Nodal Loop Research Articles

Tunable topologically nontrivial states in newly discovered graphyne allotropes: from Dirac nodal grid to Dirac nodal loop

By means of quotient-graph associated crystal prediction method, a new graphyne allotrope with unique Dirac nodal grid state is reported in this work. It is named as 191-E24Y24-1 according to its hexagonal lattice (with P6/mmm symmetry, No. 191) containing 24 sp2-hybridized carbon atoms and 24 sp-hybridized ones. The first-principles results show that the total energy of 191-E24Y24-1 is more favorable than that of recent synthesized -graphdiyne and carbon ene-yne. It is also demonstrated to be dynamically, thermally, and mechanically stable. Interestingly, the 191-E24Y24-1 harbors intrinsic semimetal features showing intriguing hexagonal Dirac nodal grid state in the reciprocal space. Such unique electronic state is stable against small external tensile strains, and it is tunable under compression strains which will transform to new triangle Dirac nodal grid state. Moreover, a new metastable graphyne allotrope named 191-E12Y36-4 with Dirac nodal loop state is also observed in the process of stretching 191-E24Y24-1 with large tensile strains. The results presented in this work reveal two novel graphyne allotropes with exotic electronic properties. These discoveries are not only physical interesting, but also provide potential material candidates for carbon-based high performance electronic nanodevices.

Electronic structure of a borophene layer in rare-earth aluminum/chromium boride and its hydrogenated derivative borophane

We studied electronic states of borophene in a crystal of rare-earth aluminum/chromium boride and the hydrogenated sheet, borophane, successfully prepared by ion exchange and liquid exfoliation of the crystal. The layer has a boron network with five-membered and seven-membered rings. Electronic structures of all the boron-derived layers are found to be gapless by soft x-ray absorption and emission spectroscopy measurements at the B K-shell absorption edge and density functional theory calculations. The present results support the existence of a Dirac nodal loop at the Fermi level in the borophane layer with five- and seven-membered rings, predicted recently by topological calculations.

Three-dimensional acetylenic modified graphene for high-performance optoelectronics and topological materials

Seeking carbon phases with versatile properties is one of the fundamental goals in physics, chemistry, and materials science. Here, based on the first-principles calculations, a family of three-dimensional (3D) graphene networks with abundant and fabulous electronic properties, including rarely reported dipole-allowed truly direct band gap semiconductors with suitable band gaps (1.07–1.87 eV) as optoelectronic/photovoltaic materials and topological nodal-ring semimetals, are proposed through stitching different graphene layers with acetylenic linkages. Remarkably, the optical absorption coefficients in some of those semiconducting carbon allotropes express possibly the highest performance among all of the semiconducting carbon phases known to date. On the other hand, the topological states in those topological nodal-ring semimetals are protected by the time-reversal and spatial symmetry and present nodal rings and nodal helical loops topological patterns. Those newly revealed carbon phases possess low formation energies and excellent thermodynamic stabilities; thus, they not only host a great potential in the application of optoelectronics, photovoltaics, and quantum topological materials etc., but also can be utilized as catalysis, molecule sieves or Li-ion anode materials and so on. Moreover, the approach used here to design novel carbon allotropes may also give more enlightenments to create various carbon phases with different applications.

Time-dependent electric transport in nodal loop semimetals

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 current first reaches a plateau $\sim E$. Then, for perpendicular directions, it increases while for in-plane directions it decreases with time to another plateau, still $\sim E$. These features arise from interband processes. For long times or strong electric fields, the current grows as $\sim E^{3/2}t$ or $\sim E^3t^2$ for perpendicular or parallel electric fields, respectively. This non-linear response represents an intraband effect where the large number of excited quasiparticles responds to the electric field. Our analytical results are benchmarked by the numerical evaluation of the current from continuum and tight-binding models of nodal loop semimetals.

Symmetry-enforced band nodes in 230 space groups

Crystallographic symmetries enforcing band touchings (BTs) in the Brillouin zone (BZ) have been utilized to classify and predict the topological semimetals. Though the early proposed topological semimetals contain isolated nodal points in the BZ, the proposed nodal line semimetals later could host various structures of several nodal lines/loops: nodal chains, nodal nets or Hopf-links, etc. In this work, using compatibility relations, we first list all possible high symmetry lines (HSLs) that can be nodal lines itself, high symmetry planes (HSPLs) that can host nodal loops, high symmetry planes (HSPLs) that are nodal surfaces for all 230 SGs, with spin-orbit coupling and time-reversal symmetry considered or not. We then show how to diagnose a nodal loop from the band crossing in an HSL, or nodal line/surface from irreducible representation (irrep) of an high-symmetry point (HSP), while the rest cases correspond to nodal points. Among our results, those essential cases, for which the nodal points/lines/loops/surfaces must exist, are highlighted since they are promising for the realizations of (nearly) ideal nodal point/line/loop/surface semimetals, as well as systems with flexible tunability owning fixed structure of topological nodal points/lines/loops/surfaces. Based on our results, SGs allowing Hopf-link structure with one straight nodal line threading a nodal loop, or two nesting nodal loops lying in two respective high symmetry planes, are highlighted, with the predicted materials being B$_5$Pb$_2$IO$_9$ in SG 34 and SrAl$_2$Au$_3$ in SG 62, respectively. Our exhaustive results could serve as a useful guide for efficiently predicting and designing materials or artificial systems owning exotic geometric nodal structures of energy bands simply based on structure symmetries.

The p-orbital magnetic topological states on a square lattice

Honeycomb or triangular lattices were extensively studied and thought to be proper platforms for realizing the quantum anomalous Hall effect (QAHE), where magnetism is usually caused by d orbitals of transition metals. Here we propose that a square lattice can host three magnetic topological states, including the fully spin-polarized nodal loop semimetal, QAHE and the topologically trivial ferromagnetic semiconductor, in terms of the symmetry and k · p model analyses that are material independent. A phase diagram is presented. We further show that the above three magnetic topological states can indeed be implemented in the two-dimensional (2D) materials ScLiCl5, LiScZ5 (Z=Cl, Br) and ScLiBr5, respectively. The ferromagnetism in these 2D materials is microscopically revealed from p electrons of halogen atoms. This present study opens a door to explore the exotic topological states as well as quantum magnetism from p-orbital electrons by means of the material-independent approach.

Spin-polarized type-II nodal loop and nodal surface states in hexagonal compounds XTiO2 ( X = Li, Na, K, Rb)

Topological semimetals and metals have been extensively studied in nonmagnetic materials, while those in magnetic materials are still quite rare. Here, based on first-principles calculations, we reveal that a family of hexagonal compounds $X{\mathrm{TiO}}_{2}$ ($X$ = Li, Na, K, Rb) with ferromagnetic ground states host excellent topological electronic structures. In their low-energy band structures, they show two Weyl nodal loops and two Weyl nodal surfaces, where each nodal loop and nodal surface is contributed by the electron states in the same spin channel, namely, spin-polarized nodal loops and nodal surfaces. In particular, the nodal loops in these materials show type-II band dispersions and are robust against spin-orbit coupling because of the protection of the mirror symmetry. These nodal loops are featured with spin-polarized drumhead surface states. We build an effective Hamiltonian, which can well describe the type-II nodal loops in the system. Besides the type-II nodal loops, these materials also possess two spin-polarized nodal surfaces, protected by specific nonsymmorphic symmetry. The findings in this paper can serve as good material platforms to study the fundamental physics of type-II nodal loop and nodal surface fermions in a time-reversal-breaking system.

Coexistence of topological nontrivial and spin-gapless semiconducting behavior in MnPO4 : A composite quantum compound

Composite quantum compounds (CQC) are classic example of quantum materials which host more than one apparently distinct quantum phenomenon in physics. Magnetism, topological superconductivity, Rashba physics etc. are few such quantum phenomenon which are ubiquitously observed in several functional materials and can co-exist in CQCs. In this letter, we use {\it ab-initio} calculations to predict the co-existence of two incompatible phenomena, namely topologically non-trivial Weyl semimetal and spin gapless semiconducting (SGS) behavior, in a single crystalline system. SGS belong to a special class of spintronics material which exhibit a unique band structure involving a semiconducting state for one spin channel and a gapless state for the other. We report such a SGS behavior in conjunction with the topologically non-trivial multi-Weyl Fermions in MnPO$_4$. Interestingly, these Weyl nodes are located very close to the Fermi level with the minimal trivial band density. A drumhead like surface state originating from a nodal loop around Y-point in the Brillouin zone is observed. A large value of the simulated anomalous Hall conductivity (1265 $\Omega^{-1} cm^{-1}$) indirectly reflects the topological non-trivial behavior of this compound. Such co-existent quantum phenomena are not common in condensed matter systems and hence it opens up a fertile ground to explore and achieve newer functional materials.

Crossdimensional universality classes in static and periodically driven Kitaev models

The Kitaev model on the honeycomb lattice is a paradigmatic system known to host a wealth of nontrivial topological phases and Majorana edge modes. In the static case, the Majorana edge modes are nondispersive. When the system is periodically driven in time, such edge modes can disperse and become chiral. We obtain the full phase diagram of the driven model as a function of the coupling and the driving period. We characterize the quantum criticality of the different topological phase transitions in both the static and driven model via the notions of Majorana-Wannier state correlation functions and momentum-dependent fidelity susceptibilities. We show that the system hosts crossdimensional universality classes: although the static Kitaev model is defined on a two-dimensional (2D) honeycomb lattice, its criticality falls into the universality class of one-dimensional (1D) linear Dirac models. For the periodically driven Kitaev model, in addition to the universality class of prototype 2D linear Dirac models, an additional 1D nodal loop type of criticality exists owing to emergent time-reversal and mirror symmetries, indicating the possibility of engineering multiple universality classes by periodic driving. The manipulation of time-reversal symmetry allows the periodic driving to control the chirality of the Majorana edge states.

Anisotropic nonlinear optical response of nodal-loop materials

Nodal line semimetals have lately aroused much experimental and theoretical interest, with their gap closing along unconventional trajectories in the 3D Brillouin zone. These trajectories or nodal lines can close into loops and trace out intricate knotted or linked configurations with complicated topologies. In this work, we investigate the semi-classical optical response of two nodal loops in linked, unlinked and touching configurations, focusing particularly on the interplay of response anisotropy and non-linearity. We provide a geometric picture that unifies these aspects and sheds light on the effects of nodal topology and geometry. Based on a model abstracted from generic multi nodal loop scenarios, both with or without linkages, our findings will be applicable for a large class of nodal semimetal materials with multiple nodal lines or loops.

Quantum oscillation and Landau–Zener transition in untilted nodal line semimetals under a time-periodic magnetic field

Nodal line semimetals (NLSM) exhibit interesting quantum oscillation (QO) characteristics when acted upon by a strong magnetic field. We study the combined effect of strong direct and alternating magnetic field, perpendicular to the nodal plane in an untilted NLSM in order to probe the behavior of the low lying Landau level states that can periodically become gapless for suitably chosen field parameters. The oscillatory field variation, as opposed to a steady one, has interesting impact on the QO phenomena with the Landau tubes crossing the Fermi surface extremally two times per cycle. Furthermore, the low energy modes can witness Landau–Zener like transitions between valence and conduction band providing further routes to conduction. We discuss such transition phenomena following the framework of adiabatic-impulse approximation for slow quenches. Next we also investigate the effect of oscillating magnetic field acting parallel to the nodal loop where topologically nontrivial magnetic oscillations at low energies can be witnessed. Therefore, with proper parameters chosen, one can engineer topological transitions to occur periodically in such systems as the oscillating field is swept through its cycles.

Topological gimbal phonons in T -carbon

The topological metal states in electronic systems have been extensively studied, but topological phonons were explored only in few examples so far. Here, we expose for the first time that the topological nodal gimbal phonons, type-I and type-II Weyl phonons are simultaneously present in T-carbon, a recently realized new allotrope of carbon. At about 15.2 THz, we find that there exist three mutually intersecting nodal loops (named as nodal gimbal phonons) around {\Gamma} point, and two pairs of type-I Weyl phonons on the boundary of Brillouin zone around each X point. In addition, there exist three pairs of type-II Weyl phonons at about 14.5 THz around each L point. It is shown that these exotic topological phonons are protected by corresponding symmetries, and lead to topologically nontrivial surface states. Our findings not only afford plenty of intriguing topological phonon states in a simple material like T-carbon but also provide a new platform to study novel properties of topological phonons, which would facilitate further both experimental and theoretical works in future.

Extremely large magnetoresistance from electron-hole compensation in the nodal-loop semimetalZrP2

Several early transition metal dipnictides have been found to host topological semimetal states and exhibit large magnetoresistance. In this study, we use angle-resolved photoemission spectroscopy (ARPES) and magneto-transport to study the electronic properties of a new transition metal dipnictide ZrP$_2$. We find that ZrP$_2$ exhibits an extremely large and unsaturated magnetoresistance of up to 40,000 % at 2 K, which originates from an almost perfect electron-hole compensation. Our band structure calculations further show that ZrP$_2$ hosts a topological nodal loop in proximity to the Fermi level. Based on the ARPES measurements, we confirm the results of our calculations and determine the surface band structure. Our study establishes ZrP$_2$ as a new platform to investigate near-perfect electron-hole compensation and its interplay with topological band structures.

Generalization of Methods for Calculating Steady-State Flow Distribution in Pipeline Networks for Non-Conventional Flow Models

This paper proposes generalized models and methods for calculating flow distribution in hydraulic circuits with lumped parameters. The main models of the isothermal steady-state flow of medium are classified by an element of the hydraulic circuit. These models include conventional, implicitly specified by flow rate, and pressure-dependent ones. The conditions for their applicability, which ensure the existence and uniqueness of a solution to the flow distribution problem, are considered. We propose generalized nodal pressure and loop flow rate methods, which can be applied regardless of the forms of specific element models. Final algorithms, which require lower computational costs versus the known approaches designed for non-conventional flow models, are substantiated. Proposed models, methods, algorithms, and their capabilities, are analytically and numerically illustrated by an example of a fragment of gas transmission network with compressor stations.

Anomalous Hall optical conductivity in tilted topological nodal-line semimetals

Nodal-line semimetals are typically characterized by a nodal ring, a closed line of Dirac points in the Brillouin zone, whose topology is characterized by a quantized Berry phase. The nodal loop is protected by the combined inversion and time-reversal ($\mathcal{PT}$) symmetry in the absence of spin-orbit coupling, exhibiting the parity anomaly. Introduction of the $\mathcal{PT}$-break mass term can lead to the transverse Hall response at each point in the nodal ring, but the net Hall current vanishes due to the existence of inversion-symmetry points that contribute to the transverse current with opposite signs. We show that the combination of an inversion-broken tilt term and finite Fermi energy can cause the nonzero anomalous Hall effect. We explore the DC and AC anomalous Hall conductivity in tilted nodal-line semimetals using Kubo formula and focus on the contribution of the free carriers to the Hall conductivity. We find that the interband transition of free carriers dominates the nonvanishing dynamic Hall conductivity over the filled band contribution. The resulting anomalous Hall conductivity exhibits the parity anomaly for the DC case, and rich characteristic frequencies for the AC case which is closely related to the change of the geometry of the Fermi surface. These features may serve as a diagnostic tool to characterize the topological property, tilting parameter, or the radius of the nodal line.

Topological properties of noncentrosymmetric superconductors TIr2B2 ( T=Nb,Ta )

A recent experiment reported two new non-centrosymmetric superconductors NbIr$_{2}$B$_{2}$ and TaIr$_{2}$B$_{2}$ with respective superconducting transition temperatures of 7.2 K and 5.2 K and further suggested their superconductivity to be unconventional [K. G\'ornicka \textit{et al}., Adv. Funct. Mater. 2007960 (2020)]. Here, based on first-principles calculations and symmetry analysis, we propose that $T$Ir$_{2}$B$_{2}$ ($T$=Nb, Ta) are topological Weyl metals in the normal state. In the absence of spin-orbit coupling (SOC), we find that NbIr$_{2}$B$_{2}$ has 12 Weyl points, and TaIr$_{2}$B$_{2}$ has 4 Weyl points, i.e. the minimum number under time-reversal symmetry; meanwhile, both of them have a nodal net composed of three nodal lines. In the presence of SOC, a nodal loop on the mirror plane evolves into two hourglass Weyl rings, along with the Weyl points, which are dictated by the nonsymmorphic glide mirror symmetry. Besides the rings, NbIr$_{2}$B$_{2}$ and TaIr$_{2}$B$_{2}$ have 16 and 20 pairs of Weyl points, respectively. The surface Fermi arcs are explicitly demonstrated. On the (110) surface of TaIr$_{2}$B$_{2}$, we find extremely long surface Fermi arcs ($\sim$0.6 ${\text{\AA}}^{-1}$) located 1.4 meV below the Fermi level, which should be readily probed in experiment. Combined with the intrinsic superconductivity and the nontrivial bulk Fermi surfaces, $T$Ir$_{2}$B$_{2}$ may thus provide a very promising platform to explore the three-dimensional topological superconductivity.

Coexistence of fully spin-polarized Weyl nodal loop, nodal surface, and Dirac point in a family of quasi-one-dimensional half-metals

Recent investigation has connected topology with magnetization, which has been attracting great research interest. Here, based on symmetry analysis and first-principle calculations, we reveal rich band crossings in ferromagnetic quasi-one-dimensional compounds ${XYZ}_{3}$ ($X=\mathrm{Cs},\mathrm{Rb}$, $Y=\mathrm{Cr},\mathrm{Cu}$, $Z=\mathrm{Cl},\mathrm{I}$). We show that these materials host hourglass Weyl loops, nodal surfaces, and Dirac points in the absence of spin-orbital coupling. Notably, these topological phases only occur in a single spin channel, leading to fully spin-polarized phases. Based on symmetry analysis, we present that the hourglass-like Weyl loop is enforced by nonsymmorphic symmetries, which give rise to drumhead surface states. Particularly, the Weyl loop can persist when a magnetic direction is normal to the plane where it lies on. Additionally, we also find the nodal surfaces protected by the combination of screw rotation and time reversal symmetry. Besides, there exists a Dirac point enabled by nonsymmorphic symmetry and time reversal symmetry. Notably, this Dirac point shows a linear dispersion in one direction and quadratic dispersion in the normal plane, showing two Fermi arcs in its surface energy spectrum. By tuning the direction of magnetization, the Dirac point can be reduced to a pair of Weyl points. Our work suggests a concrete platform for exploring the fascinating physics associated with nonsymmorphic band crossings in quasi-one-dimensional ferromagnetic systems.

Dirac Loops in a Strongly Correlated Metal: Origin of Large Magnetoresistance?

Researchers have discovered an extremely large magnetoresistance effect in a high-quality single crystal of β-ReO2, where high-mobility minority carriers relevant to the Dirac nodal loops coexist with majority carriers with a strong electron correlation.

Various half-metallic nodal loops in organic Cr2N6C3 monolayers.

Topological nodal-line semimetals, as a type of exotic quantum electronic state, have drawn considerable research interest recently. In this work, we propose a new two-dimensional covalent-organic Cr2N6C3 monolayer (ML) material, which has a combined honeycomb and effective Kagome lattice and has various half-metallic nodal loops (HMNLs). First-principles calculations show that the Cr2N6C3 ML is dynamically and thermally stable and has an out-of-plane ferromagnetic order. Remarkably, various nodal loops, including types I-III, are found coexisting in the material, all of which are rare half-metallic states. The obtained HMNLs, simultaneously possessing the merits of spintronics and semimetals, are robust against spin-orbit coupling and biaxial strain. A topological phase transition, characterized by loop-winding indexes, can be induced in the ML by applying uniaxial strain. Tight-binding model calculations show that the obtained HMNLs originate primarily from the band inversion between Cr dx2-y2/xy and N pz orbitals, accommodated on the honeycomb and Kagome sublattices, respectively. The various predicted HMNLs and topological behaviors mean that the Cr2N6C3 MLs have promisingly versatile applications in future low-power-consuming spintronics and electronics.

Newly discovered graphyne allotrope with rare and robust Dirac node loop.

Two-dimensional (2D) carbon allotropes with topologically nontrivial states are drawing considerable attention owing to their unique physical properties and great potential applications in the next generation of micro-nano devices. In contrast to the numerous Dirac points predicted in 2D carbon allotropes, systems featuring Dirac nodal lines (loops) are still quite rare. Here, by means of first-principles calculation, we report our newly discovered carbon monolayer 123-E8Y24-1 with robust Dirac nodal line states, which possesses a tetragonal lattice with P4/mmm symmetry and contains 8 sp2 carbon atoms (graphene: E8) and 24 sp carbon atoms (grapheyne: Y24) in the crystalline cell. This 2D material is as energetically stable as the recently experimentally synthesized β-graphdiyne, and it is further predicted to be dynamically, mechanically, and also thermodynamically stable. Owing to its intrinsic geometric characteristics, 123-E8Y24-1 also exhibits obvious Young's modulus anisotropy, with a sizable ratio between the maximum and minimum value of up to 5.8. Remarkably, 123-E8Y24-1 presents a semimetal nature and possesses Dirac nodal line states in the electronic band structure, and such behavior could be kept well under external strain between -10.0% and 8.0%. The electronic properties of 123-E8Y24-1 can be carefully confirmed by constructing a tight-binding (TB) model. The findings presented in this paper reveal a novel 2D Dirac nodal loop carbon sheet, providing a new candidate for carbon-based high-speed electronic devices.