Kramers Pairs Research Articles

Dislocation Majorana bound states in iron-based superconductors

We show that lattice dislocations of topological iron-based superconductors such as FeTe1−xSex will intrinsically trap non-Abelian Majorana quasiparticles, in the absence of any external magnetic field. Our theory is motivated by the recent experimental observations of normal-state weak topology and surface magnetism that coexist with superconductivity in FeTe1−xSex, the combination of which naturally achieves an emergent second-order topological superconductivity in a two-dimensional subsystem spanned by screw or edge dislocations. This exemplifies a new embedded higher-order topological phase in class D, where Majorana zero modes appear around the “corners” of a low-dimensional embedded subsystem, instead of those of the full crystal. A nested domain wall theory is developed to understand the origin of these defect Majorana zero modes. When the surface magnetism is absent, we further find that s± pairing symmetry itself is capable of inducing a different type of class-DIII embedded higher-order topology with defect-bound Majorana Kramers pairs. We also provide detailed discussions on the real-world material candidates for our proposals, including FeTe1−xSex, LiFeAs, β-PdBi2, and heterostructures of bismuth, etc. Our work establishes lattice defects as a new venue to achieve high-temperature topological quantum information processing.

Realizing Majorana Kramers pairs in two-channel InAs-Al nanowires with highly misaligned electric fields

Realizing Majorana Kramers pairs in two-channel InAs-Al nanowires with highly misaligned electric fields

Quantum dot detects Majorana modes of both chiralities

A tunneling junction between normal electrode and a topological superconducting wire, mediated by a quantum dot, is considered theoretically. We show that the presence of the dot in the junction can be advantageous to Majorana zero modes identification. Namely, we demonstrate that for the dot strongly coupled to the wire, the Majorana mode from the upper chiral sub-band ”leaks” into the dot, providing supplementary information on Majorana mode formation. Thus, both the Zeeman-split dot sub-levels detect Majorana partners of a Kramers pair, formed at the wire end. The characteristic three-peak structures in both spin sectors of the spectral density of the dot, distinguish from the trivial scenario of one Andreev resonance at Fermi energy produced exclusively by the dot’s spin sub-levels.

Systematic study for two-dimensional Z2 topological phase transitions at high-symmetry points in all layer groups

We construct a general theory of $Z_2$ topological phase transitions in two-dimensional systems with time-reversal symmetry. We investigate the possibilities of $Z_2$ topological phase transitions at band inversions at all high-symmetry points in $k$-space in all the 80 layer groups. We exclude the layer groups with inversion symmetry because the $Z_2$ topological phase transition is known to be associated with band inversions with an exchange of parities. Among the other layer groups, we find 21 layer groups with insulator-to-insulator transitions with band inversion, and this problem is finally reduced to five point groups $C_3, C_4, C_6, S_4$, and $C_{3h}$. We show how the change of the $Z_2$ topological invariant at a band inversion is entirely determined by the irreps of occupied and unoccupied bands at the high-symmetry point. For example, in the case of $C_3$, we show that the $Z_2$ topological invariants change whenever the band inversion occurs between two Kramers pairs whose $C_3$ eigenvalues are $\{e^{\pi i / 3}, e^{-\pi i / 3}\}$ and $\{-1, -1\}$. These results are not included in the theory of symmetry-based indicators or topological quantum chemistry.

Fractional second-order topological insulator from a three-dimensional coupled-wires construction

We construct a three-dimensional second-order topological insulator with gapless helical hinge states from an array of weakly tunnel-coupled Rashba nanowires. For suitably chosen interwire tunnelings, we demonstrate that the system has a fully gapped bulk as well as fully gapped surfaces, but hosts a Kramers pair of gapless helical hinge states propagating along a path of hinges that is determined by the hierarchy of interwire tunnelings and the boundary termination of the system. Furthermore, the coupled-wires approach allows us to incorporate electron-electron interactions into our description. At suitable filling factors of the individual wires, we show that sufficiently strong electron-electron interactions can drive the system into a fractional second-order topological insulator phase with hinge states carrying only a fraction $e/p$ of the electronic charge $e$ for an odd integer $p$.

Quantum Computing with Majorana Kramers Pairs

We propose a universal gate set acting on a qubit formed by the degenerate ground states of a Coulomb-blockaded time-reversal invariant topological superconductor island with spatially separated Majorana Kramers pairs: the "Majorana Kramers qubit." All gate operations are implemented by coupling the Majorana Kramers pairs to conventional superconducting leads. Interestingly, in such an all-superconducting device, the energy gap of the leads provides another layer of protection from quasiparticle poisoning independent of the island charging energy. Moreover, the absence of strong magnetic fields-which typically reduce the superconducting gap size of the island-suggests a unique robustness of our qubit to quasiparticle poisoning due to thermal excitations. Consequently, the Majorana Kramers qubit should benefit from prolonged coherence times and may provide an alternative route to a Majorana-based quantum computer.

Topological zero-dimensional defect and flux states in three-dimensional insulators

In insulating crystals, it was previously shown that defects with two fewer dimensions than the bulk can bind topological electronic states. We here further extend the classification of topological defect states by demonstrating that the corners of crystalline defects with integer Burgers vectors can bind 0D higher-order end (HEND) states with anomalous charge and spin. We demonstrate that HEND states are intrinsic topological consequences of the bulk electronic structure and introduce new bulk topological invariants that are predictive of HEND dislocation states in solid-state materials. We demonstrate the presence of first-order 0D defect states in PbTe monolayers and HEND states in 3D SnTe crystals. We relate our analysis to magnetic flux insertion in insulating crystals. We find that π-flux tubes in inversion- and time-reversal-symmetric (helical) higher-order topological insulators bind Kramers pairs of spin-charge-separated HEND states, which represent observable signatures of anomalous surface half quantum spin Hall states.

Corner- and sublattice-sensitive Majorana zero modes on the kagome lattice

In a first-order topological phase with sublattice degrees of freedom, a change in the boundary sublattice termination has no effect on the existence of gapless boundary states in dimensions higher than one. However, such a change may strongly affect the physical properties of those boundary states. Motivated by this observation, we perform a systematic study of the impact of sublattice terminations on the boundary physics on the two-dimensional kagome lattice. We find that the energies of the Dirac points of helical edge states in two-dimensional first-order topological kagome insulators sensitively depend on the terminating sublattices at the edge. Remarkably, this property admits the realization of a time-reversal invariant second-order topological superconducting phase with highly controllable Majorana Kramers pairs at the corners and sublattice domain walls by putting the topological kagome insulator in proximity to a $d$-wave superconductor. Moreover, substituting the $d$-wave superconductor with a conventional $s$-wave superconductor, we find that highly controllable Majorana zero modes can also be realized at the corners and sublattice domain walls if an in-plane Zeeman field is additionally applied. Our study reveals promising platforms to implement highly controllable Majorana zero modes.

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.

Third-order topological insulators with wallpaper fermions in Tl4PbTe3 and Tl4SnTe3

Nonsymmorphic symmetries open up horizons of exotic topological boundary states and even generalize the bulk–boundary correspondence, which, however, the third-order topological insulator in electronic materials are still unknown. Here, by means of the symmetry analysis and k · p models, we uncover the emergence of long-awaited third-order topological insulators and the wallpaper fermions in space group I4/mcm (No.140). Based on this, we present the hourglass fermion, fourfold-degenerate Dirac fermion, and Möbius fermion in the (001) surface of Tl4XTe3 (X = Pb/Sn) with a nonsymmorphic wallpaper group p4g. Remarkably, 16 helical corner states reside on eight corners in Kramers pair, rendering the real electronic material of third-order topological insulators. More importantly, a time-reversal polarized octupole polarization is defined to uncover the nontrivial third-order topology, as is implemented by the 2nd and 3rd order Wilson loop calculations. Our results could considerably broaden the range of wallpaper fermions and lay the foundation for future experimental investigations of third-order topological insulators.

Two-particle Berry phase mechanism for Dirac and Majorana Kramers pairs of corner modes

We uncover a novel two-particle Berry phase mechanism to realize exotic corner modes in second-order topological insulators (TIs) and topological superconductors (TSCs) with time-reversal symmetry. We show that the nontrivial pseudospin textures of edge states in two different types of two-dimensional TIs give rise to the novel two-particle geometric phases in the particle-hole and particle-particle channels, respectively, for which the edge mass domain walls or intrinsic $\pi$-junctions emerge across corners when an external magnetic field or $s$-wave superconductivity is considered, hosting Dirac or Majorana corner modes. Remarkably, with this mechanism we predict the Majorana Kramers pair of corner modes by directly coupling the edge of a type-II time-reversal invariant TI to a uniform $s$-wave SC, in sharp contrast to the previous proposals which rely on unconventional SC pairing or complex setting for fine-tuned SC $\pi$-junction. We find Au/GaAs(111) to be a realistic material candidate for realizing such Majorana Kramers pair of corner modes.

Higher-order topological superconductivity from repulsive interactions in kagome and honeycomb systems

We discuss a pairing mechanism in interacting two-dimensional multipartite lattices that intrinsically leads to a second order topological superconducting state with a spatially modulated gap. When the chemical potential is close to Dirac points, oppositely moving electrons on the Fermi surface undergo an interference phenomenon in which the Berry phase converts a repulsive electron–electron interaction into an effective attraction. The topology of the superconducting phase manifests as gapped edge modes in the quasiparticle spectrum and Majorana Kramers pairs at the corners. We present symmetry arguments which constrain the possible form of the electron–electron interactions in these systems and classify the possible superconducting phases which result. Exact diagonalization of the Bogoliubov-de Gennes Hamiltonian confirms the existence of gapped edge states and Majorana corner states, which strongly depend on the spatial structure of the gap. Possible applications to vanadium-based superconducting kagome metals AV3Sb5 (A = K, Rb, Cs) are discussed.

Strain-induced spin vortex and Majorana Kramers pairs in doped topological insulators with nematic superconductivity

Using Ginzburg-Landau approach we show that the strain of the nematic superconductor can generate a specific (nematic) vorticity. In the case of doped topological insulators that vorticity forms a spin vortex. We find two types of topologically different spin vortices that either enhance (type I) or suppress (type II) superconductivity far from the vortex core. We apply Bogoliubov-de Geunnes equations to study electronic states in the nematic superconductor with spin vortices. We find that in the case of the vortex of type I, zero-energy states are localized near the vortex core. These states can be identified as Majorana Kramer's pairs. In the case of the vortex of type II, zero-energy states form Majorana flat bands. Thus, we establish a non-trivial connection between the strain and Majorana fermions in the doped topological insulators with nematic superconductivity.

Fragility of Z2 topological invariant characterizing triplet excitations in a bilayer kagome magnet

The discovery by Kane and Mele of a model of spinful electrons characterized by a $\mathcal{Z}_2$ topological invariant had a lasting effect on the study of electronic band structures. Given this, it is natural to ask whether similar topology can be found in the band-like excitations of magnetic insulators, and recently models supporting $\mathcal{Z}_2$ topological invariants have been proposed for both magnon [Kondo et al. Phys. Rev. B 99, 041110(R) (2019)] and triplet [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100, 020407 (2019)] excitations. In both cases, magnetic excitations form time--reversal (TR) partners, which mimic the Kramers pairs of electrons in the Kane-Mele model but do not enjoy the same type of symmetry protection. In this paper, we revisit this problem in the context of the triplet excitations of a spin model on the bilayer kagome lattice. Here the triplet excitations provide a faithful analog of the Kane-Mele model as long as the Hamiltonian preserves the TR$\times$U(1) symmetry. We find that exchange anisotropies, allowed by the point group and typical in realistic models, break the required TR$\times$U(1) symmetry and instantly destroy the $\mathcal{Z}_2$ band topology. We further consider the effects of TR breaking by an applied magnetic field. In this case, the lifting of spin-degeneracy leads to a triplet Chern insulator, which is stable against the breaking of TR$\times$U(1) symmetry. Kagome bands realize both a quadratic and a linear band touching, and we provide a thorough characterization of the Berry curvature associated with both cases. We also calculate the triplet-mediated spin Nernst and thermal Hall signals which could be measured in experiments. These results suggest that the $\mathcal{Z}_2$ topology of band-like excitations in magnets may be intrinsically fragile compared to their electronic counterparts.

Electric Multipoles of Double Majorana Kramers Pairs

A single Majorana Kramers pair hosts only one component of the magnetic multipole. This can be used to determine the bulk Cooper-pair symmetry through surface-sensitive spectroscopic measurements, either by applying a magnetic field or by using a ferromagnet/superconductor junction. This paper proposes that the electric response, which is free from the Meissner effect, can be used an alternative method to measure the bulk Cooper-pair symmetry in time-reversal-invariant superconductors with double Majorana Kramers pairs. The relationships among electric multipoles, strain tensors and superconducting symmetries under a given wallpaper group on the surfaces of topological crystalline superconductors are shown. This study also reveals that only a specific irreducible representation of a uniform strain yields a gap in the double Majorana Kramers pairs for the topological-crystalline-superconductor candidate Sr$_3$SnO. This highlights the viability of electric detection regarding Cooper-pair symmetry.

Topological electronic states and thermoelectric transport at phase boundaries in single-layer WSe2: an effective Hamiltonian theory

Monolayer transition metal dichalcogenides in the distorted octahedral 1T′ phase exhibit a large bulk bandgap and gapless boundary states, which is an asset in the ongoing quest for topological electronics. In single-layer tungsten diselenide (WSe2), the boundary states have been observed at well ordered interfaces between 1T′ and semiconducting (1H) phases. This paper proposes an effective 4-band theory for the boundary states in single-layer WSe2, describing a Kramers pair of in-gap states as well as the behaviour at the spectrum termination points on the conduction and valence bands of the 1T′ phase. The spectrum termination points determine the temperature and chemical potential dependences of the ballistic conductance and thermopower at the phase boundary. Notably, the thermopower shows an ambipolar behaviour, changing the sign in the bandgap of the 1T′–WSe2 and reflecting its particle-hole asymmetry. The theory establishes a link between the bulk band structure and ballistic boundary transport in single-layer WSe2 and is applicable to a range of related topological materials.

Majorana multipole response: General theory and application to wallpaper groups

Whereas identification of Cooper-pair symmetry is the first and crucial step in the investigation of unconventional superconductors, only a few have been established so far because of their own difficulties. To solve this problem, we develop a theory for identification of pairing symmetry using knowledge of topological superconductivity. Establishing the multipole theory of emergent Majorana fermions in time-reversal-invariant topological superconductors, we discover a one-to-one correspondence between the electromagnetic response of Majorana fermions and Cooper-pair symmetry. The emergent Majorana fermions host magnetic structures that share the same irreducible representation with Cooper pairs under crystalline symmetry. We furthermore reveal that Majorana fermions in high-spin or nonsymmorphic superconductors may exhibit magnetic octupole responses, which give a direct evidence of these exotic superconducting states. Electric responses of multiple Majorana Kramers pairs are also clarified. Our theory provides the fundamentals for identification of unconventional Cooper pairings through surface-spin-sensitive measurements as well as that for manipulation of Majorana fermions by external electromagnetic fields.

Conductance suppression by nonmagnetic point defects in helical edge channels of two-dimensional topological insulators

We study backscattering of electrons and conductance suppression in a helical edge channel in two-dimensional topological insulators with broken axial spin symmetry in the presence of nonmagnetic point defects that create bound states. In this system the tunneling coupling of the edge and bound states results in the formation of composite helical edge states in which all four partners of both Kramers pairs of the conventional helical edge states and bound states are mixed. The backscattering is considered as a result of inelastic two-particle scattering of electrons, which are in these composite states. Within this approach we find that sufficiently strong backscattering occurs even if the defect creates only one energy level. The effect is caused by electron transitions between the composite states with energy near the bound state level. We study the deviation from the quantized conductance due to scattering by a single defect as a function of temperature and Fermi level. The results are generalized to the case of scattering by many different defects with energy levels distributed over the band gap. In this case, the conductance deviation turns out to be quite strong and comparable with experiment even at a sufficiently low density of defects. Interestingly, under certain conditions, the temperature dependence of the conductance deviation becomes very weak over a wide temperature range.

Evolution of Topological Surface States Following Sb Layer Adsorption on Bi2Se3.

Thin antimony layers adsorbed on bismuth selenide () present an exciting topological insulator system. Much recent effort has been made to understand the synthesis and electronic properties of the heterostructure, particularly the migration of the topological surface states under adsorption. However, the intertwinement of the topological surface states of the pristine substrate with the Sb adlayer remains unclear. In this theoretical work, we apply density functional theory (DFT) to model heterostructures of single and double atomic layers of Sb on a bismuth selenide substrate. We thereby discuss established and alternative structural models, as well as the hybridization of topological surface states with the Sb states. Concerning the geometry, we reveal the possibility of structures with inverted Sb layers which are energetically close to the established ones. The formation energy differences are below 10 meV/atom. Concerning the hybridization, we trace the band structure evolution as a function of the adlayer-substrate distance. By following changes in the connection between the Kramers pairs, we extract a series of topological phase transitions. This allows us to explain the origin of the complex band structure, and ultimately complete our knowledge about this peculiar system.

Magnetic response of Majorana Kramers pairs with an order-two symmetry

We study an intrinsic relation between the topology of bulk electronic states and magnetic responses of Majorana Kramers pairs, Kramers pairs of Majorana fermions, on a surface of time-reversal-invariant topological superconductors. Majorana Kramers pairs respond to an applied magnetic field anisotropically due to the interplay between time-reversal and crystal symmetries. In this paper, we propose a systematic procedure to determine such surface magnetic responses in systems with an order-two symmetry. From the analysis of topological invariants associated with an order-two symmetry, it is found that magnetic responses are classified into four types, which are attributed to different topological invariants and exhibit distinguishable, characteristic magnetic responses. For a Kramers pair of Majorana fermions protected by $\mathbb{Z}_2$ topological invariants, we clarify that types of magnetic responses are determined only from Fermi--surface topology and symmetry of Cooper pairs. Finally, we apply our theory to the topological nonsymmorphic crystalline superconducting state in UCoGe, which exhibits a biaxially anisotropic magnetic response.