Quasi-1D Topological Nodal Vortex Line Phase in Doped Superconducting 3D Dirac Semimetals

Topological superconductivity from unconventional band degeneracy with conventional pairing

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.

Toward large-scale, ordered and tunable Majorana-zero-modes lattice on iron-based superconductors

Majorana excitations are the quasiparticle analog of Majorana fermions in solid materials. Typical examples are the Majorana zero modes (MZMs) and the dispersing Majorana modes. When probed by scanning tunneling spectroscopy, the former manifest as a pronounced conductance peak locating precisely at zero-energy, while the latter behaves as constant or slowly varying density of states. The MZMs obey non-abelian statistics and are believed to be building blocks for topological quantum computing, which is highly immune to the environmental noise. Existing MZM platforms include hybrid structures such as topological insulator, semiconducting nanowire or 1D atomic chains on top of a conventional superconductor, and single materials such as the iron-based superconductors (IBSs) and 4Hb-TaS2. Very recently, ordered and tunable MZM lattice has also been realized in IBS LiFeAs, providing a scalable and applicable platform for future topological quantum computation. In this review, we present an overview of the recent local probe studies on MZMs. Classified by the material platforms, we start with the MZMs in the iron-chalcogenide superconductors where FeTe0.55Se0.45and (Li0.84Fe0.16)OHFeSe will be discussed. We then review the Majorana research in the iron-pnictide superconductors as well as other platforms beyond the IBSs. We further review recent works on ordered and tunable MZM lattice, showing that strain is a feasible tool to tune the topological superconductivity. Finally, we give our summary and perspective on future Majorana research.

Superconductor Vortex Spectrum Including Fermi Arc States in Time-Reversal Symmetric Weyl Semimetals

Using semiclassics to surmount the hurdle of bulk-surface inseparability, we derive the superconductor vortex spectrum in nonmagnetic Weyl semimetals and show that it stems from the Berry phase of orbits made of Fermi arcs on opposite surfaces and bulk chiral modes. Tilting the vortex transmutes it between bosonic, fermionic, and supersymmetric, produces periodic peaks in the density of states that signify novel nonlocal Majorana modes, and yields a thickness-independent spectrum at magic "magic angles." We propose (Nb,Ta)P as candidate materials and tunneling spectroscopy as the ideal experiment.

Topological superconducting vortex from trivial electronic bands

Superconducting vortices are promising traps to confine non-Abelian Majorana quasi-particles. It has been widely believed that bulk-state topology, of either normal-state or superconducting ground-state wavefunctions, is crucial for enabling Majorana zero modes in solid-state systems. This common belief has shaped two major search directions for Majorana modes, in either intrinsic topological superconductors or trivially superconducting topological materials. Here we show that Majorana-carrying superconducting vortex is not exclusive to bulk-state topology, but can arise from topologically trivial quantum materials as well. We predict that the trivial bands in superconducting HgTe-class materials are responsible for inducing anomalous vortex topological physics that goes beyond any existing theoretical paradigms. A feasible scheme of strain-controlled Majorana engineering and experimental signatures for vortex Majorana modes are also discussed. Our work provides new guidelines for vortex-based Majorana search in general superconductors.

Gapless vortex bound states in superconducting topological semimetals

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.

Competing Vortex Topologies in Iron-Based Superconductors

In this Letter, we establish a new theoretical paradigm for vortex Majorana physics in the recently discovered topological iron-based superconductors (TFeSCs). While TFeSCs are widely accepted as an exemplar of topological insulators (TIs) with intrinsic s-wave superconductivity, our theory implies that such a common belief could be oversimplified. Our main finding is that the normal-state bulk Dirac nodes, usually ignored in TI-based vortex Majorana theories for TFeSCs, will play a key role of determining the vortex state topology. In particular, the interplay between TI and Dirac nodal bands will lead to multiple competing topological phases for a superconducting vortex line in TFeSCs, including an unprecedented hybrid topological vortex state that carries both Majorana bound states and a gapless dispersion. Remarkably, this exotic hybrid vortex phase generally exists in the vortex phase diagram for our minimal model for TFeSCs and is directly relevant to TFeSC candidates such as LiFeAs. When the fourfold rotation symmetry is broken by vortex-line tilting or curving, the hybrid vortex gets topologically trivialized and becomes Majorana free, which could explain the puzzle of ubiquitous trivial vortices observed in LiFeAs. The origin of the Majorana signal in other TFeSC candidates such as FeTe_{x}Se_{1-x} and CaKFe_{4}As_{4} is also interpreted within our theory framework. Our theory sheds new light on theoretically understanding and experimentally engineering Majorana physics in high-temperature iron-based systems.

Fermi Arc Criterion for Surface Majorana Modes in Superconducting Time-Reversal Symmetric Weyl Semimetals

Many clever routes to Majorana fermions have been discovered by exploiting the interplay between superconductivity and band topology in metals and insulators. However, realizations in semimetals remain less explored. We ask, "Under what conditions do superconductor vortices in time-reversal symmetric Weyl semimetals-three-dimensional semimetals with only time-reversal symmetry-trap Majorana fermions on the surface?" If each constant-k_{z} plane, where z is the vortex axis, contains equal numbers of Weyl nodes of each chirality, we predict a generically gapped vortex and derive a topological invariant ν=±1 in terms of the Fermi arc structure that signals the presence or absence of surface Majorana fermions. In contrast, if certain constant-k_{z} planes contain a net chirality of Weyl nodes, the vortex is gapless. We analytically calculate ν within a perturbative scheme and provide numerical support with a lattice model. The criteria survive the presence of other bulk and surface bands and yield phase transitions between trivial, gapless, and topological vortices upon tilting the vortex. We propose Li(Fe_{0.91}Co_{0.09})As and Fe_{1+y}Se_{0.45}Te_{0.55} with broken inversion symmetry as candidates for realizing our proposals.

Majorana zero modes in impurity-assisted vortex of LiFeAs superconductor

The iron-based superconductor is emerging as a promising platform for Majorana zero mode, which can be used to implement topological quantum computation. One of the most significant advances of this platform is the appearance of large vortex level spacing that strongly protects Majorana zero mode from other low-lying quasiparticles. Despite the advantages in the context of physics research, the inhomogeneity of various aspects hampers the practical construction of topological qubits in the compounds studied so far. Here we show that the stoichiometric superconductor LiFeAs is a good candidate to overcome this obstacle. By using scanning tunneling microscopy, we discover that the Majorana zero modes, which are absent on the natural clean surface, can appear in vortices influenced by native impurities. Our detailed analysis reveals a new mechanism for the emergence of those Majorana zero modes, i.e. native tuning of bulk Dirac fermions. The discovery of Majorana zero modes in this homogeneous material, with a promise of tunability, offers an ideal material platform for manipulating and braiding Majorana zero modes, pushing one step forward towards topological quantum computation.

Despite the discovery of Majorana zero modes (MZM) in iron-based superconductors, sample inhomogeneity may destroy MZMs during braiding. Here, authors observe MZM in impurity-assisted vortices due to tuning of the bulk Dirac fermions in a homogeneous superconductor LiFeAs.

On the Remarkable Superconductivity of FeSe and Its Close Cousins

Emergent electronic phenomena in iron-based superconductors have been at the forefront of condensed matter physics for more than a decade. Much has been learned about the origins and intertwined roles of ordered phases, including nematicity, magnetism, and superconductivity, in this fascinating class of materials. In recent years, focus has been centered on the peculiar and highly unusual properties of FeSe and its close cousins. This family of materials has attracted considerable attention due to the discovery of unexpected superconducting gap structures, a wide range of superconducting critical temperatures, and evidence for nontrivial band topology, including associated spin-helical surface states and vortex-induced Majorana bound states. Here, we review superconductivity in iron chalcogenide superconductors, including bulk FeSe, doped bulk FeSe, FeTe1−xSex, intercalated FeSe materials, and monolayer FeSe and FeTe1−xSex on SrTiO3. We focus on the superconducting properties, including a survey of the relevant experimental studies, and a discussion of the different proposed theoretical pairing scenarios. In the last part of the paper, we review the growing recent evidence for nontrivial topological effects in FeSe-related materials, focusing again on interesting implications for superconductivity.

Vortex End Majorana Zero Modes in Superconducting Dirac and Weyl Semimetals

Time-reversal invariant Dirac and Weyl semimetals in three dimensions (3D) can host open Fermi arcs and spin-momentum locking Fermi loops on the surfaces. We find that when they become superconducting with s-wave pairing and the doping is lower than a critical level, straight π-flux vortex lines terminating at surfaces with Fermi arcs or spin-momentum locking Fermi loops can realize 1D topological superconductivity and harbor Majorana zero modes at their ends. Remarkably, we find that the vortex-generation-associated Zeeman field can open (when the surfaces have only Fermi arcs) or enhance the topological gap protecting Majorana zero modes, which is contrary to the situation in superconducting topological insulators. By studying the tilting effect of bulk Dirac and Weyl cones, we further find that type-I Dirac and Weyl semimetals in general have a much broader topological regime than type-II ones. Our findings build up a connection between time-reversal invariant Dirac and Weyl semimetals and Majorana zero modes in vortices.

Quantum Phase Transition of Correlated Iron-Based Superconductivity in LiFe_{1-x}Co_{x}As

The interplay between unconventional Cooper pairing and quantum states associated with atomic scale defects is a frontier of research with many open questions. So far, only a few of the high-temperature superconductors allow this intricate physics to be studied in a widely tunable way. We use scanning tunneling microscopy to image the electronic impact of Co atoms on the ground state of the LiFe_{1-x}Co_{x}As system. We observe that impurities progressively suppress the global superconducting gap and introduce low energy states near the gap edge, with the superconductivity remaining in the strong-coupling limit. Unexpectedly, the fully opened gap evolves into a nodal state before the Cooper pair coherence is fully destroyed. Our systematic theoretical analysis shows that these new observations can be quantitatively understood by the nonmagnetic Born-limit scattering effect in an s±-wave superconductor, unveiling the driving force of the superconductor to metal quantum phase transition.

Topological vortex phase transitions in iron-based superconductors

We study topological vortex phases in iron-based superconductors. Besides the previously known vortex end Majorana zero modes (MZMs) phase stemming from the existence of a three dimensional (3D) strong topological insulator state, we show that there is another topologically nontrivial phase as iron-based superconductors can be doped superconducting 3D weak topological insulators (WTIs). The vortex bound states in a superconducting 3D WTI exhibit two different types of quantum states, a robust nodal superconducting phase with pairs of bulk MZMs and a full-gap topologically nontrivial superconducting phase which has single vortex end MZM in a certain range of doping level. Moreover, we predict and summarize various topological phases in iron-based superconductors, and find that carrier doping and interlayer coupling can drive systems to have phase transitions between these different topological phases.