Z2 Index Research Articles

Appearance of topological phase in YAs semimetal under hydrostatic pressure and epitaxial strain

By means of hybrid density functional theory, we present the evolution of the topological phase in rare earth monopnictide YAs with hydrostatic pressure and epitaxial strain. This material exists in NaCl-type structure at ambient conditions and shows structural phase transition into CsCl-type structure at 56.54 GPa hydrostatic pressure. The epitaxial strain reduces the structure into a compressed tetragonal-type. The thermodynamical and dynamical stability of the material is established with the calculation of enthalpy and phonon band structure within structural phase transition, respectively. The topologically trivial phase of the material is observed at ambient pressure in agreement with previous reports. This material shows topological phase transition at 24.8 GPa applied hydrostatic pressure and 10% epitaxial strain. The band inversion at the X-point between d-Y and p-As orbitals is verified with the help of the product of parity analysis of all the filled bands. The presence of the Dirac cone in the (001) plane and the existence of topologically non-trivial states at M‾-point in the Fermi arc contour also established our claim. The Z2 indices are calculated with the help of the product of parities and a change in Z2 indices from (0; 000) to (1; 000) is also verified with the evolution of Wannier change centers under the conditions of applied hydrostatic pressure and epitaxial strain. The time reversal and inversion symmetries are preserved throughout the study and the topological phase transition at 24.8 GPa is much lower than the structural phase transition pressure i.e., 56.54 GPa.

Machine learning guided discovery of stable, spin-resolved topological insulators

Identification of a nontrivial Z2 index in a spinful two-dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, Cs=(C↑−C↓)/2. However, the statement is not biconditional. An odd spin-Chern number can survive when the familiar Z2 index vanishes. Identification of solid-state systems hosting an odd, quantized Cs and trivial Z2 index is a pressing issue due to the potential for such insulators to admit band gaps optimal for experiments and quantum devices. Nevertheless, they have proven elusive due to the computational expense associated with their discovery. In this work, a neural network capable of identifying the spin-Chern number is developed and used to identify the first solid-state systems hosting a trivial Z2 index and odd Cs. We demonstrate the potential of one such system, Ti2CO2, to support Majorana corner modes via the superconducting proximity effect. Published by the American Physical Society 2024

Nontrivial Topological Phases in Ternary Borides M2XB2 (M=W, Mo; X=Co, Ni)

The nontrivial band topologies protected by certain symmetries have attracted significant interest in condensed matter physics. The discoveries of nontrivial topological phases in real materials provide a series of archetype materials to further explore the topological physics. Ternary borides M2XB2 (M = W, Mo; X = Co, Ni) have been widely investigated as the wear-resistant and high-hardness materials. Based on first-principles calculations, we find the nontrivial topological properties in these materials. Taking W2NiB2 as an example, this material shows the nodal line semimetal state in the absence of spin-orbit coupling. Two types of nodal lines appear near the Fermi level simultaneously. One is protected by the combined space-inversion and time-reversal symmetry, and the other is by the mirror symmetry. Part of these two-type nodal lines form nodal chains. When spin-orbit coupling is included, these nodal lines are fully gapped and the system becomes a strong topological insulator with nontrivial Z 2 index (1;000). Our calculations demonstrate that a nontrivial spin-momentum locked surface Dirac cone appears on the (1¯10) surface. We also find that other isostructural ternary borides Mo2NiB2, Mo2CoB2, and W2CoB2 possess similar topological band structures. Therefore, our work not only enriches the understanding of band topology for ternary borides, but also lays the foundation for the further study of topological phases manipulation and potential spintronic applications in realistic materials.

A combined first-principles calculations and X-ray photoelectron spectroscopy of Ce2T3X9 (T = Rh, Ru, Ir; X = Al, Ga): Possible strong topological insulator state in Ce2Ir3Al9

We calculated electronic structure of Ce2Rh3Ga9, Ce2Ru3Ga9, La2Ir3Al9 and Ce2Ir3Al9 compounds and we found that Ce2Ir3Al9 may exhibit strong topological insulator state according to Z2 index estimation based on parity algorithm and shape of band structure near Fermi level. The same algorithm shows strong topological insulator state for La2Ir3Al9 and Ce2Rh3Ga9 compounds, but their band structure near Fermi level is more like semimetallic one. Ce2Ru3Ga9 is a normal metal. Our calculations show no magnetic ordering for studied compounds. We also measured the XPS (X-ray Photoelectron Spectroscopy) spectra for these materials, and compared the valence band with simulated spectra, where good agreement was found. Core level states were also investigated and compared with Mulliken population analysis results. According to our measurements and calculations, the highest Ce valence is observed for Ce2Ir3Al9 whereas for Ce2Rh3Ga9 and Ce2Ru3Ga9 the Ce valence is smaller. Hybridization between conduction electrons and 4f states is very small, and for Ce2Ru3Ga9 essentially negligible.

Coexistence of superconductivity and topological aspects in beryllenes

By employing first-principles calculations, we theoretically propose that α- and β-beryllenes are two dimensional superconductors with Dirac bands at ambient pressure. Be-p orbitals dominate the electronic structure near the Fermi level, with two-band and three-band Fermi surfaces for α- and β-beryllene, respectively. α-beryllene hosts intrinsic type-I Dirac Fermions with the existence of nontrivial edge states and topological Z2 index, while β-beryllene has both type-I and type-II Dirac cones around the Fermi level and is topologically trivial. Migdal-Eliashberg theory combined with electron-phonon calculations predicts that both α- and β-beryllene are single-gap superconductors with critical temperature Tc = 9.9 K and 12.6 K, respectively. The difference is ascribed to the structure. Specifically, β-beryllene can be viewed as a bilayer of α-beryllene with a resulting increased density of states and enhanced electron-phonon coupling. And additionally, the Fermi surface of β-beryllene is strongly nested leading to Kohn anomalies that enhance the electron-phonon coupling λ. Replacing Be with heavier group-II atoms (Mg, Ca, Sr, and Ba) leads to strongly suppressed Tc due to electronic structure changes and the ensuing lower λ as well as lower phonon frequencies that reduce the prefactor. Our findings suggest beryllenes are promising two-dimensional elemental platforms for studying superconductivity and Dirac band topology.

Rigorous index theory for one-dimensional interacting topological insulators

We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting (and possibly disordered) fermions with U(1)⋊Z2 symmetry defined on the infinite chain. The class includes the Su–Schrieffer–Heeger (SSH) model [Su et al., “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698 (1979); Su et al., “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099 (1983); and Asbóth et al., A Short Course on Topological Insulators: Band-Structure Topology and Edge States in One and Two Dimensions, Lecture Notes in Physics (Springer, 2016)] as a special case. For any locally unique gapped (fixed-charge) ground state of a model in the class, we define a Z2 index in terms of the sign of the expectation value of the local twist operator. We prove that the index is topological in the sense that it is invariant under continuous modification of models in the class with a locally unique (fixed-charge) gapped ground state. This establishes that any path of models in the class that connects the two extreme cases of the SSH model must go through a phase transition. Our rigorous Z2 classification is believed to be optimal for the class of models considered here. We also show an interesting duality of the index and prove that any topologically nontrivial model in the class has a gapless edge excitation above the ground state when defined on the half-infinite chain. The results extend to other classes of models, including the extended Hubbard model. Our strategy to focus on the expectation value of local unitary operators makes the theory intuitive and conceptually simple. This paper also contains a careful discussion about the notion of unique gapped ground states of a particle system on the infinite chain. (There are two lecture videos in which the main results of this paper are discussed [H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: A brief introduction,” online lecture (21:41), November, 2021, seehttps://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021S and https://youtu.be/ypGVb3eYrpg and H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: With a pedagogical introduction to the topological phase transition in the SSH model,” online lecture (49:07), November, 2021, see https://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021L and https://youtu.be/yxZYOevV2Y].

Emergent topological states via digital (001) oxide superlattices

Oxide heterostructures exhibit many intriguing properties. Here we provide design principles for inducing multiple topological states in (001) (AMO3)1/(AM′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AM^{\\prime}$$\\end{document}O3)1 oxide superlattices. Aided by first-principles calculations and model analysis, we show that a (SrMO3)1/(SrM′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\prime}$$\\end{document}O3)1 superlattice (M = Nb, Ta and M′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\prime}$$\\end{document} = Rh, Ir) is a strong topological insulator with Z2 index (1;001). More remarkably, a (SrMoO3)1/(SrIrO3)1 superlattice exhibits multiple coexisting topological insulator (TI) and topological Dirac semi-metal (TDS) states. The TDS state has a pair of type-II Dirac points near the Fermi level and symmetry-protected Dirac node lines. The surface TDS Dirac cone is sandwiched by two surface TI Dirac cones in the energy-momentum space. The non-trivial topological properties arise from the band inversion between d orbitals of two dissimilar transition metal atoms and a particular parity property of (001) superlattice geometry. Our work demonstrates how to induce non-trivial topological states in (001) perovskite oxide heterostructures by rational design.

Electronic Structures of Group III–V Element Haeckelite Compounds: A Novel Family of Semiconductors, Dirac Semimetals, and Topological Insulators

AbstractThe family of III–V element compounds (i.e., XY compounds; X = B, Al, Ga, In, or Tl; Y = N, P, As, or Sb) have been intensively investigated for several decades because of their enormous applications for many optoelectronic devices. Here, by employing first‐principles calculations, the electronic structures of bulk XY haeckelite compounds are examined. It is identified that InSb (TlN and TlP) is Dirac semimetal (are strong topological insulators). The other fifteen XY compounds are semiconducting. The effect of biaxial and uniaxial tensile and compressive strains on the electronic structures are studied. These materials offer diverse topological orders. The semiconducting band gaps are mainly found between the bonding and antibonding states of the mixed X(p)–Y(p) orbitals at the top of the valence band and the bottom of the conduction bands, respectively. The topological insulating nature of the XY compounds is explained based on the degenerate px + py orbitals and their orbital energies relative to the pz orbitals near the Fermi energy. The nontrivial band topologies of TlN and TlP are confirmed by calculating the Z2 (1;000) index, surface states, and Wilson loop calculations. The bands split into two branches by including spin‐orbit interaction. The results demonstrate that haeckelite compounds are fascinating materials with broad potential applications in optoelectronics and possessing the possibility of hosting emergent physical phenomena.

Structural, electronic and topological properties of NaCaBi and KBaBi compounds

Topological insulators are promising candidates for potential applications in spintronics owing to their topologically protected dissipationless surface states. Here, we have investigated the structural, electronic and topological properties of NaCaBi and KBaBi compounds without and with spin-orbit coupling effect by employing density functional theory in Quantum Espresso code. Without spin-orbit coupling, NaCaBi and KBaBi compounds behave as a type-I nodal line semimetal and form a nodal loop around the Γ-point in the kz = 0 plane of Brillouin zone. When the bulk states are projected onto the (001) surface, a drumhead-like surface state is reported inside the nodal line for both compounds. The s/p band inversion in NaCaBi and KBaBi compounds are reported due to the strong spin-orbit coupling with band inversion energy of 0.87 eV and 0.32 eV respectively. The spin-orbit coupling in NaCaBi and KBaBi compounds induced the band gap of value 0.32 eV and 0.21 eV respectively. Z2 indices for NaCaBi and KBaBi are found (1; 000) which indicates that NaCaBi and KBaBi are strong topological insulators. The topological surface states and corresponding spin-texture are also calculated for both compounds which show a single Dirac cone at Γ-point with Dirac point lying below the Fermi-level. The obtained results are in good agreement with the available results reported in the literature.

Reliable Prediction of New Quantum Materials for Topological and Renewable-Energy Applications: A High-Throughput Screening.

Half-Heusler (HH) alloys provide a general platform for searching candidate materials for various energy applications. Here, we present a high-throughput first-principles calculation of a set of 960 eight valence-electron HH alloys to search potential candidates for thermoelectric (TE), solar harvesting (SH), topological insulator (TI), and transparent conductor (TC) applications. The initial screening parameters (such as stability, bandgap (Eg), band-inversion strength) followed by application specific descriptors are used to predict promising compounds. 121 out of 960 compounds were found to be dynamically and chemically stable. Of them, 31 compounds (with Eg < 1.5 eV) were studied for TE application, 30 (with 1 < Eg < 1.8 eV) for SH application, 21 for TI application, and 29 (with Eg > 2 eV) for TC applications. Some of the compounds show reasonably high thermoelectric figure of merit (ZT ∼ 1.6) and solar efficiency (SLME) > 20%, comparable to existing state-of-the-art materials. Surface band structure and topological Z2 index reconfirms the robustness of topological behavior. We strongly believe that our calculations should leverage useful insights to experimentalists.

Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension

In this paper, we study the long time behavior of the Kirchhoff type wave equation in the space H01(Ω)×L2(Ω). We prove the existence of the global attractor for the equation which covers the case of possible generation of the stiffness coefficient. We also consider the geometrical property of the global attractor. By means of the Z2 index, we provide, under suitable assumptions, the fractal dimension of the global attractor is infinite.

Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii–Moriya interaction

The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the ‘relative’ Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii–Moriya-like (DM) interaction parameterized by that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at which belongs to the BDI class is indexed by an integer winding number while the daughter Hamiltonian which belongs to class D is specified by a Z2 index . We show that the localization length, in addition to determining Z2, can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian—which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.

Quantum oscillations and nontrivial topological state in a compensated semimetal TaP2

We report systematic magneto-transport measurements and ab initio calculations on single-crystalline TaP2, a new member of the transition-metal dipnictides. We observed unsaturated magnetoresistance (MR) reaching ~ 700% at a magnetic field (B) of 9 T at 2 K along with striking Shubnikov-de Hass (SdH) oscillations. Our analysis on the SdH oscillations reveals nonzero Berry phase, indicating nontrivial band topology. The analysis also uncovers three fundamental magnetic oscillation frequencies of 72 T, 237 T, and 356 T, consistent with the theoretical calculations which reveal one hole pocket and two electron pockets at the L point and one electron pocket at the Z point of the Brillouin zone. We also found negative longitudinal MR (n-MR) within a narrow window of the angles between B and the electric current (I). The n-MR could be fitted with the Adler-Bell-Jackiw chiral anomaly equation but the origin remains yet ambiguous. The ab inito calculations suggest TaP2 as a weak topological insulator with the Z2 indices of (0; 111), which exhibits topological surface states on the (001) surface.

Broken symmetry driven phase transitions from a topological semimetal to a gapped topological phase in SrAgAs

We show the occurence of Dirac, Triple point, Weyl semimetal and topological insulating phase in a single ternary compound using specific symmetry preserving perturbations. Based on {\it first principle} calculations, \textbf{\textit{k.p}} model and symmetry analysis, we show that alloying induced precise symmetry breaking in SrAgAs (space group P6$_3/mmc$) leads to tune various low energy excitonic phases transforming from Dirac to topological insulating via intermediate triple point and Weyl semimetal phase. We also consider the effect of external magnetic field, causing time reversal symmetry (TRS) breaking, and analyze the effect of TRS towards the realization of Weyl state. Importantly, in this material, the Fermi level lies extremely close to the nodal point with no extra Fermi pockets which further, make this compound as an ideal platform for topological study. The multi fold band degeneracies in these topological phases are analyzed based on point group representation theory. Topological insulating phase is further confirmed by calculating Z2 index. Furthermore, the topologically protected surface states and Fermi arcs are investigated in some detail.

Quantum transport in a compensated semimetal W2As3 with nontrivial Z2 indices

We report a topological semimetal W2As3 with a space group C2/m. Based on the first-principles calculations, band crossings are partially gapped when spin-orbit coupling is included. The Z2 indices at the electron filling are [1;111], characterizing a strong topological insulator and topological surface states. From the magnetotransport measurements, nearly quadratic field dependence of magnetoresistance (MR) (B || [200]) at 3 K indicates an electron-hole compensated compound whose longitudinal MR reaches 115 at 3 K and 15 T. In addition, multiband features are detected from the high-magnetic-field Shubnikov-de Haas (SdH) oscillation, Hall resistivity, and band calculations. A nontrivial pi Berry's phase is obtained, suggesting the topological feature of this material. A two- band model can fit well the conductivity and Hall coefficient. Our experiments manifest that the transport properties of W2As3 are in good agreement with the theoretical calculations.

Nonsymmorphic symmetry protected node-line semimetal in the trigonal YH3

Using ab initio calculations based on density-functional theory and effective model analysis, we propose that the trigonal YH3 (Space Group: Pbar{{bf{3}}}c1) at ambient pressure is a node-line semimetal when spin-orbit coupling (SOC) is ignored. This trigonal YH3 has very clean electronic structure near Fermi level and its nodal lines locate very closely to the Fermi energy, which makes it a perfect system for model analysis. Symmetry analysis shows that the nodal ring in this compound is protected by the glide-plane symmetry, where the band inversion of |Y+, dxz〉 and |H1−, s〉 orbits at Γ point is responsible for the formation of the nodal lines. When SOC is included, the line nodes are prohibited by the glide-plane symmetry, and a small gap (≈5 meV) appears, which leads YH3 to be a strong topological insulator with Z2 indices (1,000). Thus the glide-plane symmetry plays an opposite role in the formation of the nodal lines in cases without and with SOC. As the SOC-induced gap is so small that can be neglected, this Pbar{{bf{3}}}c1 YH3 may be a good candidate for experimental explorations on the fundamental physics of topological node-line semimetals. We find the surface states of this Pbar{{bf{3}}}c1 phase are somehow unique and may be helpful to identify the real ground state of YH3 in the experiment.

Theoretical prediction of two-dimensional functionalized MXene nitrides as topological insulators

Recently, two-dimensional (2D) transition metal carbides and nitrides, namely, MXenes have attracted lots of attention for electronic and energy storage applications. Due to a large spin-orbit coupling (SOC) and the existence of a Dirac-like band at the Fermi energy, it has been theoretically proposed that some of the MXenes will be topological insulators (TIs). Up to now, all of the predicted TI MXenes belong to transition metal carbides, whose transition metal atom is W, Mo or Cr. Here, on the basis of first-principles and Z2 index calculations, we demonstrate that some of the MXene nitrides can also be TIs. We find that Ti3N2F2 is a 2D TI, whereas Zr3N2F2 is a semimetal with nontrivial band topology and can be turned into a 2D TI when the lattice is stretched. We also find that the tensile strain can convert Hf3N2F2 semiconductor into a 2D TI. Since Ti is one of the mostly used transition metal element in the synthesized MXenes, we expect that our prediction can advance the future application of MXenes as TI devices.

Volkov-Pankratov states in topological heterojunctions

We show that a smooth interface between two insulators of opposite topological Z2 indices possesses multiple surface states, both massless and massive. While the massless surface state is non-degenerate, chiral and insensitive to the interface potential, the massive surface states only appear for a sufficiently smooth heterojunction. The surface states are particle-hole symmetric and a voltage drop reveals their intrinsic relativistic nature, similarly to Landau bands of Dirac electrons in a magnetic field. We discuss the relevance of the massive Dirac surface states in recent ARPES and transport experiments.

Majorana zero modes in a ladder of density-modulated Kitaev superconductor chains

We investigate the topological properties of a ladder model of the Kitaev superconductor chains with a periodically modulated chemical potential. It is found that the system has two different topological classes, the class BDI characterized by the Z index and the class D characterized by the Z2 index. The topological class of the system is determined by the relative phase θ between the inter- and intra-chain superconducting pairing. Based on the calculation of the topological numbers of the class BDI and the class D, we present the topological phase diagrams of the system for two different topological classes. It is shown that the topological phase transition accompanying the variation of the number of Majorana zero modes could be induced by periodical potential in both two classes. Finally, we calculate the differential conductance of normal metal-superconductor junction at zero bias, and find that it matches the topological phase diagrams in both class BDI and class D.

Time-reversal-symmetry broken quantum spin Hall in Lieb lattice

In this paper, the time-reversal (TR) symmetry broken quantum spin Hall (QSH) in Lieb lattice is investigated in the presence of both Rashba spin-orbit coupling (SOC) and uniform exchange field. The Lieb lattice has a simple cubic symmetry, and it has three different sites in each unit cell. The most distinctive feature of this model is that it contains only one Dirac-cone in the first Brillouin zone, where the upper dispersive band and the lower dispersive band touch the middle zero-energy band at M point and form a cone-like dispersion. The intrinsic SOC is essentially needed to open the full energy gap in the bulk. When the intrinsic SOC is nonzero, all the band structures are separated everywhere in the Brillouin zone and can be characterized by some topological invariants. The exact QSH first put forward by Kane and Mele in 2005 is characterized by the z2 number. The protection from the TR symmetry ensures the gapless crossing in the surface state in the bulk gap. In our model, the presence of the exchange field breaks the TR symmetry, which results in opening a small gap in the crossing point and the z2 topological order is not suitable for the system. This kind of state is a TR symmetry broken QSH, which is characterized by the spin Chern numbers. The spin Chern numbers have a much wider scope of application than z2 index. It is suitable for both TR symmetry system and the TR symmetry broken system. For Lieb lattice ribbons, the spin polarization and the wave-function distributions are obtained numerically. There exists a weak scattering between the counter-propagating states in the TR symmetry broken QSH, and the spin transport along the boundary with a low dissipation replaces the dissipationless spin current in a TR symmetry system. In experiment, such a system can be realized by the two-dimensional Fermi gases in optical lattice with Lieb symmetry. The above conclusions are expected to give theoretical guidance in the spin device and the quantum information.