Featuring exotic quantum transport and surface states, topological semimetals can be classified into nodal-point, nodal-line, and nodal-surface semimetals according to the degeneracy and dimensionality of their nodes. However, a topological semimetal that possesses both nodal points and nodal lines is rarely reported. Here, we propose a scheme to construct this type of topological semimetal, which simultaneously exhibits hinge Fermi arcs and drumhead surface states. Then, by applying periodic driving on the system, we find a hybrid-order topological semimetal with nodal points and rich nodal-line structures and its conversion into a first-order topological semimetal, which are absent in a static system. Our results enrich the family of topological semimetals, and establish a foundation for further exploration of their potential applications.
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.
The high-magnetic-field thermoelectric effect in nodal-line semimetals with straight nodal lines was investigated. Three-dimensional (3D) Dirac/Weyl semimetals exhibit constant thermoelectric Hall conductivity at the high-magnetic-field quantum limit, resulting in a boundless linear increase of Seebeck coefficient. This is known as the quantized thermoelectric Hall effect (QTHE), and is expected to lead to high-performance thermoelectricity at low temperatures. Here, in addition to Dirac/Weyl semimetals, we demonstrated that 3D semimetals with straight nodal lines can also exhibit the QTHE und…
In topological semimetals, the valence band and conduction band meet at zero-dimensional nodal points or one-dimensional nodal rings, which are protected by band topology and symmetries. In this Rapid Communication, we introduce "nodal-link semimetals", which host linked nodal rings in the Brillouin zone. We put forward a general recipe based on the Hopf map for constructing models of nodal-link semimetal. The consequences of nodal ring linking in the Landau levels and Floquet properties are investigated.
We study the electrical conductivity in a nodal-line semimetal with charged impurities. The screening of the Coulomb potential in this system is qualitatively different from what is found in conventional metals or semiconductors, with the screened potential $ϕ$ decaying as $ϕ\propto 1/r^2$ over a wide interval of distances $r$. This unusual screening gives rise to a rich variety of conduction regimes as a function of temperature, doping level and impurity concentration. In particular, nodal-line semimetals exhibit a diverging mobility $\propto 1/|μ|$ in the limit of vanishing chemical potentia…
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…
Thermoelectric materials intrigue much interest due to their wide range of application such as power generators and refrigerators. The efficiency of thermoelectric materials is quantified by the figure of merit, and a figure greater than unity is desired. To achieve this, a large Seebeck coefficient and low phonon thermal conductivity are required. We show that this can be achieved with a thin film of topological nodal line semimetals. We also discusses the correlation effect and spin current induced by a temperature gradient. The obtained results provide insight for the improvement of thermoe…
Topological nodal-line semimetals are predicted to exhibit unique drumhead-like surface states (DSS). Yet, a direct detection of such states remains a challenge. Here, we propose spin-resolved transport in a junction between a normal metal and a spin-orbit coupled nodal-line semimetal as the mechanism for their detection. Specifically, we find that in such a device, the DSS induce resonant spin-flipped reflection. This effect can be probed by both vertical spin transport and lateral charge transport between anti-parallel magnetic terminals. In the tunneling limit of the junction, both spin and…
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…
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 …
Topological nodal-line semimetals are characterized by one-dimensional lines of band crossing in the Brillouin zone. In contrast to nodal points, nodal lines can be in topologically nontrivial configurations. In this paper, we study the simplest topologically nontrivial forms of nodal line, namely, a single nodal line taking the shape of a knot in the Brillouin zone. We introduce a generic construction for various "nodal-knot semimetals", which yields the simplest trefoil nodal knot and other more complicated nodal knots in the Brillouin zone. The knotted-unknotted transitions by nodal-line re…
Weyl semimetals and nodal line semimetals are characterized by linear band touching at zero-dimensional points and one-dimensional lines, respectively. We predict that a circularly polarized light drives nodal line semimetals into Weyl semimetals. The Floquet Weyl points thus obtained are tunable by the incident light, which enables investigations of them in a highly controllable manner. The transition from nodal line semimetals to Weyl semimetals is accompanied by the emergence of a large and tunable anomalous Hall conductivity. Our predictions are experimentally testable by transport measure…
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…
Topological nodal line semimetals are characterized by the crossing of the conduction and valence bands along one or more closed loops in the Brillouin zone. Usually, these loops are either isolated or touch each other at some highly symmetric points. Here, we introduce a new kind of nodal line semimetal, that contains a pair of linked nodal loops. A concrete two-band model was constructed, which supports a pair of nodal lines with a double-helix structure, which can be further twisted into a Hopf link because of the periodicity of the Brillouin zone. The nodal lines are stabilized by the comb…
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…
The effect of short-range disorder in nodal line semimetals is studied by numerically exact means. For arbitrary small disorder, a novel semimetallic phase is unveiled for which the momentum-space amplitude of the ground-state wave function is concentrated around the nodal line and follows a multifractal distribution. At a critical disorder strength, a semimetal to compressible metal transition occurs, coinciding with a multi- to single-fractality transition. The universality class of this critical point is characterized by the correlation length and dynamical exponents. At considerably higher…
The self-consistent Born approximation quantitatively fails to capture disorder effects in semimetals. We present an alternative, simple-to-use non-perturbative approach to calculate the disorder induced self-energy. It requires a sufficient broadening of the quasiparticle pole and the solution of a differential equation on the imaginary frequency axis. We demonstrate the performance of our method for various paradigmatic semimetal Hamiltonians and compare our results to exact numerical reference data. For intermediate and strong disorder, our approach yields quantitatively correct momentum re…
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…
Weyl points with monopole charge $\pm 1$ have been extensively studied, however, real materials of multi-Weyl points, whose monopole charges are higher than $1$, have yet to be found. In this Rapid Communication, we show that nodal-line semimetals with nontrivial line connectivity provide natural platforms for realizing Floquet multi-Weyl points. In particular, we show that driving crossing nodal lines by circularly polarized light generates double-Weyl points. Furthermore, we show that monopole combination and annihilation can be observed in crossing-nodal-line semimetals and nodal-chain semi…
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…
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…
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.
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…
We propose a general framework for constructing a large set of nodal-point semimetals by tuning the number of linearly ($d_L$) and (at most) quadratically ($d_Q$) dispersing directions. By virtue of such a unifying scheme, we identify a new perturbative route to access various strongly interacting non-Dirac semimetals with $d_Q>0$. As a demonstrative example, we relate a two dimensional anisotropic semimetal with $d_L=d_Q=1$, describing the topological transition between a Dirac semimetal and a normal insulator, and its three dimensional counterparts with $d_L=1$, $d_Q=2$. We address the quant…
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…
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…
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.
Coexistence of topological elements in a topological metal/semimetal (TM) has gradually attracted attentions. However, the non-topological factors always mess up the Fermi surface and cover interesting topological properties. Here, we find that Ba3Si4 is a "clean" TM in which coexists nodal-chain network, intersecting nodal rings (INRs) and triple points, in the absence of spin-orbit coupling (SOC). Moreover, the nodal rings in the topological phase exhibit diverse types: from type-I, type-II to type-III rings according to band dispersions. All the topological elements are generated by crossin…
Based on first-principles calculation and effective model analysis, we propose that the cubic antiperovskite material Cu$_3$PdN can host a three-dimensional (3D) topological nodal line semimetal state when spin-orbit coupling (SOC) is ignored, which is protected by coexistence of time-reversal and inversion symmetry. There are three nodal line circles in total due to the cubic symmetry. "Drumhead"-like surface flat bands are also derived. When SOC is included, each nodal line evolves into a pair of stable 3D Dirac points as protected by C$_4$ crystal symmetry. This is remarkably distinguished …
We introduce a new type of topological magnon matter: the magnonic pendant to electronic nodal-line semimetals. Magnon spectra of anisotropic pyrochlore ferromagnets feature twofold degeneracies of magnon bands along a closed loop in reciprocal space. These magnon nodal lines are topologically protected by the coexistence of inversion and time-reversal symmetry; they require the absence of spin-orbit interaction (no Dzyaloshinskii-Moriya interaction). We calculate the topological invariants of the nodal lines and show that details of the associated magnon drumhead surface states depend strongl…
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…
We study what kind of topological crystalline insulator phase emerges from nodal line semimetal phases in the face-centered cubic system which is not indicated by topological indices when spin-orbit coupling (SOC) is introduced. We construct an effective model which hosts two different nodal lines phases, and calculated mirror Chern numbers in it by introducing SOC. As a result, we find that the two nodal line phases with different nodal line configurations are mapped to different topological crystalline insulator phases. This result shows that turning off SOC and checking the nodal line confi…
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…
