Time-reversal Invariant Momenta Research Articles

Zeeman splitting in doped antiferromagnetic semiconductors

Zeeman splitting of conduction electrons in doped antiferromagnetic materials in the presence of a strong spin–orbit interaction is considered. Using symmetry arguments the Hamiltonian and the spectrum of conduction electrons near some time-reversal invariant momenta were constructed. It was shown, that in the vicinity of some time-reversal invariant momenta the external magnetic field perpendicular to the Neel axis does not lead to the Zeeman splitting. The Kramers degeneracy is preserved in that case. The absence of the Zeeman splitting in antiferromagnets leads to the absence of spin zeros in quantum oscillation experiments. Recent experimental confirmation of the predicted effects in organic antiferromagnet and High-Tc Superconductors is discussed.

Rashba-like spin splitting around non-time-reversal-invariant momenta

The ubiquitous Rashba spin splitting occurring around time-reversal-invariant momenta (TRIM) reflects the fundamental interplay between spin-orbit coupling (SOC) and crystalline symmetry. Unexpectedly, recent measurements reveal novel Rashba spin splitting around the momenta lacking time-reversal symmetry (non-TRIM), whose mechanism remains elusive. Here, we theoretically elucidate the microscopic origin of Rashba spin splitting around non-TRIM and identify the design principles for its appearance. Conventionally, a Zeeman-like spin splitting dominates around non-TRIM due to the broken time-reversal symmetry and diminishes the Rashba-like features. Our results show that the Bloch wave function with specific orbital representation at certain non-TRIM could eliminate the first-order SOC effect. As a result, the corresponding little group hosts a type of quasisymmetry that commutes with the dominating SOC Hamiltonian, leading to a tiny SOC gap and helical spin texture. Furthermore, such a quasi-symmetry-induced tiny gap could manifest large Berry curvature, which is desirable for various topological applications. Our work expands the concept of Rashba physics beyond the conventional TRIM scenario, paving an avenue for designing novel spin-splitting materials through the cooperation of symmetry and orbital engineering.

Exploring strong and weak topological states on isostructural substitutions in TlBiSe_2

Topological Insulators (TIs) are unique materials where insulating bulk hosts linearly dispersing surface states protected by the Time-Reversal Symmetry. These states lead to dissipationless current flow, which makes this class of materials highly promising for spintronic applications. Here, we predict TIs by employing state-of-the-art first-principles based methodologies, viz., density functional theory and many-body perturbation theory (G_0W_0) combined with spin-orbit coupling effects. For this, we take a well-known 3D TI, TlBiSe_2 and perform complete substitution with suitable materials at different sites to check if the obtained isostructural materials exhibit topological properties. Subsequently, we scan these materials based on SOC-induced parity inversion at Time-Reversal Invariant Momenta. Later, to confirm the topological nature of selected materials, we plot their surface states along with calculation of Z_2 invariants. Our results show that GaBiSe_2 is a strong Topological Insulator, besides, we report six weak Topological Insulators, viz., PbBiSe_2, SnBiSe_2, SbBiSe_2, Bi_2Se_2, TlSnSe_2 and PbSbSe_2. We have further verified that all the reported TIs are dynamically stable, showing all real phonon modes of vibration.

Sublattice-sensitive Majorana modes

For two- and three-dimensional topological insulators whose unit cells consist of multiple sublattices, the boundary terminating at which type of sublattice can affect the time-reversal invariant momentum at which the Dirac points of helical boundary states are located. By incorporating a generic theory and a representative model, we reveal that this interesting property allows the realization of Majorana modes at sublattice domain walls forming on the boundary when the boundary Dirac points of the topological insulator are gapped by appropriate superconductivity in proximity. Remarkably, we find that the sensitive sublattice dependence of the Majorana modes allows their positions to be precisely manipulated by locally controlling the terminating sublattices or boundary potential. Our work reveals that the sublattice degrees of freedom commonly found in materials open a different route to realize and manipulate Majorana modes.

Tunable Topological Phase Transition in Two-Dimensional Ternary Transition Metal Halides TiXI (X = P and As)

With first-principles calculations we predict tunable topological phase transition in two-dimensional (2D) ternary transition metal halides α-TiXI (X = P and As) via strain engineering. Besides, changing the number of stacking layers or van der Waals interlayer spacing can also result in topological phase transition in few-layer TiXI. The on-site Coulomb U of Ti atoms is neither tunable nor empirical in this work. Instead, it is evaluated by the density functional perturbation theory and hence the results are more accurate. The tunable topological phase of 2D α-TiXI via strain engineering makes them promising in spintronics devices that exploit quantum spin Hall effect. Furthermore, the topological edge states of the single-layer TiAsI exhibit interesting feature. They do not cross at the time-reversal invariant momenta (TRIM) where the band inversion occurs. Instead, they extend over the whole one-dimensional Brillouin zone and cross at the other TRIM. While all requirements of Z2 topological phase are fulfilled, it is rare especially for those with direct gap. The linear bulk bands around and at Γ point may be the reason for the unusual crossing.

Hidden spin-orbital texture at the overline{{{Gamma }}}-located valence band maximum of a transition metal dichalcogenide semiconductor

Finding stimuli capable of driving an imbalance of spin-polarised electrons within a solid is the central challenge in the development of spintronic devices. However, without the aid of magnetism, routes towards this goal are highly constrained with only a few suitable pairings of compounds and driving mechanisms found to date. Here, through spin- and angle-resolved photoemission along with density functional theory, we establish how the p-derived bulk valence bands of semiconducting 1T-HfSe2 possess a local, ground-state spin texture spatially confined within each Se-sublayer due to strong sublayer-localised electric dipoles orientated along the c-axis. This hidden spin-polarisation manifests in a ‘coupled spin-orbital texture’ with in-equivalent contributions from the constituent p-orbitals. While the overall spin-orbital texture for each Se sublayer is in strict adherence to time-reversal symmetry (TRS), spin-orbital mixing terms with net polarisations at time-reversal invariant momenta are locally maintained. These apparent TRS-breaking contributions dominate, and can be selectively tuned between with a choice of linear light polarisation, facilitating the observation of pronounced spin-polarisations at the Brillouin zone centre for all kz. We discuss the implications for the generation of spin-polarised populations from 1T-structured transition metal dichalcogenides using a fixed energy, linearly polarised light source.

Structural, electronic and topological properties of 3D TmBi compound

Using density functional theory based methods we report the structural, electronic and topological properties of the FCC crystal compound TmBi. This material is found to be dynamically stable and shows a non magnetic semimetalic character. By tuning the spin-orbit coupling, we observe a significant change in the band structure, and the occurrence of band inversion along $\Gamma -X$ direction. The parity product at time-reversal invariant momentum points and the Wannier charge center calculations provide a topological index $Z_{2}=1$ on the $k_{j}=0$ plane (with $j=1,2$ and $3$) revealing the non trivial topological character of TmBi. The existence of topologically protected surface states of TmBi through the observation of a Dirac cones at $\bar{X}$ point confirm our finding. The present work could inspire platforms for exploring novel topological states within the family of rare-earth monobismuthides.

Topological multiband s -wave superconductivity in coupled multifold fermions

We study three-dimensional time-reversal-invariant topological superconductivity in noncentrosymmetric materials such as RhSi, CoSi, and AlPt which host coupled multifold nodes energetically split by the spin-orbit coupling at the same time-reversal-invariant momentum (TRIM). The topological superconductivity arises from the $s_{+} \oplus s_{-}$ gap function, which is $\boldsymbol{k}$ independent, but with opposite signs for the two nodes split at the same TRIM. We consider various electron-electron interactions in the tight-binding model for RhSi and find that the topological superconducting phase supporting a surface Majorana cone and topological nodal rings is favored in a wide range of interaction parameters.

On Strong f-Electron Localization Effect in a Topological Kondo Insulator

We study a strong f-electron localization effect on the surface state of a generic topological Kondo insulator (TKI) system by performing a mean-field theoretic (MFT) calculation within the framework of the periodic Anderson model (PAM) using the slave boson technique. The surface metallicity, together with bulk insulation, requires this type of localization. A key distinction between surface states in a conventional insulator and a topological insulator is that, along a course joining two time-reversal invariant momenta (TRIM) in the same BZ, there will be an intersection of these surface states, an even/odd number of times, with the Fermi energy inside the spectral gap. For an even (odd) number of surface state crossings, the surface states are topologically trivial (non-trivial). The symmetry consideration and the pictorial representation of the surface band structure obtained here show an odd number of crossings, leading to the conclusion that, at least within the PAM framework, the generic system is a strong topological insulator.

Anisotropic phonon-mediated electronic transport in chiral Weyl semimetals

Discovery and observations of exotic, quantized optical and electrical responses have sparked renewed interest in nonmagnetic chiral crystals. Within this class of materials, six group V transition metal ditetrelides, that is, XY$_2$ (X = V, Nb, Ta and Y = Si, Ge), host composite Weyl nodes on high-symmetry lines, with Kramers-Weyl fermions at time-reversal invariant momenta. In addition, at least two of these materials, NbGe$_2$ and NbSi$_2$, exhibit superconducting transitions at low temperatures. The interplay of strong electron-phonon interaction and complex Fermi surface topology present an opportunity to study both superconductivity and hydrodynamic electron transport in these systems. Towards this broader question, we present an ab initio theoretical study of the electronic transport and electron-phonon scattering in this family of materials, with a particular focus on NbGe$_2$ vs. NbSi$_2$, and the other group V ditetrelides. We shed light on the microscopic origin of NbGe$_2$'s large and anisotropic room temperature resistivity and contextualize its strong electron-phonon scattering with a presentation of other relevant scattering lifetimes, both momentum-relaxing and momentum-conserving. Our work explores the intriguing possibility of observing hydrodynamic electron transport in these chiral Weyl semimetals.

Kramers nodal line metals

Recently, it was pointed out that all chiral crystals with spin-orbit coupling (SOC) can be Kramers Weyl semimetals (KWSs) which possess Weyl points pinned at time-reversal invariant momenta. In this work, we show that all achiral non-centrosymmetric materials with SOC can be a new class of topological materials, which we term Kramers nodal line metals (KNLMs). In KNLMs, there are doubly degenerate lines, which we call Kramers nodal lines (KNLs), connecting time-reversal invariant momenta. The KNLs create two types of Fermi surfaces, namely, the spindle torus type and the octdong type. Interestingly, all the electrons on octdong Fermi surfaces are described by two-dimensional massless Dirac Hamiltonians. These materials support quantized optical conductance in thin films. We further show that KNLMs can be regarded as parent states of KWSs. Therefore, we conclude that all non-centrosymmetric metals with SOC are topological, as they can be either KWSs or KNLMs.

Fermi surfaces of the topological semimetal CaSn3 probed through de Haas van Alphen oscillations

In the search of topological superconductors, nailing down the Fermiology of the normal state is as crucial a prerequisite as unraveling the superconducting pairing symmetry. In particular, the number of time-reversal-invariant momenta (TRIM) in the Brillouin zone enclosed by Fermi surfaces is closely linked to the topological class of time-reversal-invariant systems, and can experimentally be investigated. We report here a detailed study of de Haas van Alphen quantum oscillations in single crystals of the topological semimetal CaSn3 with torque magnetometry in high magnetic fields up to 35 T. In conjunction with density functional theory based calculations, the observed quantum oscillations frequencies indicate that the Fermi surfaces of CaSn3 enclose an odd number of TRIM, satisfying one of the proposed criteria to realize topological superconductivity. Nonzero Berry phases extracted from the magnetic oscillations also support the nontrivial topological nature of CaSn3.

Dirac states in the noncentrosymmetric superconductor BiPd

Quantum materials having Dirac fermions in conjunction with superconductivity is believed to be the candidate material to realize exotic physics as well as advanced technology. Angle-resolved photoemission spectroscopy (ARPES), a direct probe of the electronic structure, has been extensively used to study these materials. However, experiments often exhibit conflicting results on dimensionality and momentum of the Dirac fermions (e.g., Dirac states in BiPd, a novel noncentrosymmetric superconductor), which is crucial for the determination of the symmetry, time-reversal invariant momenta, and other emerging properties. Employing high-resolution ARPES at varied conditions, we demonstrated a methodology to identify the location of the Dirac node accurately and discover that the deviation from two dimensionality of the Dirac states in BiPd proposed earlier is not a material property. These results helped to reveal the topology of the anisotropy of the Dirac states accurately. We have constructed a model Hamiltonian considering higher-order spin-orbit terms and demonstrate that this model provides an excellent description of the observed anisotropy. Intriguing features of the Dirac states in a noncentrosymmetric superconductor revealed in this study are expected to have significant implications regarding the properties of topological superconductors.

Kramers Weyl semimetals as quantum solenoids and their applications in spin-orbit torque devices

Kramers Weyl semimetals are Weyl semimetals that have Weyl points pinned at the time reversal invariant momenta. Recently it has been discovered that all chiral crystals host Weyl points at time reversal invariant momenta, so metals with chiral lattice symmetry all belong to the category of Kramers Weyl semimetals. In this work, we show that due to the chiral lattice symmetry, Kramers Weyl semimetals have the unique longitudinal magnetoelectric effect in which the charge current induced spin and orbital magnetization is parallel to the direction of the current. This feature allows Kramers Weyl semimetals to act as nanoscale quantum solenoids with both orbital and spin magnetization. As the moving electrons of Kramers Weyl semimetal can generate longitudinal magnetization, Kramers Weyl semimetals can be used for new designs of spin-orbit torque devices with all electric control of magnetization switching for magnets with perpendicular magnetic anisotropy.

Probing mirror anomaly and classes of Dirac semimetals with circular dichroism

We theoretically investigate the optical activity of three dimensional Dirac semimetals (DSMs) using circular dichroism (CD). We show that DSMs in the presence of a magnetic field in any one of the mirror-symmetric planes of the materials exhibit a notable dichroic behavior. In particular, for different orientations of the light field with respect to the mirror-symmetric plane, the CD in type-II DSMs can detect the presence of mirror anomaly by showing sharply distinct patterns at the mirror-symmetric angle. Interestingly, CD can also distinguish type-II DSMs having only one Dirac point at a time-reversal invariant momentum from type-I DSMs with a pair of Dirac points on the rotation axis of the crystals.

Qeirreps: An open-source program for Quantum ESPRESSO to compute irreducible representations of Bloch wavefunctions

Bloch wavefunctions in solids form a representation of crystalline symmetries. Recent studies revealed that symmetry representations in band structure can be used to diagnose the topological properties of weakly interacting materials. In this work, we introduce an open-source program qeirreps that computes the representation characters in a band structure based on the output file of Quantum ESPRESSO. Our program also calculates the Z4 index, i.e., the sum of inversion parities at all time-reversal invariant momenta, for materials with inversion symmetry. When combined with the symmetry indicator method, this program can be used to explore new topological materials. Program summaryProgram title: qeirrepsCPC Library link to program files:https://doi.org/10.17632/dfgw6wgvmt.1Licensing provisions: GNU General Public Licence 3.0Programming language: Fortran 90External routines/libraries:•BLAS (http://www/netlib.org/blas)•LAPACK (http://www.netlib.org/lapack)Nature of problem: Irreducible representations of Bloch wavefunctionsSolution method: Linear algebra calculation for Bloch wavefunctions

Expansive open Fermi arcs and connectivity changes induced by infrared phonons in ZrTe5

Expansive open Fermi arcs covering most of the surface Brillouin zone (SBZ) are desirable for detection and control of many topological phenomena, but they have generally been reported for Kramers-Weyl points, or unconventional chiral fermions, pinned at time-reversal invariant momentum in chiral materials. Here using first-principles band structure calculations, we show that for conventional Weyl points in $\mathrm{Zr}{\mathrm{Te}}_{5}$ with the chirality of +1/\ensuremath{-}1 near the BZ center at general momentum induced by one of the infrared phonons---the second lowest ${B}_{1u}$ mode for breaking inversion symmetry---they can also form expansive open Fermi arcs across the SBZ boundary to occupy most of the SBZ when projected on the (001) surface. We reveal that such expansive open Fermi arcs are evolved from the topological surface states that connect multiple surface Dirac points on the (001) surface of the topological insulator phases without lattice distortion in $\mathrm{Zr}{\mathrm{Te}}_{5}$. Furthermore, we find that the connectivity of the induced open Fermi arcs can be changed by the magnitude of the lattice distortion of this infrared phonon mode. Thus, we propose that using coherent optical phonons to modulate lattice parameters can offer ways to induce unique topological features including expansive open Fermi arcs and to dynamically control Fermi arc connectivity in $\mathrm{Zr}{\mathrm{Te}}_{5}$.

Electronic Structure of a Single-Component Molecular Conductor [Pd(dddt)2] (dddt = 5,6-dihydro-1,4-dithiin-2,3-dithiolate) under High Pressure

We examined high-pressure electronic structure of a single-component molecular conductor [Pd(dddt)$_2$] (dddt = 5,6-dihydro-1,4-dithiin-2,3-dithiolate) at room temperature, based on the crystal structure determined by single crystal synchrotron X-ray diffraction measurements at 5.9 GPa. The monoclinic unit cell contains four molecules that form two crystallographically independent molecular layers. A tight-binding model of 8 $\times$ 8 matrix Hamiltonian gives an electronic structure as a Dirac electron system. The Dirac point describes a loop within the first Brillouin zone, and a nodal line semimetal is obtained. The noticeable property of the Dirac cone with a linear dispersion is shown by calculating density of states (DOS). The Dirac cone in this system is associated with the crossing of HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) bands, which originates from the direct interaction between different molecular layers. This is a newly found mechanism in addition to the indirect one [J. Phys. Soc. Jpn., {\bf 86}, 064705 (2017)]. The Dirac points emerge as a line, when the HOMO and LUMO bands meet on the surface and the HOMO-LUMO couplings are absent. Such a mechanism is verified using a reduced model of 4 $\times$ 4 matrix Hamiltonian. The deviation of the band energy ($\delta E$) at the Dirac point from the Fermi level is very small ($\delta E < $ 0.4meV). The nodal line is examined by calculating the parity of the occupied band eigen states at TRIM (Time Reversal Invariant Momentum) showing that the topological number is 1.

Higher-Order Topological Spin Hall Effect of SoundSupported by the National Natural Science Foundation of China (Grant Nos. 11675116 and 11904060), the Jiangsu Distinguished Professor Fund, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

We theoretically propose a reconfigurable two-dimensional (2D) hexagonal sonic crystal with higher-order topology protected by the six-fold, C6, rotation symmetry. The acoustic band gap and band topology can be controlled by rotating the triangular scatterers in each unit cell. In the nontrivial phase, the sonic crystal realizes the topological spin Hall effect in a higher-order fashion: (i) the edge states emerging in the bulk band gap exhibit partial spin-momentum correlation and are gapped due to the reduced spatial symmetry at the edges. (ii) The gapped edge states, on the other hand, stabilize the topological corner states emerging in the edge band gap. The partial spin-momentum correlation is manifested as pseudo-spin-polarization of edge states away from the time-reversal invariant momenta, where the pseudospin is emulated by the acoustic orbital angular momentum. We reveal the underlying topological mechanism using a corner topological index based on the symmetry representation of the acoustic Bloch bands.

A Tight-binding Model of an Ambient-pressure Molecular Dirac Electron System with Open Nodal Lines

By deriving a tight-binding model, we demonstrate a mechanism of forming a nodal line of Dirac points in a single-component molecular conductor [Pt(dmtd)$_2$] [Zhou {\it et al.}, Chem. Commun. {\bfseries 55}, 3327 (2019)], consisting of HOMO and LUMO. The nodal line is obtained as the intersection of two surfaces, where one corresponds to the HOMO-LUMO band crossing and another is vanishing of the HOMO-LUMO couplings due to their different symmetries. The latter property is essential for the Dirac electron in molecular conductors. The nature of the open nodal line is discussed in terms of the parity of the wavefunctions at eight TRIMs (time reversal invariant momenta).