We investigate the electron spectra, Fermi surfaces and their characteristics near crossing points of two band-contact lines in nodal-line semimetals. In particular, the extremal cross-sectional areas, and the appropriate cyclotron masses are calculated. We also find the phase of the quantum oscillations associated with the electron orbits near the crossing point. The analysis of all these quantities is carried out both without and with consideration of the spin-orbit interaction. To illustrate the obtained results, we apply them to ZrSiS in which the crossing of the nodal lines occurs.
Unveiling new topological phases of matter is one of the current objectives in condensed matter physics. Recent experimental discoveries of Dirac and Weyl semimetals prompt to search for other exotic phases of matter. Here we present a systematic angle-resolved photoemission spectroscopy (ARPES) study of ZrSiS, a prime topological nodal semimetal candidate. Our wider Brillouin zone (BZ) mapping shows multiple Fermi surface pockets such as the diamond-shaped Fermi surface, ellipsoidal-shaped Fermi surface, and a small electron pocket encircling at the zone center (G) point, the M point and the …
ZrSiS is one of the strong candidates for realistic nodal-line semimetal. We theoretically investigate the dynamical conductivity in ZrSiS by using a multi-orbital theoretical model based on the first-principles band calculation. We find that the dynamical conductivity in the clean limit is actually not frequency independent unlike the ideal Dirac model, while nearly flat dependence is achieved by introducing the energy broadening possibly induced by the disorder. The results can be applied to other compounds with the similar crystal structure, such as ZrSiSe, ZrSiTe, and HfSiS.
The discovery of topological semimetal phase in three-dimensional (3D) systems is a new breakthrough in topological material research. Dirac nodal-line semimetal is one of the three topological semimetal phases discovered so far; it is characterized by linear band crossing along a line/loop, contrasted with the linear band crossing at discrete momentum points in 3D Dirac and Weyl semimetals. The study of nodal-line semimetal is still at initial stage; only three material systems have been verified to host nodal line fermions until now, including PbTaSe2, PtSn 4and ZrSiS. In this letter, we rep…
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 $k_z=π$ (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 …
The ultrafast optical response of two nodal-line semimetals, ZrSiS and ZrSiSe, was studied in the near-infrared using transient reflectivity. The two materials exhibit similar responses, characterized by two features, well resolved in time and energy. The first transient feature decays after a few hundred femtoseconds, while the second lasts for nanoseconds. Using Drude-Lorentz fits of the materials' equilibrium reflectance, we show that the fast response is well-represented by a decrease of the Drude plasma frequency, and the second feature by an increase of the Drude scattering rate. This di…
Despite being the most studied nodal line semimetal, a clear understanding of the transient state relaxation dynamics and the underlying mechanism in ZrSiS is lacking. Using time and angle resolved photoemission spectroscopy, we study the ultrafast relaxation dynamics in ZrSiS and reveal a unique relaxation in the bulk nodal-line state which is well captured by a simple model based on optical and acoustic phonon cooling. We find linear decay processes for both optical and acoustic phonon relaxations with acoustic cooling suppressed at high temperatures. Our results reveal different decay mecha…
The topological properties of fermions arise from their low-energy Dirac-like band dispersion and associated chiralities. Initially confined to points, extensions of the Dirac dispersion to lines and even loops have now been uncovered and semimetals hosting such features have been identified. However, experimental evidence for the enhanced correlation effects predicted to occur in these topological semimetals has been lacking. Here, we report a quantum oscillation study of the nodal loop semimetal ZrSiS in high magnetic fields that reveals significant enhancement in the effective mass of the q…
In this work, we present a model of the surface states of nonsymmorphic semimetals. These are derived from surface mass terms that lift the high degeneracy imposed in the band structure by the nonsymmorphic bulk symmetries. Reflecting the reduced symmetry at the surface, the bulk bands are strongly modified. This leads to the creation of two-dimensional floating bands, which are distinct from Shockley states, quantum well states or topologically protected surface states. We focus on the layered semimetal ZrSiS to clarify the origin of its surface states. We demonstrate an excellent agreement b…
ZrSiS has recently gained attention due to its unusual electronic properties: nearly perfect electron-hole compensation, large, anisotropic magneto-resistance, multiple Dirac nodes near the Fermi level, and an extremely large range of linear dispersion of up to 2 eV. We have carried out a series of high pressure electrical resistivity measurements on single crystals of ZrSiS. Shubnikov-de Haas measurements show two distinct oscillation frequencies. For the smaller orbit, we observe a change in the phase of 0.5, which occurs between 0.16 - 0.5 GPa. This change in phase is accompanied by an abru…
Unambiguous and complete determination of the Fermi surface is a primary step in understanding the electronic properties of topical metals and semi-metals, but only in a relatively few cases has this goal been realized. In this work, we present a systematic high-field quantum oscillation study up to 35 T on ZrSiS, a textbook example of a nodal-line semimetal with only linearly dispersive bands crossing the Fermi energy. The topology of the Fermi surface is determined with unprecedented precision and all pockets are identified by comparing the measured angle dependence of the quantum oscillatio…
We theoretically investigate the phase of the de Haas - van Alphen oscillations in topological line-node semimetals. In these semimetals the chemical potential of charge carriers can essentially depend on the magnetic field, and this dependence changes the phase of the oscillations as compared to the phase in a three-dimensional metal with a band-contact line. Our results elucidate recent experimental data on the Berry phase for certain electron orbits in ZrSiS, ZrSiTe, and ZrSiSe.
We present a far-infrared magneto-optical study of the gapped nodal-line semimetal ZrSiS in magnetic fields $B$ up to 7 T. The observed field-dependent features, which represent intra- (cyclotron resonance) and interband transitions, develop as $\sqrt{B}$ in increasing field and can be consistently explained within a simple 2D Dirac band model with a gap of 26 meV and an averaged Fermi velocity of $3\times10^{5}$ m/s. This indicates a rather narrow distribution of these parameters along the in-plane portions of the nodal line in the Brillouin zone. A field-induced feature with an energy positi…
We study a general phase transition between spinless topological nodal-line semimetal and Weyl semimetal phases. We classify topological nodal lines into two types based on their positions and shapes, and their phase transitions depends on their types. We show that the topological nodal-line semimetal becomes the Weyl semimetal by breaking time-reversal symmetry when the nodal lines enclose time-reversal invariant momenta (type-A nodal lines). We also discuss an effect of crystallographic symmetries determining the band structure of the topological nodal-line semimetals. Thanks to protection b…
We theoretically study the electric polarization in magnetic topological nodal semimetal thin films. In magnetically doped topological insulators, topological nodal semimetal phases emerge once the exchange coupling overcomes the band gap. Changing the magnetization direction, nodal structure is modulated and the system becomes topological nodal point or line semimetals. We find that nodal line semimetals are characterized by non-linear electric polarization, which is not observed in nodal point semimetals. The non-linear response originates from the existence of the surface states. Screening …
Hosting unique drumhead surface states enclosed by nodal lines, topological nodal-line semimetals exhibit novel transport phenomena. Thus, the exploration of topological semimetals with different nodal-line structures has attracted much attention. In this paper, we first find a second-order nodal line semimetal (SONLS), which has coexisting hinge Fermi arcs and drumhead surface states, in a $\mathcal{PT}$-symmetry broken system. Then, without changing the intrinsic parameters, we artificially create exotic hybrid-order nodal-line semimetals hosted by different quasienergy gaps and rich nodal-l…
The discovery of a topological nodal-line (TNL) semimetal phase in ZrSiS has invigorated the study of other members of this family. Here, we present a comparative electronic structure study of ZrSiX (where X = S, Se, Te) using angle-resolved photoemission spectroscopy (ARPES) and first-principles calculations. Our ARPES studies show that the overall electronic structure of ZrSiX materials comprises of the diamond-shaped Fermi pocket, the nearly elliptical-shaped Fermi pocket, and a small electron pocket encircling the zone center ($Γ$) point, the M point, and the X point of the Brillouin zone,…
We study a topological phase diagram of a ferromagnetic topological nodal semimetal. We consider a lattice model for three-dimensional topological insulators with ferromagnetic ordering. The exchange coupling between the magnetization and the electron spin leads to the nodal band structure. The topology of the nodal band structure depends on the direction of the magnetization and both the Weyl points and the nodal line emerge. We find that the nodal line structure is stable under an easy-plane magnetization in appropriate model parameters. In this case, the nodal line phase emerges as a phase …
The ZrSiS-type materials have attracted intensive attention due to the existence of various topological fermions. The magnetic version of the ZrSiS-type materials, LnSbTe (Ln = lanthanides), is an ideal candidate to explore novel exotic states due to the interaction between magnetism and topology. In this work, we report the experimental study on structural, magnetic, thermodynamic, and electronic properties for SmSbxTe2-x with various Sb content. The revealed evolutions of these properties with tuning the compositions would provide useful insights for the fundamental topological physics and t…
A nodal-line semimetal is a topological gapless phase containing one-dimensional degeneracies called nodal lines. The nodal lines are deformed by a continuous change of the system such as pressure and they can even change their topology, but it is not systematically understood what kind of changes of topology of nodal lines are possible. In this paper, we classify the events of topology change of nodal lines by the Morse theory and reveal that only three types of topology changes of nodal lines, i.e., creation, reconnection, and annihilation, are possible in the spinless nodal-line semimetal p…
We theoretically show that IV-VI semiconducting compounds with low-temperature rhombohedral crystal structure represent a new potential platform for topological semimetals. By means of minimal $\mathbf{k}\cdot\mathbf{p}$ models we find that the two-step structural symmetry reduction of the high-temperature rocksalt crystal structure, comprising a rhombohedral distortion along the [111] direction followed by a relative shift of the cation and anion sublattices, gives rise to topologically protected Weyl semimetal and nodal line semimetal phases. We derive general expressions for the nodal featu…
Topological semimetals is a new class of condensed matter systems with nontrivial electronic structure topology. Their unusual observable properties may often be understood in terms of quantum anomalies. In particular, Weyl and Dirac semimetals, which have point band touching nodes, are characterized by the chiral anomaly, which leads to the Fermi arc surface states, anomalous Hall effect, negative longitudinal magnetoresistance and planar Hall effect. In this paper we explore analogous phenomena in nodal line semimetals. We demonstrate that such semimetals realize a three dimensional analog o…
Nodal-line semimetals with magnetic orders have been theoretically predicted and experimentally observed in only few compounds. We theoretically explore the electronic structure in bulk and boundary of such a magnetic nodal-line state by introducing magnetism in topological Dirac semimetal (TDSM). TDSMs, such as $\mathrm{Cd_3 As_2}$ and $\mathrm{Na_3 Bi}$, are characterized by a pair of spin-degenerate Dirac points protected by rotational symmetries of crystals. By introducing local magnetic moments coupled to the electron spins in the lattice model of TDSM, we show that the TDSM can turn into…
We theoretically study three-dimensional topological semimetals (TSMs) with nodal lines protected by crystalline symmetries. Compared with TSMs with point nodes, e.g., Weyl semimetals and Dirac semimetals, where the conduction and the valence bands touch at discrete points, in these new TSMs the two bands cross at closed lines in the Brillouin zone. We propose two new classes of symmetry protected nodal lines in the absence and in the presence of spin-orbital coupling (SOC), respectively. In the former, we discuss nodal lines that are protected by the combination of inversion symmetry and time…
In this work, we investigate collective modes in a nodal line semimetal with two nodal-lines that have opposite spin polarization in the presence of spin population imbalance. We find the components of polarization operator taking into account the electron-electron exchange interaction and obtain the dispersion relations of collective modes for the bulk and surface states. There exist four modes in the bulk, among which one is gapless and other three are gapped. The gapless surface mode is sensitive to the boundary conditions.
We study the transport properties in nodal line semimetals with short-ranged impurity potentials at zero temperature. By computing the Drude conductivity and the corrections from the interference of particle and hole trajectories, we find that the electrons are localized in directions both parallel and perpendicular to the plane of nodal ring. We further calculate the conductivity in a weak magnetic field, and find that the perpendicular magnetic field totally destroys the weak localization gives a positive quantum interference correction, which is similar to the result in Weyl semimetals. But…
Honeycomb structure has a natural extension to the three dimensions. Simple examples are hyperhoneycomb and stripy-honeycomb lattices, which are realized in $β$-Li$_{2}$IrO$_{3}$ and $γ$-Li$_{2}$IrO$_{3}$, respectively. We propose a wide class of three-dimensional (3D) honeycomb lattices which are loop-nodal semimetals. Their edge states have intriguing properties similar to the two-dimensional honeycomb lattice in spite of dimensional difference. Partial flat bands emerge at the zigzag or beard edge of the 3D honeycomb lattice, whose boundary is given by the Fermi loop in the bulk spectrum. A…
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 invari…
We study an atomic chain in the presence of modulated charge potential and modulated Rashba spin-orbit coupling (RSOC) of equal period. We show that for commensurate periodicities $λ=4 n$ with integer $n$, the three-dimensional synthetic space obtained by sliding the two phases of the charge potential and RSOC features a topological nodal line semimetal protected by an antiunitary particle-hole symmetry. The location and shape of the nodal lines strongly depend on the relative amplitude between the charge potential and RSOC.
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 gener…
Creating and manipulating topological states is a key goal of condensed matter physics. Periodic driving offers a powerful method to manipulate electronic states, and even to create topological states in solids. Here, we investigate the tunable Floquet states in a periodically driven higher-order nodal line semimetal with both spatial inversion and time-reversal symmetries. We found that the Floquet Weyl semimetal states, which support both one-dimensional hinge Fermi arc and two-dimensional surface Fermi arc states, can be induced in the higher-order nodal-line semimetal by shining circularly…
Spatial modulations, such as multilayers, to realize topological materials have recently been studied in theoretical and experimental works. In this paper, we investigate properties of the multilayers of the nodal-line semimetal (NLS) and the normal insulator. We consider two types of multilayers, with the stacking direction being perpendicular or parallel to the plane where the nodal line lies. In particular, we show that when the stacking direction is parallel to the plane, the nodal lines remain but they change their shapes because of the folding of the Brillouin zone. We also study the mul…
We study the electronic band structure of three-dimensional ABC-stacked (rhombohedral) graphdiyne, which is a new planar carbon allotrope recently fabricated. Using the first-principles calculation, we show that the system is a nodal-line semimetal, in which the conduction band and valence band cross at a closed ring in the momentum space. We derive the minimum tight-binding model and the low-energy effective Hamiltonian in a $4\times 4$ matrix form. The nodal line is protected by a non-trivial winding number, and it ensures the existence of the topological surface state in a finite-thickness …
We theoretically investigate the barrier tunneling in the three-dimensional model of the hyperhoneycomb lattice, which is a nodal-line semimetal with a Dirac loop at zero energy. In the presence of a rectangular potential, the scattering amplitudes for different injecting states around the nodal loop are calculated, by using analytical treatments of the effective model, as well as numerical simulations of the tight binding model. In the low energy regime, states with remarkable transmissions are only concentrated in a small range around the loop plane. When the momentum of the injecting electr…
The existence and topological classification of lower-dimensional Fermi surfaces is often tied to the crystal symmetries of the underlying lattice systems. Artificially engineered lattices, such as heterostructures and other superlattices, provide promising avenues to realize desired crystal symmetries that protect lower-dimensional Fermi surface, such as nodal lines. In this work, we investigate a Weyl semimetal subjected to spatially periodic onsite potential, giving rise to several phases, including a nodal-line semimetal phase. In contrast to proposals that purely focus on lattice symmetri…
We investigate the low-temperature thermoelectric properties of three-dimensional nodal-line semimetals within the semiclassical Boltzmann formalism. Considering short-range interaction between electrons and scattering agents, we calculate the anisotropic relaxation times and then obtain the charge conductivity and thermopower along the radial and the axial directions with respect to the nodal-line plane. Increasing the carrier concentration, a topological transition in the shape of the Fermi surface from a torus into an ellipsoid signal as a sharp change in the thermopower. An adequate treatm…
A three-dimensional (3D) nodal-loop semimetal phase is exploited to engineer a number of intriguing phases featuring different peculiar topological surface states. In particular, by introducing various two-dimensional gap terms to a 3D tight-binding model of a nodal-loop semimetal, we obtain a rich variety of topological phases of great interest to ongoing theoretical and experimental studies, including chiral insulator, degenerate-surface-loop insulator, second-order topological insulator, as well as Weyl semimetal with tunable Fermi arc profiles. The unique concept underlying our approach is…
Close to the Fermi energy, nodal loop semimetals have a torus-shaped, strongly anisotropic Fermi surface which affects their transport properties. Here we investigate the non-equilibrium dynamics of nodal loop semimetals by going beyond linear response and determine the time evolution of the current after switching on a homogeneous electric field. The current grows monotonically with time for electric fields perpendicular to the nodal loop plane however it exhibits non-monotonical behavior for field orientations aligned within the plane. After an initial non-universal growth $\sim Et$, the cur…
We identify a topological Z index for three dimensional chiral insulators with P*T symmetry where two Hamiltonian terms define a nodal loop. Such systems may belong in the AIII or DIII symmetry class. The Z invariant is a winding number assigned to the nodal loop and has a correspondence to the geometric relation between the nodal loop and the zeroes of the gap terms. Dirac cone edge states under open boundary conditions are in correspondence with the winding numbers assigned to the nodal loops. We verify our method with the low-energy effective Hamiltonian of a three-dimensional material of t…
Starting from a minimal model for a 2D nodal loop semimetal, we study the effect of chiral mass gap terms. The resulting Dirac loop anomalous Hall insulator's Chern number is the phase winding number of the mass gap terms on the loop. We provide simple lattice models, analyze the topological phases and generalize a previous index characterizing topological transitions. The responses of the Dirac loop anomalous Hall and quantum spin Hall insulators to a magnetic field's vector potential are also studied both in weak and strong field regimes, as well as the edge states in a ribbon geometry.
We study the magnetic susceptibility in various three-dimensional gapless systems, including Dirac and Weyl semimetals and a line-node semimetal. The susceptibility is decomposed into the orbital term, the spin term and also the spin-orbit cross term which is caused by the spin-orbit interaction. We show that the orbital susceptibility logarithmically diverges at the band touching energy in the point-node case, while it exhibits a stronger delta-function singularity in the line node case. The spin-orbit cross term is shown to be paramagnetic in the electron side while diamagnetic in the hole s…
We introduce a two-band model of three-dimensional nodal line semimetals, the Fermi surface of which at half-filling may form various one-dimensional configurations of different topology. We study the symmetries and "drumhead" surface states of the model, and find that the transitions between different configurations, namely, the Lifshitz transitions, can be identified solely by the number of gap-closing points on some high-symmetry planes in the Brillouin zone. A global phase diagram of this model is also obtained accordingly. We then investigate the effect of some extra terms analogous to a …
Among different topological and related phases of condensed matter, nodal semimetals occupy a special place - the electronic band topology in these materials is related to three-dimensional bulk, rather than to surface, states. A great variety of different realizations of electronic band crossings (the nodes) leads to a plethora of different electronic properties, ranging from the chiral anomaly to solid-state realizations of a black-hole horizon. The different nodal phases have similar low-energy band structure and quasiparticle dynamics, which both can be accessed experimentally by a number …
Band-crossings occurring on a mirror plane are compelled to form a nodal loop in the momentum space without spin-orbit coupling (SOC). In the presence of other equivalent mirror planes, multiple such nodal loops can combine to form interesting nodal-link structures. Here, based on first-principles calculations and an effective $\mathbf{k.p}$ model analysis, we show that CaAuAs hosts a unique starfruit-like nodal-link structure in the bulk electronic dispersion in the absence of SOC. This nodal-link is comprised of three nodal loops, which cross each other at the time-reversal-invariant momentu…
Nodal-line semimetals are topological semimetals in which band touchings form nodal lines or rings. Around a loop that encloses a nodal line, an electron can accumulate a nontrivial $π$ Berry phase, so the phase shift in the Shubnikov-de Haas (SdH) oscillation may give a transport signature for the nodal-line semimetals. However, different experiments have reported contradictory phase shifts, in particular, in the WHM nodal-line semimetals (W=Zr/Hf, H=Si/Ge, M=S/Se/Te). For a generic model of nodal-line semimetals, we present a systematic calculation for the SdH oscillation of resistivity unde…
Recently, the nodal line semimetals have attracted considerable interests in condensed matter physics. We show that their distinct band structure can be detected by measuring the collective modes. In particular, we find that the dependence of the plasmon frequency $ω_p$ on the electron density $n$ follows a $ω_p \sim n^{1/4}$ law in the long wavelength limit. Our results will be useful in the ongoing search for new candidates of nodal line semimetals.
Nodal-line semimetals, one of the topological semimetals, have degeneracy along nodal lines where the band gap is closed. In many cases, the nodal lines appear accidentally, and in such cases it is impossible to determine whether the nodal lines appear or not, only from the crystal symmetry and the electron filling. In this paper, for spinless systems, we show that in specific space groups at $4N+2$ fillings ($8N+4$ fillings including the spin degree of freedom), presence of the nodal lines is required regardless of the details of the systems. Here, the spinless systems refer to crystals where…
We investigate the optical activity of tilted nodal loop semimetals. We calculate the full conductivity matrix for a band structure containing a nodal loop with possible tilt in the $x-y$ plane, which allows us to study the Kerr rotation and ellipticity both for a thin film and a bulk material. We find signatures in the Kerr signal that give direct information about the tilt velocity and direction, the radius of the nodal loop and the internal chemical potential of the system. These findings should serve as guide to understanding optical measurements of nodal loop semimetals and as an addition…
