Bulk Gap Research Articles

Robust propagating in-gap modes due to spin-orbit domain walls in graphene

Recently, great experimental efforts towards designing topological electronic states have been invested in layered incommensurate heterostructures which form various nano- and meso-scale domains. In particular, it has become clear that a delicate interplay of different spin-orbit terms is induced in graphene on transition metal dichalcogenide substrates. We therefore theoretically study various types of domain walls in spin-orbit coupling in graphene looking for robust one-dimensional propagating electronic states. To do so, we use an interface Chern number and a spectral flow analysis in the low-energy theory and contrast our results to the standard arguments based on valley-Chern numbers or Chern numbers in continuum models. Surprisingly, we find that a sign-changing domain wall in valley-Zeeman spin-orbit coupling binds two robust Kramers pairs, within the bulk gap opened due to a simultaneous presence of Rashba coupling. We also study the robustness to symmetry breaking and lattice backscattering effects in tight-binding models. We show an explicit mapping of our valley-Zeeman domain wall to a domain wall in gated spinless bilayer graphene. We discuss the possible spectroscopic and transport signatures of various types of spin-orbit coupling domain walls in heterostructures.

Probing Majorana modes via local spin dynamics

We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here, the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for p-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding frequencies in the two-time correlation function. We derive perturbative interactions that map the KH spin chain onto the topological regime of Kitaev's fermionic model, thus opening up a bulk gap whilst retaining almost degenerate modes in the mesoscopic regime, i.e., for finite system sizes. This showcases the emergence of Majorana modes in a chain of effective dimers. Here, the binding energy within each unit cell competes with the inter-dimer coupling to generate a finite size energy gap, in analogy with local energy terms in the transverse-field Ising model. These modes give rise to long coherence times of local spins located at the system edges. By breaking the local symmetry in each dimer, one can also observe a second class of Majorana modes in terms of a beating frequency in the two-time correlations function of the edge spin. Furthermore, we develop a scenario for realizing these model predictions in ion-trap quantum simulators with collective addressing of the ions.

Thermodynamic properties of perturbed monolayer PbBiI

In this paper, we theoretically investigate the influence of external electric field, Zeeman splitting field, and dilute charged impurity on the thermodynamic features of the PbBiI monolayer. Green’s function approach, the Born approximation, and the tight-binding model are employed to address these effects. We find the responses of the electronic heat capacity (HC) and Pauli spin paramagnetic susceptibility (PSPS) to the symmetry breaking originated from the presence of the perturbations. The results show that the electric (Zeeman) field leads to an increase (decrease) in the bulk gap of the system, resulting in a gapless phase at a critical Zeeman field. Correspondingly, HC and PSPS results confirm almost zero responses at the appeared critical Zeeman field. On the other hand, critical impurity characters emerge within the validity regime of the Born approximation. Finally, the combined effects of these perturbations are compared with the findings from the individual ones.

Optical refraction and absorption spectra in perturbed monolayer PbBiI

We investigate the optical properties of a noncentrosymmetric quantum spin Hall insulator in the presence of electric and magnetic fields. We employ the Kubo formula to engineer the optical redshift and blueshift phenomena through the interplay between the bulk gap, the Rashba-like gap, and the external fields. Intriguingly, for a range of electric (magnetic) fields, the refracted and absorbed lights exhibit direct (indirect) interband optical transitions. Accordingly, for a critical electric field, optical refraction and absorption spectrum decreases gradually and becomes constant, respectively. While for a critical magnetic field, step-like jumps appear in the optical coefficients. All these findings are discussed in the context of symmetry breaking.

A Bulk Spectral Gap in the Presence of Edge States for a Truncated Pseudopotential

We study the low-energy properties of a truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry. For this Hamiltonian with either open or periodic boundary conditions, we prove a spectral gap above the highly degenerate ground-state space which is uniform in the volume and particle number. Our proofs rely on identifying invariant subspaces to which we apply gap-estimate methods previously developed only for quantum spin Hamiltonians. In the case of open boundary conditions, the lower bound on the spectral gap accurately reflects the presence of edge states, which do not persist into the bulk. Customizing the gap technique to the invariant subspace, we avoid the edge states and establish a more precise estimate on the bulk gap in the case of periodic boundary conditions.

Acoustic Realization of Surface-Obstructed Topological Insulators.

Recently, higher-order topological insulators have been attracting extensive interest. Unlike the conventional topological insulators that demand bulk gap closings at transition points, the higher-order band topology can be changed without bulk closure and exhibits as an obstruction of higher-dimensional boundary states. Here, we report the first experimental realization of three-dimensional surface-obstructed topological insulators with using acoustic crystals. Our acoustic measurements demonstrate unambiguously the emergence of one-dimensional topological hinge states in the middle of the bulk and surface band gaps, as a direct manifestation of the higher-order band topology. Together with comparative measurements for the trivial and phase-transition-point insulators, our experimental data conclusively evidence the unique bulk-boundary physics for the surface-obstructed band topology. That is, the topological phase transition is determined by the closure of the surface gap, rather than by closing the bulk gap. Our study might spur on new activities to deepen the understanding of such elusive topological phases.

Edge states in proximitized graphene ribbons and flakes in a perpendicular magnetic field: Emergence of lone pseudohelical pairs and pure spin-current states

We investigate the formation of edge states in graphene ribbons and flakes with proximity-induced valley-Zeeman and Rashba spin-orbit couplings in the presence of a perpendicular magnetic field $B$. Two types of edge states appear in the spin-orbit gap at the Fermi level at zero field: strongly localized pseudohelical (intervalley) states and weakly localized intravalley states. We show that if the magnetic field is stronger than a crossover field ${B}_{c}$, which is a few milliteslas for realistic systems such as $\mathrm{graphene}/\mathrm{W}{\mathrm{Se}}_{2}$, only the pseudohelical edge states remain in zigzag graphene ribbons; the intravalley states disappear. The crossover is directly related to the closing and reopening of the bulk gap formed between nonzero Landau levels. Remarkably, in finite flakes the pseudohelical states undergo perfect reflection at the armchair edges if $B>{B}_{c}$, forming standing waves at the zigzag edges. These standing waves comprise two counterpropagating pseudohelical states; so while they carry no charge current, they do carry (pure) spin current.

Boundary Modes from Periodic Magnetic and Pseudomagnetic Fields in Graphene.

Single-layer graphene subject to periodic lateral strains is an artificial crystal that can support boundary spectra with an intrinsic polarity. This is analyzed by comparing the effects of periodic magnetic fields and strain-induced pseudomagnetic fields that, respectively, break and preserve time-reversal symmetry. In the former case, a Chern classification of the superlattice minibands with zero total magnetic flux enforces single counterpropagating modes traversing each bulk gap on opposite boundaries of a nanoribbon. For the pseudomagnetic field, pairs of counterpropagating modes migrate to the same boundary where they provide well-developed valley-helical transport channels on a single zigzag edge. We discuss possible schemes for implementing this situation and their experimental signatures.

Tuning the surface states of TiO2 using Cu5 atomic clusters

If the ability of Cu5 atomic quantum clusters (AQCs) to create states within the bulk gap of TiO2 persists in the presence of oxygen, then the absorption spectrum of the AQC-TiO2 complex could be matched to the peak of the solar spectrum, thereby enhancing its photocatalytic activity under ambient conditions. Here we demonstrate that this is indeed the case, by examining the electronic structure of Cu5 AQCs adsorbed on a rutile (110) TiO2 surface in the presence of oxygen. We find that adsorbed oxygen changes the most stable AQC isomer from a 2-dimensional trapezoidal structure to a 3-dimensional bipyramidal structure. Furthermore, the bipyramidal AQC creates two ferromagnetically coupled polarons in the TiO2. In contrast, the trapezoidal AQC only creates a single polaron. In the presence of oxygen, these polaronic states associated with the bare cluster are joined by further mid-gap energy levels, formed from hybrid states located on copper and oxygen atoms. Dissociated oxygen-dimer adsorption on Cu5 generally leads to significant deformation of the whole cluster, which accounts for the dramatic drop of the total energy compared to molecular oxygen dimer adsorption. The ability of Cu5 AQCs to create states within the bulk gap of TiO2 under ambient conditions demonstrates that the absorption spectrum of the AQC-TiO2 complex is more closely matched to the peak of the solar spectrum, than either the AQC or TiO2 alone.

From atomic semimetal to topological nontrivial insulator

Topological band insulators and (semi-) metals can arise out of atomic insulators when the hopping strength between electrons increases. Such topological phases are separated from the atomic insulator by a bulk gap closing. In this work, we show that in many (magnetic) space groups, the crystals with certain Wyckoff positions and orbitals being occupied must be semimetal or metals in the atomic limit, e.g. the hopping strength between electrons is infinite weak but not vanishing, which then are termed atomic (semi-)metals (ASMs). We derive a sufficient condition for realizing ASMs in spinless and spinful systems. Remarkably, we find that increasing the hopping strength between electrons may transform an ASM into an insulator with both symmetries and electron fillings of crystal are preserved. The induced insulators inevitably are topologically non-trivial and at least are obstructed atomic insulators (OAIs) that are labeled as trivial insulator in topological quantum chemistry website. Particularly, using silicon as an example, we show ASM criterion can discover the OAIs missed by the recently proposed criterion of filling enforced OAI. Our work not only establishes an efficient way to identify and design non-trivial insulators but also predicts that the group-IV elemental semiconductors are ideal candidate materials for OAI.

Crossed Andreev reflection in zigzag phosphorene nanoribbon based ferromagnet/superconductor/ferromagnet junctions

We study the crossed Andreev reflection in zigzag phosphorene nanoribbon based ferromagnet/superconductor/ferromagnet junction. Only edge states, which are entirely detached from the bulk gap, involved in the transport processes. The perfect crossed Andreev reflection, with the maximal nonlocal conductance -2e^{2}/h, is addressed by setting the electric potentials of the leads and device properly. At this situation, the local Andreev reflection and the electron tunneling are completely eliminated, the incoming electrons can only be reflected as electrons or transmitted as holes, corresponding to the electron reflection and the crossed Andreev reflection respectively.The nonlocal conductance oscillates periodically with the length and the electric potential of the superconductor. Our study shows that the phosphorene based junction can be used as the quantum device to generate entangled-electrons.

Vortex core near planar defects in a clean layered superconductor

We investigate the structure of quasiparticle states localized in a core of an Abrikosov vortex in a clean layered superconductor in the presence of planar defects. It is shown that even a highly transparent defect opens a minigap at the Fermi energy. Its magnitude, $E_g\sim\Delta\sqrt{R}$, exceeds the mean level spacing for the chiral branch, $\omega_0\sim\Delta^2/E_F$, already for very small values of the reflection coefficient off the defect, $R\ll1$ ($\Delta$ is the bulk gap). For $R\gtrsim\sqrt{\Delta/E_F}$, formation of the minigap is accompanied by the appearance of subgap states localized along the defect, in accordance with [A. V. Samokhvalov \emph{et al.}, Phys.\ Rev.\ B \textbf{102}, 174501 (2020)]. The minigap takes its maximal value for the vortex located right at the defect, decreases with increasing the distance $b$ from the defect, and closes when $k_Fb\sim (\Delta/\omega_0)\sqrt{R}$. We also study various configurations of several planar defects (few crossing planes, stars, periodic structures). Although the minigap remains, a strong commensurability effect is observed. For two crossing planar defects, the magnitude of the minigap strongly depends on how close the intersection angle is to a rational fraction of $\pi$.

Nanoscale studies of electric field effects on monolayer 1T\u2032-WTe2

Monolayer 1 T′-WTe2 is a quantum spin Hall insulator with a gapped 2D-bulk and gapless helical edge states persisting to temperatures ~100 K. Despite the far-ranging interest, the magnitude of the bulk gap, the effect of gating on the 2D-band structure, as well the role interactions are not established. In this work we use STM spectroscopy to measure the intrinsic bulk gap of monolayer 1 T′-WTe2 and show that gate induced electric fields cause large changes of the gap magnitude. Our first-principles DFT-derived tight-binding model reveal that a combination of spatial localization of the conduction and valance bands and Rashba-like spin-orbit coupling leads to a gating induced spin-splitting of the 2D-bulk bands in the tens of meV, thereby reducing the band gap. Our work explains the large sensitivity of the band structure to electric fields and suggests a new avenue for realizing proximity induced non-trivial superconductivity in monolayer 1 T′-WTe2.

Response of quantum spin Hall insulators to Zeeman fields, and device design based on stanene

The nontrivial topology of materials has been one of the keys for breaking through to a new stage of electronics involving phenomena such as topologically protected quantum spin Hall (QSH) states and quantum anomalous Hall (QAH) states. In this work, we study the Kane-Mele model with next-nearest-neighbor Rashba spin-orbit coupling (SOC), which accurately describes the QSH effect of two-dimensional group IV A materials. By investigating electronic structures with different SOCs and Zeeman fields, we obtain various phase transitions and plot rich phase diagrams where a time-reversal-symmetry-broken QSH insulator and three types of QAH insulators are identified based on calculations of topological invariants, including total and spin Chern numbers and Wannier function centers. In addition, the edge modes and spin texture within the bulk gap are also nontrivial. Based on the results of the model, we also study a real QSH material, stanene, with a large bulk gap and propose a scheme based on its transport properties to design a spin-resolved filter.

Topological edge and corner states and fractional corner charges in blue phosphorene

We theoretically study emergent edge and corner states in monolayer blue phosphorus (blue phosphorene) using the first-principles calculation and tight-binding model. We show that the existence of the Wannier orbitals at every bond center yields edge states both in zigzag and armchair nanoribbons. The properties of the edge states can be well described by a simple effective Hamiltonian for uncoupled edge orbitals, where the structural relaxation near the boundary significantly affects the edge band structure. For corner states, we examine two types of corner structures consisting of zigzag and armchair edges, where we find that multiple corner states emerge in the bulk gap as a consequence of hybridization of edge and corner uncoupled orbitals. In the armchair corner, in particular, we demonstrate that corner states appear right at the Fermi energy, which leads to the emergence of fractional corner charge due to filling anomaly. Finally, we discuss the relationship between blue phosphorene and black phosphorene, and show that two systems share the equivalent Wannier orbital positions and similar edge/corner state properties even though their atomic structures are totally different.

Higher-order topological superconductors from Weyl semimetals

We propose that doped Weyl semimetals with {time-reversal and certain crystalline symmetries} are natural candidates to realize higher-order topological superconductors, which exhibit a fully gapped bulk while the surface hosts robust gapless chiral hinge states. We show that in such a doped Weyl semimetal, a featureless finite-range attractive interaction favors a p+ip pairing symmetry. By analyzing its topological properties, we identify such a chiral pairing state as a higher-order topological superconductor, which depending on the existence of a four-fold rotoinversion symmetry, is either intrinsic, {meaning that the corresponding hinge states can only be removed by closing the bulk gap, rather than modifying the surface states}, or extrinsic. We achieve this understanding via various methods recently developed for higher-order topology, including Wannier representability, Wannier spectrum, and defect classification approaches. For the four-fold rotoinversion symmetric case, we provide a complete classification of the higher-order topological superconductors. We show that such second-order topological superconductors exhibit chiral hinge modes that are robust in the absence of interaction effects but can be eliminated at the cost of introducing surface topological order.

Kekulé Lattice in Graphdiyne: Coexistence of Phononic and Electronic Second-Order Topological Insulator.

Topological physics has been extensively studied in different kinds of bosonic and Fermionic systems, but the coexistence of topological phonons and electrons in one single material has seldom been reported. Recently, graphdiyne has been proposed as a two-dimensional (2D) electronic second-order topological insulator (SOTI). In this work, we found that graphdiyne is equivalent to Kekulé lattice, also realizing a 2D phononic SOTI in both out-of-plane and in-plane modes. Depending on edge terminations, the characterized topological corner states can be either inside or outside the bulk gap and are tunable by the local corner potential. Most remarkably, a unique selectivity of space and symmetry is revealed in the electron-phonon coupling between the localized phononic and electronic topological corner states. Our results not only demonstrate the phononic higher-order band topology in a real carbon material but also provide an opportunity to investigate the interplay between phononic and electronic higher-order topological states.

Helical conducting edge states in narrow-gap semiconductors without band inversion

Band inversion accompanied by the emergence of gapless and helical edge states at the boundary is a hallmark of 2D topological insulators. However, a recent experiment reports the existence of edge states before band inversion in InAs/GaSb quantum wells [F. Nichele et al., New J. Phys. 18, 083005 (2016)]. The underlying physics remains an open problem. Here we provide a possible solution by showing that helical edge states emerge before band inversion as long as the bulk gap falls below a positive threshold determined by the particle-hole asymmetry and interband coupling. In the presence of quantum confinement or local electrostatic potential near the boundary, these edge states may enter the gap of bulk states and be detected as edge conductance in transport measurements. Our work reveals the possible existence of the helical conducting edge states in a large class of narrow-gap semiconductors.

Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires

Majorana bound states (MBS) and Andreev bound states (ABS) in realistic Majorana nanowires setups have similar experimental signatures which make them hard to distinguishing one from the other. Here, we characterize the continuous Majorana/Andreev crossover interpolating between fully-separated, partially-separated, and fully-overlapping Majorana modes, in terms of global and local topological invariants, fermion parity, quasiparticle densities, Majorana pseudospin and spin polarizations, density overlaps and transition probabilities between opposite Majorana components. We found that in inhomogeneous wires, the transition between fully-overlapping trivial ABS and nontrivial MBS does not necessarily mandate the closing of the bulk gap of quasiparticle excitations, but a simple parity crossing of partially-separated Majorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate that fully-separated and fully-overlapping Majorana modes correspond to the two limiting cases at the opposite sides of a continuous crossover: the only distinction between the two can be obtained by estimating the degree of separations of the Majorana components. This result does not contradict the bulk-edge correspondence: indeed, the field inhomogeneities driving the Majorana/Andreev crossover have a length scale comparable with the nanowire length, and therefore correspond to a nonlocal perturbation which breaks the topological protection of the MBS.

Comparative study on epitaxial growth of stanene and bismuthene on InSb(111) substrate

Two-dimensional topological insulator (2DTI) with a large bandgap is prerequisite for potentially observing quantum spin Hall and other quantum phenomena at room-temperature. At present, the synthesis of such materials possesses formidable challenge. In this work, we report our experimental results on synthesis of large-gap 2DTI stanene and bismuthene on B-faced InSb(111) substrate by using molecular beam epitaxy technology. We find that both the stanene and bismuthene can be synthesized by following the forming of a wetting layer on InSb(111) substrate, but with different prospects. On the one hand, it is found that the binding energy between Sn and the substrate is not strong enough to compete the binding force between Sn atoms during the post annealing, thus resulting in a wetting layer composed of many small domains. It significantly restricts the quality of the stanene epilayers. On the other hand, the Bi atoms on InSb(111) are found more stable than the Sn atoms on InSb(111), resulting in a uniform wetting layer which can be optimized by adjusting substrate temperature and post-annealing conditions. Large size and single crystal bismuthene domains have been observed under the STM measurement, which also indicates a bulk gap of ～0.15 eV and metallic edge states.