Topological semimetal phases in two-dimensional (2D) materials have gained widespread interest due to their potential applications in developing nanoscale devices. Despite the prediction of the Dirac/Weyl points in a wide variety of 2D candidates, materials featuring topological nodal lines are still in great scarcity. Herein, we predict two stable thinnest films of aluminum diboride with hyper- and hypo-stoichiometries of Al$_2$B$_2$ and AlB$_4$ as new 2D nonmagnetic Dirac nodal line semimetals (NLSMs) which promise to offer many novel features. Our elaborate electronic structure calculations combined with analytical studies reveal that, in addition to the multiple Dirac points, these 2D configurations host various type-I closed nodal lines (NLs) around the Fermi level, all of which are semimetal states protected by the time-reversal and in-plane mirror symmetries. The most intriguing NL in Al$_2$B$_2$ encloses the K point and crosses the Fermi level with a considerable dispersion, thus providing a fresh playground to explore exotic properties in dispersive Dirac nodal lines. More strikingly, in the case of 2D superconductor AlB$_4$ which exhibits a high transition temperature, we provide the first evidence for a set of 2D nonmagnetic open type-II NLs in weak spin-orbit coupling limit, coinciding with closed type-I NLs near the Fermi level. The coexistence of superconductivity and nontrivial band topology in AlB$_4$ not only makes it a promising material to exhibit novel topological superconducting phases, but also the rather large energy dispersion of type-II nodal lines in this configuration, may offer a distinguished platform for realization of novel topological features in two-dimensional limit.
We theoretically study the electric polarization in magnetic topological nodal semimetal thin films. In magnetically doped topological insulators, topological nodal semimetal phases emerge once the exchange coupling overcomes the band gap. Changing the magnetization direction, nodal structure is modulated and the system becomes topological nodal point or line semimetals. We find that nodal line semimetals are characterized by non-linear electric polarization, which is not observed in nodal point semimetals. The non-linear response originates from the existence of the surface states. Screening …
Hosting unique drumhead surface states enclosed by nodal lines, topological nodal-line semimetals exhibit novel transport phenomena. Thus, the exploration of topological semimetals with different nodal-line structures has attracted much attention. In this paper, we first find a second-order nodal line semimetal (SONLS), which has coexisting hinge Fermi arcs and drumhead surface states, in a $\mathcal{PT}$-symmetry broken system. Then, without changing the intrinsic parameters, we artificially create exotic hybrid-order nodal-line semimetals hosted by different quasienergy gaps and rich nodal-l…
We study a topological phase diagram of a ferromagnetic topological nodal semimetal. We consider a lattice model for three-dimensional topological insulators with ferromagnetic ordering. The exchange coupling between the magnetization and the electron spin leads to the nodal band structure. The topology of the nodal band structure depends on the direction of the magnetization and both the Weyl points and the nodal line emerge. We find that the nodal line structure is stable under an easy-plane magnetization in appropriate model parameters. In this case, the nodal line phase emerges as a phase …
A nodal-line semimetal is a topological gapless phase containing one-dimensional degeneracies called nodal lines. The nodal lines are deformed by a continuous change of the system such as pressure and they can even change their topology, but it is not systematically understood what kind of changes of topology of nodal lines are possible. In this paper, we classify the events of topology change of nodal lines by the Morse theory and reveal that only three types of topology changes of nodal lines, i.e., creation, reconnection, and annihilation, are possible in the spinless nodal-line semimetal p…
We theoretically show that IV-VI semiconducting compounds with low-temperature rhombohedral crystal structure represent a new potential platform for topological semimetals. By means of minimal $\mathbf{k}\cdot\mathbf{p}$ models we find that the two-step structural symmetry reduction of the high-temperature rocksalt crystal structure, comprising a rhombohedral distortion along the [111] direction followed by a relative shift of the cation and anion sublattices, gives rise to topologically protected Weyl semimetal and nodal line semimetal phases. We derive general expressions for the nodal featu…
Nodal-line semimetals with magnetic orders have been theoretically predicted and experimentally observed in only few compounds. We theoretically explore the electronic structure in bulk and boundary of such a magnetic nodal-line state by introducing magnetism in topological Dirac semimetal (TDSM). TDSMs, such as $\mathrm{Cd_3 As_2}$ and $\mathrm{Na_3 Bi}$, are characterized by a pair of spin-degenerate Dirac points protected by rotational symmetries of crystals. By introducing local magnetic moments coupled to the electron spins in the lattice model of TDSM, we show that the TDSM can turn into…
In this work, we investigate collective modes in a nodal line semimetal with two nodal-lines that have opposite spin polarization in the presence of spin population imbalance. We find the components of polarization operator taking into account the electron-electron exchange interaction and obtain the dispersion relations of collective modes for the bulk and surface states. There exist four modes in the bulk, among which one is gapless and other three are gapped. The gapless surface mode is sensitive to the boundary conditions.
We study the transport properties in nodal line semimetals with short-ranged impurity potentials at zero temperature. By computing the Drude conductivity and the corrections from the interference of particle and hole trajectories, we find that the electrons are localized in directions both parallel and perpendicular to the plane of nodal ring. We further calculate the conductivity in a weak magnetic field, and find that the perpendicular magnetic field totally destroys the weak localization gives a positive quantum interference correction, which is similar to the result in Weyl semimetals. But…
We study an atomic chain in the presence of modulated charge potential and modulated Rashba spin-orbit coupling (RSOC) of equal period. We show that for commensurate periodicities $λ=4 n$ with integer $n$, the three-dimensional synthetic space obtained by sliding the two phases of the charge potential and RSOC features a topological nodal line semimetal protected by an antiunitary particle-hole symmetry. The location and shape of the nodal lines strongly depend on the relative amplitude between the charge potential and RSOC.
Creating and manipulating topological states is a key goal of condensed matter physics. Periodic driving offers a powerful method to manipulate electronic states, and even to create topological states in solids. Here, we investigate the tunable Floquet states in a periodically driven higher-order nodal line semimetal with both spatial inversion and time-reversal symmetries. We found that the Floquet Weyl semimetal states, which support both one-dimensional hinge Fermi arc and two-dimensional surface Fermi arc states, can be induced in the higher-order nodal-line semimetal by shining circularly…
Spatial modulations, such as multilayers, to realize topological materials have recently been studied in theoretical and experimental works. In this paper, we investigate properties of the multilayers of the nodal-line semimetal (NLS) and the normal insulator. We consider two types of multilayers, with the stacking direction being perpendicular or parallel to the plane where the nodal line lies. In particular, we show that when the stacking direction is parallel to the plane, the nodal lines remain but they change their shapes because of the folding of the Brillouin zone. We also study the mul…
We investigate the electron spectra, Fermi surfaces and their characteristics near crossing points of two band-contact lines in nodal-line semimetals. In particular, the extremal cross-sectional areas, and the appropriate cyclotron masses are calculated. We also find the phase of the quantum oscillations associated with the electron orbits near the crossing point. The analysis of all these quantities is carried out both without and with consideration of the spin-orbit interaction. To illustrate the obtained results, we apply them to ZrSiS in which the crossing of the nodal lines occurs.
We investigate the low-temperature thermoelectric properties of three-dimensional nodal-line semimetals within the semiclassical Boltzmann formalism. Considering short-range interaction between electrons and scattering agents, we calculate the anisotropic relaxation times and then obtain the charge conductivity and thermopower along the radial and the axial directions with respect to the nodal-line plane. Increasing the carrier concentration, a topological transition in the shape of the Fermi surface from a torus into an ellipsoid signal as a sharp change in the thermopower. An adequate treatm…
Close to the Fermi energy, nodal loop semimetals have a torus-shaped, strongly anisotropic Fermi surface which affects their transport properties. Here we investigate the non-equilibrium dynamics of nodal loop semimetals by going beyond linear response and determine the time evolution of the current after switching on a homogeneous electric field. The current grows monotonically with time for electric fields perpendicular to the nodal loop plane however it exhibits non-monotonical behavior for field orientations aligned within the plane. After an initial non-universal growth $\sim Et$, the cur…
Chiral topological insulating phases from three-dimensional nodal loop semimetals
我们确定了一个拓扑Z指数的三维手征绝缘体的P*T对称性，其中两个哈密顿项定义了一个节环。这样的系统可能属于AIII或DIII对称类。Z不变量是分配给节环的缠绕数，并且对应于节环与间隙项的零点之间的几何关系。开边界条件下的狄拉克锥边缘态与节点环的缠绕数相对应。我们验证了我们的方法与低能量的有效哈密顿量的三维材料的拓扑绝缘体的Bi $_2$ Te $_3$家庭。
Among different topological and related phases of condensed matter, nodal semimetals occupy a special place - the electronic band topology in these materials is related to three-dimensional bulk, rather than to surface, states. A great variety of different realizations of electronic band crossings (the nodes) leads to a plethora of different electronic properties, ranging from the chiral anomaly to solid-state realizations of a black-hole horizon. The different nodal phases have similar low-energy band structure and quasiparticle dynamics, which both can be accessed experimentally by a number …
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 participat…
For conventional topological phases, the boundary gapless modes are determined by bulk topological invariants. Based on developing an analytic method to solve higher-order boundary modes, we present $PT$-invariant $2$D topological insulators and $3$D topological semimetals that go beyond this bulk-boundary correspondence framework. With unchanged bulk topological invariant, their first-order boundaries undergo transitions separating different phases with second-order-boundary zero-modes. For the $2$D topological insulator, the helical edge modes appear at the transition point for two second-or…
In this work, we propose a ferromagnetic Bi$_2$Se$_3$ as a candidate to hold the coexistence of Weyl- and nodal-line semimetal phases, which breaks the time reversal symmetry. We demonstrate that the type-I Weyl semimetal phase, type-I-, type-II- and their hybrid nodal-line semimetal phases can arise by tuning the Zeeman exchange field strength and the Fermi velocity. Their topological responses under U(1) gauge field are also discussed. Our results raise a new way for realizing Weyl and nodal-line semimetals and will be helpful in understanding the topological transport phenomena in three-dim…
We study the electronic structure and topological properties of monolayer and ABC-stacked multilayer of graphyne-$N$, which are a family of planar carbon sheets consisting of $sp$ and $sp_2$-bonding. By using the density-functional theory and the effective continuum model, we find a striking even-odd effect in the dependence of the band structure on $N$ (the number of carbon-carbon triple bonds between neighboring benzene rings). Specifically, even-$N$ graphyne monolayer has doubly-degenerate conduction and valence bands near the Fermi energy, and in its ABC multilayer, the band inversion of t…
Recently, topological quantum states of non-Hermitian systems, exhibiting rich new exotic states, have attracted great attention in condensed-matter physics. As for the demonstration, most of non-Hermitian topological phenomena previously focused on are in one- and two-dimensional systems. Here, we investigate three-dimensional non-Hermitian nodal-line semimetals in the presence of a particle gain-and-loss perturbation. It is found that this perturbation will split the original nodal ring into two exceptional rings (ERs). The topological nature of the bulk electronic structure is characterized…
We consider the impact of disorder on the spectrum of three-dimensional nodal-line semimetals. We show that the combination of disorder and a tilted spectrum naturally leads to a non-Hermitian self-energy contribution that can split a nodal line into a pair of exceptional lines. These exceptional lines form the boundary of an open and orientable bulk Fermi ribbon in reciprocal space on which the energy gap vanishes. We find that the orientation and shape of such a disorder-induced bulk Fermi ribbon is controlled by the tilt direction and the disorder properties, which can also be exploited to …
We investigate the dynamical conductivity in multiply-degenerate point-nodal semimetal CoSi. In the semimetal, the band structure holds point nodes at the $Γ$ and R points in the Brillouin zone and more than three bands touch at the nodes. Around the nodes, electronic states are predicted to be described as the multifold chiral fermion, a new class of fermion. We show that the dynamical conductivity exhibits a characteristic spectrum corresponding to the band structure and the chiral fermionic states. The dynamical conductivity of CoSi is calculated as a function of photon energy by using the …
We investigate the optical activity of tilted nodal loop semimetals. We calculate the full conductivity matrix for a band structure containing a nodal loop with possible tilt in the $x-y$ plane, which allows us to study the Kerr rotation and ellipticity both for a thin film and a bulk material. We find signatures in the Kerr signal that give direct information about the tilt velocity and direction, the radius of the nodal loop and the internal chemical potential of the system. These findings should serve as guide to understanding optical measurements of nodal loop semimetals and as an addition…
