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Vanadium-based kagome systems AV$_3$Sb$_5$ (A = K, Rb, Cs) have emerged as paradigmatic examples exhibiting unconventional charge density waves (CDWs) and superconductivity linked to van Hove singularities (VHSs). Despite extensive studies, the three-dimensional (3D) nature of CDW states in these systems remains elusive. This study employs first-principles density functional theory and a tight-binding model to investigate the stacking-dependent electronic structures of 3D CDWs in AV$_3$Sb$_5$, emphasizing the significant role of interlayer coupling in behaviors of the VHSs associated with diverse 3D CDW orders. We develop a minimal 3D tight-binding model and present a detailed analysis of band structures and density of states for various 3D CDW stacking configurations, including those with and without a $\pi$-phase shift stacking of the inverse star of David, as well as alternating stacking of the inverse star of David and the star of David. We find that VHSs exist below the Fermi level even in 3D CDWs without $\pi$-phase shift stackings, and that these VHSs shift downward in the $\pi$-phase shift stacking CDW structure, stabilizing the $2\times2\times2$ $\pi$-shifted inverse star of David distortions in alternating vanadium layers as the ground state 3D CDW order of AV$_3$Sb$_5$. Our work provides the electronic origin of 3D CDW orders, paving the way for a deeper understanding of CDWs and superconductivity in AV$_3$Sb$_5$ kagome metals., Comment: 4 figures + Supplementary information

Graphene, with spin and valley degrees of freedom, fosters unexpected physical and chemical properties for the realization of next-generation quantum devices. However, the spin symmetry of graphene is rather robustly protected, hampering manipulation of the spin degrees of freedom for the application of spintronic devices such as electric gate tunable spin filters. We demonstrate that a hybrid heterostructure composed of graphene and LaCoO3 epitaxial thin film exhibits an electrically tunable spin exchange splitting. The large and adjustable spin exchange splitting of 155.9 - 306.5 meV was obtained by the characteristic shifts in both the spin symmetry broken quantum Hall states and the Shubnikov-de-Haas oscillations. Strong hybridization induced charge transfer across the hybrid heterointerface has been identified for the observed spin exchange splitting. The substantial and facile controllability of the spin exchange splitting provides an opportunity for spintronics applications with the electrically-tunable spin polarization in hybrid heterostructures., Comment: 20 pages, 5 figures, 1 table

The Hofstadter energy spectrum of twisted bilayer graphene is found to have recursive higher-order topological properties. We demonstrate that higher-order topological insulator (HOTI) phases, characterized by localized corner states, occur as replicas of the original HOTIs to fulfill the self-similarity of the Hofstadter spectrum. We show the existence of the exact flux translational symmetry of twisted bilayer graphene at all commensurate angles. Based on this result, we carefully identify that the original HOTI phase at zero flux is re-entrant at a half-flux periodicity, where the effective twofold rotation is preserved. In addition, numerous replicas of the original HOTIs are found for fluxes without protecting symmetries. Similar to the original HOTIs, replica HOTIs feature both localized corner states and edge-localized real-space topological markers. The replica HOTIs originate from the different interaction scales, namely, intralayer and interlayer couplings, in twisted bilayer graphene. The topological aspect of Hofstadter butterflies revealed in our results highlights symmetry-protected topology in quantum fractals., Comment: 5 figures + Supplementary Information, published version

We study topological phase transitions in tetragonal NaZnSb$_{1-x}$Bi$_x$, driven by the chemical composition $x$. Notably, we examine mirror Chern numbers that change without symmetry indicators. First-principles calculations are performed to show that NaZnSb$_{1-x}$Bi$_x$ experiences consecutive topological phase transitions, diagnosed by the strong $\mathbb Z_{2}$ topological index $\nu_{0}$ and two mirror Chern numbers $\mu_{x}$ and $\mu_{xy}$. As the chemical composition $x$ increases, the topological invariants ($\mu_{x}\mu_{xy}\nu_{0}$) change from $(000)$, $(020)$, $(220)$, to $(111)$ at $x$ = 0.15, 0.20, and 0.53, respectively. A simplified low-energy effective model is developed to examine the mirror Chern number changes, highlighting the central role of spectator Dirac fermions in avoiding symmetry indicators. Our findings suggest that NaZnSb$_{1-x}$Bi$_{x}$ can be an exciting testbed for the exploration of the interplay between the topology and symmetry.

Recently, Lee \textit{et. al.} [Nano Lett. \textbf{21}, 4305 (2021)] newly synthesized monochalcogenide GeSe in a polar phase, referred to as $\gamma$-phase. Motivated by this work, we study shift current of $\gamma$-GeSe and its tunability via an in-plane uniaxial strain. Using first-principles calculations, we uncover the electronic structure of the strained $\gamma$-GeSe systems. We then calculate frequency-dependent shift current conductivities at various strains. The tunability is demonstrated to enhance the shift current up to $\sim$ 20 $\mu$A/V$^2$. Moreover, the direction of shift current can be inverted by a light strain. Markedly, an anomalous behavior is found in the zero-frequency limit, which can be an indicative of band inversion and a potential topological phase transition driven by the strain. Our results suggest that shift current can be a tangible prove of bulk electronic states of $\gamma$-GeSe.

Recently, layered kagome metals AV$_3$Sb$_5$ (A = K, Rb, and Cs) have emerged as a fertile platform for exploring frustrated geometry, correlations, and topology. Here, using first-principles and mean-field calculations, we demonstrate that AV$_3$Sb$_5$ can crystallize in a mono-layered form, revealing a range of properties that render the system unique. Most importantly, the two-dimensional monolayer preserves intrinsically different symmetries from the three-dimensional layered bulk, enforced by stoichiometry. Consequently, the van Hove singularities, logarithmic divergences of electronic density of states, are enriched, leading to a variety of competing instabilities such as doublets of charge density waves and s-and d-wave superconductivity. We show that the competition between orders can be fine-tuned in the monolayer via electron-filling of the van Hove singularities. Thus, our results suggest the monolayer kagome metal AV$_3$Sb$_5$ as a promising platform for designer quantum phases., Comment: 6 figures + Supplementary Information, published version

The Su-Schrieffer-Heeger (SSH) chain is an one-dimensional lattice that comprises two dimerized sublattices. Recently, Zhu, Prodan, and Ahn (ZPA) proposed in [L. Zhu, E. Prodan, and K. H. Ahn, Phys. Rev. B \textbf{99}, 041117 (2019)] that one-dimensional flat bands can occur at topological domain walls of a two-dimensional array of the SSH chains. Here, we newly suggest a two-dimensional topological insulator that is protected by inversion and time-reversal symmetries without spin-orbit coupling. It is shown that the two-dimensional SSH chains realize the proposed topological insulator. Utilizing the first Stiefel-Whitney numbers, a weak type of $\mathbb{Z}_2$ topological indices are developed, which identify the proposed topological insulator, dubbed a two-dimensional Stiefel-Whitney insulator (2DSWI). The ZPA model is employed to study the topological phase diagrams and topological phase transitions. It is found that the phase transition occurs via the formation of the massless Dirac points that wind the entire Brillouin zone. We argue that this unconventional topological phase transition is a characteristic feature of the 2DSWI, manifested as the one-dimensional domain wall states. The new insight from our work could help efforts to realize topological flat bands in solid-state systems.

The ordinary Hall effect refers to generation of a transverse voltage upon exertion of an electric field in the presence of an out-of-plane magnetic field. While a linear Hall effect is commonly observed in systems with breaking time-reversal symmetry via an applied external magnetic field or their intrinsic magnetization1, 2, a nonlinear Hall effect can generically occur in non-magnetic systems associated with a nonvanishing Berry curvature dipole3. Here we report, observations of a nonlinear optical Hall effect in a Weyl semimetal WTe2 without an applied magnetic field at room temperature. We observe an optical Hall effect resulting in a polarization rotation of the reflected light, referred to as the nonlinear Kerr rotation. The nonlinear Kerr rotation linearly depends on the charge current and optical power, which manifests the fourth-order nonlinearity. We quantitatively determine the fourth-order susceptibility, which exhibits strong anisotropy depending on the directions of the charge current and the light polarization. Employing symmetry analysis of Berry curvature multipoles, we demonstrate that the nonlinear Kerr rotations can arise from the Berry curvature hexapole allowed by the crystalline symmetries of WTe2. There also exist marginal signals that are incompatible with the symmetries, which suggest a hidden phase associated with the nonlinear process., Comment: 21 pages, 4 figures

The coexistence of ferroelectric and topological orders in two-dimensional (2D) atomic crystals allows non-volatile and switchable quantum spin Hall states. Here we offer a general design principle for 2D bilayer heterostructures that can host ferroelectricity and nontrivial band topology simultaneously using only topologically trivial building blocks. The built-in electric field arising from the out-of-plane polarization across the heterostrucuture enables a robust control of the band gap size and band inversion strength, which can be utilized to manipulate topological phase transitions. Using first-principles calculations, we demonstrate a series of bilayer heterostructures are 2D ferroelectric topological insulators (2DFETIs) characterized with a direct coupling between band topology and polarization state. We propose a few 2DFETI-based quantum electronics including domain-wall quantum circuits and topological memristor.

We propose a new type of instanton interference effect in two-dimensional higher-order topological insulators. The intercorner tunneling consists of the instanton and the anti-instanton pairs that travel through the boundary of the higher-order topological insulator. The Berry phase difference between the instanton pairs causes the interference of the tunneling. This topological effect leads to the gate-tunable oscillation of the energy splitting between the corner states, where the oscillatory nodes signal the perfect suppression of the tunneling. We suggest this phenomenon as a unique feature of the topological corner states that differentiate from trivial bound states. In the view of experimental realization, we exemplify twisted bilayer graphene, as a promising candidate of a two-dimensional higher-order topological insulator. The oscillation can be readily observed through the transport experiment that we propose. Thus, our work provides a feasible route to identify higher-order topological materials., Comment: 4+9 pages, 4 figures

Higher-order topological insulators are newly proposed topological phases of matter, whose bulk topology manifests as localized modes at two- or higher-dimensional lower boundaries. In this work, we propose the twisted bilayer graphenes with large angles as higher-order topological insulators, hosting topological corner charges. At large commensurate angles, the intervalley scattering opens up the bulk gap and the corner states occur at half filling. Based on both first-principles calculations and analytic analysis, we show the striking results that the emergence of the corner states do not depend on the choice of the specific angles as long as the underlying symmetries are intact. Our results show that the twisted bilayer graphene can serve as a robust candidate material of two-dimensional higher-order topological insulator.

In this article, we review the recent progress in the study of topological phases in systems with space-time inversion symmetry $I_{\text{ST}}$. $I_{\text{ST}}$ is an anti-unitary symmetry which is local in momentum space and satisfies $I_{\text{ST}}^2=1$ such as $PT$ or $C_{2}T$ symmetry where $P$, $T$, $C_2$ indicate inversion, time-reversal, and two-fold rotation symmetries, respectively. Under $I_{\text{ST}}$, the Hamiltonian and the Bloch wave function can be constrained to be real-valued, which makes the Berry curvature and the Chern number to vanish. In this class of systems, gapped band structures of real wave functions can be topologically distinguished by Stiefel-Whitney numbers instead. The first and second Stiefel-Whitney numbers $w_1$ and $w_2$, respectively, are the corresponding invariants in 1D and 2D, which are equivalent to the quantized Berry phase and the $Z_2$ monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel-Whitney number, including 1D topological insulators with quantized charge polarization, 2D Dirac semimetals, and 3D nodal line semimetals. Next we review how the second Stiefel-Whitney class characterizes the 3D nodal line semimetals carrying a $Z_{2}$ monopole charge. In particular, we explain how the second Stiefel-Whitney number $w_2$, the $Z_{2}$ monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2D and 3D topological insulators characterized by the nontrivial second Stiefel Whitney class., Comment: A review article; 17 pages, 12 figures; suggestions and comments are highly appreciated

Previously known three-dimensional Dirac semimetals (DSs) occur in two types -- topological DSs and nonsymmorphic DSs. Here we present a novel three-dimensional DS that exhibits both features of the topological and nonsymmorphic DSs. We introduce a minimal tight-binding model for the space group 100 that describes a layered crystal made of two-dimensional planes in the $p4g$ wallpaper group. Using this model, we demonstrate that double glide-mirrors allow a noncentrosymmetric three-dimensional DS that hosts both symmetry-enforced Dirac points at time-reversal invariant momenta and twofold-degenerate Weyl nodal lines on a glide-mirror-invariant plane in momentum space. The proposed DS allows for rich topological physics manifested in both topological surface states and topological phase diagrams, which we discuss in detail. We also perform first-principles calculations to predict that the proposed DS is realized in a set of existing materials BaLa$X$B$Y_5$, where $X$ = Cu or Au, and $Y$ = O, S, or Se.

Based on first-principles calculations, we predict a new two-dimensional ferromagnetic material that exhibits exotic Fermi surface topology. We show that monolayer hexagonal indium carbide ({\em h}-InC) is thermodynamically and dynamically stable, and it energetically favors the ferromagnetic ordering of spins. The perfectly planar geometry in two dimensions, together with ferromagnetism, gives rise to a unique opportunity to encounter intriguing electronic properties, captured in the Fermi surface and band topology. We show that multiple nodal lines coexist in momentum space, accompanied by the electron and hole pockets that touch each other linearly at the nodal lines. Inclusion of spin-orbit coupling enriches the magnetic and electronic properties of {\em h}-InC. Spin-orbit coupling leads to an easy-plane type magnetocrystalline anisotropy, and the nodal lines can be tuned into topological nodal points, contingent upon the magnetization direction. Symmetry analysis and a tight-binding model are provided to explain the nodal structure of the bands. Our findings suggest {\em h}-InC as a new venue for supporting carbon-based magnetism and exotic band topology in two dimensions.

We review recent theoretical progress in the understanding and prediction of novel topological semimetals. Topological semimetals define a class of gapless electronic phases exhibiting topologically stable crossings of energy bands. Different types of topological semimetals can be distinguished based on the degeneracy of the band crossings, their codimension (e.g., point or line nodes), as well as the crystal space group symmetries on which the protection of stable band crossings relies. The dispersion near the band crossing is a further discriminating characteristic. These properties give rise to a wide range of distinct semimetal phases such as Dirac or Weyl semimetals, point or line node semimetals, and type-I or type-II semimetals. In this review we give a general description of various families of topological semimetals with an emphasis on proposed material realizations from first-principles calculations. The conceptual framework for studying topological gapless electronic phases is reviewed, with a particular focus on the symmetry requirements of energy band crossings, and the relation between the different families of topological semimetals is elucidated. In addition to the paradigmatic Dirac and Weyl semimetals, we pay particular attention to more recent examples of topological semimetals, which include nodal line semimetals, multifold fermion semimetals, triple-point semimetals. Less emphasis is placed on their surface state properties, responses to external probes, and recent experimental developments., Comment: 29 pages, 11 figures; (short) review article for Annual Reviews of Materials Research; comments are most welcome

Dirac line node (DLN) semimetals are a class of topological semimetals that feature band-crossing lines in momentum space. We study the type-I and type-II classification of DLN semimetals by developing a criterion that determines the type using band velocities. Using first-principles calculations, we also predict that Na3N under an epitaxial tensile strain realizes a type-II DLN semimetal with vanishing spin-orbit coupling (SOC), characterized by the Berry phase that is Z2-quantized in the presence of inversion and time-reversal symmetries. The surface energy spectrum is calculated to demonstrate the topological phase, and the type-II nature is demonstrated by calculating the band velocities. We also develop a tight-binding model and a low-energy effective Hamiltonian that describe the low-energy electronic structure of strained Na3N. The occurrence of a DLN in Na3N under strain is captured in the optical conductivity, which we propose as a means to experimentally confirm the type-II class of the DLN semimetal., Comment: 14 pages, 9 figure

We study the band topology and the associated linking structure of topological semimetals with nodal lines carrying $Z_{2}$ monopole charges, which can be realized in three-dimensional systems invariant under the combination of inversion $P$ and time reversal $T$ when spin-orbit coupling is negligible. In contrast to the well-known $PT$-symmetric nodal lines protected only by $\pi$ Berry phase in which a single nodal line can exist, the nodal lines with $Z_{2}$ monopole charges should always exist in pairs. We show that a pair of nodal lines with $Z_{2}$ monopole charges is created by a {\it double band inversion} (DBI) process, and that the resulting nodal lines are always {\it linked by another nodal line} formed between the two topmost occupied bands. It is shown that both the linking structure and the $Z_{2}$ monopole charge are the manifestation of the nontrivial band topology characterized by the {\it second Stiefel-Whitney class}, which can be read off from the Wilson loop spectrum. We show that the second Stiefel-Whitney class can serve as a well-defined topological invariant of a $PT$-invariant two-dimensional (2D) insulator in the absence of Berry phase. Based on this, we propose that pair creation and annihilation of nodal lines with $Z_{2}$ monopole charges can mediate a topological phase transition between a normal insulator and a three-dimensional weak Stiefel-Whitney insulator (3D weak SWI). Moreover, using first-principles calculations, we predict ABC-stacked graphdiyne as a nodal line semimetal (NLSM) with $Z_{2}$ monopole charges having the linking structure. Finally, we develop a formula for computing the second Stiefel-Whitney class based on parity eigenvalues at inversion invariant momenta, which is used to prove the quantized bulk magnetoelectric response of NLSMs with $Z_2$ monopole charges under a $T$-breaking perturbation., Comment: 4+28 pages, 3+17 figures

We propose that the noncentrosymmetric LiGaGe-type hexagonal $ABC$ crystal SrHgPb realizes a new type of topological semimetal that hosts both Dirac and Weyl points in momentum space. The symmetry-protected Dirac points arise due to a band inversion and are located on the sixfold rotation $z$-axis, whereas the six pairs of Weyl points related by sixfold symmetry are located on the perpendicular $k_z=0$ plane. By studying the electronic structure as a function of the buckling of the HgPb layer, which is the origin of inversion symmetry breaking, we establish that the coexistence of Dirac and Weyl fermions defines a phase separating two topologically distinct Dirac semimetals. These two Dirac semimetals are distinguished by the $\mathbb{Z}_2$ index of the $k_z=0$ plane and the corresponding presence or absence of 2D Dirac fermions on side surfaces. We formalize our first-principles calculations by deriving and studying a low-energy model Hamiltonian describing the Dirac-Weyl semimetal phase. We conclude by proposing several other materials in the non-centrosymmetric $ABC $ material class, in particular SrHgSn and CaHgSn, as candidates for realizing the Dirac-Weyl semimetal., Comment: 6 pages, 4 figures + 5 pages of supplemental material

Recent developments in the relationship between bulk topology and surface crystal symmetry have led to the discovery of materials whose gapless surface states are protected by crystal symmetries. In fact, there exists only a very limited set of possible surface crystal symmetries, captured by the 17 "wallpaper groups." We show that a consideration of symmetry-allowed band degeneracies in the wallpaper groups can be used to understand previous topological crystalline insulators, as well as to predict new examples. In particular, the two wallpaper groups with multiple glide lines, $pgg$ and $p4g$, allow for a new topological insulating phase, whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion. Like the surface state of a conventional topological insulator, the surface Dirac fermion in this "nonsymmorphic Dirac insulator" provides a theoretical exception to a fermion doubling theorem. Unlike the surface state of a conventional topological insulator, it can be gapped into topologically distinct surface regions while keeping time-reversal symmetry, allowing for networks of topological surface quantum spin Hall domain walls. We report the theoretical discovery of new topological crystalline phases in the A$_2$B$_3$ family of materials in SG 127, finding that Sr$_2$Pb$_3$ hosts this new topological surface Dirac fermion. Furthermore, (100)-strained Au$_2$Y$_3$ and Hg$_2$Sr$_3$ host related topological surface hourglass fermions. We also report the presence of this new topological hourglass phase in Ba$_5$In$_2$Sb$_6$ in SG 55. For orthorhombic space groups with two glides, we catalog all possible bulk topological phases by a consideration of the allowed non-abelian Wilson loop connectivities, and we develop topological invariants for these systems. Finally, we show how in a particular limit, these crystalline phases reduce to copies of the SSH model., Comment: Final version, 6 pg main text + 29 pg supplement, 6 + 13 figures

We study a class of Dirac semimetals that feature an eightfold-degenerate double Dirac point. We show that 7 of the 230 space groups can host such Dirac points and argue that they all generically display linear dispersion. We introduce an explicit tight-binding model for space groups 130 and 135, showing that 135 can host an intrinsic double Dirac semimetal -- one with no additional degeneracies at the Fermi energy. We consider symmetry-lowering perturbations and show that uniaxial compressive strain in different directions leads to topologically distinct insulating phases. In addition, the double Dirac semimetal can accommodate topological line defects that bind helical modes. Potential materials realizations are discussed., Comment: 5 pages + supplementary information

Based on first-principles calculations, we predict a substantial increase in the optical dielectric function of LiAsSe$_2$ under pressure. We find that the optical dielectric constant is enhanced threefold under volume compression. This enhancement is mainly due to the dimerization strength reduction of the one-dimensional (1D) As--Se chains in LiAsSe$_2$, which significantly alters the wavefunction phase mismatch between two neighboring chains and changes the transition intensity. By developing a tight-binding model of the interacting 1D chains, the essential features of the low-energy electronic structure of LiAsSe$_2$ are captured. Our findings are important for understanding the fundamental physics of LiAsSe$_2$ and provide a feasible way to enhance the material optical response that can be applied to light harvesting for energy applications., Comment: 13 pages, 6 figures