When the surface states of a topological insulator becomes superconducting, topological superconductivity can be obtained, and each vortex on the surface can host one single Majorana zero-energy mode which is usually a wave packet decaying exponentially off the vortex core. Here, we predict stable Majorana zero-energy mode whose wave function is ring-shape, dubbed as annular Majorana mode, in the superconducting vortex in topological insulators respecting 3-fold or 6-fold rotational symmetry. Such topological insulators are featured with a single nonlinear Dirac cone located at or three linear Dirac cones at M̅ in the surface Brillouin zone. The annular Majorana mode originates from the effective chiral f-wave superconductivity on the nonlinear Dirac cone in the former case and the interference of the effective chiral p-wave superconductivity on the three linear Dirac cones in the latter. In both cases, the annular Majorana mode is stabilized by the rotational symmetry and the winding number 3 carried by the surface states. Candidate materials supporting the annular Majorana mode are predicted. Our work provides new insights into the topological superconductivity in superconducting topological insulators.

Topological superconductivity has attracted great interest due to its fundamental significance for realizing Majorana quasiparticles and fault-tolerant quantum computation. Nonsymmorphic superconductors, with symmetry-protected nontrivial electronic structures, offer a promising route to exotic topological superconducting states, yet experimental realizations remain scarce. Here we identify nonsymmorphic compound PtPb4 as a robust platform hosting superconductivity with spontaneous rotational symmetry breaking and nontrivial zero-energy modes. PtPb4 crystallizes in a frustrated Shastry-Sutherland lattice and exhibits nontrivial band topology. By combining in-plane and out-of-plane resistivity measurements, pronounced twofold anisotropy is observed in both the superconducting state and the upper critical field, evidencing spontaneous rotational symmetry breaking. Scanning tunneling microscopy/spectroscopy further reveal twofold-symmetric magnetic vortices, providing direct real-space evidence for the symmetry-broken superconducting state. Notably, a robust zero-energy vortex bound state emerges and persists without spatial splitting over extended distances, consistent with the characteristics expected for Majorana bound state. These findings uncover an exotic superconducting state in PtPb4 and establish a promising platform for exploring topological superconductivity and superconducting quantum devices.

Sublattice dichotomy in monolayer FeSe/SrTiO 3 , signaling the breaking of symmetries exchanging the two Fe sublattices, has recently been reported. We propose that interband pairing serves as the origin of this dichotomy, regardless of whether the symmetries are broken in the normal state or in the pairing state. If symmetry breaking occurs in the normal state, the Fermi surfaces are sublattice-polarized, and the intersublattice d-wave pairing naturally acts as interband pairing, reproducing the observed dichotomy in the spectra. Alternatively, if symmetry breaking takes place in the pairing state, the dichotomy arises from the coexistence of intraband and interband pairing, with the constraint that interband pairings share the same sign while intraband pairings carry opposite signs. In both cases, interband pairing is indispensable, establishing it as a key ingredient for understanding superconductivity in monolayer FeSe/SrTiO 3 .

Sublattice dichotomy in monolayer FeSe/SrTiO3,signaling the breaking of symmetries exchanging the two Fe sublattices,has recently been reported.We propose that interband pairing serves as the origin of this dichotomy,regardless of whether the symmetries are broken in the normal state or in the pairing state.If symmetry breaking occurs in the normal state,the Fermi surfaces are sublattice-polarized,and the intersublattice d-wave pairing naturally acts as interband pairing,reproducing the observed dichotomy in the spectra.Alternatively,if symmetry breaking takes place in the pairing state,the dichotomy arises from the coexistence of intraband and interband pairing,with the constraint that interband pairings share the same sign while intraband pairings carry opposite signs.In both cases,interband pairing is indispensable,establishing it as a key ingredient for understanding superconductivity in monolayer FeSe/SrTiO3.

We unveil a mechanism that enables a robust supercurrent diode effect in Josephson junctions based on multiband superconductors. We predict that interband pairing can significantly amplifies this effect, even under weak spin-orbit coupling while intraband pairing alone would render it negligible. To illustrate this, we examine monolayer FeSe/STO, a system where recent experiments suggest interband pairing in either a nodeless d-wave or η pairing state. Using experimentally derived parameters, we predict that FeSe/STO can serve as a high-temperature platform for realizing a substantial supercurrent diode effect, with efficiencies reaching up to 30% for d-wave pairing and 12% for η pairing. These results demonstrate that measuring the supercurrent diode effect can provides a powerful probe of the pairing symmetry in monolayer FeSe/STO, offering critical insights into its superconducting state.

Vortex lines, known as topological defects, are capable of trapping Majorana modes in superconducting topological materials. Previous studies have primarily focused on topological bands with conventional s-wave pairing. However, topological Dirac semimetals exhibiting a unique orbital texture can favor unconventional pairing when electronic correlations are significant. The topology of vortices in these systems remains elusive and unexplored. In this work, we investigate the vortex bound states in C4z-symmetric superconducting Dirac semimetals, with a particular focus on the orbital-singlet unconventional pairing, which generates higher-order Majorana hinge modes. We identify robust doubly-degenerate Majorana vortex flat bands at zero energy in both type-I and type-II Dirac semimetals. These doubly-degenerate flat bands arise from a nontrivial Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}}}_{2}$$\end{document} topology defined by an effective particle-hole symmetry and are protected by the four-fold rotational symmetry. Additionally, we observe that moving the vortex line close to a hinge can trivialize the higher-order Majorana arc on the hinge, leaving a single Majorana mode at the vortex core due to the hybridization of Majorana modes. Finally, we discuss the potential experimental implications for correlated Dirac semimetals, such as electron-doped iron-based superconductors.

The topological properties observed in iron-based superconductors extend our understanding of vortex Majorana quasiparticle excitations in unexpected ways. Vortex Majorana physics has been extensively studied within the context of the topologically protected surface Dirac state. By employing an in-situ strain device, we demonstrate that uniaxial strain can generate Majorana zero modes out of the topological Dirac semimetal bulk state in LiFeAs. Uniaxial strain along [100] direction is found to enhance the band renormalization of LiFeAs, effectively reducing the energy separation between the Fermi level and the topological Dirac semimetal state, and breaking C4 symmetry. Using scanning tunneling microscopy, we observe the evolution of vortex bound states in the topological Dirac semimetal state region, accompanied by the emergence of Majorana zero modes and vortex bound states contributed by the bulk band. Our work provides a controllable method for experimentally engineering Majorana physics in iron-based superconductors, and offers valuable insights into the topological Dirac semimetal state with intrinsic s-wave superconductivity.

Intrinsic Rashba spin-orbital coupling (SOC) can exist in centrosymmetric materials with local inversion symmetry breaking. Here we show that such a SOC can induce topological superconductivity together with an in-plane Zeeman field in planar Josephson junctions formed by the centrosymmetric materials. A single Majorana mode can be created at each end of the junction. We demonstrate this result in a model based on iron-based superconductors. We derive the necessary Fermi surface condition for the topological planar junction and calculate the topological phase diagram with respect to the in-plane Zeeman field and the phase difference between the two superconductors. We provide experimental characteristics for the topological superconductivity, including the differential conductance and the Fano factor tomography which can be measured in the scanning tunneling spectroscopy. Our study reveals that the centrosymmetric systems with local-inversion-symmetry breaking can serve as new platforms for the topological planar Josephson junctions, and help to find more experimentally feasible materials for the topological superconductors.

We present a new scheme for Majorana modes in systems with nonsymmorphic-symmetry-protected band degeneracy. We reveal that when the gapless fermionic excitations are encoded with conventional superconductivity and magnetism, which can be intrinsic or induced by proximity effect, topological superconductivity and Majorana modes can be obtained. We illustrate this outcome in a system which respects the space group P4/nmm and features a fourfold-degenerate fermionic mode at (π,  π) in the Brillouin zone. We show that in the presence of conventional superconductivity, different types of topological superconductivity, i.e., first-order and second-order topological superconductivity, with coexisting fragile Wannier obstruction in the latter case, can be generated in accordance with the different types of magnetic orders; Majorana modes are shown to exist on the boundary, at the corner and in the vortices. To further demonstrate the effectiveness of our approach, another example related to the space group P4/ncc based on this scheme is also provided. Our study offers insights into constructing topological superconductors based on bulk energy bands and conventional superconductivity and helps to find new material candidates and design new platforms for realizing Majorana modes.

Determining the pairing symmetry of single-layer FeSe on SrTiO 3 is the key to understanding the enhanced pairing mechanism. It also guides the search for superconductors with high transition temperatures. Despite considerable efforts, it remains controversial whether the symmetry is the sign-preserving s - or the sign-changing s ± -wave. Here, we investigate the pairing symmetry of single-layer FeSe from a topological point of view. Using low-temperature scanning tunneling microscopy/spectroscopy, we systematically characterize the superconducting states at edges and corners of single-layer FeSe. The tunneling spectra collected at edges and corners show a full energy gap and a substantial dip, respectively, suggesting the absence of topologically non-trivial edge and corner modes. According to our theoretical calculations, these spectroscopic features can be considered as strong evidence for the sign-preserving s -wave pairing in single-layer FeSe.

Presence of the Dzyaloshinskii-Moriya (DM) interaction in limited noncentrosymmetric materials leads to novel spin textures and exotic chiral physics. The emergence of DM interaction in centrosymmetric crystals could greatly enrich material realization. Here we show that an itinerant centrosymmetric crystal respecting a nonsymmorphic space group is a new platform for the DM interaction. Taking the P4/nmm space group as an example, we demonstrate that the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction induces the DM interactions, in addition to the Heisenberg exchange and the Kaplan-Shekhtman-Entin-wohlman-Aharony (KSEA) interaction. The direction of DM vector depends on the positions of magnetic atoms in the real space, and the amplitude depends on the location of the Fermi surface in the reciprocal space. The diversity stems from the position-dependent site groups and the momentum-dependent electronic structures guaranteed by the nonsymmorphic symmetries. Our study unveils the role of the nonsymmorphic symmetries in affecting magnetism, and suggests that the nonsymmorphic crystals can be promising platforms to design magnetic interactions.

Spin-triplet superconductivity is known to be a rare quantum phenomenon. Here we show that nonsymmorphic crystalline symmetries can dramatically assist spin-triplet superconductivity in the presence of spin-orbit coupling. Even with a weak spin-orbit coupling, the spin-triplet pairing can be the leading pairing instability in a lattice with a nonsymmorphic symmetry. The underlining mechanism is the spin-sublattice-momentum lock on electronic bands that are protected by the nonsymmorphic symmetry. We use the nonsymmorphic space group P4/nmm to demonstrate these results and discuss related experimental observables. Our work paves a new way in searching for spin-triplet superconductivity.

In the presence of both space and time reversal symmetries, an s-wave A1g superconducting state is usually topologically trivial. Here we demonstrate that an exception can take place in a type of nonsymmorphic lattice structures. We specify the demonstration in a system with a centrosymmetric space group P4/nmm, the symmetry group that governs iron-based superconductors, by showing the existence of a second-order topological state protected by a mirror symmetry. The topological superconductivity is featured by 2Z degenerate Dirac cones on the (1,0) edge, and Z pairs of Majorana modes at the intersection between the (1,1) and (1,-1) edges. The topological invariance and Fermi surface criterion for the topological state are provided. Moreover, we point out that the previously proposed s-wave state in iron-based superconductors, which features a sign-changed superconducting order parameter between two electron pockets, is such a topological state. Thus, these results not only open a new route to pursue topological superconductivity, but also establish a measurable quantity to settle one long-lasting debate on the pairing nature of iron-based superconductors.

We find that the vortex bound states in superconducting topological semimetals are gapless owing to topological massless excitations in their normal states. We demonstrate this universal result in a variety of semimetals, including Dirac and Weyl semimetals, three-fold degenerate spin-1 fermions, spin-3/2 Rarita-Schwinger-Weyl fermion semimetals and other exotic fermion semimetals. The formation of these gapless bound states is closely related to their Andreev specular reflection and propagating Andreev modes in π-phase superconductor-normal metal-superconductor junctions. We further demonstrate that these gapless states are topologically protected and can be derived from a topological pumping process.

We theoretically study potential unconventional superconductivity in doped AB-type IV-VI semi-conductors, based on a minimal effective model with interaction up to the next-nearest neighbors. According to the experimental implications, we focus on the spin-triplet channels and obtain the superconducting phase diagram with respect to the anisotropy of the Fermi surfaces and the interaction strength. Abundant nodal and nodeless states with different symmetry breaking appear in the phase diagram, and all the states are time reversal invariant and topologically nontrivial. Specifically, the various nodal superconducting ground states, dubbed as the topological Dirac superconductors, are featured by Dirac nodes in the bulk and Majorana arcs on the surface; among the full-gap states, there exist a mirror-symmetry-protected second-order topological superconductor state favoring helical Majorana hinge cones, and different first-order topological superconductor states supporting 4 surface Majorana cones. The experimental verification of the different kinds of superconducting ground states is also discussed.

We show that there are diversified Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions between magnetic impurities, mediated by itinerant electrons, in a centrosymmetric crystal respecting a nonsymmorphic space group. We take the $P4/nmm$ space group as an example. We demonstrate that the different type of interactions, including the Heisenberg-type, the Dzyaloshinskii-Moriya (DM)-type, the Ising-type and the anisotropic interactions, can appear in accordance with the positions of the impurities in the real space. Their strengths strongly depend on the location of the itinerant electrons in the reciprocal space. The diversity stems from the position-dependent site groups and the momentum-dependent electronic structures guaranteed by the nonsymmorphic symmetries. Our study unveils the role of the nonsymmorphic symmetries in affecting magnetism, and suggests that the nonsymmorphic crystals can be promising platforms to design magnetic interactions.

We show that topological superconductivity can be generally induced in a magnetic metal through the superconducting proximity effect. In this case, the topological superconductivity originates from band degeneracies controlled by crystalline symmetries. We demonstrate this general scheme in a model with a 4-fold band degeneracy protected by the space group P4=nmm. We show that first order or second-order topological superconductivity can be realized in the presence of a ferromagnetic order or an antiferromagnetic order respectively. We derive the corresponding topological invariants and analyze Majorana modes. Our study provides a general method to realize topological superconductivity and help to identify new platforms to generate Majorana zero modes.

The recently discovered cuprate superconductor ${\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{3+\ensuremath{\delta}}$ exhibits a high ${T}_{c}\ensuremath{\simeq}73\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ at $\ensuremath{\delta}\ensuremath{\simeq}0.2$. The polycrystal grown under high pressure has a structure similar to ${\mathrm{La}}_{2}\mathrm{Cu}{\mathrm{O}}_{4}$ but with dramatically different lattice parameters due to the $\mathrm{Cu}{\mathrm{O}}_{6}$ octahedron compression. The crystal field in the compressed ${\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{4}$ leads to an inverted Cu $3d\phantom{\rule{4pt}{0ex}}{e}_{g}$ complex with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ orbital sitting below the ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ and an electronic structure highly unusual compared to the conventional cuprates. We construct a two-orbital Hubbard model for the Cu ${d}^{9}$ state at hole doping $x=2\ensuremath{\delta}$ and study the orbital-dependent strong correlation and superconductivity. For the undoped case at $x=0$, we found that strong correlation drives an orbital-polarized Mott-insulating state with the spin-$1/2$ moment of the localized ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ orbital. In contrast to the single-band cuprates where superconductivity is suppressed in the overdoped regime, hole doping the two-orbital Mott insulator leads to orbital-dependent correlations and the robust spin and orbital exchange interactions produce a high-${T}_{c}$ antiphase $d$-wave superconductor even in the heavily doped regime at $x=0.4$. We conjecture that ${\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{3+\ensuremath{\delta}}$ realizes mixtures of such heavily hole-doped superconducting ${\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{4}$ and disordered ${\mathrm{Ba}}_{2}\mathrm{Cu}{\mathrm{O}}_{3}$ chains in a single-layer or predominately separated bilayer structure. Our findings suggest that unconventional cuprates with liberated orbitals as doped two-band Mott insulators can be a direction for realizing high-${T}_{c}$ superconductivity with enhanced transition temperature ${T}_{c}$.

Here we report a pressure-induced reemergence of superconductivity in recently discovered superconductor K2Mo3As3, which is the first experimental case observed in quasi-one-dimensional superconductors. We find that, after full suppression of the ambient-pressure superconducting (SC-I) state at 8.7 GPa, an intermediary non-superconducting state sets in and prevails to the pressure up to 18.2 GPa, however, above this pressure a new superconducting (SC-II) state appears unexpectedly. High pressure x-ray diffraction measurements demonstrate that the pressure-induced dramatic change of the lattice parameter c contributes mainly to the emergence of the SC-II state. Combined with the theioretical calculations on band strcture, our results suggest that the reemergemce of superconductivity is associated with the change of the complicated interplay among different orbital electrons, driven by the pressure-induced unisotropic change of the lattice.

Yi Zhang, ∗ Shengshan Qin, ∗ Kun Jiang, † and Jiangping Hu 3, 1, ‡ Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China (Dated: February 15, 2021)