Nodal Loop Research Articles

Designing xenes with two-dimensional triangular lattice

Xenes, graphene-like two-dimensional (2D) monoelemental crystals with a honeycomb symmetry, have been the focus of numerous experimental and theoretical studies. In comparison, single-element 2D materials with a triangular lattice symmetry have not received due attention. Here, taking Pb as an example, we investigate the triangular-lattice monolayer made of group-IV atoms employing first-principles density functional theory calculations. The flat Pb monolayer supports a mirror-symmetry-protected spinless nodal line in the absence spin-orbit coupling (SOC). The introduction of an out-of-plane buckling creates a glide mirror, protecting an anisotropic Dirac nodal loop. Both flat and buckled Pb monolayers become topologically trivial after including SOC. A large buckling will make the Pb sheet a 2D semiconductor with symmetry-protected Dirac points below the Fermi level. The electronic structures of other group-IV triangular lattices such as Ge and Sn demonstrate strong similarity to Pb. We further design a quasi-3D crystal PbHfO$_2$ by alternately stacking Pb and 1T-HfO$_2$ monolayers. The new compound PbHfO$_2$ is dynamically stable and retains the properties of Pb monolayer. By applying epitaxial strains to PbHfO$_2$, it is possible to drive an insulator-to-metal transition coupled with an anti-ferroelectric-to-paraelectric phase transition. Our results suggest the potential of the 2D triangular lattice as a complimentary platform to design new type of broadly-defined Xenes.

Competing orders in a frustrated Heisenberg model on the Fisher lattice

We investigate the Heisenberg model on a decorated square (Fisher) lattice in the presence of first-neighbor $J_{1}$, second-neighbor $J_{2}$, and third-neighbor $J_{3}$ exchange couplings, with antiferromagnetic $J_{1}$. The classical ground-state phase diagram obtained within a Luttinger-Tisza framework is spanned by two antiferromagnetically ordered phases, and an infinitely degenerate antiferromagnetic chain phase. Employing classical Monte Carlo simulations we show that thermal fluctuations fail to lift the degeneracy of the antiferromagnetic chain phase. Interestingly, the spin-wave spectrum of the N\'eel state displays three Dirac nodal loops out of which two are symmetry protected while for the antiferromagnetic chain phase we find symmetry-protected Dirac lines. Furthermore, we investigate the spin $S=\frac{1}{2}$ limit employing a bond operator formalism which captures the singlet-triplet dynamics, and find a rich ground-state phase diagram host to a variety of valence bond solid orders in addition to antiferromagnetically ordered phases.

Hexagonal Zr3X (X = Al, Ga, In) Metals: High Dynamic Stability, Nodal Loop, and Perfect Nodal Surface States.

In recent years, topological semimetals/metals, including nodal point, nodal line, and nodal surface semimetals/metals, have been studied extensively because of their potential applications in spintronics and quantum computers. In this study, we predict a family of materials, Zr3X (X = Al, Ga, In), hosting the nodal loop and nodal surface states in the absence of spin–orbit coupling. Remarkably, the energy variation of the nodal loop and nodal surface states in Zr3X are very small, and these topological signatures lie very close to the Fermi level. When the effect of spin–orbit coupling is considered, the nodal loop and nodal surface states exhibit small energy gaps (<25 and 35 meV, respectively) that are suitable observables that reflect the spin-orbit coupling response of these topological signatures and can be detected in experiments. Moreover, these compounds are dynamically stable, and they consequently form potential material platforms to study nodal loop and nodal surface semimetals.

Possibility of fully spin-polarized nodal chain state in several spinel half metals

Nodal-chain fermions, as novel topological states of matter, have been hotly discussed in non-magnetic materials. Here, by using first-principles calculations and symmetry analysis, we propose the realization of fully spin-polarized nodal chain in the half-metal state of LiV$_2$O$_4$ compound. The material naturally shows a ferromagnetic ground state, and takes on a half-metal band structure with only the bands from the spin-up channel present near the Fermi level. The spin-up bands cross with each other, which form two types of nodal loops. These nodal loops arise from band inversion and are under the protection of the glide mirror symmetries. Remarkably, we find the nodal loops conjunct with each other and form chain-like nodal structure. Correspondingly, the w-shaped surface states are also fully spin-polarized. The fully spin-polarized nodal chain identified here has not been proposed in realistic materials before. An effective model is constructed to describe the nature of nodal chain. The effects of the electron correlation, the lattice strains, and the spin-orbit coupling are discussed. The fully spin-polarized bulk nodal-chain and the associated nontrivial surface states for a half-metal may open novel applications in spintronics.

Filling-enforced Dirac nodal loops in nonmagnetic systems and their evolutions under various perturbations

Based on symmetry analysis, we propose that filling-enforced Dirac loops (FEDLs) in non-magnetic systems exist and only exist in only five space groups (SGs), namely, SG.57, SG.60, SG.61, SG.62 and SG.205. %, respectively. We explore all possible configurations of the FEDLs in these space groups, and classify them accordingly. Furthermore, we study the evolutions of the FEDLs under various types of symmetry-breaking perturbations, such as an applied strain or an external field. The results show that FEDL materials can serve as parent materials of both topological semimetals hosting nodal points/loops, and topological insulators/topological crystalline insulators. By means of first-principles calculations, many materials possessing FEDLs are predicted.

Topological phases in the α−Li3N -type crystal structure of light-element compounds

Materials with tunable topological features, simple crystal structure and flexible synthesis, are in extraordinary demand towards technological exploitation of unique properties of topological nodal points. The controlled design of the lattice geometry of light elements is determined by utilizing density functional theory and the effective Hamiltonian model together with the symmetry analysis. This provides an intriguing venue for reasonably achieving various distinct types of novel fermions. We, therefore, show that a nodal line (type-I and II), Dirac fermion, and triple point (TP) fermionic excitation can potentially appear as a direct result of a band inversion in group-I nitrides with $\alpha$-Li$_{\rm 3}$N-type crystal structure. The imposed strain is exclusively significant for these compounds, and it invariably leads to the considerable modification of the nodal line type. Most importantly, a type-II nodal loop can be realized in the system under strain. These unique characteristics make $\alpha$-Li$_{\rm 3} $N-type crystal structure an ideal playground to achieve various types of novel fermions well-suited for technological applications.

Strain-tunable magnetism and nodal loops in monolayer MnB

Designing two-dimensional materials with magnetic and topological properties has continuously attracted intense interest in fundamental science and potential applications. Here, on the basis of first-principles calculations, we predict the coexistence of antiferromagnetism and Dirac nodal loops (NLs) in monolayer MnB, where the band crossing points are very close to the Fermi level. Remarkably, a moderate strain can induce an antiferromagnetic to ferromagnetic phase transition, driving monolayer MnB to a ferromagnetic metal with Weyl NLs. Such a type of topological quantum phase transition has not been observed before. In addition, the symmetry-protected properties of the two types of NLs as well as the magnetic critical temperatures are investigated. The controllable magnetic and topological order in monolayer MnB offers a unique platform for exploring topological quantum phase transitions and realizing nanospintronic devices.

Ferromagnetic hybrid nodal loop and switchable type-I and type-II Weyl fermions in two dimensions

As a novel type of fermionic state, hybrid nodal loop with the coexistence of both type-I and type- II band crossings has attracted intense research interest. However, it remains a challenge to realize hybrid nodal loop in both two-dimensional (2D) materials and in ferromagnetic (FM) materials. Here, we propose the first FM hybrid nodal loop in 2D CrN monolayer. We show that the material has a high Curie temperature (> 600 K) FM ground state, with the out-of-plane [001] magnetization. It shows a half-metallic band structure with two bands in the spin-up channel crossing each other near the Fermi level. These bands produce both type-I and type-II band crossings, which form a fully spin-polarized hybrid nodal loop. We find the nodal loop is protected by the mirror symmetry and robust against spin-orbit coupling (SOC). An effective Hamiltonian characterizing the hybrid nodal loop is established. We further find the configuration of nodal loop can be shifted under external perturbations such as strain. Most remarkably, we demonstrate that both type-I and type-II Weyl nodes can be realized from such FM hybrid nodal loop by simply shifting the magnetization from out-of-plane to in-plane. Our work provides an excellent candidate to realize FM hybrid nodal loop and Weyl fermions in 2D material, and is also promising for related topological applications with their intriguing properties.

Magnetotransport properties of the topological nodal-line semimetal CaCdSn

Topological nodal-line semimetals support protected band crossings which form nodal lines or nodal loops between the valence and conduction bands and exhibit novel transport phenomena. Here we address the topological state of the nodal-line semimetal candidate material, CaCdSn, and report magnetotransport properties of its single crystals grown by the self-flux method. Our first-principles calculations show that the electronic structure of CaCdSn harbors a single nodal loop around the $\Gamma$ point in the absence of spin-orbit coupling (SOC) effects. The nodal crossings in CaCdSn are found to lie above the Fermi level and yield a Fermi surface that consists of both electron and hole pockets. CaCdSn exhibits high mobility ($\mu \approx 3.44\times 10^4$ cm$^2$V$^{-1}$s$^{-1}$) and displays a field-induced metal-semiconductor like crossover with a plateau in resistivity at low temperature. We observe an extremely large and quasilinear non-saturating transverse as well as longitudinal magnetoresistance (MR) at low temperatures ($\approx 7.44\times 10^3 \%$ and $\approx 1.71\times 10^3\%$, respectively, at 4K). We also briefly discuss possible reasons behind such a large quasilinear magnetoresistance and its connection with the nontrivial band structure of CaCdSn.

Symmetry-protected spinful magnetic Weyl nodal loops and multi-Weyl nodes in 5dn cubic double perovskites (n=1,2)

Using both an effective three-band model and {\it ab initio} calculations, we have investigated various topological features in the cubic ferromagnetic $5d^{1,2}$ systems showing large spin-orbit coupling (SOC): Ba$_2$NaOsO$_6$, Sr$_2$SrOsO$_6$, and Ba$_2$$B$ReO$_6$ ($B$= Mg, Zn). In the presence of time-reversal symmetry (${\cal T}$), spinless Dirac nodal loops linked to each other at the $W$ points appear in the mirror planes. Remarkably, breaking ${\cal T}$ leads to spinful magnetic Weyl nodal loops (MWNLs) that are robust even at large SOC and correlation strength $U$ variation due to the combination of mirror symmetry and broken ${\cal T}$. Additionally, there are two types of magnetic Weyl points with chiral charges $|\chi|=1, 2$ along the $C_{4v}$ symmetry line, and another type-II MWNL encircling the zone center, that are dependent on $U$. Furthermore, the ferromagnetic Ba$_2$ZnReO$_6$ is an ideal half semimetal with MWNLs and magnetic Weyl nodes at the Fermi level without the interference of topologically trivial bulk states. These systems give rise to a remarkably large anomalous Hall conductivity $\sigma_{xy}$ of up to 1160 ($\Omega$cm)$^{-1}$. Our findings may apply widely for $t_{2g}$ systems with cubic (or slightly distorted) fcc-like structures.

Enhanced higher harmonic generation from nodal topology

Among topological materials, nodal loop semimetals (NLSMs) are arguably the most topologically sophisticated, with their valence and conduction bands intersecting along arbitrarily intertwined nodes. But unlike the well-known topological band insulators with quantized edge conductivities, nodal loop materials possess topologically nontrivial Fermi surfaces, not bands. Hence an important question arises: Are there also directly measurable or even technologically useful physical properties characterizing nontrivial nodal loop topology? In this work, we provide an affirmative answer by showing, for the first time, that nodal linkages protect the higher harmonic generation (HHG) of electromagnetic signals. Specifically, nodal linkages enforce non-monotonicity in the intra-band semi-classical response of nodal materials, which will be robust against perturbations preserving the nodal topology. These nonlinearities distort incident radiation and produce higher frequency peaks in the teraHertz (THz) regime, as we quantitatively demonstrate for a few known nodal materials. Since THz sources are not yet ubiquitous, our new mechanism for HHG will greatly aid applications like material characterization and non-ionizing imaging of object interiors.

Quantum Transport Signatures of a Close Candidate for a Type II Nodal-Line Semimetal

The nodal-line semimetal is a new type of topological state of matter in which the crossing of two energy bands forms a nodal loop. In the absence of spin-orbit coupling, Mg3Bi2 is predicted as a type II nodal-line semimetal, which may evolve to a topological insulator with a small energy gap of ∼35 meV in the presence of spin-orbit coupling. However, the transport evidence is still lacking. Here, we measure the magneto-transport in Mg3Bi2. At low temperatures, the magnetoconductivity exhibits a weak antilocalization behavior. We fit the experimental data with a magnetoconductivity formula for the weak antilocalization effect of three-dimensional nodal-line semimetals as well as the well-known Hikami-Larkin-Nagaoka formula for two-dimensional weak (anti)localization effects. By comparing the fitting results of these two theories, we demonstrate that the weak antilocalization in Mg3Bi2 is better described by the theory for nodal-line semimetals. Our work will inspire more explorations to use the new weak localization theory to identify a large spectrum of nodal-line semimetals.

Emerging Superconductivity and the Origin of Its Enhancement in Pressurized Topological Nodal‐Line Semimetal SrAs3

AbstractTopological nodal‐line semimetal SrAs3 hosts a simple nodal loop on the verge of Fermi level without distracting trivial states, providing a good platform for further investigations of electronic quantum states. Here, a report on pressure‐driven superconductivity in SrAs3 single crystal is provided and the interplay between the crystal structure, electronic state, and superconductivity is clarified therein. The comprehensive high‐pressure experiments, i.e., synchrotron X‐ray diffraction, electrical transport, and Raman spectroscopy, consistently reveal a critical pressure Pc ≈ 25 GPa. Around Pc, a superconducting transition appears, concomitant with the formation of high‐pressure cubic Pm‐3m phase. Meanwhile, the superconducting transition temperature Tc and hole carrier density increase synchronously above Pc. This finding demonstrates that the emerging superconductivity in SrAs3 stems from the high‐pressure cubic phase on the one hand, and shows that the evolution of Tc(P) is intimately correlated with the enhanced density of states near the Fermi level on the other hand.

Lorentz-violating type-II Dirac fermions in full-Heusler compounds XMg2Ag (X = Pr, Nd, Sm)

Lorentz-violating type-II Dirac fermion, as a new type of quasiparticles beyond the high-energy physics, has received intense attention recently. However, excellent candidate materials which contain sufficiently more type-II Dirac points near the Fermi level are still in scarcity. Here, we report a family of existing full-Heusler compounds, namely XMg2Ag (X = Pr, Nd, Sm), can serve as excellent Lorentz-violating type-II Dirac semimetals. We find they show several symmetry-protected nodal loops and triply degenerate nodal points (TDNPs) when spin–orbit coupling (SOC) is not considered. These fermions show clear nontrivial surface states. When SOC is included, the TDNPs transform into type-II Dirac points, characterized by Fermi arc surface states. The type-II DPs are protected by the C4v symmetry in the Γ–X path. Comparing with other type-II Dirac semimetals, XMg2Ag compounds have additional advantages including: (i) they contain as much as three pairs of type-II Dirac points; (ii) all the Dirac points locate very close to the Fermi level. These advantages make XMg2Ag compounds are suitable for studying the novel properties of type-II Dirac fermions in future experiments.

Large Josephson current in Weyl nodal loop semimetals due to odd-frequency superconductivity

Weyl nodal loop semimetals (WNLs) host a closed nodal line loop Fermi surface in the bulk, protected zero-energy flat band, or drumhead, surface states, and strong spin-polarization. The large density of states of the drumhead states makes WNL semimetals exceedingly prone to electronic ordering. At the same time, the spin-polarization naively prevents conventional superconductivity due to its spin-singlet nature. Here we show the complete opposite: WNLs are extremely promising materials for superconducting Josephson junctions, entirely due to odd-frequency superconductivity. By sandwiching a WNL between two conventional superconductors we theoretically demonstrate the presence of very large Josephson currents, even up to orders of magnitude larger than for normal metals. The large currents are generated both by an efficient transformation of spin-singlet pairs into odd-frequency spin-triplet pairing by the Weyl dispersion and the drumhead states ensuring exceptionally proximity effect. As a result, WNL Josephson junctions offer unique possibilities for detecting and exploring odd-frequency superconductivity.

IrSi as a Superior Electronic Material with Novel Topological Properties and Nice Compatibility with Semiconductor Si

To favor the applications in electronic devices, topological semimetals require not only excellent topological electronic structures but also nice compatibility with the traditional Si‐based semiconductor. Herein, IrSi compound is revealed as a superior electronic material which satisfies both criteria proposed earlier. First, IrSi hosts novel band topology with multiple fermionic states. Without spin–orbit coupling (SOC), IrSi shows an accidental node loop, protected by two independent mechanisms. With SOC included, it shows two symmetry‐enforced nodal lines (NLs) and Dirac points (DPs). The presence of NLs and DPs under SOC has been verified by symmetry analysis. Remarkably, the DPs in IrSi locate quite near the Fermi level and possess type‐II band dispersions, which are rarely identified in Dirac semimetals. Second, IrSi has already been prepared on epitaxial Si wafers and thereby has been fabricated as IrSi/Si devices with excellent compatibility in previous experiments. This work further shows that the nodal loop and DPs are robust against potential lattice distortion during its epitaxial growth.

Determination of the Fermi surface and field-induced quasiparticle tunneling around the Dirac nodal loop in ZrSiS

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 oscillations to density functional theory calculations. Comparison of the Shubnikov-de Haas and de Haas-van Alphen oscillations at low temperatures and analysis of the respective Dingle plots reveal the presence of significantly enhanced scattering on the electron pocket. Above a threshold field that is aligned along the c-axis of the crystal, the specific cage-like Fermi surface of ZrSiS allows for electron-hole tunneling to occur across finite gaps in momentum space leading to quantum oscillations with a complex frequency spectrum. Additional high-frequency quantum oscillations signify magnetic breakdown orbits that encircle the entire Dirac nodal loop. We suggest that the persistence of quantum oscillations in the resistivity to high temperatures is caused by Stark interference between orbits of nearly equal masses.

Topological Dirac nodal loops in nonsymmorphic hydrogenated monolayer boron

The existence of a topological Dirac nodal loop is predicted for hydrogenated monolayer boron sheets with a nonsymmorphic symmetry. The three-center two-electron bonds in boron compounds restrict the electronic system to be insulating or semimetallic with a Dirac nodal loop. Two types of electronic structures are distinguished by the Z_2 topological index and confirmed by first-principles total-energy calculations. The topological taxonomy, developed in this research, can be applied to other two-dimensional materials and to seek novel Dirac fermions.

Termination-dependent topological surface states in nodal-loop semimetal HfP2

Symmetry plays a major role in all disciplines of physics. Within the field of topological materials there is a great interest in understanding how the mechanics of crystalline and internal symmetries protect crossings between the conduction and valence bands. Additionally, exploring this direction can lead to a deeper understanding on the topological properties of crystals hosting a variety of symmetries. For the first time, we report the experimental observation of topological surface states in the nodal loop semimetal HfP2 using angle resolved photoemission spectroscopy (ARPES) which is supported by our first principles calculations. Our study shows termination dependent surface states in this compound. Our experimental data reveal surface states linked to three unique nodal loops confirmed by theoretical calculation to be topologically non-trivial. This work demonstrates that transition metal dipnictides provide a good platform to study non-trivial topological states protected by nonsymmorphic symmetry.

Experimental Realization of Two-Dimensional Buckled Lieb Lattice.

Two-dimensional (2D) materials with a Lieb lattice host exotic electronic band structures. Such a system does not exist in nature, and it is also difficult to obtain in the laboratory due to its structural instability. Here, we experimentally realized a 2D system composed of a tin overlayer on an aluminum substrate by molecular beam epitaxy. The specific arrangement of Sn atoms on the Al(100) surface, which benefits from favorable interface interactions, forms a stabilized buckled Lieb lattice. Theoretical calculations indicate a partially broken nodal line loop and a topologically nontrivial insulating state with a spin-orbital coupling effect in the band structure of this Lieb lattice. The electronic structure of this system is experimentally characterized by angle-resolved photoemission spectroscopy, in which the hybridized states between topmost Al atoms and Sn atoms are revealed. Our work provides an appealing method for constructing 2D quantum materials based on the Lieb lattice.