Dirac Nodal Line Research Articles

Dimensional crossover of effective orbital dynamics in polar distorted He3−A : Transitions to antispacetime

Topologically protected superfluid phases of $^3$He allow one to simulate many important aspects of relativistic quantum field theories and quantum gravity in condensed matter. Here we discuss a topological Lifshitz transition of the effective quantum vacuum in which the determinant of the tetrad field changes sign through a crossing to a vacuum state with a degenerate fermionic metric. Such a transition is realized in polar distorted superfluid $^3$He-A in terms of the effective tetrad fields emerging in the vicinity of the superfluid gap nodes: the tetrads of the Weyl points in the chiral A-phase of $^3$He and the degenerate tetrad in the vicinity of a Dirac nodal line in the polar phase of $^3$He. The continuous phase transition from the $A$-phase to the polar phase, i.e. in the transition from the Weyl nodes to the Dirac nodal line and back, allows one to follow the behavior of the fermionic and bosonic effective actions when the sign of the tetrad determinant changes, and the effective chiral space-time transforms to anti-chiral "anti-spacetime". This condensed matter realization demonstrates that while the original fermionic action is analytic across the transition, the effective action for the orbital degrees of freedom (pseudo-EM) fields and gravity have non-analytic behavior. In particular, the action for the pseudo-EM field in the vacuum with Weyl fermions (A-phase) contains the modulus of the tetrad determinant. In the vacuum with the degenerate metric (polar phase) the nodal line is effectively a family of $2+1$d Dirac fermion patches, which leads to a non-analytic $(B^2-E^2)^{3/4}$ QED action in the vicinity of the Dirac line.

First-principles design of low-dimensional quantum materials with nontrivial band topology

We discover new class of electrides [1,2], the first electride with nontrivial band topology, based on 1D building blocks by coupling materials database searches and first-principles-calculations-based analysis. This new class of electrides, composed of 1D nanorod building blocks, has crystal structures that mimic β-TiCl3 with the position of anions and cations exchanged. Unlike the weakly coupled nanorods of β-TiCl3 , Cs3 O and Ba3 N retain 1D anionic electrons along the hollow interrod sites; additionally, strong inter-rod interaction in C3 O and Ba3 N induces band inversion in a 2D superatomic triangular lattice, resulting in nontrivial band topology of Dirac nodal lines [2]. This new material could be served as an ideal template to explore various quantum phases. Using a tight-binding Hamiltonian based on twodimensional (2D) honeycomb lattices, we construct a phase diagram in terms of exchange coupling and spinorbit coupling (SOC), which spans four different phases, such as topological insulator, large/small gap quantum anomalous Hall (QAH) insulator, and ferromagnetic semiconductor [3]. We reveal that 2D honeycomb lattices consisting of some post-transition metals, such as Sn, Pb, and Bi, undertake ferromagnetic transition as the lattice constant increases and significant changes in SOC strength, which makes them an ideal material to explore the versatile phases solely by changing lattice constants. First-principles density functional calculations demonstrate that 2D honeycomb SnF can show QAH effect with SOC gap of ~0.25eV and Curie temperature (Tc ) of ~780K [3]. Our calculations propose a new avenue to the room-temperature QAH effect in realistic 2D materials systems.

Prediction of TiRhAs as a Dirac nodal line semimetal via first-principles calculations

Using first-principles calculations we predict that $\mathrm{TiRhAs}$, a previously synthesized compound, is a Dirac nodal line (DNL) semimetal. The DNL in this compound is found to be protected both by the combination of inversion and time-reversal symmetry, and by a reflection symmetry, in the absence of spin-orbit coupling (SOC). Our calculations show that band velocities associated with the nodal line have a high degree of directional anisotropy, with in-plane velocities $v_\perp$ perpendicular to the nodal line between $1.2-2.8\times10^5$ m/s. The crossings along the DNL are further found to exhibit a prominent and position-dependent tilt along directions perpendicular to the nodal line. We calculate $\mathbb{Z}_2$ indices based on parity eigenvalues at time-reversal invariant momenta and show that $\mathrm{TiRhAs}$ is topological. A tight-binding model fit from our first-principles calculations demonstrates the existence of two-dimensional drumhead surface states on the surface Brillouin zone. Based on the small gapping of the DNL upon inclusion of SOC and the clean Fermi surface free from trivial bands, $\mathrm{TiRhAs}$ is a promising candidate for further studies of the properties of topological semimetals.

Topological quantum catalyst: Dirac nodal line states and a potential electrocatalyst of hydrogen evolution in the TiSi family

Topological nodal line (DNL) semimetals, a closed loop of the inverted bands in its bulk phases, result in the almost flat drumhead-like non-trivial surface states (DNSSs) with an unusually high electronic density near the Fermi level. High catalytic active sites generally associated with high electronic densities around the Fermi level, high carrier mobility and a close-to-zero free energy of the adsorbed state of hydrogen (ΔGH*≈0) are prerequisite to design alternative of precious platinum for catalyzing electrochemical hydrogen production from water. By combining these two aspects, it is natural to consider if the DNLs are a good candidate for the hydrogen evolution reaction (HER) or not because its DNSSs provide a robust platform to activate chemical reactions. Here, through first-principles calculations we reported a new DNL TiSi-type family, exhibiting a closed Dirac nodal line due to the linear band crossings in k y =0 plane. The hydrogen adsorbed state on the surface yields ΔGH* to be almost zero and the topological charge carries participate in HER. The results highlight a new routine to design topological quantum catalyst utilizing the topological DNL-induced surface bands as active sites, rather than edge sites-, vacancy-, dopant-, strain-, or heterostructure-created active sites.

Dirac and Nodal Line Magnons in Three-Dimensional Antiferromagnets.

We study the topological properties of magnon excitations in three-dimensional antiferromagnets, where the ground state configuration is invariant under time reversal followed by space inversion (PT symmetry). We prove that Dirac points and nodal lines, the former being the limiting case of the latter, are the generic forms of symmetry-protected band crossings between magnon branches. As a concrete example, we study a Heisenberg spin model for a "spin-web" compound, Cu_{3}TeO_{6}, and show the presence of the magnon Dirac points assuming a collinear magnetic structure. Upon turning on symmetry-allowed Dzyaloshinsky-Moriya interactions, which introduce a small noncollinearity in the ground state configuration, we find that the Dirac points expand into nodal lines with nontrivial Z_{2}-topological charge, a new type of nodal line not predicted in any materials so far.

Similar ultrafast dynamics of several dissimilar Dirac and Weyl semimetals

Recent years have seen the rapid discovery of solids whose low-energy electrons have a massless, linear dispersion, such as Weyl, line-node, and Dirac semimetals. The remarkable optical properties predicted in these materials show their versatile potential for optoelectronic uses. However, little is known of their response in the picoseconds after absorbing a photon. Here, we measure the ultrafast dynamics of four materials that share non-trivial band structure topology but that differ chemically, structurally, and in their low-energy band structures: ZrSiS, which hosts a Dirac line node and Dirac points; TaAs and NbP, which are Weyl semimetals; and Sr1–yMn1–zSb2, in which Dirac fermions coexist with broken time-reversal symmetry. After photoexcitation by a short pulse, all four relax in two stages, first sub-picosecond and then few-picosecond. Their rapid relaxation suggests that these and related materials may be suited for optical switches and fast infrared detectors. The complex change of refractive index shows that photoexcited carrier populations persist for a few picoseconds.

Measurement of the bulk and surface bands in Dirac line-node semimetal ZrSiS**Project supported by the National Key R&D Program of China (Grant No. 2017YFA0305400), Chinese Academy of Science–Shanghai Science Research Center (Grant No. CAS-SSRC-YH-2015-01), the National Natural Science Foundation of China (Grant No. 11674229), the Engineering and Physical Sciences Research Council Platform (Grant No. EP/M020517/1), and the Hefei Science–Center Chinese

Dirac semimetals are materials in which the conduction and the valence bands have robust crossing points protected by topology or symmetry. Recently, a new type of Dirac semimetals, so called the Dirac line-node semimetals (DLNSs), have attracted a lot of attention, as they host robust Dirac points along the one-dimensional (1D) lines in the Brillouin zone (BZ). In this work, using angle-resolved photoemission spectroscopy (ARPES) and first-principles calculations, we systematically investigated the electronic structures of non-symmorphic ZrSiS crystal where we clearly distinguished the surface states from the bulk states. The photon-energy-dependent measurements further prove the existence of Dirac line node along the X–R direction. Remarkably, by in situ surface potassium doping, we clearly observed the different evolutions of the bulk and surface electronic states while proving the robustness of the Dirac line node. Our studies not only reveal the complete electronic structures of ZrSiS, but also demonstrate the method manipulating the electronic structure of the compound.

Experimental realization of two-dimensional Dirac nodal line fermions in monolayer Cu2Si

Topological nodal line semimetals, a novel quantum state of materials, possess topologically nontrivial valence and conduction bands that touch at a line near the Fermi level. The exotic band structure can lead to various novel properties, such as long-range Coulomb interaction and flat Landau levels. Recently, topological nodal lines have been observed in several bulk materials, such as PtSn4, ZrSiS, TlTaSe2 and PbTaSe2. However, in two-dimensional materials, experimental research on nodal line fermions is still lacking. Here, we report the discovery of two-dimensional Dirac nodal line fermions in monolayer Cu2Si based on combined theoretical calculations and angle-resolved photoemission spectroscopy measurements. The Dirac nodal lines in Cu2Si form two concentric loops centred around the Γ point and are protected by mirror reflection symmetry. Our results establish Cu2Si as a platform to study the novel physical properties in two-dimensional Dirac materials and provide opportunities to realize high-speed low-dissipation devices.

Topological Dirac line nodes and superconductivity coexist in SnSe at high pressure

Due to fundamental interest and potential applications in quantum computation, tremendous efforts have been invested to study topological superconductivity. However, bulk topological superconductivity seems to be difficult to realize and its mechanism is still elusive. Several possible routes to induce topological superconductivity have been proposed, including proximity efforts, doping or pressurizing a topological insulator or semimetal. Among them, the pressurizing is considered to be a clean way to tune the electronic structures. Here we report the discovery of a pressure-induced topological and superconducting phase of SnSe, a material which is highly focused recently due to its superior thermoelectric properties. In situ high-pressure electrical transport and synchrotron X-ray diffraction measurements show that the superconductivity emerges along with the formation of a CsCl-type structural symmetry of SnSe above around 27 GPa, with a maximum critical temperature of 3.2 K at 39 GPa. Based on ab initio calculations, this CsCl-type SnSe is predicted to be a Dirac line nodes (DLN) semimetal in the absence of spin-orbit coupling, whose DLN states are protected by the coexistence of time-reversal and inversion symmetries. These results make CsCl-type SnSe an interesting model platform with simple crystal symmetry to study the interplay of topological physics and superconductivity.

Topological phase transition under pressure in the topological nodal-line superconductor PbTaSe2

A first-order-like resistivity hysteresis is induced by a subtle structural transition under hydrostatic pressure in the topological nodal-line superconductor PbTaSe$_2$. This structure transition is quickly suppressed to zero at pressure $\sim$0.25 GPa. As a result, superconductivity shows a marked suppression, accompanied with fundamental changes in the magnetoresistance and Hall resistivity, suggesting a Lifshitz transition around $\sim$0.25 GPa. The first principles calculations show that the spin-orbit interactions partially gap out the Dirac nodal line around $K$ point in the Brillouin zone upon applying a small pressure, whilst the Dirac states around $H$ point are completely destroyed. The calculations further reveal a second structural phase transition under a pressure as high as $\sim$30 GPa, through which a transition from a topologically nontrivial phase to a trivial phase is uncovered, with a superconducting dome emerging under this high-pressure phase.

Classical spin spirals in frustrated magnets from free-fermion band topology

The formation of coplanar spin spirals is a common motif in the magnetic ordering of many frustrated magnets. For classical antiferromagnets, geometric frustration can lead to a massively degenerate ground state manifold of spirals whose propagation vectors can be described, depending on the lattice geometry, by points (triangular), lines (fcc), surfaces (frustrated diamond) or completely flat bands (pyrochlore). Here we demonstrate an exact mathematical correspondence of these spiral manifolds of classical antiferromagnets with the Fermi surfaces of free-fermion band structures. We provide an explicit lattice construction relating the frustrated spin model to a corresponding free-fermion tight-binding model. Examples of this correspondence relate the 120$^\circ$ order of the triangular lattice antiferromagnet to the Dirac nodal structure of the honeycomb tight-binding model or the spiral line manifold of the fcc antiferromagnet to the Dirac nodal line of the diamond tight-binding model. We discuss implications of topological band structures in the fermionic system to the corresponding classical spin system.

Dirac nodal lines and induced spin Hall effect in metallic rutile oxides

We have found Dirac nodal lines (DNLs) in the band structures of metallic rutile oxides IrO$_2$, OsO$_2$, and RuO$_2$ and revealed a large spin Hall conductivity contributed by these nodal lines, which explains a strong spin Hall effect (SHE) of IrO$_2$ discovered recently. Two types of DNLs exist. The first type forms DNL networks that extend in the whole Brillouin zone and appears only in the absence of spin-orbit coupling (SOC), which induces surface states on the boundary. Because of SOC-induced band anti-crossing, a large intrinsic SHE can be realized in these compounds. The second type appears at the Brillouin zone edges and is stable against SOC because of the protection of nonsymmorphic symmetry. Besides reporting new DNL materials, our work reveals the general relationship between DNLs and the SHE, indicating a way to apply Dirac nodal materials for spintronics.

Topological nodal line semimetals predicted from first-principles calculations

Topological semimetals are newly discovered states of quantum matter, which have extended the concept of topological states from insulators to metals and attracted great research interest in recent years. In general, there are three kinds of topological semimetals, namely Dirac semimetals, Weyl semimetals, and nodal line semimetals. Nodal line semimetals can be considered as precursor states for other topological states. For example, starting from such nodal line states, the nodal line structure might evolve into Weyl points, convert into Dirac points, or become a topological insulator by introducing the spin–orbit coupling (SOC) or mass term. In this review paper, we introduce theoretical materials that show the nodal line semimetal state, including the all-carbon Mackay–Terrones crystal (MTC), anti-perovskite Cu3PdN, pressed black phosphorus, and the CaP3 family of materials, and we present the design principles for obtaining such novel states of matter.

Boosting the discovery of 3D topological materials: mixing chemistry with physics via a two-step computational screening strategy

Topological materials in crystal solids, including topological insulators (TIs), topological crystalline insulators (TCIs), topological Dirac semimetals (DSMs), topological Weyl semimetals (WSMs), topological Dirac or Weyl nodal line semimetals (NLSMs) and beyond, are mainly featured with topological, protected non-trivial surface states, and their bulk phases are insulators or semimetals with the proper presence of Dirac cones, Weyl nodes or Dirac nodal lines around the Fermi level. The author suggests a two-step computational screening strategy of 3D topological materials by mixing chemistry with physics with the considerations of fully filled bands and band inversion.

Dirac Nodal Lines and Tilted Semi-Dirac Cones Coexisting in a Striped Boron Sheet.

The enchanting Dirac fermions in graphene stimulated us to seek other 2D Dirac materials, and boron monolayers may be a good candidate. So far, a number of monolayer boron sheets have been theoretically predicted, and three have been experimentally prepared. However, none of intrinsic sheets possess Dirac electrons near the Fermi level. Herein, by means of density functional theory computations, we identified a new boron monolayer, namely, hr-sB, with two types of Dirac fermions coexisting in the sheet: One type is related to Dirac nodal lines traversing Brillouin zone (BZ) with velocities approaching 106 m/s, and the other is related to tilted semi-Dirac cones with strong anisotropy. This newly predicted boron monolayer consists of hexagon and rhombus stripes. With an exceptional stability comparable to the experimentally achieved boron sheets, it is rather optimistic to grow hr-sB on some suitable substrates such as the Ag (111) surface.

Dirac line nodes and effect of spin-orbit coupling in the nonsymmorphic critical semimetals MSiS(M=Hf,Zr)

Topological Dirac semimetals (TDSs) represent a new state of quantum matter recently discovered that offers a platform for realizing many exotic physical phenomena. A TDS is characterized by the linear touching of bulk (conduction and valance) bands at discrete points in the momentum space (i.e. 3D Dirac points), such as in Na3Bi and Cd3As2. More recently, new types of Dirac semimetals with robust Dirac line-nodes (with non-trivial topology or near the critical point between topological phase transitions) have been proposed that extends the bulk linear touching from discrete points to 1D lines. In this work, using angle-resolved photoemission spectroscopy (ARPES), we explored the electronic structure of the non-symmorphic crystals MSiS (M=Hf, Zr). Remarkably, by mapping out the band structure in the full 3D Brillouin Zone (BZ), we observed two sets of Dirac line-nodes in parallel with the kz-axis and their dispersions. Interestingly, along directions other than the line-nodes in the 3D BZ, the bulk degeneracy is lifted by spin-orbit coupling (SOC) in both compounds with larger magnitude in HfSiS. Our work not only experimentally confirms a new Dirac line-node semimetal family protected by non-symmorphic symmetry, but also helps understanding and further exploring the exotic properties as well as practical applications of the MSiS family of compounds.

First Brillouin Polytope and Band Structure of Diamond Lattice in Four Dimensions

We study the diamond lattice in four dimensions — a descendant of the three-dimensional diamond lattice. As a four-dimensional polytope, we determine the first Brillouin zone and draw the band structure of the corresponding tight-binding model on two-dimensional paper in the usual manner. In the polyhedral decomposition, we find the zone boundary of the first Brillouin zone in four dimensions to be the omnitruncated 5-cell, which comprises ten truncated octahedra glued to 20 hexagonal prisms. We find Dirac line nodes inside the hexagonal prisms.

PT-Symmetric Real Dirac Fermions and Semimetals.

Recently, Weyl fermions have attracted increasing interest in condensed matter physics due to their rich phenomenology originated from their nontrivial monopole charges. Here, we present a theory of real Dirac points that can be understood as real monopoles in momentum space, serving as a real generalization of Weyl fermions with the reality being endowed by the PT symmetry. The real counterparts of topological features of Weyl semimetals, such as Nielsen-Ninomiya no-go theorem, 2D subtopological insulators, and Fermi arcs, are studied in the PT symmetric Dirac semimetals and the underlying reality-dependent topological structures are discussed. In particular, we construct a minimal model of the real Dirac semimetals based on recently proposed cold atom experiments and quantum materials about PT symmetric Dirac nodal line semimetals.

Crystal growth of Dirac semimetal ZrSiS with high magnetoresistance and mobility

High quality single crystal ZrSiS as a theoretically predicted Dirac semimetal has been grown successfully using a vapor phase transport method. The single crystals of tetragonal structure are easy to cleave into perfect square-shaped pieces due to the van der Waals bonding between the sulfur atoms of the quintuple layers. Physical property measurement results including resistivity, Hall coefficient (RH), and specific heat are reported. The transport and thermodynamic properties suggest a Fermi liquid behavior with two Fermi pockets at low temperatures. At T = 3 K and magnetic field of Hǁc up to 9 Tesla, large magneto-resistance up to 8500% and 7200% for Iǁ(100) and Iǁ(110) were found. Shubnikov de Haas (SdH) oscillations were identified from the resistivity data, revealing the existence of two Fermi pockets at the Fermi level via the fast Fourier transform (FFT) analysis. The Hall coefficient (RH) showed hole-dominated carriers with a high mobility of 3.05 × 104 cm2V−1s−1 at 3 K. ZrSiS has been confirmed to be a Dirac semimetal by the Dirac cone mapping near the X-point via angle resolved photoemission spectroscopy (ARPES) with a Dirac nodal line near the Fermi level identified using scanning tunneling spectroscopy (STS).

Topological Dirac nodal lines and surface charges in fcc alkaline earth metals

In nodal-line semimetals, the gaps close along loops in k space, which are not at high-symmetry points. Typical mechanisms for the emergence of nodal lines involve mirror symmetry and the π Berry phase. Here we show via ab initio calculations that fcc calcium (Ca), strontium (Sr) and ytterbium (Yb) have topological nodal lines with the π Berry phase near the Fermi level, when spin–orbit interaction is neglected. In particular, Ca becomes a nodal-line semimetal at high pressure. Owing to nodal lines, the Zak phase becomes either π or 0, depending on the wavevector k, and the π Zak phase leads to surface polarization charge. Carriers eventually screen it, leaving behind large surface dipoles. In materials with nodal lines, both the large surface polarization charge and the emergent drumhead surface states enhance Rashba splitting when heavy adatoms are present, as we have shown to occur in Bi/Sr(111) and in Bi/Ag(111).