Nodal Loop Research Articles

Spin and charge transport in topological nodal-line semimetals

We study transport properties of topological Weyl nodal-line semimetals(NLSs). Starting from a minimal lattice model with a single nodal loop, and by focusing on a normal-metal-NLS-normal-metal junction, we investigate the dependence of the novel transport behavior on the orientation of the nodal loop. When the loop is parallel to the junction interfaces, the transmitted current is found to be nearly fully spin-polarized. Correspondingly, there exists a spin orientation, along which the incident electrons would be totally reflected. An unusual resonance of half transmission with the participation of surface states also occurs for a pair of incident electrons with opposite spin orientations. All these phenomena have been shown to originate from the existence of a single forward-propagating mode in the NLS of the junction, and argued to survive in more generic multi-band Weyl NLSs.

Photoemission Spectroscopic Evidence for the Dirac Nodal Line in the Monoclinic Semimetal SrAs_{3}.

Topological nodal-line semimetals with exotic quantum properties are characterized by symmetry-protected line-contact bulk band crossings in the momentum space. However, in most of identified topological nodal-line compounds, these topological nontrivial nodal lines are enclosed by complicated topological trivial states at the Fermi energy (E_{F}), which would perplex their identification and hinder further applications. Utilizing angle-resolved photoemission spectroscopy and first-principles calculations, we provide compelling evidence for the existence of Dirac nodal-line fermions in the monoclinic semimetal SrAs_{3}, which possesses a simple nodal loop in the vicinity of E_{F} without the distraction from complicated trivial Fermi surfaces. Our calculations revealed that two bands with opposite parities were inverted around Y near E_{F}, resulting in the single nodal loop at the Γ-Y-S plane with a negligible spin-orbit coupling effect. The band crossings were tracked experimentally and the complete nodal loop was identified quantitatively, which provide a critical experimental support for the existence of nodal-line fermions in the CaP_{3} family of materials. Hosting simple topological nontrivial bulk electronic states around E_{F} and without complication from the trivial states, SrAs_{3} is expected to be a potential platform for topological quantum state investigation and applications.

Phononic Weyl nodal straight lines in MgB2

Based on first-principles calculations, we predict that the superconducting MgB$_2$ with a AlB$_2$-type centrosymmetric lattice host the so-called phononic topological Weyl nodal lines (PTWNLs) on its bulk phonon spectrum. These PTWNLs can be viewed as countless Weyl points (WPs) closely aligned along the straight lines in the $-$H-K-H direction within the three-dimensional Brillouin zone (BZ). Their topological non-trivial natures are confirmed by the calculated Berry curvature distributions on the planes perpendicular to these lines. These lines are highly unique, because they exactly locate at the high-symmetry boundary of the BZ protected by the mirror symmetry and, simultaneously, straightly transverse the whole BZ, in different from known classifications including nodal rings, nodal chains or nets, and nodal loops. On the (10$\bar{1}$0) crystal surface, the PTWNLs-induced drumhead-like non-trivial surface states appear within the rectangular area confined by the projected lines of the PTWNLs with opposite chirality. Moreover, when the mirror symmetry is broken, the double-degenerate PTWNLs are further lifted to form a pair WPs with opposite chirality. Our results pave the ways for future experimental study on topological phonons on MgB$_2$ and highlights similar results in a series of isostructural AlB$_2$-type metallic diborides.

Glide Mirror Plane Protected Nodal-Loop in an Anisotropic Half-Metallic MnNF Monolayer.

Two-dimensional (2D) nodal-loop (NL) semimetals have attracted tremendous attention for their abundant physics and potential device applications, whereas the realization of gapless NL semimetals robust against spin-orbit coupling (SOC) remains a big challenge. Recently, breakthroughs have been made with the realization of gapless NL semimetals in 2D half-metallic materials, where NLs were protected by a horizontal mirror plane symmetry. Here we first propose an alternative nonsymmorphic horizontal glide mirror plane symmetry which could protect the NLs in 2D materials. On the basis of comprehensive first-principles calculations and symmetry analysis, we found that the glide mirror symmetry together with intrinsic out-of-plane spin polarization can protect the NL against SOC in a half-metallic semimetal, namely, the MnNF monolayer. Moreover, we predict that the MnNF monolayer has strong anisotropic characteristics, tunable band structure by changing the magnetization direction, and 100% spin-polarized transport properties. Our work not only provides a novel 2D half-metallic semimetal with strong anisotropy but also broadens the scope of 2D nodal-loop materials.

Role of Velocity Field and Principal Axis of Tilted Dirac Cones in Effective Hamiltonian of Non-Coplanar Nodal Loop

Nodal line in single-component molecular conductor [Pd(dddt)_2] has been examined to understand the tilted Dirac cone on the non-coplanar loop. In the previous work [J. Phys. Soc. Jpn. 87, 113701 (2018)], the velocity of the cone was calculated at respective Dirac points on the nodal loop based on our first-principles band structure calculations, which was a new method to derive an effective Hamiltonian with a 2 x 2 matrix. However, the Dirac cones on the nodal line are fully reproduced only at symmetric points. In the present paper, we show that our improved method well reproduces reasonable behaviors of all the Dirac cones and a very small energy dispersion of 6~meV among the Dirac points. The variation of velocities along the nodal line are shown by using principal axes of the gap function between the conduction and valence bands. Further, the density of states close to the chemical potential and orbital magnetic susceptibility are calculated using such an effective Hamiltonian.

Si-Cmma: A silicon thin film with excellent stability and Dirac nodal loop

Silicon thin film with both remarkable stability and graphenelike properties attract many research interests in materials science and condensed matter physics. In this paper, two new two-dimensional silicon allotropes (Si-Cmma and Si-Pmma) with excellent energetic stability and positive dynamical stability are proposed and investigated by first-principles calculations. Si-Pmma possesses the energy of about 203 meV/atom lower than that of silicene and it is a normal metallic phase. Si-Cmma is of about 198 meV/atom lower than silicene in energy and it is confirmed to be a nodal loop semimetallic material. Symmetry analysis reveals that the nodal loop in Si-Cmma is protected by the glide mirror and it belongs to class AI $+$ R of Altland-Zirnbauer symmetry classes. A further three-band $k\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{p}$ model is constructed to understand the formation of the nodal loop and its strong anisotropy.

Fully spin-polarized open and closed nodal lines in β -borophene by magnetic proximity effect

Recently, the experimental realization of Dirac fermions in ${\ensuremath{\beta}}_{12}$- and ${\ensuremath{\chi}}_{3}$-boron sheets has attracted tremendous attention and simulated a broad exploration for the novel topologically nontrivial states in two-dimensional materials. Herein, by combining first-principles and tight-binding calculations, we discover a coexistence of open nodal arc and closed nodal loop produced by three-band touching in the recently synthesized $\ensuremath{\beta}$-boron sheet [Q. Zhong et al., Phys. Rev. Mater. 1, 021001 (2017)]. The symmetry analysis reveals that the nodal lines are protected by both time-reversal symmetry (TRS) and mirror-reflection symmetry (MRS). Intriguingly, when TRS is broken, e.g., via introducing magnetic proximity effect, a topological phase transition occurs, namely, the original spin-degenerate nodal line (loop) decays into fully spin-polarized nodal line (loop). Furthermore, when Rashba spin-orbit coupling (SOC) is turned on to break the MRS, spin-up and -down states are shifted toward opposite momentum directions, leading to the emergence of two new nodal loops around the $\mathrm{\ensuremath{\Gamma}}$ point. In addition, inclusion of both Rashba SOC and exchange field leads to band-gap opening of the Dirac points and nodal lines. Our findings not only reveal a form of nodal line in the $\ensuremath{\beta}$-boron sheet, but also offer an alternative approach to realize spin-polarized nodal line semimetals with promising applications in spintronic devices.

Signature of odd-frequency equal-spin triplet pairing in the Josephson current on the surface of Weyl nodal loop semimetals

We theoretically predict proximity-induced odd-frequency (odd-$\omega$) pairing on the surface of a Weyl nodal loop semimetal, characterized by a nodal loop Fermi surface and drumhead-like surface states (DSSs), attached to conventional spin-singlet $s$-wave superconducting leads. Due to the complete spin polarization of the DSS, odd-$\omega$ equal-spin triplet pairing is present, and we show that it gives rise to a finite Josephson current. Placing an additional ferromagnet in the junction can also generate odd-$\omega$ mixed-spin triplet pairing, but the pairing and current is not affected if the magnetization is orthogonal to the DSS spin-polarization, which further confirms the equal-spin structure of the pairing.

Topological hybrid nodal-loop semimetal in a carbon allotrope constructed by interconnected Riemann surfaces

We predict by ab initio calculations an $\mathrm{all}\ensuremath{-}s{p}^{2}$ nanostructured carbon phase of a topological hybrid nodal-loop semimetal with momentum-dependent crossings of linear bands near the Fermi level. This carbon phase is assembled from interconnected Riemann surfaces, naturally occurring as screw dislocations in graphitic carbons, which not only result in notably higher stability than most previously reported allotropes but also sustain a strain-robust nodal loop up to 75% elastic stretch due to mirror symmetry. The design strategy, based on the elegant topology of Riemann surfaces, can yield a family of topological carbon materials with fascinating mechanical and electronic properties.

Dirac nodal lines and large spin Hall effect in the 6H -perovskite iridate Ba3TiIr2O9

Nodal line semimetals with symmetry protected band structure offer a platform for the investigation of a number of emerging quantum phenomena. Using first-principles calculation combined with symmetry analysis, we show that ${\mathrm{Ba}}_{3}{\mathrm{TiIr}}_{2}{\mathrm{O}}_{9}$ hosts a Dirac nodal line (DNL) along the $A\text{\ensuremath{-}}L$ direction of the Brillouin zone, protected by the glide reflection symmetry. In the presence of the spin-orbit coupling, even though the DNL along the $A\text{\ensuremath{-}}L$ direction is protected, DNLs along other directions as well as multiple nodal loops present in the system gap out. The gapped out Dirac nodal loops act as a source of spin Berry curvature, resulting in a large spin Hall conductivity ($\ensuremath{\approx}300$ $\frac{\ensuremath{\hbar}}{e}\phantom{\rule{4pt}{0ex}}{\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}{\phantom{\rule{0.16em}{0ex}}\mathrm{cm}}^{\ensuremath{-}1}$). This suggests the possible application of the material as a spin current detector.

Two-dimensional nodal-loop half-metal in monolayer MnN

Two-dimensional (2D) materials with nodal-loop band crossing have been attracting great research interest. However, it remains a challenge to find 2D nodal loops that are robust against spin-orbit coupling (SOC) and realized in magnetic states. Here, based on first-principles calculations and theoretical analysis, we predict that monolayer MnN is a 2D nodal-loop half metal with fully spin polarized nodal loops. We show that monolayer MnN has a ferromagnetic ground state with out-of-plane magnetization. Its band structure shows half metallicity with three low-energy bands belonging to the same spin channel. The crossing between these bands forms two concentric nodal loops centered around the $\Gamma$ point near the Fermi level. Remarkably, the nodal loops and their spin polarization are robust under SOC, due to the protection of a mirror symmetry. We construct an effective model to characterize the fully polarized emergent nodal-loop fermions. We also find that a uniaxial strain can induce a loop transformation from a localized single loop circling around $\Gamma$ to a pair of extended loops penetrating the Brillouin zone.

Tuning nodal line semimetals in trilayered systems

We investigate two-dimensional trilayered quantum systems with multi-orbital conduction bands by focusing on the role played by the layer degree of freedom in setting the character of nodal line semimetals. The layer index can label the electronic states where the electrons reside in the unit cell and can enforce symmetry constraints in the electronic structure by protecting bands crossing. We demonstrate that both the atomic spin-orbit coupling and the removal of local orbital degeneracy can lead to different types of electronic transitions with nodal lines that undergo a changeover from a loop structure enclosing the center of the Brillouin zone to pockets winding around multiple high symmetry points. We introduce and employ a criterion to find the nodal lines transitions. On the basis of a zero-dimensional topological invariant that, for a selected electronic and energy manifold, counts the number of bands below the Fermi level with a given layer inversion eigenvalue in high symmetry points of the Brillouin zone, one can determine the structure of the nodal loops and the ensuing topological transitions.

Evidence for bulk nodal loops and universality of Dirac-node arc surface states in ZrGeXc ( Xc=S , Se, Te)

We have performed angle-resolved photoemission spectroscopy (ARPES) on layered ternary compounds ZrGeXc (Xc = S, Se, and Te) with square Ge lattices. ARPES measurements with bulk-sensitive soft-x-ray photons revealed a quasi-two-dimensional bulk-band structure with the bulk nodal loops protected by glide mirror symmetry of the crystal lattice. Moreover, high-resolution ARPES measurements near the Fermi level with vacuum-ultraviolet photons combined with first-principles band-structure calculations elucidated a Dirac-node-arc surface state traversing a tiny spin-orbit gap associated with the nodal loops. We found that this surface state commonly exists in ZrGeXc despite the difference in the shape of nodal loops. The present results suggest that the spin-orbit coupling and the multiple nodal loops cooperatively play a key role in creating the exotic Dirac-node-arc surface states in this class of topological line-node semimetals.

Fully Spin-Polarized Nodal Loop Semimetals in Alkaline Metal Monochalcogenide Monolayers.

Topological semimetals in ferromagnetic materials have attracted an enormous amount of attention due to potential applications in spintronics. Using first-principles density functional theory together with an effective lattice model, here we present a new family of topological semimetals with a fully spin-polarized nodal loop in alkaline metal monochalcogenide MX (M = Li, Na, K, Rb, or Cs; X = S, Se, or Te) monolayers. The half-metallic ferromagnetism can be established in MX monolayers, in which one nodal loop formed by two crossing bands with the same spin components is found at the Fermi energy. This nodal loop half-metal survives even when considering the spin-orbit coupling owing to the symmetry protection provided by the mirror plane. The quantum anomalous Hall state and Weyl-like semimetal in this system can be also achieved by rotating the spin from the out-of-plane to the in-plane direction. The MX monolayers hosting rich topological phases thus offer an excellent platform for realizing advanced spintronic concepts.

Weak Localization and Antilocalization in Nodal-Line Semimetals: Dimensionality and Topological Effects.

New materials such as nodal-line semimetals offer a unique setting for novel transport phenomena. Here, we calculate the quantum correction to conductivity in a disordered nodal-line semimetal. The torus-shaped Fermi surface and encircled π Berry flux carried by the nodal loop result in a fascinating interplay between the effective dimensionality of electron diffusion and band topology, which depends on the scattering range of the impurity potential relative to the size of the nodal loop. For a short-range impurity potential, backscattering is dominated by the interference paths that do not encircle the nodal loop, yielding a 3D weak localization effect. In contrast, for a long-range impurity potential, the electrons effectively diffuse in various 2D planes and the backscattering is dominated by the interference paths that encircle the nodal loop. The latter leads to weak antilocalization with a 2D scaling law. Our results are consistent with symmetry consideration, where the two regimes correspond to the orthogonal and symplectic classes, respectively. Furthermore, we present weak-field magnetoconductivity calculations at low temperatures for realistic experimental parameters and predict that clear scaling signatures ∝sqrt[B] and ∝-lnB, respectively. The crossover between the 3D weak localization and 2D weak antilocalization can be probed by tuning the Fermi energy, giving a unique transport signature of the nodal-line semimetal.

Hourglass Weyl loops in two dimensions: Theory and material realization in monolayer GaTeI family

Nodal loops in two-dimensional (2D) systems are typically vulnerable against spin-orbit coupling (SOC). Here, we explore 2D systems with a type of doubly degenerate nodal loops that are robust under SOC and feature an hourglass type dispersion. We present symmetry conditions for realizing such hourglass Weyl loops, which involve nonsymmorphic lattice symmetries. Depending on the symmetry, the loops may exhibit different patterns in the Brillouin zone. Based on first-principles calculations, we identify the monolayer GaTeI-family materials as a realistic material platform to realize such loops. These materials host a single hourglass Weyl loop circling around a high-symmetry point. Interestingly, there is also a spin-orbit Dirac point enabled by an additional screw axis. We show that the hourglass Weyl loop and the Dirac point are robust under a variety of applied strains. By breaking the screw axis, the Dirac point can be transformed into a second Weyl loop. Furthermore, by breaking the glide mirror, the hourglass Weyl loop and the spin-orbit Dirac point can both be transformed into a pair of spin-orbit Weyl points. Our work offers guidance and realistic material candidates for exploring fascinating physics of several novel 2D emergent fermions.

High-pressure lithium as an elemental topological semimetal

Topological semimetals generally contain heavy elements. Using density-functional theoretic calculations, we predict that three dense lithium polymorphs in the pressure range 200--360 GPa display nontrivial semimetallic electronic structure. Specifically, these high-pressure phases exhibit Fermi pockets which are degenerate over a loop in $\mathbit{k}$ space, around which an encircling $\mathbit{k}$-space path is threaded by $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ Berry phase. Accordingly, these dense lithium phases are topological nodal loop semimetals involving a single light element.

Topological node line semimetal state in two-dimensional tetragonal allotrope of Ge and Sn

Realizing semimetal states in two-dimensional (2D) materials are of great importance for their future application in novel quantum devices at the nanoscale. Using first-principles calculations based on density functional theory, we propose a new 2D tetragonal allotrope of Ge and Sn, 2D T-Ge and T-Sn, which consists of repeated square and octagon rings. The calculated cohesive energy, ab initio molecular dynamic simulations and phonon dispersions indicate that 2D T-Ge and T-Sn are stable at room temperature and possibly synthesized in the experiments under appropriate growth conditions. In the absence of spin–orbital coupling (SOC), 2D T-Ge and T-Sn are node line semimetals and the nodal loop in 2D T-Ge and T-Sn are protected by the combination of the spatial inversion P and time-reversal T symmetries. Remarkably, when SOC is included, 2D T-Ge and T-Sn are still node line semimetals although small gap is opened along the nodal loop, which are identified by nontrivial Z2 invariant and topological edge states at the sample boundaries. Furthermore, our calculations show that node line semimetal states in 2D T-Ge and T-Sn are robust with the biaxial strains in the range of −3% to 3%. Our results provide a new 2D material platform for realization of 2D topological node line semimetals and innovative applications of topological node line semimetals.

Two-dimensional honeycomb borophene oxide: strong anisotropy and nodal loop transformation.

The search for topological semimetals is mainly focused on heavy-element compounds by following the footsteps of previous research on topological insulators, with less attention on light-element materials. However, the negligible spin orbit coupling with light elements may turn out to be beneficial for realizing topological band features. Here, using first-principles calculations, we propose a new two-dimensional light-element material-the honeycomb borophene oxide (h-B2O), which has nontrivial topological properties. The proposed structure is based on the recently synthesized honeycomb borophene on an Al (111) substrate [W. Li, L. Kong, C. Chen, J. Gou, S. Sheng, W. Zhang, H. Li, L. Chen, P. Cheng and K. Wu, Sci. Bull., 2018, 63, 282-286]. The h-B2O monolayer is completely flat, unlike the oxides of graphene or silicene. We systematically investigate the structural properties of h-B2O, and find that it has very good stability and exhibits significant mechanical anisotropy. Interestingly, the electronic band structure of h-B2O hosts a nodal loop centered around the Y point in the Brillouin zone, protected by the mirror symmetry. Furthermore, under moderate lattice strain, the single nodal loop can be transformed into two loops, each penetrating through the Brillouin zone. The loops before and after the transition are characterized by different [Doublestruck Z] × [Doublestruck Z] topological indices. Our work not only predicts a new two-dimensional material with interesting physical properties, but also offers an alternative approach to search for new topological phases in 2D light-element systems.

Emergence of topological phases by stacking of two-dimensional lattices with nonsymmorphic symmetry

Topological semimetals have a variety of phases, whose Fermi surfaces can be nodal points, nodal lines and nodal loops. Here we construct four classes of 3D minimal models via vertically stacking a 2D nonsymmorphic lattice with and without breaking crystalline symmetries. As a result, four distinct topological phases can be generated in our minimal model, such as Dirac nodal line semimetals, Weyl nodal line semimetals, unconventional Weyl semimetals with topological charge , and weak topological insulators. Unexpectedly, Weyl nodal loops are generated without mirror symmetry protection, where nontrivial ‘drumhead’ surface states emerge within the loops.