In some kind of three-dimensional higher-order topological insulators, bulk and surface electronic states are gapped, while there appear gapless hinge states protected by spatial symmetry. Here we show by ab initio calculations that the La apatite electride is a higher-order topological crystalline insulator. It is a one-dimensional electride, in which the one-dimensional interstitial hollows along the $c$ axis support anionic electrons, and the electronic states in these one-dimensional channels are well approximated by the one-dimensional Su-Schrieffer-Heeger model. When the crystal is cleaved into a hexagonal prism, the ${120}^{\circ }$ hinges support gapless hinge states, with their filling quantized to be 2/3. This quantization of the filling comes from a topological origin. We find that the quantized value of the filling depends on the fundamental blocks that constitute the crystal. The apatite consists of the triangular blocks, which is crucial for giving nontrivial fractional charge at the hinge.

A IV-VI compound SnTe is proposed to be a topological crystalline insulator (TCI), and the band structures have been studied by using angle-resolved photoemission spectroscopy (ARPES). However, the topological surface states (TSSs) that hallmark the nontrivial topology disperse mostly in the unoccupied side where access via ARPES is limited. Here we investigate the (111) face of a SnTe film by using pump-probe ARPES. We find distinct energy sections in the unoccupied side that behave differently in terms of the excitation and recovery dynamics. From these different behaviors, the boundaries of these sections are attributed to the edges of the conduction and valence bands. High statistics data reveal that the TSSs are traversing the band gap, which evidences that SnTe belongs to a TCI. We also find that the bulk bands near the gap show splitting, which is attributed to the Rashba effect occurring in the dipolar surface region.

We study what kind of topological crystalline insulator phase emerges from nodal line semimetal phases in the face centered cubic system, which is not indicated by topological indices when spin-orbit coupling (SOC) is introduced. We construct an effective model, which hosts two different nodal lines phases, and calculated mirror Chern numbers in it by introducing SOC. As a result, we find that the two nodal line phases with different nodal line configurations are mapped to different topological crystalline insulator phases. This result shows that turning off SOC and checking the nodal line configuration can distinguish the two topological crystalline insulator phases, which have not been distinguished by previous methods.

We calculate the electronic structure of the narrow gap semiconductors PbTe, SnTe, and GeTe in the cubic phase using density functional theory (DFT) and the $G_0{W}_0$ method. Within DFT, we show that the band ordering obtained with a conventional semilocal exchange-correlation approximation is correct for SnTe and GeTe but wrong for PbTe. The correct band ordering at the high-symmetry point $L$ is recovered adding $G_0{W}_0$ quasiparticle corrections. However, one-shot $G_0{W}_0$ produces artifacts in the band structure due to the wrong orbital character of the DFT single-particle states at the band edges close to $L$. We show that in order to correct these artifacts it is enough to consider the off-diagonal elements of the $G_0{W}_0$ self-energy corresponding to these states. We also investigate the pressure dependence of the band gap for these materials and the possibility of a transition from a trivial to a nontrivial topology of the band structure. For PbTe, we predict the band crossover and topological transition to occur at around 4.8 GPa. For GeTe, we estimate the topological transition to occur at 1.9 GPa in the constrained cubic phase, a pressure lower than that of the structural phase transition from rombohedral to cubic. SnTe is a crystalline topological insulator at ambient pressure, and the transition into a trivial topology would take place under a volume expansion of approximately $10\%$.

A topological mechanism is a zero-elastic-energy deformation of a mechanical structure that is robust against smooth changes in system parameters. Here, we map the nonlinear elasticity of a paradigmatic class of topological mechanisms onto a supersymmetric field theory introduced by Witten and Olive. Heuristically, this approach entails taking the square root of a nonlinear Hamiltonian. It generalizes the standard procedure of obtaining two copies of the Dirac equation by taking the square root of the linear Klein-Gordon equation. Our real-space formalism goes beyond topological band theory by incorporating nonlinearities and spatial inhomogeneities, such as domain walls (i.e., kinks), where topological states are typically localized. We interpret the two components of the real fermionic field as site and bond displacements, respectively. The constraint of zero elastic energy insures that kinks in the mechanical system saturate the Bogomolny-Prasad-Sommerfield bound, while forbidding antikinks. This mechanism can be viewed as a manifestation of the underlying supersymmetry being half broken.

In recent theoretical and experimental investigations, researchers have linked the low-energy field theory of a Weyl semimetal gapped with a charge-density wave (CDW) to high-energy theories with axion electrodynamics. However, it remains an open question whether a lattice regularization of the dynamical Weyl-CDW is in fact a single-particle axion insulator (AXI). In this Rapid Communication, we use analytic and numerical methods to study both lattice-commensurate and incommensurate minimal (magnetic) Weyl-CDW phases in the mean-field state. We observe that, as previously predicted from field theory, the two inversion ($\mathcal{I}$)-symmetric Weyl-CDWs with $\varphi =0,\pi$ differ by a topological axion angle $\delta {\theta }_{\varphi }=\pi$. However, we crucially discover that neither of the minimal Weyl-CDW phases at $\varphi =0,\pi$ is individually an AXI; they are instead quantum anomalous Hall (QAH) and “obstructed” QAH insulators that differ by a fractional translation in the modulated cell, analogous to the two phases of the Su-Schrieffer-Heeger model of polyacetylene. Using symmetry indicators of band topology and non-Abelian Berry phase, we demonstrate that our results generalize to multiband systems with only two Weyl fermions, establishing that minimal Weyl-CDWs unavoidably carry nontrivial Chern numbers that prevent the observation of a static magnetoelectric response. We discuss the experimental implications of our findings and provide models and analysis generalizing our results to nonmagnetic Weyl- and Dirac-CDWs.

Certain patterns of symmetry fractionalization in (2+1)-dimensional [(2+1)D] topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group $G$, one can define a (3+1)-dimensional [(3+1)D] topologically invariant path integral in terms of a state sum for a $G$-symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D $G$-symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and antiunitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a by-product, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the ${\mathcal{H}}^4(G,U\left(1\right))$ obstruction that arises in the theory of $G$-crossed braided tensor categories, for which no general method has been presented to date.

Higher-order topological materials with topologically protected states at the boundaries of their boundaries (hinges or corners) have attracted attention in recent years. In this paper, we utilize time-periodic driving to generate second-order topological superconductors out of systems which otherwise do not even allow second-order topological characterization. This is made possible by the design of periodic drives which inherently exhibit nontrivial winding in the time domain. Through the interplay of topology in both spatial and temporal dimensions, nonchiral Majorana modes may emerge at the systems' corners and sometimes even coexist with chiral Majorana modes. Our proposal thus presents an opportunity for Floquet engineering with minimal system complexity and its application in quantum information processing.

The theoretical analysis of topological insulators (TIs) has been traditionally focused on infinite homogeneous crystals with band gap in the bulk and nontrivial topology of their wave functions, or infinite wires whose boundaries host surface or edge metallic states. Such infinite-length edge states exhibit quantized conductance which is insensitive to edge disorder, as long as it does not break the underlying symmetry or introduce energy scale larger than the bulk gap. However, experimental devices contain finite-size topological region attached to normal metal (NM) leads, which poses a question about how precise is quantization of longitudinal conductance and how electrons transition from topologically trivial NM leads into the edge states. This particularly pressing issue for recently conjectured two-dimensional (2D) Floquet TI where electrons flow from time-independent NM leads into time-dependent edge states, the very recent experimental realization [J. W. McIver et al., Nat. Phys. 16, 38 (2020)] of Floquet TI using graphene irradiated by circularly polarized light did not exhibit either quantized longitudinal or Hall conductance. Here, we employ a charge-conserving solution for Floquet-nonequilibrium Green functions of irradiated graphene nanoribbon to compute longitudinal two-terminal conductance, as well as spatial profiles of local current density as electrons propagate from NM leads into the Floquet TI. For comparison, we also compute conductance of graphene-based realization of 2D quantum Hall, quantum anomalous Hall, and quantum spin Hall insulators. Although zero-temperature conductance within the gap of these three conventional time-independent 2D TIs of finite length exhibits small oscillations due to reflections at the NM-lead/2D-TI interface, it remains very close to perfectly quantized plateau at $2e^2/h$ and completely insensitive to edge disorder. This is due to the fact that inside conventional TIs there is only edge local current density which circumvents any disorder. In contrast, in the case of Floquet TI both bulk and edge local current densities contribute equally to total current, which leads to longitudinal conductance below the expected quantized plateau that is further reduced by edge vacancies. We propose two experimental schemes to detect coexistence of bulk and edge current densities within Floquet TI: (i) drilling a nanopore in the interior of irradiated region of graphene will induce backscattering of bulk current density, thereby reducing longitudinal conductance by $\sim 28\%$; (ii) imaging of magnetic field produced by local current density using diamond nitrogen-vacancy centers.

We propose the concept of a device based on a square-shaped sample of a two-dimensional second-order topological helical superconductor which hosts two zero-dimensional Majorana quasiparticles at the corners. The two zero-energy modes rely on particle-hole symmetry (PHS) and their spatial position can be shifted by rotating an in-plane magnetic field and tuning proximity-induced spin-singlet pairing. We consider an adiabatic cycle performed on the degenerate ground-state manifold and show that it realizes the braiding of the two modes whereby they accumulate a nontrivial statistical phase $\pi$ within one cycle. Alongside the PHS-ensured operator algebra, the fractional statistics confirms the Majorana nature of the zero-energy excitations. A schematic design for a possible experimental implementation of such a device is presented, which could be a step towards realizing non-Abelian braiding.

We propose a general scheme to construct a Hamiltonian $H_{\text{root}}$ describing a square root of an original Hamiltonian $H_{\text{original}}$ based on the graph theory. The square-root Hamiltonian is defined on the subdivided graph of the original graph of $H_{\text{original}}$, where the subdivided graph is obtained by putting one vertex on each link in the original graph. When $H_{\text{original}}$ describes a topological system, there emerge in-gap edge states at nonzero energy in the spectrum of $H_{\text{root}}$, which are the inherence of the topological edge states at zero energy in $H_{\text{original}}$. In this case, $H_{\text{root}}$ describes a square-root topological insulator or superconductor. Typical examples are square roots of the Su-Schrieffer-Heeger (SSH) model, the Kitaev topological superconductor model, and the Haldane model. Our scheme is also applicable to non-Hermitian topological systems, where we study an example of a nonreciprocal non-Hermitian SSH model.

Spectra of bulk or edges in topological insulators are often made complex by non-Hermiticity. Here, we show that symmetry protection enables entirely real spectra for both bulk and edges even in non-Hermitian topological insulators. In particular, we demonstrate the entirely real spectra without non-Hermitian skin effects due to a combination of pseudo-Hermiticity and Kramers degeneracy. This protection relies on nonspatial fundamental symmetry and has stability against disorder. As an illustrative example, we investigate a non-Hermitian extension of the Bernevig-Hughes-Zhang model. The helical edge states exhibit oscillatory dynamics due to their nonorthogonality as a unique non-Hermitian feature.

Kondo insulators are strongly correlated systems in which a clean insulating gap emerges only at very low temperature due to many-body effects involving localized $f$ electrons. However, certain Kondo insulators, like $\mathrm{Sm}{\mathrm{B}}_6$ and ${\mathrm{Ce}}_3\mathrm{Bi}{}_4\mathrm{Pt}{}_3$, display metallic behaviors at extremely low temperature and have defied current understanding. Recent advances in topological effects in materials have raised attention on the protected surface states in these “topological Kondo insulators” as a potential resolution to some of the puzzling behaviors. Here we resolve these puzzles via a different route, by showing that the emergent Kondo insulating scale is extremely vulnerable against a moderate degree of disorder, such that the gap is filled with a small number of states. Therefore, the real samples are probably never truly insulating and this in turn compromises the essential building block of topological considerations. Our results suggest strongly that systems like the Slater insulators would be a more promising direction to extend the realm of topology to strongly correlated systems.

We have performed angle-resolved photoemission spectroscopy on ${\mathrm{EuIn}}_2{\mathrm{As}}_2$ which is predicted to be an axion insulator in the antiferromagnetic state. By utilizing soft-x-ray and vacuum-ultraviolet photons, we revealed a three-dimensional hole pocket centered at the $\mathrm{\Gamma }$ point of the bulk Brillouin zone together with a heavily hole-doped surface state in the paramagnetic phase. Upon entering the antiferromagnetic phase, the band structure exhibits a marked reconstruction characterized by the emergence of an “M”-shaped bulk band near the Fermi level. The qualitative agreement with first-principles band-structure calculations suggests the occurrence of bulk-band inversion at the $\mathrm{\Gamma }$ point in the antiferromagnetic phase. We suggest that ${\mathrm{EuIn}}_2{\mathrm{As}}_2$ provides a good opportunity to study the exotic quantum phases associated with a possible axion-insulator phase.

We study a three-dimensional chiral second-order topological insulator (SOTI) subject to a magnetic field. Via its gauge field, the applied magnetic field influences the electronic motion on the lattice, and via the Zeeman effect, the field influences the electronic spin. We compare two approaches to the problem: an effective surface theory, and a full lattice calculation. The surface theory predicts a massive Dirac spectrum on each of the gapped surfaces, giving rise to Landau levels once the surfaces are pierced by magnetic flux. The surface theory qualitatively agrees with our lattice calculations, accurately predicting the surface gap as well as the spin and orbital components of the states at the edges of the surface Dirac bands. In the context of the lattice theory, we calculate the spectrum with and without magnetic field and find a deviation from the surface theory when a gauge field is applied. The energy of the lowest-lying Landau level is found closer to zero than is predicted by the surface theory, which leads to an observable magnetotransport signature: inside the surface gap, there exist different energy regions where either one or two chiral hinge modes propagate in either direction, quantizing the differential conductance to either one or two conductance quanta.

Using Abelian and non-Abelian topological orders in two-dimensional (2D) space and the different ways to glue them together via their gapped boundaries, we propose a systematic way to construct three-dimensional (3D) gapped states (and in other dimensions). The resulting states are called cellular topological states, which include gapped nonliquid states, as well as gapped liquid states in some special cases. Some new fracton states with fractal excitations are constructed even using 2D $Z_2$ topological order. More general cellular topological states can be constructed by connecting 2D domain walls between different 3D topological orders. The constructed cellular topological states can be viewed as fixed-point states for a reverse renormalization of gapped nonliquid states.

The study of symmetry-protected topological states in the presence of electron correlations has recently aroused great interest as rich and exotic phenomena can emerge. Here we report a concrete example by employing a large-scale unbiased quantum Monte Carlo study of the Kane-Mele model with cluster charge interactions. The ground-state phase diagram for the model at half filling is established. Our simulation identifies the coexistence of a symmetry-protected topological order with a symmetry-breaking Kekulé valence bond order and shows that the spontaneous symmetry breaking is accompanied by an interaction-driven topological phase transition (TPT). This TPT features the appearance of zeros of a single-particle Green's function and gap closing in a spin channel rather than a single-particle excitation spectrum and thus has no mean-field correspondence.

In flat-band systems, destructive interference leads to the localization of noninteracting particles and forbids their motion through the lattice. However, in the presence of interactions the overlap between neighboring single-particle localized eigenstates may enable the propagation of bound pairs of particles. In this work, we show how these interaction-induced hoppings can be tuned to obtain a variety of two-body topological states. In particular, we consider two interacting bosons loaded into the orbital angular momentum $l=1$ states of a diamond-chain lattice, wherein an effective $\pi$ flux may yield a completely flat single-particle energy landscape. In the weakly interacting limit, we derive effective single-particle models for the two-boson quasiparticles which provide an intuitive picture of how the topological states arise. By means of exact diagonalization calculations, we benchmark these states and we show that they are also present for strong interactions and away from the strict flat-band limit. Furthermore, we identify a set of doubly localized two-boson flat-band states that give rise to a special instance of Aharonov-Bohm cages for arbitrary interactions.

Motivated by recent experimental progress in the study of quantum systems far from equilibrium, we investigate the relation between several dynamical signatures of topology in the coherent time evolution after a quantum quench. Specifically, we study the conditions for the appearance of entanglement spectrum crossings, dynamical quantum phase transitions, and dynamical Chern numbers. For noninteracting models, we show that in general there is no direct relation between these three quantities. Instead, we relate the presence of level crossings in the entanglement spectrum to localized boundary modes that may not be of topological origin in the conventional sense. We exemplify our findings with explicit simulations of one-dimensional two-banded models. Finally, we investigate how interactions influence the presence of entanglement spectrum crossings and dynamical quantum phase transitions, by means of time-dependent density matrix renormalization group simulations.

A central tenet in the theory of quantum phase transitions (QPTs) is that a nonanalyticity in a ground-state energy implies a QPT. Here we report on a finding that challenges this assertion. As a case study we take a phase diagram of a one-dimensional band insulator with spin-orbit coupled electrons, supporting trivial, and topological gapped phases separated by intersecting critical surfaces. The intersections define multicritical lines across which the ground-state energy becomes nonanalytical, concurrent with a closing of the band gap, but with no phase transition taking place.