets encapsulating the nodal line. Moreover, the underlying two-band model to describe a nodal line is not unique, in that there are two classes of Hamiltonian with distinct band topology giving rise to the same Fermi surface geometry. After identifying the extremal cyclotron orbits in various magnetic field directions, we study their concomitant Landau levels and resulting quantum oscillation signatures. By Landau-fan-diagram analyses we extract the non-trivial $pi$ Berry phase signature for extremal orbits linking the nodal line.
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We report a study of quantum oscillations (QO) in the magnetic torque of the nodal-line Dirac semimetal ZrSiS in the magnetic fields up to 35 T and the temperature range from 40 K down to 2 K, enabling high resolution mapping of the Fermi surface (FS) topology in the $k_z=pi$ (Z-R-A) plane of the first Brillouin zone (FBZ). It is found that the oscillatory part of the measured magnetic torque signal consists of low frequency (LF) contributions (frequencies up to 1000 T) and high frequency (HF) contributions (several clusters of frequencies from 7-22 kT). Increased resolution and angle-resolved measurements allow us to show that the high oscillation frequencies originate from magnetic breakdown (MB) orbits involving clusters of individual $alpha$ hole and $beta$ electron pockets from the diamond shaped FS in the Z-R-A plane. Analyzing the HF oscillations we have unequivocally shown that the QO frequency from the dog-bone shaped Fermi pocket ($beta$ pocket) amounts $beta=591(15)$ T. Our findings suggest that most of the frequencies in the LF part of QO can also be explained by MB orbits when intraband tunneling in the dog-bone shaped $beta$ electron pocket is taken into account. Our results give a new understanding of the novel properties of the FS of the nodal-line Dirac semimetal ZrSiS and sister compounds.
We review the recent, mainly theoretical, progress in the study of topological nodal line semimetals in three dimensions. In these semimetals, the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone, and any perturbation that preserves a certain symmetry group (generated by either spatial symmetries or time-reversal symmetry) cannot remove this crossing line and open a full direct gap between the two bands. The nodal line(s) is hence topologically protected by the symmetry group, and can be associated with a topological invariant. In this Review, (i) we enumerate the symmetry groups that may protect a topological nodal line; (ii) we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface, establishing a topological classification; (iii) for certain classes, we review the proposals for the realization of these semimetals in real materials and (iv) we discuss different scenarios that when the protecting symmetry is broken, how a topological nodal line semimetal becomes Weyl semimetals, Dirac semimetals and other topological phases and (v) we discuss the possible physical effects accessible to experimental probes in these materials.
Recently, it was pointed out that all chiral crystals with spin-orbit coupling (SOC) can be Kramers Weyl semimetals (KWSs) which possess Weyl points pinned at time-reversal invariant momenta. In this work, we show that all achiral non-centrosymmetric materials with SOC can be a new class of topological materials, which we term Kramers nodal line metals (KNLMs). In KNLMs, there are doubly degenerate lines, which we call Kramers nodal lines (KNLs), connecting time-reversal invariant momenta. The KNLs create two types of Fermi surfaces, namely, the spindle torus type and the octdong type. Interestingly, all the electrons on octdong Fermi surfaces are described by two-dimensional massless Dirac Hamiltonians. These materials support quantized optical conductance in thin films. We further show that KNLMs can be regarded as parent states of KWSs. Therefore, we conclude that all non-centrosymmetric metals with SOC are topological, as they can be either KWSs or KNLMs.
We report a study of quantum oscillations in the high-field magneto-resistance of the nodal-line semimetal HfSiS. In the presence of a magnetic field up to 31 T parallel to the c-axis, we observe quantum oscillations originating both from orbits of individual electron and hole pockets, and from magnetic breakdown between these pockets. In particular, we find an oscillation associated with a breakdown orbit enclosing one electron and one hole pocket in the form of a `figure of eight. This observation represents an experimental confirmation of the momentum space analog of Klein tunneling. When the c-axis and the magnetic field are misaligned with respect to one another, this oscillation rapidly decreases in intensity. Finally, we extract the cyclotron masses from the temperature dependence of the oscillations, and find that the mass of the figure of eight orbit corresponds to the sum of the individual pockets, consistent with theoretical predictions for Klein tunneling in topological semimetals.
We study the frequency-dependent conductivity of nodal line semimetals (NLSMs), focusing on the effects of carrier density and energy dispersion on the nodal line. We find that the low-frequency conductivity has a rich spectral structure which can be understood using scaling rules derived from the geometry of their Dupin cyclide Fermi surfaces. We identify different frequency regimes, find scaling rules for the optical conductivity in each, and demonstrate them with numerical calculations of the inter- and intraband contributions to the optical conductivity using a low-energy model for a generic NLSM.
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